FOREWORD IT IS less than 20 years ago that nuclear magnetic resonance was first observed
in bulk matter. This new form of spectroscopy quickly proved to have useful applications in physics and chemistry, and with the discovery of the “chemical shift” in 1949 it became clear that this was to be a most important field of study for the chemist. As the design of magnets reached quite remarkable performances, more and more subtle effects became apparent, and to&y nuclear magnetic resonance is applied to a quite astonishing range of chemical and physical problems. The most extensive applications to chemistry come from the so-called “high resolution spectroscopy”, and this important part of the subject is reviewed most thoroughly in this book. Although much useful information can be derived from purely empirical applications of NMR, a great wealth of data can often be derived only from a proper analysis and understanding of the spectra. A detailed account of the theory and analysis of high resolution spectra is given at length in this book, which will be of great value to all magnetic resonance spectroscopists, There is also a full discussion of other factors which can affect the spectra, profusely illustrated with examples from the literature. In addition to the theoretical part, this book contains a remarkable collection of data for reference purposes. A considerable proportion of the work done so far has been summarised in compact form, which is most useful for reference. Nuclear resonance work is now being published at an ever-increasing rate, which has now reached about 2000 papers a year, and the authors are to be congratulated for having encompassed the work so far in a single book. R. E. RI-S
xvii
PREFACE IN WRVINGthis monograph we have been bold enough to attempt to provide a detailed account of the basic theory underlying high resolution nuclear magnetic resonance (NMR) spectroscopy and also to present a survey of the major applications to problems in physics and chemistry. We have aimed at being as comprehensive as possible with the intention that almost all the text will interest everyone actively engaged in NMR spectroscopy. Because the subject is so vast, the authors are not competent to be critical throughout. Indeed, the phenomenal increase in the published work on NMR will soon make impracticable the task of containing full coverage of the topic within a single text. Already, several aspects of the subject are so well developed that they merit individual presentation in monographs written by the appropriate specialists. It has been necessary to divide our monograph into two volumes, the first one of which is concerned primarily with basic theory and spectral analysis (Chapters 1 to 9) while the other contains most of the published work on struo tural applications of high resolution NMR spectroscopy (Chapters 10 to 12). From the extensive cross-referencing, the reader will be aware that the book has been written as a single entity. A prerequisite for the successful application of NMR is the ability to analyse a spectrum in order to obtain the chemical shift and spin coupling constantparameters, Therefore, we have made the chapter on the analysis of spectra, Chapter 8, the mainspring of the book and we have covered the subject as fully as possible. This section has been written keeping in mind the reader who does not have a strong background in quantum mechanics,hencea ratherdetailedaccount has been given of the theory behind spectral analysis. Nearly all the spectral systems which have been analysed are described and the analyses of some of the spectral types are discussed in great detail to illustrate the general method. This chapter provides an introduction to the analysis of NMR spectra and also serves as a reference source for the practising spectroscopist. The basic theory of NMR is dealt with in Chapters 2,3 and 9, and the aim has been to cover the background of all major applications of NMR rather than to attempt a unified and mathematically rigorous treatment. Chapters 4 and 5 review the progress made in relating chemical shifts and coupling constants to the electronic structure of molecules, while Chapter 9 covers the applications of NMR to kinetic processes. Chapters 10, 11 and 12 contain a survey of the applications of NMR to the determination of molecular structure: Chapter 10 deals with hydrogen resonance, Chapter 11 with fluorine resonance and Chap ter 12 covers all the remaining suitable magnetic nuclei. All three chapters hold many reproductions of spectra and many compilations of chemical shifts and coupling constants. This particular volume should be invaluable to those who
XX
PREFACE
have some knowledge of the fundamentals of NMR and to wish use the technique as an analytical tool. The practicalaspects of NMR spectroscopy are dealt with in Chapters 6 and 7. Chapter 6 covers the theory of the instrumentation required for the measuremeat of high resolution NMR spectra and also describes the types of spectrometer available commercially. In Chapter 7 is given a discussion of all the practical factors which need to be considered in obtaining NMR spectra. This chapter is intended primarily for those new to the subject and includes preparation of the sample, tuning the spectrometer and measurement of spectra. The material for this chapter has been drawn not only from the authors’ own experience but also from many discussions with other workers in the field. In particular, much valuable information has been obtained from the contributions to the monthly letters on NMR (MELLONMR) edited by Drs. A. A. Bothner-By and B. A. Shapiro of the Mellon Institute, Pittsburg, U.S.A. We would like to acknowledge the many helpful discussions we have had with Drs. R. J. Abraham and J. Lee. We are also indebted to Drs. R. J. Abraham, T. B. Grimley and G. Skirrow, who have given up much of their time to read the manuscript, for their suggestions and constructive criticisms of the text. In the course of preparing this monograph we have been particularly fortunate in the help given by those who have supplied us with unpublished data and with manuscripts prior to publication. Their help has enabled us to cover the scientific literature up to the end of 1963. We would like to thank especially Dr. G. V. D. Tiers for allowing us to reproduce his compilation of t-values and also Professors H. S. Gutowsky, G. G. Hall and J. D. Roberts and Drs. R. J. Abraham, J. D. Baldeschwieler, C. N. Baawell, A. A. Bothner-By, F. A. Bovey, P. L. Corio, R. Freeman, P. C. Lauterbur, A. Melera, W. G. Schneider, G. W. Smith, F. C. Stehhng, G. V. D. Tiers and the Office of Naval Research of the U.S.A. We wish to thank Professors C. E. H. Bawn, C.B.E., F.R.S. and W. K. R. Musgrave for their encouragement at all times. We also thank all the authors and editors who have given us permission to reproduce figures and diagrams from their publications. We would be grateful if readers would draw our attention to any errors which they may encounter in the text.
CHAPTER
1
INTRODUCTION HISTORICAL UNTILrecently, our understanding of atomic and molecular structure was based to a large extent on the findings of optical spectroscopy. New sources of structural information have been forthcoming from the development of experimental techniques which have extended spectroscopic measurements to the microwave (roughly 10’ to lo3 MC see-‘) and the radiofrquency (roughly lo2 to IO-” MC set-r) regions. In these regions, radiation is absorbed and emitted by the same basic process as at other wavelengths in the electromagnetic spectrum, namely a quantum of energy is transferred tb or from a system undergoing a transition between two levels. The separation of the levels is equal to the amount of energy transferred, the values encountered being about a tenmillionth of an electron volt and about a ten-thousandth of an electron volt for the radiofrequency and microwave regions respectively. In two of the principal branches of radiofrequency spectroscopy, nuclear magnetic and electric quadrupole resonance, this separation is associated with different orientations of, respectively, the nuclear magnetic dipole moments in applied magnetic fields and of nuclear electric quadrupole moments in non-spherical molecular electric fields. The occurmnce of nuclear moments was first detected from a study of the hype&e structure observed in the electronic spectra of certain atoms using optical spectrographs having very high resolving power’l). The presence of the hypetie structure in atomic spectra led PatW) to suggest, in 1924,that certain nuclei possess angular momentum and thus a magnetic moment which interacts with the atomic orbital electrons. This was confirmed later by spectroscopic work which enabled values of the angular momentum and magnetic moment to be determined for many nuclei. In the presence of an applied magnetic field, such magnetic moments take up specific orientations and it is possible to observe transitions occurring between the mrclear energy levels associated with these orientations by irradiating with energy of a suitable frequency. The nuclear energy levels are quantised as a direct consequence of the quantised nature of the angular momentum operator of a nucleus which has 2Z+ 1 values. The spin number Z can assume any value which is a multiple of +; the highest value known is that of ‘?f Lu having Z = 6 (see Appendix A). The largest measurable component of the angular momentum is Zh, where li = h/2rc and h is Planck’s constant. Values of Z for a particular nucleus (see Appendix A) cannot be predicted but it has been noticed that isotopes having mass and atomic numbers which are both even have Z = 0, and isotopes having odd mass numbers have half integral spin values. Even and equal numbers of protons HB8.1
1
2
HIGH
RESOLUTION
NMR
SPECTROSCOPY
and neutrons in a nucleus (I= 0) may be regarded as a state of complete “pairing” analogous to the complete “pairing” of electrons in a diamagnetic molecule. In 1921, Stem and Gerlacht3), using a beam method, showed that the measurable values of an atomic magnetic moment are discrete, corresponding to the space quantisation of an atom when placed in an inhomogeneous magnetic field. In later experiments a beam of hydrogen molecules was directed through a steady magnetic field thus enabling the small magnetic moment of the hydrogen nucleus to be measured. The method has been developed further by subjecting the beam to an additional magnetic field oscillating at the frequency required to induce transitions between the nuclear energy levels corresponding to the quantised values of the nuclear magnetic momentC4). For a nuclear spin number I there will be 21+ 1 equally spaced energy levels, and in a steady field of strength H, the separation between the highest and the lowest of these levels is 2,~H where p is the maximum measurable value of the nuclear magnetic moment. Thus the separation between successive sub-levels is pH/I and the frequency of the oscillating magnetic field which can excite transitions between these levels is pH/Ih. In the molecular beam experiment, only the molecules which do n,ot change their energy reach the detector, hence the frequency of the resonant exchange between levels is determined by sweeping through a range of frequencies. At some particular frequency value there is a sudden reduction in the number of molecules arriving at the detector. These first successful demonstrations of nuclear resonance were accomplished with main magnetic fields of several kilogauss which lead to oscillating field frequencies in the range lo5 to lo8 cycles set-i. The resonance exchange of energy is not restricted to molecular beams and is detectable in all forms of matter. In 1936, GorteP) attempted to observe the resonances of ‘Li nuclei in lithium fluoride and ‘H nuclei in potassium aluminium sulphate. A further fruitless attempt was made by Gorter and 3roeP) in 1942. In the two attempts, the detection of the absorption of radiofrequency power was to be made by calorimetric methods and the occurrence of anomalous dispersion respectively. The use of unfavourable materials was the main cause of failure. It was not until late in 1945 that the fist actual nuclear magnetic resonance signals were observed. Bloch (‘I) at Stanford University observed the hydrogen resonance absorption in water and Purcell(*) working at Harvard University was successful in detecting hydrogen nuclear resonance absorption in paraffin wax. They received jointly in 1952 a Nobel prize for their discovery. It is interesting to consider why the nuclear paramagnetic susceptibility is too small to be of value in static susceptibility measurements. An important fact is that the nuclear contribution to the paramagnetic susceptibility of a substance is negligible compared with that from the electrons, as the following discussion will show. The magnetic moment of an electron assembly in an atom or molecule is given by (1.1) where g, the Landd or spectroscopic splitting factor, is a measure of the contri-
INTRODUCTION
3
bution of the spin and orbital motion of the electron to its total orbital angular momentum and has the value of 2.0023 for a spinning free electron. J is the quantum number representing total angular momentum and fl is the Bohr magneton defined by B = eh/4n MC (l-2) where e is the charge on the electron, M is the electron mass and c is the velocity of light. In a similar way the magnetic moment of a nucleus is given by PN = gN
MH ,$,J[I(1 N M
+
1)]“2
(l-3)
/IN, the nuclear magneton, is defined in terms of the mass of the hydrogen nucleus bN = eh/4n MHc. (1.4) Now an assembly of paramagnetic nuclei when placed in a steady magnetic field will orientate themselves according to the Boltxmann energy distribution, and the resultant volume susceptibility XNwill be given by the classical hq$vinBrillouin relation XN =
N/JN2/3
kT
WI
where N is the nuinber of magnetic nuclei in 1 cm3. Since g, and gNare approximately equal and ,$# = l/1838 then the macroscopic nuclear susceptibility XNis less by a factor of about (1838)2 than the susceptibility of a substance having paramagnetism originating from unpaired electrons. & is also masked by the electronic diamagnetic susceptibility even in the neighbourhood of the absolute zero of temperature. These considerations indicate the importance of eliminating the electronic diamagnetic factor by means of a resonance experiment employing an oscillating magnetic field. However, despite the intrinsic dif&ulty, in 1937 Lasarev and SchubnikovCg) were able to detect the static nuclear susceptibility of solid hydrogen by using a non-resonance technique in the temperature range 1.76 to 422°K. They were able to separate the diamagnetic electronic contribution from the paramagnetic nuclear contribution because of the temperature dependence of the latter. These experiments led to the determination of the magnetic moment of the hydrogen nucleus to an accumcy of about 10 per cent. Nuclear magnetic resonance absorption is proportional to the static nuclear paramagnetic susceptibility but is far easier to detect. From the above discussion it will be clear that a similar resonance effect will occur for atoms or molecules possessing one or more unpaired electrons. Since the electron has spin + there are basically only two energy levels but these are much farther apart (for a given applied magnetic field) than those of say the hydrogen nucleus, because their magnetic momenta have the magnitudes 9270 x 102*and 14-l x 1O24erg gauss-’ respectively. This means that for applied magnetic fields of the order of kilogauss, the oscillating field required for 1*
4
HIGH
RESOLUTION
NMR
SPECTROSCOPY
electron resonance must have a frequency in the microwave region. It is not surprising then that electron spin resonance (ESR) has been known for as long a period as nuclear magnetic resonance (NMR). The first successful demonstration of the former type of resonance was made by Zavoisky(lo) who used cupric chloride dihydrate; the first free radical spectrum was observed by Kozyrev and Salikhov(l i) who studied pentaphenylcyclopentadienyl. The hyperfine splitting that can be observed in the electron spin resonance spectra of some free radicals arises because some of the nuclei present possess a magnetic moment. Electron spin resonance(12 -I’) now forms one of the main branches of microwave spectroscopy. Nuclear magnetic resonance studies are divided into three main types (i) high resolution (ii) low resolution and (iii) spin echo measurements. These divisions arise mainly from the very different experimental arrangements employed. High resolution measurements are confined to liquids and occasionally gases and the spectra occupy less than O-1per cent of the applied static magnetic field. Low resolution techniques are applied to solids and occasionally to liquids and the spectra occupy between 1 and 10 per cent of the steady magnetic field. This monograph is devoted entirely to high resolution NMR spectroscopy but an outline of spin echo methods is included. It would not be out of place, however, to draw the reader’s attention to the considerable amount of information which is obtainable by the low resolution method”*). The method provides a useful adjunct to X-ray crystallography in the determination of the position of hydrogen atoms in crystaWg’. The onset of various types of molecular motion can be detected as solids are taken through their transition points, thus providing insight into the molecular structure and solid state of small molecules and of po1ymers(20).
HIGH
RESOLUTION
NMR SPECTROSCOPY
The power of the high resolution method derives from the fact that nuclei of the same species in different chemical environments absorb energy at di& erent radiofrequencies (for a given applied field) due to the irbeing shielded to different extents from the applied magnetic field. This discovery was made in 1949 when it was found that the resonance frequencies of the nuclei of phosphorus, nitrogen and fluorine depended upon the chemical compound in which they were located. The separations between the various resonance frequencies are called chemicalshifts: the values of the shifts can be used to obtain information about the electronic environment of a particular nucleus in the molecule under examination. If, say, ‘H resonance is being studied then it is found that a mokule containing only one hydrogen nucleus or a set of ‘H nuclei having the same environment (as for example in water, benzene or cyclohexane) gives a single absorption band. The position of the band is a characteristic of the molecule. Since chemical shifts cannot be measured absolutely, that is, from a nucleus stripped of its electrons, the signal from a reference compound is used as an arbitrary zero : a suitable compound for ‘H resonance is tetramethyl-
5
INTRODUCTION
silane. It is customary to quote chemical shifts in tbe dimensionless parameter B (in units of parts per million) defined by
6=
106
where H - HP is the chemical shift difference between the sample and tbe reference, H, being the absolute position of tbe reference signal in tbe main magnetic field. These field measurements cannot be made with certainty hence it is more convenient to calculate 6 from tbe expression 6 =-
( ) 1’ -
v,
106
vo
Benzene
/I -7.3
FIG. 1.1 Chunid
Water
-5.2
Cyclohrxonc
Tetromethylsilanc
-1.4
shifts of benzene, water and cyclohexanc measured from tetramethylsilane
where v - v, is tbe chemical sbift difference between tbe sample and reference in frequency units (cycles se&)-these are the units in wbicb the spectnun is calibrated. Strictly, one should not use vo-tbe radiofrequency (usually tied) at wbicb the spectrometer operates-but v,, the absolute frequency at wbicb the reference signal appears. The error introduced, however, is very small because y. and V, are almost identical. Tbe obvious need for expressing S in dimensionless units can now be appreciated because different NMR spectrometers are operated at a variety of frequencies vo. Figure 1.1 illustrates the relative positions of the signals of benzene, water and cyclobexane measured from tetramethylsilane. A theoretical treatment of tbe origin of chemical shifts is given in chapter 3. Some molecules can give rise to more than one absorption band, each of which is characteristic of the electronic environment of tbe magnetic nucleus under examination. For example, toluene has a ‘H resonance spc&um consisting of two absorption bands 4.85 ppm apart. Figure 1.2(a) shows tbe spectrum obtained at 60 MC set-l, also included is a small band from the small amount of tetrametbylsilane added as a reference. Tbe low field band corresponds to tbe hydrogen nuclei of tbe benzene ring while the other band originates from the hydrogen nuclei in tbe methyl group. Tbe ratio of tbe band areas (intensities) is 5 : 3 corresponding to tbe number of ‘H nuclei in each environment; tbis property can be used for diagnostic purposes.
6
HIGH RESOLUTION
NMR SPECTROSCOPY
Compounds having magnetic nuclei in more than one chemical environment can have absorption bands showing fine structure (see Fig. 1.2(b)). This structure, know nas multiplet splitting, has its origin in the coupling between magnetic nuclei taking place via bonding electrons (a theoretical discussion is present400 I
r
300 I
200 1
I00 I
0
(0)
H
1
!6-O
7.0
6.0
5-O
Tetromethylsilone
4-o
30
I-0
3
mm FIO. 1.2 (a) The ‘H resonance spectrum of toluene at 60 MC see-‘. (b) The ‘H resonance spectrum of isopropylbenzene at 60 MC se& ; the septet of bands has been increased in intensity to fully reveal its structure. A little tetramethylsilane is present in both spectra to provide a reference. By courtesy of Varian Associates
ed in Chapter 3). The magnitude of the coupling can be determined by measuring the separation of the multiplet components providing that the chemical shift difference of the coupled nuclei is much greater than the magnitude of the multiplet splittings. This band separation is independent of the field strength hence there is no need to quote the splitting in dimensionless units-cycles set-’ are the units normally used. The simplest example of multiplet splitting one can encounter is to be found in the resonance spectrum of a molecule having
INTRODUCTION
7
amagnetic nucleus A and a magnetic nucleusX of another isotopic species. Figure 1.3 is a representation of the resonance spectrum that would be obtained if A and X each have a spin number I = 3. The separation of the lines within each daublet is the spin couphg cons&nr JAx. The doublets appear because each nucleus splits the resonance of its neighbour into 21+ 1 components. The energy differences between the various spin states are so small that, at thermal equilibrium, the Boltzmann distribution makes them all equally probable. Hence, all the lines in a multiplet resulting from coupling to a single nucleus are equally intense. If, however, there are n indistinguishable nuclei, the neighbour is split x _____--_________y ~
nucleus K-----------8A, A
nucleus
FIG. 1.3 Repmentation of the NMR spectnun of a system containing the magnetic nuclei A and X each having Z = l/2 and the condition 6m s JAX is fumed
into 2nl+ 1 lines. For nuclei having I = +, the intensities of the lines in a multiplet follow the binomial coefficients given in the table below: n
Relative Intensities of Multiplet Components
1
1
2
12
1
3
13
3
1
4
14
6
4
1
5
1 5
10
10
5
1
6
1 6
15 20
15
6
1
7
1
7
21
35
35
21
7
1
8
1 8
28
56
70
56 28
8
1
1
Figure 1.2(b) shows the hydrogen resonance spectrum of isopropylbenzene. The single band located at 725 ppm from tetramethylsilane (designated 0 ppm) originates from the ring hydrogen nuclei : like the corresponding toluene band, no multiplet splitting is observed. The most intense group of bands (a doublet) is that centred at l-25 ppm so these may be assigned to the two methyl groups. The remaining group forms a septet centred at 29Oppm which originates from the CH group. The doublet is caused by this group since the number of
8
HIGH
RESOLUTION
NMR
SPECTROSCOPY
hyperfine lines is 2 x + + 1 = 2. The magnitude of the splitting (J7 cycles se&) is common to both the doublet and the septet. The septet appears because of the mutual coupling between the CH group and the two methyl groups. The latter have n = 6, thus they split the band from the CH group into seven components having relative intensities 1: 6 : 15 : 20 : 15 : 6 : 1. From the foregoing it is clear that the magnitudes and types of band splittings can be used to decide the relative positions of functional groups in a molecule. Further assistance may be had with the aid of isotopic substitution. For example, if hydrogen is replaced with deuterium (I = l), the number of lines in the spin multiplet will change according to the change in I. Also the spacing will diminish because of the nearly sevenfold decrease in the magnetogyric ratio y. The treatment given here applies only to molecules having the property ~,,b % J; unfortunately, such a condition is not realisable very often in practice. Spectra governed by this condition are referred to as beingfirst order in type. The spectra normally obtained are more complicated and the evaluation of the variousv,,8 and Jvalues can be a lengthy process. The whole of Chapter 8 is devoted to describing the various methods available for doing this. Considerable effort is expended on analysing spectra fully as a consequence of the wide theoretical interest in the way in which chemical shifts and coupling constants vary from molecule to molecule. Kinetic and thermodynamic data can be obtained from line width or line position variations with reactant concentration and with temperature: typical applications are hydrogen bonding and molecular conformation studies. Another useful feature of a NMR spectrum is that the relative intensities of the absorption bands indicate the relative numbers of differently shielded magnetic nuclei. However, the application to quantitative analysis has not been exploited yet. From this brief outline it is clear that nuclear magnetic resonance is of considerable importance to physical, organic and inorganic chemists. The two nuclei which have ideal properties of high natural abundance, high magnetic moment and nuclear spin + are ‘H and lgF and most investigations concern them. Considerable work has also been carried out on 31P llB and 13C resonances. Further improvements in experimental techniques &ll probably result in the method being used more widely for other nuclei, particularly for the 13C nucleus. REFERENCES 1. S. TOLANSKY,H&h Resolution Spectroswpy, Methuen, London (1947). 2. W. PAUL&Naturwisschqfh, 12,741 (1924). 3. 0. STERN, Z. Phys., 7,249 (1921); W. GBRLXH and 0. Smm, Ann. Phys. Leipzig, 74,673 (1924). 4. N. F. RUSEY, MoIeahr Beams, Oxford University Press (1956). 5. C. J. GARTER,Physica, 3,995 (1936). 6. C. J. GARTERand L. F. J. BROW, Physicu, 9, 59 1 (1942). 7. F. BLOCH,W. W. HANSENand M. PACKARD,Phys. Rev., 69, 127 (1946). 8. E. M. PURCELL, H. C. TORREYand R. V. POUND,Phys. Rev., 69, 37 (1946). 9. B. G. LASAIUVand L. V. SCHUBNIKOV, Phys. Z. Sowjetunion, 11, 445 (1937).
INTRODUCTION
9
10. E. ZAVOISKY, J. Phys. (U.S.S.R), 9,245,447, (1945). 11. B. M. KOZYRJZVaud S. G. SNZHOV, Dokl. Akad. Nat& S.S.S.R., 58, 1023 (1947). 12. D. J. E. INGRAM,(1958), Free Radicals as Studied by Electron Spin Resonance, Butterworths, London. 13. D. Ii. Wmmm, Qaart. B-m., 12,250 (1958). 14. D. J. E. INQRAM,Spectroscopy at ZWio and Microwave frequencies, Butterworths, London (1955). 15. B. BLEANEYand K. W. H. STEVENS, Rep. Prog. Phys., 16, 108 (1953). 16. A. CARRINGTON and H. C. LONGUEPHIGC~INS, Qua. Rev., 14,427 (1960). 17. G. E. PAKE,Paramagnetic Resonance, W. A. Bajamin, New York (1962). 18. E. R. ANDREW,Nuclem Mugnetic Be.mnmx, Cambridge University Press (1956). 19. R. E. RI-, Qrum. Rev., 10,480 (1956). 20. J. G. POWLES,Polymer, 1,219 (1960).
BIBLIOGRAPHY 1. High ZZesolution N&ear Magnetic Resonance, J. A P~PL.E,W. G. Scsmamaa and H. 3. Bnunw, MeGraw-Hill, New York (1959). 2. Applications of Nuclear Magnetic R esonance Spectroswpy in Organic Chemistry, L. M. JA-, Pergamon Press, London (1959). 3. NMR and EPR Spectroscopy, VAIUN Assoa~m, Pergamon Press, London (1960). 4. NMR Spectra CataZog, N. S. BHACCA, L. F. JOHNSON and J. N. SEOOLZRY, National Press, New York Volume 1 (1962), Volume 2 (1963). 5. llw Principles of Nuclear Magnetism, A. h3RAt%M, Clarendon Press, Oxford (1961). 6. Nuclear Ztuhrction, A. K. SN-IAand T. P. DAS, Institute of Nuclear Physics, Calcutta (1957). 7. Nuc&ar Magnetic Relaxation, N. BLOEMBEROEN, W. A. Benjamin, New York (1961).
8. La Rpsomaze Paramagndtique NucZeaire, P. Ganm, Centre National de la Reeherehe SeientZqw, Paris (1955); 9. Nuclear Magnetic Resonance Applications to Organic Chemistry, J. D. Rosaam, McGrawHill, New York (1959). 10. Determination of Organic Structures by Physical Met&,
Vol. 2. Edited by F. C. NACHOD and W. D. PRILLIPS,Academic Press, New York (1962). 11. An Zntroduction to the A~Iysis of Spin-spin Splitting in High Resolution Magnetic Z&sonance Spectra, J. D. ROBERTS,W. A. Bmjamin, New York (1961). 12. Interpretation of NMR Spectra, K. B. WIBERGand B. J. Nm, W. A. Benjamin, New York (1962). 13. Principles of Magnetic Resonance, C. P. SL~CHTER, Harper and Row, New York (1963). 14. Appburtions of NMR Spectroswpy to Organic Chemistry: Zllustratt&ts from the Steroid Field, N. S. BHACCAand D. H. Wm, Holden-Day Inc., New York (1964). 15. The Theory of Nuclear Magnetic Resonance, I. V. Am.~mRov, Academic Press, New York (1966).
16. Structure of High Resolution NMR Spectra, P. L. CURIO, Academic Press, New York (1966). 17. Die Rernmagnetische Resonanr undihre Anwendung in der Anorganischen Chemie, E. FLUCK, Springer-&&g, Berlin (1963). 18. Thories Mol&daires de ha ZZ&onance MagnPtique Nucleaire, G. MAVEL,Duuod, Paris (1965). 19. Interpretation of NMR Spectra, an Empirical Approach, R. H. BIELE,Consultants Bureau, Plenum Press, New York (1965).
EBB.
1s
CHAPTER
2
GENERAL THEORY OF NUCLEAR MAGNETIC RESONANCE TWEfirst two sections of this chapter are concerned with classical and quantum mechanical descriptions of the resonance condition. Both approaches are presented because the former is helpful in dealing with transient phenomena, while the latter is preferable when steady-state processes are being considered. 2.1 Tma CLASSICALME-CAL
DESCRIPTION OF TBBRESONANCE CONDITION
A rotating charge can be regarded as a current circulating in a loop and will therefore behave like a magnetic dipole whose moment p is given by B = iA
(24
‘where i is the equivalent current and A is the cross-sectional area of the enclosed loop. A charge q rotating at v/2zr revolutions a second is equivalent to a current i = qv/2xr. (2.2) On converting the charge to electromagnetic units by dividing by the velocity of light c, the magnetic moment can be found from p = qvr/2c.
(2.3)
Now the circulation of themass M of theparticlegeneratesanangularmomentum and having the value Mu r (see Fig. 2.1). The spinning motion is common to both charge and mass hence the magnetic moment vector is collinear with and directly proportional to that of the angular momentum according to the relationship
p, a vector directed along the axis of rotation
iu = W-W
P.
(2.4)
The direction of the vectors is determined by the sign of the charge on the particle. This model, of course, provides no explanation for the magnetic moment of a neutral particle such as the neutron, nor does it explain why some nuclei have negative magnetic moments. The deficiencies of the model suggest that nuclear structure is complex due to nuclear components having orbital as well as spin motions combining in various ways to give a resultant angular momentum for the nucleus, similar to the coupling of electronic spin and orbital moments 10
GENERAL
THEORY
OF NUCLEAR
MAGNETIC
RESONANCE
11
in atoms and molecules. The origin of nuclear spin is outside the scope of this book; for further information the reader is referred elsewhere”‘. The ratio p/p is called the magnetogyric ratio (y) and is a characteristic property of a particle. A gyroscope will precess about a vertical axis due to the
. FIG. 2.T Classical model of a spinning charged particle
HO
/-
------
I
------
t
\ ----__
,
__i_-jp
Fm. 2.2 The Larmor precession of the nuclear spin axis about the direction of the steady applied magnetic hid I&,
12
HIGH
RESOLUTION
NMR SPECTROSCOPY
torque arising from the earth’s gravitational field. Similarly, if a spinning charged particle is placed in a magnetic field of strength H,, with its magnetic moment at an angle 8 to the direction of this field then it will experience a torque L tending to align it parallel to the field. Newton’s law states that the rate of change of angular momentum p with time is equal to the torque, that is
FIG. 2.3 Vectorial representationof the classical Larmor precession
From simple magnetic theory L = p x Ho.
(2.6)
Substituting equations (2.4) and (2.6) into (2.5)
9 dt-
q
--pxH, 2Mc =yp
(2.7)
x H,,.
(2.8)
This equation of motion describes the precession of p about Ho with an angular velocity w,, defined by dp dt=pwo
and hence
w. = yHo.
(2.9)
GENERAL
THEORY
OF NUCLEAR
MAGNETIC
RESONANCE
13
This is the Larmor equation which may be re-written in terms of the precession frequency v. (2.10) An important feature of this equation is that the precession frequency is independent of the angle of inclination of the particle axis to the field direction. If a small magnetic field H1 is applied at right angles to the main field Ho, then at a particular point on the precessional path the nuclear dipole will experience a combination of Ho and H, which will tend to change the angle 8 by an amount 86. When the dipole has moved a further 180” around the precessional path, the combination of the two magnetic fields will change 0 by - 60, hence the total change in 19is zero. In order to change the orientation and thereby the magnetic energy of the particle, the secondary field Hz must rotate in synchronisation with the precession of the magnetic moment about Ho. In other words, the rotation of H1 must be in resonance with the Larmor precession about Ho. A rotating magnetic field of this type is associated with circularly polarised radiation of frequency vo. The sense of the Larmor precession depends upon the sign of the magnetic moment ; for example, a positive moment requires a left circularly polarised H1. For most purposes linear polarisation is quite adequate since a linearly oscillating field may be regarded as the superimposition of two fields rotating in opposite directions. Only the component
having the correct sense will synchronise with the precessing magnetic moment; the other component will have no effect In practice, an alternating current is passed through a coil mounted perpendicular to Ho so as to produce a magnetic field oscillating along the axis of the coil, say, along the x direction. The voltage applied to the coil at frequency a, produces the equivalent of two counter-rotating fields having the magnitudes (H1 COW t + H1 sine t) and (H, cosc0t - HI sina, t). When o corresponds to the resonance frequency, the magnetic dipole will absorb energy from the coil thereby causing the magnetic moment vector to dip towards the x y plane and an e.m.f. to be induced in a second (receiver) coil having its axis along the y direction. A purely classical model, as presented here, cannot predict more than this continuous absorption of energy. Having seen the nature of the resonance condition, some attention should now be paid to the orders of magnitude involved when these ideas are applied to electrons and magnetic nuclei. Spin angular momentum is quantised and
14
HIGH
RESOLUTION
NMR SPECTROSCOPY
has a value of [I(1 + l)p for nuclei and the projection of it on the main field axis is m It. The magnetic quantum number m can have the values 1, I - 1, . . . - I and transition between these levels is governed by the selection rule flm= f 1. On replacingq by the electronic charge e in equation (2.4) the following classical relationship is obtained P
pe
=2Mc*
This can be applied to atomic particles provided that p is replaced by m h and a factor g is introduced. The Land6 or spectroscopic splitting factor, g, is a measure of the spin and orbital motion of a particle in relation to its total angular momentum. For a completely free electron, g has a value of 20023 while g for a hydrogen nucleus is 558490. The magnetic moments of the electron and a nucleus are given by (2.12)
P
=
p=gNz
gN IbN
(2.13)
where 6 and &. are the Bohr and the nuclear magneton having the numerical values of 9.2712 x 1O-21 and 5.0493 x 1O-24 erg gauss-l respectively. The hydrogen nucleus is taken as the standard for calculation of the latter value. Now the magnetogyric ratio y for sub-atomic particles is defined by (2.14) and from equations (2.10), (2.12), (2.13) and (2.14), the resonance conditions for the electron and magnetic nucleus are found to be IIvg = g,B&
(2.15)
hv, = &al% Ho*
(2.16)
For a steady field of 10,000 gauss, the alternating current providing the oscillating field iY1 would need to have a frequency of 28,000 MC se& for electron resonance to be observed while the corresponding frequency for hydrogen nuclear resonance would have to be 42 MC set+. In practice, electron spin resonance is usually studied at lower main magnetic fields to enable X-band (3 cm; 9,000 MC se&) microwave components to be employed. For nuclear magnetic resonance work it is desirable to have a range of frequencies available to enable each magnetic nucleus to be examined at as high a field strength as adequate homogeneity will permit. Typical frequency values for a field of 10,000 gauss are
GENERAL
THEORY
OF NUCLEAR
MAGNETIC
‘H
42576 MC set-’
2I-l IlB
6.536 MC set- l 13.66OMc set-’
‘SC
10.705 MC set-l
l4N
3.077 MC set-’
‘9F
40.055 MC set-i
RESONANCE
15
2.2 THEQUANTUM MECWWALDESCXIPTIONOFNUCLEARMAGNFI~CRESONANCE When a nucleus of magnetic moment EL is placed in a magnetic field the Hamiltonian for the system is given by
%%= -p-H
(2.17)
and since p=yiI,
then
3P-_ - yi H-1.
(2.18)
The expectation values of the operatorI,are m, hence the expectation values of 2, that is the energy levels of the system, are E = y&mH.
(2.19)
In order to induce transitions between the energy states, a perturbation of some kind must be introduced. A suitable perturbation is the application of an oscillating magnetic field, and the necessary direction of this field can be decided from the properties of spin operators and eigenfunctions appropriate to a nucleus of spin I. A set of spin angular momentum operators Z,, I,,, Z, , I2 can be defined for a nucleus of spin Z which are analogous to the more familiar operators for electron spin angular momentum, namely S,, S,, S, and S2. I2 has eigenvalues Z(Z + 1) h, and Z, has eigenvalues m A.Restricting the argument to nuclei of spin Z = + for simplicity, the eigenvalues of Z, are simply f +h, hence the two possible energy levels are f 3y li H and if the spin eigenfunctions are denoted by the symbols ocand 8, then z,a = + *Aa;
z,/3 = - 31 /I.
(2.20)
The functions (Yand #Iare orthonormal so that
s
a*a&
I
=
a*j%h =
I p/M-r s
= 1
/!?*adr=o
(2.21)
The operators Z, and Z, have the properties ZXa==+1t9; Z,a = *ittttl;
I,/?-+ia Iy(? = -+iia
(2.22)
16
HIGH RESOLUTION NMR SPECTROSCOPY
The probability W’ of a transition between the two levels when an oscillating field is applied along the z-axis is given by W’ = yZHfI (411,18)~* = 0 hence transitions cannot be induced with this arrangement oscillating magnetic fields. If an oscillating field is applied along the x-axis, then (alZ,lB) = tn
(2.23) of the steady and
(2.24)
and a similar tite transition probability is obtained if the oscillating field is regarded as applied along the y-axis. The change in energy when a transition takes place is AE=yiH (2.25) thus the frequency of the oscillating magnetic field must be
AE
v=-=_ h
yH 2n
(2.26)
which is the resonance condition. For the general case of a nucleus of spin Z the transition probability between two energy levels m and m1 is given by . W’ =
r*H:l(~mll;l~m~) I*
(2.27)
which is non-zero only when m = ml f 1; that is, transitions between nuclear energy levels are governed by the selection rule Am = f 1. Transition probabilities for the general case of a nucleus of spin Z have also been calculated by a perturbation method (*). However, such an approximation method is not necessary and both steady state and transient phenomena of nuclear magnetic resonance have been treated rigorously by Schwingefi3) and also by Bloch and Rabi(*). 2.3 THE POPULATION OF SPM STATES The absorption coefficients obtained in optical spectroscopy are independent of the intensity of the radiation source. Normally, a rapid return is made from the excited to the ground state, the liberated energy being dissipated as heat. In nuclear magnetic resonance spectroscopy, the intensity of irradiation (that is, the amplitude of the radiofrequency field) may weaken the absorption signals or cause them to disappear entirely. The situation is a consequence of the isolation of a nucleus from its surrounding lattice. In the case of a liquid, the nucleus may remain isolated for periods of a few seconds while this can be extended to hours for solids at low temperatures. However, fluctuating magnetic fields associated with either inter- or intramolecular motion in the sample will have components at the resonance frequency and they will induce a return to the ground state. Whilst for most of the electromagnetic spectrum the probability of stimulated emission (or absorption)induced by irradiation is usuallynegligible,
GENERAL
THEORY
OF
NUCLEAR
MAGNETIC
RESONANCE
17
for nuclear magnetic resonance spectroscopy it is large’“’ and in comparison the probability of spontaneous emission is negligible(6) (the excited state would have a lifetime of about 10”’ years for hydrogen nuclei”)). Because of this, irradiation with an intense radiofrequency field would rapidly equalise the population of the energy levels, A dynamic equilibrium would be set up between the levels but no absorption would be detected. If there is a Boltzmann distribution of nuclei among the spin states, a net absorption of energy may be observed upon application of low intensity radiation at the resonance frequency. For an assembly of weakly coupled identical nuclei of spin value + the energy levels derived for an isolated nucleus may be applied to the assembly as a whole. It must be assumed that the nuclei do not interact with any other part of the system. When there is thermal equilibrium throughout the assembly then the relative populations of the two energy levels is given by (2.28) where N2 and Nt are the number of nuclei in the high and the low energy levels respectively, k is the Boltxmann constant and Tis theabsolute temperature. The separation of the energy levels is given by hence 2PH
(2.30)
-kT*
Thus, hydrogen nuclei in a field of 10,ooOgauss have an excess population in the lower state of
wf
-=7x kT
10-6.
On examination of equation (2.30) the desirability of the main magnetic field being as large as possible becomes apparent : not only are the energy levels more widely spaced but also the sensitivity is increased since the excess population is increased. The observation of nuclear magnetic resonance absorption depends upon the net absorption of energy by this small excess of population. With continued absorption, the fractional excess dwindles. We may still describe the ratio of populations by a Boltxmann factor but now there is a corresponding rise in temperatnre(*). For this purpose, T is defined as the spin temperature and the spin system may be regarded as undergoing radiofrquency heating. (Since it has been assumed that the nuclei do not interact with any other part of the system, the temperature of the lattice will not be altered. In the next section, the nature of spin-lattice interactions will be discussed.) It is worth digressing for a moment on the concept of spin temperature because when radiofrequency cooling occurs and there is an excess of nuclei in the higher energy level then
18
HIGH
RESOLUTION
NMR
SPECTROSCOPY
the spin temperature can have negative values. Ramsey cg) has discussed how the first and second laws of thermodynamics may be formulated so as to include systems characterised by negative temperature. The line dividing positive and negative temperatures corresponds to all energy levels being equally populated; statistically this is an infinite temperature. Therefore, systemsdonotpass through absolute zero when going from positive to negative temperatures. Since the negative absolute temperature may be defined as full occupation of the top energy level, it is just as inaccessible as the usual positive absolute zero. The probabilities of a given nucleus being in either the upper or lower states are +(l - pH/kT) and +(l + ,uH/kT) respectively. Because of the unequal distribution of nuclei there will be a resultant macroscopic moment per unit volume in the direction of the main magnetic field. The mean nuclear magnetic moment in this direction is given by
;+(l
-g)]=g.
+$$+(I
When there are N nuclei per unit volume the paramagnetic bility is
(2.31) volume suscepti-
NP x0 = 7
(2.32) At room temperature x0 has a value of only about 3 x lo-lo hence the nuclear effect is negligible compared with the electronic diamagnetic susceptibility which has a value of about 10e6. For the general case of a nucleus of spin value Z, (I + l)N# (2.33) x0 = 3ZkT
2.4 SPIN-LATTICE
’
hAXATION
Nuclear spins invariably interact with their surroundings but because the interaction is usually small it is permissible to distinguish between spin temperature and lattice temperature. However, the small interaction does enable thermal equilibrium to be established eventually between the two systems. The resultant temperature will be close to that of the lattice since the heat capacity of the spin system is negligible compared with that of the lattice, except at very low temperatures. Although absorption of radiofrequency energy tends to reduce the excess population in the lower state, flow of heat from the nuclei to the lattice will tend to oppose the shift in population. Because spontaneous emission is unimportant the only means of nuclear magnetic relaxation are the induced downward transitions stimulated by magnetic fields oscillating at
GBNBRAL
THBORY
OF NUCLEAR
MAGNETIC
RESONANCE
19
‘the Larmor frquency. Clearly, an important factor is the rate at which thermal equilibrium can be restored. When hydrogen nuclei in ice at - 180°C were exposed to a high intensity radiofrquency field and then this was replaced by a radiofrquency field small enough to allow resonance absorption to be observed, an exponential growth of the absorption signalwithtime was found”*). The time constant for the hydrogen nuclear spin-lattice relaxation process, that is the spin-lattice time Tl, was found to be 10 min. This is the half-life of the excited nucleus. For solids, T1 .has beenfound to vary from 1O-4 to 104 set while for pure liquids, the range is lo-3 to 10 set but the presence of paramagnetic species (see Section 2.5.1) may reduce the lower end of this range to 1O-s sec. In the previous section, it was stated that the transition probabilities of ab sorption and emission are equal; this is no longer true when spin-lattice interaction is present. If the upward and downward transition probabilities are denoted W, and W, respectively, then at equilibrium W,N,
= WIN,.
(234)
Hence
(2.35)
Provided the interaction energy is small compared with the total energy of the system, that is, the temperature.is stationary, then W,/ WI is independent of N2 and N1. Let
w, = (2.36) w, =
and
where W is the mean of WI and W,. The rates of change of the populations
dN1 -= dt
--=
dN2 dt
are N2Wz - N,W,.
(2.37)
On expansion of the exponential in equation (2.37) as allowed by the assumption that @,/kT < 1 we obtain
Wl
- Nz) dt
= -2W
(Nl -Nzl-(N~+Nzlk~.
MO
1
Multiplying both sides of this equation by cc and using the relationships (Nl + N,)p2$$
= XOHO= MO and
AN,-NJ==4
(2.38)
20
HIGH RESOLUTION
NMR
SPECTROSCOPY
the fundamental equation of paramagnetic relaxation is obtained, namely
dMz
-= dt
-2W(M,
(2.39)
- MO)
where x0 is the static volume nuclear paramagnetic susceptibility, MO is the macroscopic nuclear magnetic moment per unit volume and M, represents the component of the magnetic moment per unit volume which is parallel to the steady field Ho. The exponential relaxation process found in practice is described by equation (2.39) and is characterised by the time constant Tl which is equal to (2FV)-‘. Bloch’“) named Tl the longitudinal relaxation time because it determines the approach to equilibrium of the component of M lying parallel to Ho. The factors which influence Tl must now be considered. 2.5 MECHANLW OF SPIN-LAITICERELAXATION When molecules undergo random translation and rotational motion, any nuclear magnetic moment which may be present will experience a rapidly fluctuating magnetic field produced by neighbouring nuclear magnetic moments (and electron magnetic moments if they are also present). The Fourier component of this time-variant field, having a frequency equal to the resonance fre quency yo, causes transitions between energy levels in the same manner as the applied radiofrequency field. Bloembergen, Purcell and Pound(rz) have calculated the probability of transitions induced in this manner, as follows. The energy for each neighbouring magnetic moment involves three functions of the distance rI, between the nucleus i and its neighbourj, the angle &, between rl, and Ho, and the azimuthal angle vl, of rljmeasured from some fixed reference direction. The three functions are Yo, = (1 - 3cos*e*,)r*;3 YlJ
=
sin&J
Y*J = sin*
8,J
cost&J
(2.40)
exp(i(ptJ)ri’,3
exp(2itptJ)r;3
(2.41) (2.42)
The intensities of the Fourier spectra of these three position functions are denoted by and respectively at the frequency Y. The total intensities for all neighbours for the functions YoJ, YlJ and are represented by Jo (Y), J1 (Y) and Jz (Y) respectively. The motion of the neighbour j at the resonance frequency y. produces at nucleus i an oscillating magnetic field of this frequency which is capable of inducing transitions. Furthermore, fluctuations of Y2J at frequency 2 y. can induce transitions ; in the general case, when i and j are not identical nuclei, the frequency is not 2~~ but is equal to the sum of the precession frequencies of i and j. Fluctuations of the remaining position function Y,,, do not induce transitions. The standard methods of handling random phenomena of the type encountered here are reviewed in Appendix C, Chap ter 4 of reference 2. The calculation of the total probability W of transitions JoJ(v),
J1
J(v)
J2J(v)
Y2J
GENERAL
THEORY
OF
NUCLEAR
MAGNETIC
RESONANCE
21
induced by molecular fluctuations is presented in Appendix 3 of reference 13. The resulting expression for the spin-lattice relaxation time Tl is (2.43)
If unpaired electrons or other magnetic nuclei are present then l/T, is given by summation over all types of magnetic moment, the additional terms for other nuclei being
+
%k)
1
(2.44)
and for paramagnetic ions of magnetic moment p,,
Here, the assumption is made that the spatial quantisation of the ion does not change in a time of the order of the correlation time r,-this is not always valid since the relaxation times for paramagnetic ions can be as low as 10Vgsec. Equation (2.43) holds for.1 > + provided that the populations of successive levels are related by the same factor, thus enabling a spin temperature to be defined. In addition, nuclei having I > 3 possess an electric quadrupole’moment which may interact with the fluctuating gradient of the local electric field and thus provide another mechanism for spin-lattice relaxation in liquids. In order to estimate Tl , it is necessary to evaluate the spectral density functions J(Y) which may be calculated as follows Jo(V) = CJ&) J
= cIE;111’ 2r,,(l I
+ 4n5%)-‘.
(2.46)
te is the correlation time which is of the order of the time a molecule takes to turn through a radian or to move over a distance comparable with its dimensions. m is the time average of 1Yol 12.Taking water as an example, there are two interactions to consider: (i) that between nearest neighbours of hydrogen nuclei, that is, the two belonging to the same molecule and (ii) that between hydrogen nuclei of other water molecules. Hence T;’ = T;& f T;lmj. The intramolecular effect(i) produces a random rotational motion and may be calculated by regarding the molecule as rigid. The vector joining the two hydrogen nuclei has no preferred orientation, thus only the angular coordinates vary with time hence Y1 = case sin0 exp(i@r,‘j y2
=
sin20
exp(2iQ ri3
(2.47) (2.48)
22
HIG-H
RESOLUTION
where r. is the interhydrogen integration to be
NMR
SPECTROSCOPY
distance. The time averages of IYl’ are found by (2.49)
-Iy27z= $r;6. Combination
(2.50)
of these values with equations (2.46) and (2.43) gives 3y4b2
rc
1 + 47c%$t5
=x
+
2%
1 + 163t2~:t:
1-
(2.51)
The correlation time rc is related to rn required in the Debye theory of dielectric dispersion of polar liquids”*), rn being the time required for a dielectric to lose its charge in a preferred direction after the electric field has been switched off. -CD.
-Cc=--=_ 3
4nrj a3
(2.52)
3kT
a represents the radius of a molecule regarded as a sphere embedded in a viscous liquid of viscosity q: a can be calculated from the molar volume. For water, tc = 2.7 x lo-l2 set at 20°C; hence at the radiofrequencies usually employed the term in the square brackets in equation (2.51) reduces to 3t,. Since r. is approximately 15 A then (1 /T,),,,, = 0.12 se&. The time elapsing before relaxation is complete can be very long if the nucleus is sterically shielded from other magnetic nuclei within the same molecule. An example is the resonance of 13C at natural abundance (about 1 per cent) in neopentane; in most molecules the central carbon atom will be linked to non-magnetic 12C atoms and thus will be shielded from other magnetic nuclei, that is, the hydrogen nuclei present in the methyl groups. The intermolecular effect arises from fluctuations in the local field caused by random translational motion of neighbours. Nucleus i may be regarded as stationary while nucleus j moves by diffusion in a liquid of diffusion coefficient D. The time taken forj to move distance r in any direction from i is tcj which can be evaluated from the theory of Brownian motiorP5) r2 t,j = 120 327
ar2
=2kT, The intermolecular
contribution
(2.53)
I
is given by tcJ 1 + 4x2 &rc,
+
2% 1 + 167#t,,
1’
(2.54)
GENERAL
THEORY
OF NUCLEAR
MAGNETIC
RESONANCE
23
Because of the factor r6, only close neighbours are of importance hence the term in square brackets reduces to 3rCl. By assuming that all neighbours are independent and by summin g-their contributions by integrating from infinity to the radius of closest approach (r = 2a), (l/Z’,),,,,, is given by = 9nyil2 ?j&
(2.55)
m
where N is the number of molecules in 1 cm3. The value of (1/Ti)i,,er calculated in this manner is 0.08 set- 1 at 20% therefore the total value of l/T, is 0.20 XC-~. An experimental value (M)of 1/TXfor oxygen-free water is 0.27 se&. In the Bloembergen, Purcell and Pound theory outlined above, the main relaxation mechanism for water is assumed to be a nuclear dipole interaction. Anderson and Arnold(“) have tested the validity of this assumption by measuring Tl of hydrogen nuclei in mixtures of Hz0 and D20. The general expression for such a situation is
0
where the primes refer to non-identical nuclei interacting with the unprimed nuclei whose resonance is to be observed; rr’is the number of nuclei in the molecule. Due to its smaller magnetic moment, which enters quadratically into the relaxation process, a deuteron should be about ten times less effective than a hydrogen nucleus in promoting the relaxation of neighbouring hydrogen nuclei. The assumption would be verified if TI increased quantitatively upon the addition of heavy water. This was shown to be so experimentally; Tl depended upon X, the volume concentration of H20 in the mixture, in the following way ($)=
‘i($,[”
+ (1 - ml.
(2.57)
7’ is the ratio of the viscosity of the mixture to that of H20 and R is given by R
ID
=G
+1
.-.-=(-Jo42
z,
zt, Z” + 1
(2.58)
where ,un and pD are the magnetic moments and ZHand Zn are the spin values of hydrogen and deuterium nuclei respectively. l/q’T, was found to be a linear function of x. Bovey’l *) has questioned the importance of the intermolecular contribution to Tl when non-associated liquids are being considered. Liquids of this type
24
HIGH
RESOLUTION
NMR
SPECTROSCOPY
might have negligible intermolecular relaxation because of the r6 factor in the interaction of two dipoles. Bovey measured the spin-lattice relaxation times of mesitylene, tetramethylsilane, cyclohexane and benzene at various concen-, trations in carbon disulphide. From equation (2.55) it may be seen that qT, should be constant if intermolecular dipole-dipole contributions are unimportant-this was shown to be so within experimental error. The above theory of Bloembergen, Purcell and Pound(12) has been re-examined by Kubo and Tomita(lg) who produced a more rigorous theory based upon density matrix formalism. Gutowsky and Woessner(zo) have generalised the results so that they apply to systems of many nuclei. The relaxation time for the ith nucleus may be written
wherex are summations over nuclei df the same type as the ith and C* are over all other magnetic mklei. In equation (2.59), the’summations are over nuclei in the same molecule while in equation (2.60), they are over nuclei of the neighbouring molecule, l/r& being the mean value of l/rll for two molecules in contact. In obtaining these results, Gutowsky and Woessner have assumed that identical nuclei within a molecule are distinguishable : interference effects have been shown to be small in three and four spin systems(21). Mitchell and Eisner(22) have measured Tl for hydrogen nuclei in solutions of C6H6, C6HdCl and C6H12 in CS2 or C&. From their data they were able to choose the best model giving expressions for the correlation times, trot and zlnnr, of solutions. The Hill theoryc23) provided the superior model : in this, the mutual viscosity TAB is deduced from the binary solutions of viscosity q, given by
,.=(,,,,~)+(f~,.~)+(,,~)
(2.61)
where fAand fBare mole fractions of A and B respectively, a, and CBare the average distances between A and B type molecules, and a,,, is the average distance between molecules in the mixture. The rotational correlation time was deduced to be (2.62) Here
K: =
UABIB) . (MA+ mB> + I,) (mAmB)
(?4B
(2.63)
GENERAL
THEORY
OF
NUCLEAR
MAGNETIC
RESONANCE
25
where Z, and Za are the respective moments of inertia of the molecules A and B about their centres and ZABis the moment of inertia of A about the centre of B during collision. Hill’s theory leads to the following expression for T,-
Mitchell and Eisner have pointed out that the important factor of the Hill theory is the K: term. This term is of the form of a “reduced m,oment of inertia” divided by the reduced mass of the solvent-solute system, hence an approximate expression for the Hill correlation time is 2Zqa z, = FkT where Z is the average of the principal moments of inertia of the molecule of interest, a is the average of the semi-axes of the molecule, 3 is the solvent viscosity and p is the reduced mass of the solvent-solute system. The results obtained by Mitchell and Eisner indicate that the translational mechanism for relaxation is dominant for C6Hs and &H&l, while the rotational and translational mechanisms are about equally important for C6Hz2. It has been shown(24* 25), however, that different types of motion are not independent of one another. 2.5.1 The Eflect of Paramagnetic Substances The effects of electronic paramagnetism are so large compared with those of the nuclear moments that nuclear resonance is often undetectable in paramagnetic samples. If an atom has an unpaired electron as well as a magnetic nucleus then the local magnetic field produced at the nucleus by the electron is in the range lo3 to lo5 gauss. Splitting of the nuclear magnetic levels by these local fields may be accompanied by a more efficient spin-lattice relaxation mechanism but fluctuations in the local fields are great enough to split and/or to broaden the absorption line sufficiently to prevent detection. The observation of nuclear magnetic resonance is thus restricted to diamagnetic or weakly paramagnetic systems. Early workers(26m27) used dilute solutions (-1O-3 M) of ferric or manganous salts to reduce the spin-lattice relaxation time in order to obtain stronger signals. Bloembergen, Purcell and Pound (r2) have calculated Ti in terms of the number N of paramagnetic ions of effective magnetic moment bff in 1 cm3 in a solution of viscosity 7j +
= 123~~y’qNp,2,,/5kT.
(2.65)
1
Pople, Schneider and BemsteS2*) have replaced the factor 12/S by 4 in this equation. Further details of the effect of paramagnetic substances on Tr are given in reference(2g) and in the references quoted therein.
26
HIGH
RESOLUTION
NMR SPECTROSCOPY
If the ion forms complexes in solution then the apparent value of peff is concentration dependent and can be used as a means of studying such equilibria. Another effect which may be used for the same purpose is the concentration dependent shift of the nuclear resonance signal caused by the presence of paramagnetic ions(30* 31). It should be mentioned that diamagnetic ions lower the spin-lattice relaxation time of hydrogen nuclei in water by as much as a factor of fou+). 2.5.2 Quadrupole Effects Nuclei of spin greater than + may possess non-spherically symmetrical nuclear charge distributions, resulting in their having quadrupole moments Q. Positive and negative signs associated with Q imply that the shapes of the charge distribution about the spin axis are respectively those of prolate and oblate spheroids. Nuclei do not have electric dipole moments and so the energy of any nucleus is independent of its orientation in a uniform electric field. However, when an electric field gradient exists, the quadrupoles undergo precession which displaces the nuclear magnetic levels. Quadrupole energies range from negligibly small values to those much larger than nuclear dipolar magnetic interactions. For an isolated nucleus having I > 3 placed in a strong magnetic field?, the energy levels are given byCg3) E=
-W& I
e Q[3m2 - I(I+ + 41(211)
l)] a2v dzZ’
.
(2.66)
The magnetic quantum number m has the values - I to + Jand V is the electrostatic potential at the nucleus caused by external charges. The electric field gradient a2 V/i?z” may arise both from within a molecule (along a bond) and from its crystalline environment. In liquids or gases, Brownian motion tends to reduce intermolecular electric field gradients to zero but this is not true for intramolecular gradients as found in covalent bonds. The time-averaged interaction is generally larger than the magnetic interactions and thus provides an efficient relaxation mechanism from the fluctuating electric field gradients(34). Spin-lattice relaxation times for solids may be of the order of hours if quadrupolar effects are negligible but if large, TX has been found to be as small as 1O-4 sec. The fluctuating gradients originate mainly from lattice vibrations. The expression for the relaxation time Tl would be expected to resemble equation (2.51) and takes the form 1 1 1 3 e4q2 Q2t, + 1 + 163z2v;t,2 ~=10’h212(211)’ 1 +4&v;t:
1(2.67)
where
a2v
q = 7 a2- * The most notable results for liquids have been obtained with the hydrogen and deuterium nuclear resonances of mixtures of Hz0 and D20. In a 50 per cent solution of heavy water values of TXfor the hydrogen and deuterium nuclei are 3.0 and 0.5 set respectively. Now the nuclear magnetic moments of hydrot The magnetic field is parallel to the electric field gradient.
GENERAL
THEORY
OF
NUCLEAR
MAGNETIC
RESONANCE
27
gen and deuterium are 2.79 and O-857nuclear magnetons respectively, therefore, in the absence of quadrupolar effects, the deuterium relaxation time would be the longer. Bloembergen, Purcell and Pound(12) found that the observed Tl for the deuterium nuclear resonance could be accounted for by assuming a reasonable value for a2 V/W, that is, equivalent to a unit charge placed 1 A away from the deuterium nucleus. The quadrupole moment of this nucleus is small (2.77 x 1O-3 e x 1O-24 cm2) but nuclei having large values may have very short relaxation times thus causing the resonance line to be broad. Pound(3s) found absorption lines 10 gauss wide for 79Br and *lBr-whose quadrupole moments are O-33 x e x 1O-24 cm2 and 0.28 x e x 1O-24 cm2 respectivelypresent in aqueous solutions of lithium and sodium bromides. This particular line width is indicative of a relaxation time of 3 x IO-” sec. If the environment of a quadrupolar nucleus is sufkiently symmetrical, the electric Geld gradient will be zero at the nucleus thus eliminating quadrupole coupling effects. Because of this, nuclei havingappreciable quadrupolemoments, such as Na, Rb, Cs, Br and I, can have their magnetic resonance spectra observed in cubic crystals and in ionic solutions. Solvation effects can distort the spherical symmetry of the ions and give rise to line broadening because of quadrupolar contributions to Tl. Quadrupolar relaxation is responsible for wide hydrogen and nitrogen nuclear resonance lines in NH3 and in unsymmetrical ammonium salts, while narrow lines are observed for tetrahedrally symmetrical NH: and (CH3)4N+ ionF) (see Section 12.3.2). In covalent compounds such as Ccl,, the chlorine nuclei have a quadrupo1a.r coupling of about 35 MC set-’ with the field gradient of the C-Cl bond and this prevents observation of chlorine nuclear magnetic resonance. 2.5.3 The Anisotropic Electronic Shielding Efect The electronic shielding of magnetic nuclei is discussed in Chapter 3 but it is pertinent to mention here the relaxation mechanism which can occur when the chemical shielding is not isotropic. Unlike the other mechanisms’of spin-lattice relaxation presented above, this effect requires the presence of an appliedmagnetic field, the magnitude of which determines the relaxation time. The secondary magnetic field due to electronic currents will not generally be parallel to the direction of the external field, hence the perpendicular component so produced will oscillate as the molecule rotates thus causing transitions between nuclear energy levels. McConnell and Holm(37) have derived the relationship .
1 T,=
$ (
Y2H2(46)2T,
(2.68)
>
where da is the difference between the shielding constants parallel and perpendicular to the axis of symmetry. They obtained some evidence for the relaxation process by measuring Tl for 13C in natural abundance in carbon disulphide and in carbon tetrachloride. Under these conditions the dipole interaction is unimportant, therefore the longer relaxation time of 13CCl,, compared with 13CS2 can be attributed to the lower symmetry of the latter molecule.
28
HIGH
RESOLUTION
NMR
SPECTROSCOPY
Gutowsky and Woessner(20) have demonstrated the field dependence predicted by equation (2.68) and they have estimated Au for 1,3,%itluorobenxene and obtained agreement with the theoretical value. They also showed that Tl is longer for’ H than for rgF as expected from the anisotropic shielding mechanism. More recent work of Gutowsky, Lawrenson and Shimomura(38) cor&med this finding and it was also proposed that at room temperature there are three contributions to Tl for lgF nuclei, namely dipole-dipole interactions, Au and a spin-rotation interaction. 2.5.4 The E#ect of Pressure In an investigation of several saturated organic liquids it was found(3g’ 40) that Tl (‘H resonance) decreases with pressure (up to 1000 kg cm-*) much less rapidly than expected from the changes in viscosity. A similar result was found for aromatic liquids unless they are degassed, when Tl varies roughly linearly with the viscosity reciprocal. Available theories of the liquid state are inadequate to allow a detailed theoretical treatment of the effect of pressurt on Tl. Waugh and Johnson;41) have found that for gases, Tl is proportional to pressure (really to density) thus proving that intermolecular magnetic perturbations occurring in collisions are unimportant.. 2.6 SPIN-SPIN RELAXATION In addition to interacting with the lattice, magnetic nuclei can also interact among themselves. Each nuclear magnet is acted upon not only by the steady magnetic field Ho but also by the small local magnetic field H,,, produced by the neighbouring nuclear magnets. Since a magnetic dipole produces a field of p/r3 gauss at a distance r, a hydrogen nucleus will produce a field of about 14 gauss at another hydrogen nucleus at a distance of 1 A from the former. H,, falls off rapidly with increase of r so that only nearest neighbours have an important effect. It will be seen readily that the nuclei will experience different values of the steady magnetic field and hence there will be a broadening of the energy levels of the dipolar assembly and thus of the absorption lines. The re-orientation and diffusion of molecules in liquids, gases and some solids is usually so rapid that the local magnetic field is averaged to a very small value hence a narrow resonance line is observed. Taking HIOEas the spread of the local field, the resonance equation gives the range of frequencies of Larmor precession as
Lf the nuclear dipoles are precessing in phase at a particular instant, then the time required to get out of phase, (A v)-~, is about lO-4 sec. This may be regarded as part of the spin-spininteraction time Tz. In addition, the local fields from
GENERAL
THEORY
OF NUCLEAR
MAGNETIC
RESONANCE
29
magnetic neighbours may have oscillating (H-) and static (H,,,) components, thus nucleus j producing a magnetic field oscillating at its Larmor frequency (see Fig. 2.4) may induce a transition in nucleus k. The energy for the process comes fromj and simultaneous reorientation or flip-flop of both nuclei results, there being interchange of energy while conserving the total energy of the pair. Since the relative phases of the nuclei change in a time (d ~)-l, then a simple time interval will be required for spin exchange. This process causes a further broadening of the resonance line, observed at a tied frequency, by an amount
--. _--_--__ ;-_ ______ A _e--
___----
----_____
I
----_-______----i
H
___---,_--
I
Ii!2 n is<
--em__
/
--.
f
‘.__
1
Ifot
---~_~_~~~‘~---
norc
/
_-’
FIG.2.4 The static and oscillating components of the local iield produced by lmclcar magnets
comparable with Hi,. Only identical nuclei are capable of undergoing spin exchange whereas unlike nuclei can still contribute to Hi,. For a system containing nuclear species A and B, the local field A can be reduced by causing B to make rapid re-orientations (42). Line broadening will also result from an inhomogeneous main field since the gradients present will produce their own Hi,,. The spin-spin interaction time T2 may be defined approximately as the lifetime or phase memory time of a nuclear spin state. Bloch”‘) has assumed the rate of loss of phase memory to be exponential, so that
dMx -=-dt
MS T2
and
dM,
-_=-_
dt
M,
(2.70)
T2
where iW, and MY represent the resultant magnetic moment per unit volume at right angles to the steady field H,,. These are the transverse components of the pressing magnetisation vector M. In the absence of an applied radiofrequency magnetic field, any nuclear phase coherence will disappear in a time of the order of T2 (also known as the transverse relaxation time) until iU, and My are zero.
30
HIGH
RESOLUTION
NMR
SPECTROSCOPY
The theories of Kubo and TomitaC19) and of Solomon(43) show that for the condition t, < CO-l
1-
5% 2% 1 + fD2t2 1+0’2,2 (2.7 1) + f However, this theory neglected the intermolecular interaction which has since been estimated to be equal to the intramolecular contribution if the nuclei in the different molecules are assumed to be separated by distances of the order of bond lengths. From the theory given so far, it will be seen that Tl and T2 are approximately equal for liquids; the Bloch theory presented in section 2.9 predicts that they are exactly equal. The equality of Tl and Tz has been verified experimentally for pure liquids and is also found to be so for the relaxation of hydrogen nuclei in aqueous solutions of most paramagnetic ions, provided that complex formation is negligible. Exceptions are Gd3+ and Mn2+ which have T,/T, ratios of25 and 7.1 respectively. Bloembergen(44) has explained the anomaly in terms of a spin exchange interaction A. 1.S between paramagnetic ions and the hydrogen nuclei in neighbouring water molecules. The contribution of the exchange to Tl and T2 is = &(S and
+ l)k2fPt,p
[l + (WI - ws)’ r,‘]-1
l)h-2AZr,p.
(2.72) (2.73)
A is the coupling constant and p is the probability of a water molecule being adjacent to a paramagnetic ion, and is proportional to the ion concentration. ts is the spin-lattice relaxation of the electron spin. CD,and or are the resonance frequencies of electron and nuclear spins respectively. Since the development of many of the ideas discussed above, Bloch(45) has put forward a generalised theory of relaxation(46). 2.7 LINE wnsrrrs FOR LIQUIDS There are a number of definitions of line widths of a resonance absorption signal. For Gaussian and Lorentxian absorption line shapes@), there are the definitions (a) (b) (c) (In 2~‘)~‘~ T;’ (2~)“~ T;’ (Bzz)-~/~ T;’ Gaussian (3~)-“~ T;’ co W,)-’ (a) represents the situation where the line width is measured between half-maximum points, (b) is the line width measured between the points of maximum and minimum slope and (c) is the root mean square line width co g(v)dv [I0 whereg(v)isashapefunction.TheLorentzianshapeisreasonablywell-established
Lorentian
x 1112 (% -v)’
GENERAL
THEORY
OF
NUCLEAR
MAGNETIC
RESONANCE
31
for a single line in the high resolution experimentC4’). Tz is given by *g(v), (see Section 2.9) and is concerned only with the peak value of the normalised line shape curve(l’). One of the main sources of line broadening is that caused by the static local field. The component of the local magnetic field due to the nuclear dipole i of magnetic moment ,u aligned with the main field Ho at a point joined by the . vector rfl is (2.74) p rG3(3 co? 8,J - 1) where Brl is the angle between riJ and Ho. Since’ the magnitude y fi [I(1 + l)p, then the local field is y h [I(Z + I)]“2 r;: (3 cos~e,
- 1)
of p is (2.75)
which, by the use of the position function Ye, used in section 2.5, may be simplified to y h[l(Z + I)]“2 Yo,. (2.76) With liquids, the local field is fluctuating and only the frequency components of the spectral density Jo(v) (which are almost zero) contribute to the line width. Andrew(48) gives a rigorous derivation leading to the mean square line width Tl +‘I(z+
1) s
Job) dv
(2.77)
where f vI are the limits of the near-xero yrequencies. These limits should be of the order of the line width itself, that is, k A v = vl, where k is a numerical coefficient near unity and whose value depends upon the definitions of line width used. The square root of equation (2.77) gives the root mean square line width in units of magnetic field; frequency units are obtained by multiplication by y/23t, the line width A v is given implicitly by kAv
(Av)? =
(&)
y4 h2Z(Z + 1) 1’ Jo(v) dv.
(2.78)
-LAW
Since this equation only applies to identical nuclear neighbours, an extra term needs to be added if nuclei having different magnetic moments are present. Equation (2.46) is an expression for Jo(v), hence substitution followed by in* gration gives
(A4’ =
-& y’
h2 Z(Z + 1) 1 I~taa-1(2nkAvzc,). /
(2.79)
If all the correlation times rCI are su&iently long, the terms (2nk A v Q) tend to itinity, and tan-’ (2nk Av tc,) has the constant value n/2 and (A v)’ is given by
(A9” =
-& b4 h2w + 1) c Y&l
(2.80)
32
‘HIGH
RESOLUTION
NMR
SPECTROSCOPY
which is the mean square line width for a rigid system of nuclei. If all the correlation times rCJ are short so that all (2zk d YzC,) values tend to zero then (LIY) is zero. Usually for liquids, the rej values are sufficiently short to make (2nkd YzCj) 4 1, hence one obtains the explicit relation (2.81) It was seen in section (2.5) that tCj is proportional to q/T and since the viscosity, q, normally decreases rapidly with increase of temperature T, the line width should also decrease rapidly with increase of temperature. IO’
16' u %
c L
cl
I=
I6
FIO. 2.5 The relaxation timea Tland T2asa function of the correlation time t,. Bloembergen, Purcell and Poundtlz)
Another source of line broadening arises from the limitation imposed upon the lifetime of a nucleus in a given state by spin-lattice relaxation. This lifetime is of the order of 2T,, hence the energy levels are spread over the range h/2T, or alternatively, the resonance frequency can vary over the range (2nT,)-‘. For liquids having very short Tl values, this type of broadening is often roughly equal to that originating from magnetic dipolar broadening as discussed above. This near equality results from the fact that the intensity of the local field, given by equations like (2.46), is about the same for frequencies ranging from r. to almost zero when 2n y. r, Q 1. It is also implied that Tl and T2 are approximately equal in this region. Other regions, not normally encountered in high resolution nuclear magnetic resonance spectroscopy, have been discussed by Saha and DasC2),WertxC4g)and Purcell Cso).A further cause of line broadening is
GENERAL
THEORY
OF
NUCLEAR
MAGNETIC
RESONANCE
33
due to the inhomogeneity of Ho giving, in effect, the superimposition of absorption bands from molecules in different parts of the applied field. The effect of chemical exchange on line width is discussed in Chapter 9. Line shapes can be distorted by saturation, by transient effects, and by inadequate response of the amplifier following the detecting circuit. These factors are discussed in Sections 2.8, 2.10 and 6.5 respectively. 2.8 SATURATION In Section 2.3, it was pointed out that in nuclear magnetic resonance spectroscopy the intensity of the irradiating radiofrequency field can affect the observed resonance signal. In particular, line widths can be affected-a quantitative treatment will now be presented for an assembly of nuclei having I = 3. Before the radiofrequency is applied, the rate of change of the excess nuclei n per cm3 in the ground state is given by dn
--
n, - n
dt-
(2.82)
T,
where no is the value of n when the spin system is in thermal equilibrium with its surroundings. When the radiofrequency is switched on, the amount of energy absorbed is 2n P, where P is the probability per unit time for a transition between the two energy levels under the influence of irradiation. Hence, dn
n,-n - dtTI
(2.83)
- 211P.
The steady state value of the excess number n, is given by (2.84)
n, = (1 + 2PTJ,‘.
A value for P can be obtained from standard radiation theory(12) ; the probability of a transition in unit time between two states having magnetic quantumnumbers m and m’ is (2.85) P m-km’= +yZHf)(m 111m’)I” B(Y,,~ - v). HI is the amplitude of a radiofrequency field rotating in the correct sense and in a plane at right angles to the main field Ho, and (m 1I( m’) is the appropriate matrix element of the nuclear spin operator. S (vmnr’- v) is the Dirac delta function which is strictly zero at all values except v,,,,,,’= v, thus giving rise to an inlkritely sharp absorption or emission line. Since this is unreal, the function is replaced by a shape function g(v) normal&d as follows G(v)dv= Utilising the selection rule(51) d m = -
1. 1
(m Ill m’)2 = *(I + Ml fTRL3.
!?
(2.86)
- m+
1).
(2.87)
34
HIGH RESOLUTION NMR SPECTROSCOPY
Hence Pm+m-1 = iY2Hi(Z
+ m)U - m + l)g(v)
(2.88)
which for Z = +, reduces to P = *pa; Therefore na
-=
no
g(v).
[
1.
(2.89)
-1
1 +$y’H:g(v)T,
(2.90)
The right-hand side of equation (2.90) is usually denoted by 2 and called the saturation factor. When a radiofrequency field of large amplitude is applied, the excess number of nuclei n, in the lower energy level will become very small and the spin system is then said to be saturated. The spin temperature T, will be very high when this situation obtains. Saturation is greatest for the radiofrequency which gives the shape function g(v) its maximum value. In the previous section it was seen that (2.9 1)
Tz = 3g(%., therefore
2, = [l +
~‘H:TIT~I-~
where Z. is the saturation factor for the maximum value of g(y). In Section 2.4, the approach to equilibrium,of a spin system was seen to be governed by the characteristic time Tl. However, when the irradiation intensity is appreciable, this needs to be modified to T,Z (found by,solving equation (2.83)), which in effect means that equilibrium is reached more rapidly and is accompanied by broadening of the resonance line. When the spin-lattice relaxation process is inadequate, the signal may diminish in strength with time and in some cases may disappear entirely. Non-uniform saturation effects can be observed for a spectrum consisting of several lines each of which has a different relaxation time. A theoretical treatment of nuclear relaxation processes of a system containing two non-equivalent spins has been published (s2). Under saturation conditions the signal not only decreases in size but the line broadens because the diminution ofg(v) is most effective at the frequency corresponding to the line centre. Saturation is important for the observation of multiple quantum transitions(s3). These can be utilised in the analysis of high resolution NMR spectra (see Section 8.19.4). 2.9 RADIOPREQUENCY MAGNETICSUSCEPT~ILITYAND THR BLOCH EQUATIONS As a result of the various spin states being occupied to different extents, an assembly of magnetic nuclei in an applied field has a paramagnetic susceptibility. If M is the magnetic moment per unit volume of a substance placed in a magnetic field of strength Ho, then the static susceptibility x0 is defined M x0=x.
0
(2.93)
GENERAL
THEORY
OF NUCLEAR
The variation of x0 with temperature
MAGNETIC
35
RESONANCE
is represented by the equation
Np2(I + 1) x0 =
3kTI
(2.94)
’
Equation (2.94) may be derived in a few steps as follows. For an assembly of weakly interacting nuclei placed in a static magnetic field, the nuclei having quantum number m will occupy the level -m p Ho/I to the extent Czl+-
ew
mpH0 IkT,
(2.95)
where T, is the spin temperature at which equilibrium has been established. The population of each level per cm3 is given by
(2.96) The constant N/(2I + 1) gives the total population per cm3 in all the (2I+ levels, therefore i N(m) = N m--r
1)
(2.97)
M then is given by M=%
’ N(m)mp I
N/J’ Ho
irn2 = 12(21 -I- 1) kT, _I and since
= f $m2 M=
(2.98)
1(1 + 1)(2I + 1)
Np2 Ho(I + 1) 3kTJ
’
(2.99)
The static susceptibility is given by xo(T,) =
Np’(I + 1) 3kT,I
(2.100) l
The classical Langevin equation is obtained by writing ,u’ = (I + l)/I*p where $ is the length of the nuclear magnetic moment vector. At room temperature, x0 = 3.2 x lo-lo ergs gauss-2 for hydrogen nuclei in water. Since diamagnetic susceptibilities are of the order of 10m6ergs gauss-2, it is not possible to observe the effect of nuclearparamagnetism at room temperature by static susceptibility measurements. Even at 2”K, the paramagnetic contribution to the susceptibility of hydrogen is only 20 per cent of the totaP*). 2*
36
HIGH
RESOLUTION
NMR
SPECTROSCOPY
Bloch(“* w has introduced a set of equations to describe the interaction of the nuclear magnetisation, A4, with the radiofrequency field. In treating the radio: frequency phenomena, it is mathematically convenient to use complex quantities-these will be identified by script letters. The complex magnetic suscepti(2.101) The oscillating magnetic field applied to the sample may be identified as the real part of the complex field 2H, exp(i w t). The physically observable effects of the interactions between the radiofrequency field and the nuclear susceptibility are associated therefore with the real part of the magnetisation A4, which is derived from the complex magnetisation A as follows A = (x’ - i x”) 2 H1
= 2H,k’ Therefore
COSOJt + x” sino r) - 2iH1 (x” cost
t - x’ sinw t). (2.102)
A!, = 2H, (2’ cosw t + x” sino t).
(2.103)
In practice, a linearly oscillating radiofrequency magnetic field, 2H1 cost t, is applied so that the first term in the equation is the in-phase component of the magnetisation while the second term is the out-of-phase component. These terms involve x’ and ;d” respectively. The absorption of energy per unit volume per second is given by the real quantity A=v
s
0 I/V A = v 1’ 2H,cos(2~vt)2;~v(-~‘sin2?cvt 6
Hence
=2wf’Hf.
dt.
+ X”cos2ot)dt (2.105)
This shows that absorption by the spin system is proportional to the out-ofphase component 2“ of the nuclear magnetisation. It will also be noted that the sensitivity of the resonance method is enhanced by using higher radiofrequencies and by using large amplitudes providing that saturation is avoided. The frequency dependence of x’ and x” now requires investigation. The equation of motion for a single nuclear magnetic moment is
which, for constant H, represents the precession of the vector y about the vector H with angular frequency y H. Similarly, for an assembly of weakly interacting moments contained in unit volume, the equation of motion is dM -=yMxH. dt
(2.107)
GENERAL
THEORY
OF NUCLEAR
MAGNETIC
RESONANCE
37
In the usual magnetic resonance experiment, the three Cartesian components of H are not all constant but have the values H, = HI cosw t,
H, = -HI
sinw t
and
Hz = HO.
(2.108)
H, and H, together represent a radiofrequency field of amplitude HI rotating in a plane normal to the static field HO. Assuming that there are no other interactions involving the nuclei, then
(2.109)
d MY
-=y(-M,H, dt
-
dM, dt
+M,H,coswt)
= y(-
M, N, sinw t - M,, HI cosw t).
(2.110)
(2.111)
An important difference exists between these equations for the bulk magnetisation M and that for a single nuclear moment P. Without an applied radiofrequency field, M has only a z component while cc has x, y and z components. The individual nuclei pregss about the z-axis and have random phases hence the x and y components of f~ average to zero in forming M. In the absence of the rotating magnetic field and with the nuclei and the lattice in thermal equilibrium, then M, is equal to the static magnetisation MO. If the spin system and the lattice are not in thermal equilibrium, then M, approaches MO exponentially with the characteristic time Tl (see Section 2.4) the rate being
dMz -= dt
Wi, - MA T;
’
(2.112)
M, is proportional to the excess population of nuclei in the lower energy level. Bloch has called Tl the longitudinal relaxation time because it determines the approach to equilibrium of the component MI which is parallel to HO. M, and M, only differ from zero if a group of nuclei happen to be in phase, but any phase-coherence will be lost in a time of the order of the spin-spin interaction time T2 (neglecting any line shape peculiarities associated with the local fields). For simplicity, Bloch assumed that the rate of approach of M,or M, to equilibrium is exponential giving d& -= dt
dMY
-c-m*
dt
-- M, T2
MY T2
(2.113)
(2.114)
Bloch has used the term transversal relaxation time for T2 because it governs the time dependence of the transverse magnetisation components, M* and My.
38
HIGH
RESOLUTION
NMR
SPECTROSCOPY
In practice the time dependence of the magnetisation reflects the combined effects of the external applied fields and the interactions within the sample. Thus the actual behaviour of the system is described by adding together the appropriate magnetisation components as follows -3
dMs
y(M,H,
dt
y(M,H~coswt
dt
dt
(2.115)
sinw t) - + 2
d% -=
dMz
+ M,H,
- h&Hz)
(2.116)
-$ 2
=y(-M,H,sinwt-M,H,coscot)+
(Mo
T -
Mz).
(2 . 117)
1
These are the Bloch equations.Their derivation is based on macroscopic considerations in contrast to the arguments of Bloembergen, Purcell and Pound(12) quoted previously, which start from the magnetic properties of the nucleus and lead to macroscopic predictions.. The solution of the differential equations (2.115), (2.116) and (2.117) depends upon the boundary conditions employed. Transformation to a rotating system of axes (see Section 2.12.3) simplifies the procedure. The solution obtained by assuming that the resonance signal is observed under steady state or slow passage conditions will now be considered. Under these conditions, the absorption of radiofrequency energy is just balanced by transfer of energy from the nuclei to the lattice so that d M,/d t = 0, hence (2H, cosa, t)(oo - 4 T2 + 2Hl sinw t (2.118) % =$omoT2 co>2 Tf + y2 H: Tl T2 1 + (wo 1 MY = -z_xo~o M, = xo Ho
2 H1 cosw t - (2 H1 sina, t)(cao - CD)T2 7’2
1 + (00 - w)~ Tf + y2 H: Tl T2 1 + (coo - ~0)~T:
1 + ((00- w)~ Tf + y2 H: T1 T2 -
(2.119) (2.120)
In an actual experiment the oscillating field is linearly polarised (having the form H, = 2H1 cost t) as opposed to being circularly polarised. x’ and x” are identified as the coefficients of 2H1 cosw t and 2H1 sino t in M, (equation (2.103)) and taking these coefficients from equation (2.118) x’ and x” are found to be XI
=
&,2
coo
rf - 1 X - -zxowo
T,
6-o
1+
(Qg
-
CO)~
-
Q)> T2
Tf + ‘/’ H: Tl T2
1 7’2
1 + (00 - a)2 Tf + y2 H: Tl T2 *
(2.121) (2.122)
Both susceptibility components are seen to be frequency dependent and to contain the dimensionless parameter y2 Hf Tl T2 which determines the degree of
GENERAL
THEORY
OF NUCLEAR
MAGNETIC
RESONANCE
39
saturation, as explained in the previous section. When H1 is sufficiently small to avoid appreciable saturation, that is, when yz H: Ti T2 4 1, the Bloch susceptibilities become
x’ x
ho - 4 T2
=$oooT2
1
‘I -
(2.123)
1 + (CO,,- OJ)‘T; 1
-_i:xowo7’2
1 + (w. - co)“T;’
Plots of these susceptibilities against the dimensionless product (w. - o)T2 are shown in Figure 2.6. The curve for%” shows the resonant nature of absorption; this Lorentxian curve is the characteristic line shape for a damped oscillator as encountered in the theory of radiation and collision-broadened spectral lines. The curve for 2’ corresponds to the dispersion which accompanies an absorption band. If the spin temperature is equal to that of the lattice, 2” may also be written in terms of the shape function g(v) and the static susceptibility x0 = ;
f
x0
yogm
(2.125)
hence _ g(y)
=
2T2
(2.126)
1 + 4n2(ro - v)“T;’
If the actual absorption line shape g(v) is different from the Lorentxian-type assumed above, then the dispersion component can be calculated by use of the Kronig-Kramers relations(S6) which express a connection between the real and imaginary parts of the susceptibility. From equation (2.126) it will be seen that
0.8
$
-
0.4-
;
two-w
)
T,
is
-IN z Y) .Z
___--,
0’
-8
---
I -4
8
s -0.4
-0.8
-
‘-
FIG. 2.6 Absorption (x”) and dispersion (x’) Lorentzian curves represented by the susceptibility components f and:%’
40
HIGH RESOLUTION NMR SPECTROSCOPY
gwmnx =
2T,-a relation which was required in Section 2.7. The sharply resonant character of the absorption may be seen by comparing the line width with Q, the frequency at the line centre. The full width of the absorption line at half its maximum height is 2/T, which may be as small as 10-l set-’ and is to be compared with y0 having a value in the range 1 to 100 MC set-‘. The increased sensitivity of the radiofrequency resonance experiment is apparent by inspection of equation (2.125), after insertion of g(v) = 2 Tz the radiofrequency susceptibility is of the order of o. T2 or 103-10’ times greater than the static susceptibility. It follows, of course, that the out-of-phase susceptibility, x”, is extremely small except for values of w very close to oo. An optimum value for the radiofrequency field required to give the maximum signal strength, $=, can be calculated from equation (2.118) by putting o = o. and by setting d M&,)/d H, = 0, when y2 H: Tl T2 = 1.
(2.127)
Inserting into this equation equal and typical values of Tl and T2 of O-1 to 10 se for hydrogen nuclear resonance of liquids, the optimum range of HI is 1O-3 to IO-” gauss. Saturation broadens the absorption line as can be seen by substitution of equation (2.127) into (2.122) since the value of x” will be less than its maximum value. 2.10 TRANSIENTEFFECTS There are.nuclear resonance experiments which are concerned not with the steady state of the system but with the transient response. Examples are the rapid passage experiments described by Bloch’“) and used in relaxation investigations, the pulse method used by Torreycs7) and the spin-echo methods of HahrF). However, transient effects need to be considered since they have some bearing on the usual high resolution experiment. The steady state solution of the Bloch equation presented above assumes that equilibrium has been attained between the radiofrequency field and the nuclear magnetisation. In practice, the time taken to reach equilibrium is appreciable and thus considerable restriction is placed upon the rate at which resonance is traversed. If resonance is passed through too rapidly, several transient effects can occur; these will now be discussed. The static nuclear magnetisation at thermal equilibrium is MO which is reduced to MO/(1 + y2 Hf T,T2) in the steady state. Thus if the resonance line is entered too rapidly the signal will be strong at first and then become weaker as the magnetisation is reduced. The effect will cause distortion of the line shape. Another transient effect encountered with narrow resonance bands, as observed for liquids in a homogeneous magnetic field, manifests itself as relaxation “wiggles” or “ringing”. These are the damped oscillations which follow the resonance line when it is swept through rapidly. The origin of the effect is to be found in the inability of the nuclear magnetisation to follow the changing applied field. After the resonance has been traversed, a magnetic moment per-
GENERAL
THEORY
OF NUCLEAR
MAGNETIC
RESONANCE
41
sists in the plane at right angles to the main field for as long as a group of nuclei continue to precess in phase. Precession about H,, induces a signal at the precession frequency, the signal amplitude decaying at a rate governed by the phase memory time, T2. After the resonance line has been passed, the precession frequency and the radiofrequency differ slightly so that the two signals interfere with one another to give rise to a low frequency beat signal as shown in Fig. 2.7. BloembergerV) has shown that the phase difference between the resonance signal and the applied radiofrequency after passing through resonance at time t = 0 is given by I
(2.128) For a linear sweep of Ho (or the centre of a sinusoidal modulation v becomes 0;
dHo
L;
sweep) (2.129)
Ydtf2
so that the beat signal should be of the form cos(+y t Z d H,/d t). The beat signal remains as long as sufficient nuclei can remain in phase with one another to form an appreciable magnetic moment. From the exponential decay of the beat signal, the beats should follow the expression
(+y[$]i’>.
exp(2)cos
(2.130)
When conditions are such that the decay of wiggles is relatively slow, then the separation between wiggles well away from the start is approximately given by S(v, - v) =
Y
WJdt 2nv
*
(2.131)
It will be seen that the separation is inversely proportional to the distance from resonance. Jacobsohn and WangsnesPg) have deduced more rigorously from the Bloch equations an expression for the out-of-phase component u of the magnetic moment
cos38
+ IrlHl al’2
I
(A2
+
X2)3/2
+
‘**
(2.132) I
Ol; 0 = arctan
A
0 X
$ ld.
the deviation from the resonance angular frequency 1yl (dH,/dt), x = a112t and A = 1/(a1j2T2). HRS.
20
where do = a t, a =
42
HIGH RESOLUTION
NMR SPECTROSCOPY
Provided that the relaxation time Tz is less than about one-tenth of the sweep period, the periodicity and the variation of dAw/dt can be neglected and the sweep rate assumed to be linear. Figure 2.7 is a plot of - u/lrl H1 Tz against A w T2 ( = x/A) for various values of a 1’zT3. The free oscillations are completely S 3. For a’12T3 - 1, the ratio of the first maximum of damped out for 4 1~3Tz _ -v to the first minimum serves as a fairly sensitive measurement of T2. For values of a’12T2 - 2, the free oscillations about the slow passage tail appear sufficiently far enough out to enable the relaxation time T2 to be measured from their damping rate. Similar results are obtained for the in-phase component u which is associated with the dispersion mode. Equation (2.132) is of a similar form to equation (2.130).
0 -0.2 -0.4 -6
, -4
, -2
! 0
, 2
; 4
! 6
iI
! IO
!;
I 14
AwT, FIG. 2.7 Out-of-phase resonance line shapes for various values of a1i2T2: A = 0.0, B = 1 .O and C II 2-O. The field sweep is linear and the radiofrequency field is kept small. Jacobsohn and Wangsness(5g)
The true line shape is observed only when (2.133) where dH, is the line width expressed in field units. From the resonance condition, y. = y Ho/2n, it is clear that the experimental arrangement of a fixed field and a swept frequency could be used as an alternative to the method assumed in the above discussion. Whichever procedure is adopted, the static partner must be kept constant to a degree dictated by the precision desired in the experiment; for high resolution work on liquids such a requirement can be formidable. If a 5 milligauss separation between two lines arising from hydrogen nuclei is to be measured with an accuracy of 1 per cent at a sweep rate of 5 milligauss see-l with Ho = 5000 gauss, both the radiofrequency y. and Ho would have to be stable to 1 part in lOa set-’ with the linear sweep having an equivalent stability. The experimental difficulties of the
GENERAL
THEORY
OF NUCLEAR
MAGNETIC
43
RESONANCE
high resolution experiment thus become apparent. When the inhomogeneity of the main field is the controlling factor, the envelope of the decaying wiggles is not necessarily exponential. If a sample is regarded as being composed of two small volume elements separated by an appreciable distance in an inhomogeneous field then each produces its own pattern of wiggles which will be superimposed to give beating of beats (or “wiggle beats”) as shown in Fig. 2.8. Gabillard(60-62) has shown that the beat envelope is given by the Fourier transform of the distribution function of the inhomogeneous magnetic field over the sample. For a long specimen in a magnetic field having a linear gradient, the distribution function is approximately square, being constant for field strengths between the values at the ends of the specimen and zero for other field values
FIG. 2.8 The beating of beats pattern inhomogeneous
obtained from a sample placed in an magnetic field
This situation can be realised in practice when beating of beats is observed (as shown in Fig. 2.8). The effect has been applied to the measurement of small magnetic field gradients. As far as high resolution nuclear magnetic resonance is concerned, such a pattern is encountered when nuclei of the same species are located in different parts of a molecule, the field difference being due to slightly different chemical environments, so that the resonant frequencies are slightly different. In order to observe this particular phenomenon the steady magnetic field has to be very uniform; indeed, the effect can be utilised to check the uniformity. Also, wiggle beat formation provides a useful means of measuring the separation of closely spaced spin-spin components of a multiplet resonance signal. These two practical applications are discussed further in Chapter 7. Considering now two signals of equal intensity and separated by frequency d v, the signals can be represented by (see equation (2.130)) V,(r)=
V, exp
( V2(t)= 2a*
cos + al2
-$
Voexp[-‘@
\
>
(2.134) GF)]cos+=(f
- toj2. ,
HIGH
44
RESOLUTION
NMR SPECTROSCOPY
The time to between the two resonant frequencies is given by t _ 2nd~ 0
--.
(2.135)
a
Provided that the time to is small compared with Tz then the combined signal can be written V(f) = VI(t)
+ Vz(t) = 2Voexp(~)cos(+atz
1 x cos ?Olf,
(-
1
--al: 4
- iart
+$atjj)
(2.136)
. >
The first cosine factor leads to the usual wiggles and the second cosine factor provides the modulation of these wiggles. The time between the maxima of the beats is 2n = (Ay)-1. a to
(2.137)
The general case of a signal consisting of n equally spaced multiplets has been treated by Reilly(63). Since the intensities are proportional to the corresponding binomial coefficients (see Chapter 1), when to is sufficiently small the combined signal may be represented by V(t) =2”-l
Y,exp($f)cos[:aP
- (tt - l)nAvt]cos”-‘nAvt.
(2.138)
As before, the time interval between beat maxima is (A v)-‘. 2.11 NUCLEAR INDUCTION From equations (2.118) and (2.119) it may be seen that the components of the magnetisation, MZ and MY,constitute a magnetic moment of constant amplitude rotating in the x y plane with angular frequency w. Under the resonance conditions the component of the magnetic moment is oscillating therefore at a frequency o. If a coil of A area-turns containing the sample is placed with its axis in the xy plane then the oscillating magnetic flux will induce an electromotive force in the coil. The magnitude of this electromotive force is independent of the orientation of the coil axis, only the phase will be affected. Placing the axis of the coil along that of y will cause it to be at right angles both to the static magnetic field Ho (z-axis) and the radiofrequency field (x-axis). The magnetic induction A?,,is given by
= 8.z H,(;C’
sino
t - 2” cosw t).
(2.139)
Therefore the magnetic flux through the coil is Fo, = EAB,
(2,140)
GENERAL
THEORY
OF NUCLEAR
MAGNETIC
RESONANCE
45
where t is the filling factor giving the fraction of the effective coil area filled by the sample. The e.m.f., V, induced in the coil by the resonant nuclei may be calculated by applying Faraday’s law of magnetic induction to equation (2.139) to give d@p V =-dt = - 5 A 8n H, ~001’cosw t + 2” sino t).
(2.141)
The amplitude of the in-phase and quadrature components are thus proportional to 2’ and 2” respectively and may be amplified to permit observation of the signal. Bloch named the phenomenon nuclear induction because the e.m.f. is obtained directly by electromagnetic induction from the precessing resultant of the nuclear magnetic moments. Experimentally, two coils need to be wound round the sample. One has its axis along the x direction and is fed with radiofrequency current to give the radiofrequency field. The other coil, just discussed, having its axis along y is the receiver coil. When the planes of the two coils are mutually at right angles, the receiver coil will not pick up any e.m.f. by direct induction from the transmitter coil. It has been pointed out that the axis of the receiver coil can lie anywhere in the xy plane hence it is possible to make the transmitting and receiving coils one and the same coil. The experimental arrangement is then that for nuclear magnetic resonance absorption. The tendency is to restrict the term nuclear induction to those experiments conducted using a crossed-coil system and observing transient phenomena in which nuclear sigiials- are introduced after the external radiofrequency field has been switched off. The crossed-coil system has found favour only for high resolution studies while the single coil method is used for low resolution experiments with solids as well as for high resolution work. The crossed coil system has to be used when the relative signs of nuclear magnetic moments are to be determined. Calculation of the potential difference across a single coil (supplying the radiofrequency field and receiving the signal) shows its identity with the two coil system as follows. If a radiofrequency field is generated by loosely coupling the coil to a radiofrequency source, then a proportionally small amount of power is taken from the latter, which may be regarded as a source of constant current. For simplicity, it is assumed that the coil has no tuning condenser in parallel with it: in practice, a condenser is present but the conclusions are not affected. Since the inductance of a coil is proportional to the magnetic flux linking it, the change of inductance of the coil near resonance is AL = 43z(f - ix”) ,$L.
(2.142)
The change of impedance is therefore i w AL and as a result there is a change of potential difference across the coil given by AV-
4n05Llo(-X’sinot
where 1, cost t is the radiofrequency gives
+ X”cowt)
(2.143)
current. The definition of inductance
46
HIGH
RESOLUTION
NMR
SPECTROSCOPY
(2.144)
LI,, = 2H,A. Substitution of equation (2.144) into (2.143) gives d V = 58nAH,w(-~‘sinu,?
+ $‘cos~t).
(2.145)
The potentialdifference V is identical with the nuclear induction e.m.f. Y given by equation (2.141) apart from a phase difference of z/2 caused by placing the nuclear induction receiving coil at right angles to the transmitting coil. This equality of voltages expresses the fact that absorption in any system at a given frequency may be simulated by replacing the absorber with an oscillator of the same frequency but of opposite phase, whilst dispersion can be simulated by a similar oscillator with the phase in quadrature. Bloch’s treatment considers that such oscillators are actually present in the form of the forced precession of the resultant magnetic moment about H,,. Using the single coil method, a typical value of the applied radiofrequency voltage is 10V3volts which for hydrogen nuclei in water gives a readily detectable d V value of about 7 x 1O-5 volts. 2.12 THE MEASUREMENT
OF RELAXATION TIMES
High resolution nuclear resonance spectrometers are used occasionally for relaxation time measurements!84~ 8s!: a brief outline of the techniques and instrumentation usually employed is given here for the sake of completeness. 2.12.1 The Spin-Lattice Relaxation Time T1 (i) The direct method is applicable to the measurement of relaxation times of about 1 stx: an upper limit is set by the experimental difficulty of providing a large enough radiofrequency field. Several types of nuclear magnetic resonance equipment can be used ces); for example, Rollin’s circuit(64) which provides a direct measure of x” and utilises a single coil incorporated in a bridge circuit by Bloembergen, Purcell and Pound(12) (other bridge circuits have been reviewed elsewhere(65)), and th e two coil system of nuclear induction as devised by Bloch, Hansen and Packardt6@ is also applicable. The resonance signal is first observed with a radiofrequency field, H, , small enough to avoid saturation, then HI is increaseduntil the saturation factor Zis reduced to between IO-’ and 10e2. Thus, the susceptibility is reduced by this factor and the spin temperature is raised to a value 2-l times the lattice temperature. If HI is quickly returned to its original value, the signal then gradually increases in magnitude exponentially with the characteristic time T1 to its original level. Should it not be possible to avoid saturation when H, is suddenly decreased in value, then 2 will not be unity; its magnitude may be calculated and a correction applied. For liquids, signals are strong enough for oscilloscope presentation, but it is rarely possible to make direct signal versus time measurements because T, is usually less than 10 set: in this case, recourse has to be made to photographing the oscilloscope trace with a
GENERAL
THEORY
OF
NUCLEAR
MAGNETIC
RESONANCE
47
tine-camera. For weak signals it is necessary to use a phase-sensitive detector. Experimental details for the use of a high resolution instrument are given in reference 67. (ii) The progressive saturation method may be used(84) when Tl is too short (in the range 1O-2 to 1 set) for the direct method to be applied. From equation (2.90) 4 -= n0
-1
I
++‘H:gOT, 1
it may be seen that Tl can be obtained from a study of the saturation process. If a field modulation system is used in conjunction with a phase-sensitive amplier, then a signal proportional to the derivative d X”/dH will be observed. On differentiation
becomes dx”
d,”
-=YT dH
d [z
=;xovo
s(o)1
do
’
(2.147)
The magnitude of the observed signal is not’only proportional to dX”/dH but also to both Hi and the ampl%er gain. In practice, the product of the last two parameters-is kept constant, hence the reading R of the output meter is
R= c dlZd41 do
(2.148)
where c is a constant. Two important cases need to be distinguished: Case 1, when o,T, 4 1, where 0,/(27c) is the modulation frequency, the saturation factor varies during the modulation cycle. Case 2, when amTl % 1, Z cannot change during the modulation cycle, hence (2.149) Assuming that the absorption curve is Lorentxian then 2T2 &!@I
=
1 + (o -o~)~T;
(2.150)
For case 1, &
cc (1 + a)-3’2
(2.151)
where a is defined by a = y2H:TlT2
= Z;’ - 1.
(2.152)
48
HIGH
RESOLUTION
NMR
SPECTROSCOPY
For case 2, R maxcc [{16 + 16~ + c?}~‘~ - 2~11”~[S + 8a - U2
+ (a + 2) (16 + 16~ + ~~}l’~]-l.
(2.153)
The two forms of R rmx given in equations (2.151) and (2.153) are plotted in Fig. 2.9. Values of o,T, lying between the two limiting cases will give values of R,, lying between the two curves. Since line widths may be very narrow for liquid samples, line broadening from the inhomogeneity of the steady magnetic field may occur. If the line width exceeds the field modulation amplitude, each part of the specimen is in a slightly different steady field, and one gets integration over the whole specimen of
FIG. 2.9 The
effect of the saturation parameter a on the output meter reading
curves of the kind analysed in cases 1 or 2. In general, the decrease of R,,
with
a will follow much the same course as in these cases.
When liquids have relatively long relaxation times (that is, the natural line width is very narrow) the inhomogeneity of the steady field will again determine the line width. In this case it is the modulation amplitude, I$,,,, which determines the degree of saturation in various parts of the sample. Bloembergen(46) has shown that the degree of saturation is governed by the quantity y Hf T, fH,,, rather than by y2 Hf T, T,. The practical procedure in applying the progressive saturation method is to measure R,,, for a progressive increase of HI from low values. R,,, is then plotted against log HI, or, as is more usual, the logarithm of the radiofrequency voltage, when absolute values of T1 can be obtained only if the proportionality constant relating this voltage to HI is known. This constant can be estimated with difficulty either from the circuit parameters or more usually by calibrating the apparatus using the direct method, providing the sample has a sufficiently long T1 value. Relative value.; of T, for different substances or for the same substance at different temperatures can be obtained with greater accuracy if
GENERAL
THEORY
OF
NUCLEAR
MAGNETIC
RESONANCE
49
we are dealing with the same case throughout. Thus equal values of R_ means the saturation functions u are the same, hence for substances A and B (2.154) or
(2.155)
(H,),/(H,), is derived from the antilogarithms of the radiofrequency voltages and (T,),/(T,), is the ratio of the line widths aH,,/aH, provided that the line shapes are geometrically similar. Errors of up to a factor of two may occur if the two assumptions are not satisfied, (a) that the same case is operative throughout and (b) that the resonance line retains the same shape. A variation of the method has been suggested by SuryarP*). If the liquid is made to flow through a tube around which the specimen coil is wound, the nuclei enter the coil in an unsaturated condition and are progressively saturated as they proceed through the coil. The signal strength is therefore greater than would be the case if the liquid had been stationary and subjected to the radiofrequency field for a long time. With certain approximations, the fractional increase in signal strength resulting from flow is given by v TJd, where v is the velocity of liquid flow and d is the coil length; hence Tl can be calculated. The continuous flow method offers a means of improving the signal-to-noise ratio for weak signals. 2.12.2 The Spin-Spill Relaxation Time T2 When lines are wider than the field variation over the sample, T2 may be obtained from the line width A H by means of the expression T,=
2(y A H)-‘.
(2.156)
Thus if the main magnetic field is sufficiently uniform, Tz may be determined by measuring the width of the *mode (absorption) line under the condition y2H:TlT2
Q 1.
That this condition prevails may be ensured by employing a value of HI less than the saturation value by a factor of about ten. A check on the uniformity of the main field may be made by substituting a sample having a natural line width much narrower than the one to be measured. The damping time of the wiggles is equal to T2 if the main magnetic field is sticiently homogeneous. If the time between successive sweeps of a line is greater than T2, the recording shows no dependence on past history but when the sweep period equals T2, beats occur because of interference between succe+ sive sweeps, irrespective of whether a linear or a sinusoidal sweep is used(6g). Depending upon the experimental conditions, T2 can be determined from the damping constant or from signal amplitudes. However, for extremely narrow
50
HIGH
RESOLUTION
NMR SPECTROSCOPY
lines, such as those usually found for hydrogen nuclei in liquids, the required homogeneity is greater than has yet been attained and thus the observed damping constant is not related simply to T2. Solomon(70) has described a method of measuring intermediate or long spinspin relaxation times even when the natural line width is obscured by magnetic field inhomogeneity. The method is based on the fact that a decrease of the transverse polarisation is observed in the presence of the radiofrequency field H1 : the decay of this transverse polarisation is a measure of T2. 2.12.3 Pulse Methods Although transient methods are not well suited to the study of problems of molecular structure, they are used extensively for the measurement of relaxation times. However, coupling constants and chemical shifts have been measured using such techniqueP3’. Both Tl and T2 can be measured by the use of pulse techniques. A brief account of these methods will be presented after discussing the introduction of the use of rotating coordinates to problems in magnetic resonance(“). Consider a system consisting of one or more nuclei or atoms all of which have the same magnetogyric ratio. If I is the nuclear angular momentum in units of h, the nuclear magnetic moment is y h I and the equation of motion in a stationary coordinate system is h$
=/.lxH’
(2.157)
=yhIxH. If a/at represents differentiation with angular velocity o then
with respect to a coordinate
d1 dt-
d1 --+wxI
system rotating
(2.158)
at
where I is the angular momentum as measured by a stationary observer, and aI/at represents the variation of the stationary observer’s I with time as observed by the rotating observer. Re-arrangement of equation (2.158) gives (2.159)
where &, by
is the effective field in the rotating coordinate system and is defined H ell = H + ” . Y
Thus the effect of rotating the coordinate field by the extra term w/y.
(2.160)
system is to change the effective
GENERAL
THEORY
OF NUCLEAR
MAGNETIC
RESONANCE
51
In the nuclear magnetic resonance experiment there is a constant field Ho, and at right angles to & we have a much weaker field HI, rotating with the angular velocity -CO. From the point of view of the rotating coordinate system, however, none of the magnetic fields is changing with time. The axes of the system can be selected so that H,, = H,k,
HI = H,i,
O=
-ok
(2.161)
A
s! Y
H,0
3
-____________-__
A H ctf
FIG. 2.10 The ctkctive
I
magnetic field in a rotating
tzwrdhate
system
where i and k are unit vectors. Then, on the rotating coordinate obtain
system we
(2.162) Since this field is constant in time, the solution of the equation of motion of the system is simpler than in the stationary system of coordinates. From equation (2.162), the magnitude of the effective magnetic field is
=-
L? Y
where
Q = K% - CO)’+ (yHl)2]“2 =
By definition, CO,,= y Ho
(%
I
(2.163)
1 (2.164)
52
HIGH
RESOLUTION
NMR
SPECTROSCOPY
The angle 19between H,rr and H,, is obtained from case = (Wg - 0)/Q
sin6 = (oO H,/H,)/Q.
(2.165)
Hence, when w = oo, 6 = 90” and a magnetic moment initially parallel to Ho can precess about Hfr until it becomes antiparallel to Ho. Thus, such a moment can have its orientation relative to Ho changed by the greatest amount when o = wo, so that w. can be considered as being the resonance frequency of the system. Pulse methods can be divided into two categories : those concerned with transient behaviour during the time of application of the radiofrequency pulses, and those concerned with transient phenomena observed in the time between radiofrequency pulses. The two principal methods were developed by Torrey(57) and Hah#*) respectively. Torrey’s method will be described first. In principle, one can modulate either the steady field Ho or the radiofrequency field, H, (frequency or amplitude modulation) but since a flat-topped pulse is required only the radiofrequency field amplitude can be modulated conveniently. Using a sharp-fronted pulse, the approach to resonance takes place in a time short compared with the times (r HI)-‘, Tl and T2. The approach to resonance is more rapid than that in the Bloch “rapid passage” experiment which is made in a time long compared with (yH,)-’ but short compared with Tl and Tz. In the absence of relaxation effects the motion of the magnetic moment vector is described by dM -=y(M xH) dt where H is the resultant applied magnetic field, which has a steady component Ho along the z-axis and a component of amplitude HI rotating in the x y plane with angular velocity w. Transforming to a system of axes rotating about the z-axis in the same sense as HI and with the same angular velocity w then, if dM,,/d t is the rate of change of M in the new system, we get
d&f -=_ dt
dM+Mxw dt
(2.167)
or by equation (2.166)
dMerr
-
dt
= Y(M x He,,).
(2.168)
In the rotating system HclI is constant in time and thus equation (2.168) corresponds to the precession of M about H,, with angular velocity 52determined by equation (2.164). Near resonance, D 4 w and the motion as viewed from the laboratory will appear as a slow nutation (change of the angle 6) superimposed upon a rapid precession about the z direction (see Fig. 2.11). The effect of relaxation processes is to damp the nutation. When the absorption component (v-mode) of resonance is being observed, a decaying oscillation is seen during the “on” periods, the angular frequency
GENERAL THEORY OF NUCLEAR
MAGNETIC
53
RESONANCE
of oscillation being Q. Torrey (57)has used the Bloch equations to show that for the case (y Hi)-’ $ Tl , T2, the decay time near resonance is
2(-&+$)-‘.
(2.169)
When the pulse duration is long compared with this time, the final signal is small corresponding to saturation by the relatively intense radiofrequency field. The resultant magnetic moment of the nuclear assembly recovers exponentially with the characteristic time Tl between pulses. Observation of the
to:
ib)
FIG. 2.11 (a) Combined precession and nutation of the net magnetic moment, (b) pure nutation of the net magnetic moment as viewed from the rotating coordinates x’, y’, z’
dependence of the initial amplitude on the duration of the “off” time leads to the evaluation of the spin-lattice relaxation time. Once Tl is known, T2 can be calculated from the decay time of the nutationaloscillation givenby equation (2.169). In many cases Ti % T2 and then a knowledge of TXis unnecessary. This method of “nutational resonance” is mainly of value for the determination of relaxation times, and because of its fast response also has advantages in scanning for unknown spectra. Its resolving power is less than that of the more conventional methods hence it is unsuitable for studying the structure and shape of resonance lines. Concurrent with the pulse studies of Torrey, HahrP) was investigating the application of pulses in the nuclear induction experiment-the resultant signal being recorded after the pulse is switched off. Nutational resonance could be detected and Hahn obtained nutational frequencies down to IQ = 5 cycles see-’ for ‘H and lgF nuclei. Two strong pulses separated by time t are applied and then at time 2 t one observes a symmetrical signal similar to those obtained under steady state conditions. The first pulse is of sufficient magnitude and duration
54
HIGH
RESOLUTION
NMR SPECTROSCOPY
(time tP) to rotate the macroscopic polarisation vector by 90” from the z-axis into the xy plane. To an observer in the coordinate system rotating about the z-axis, the vector would appear to turn about the HI axis. Because of the spread in precession frequencies, the vectors corresponding to the nuclei of
obc
FIG. 2.12 The formation
d
*
f
of an echo after successive
h
90” and 180’ pulses.
(a) system in thermal equilibrium in a static field. (b) fkst pulse (90”) turns the magnetic moment vector into the x’y’ plane. (c) at conclusion of the pulse the moments will be bunched together provided that r, is small. (d) the moments in the x’y’ plane begin to fan out due to free induction decay. (e) 1800 pulse applied after time t which reflects the fanned system of vectors in the x’z’ plane. (f) the vectors continue to rotate as before but now they coalesce. (g) at time 2t the vectors have recombined and hence produce a signal. (h) the echo signal has decayed
a particular frequency would fan out in the y’ x’ plane. A second 180” pulse of the same intensity applied at time t for a duration of 2t, will have the effect of reflecting each of these vectors in the x’z’ plane which causes them to come together so that at time 2 t they will again be in phase and induce an echo signal. In practice then, one observes not just two signals on the oscilloscope trace corresponding to the pulses but a third also appears as an echo. The entire sequence
GENERAL
THEORY
OF NUCLEAR
MAGNETIC
RESONANCE
55
and its effect is shown in Fig. 2.12. A detailed mathematical analysis of the phenomenon has been made by HahrP) and by Das and Saha(72) but a simple treatment will sufke here. The first pulse changes the orientation of the resultant magnetic moment vector of each volume element of the nuclear assembly, and a nuclear induction signal decaying with time T2 is obtained from each volume element as it precesses freely. Because of inhomogeneity of the magnetic field, the volume elements each have a different precession frequency. The phase coherence of the precessing magnetic moments is lost in a time 6~ 8&)-‘, if this is supposed to be less than T2, and the first signal appears. The second pulse applied at a time t which is less than T2 may be considered as turning the magnetisation through 180”. The vectors are turned round and while precessing resume their phase
. FIG. 213
90.
tee*
180’
18V
180’
pulse sequence and shoes produced Gur and PurccW7’)
180.
when Tz is to be measured.
differences at the same rate as they previously lost them. They finally find themselves in phase again after a further time t giving rise to a third nuclear induction signal which in turn decays as the phase coherence is lost once again. If the magnetic field is very inhomogeneous then the loss of phase coherence is very rapid and all three signals are narrow having a width of the order of (y dHc)-I. Thus the spin echo mechanism eliminates the normally adverse effect of field inhomogeneity. The strength of the echo signal (observed at time 22) is determined by the factor exp( - 22/T,) hence a plot of the logarithm of the echo amplitude against various values of 2t enables the spin-spin relaxation time T2 to be measured. When diffusion processes are important the amplitude is given by@*) (2.170) For the 90” - 180” echo(73) k
(2.171)
where D is the diffusion constant and G is the mean value of the magnetic field gradient.
56
HIGH
RESOLUTION
NMR
SPECTROSCOPY
Figure 2.13 illustrates how Tz may be obtained from a succession of pulses. In a similar way the value of Tl can be obtained by the variation in amplitude of stimulated (secondary) echoes (58). These echoes are produced by 90“ pulses: a third pulse is applied at a time ti > 2t after the first pulse, that is at a time t1 - 2t after the echo. Under these conditions echoes are also obtained at times tl + t, 2t, - 2t, 2t, - t and 2r,, there being a total of seven observable signals. The signal at time t1 + t falls in amplitude according to the function exp( - t,/T,) thus enabling the spin-lattice relaxation time, Tl, to be estimated. General methods of treating any sequence of pulses have been developed(74-77). The spin echo technique has been developed(73* 78-8o1to give more convenient ways of measuring Tl and Tz and also to allow direct measurement of selfdiffusion coefficients in liquids. Some applications to studies of chemical equilibria are given in Chapter 9. A versatile and simple nuclear resonance pulse apparatus has been described by Luszczynski and Powles(81) which will measure spin-lattice times from 500 microseconds to many seconds, and spin-spin relaxation times ranging from 20 microseconds to several seconds with an accuracy of f 5 per cent. A spin echo attachment has been used in conjunction with a high resolution NMR spectrometer(*2’.
REFERENCES 1. N. F. RAMSEY,Nuclear Moktwts, Chapman and Hall, London (1953). 2. A. K. SAHA and T. P. DAS, Theory and Applications of NuclearInduction, Saha Institute of Nuclear Physics, Calcutta (1957). 3. J. SCHWINGER, Phys. Rev., 51, 648 (1937). 4. F. Beocw and I. I. RAEI, Rec. Mod. Phys., 17, 237 (1945). 5. A. E~~TQN, Phys. Z., 18, 121 (1917). 6. E. M. PER&, Phys. Rat., 69, 681 (1946). 7. Reference 2, page 166. 8. A. A~IUQAM, The Principles of hcfear Magnetism, Clarendon Press, Oxford (1961). 9. N. F. RAWEY, Phys. Reu., 103, 20 (1956). 10. G. E. PAKE,Phys. Rat., 74, 979 (1948). 11. F. BLXH, Phys. Rev., 70,460 (1946). 12. N. BL~~ERQEN, E. M. PURCELLand R. V. POUND,Phys. Reo., 73, 679 (1948). 13. E. R. ANDREW,Nuclear Magnetic Resonance, Cambridge University Press (1956). 14. P. DEBYE,Polar Molecules, Dover, New York (1945). 15. S. CHANDRASEKHAR, Reo. Mod. Phys., 15, 1 (1943). 16. G. CHIAROTI-I and L. GIULOTTO, Phys. Reo., 93, 1241 (1954). 17. W. A. ANDERSONand J. T. ARNOLD,Phys. Rev., 101,511 (1956). 18. F. A. BOVEY,J. Chem. Phys., 32,1877 (1960). 19. R. Kuao and K. TOMITA,J. Phys. .Soc. Japan, 9, 888 (1954). 20. H. S. GUT~W~KYand D. E. Woeswe~, Phys. Rev., 104, 843 (1956). 21. P. S. HUBBARD,Phys. Rev., 109, 1153 (1958). 22. R. W. Mrrcm~ and M. EISNER,J. Chem. Phys., 33, 86 (1960). 23. N. E. HILL, Proc. Phys. Sot., B 67, 149 (1954). 24. E. 0. SNSKAL, D. E. WOESSNER, T. C. FARRARand H. S. GUTOWSKY, J. Chem. Phys., 31, 55 (1959). 25. D. E. WOESSNER, J. C/tern. Phys., 36, 1 (1962).
GENERAL
THEORY
OF NUCLEAR
MAGNMIC
26. F. BLOCH,W. W. HANSENand M. PACKARD,Phys. Rev., 70.414
RESONANCE
57
(1946). 27. W. G. PROCTOR and F. C. Yu, Phys. Rev., 76, 1728 (1949). 28. J. A. POPLE,W. G. E~HNEIDER and H. J. BERNSTEIN,High-Resolution Nu&ar Magnetic Resonance, McGraw-Hill, New York (1959). 29. L. 0. MORGAN,J. MURPHYand P. F. Cm, J. Amer. Chem. Sot., 81, 5043 (1959). 30. P. GRIVJZand Y. AYE, Cbmpt. Rend., 232, 1094 (1951). 31. D. F. EVANS,J. Chem Sot., 2003 (1959). 32. S. BROER~MA, Bull. Amer. Phys. Sot., 30, No. 1, 43 (1955). 33. R. V. POUND,Phys. Rev., 79, 685 (1950). 34. Y. AYANT, Compt. Rend., 238, 1876 (1954). 35. R. V. POUND, Phys. Rev., 73, 1112 (1948). 36. R. A. GGC and J. D. RAY, J. Chem. Phys., 26,1339,1340 (1957). 37. H. M. MCCONNELL and C. H. HOLM,J. C/tern. Plr_vs., 25, 1289 (1956). 38. H. S. G~TOWSKY,I. J. LAWRENSON and K. SHIMO~A, Phys. Rev. Letters, 6,349 (1961). 39. G. B. &NEDEK and E. M. PURCELL,J. Ckem. Phys., 22,2003 (1954). 40. A. W. NOLLEand P. P. MAHENDROO, J. Chem. Phys., 33,863 (1960). 41. J. S. WAUGHand C. S. JOHNSON,Discuss. Faraaby Sot., 34, 191 (1962). 42. B. HERZOGand E. L. HAHN, E&f. Amer. Phy.s. Sot., 29, No. 7, 11 (1954). I 43. I. SOUIMON,Phys. Rev., 99, 559 (1955). 44. N. BLOEMBERGEN, J. Chem. Phys., 27,572 (1957). 45. F. BLXX-I,Phys. Rev., 105,1206 (1957). 46. N. BLCZ~ERGEN,Nuclear Magnetic Reklxatiotr, W. A. Benjamin, New York (1961). 47. G. V. D. TIERS,J. Phys. Chem., 65, 1916 (1961). 48. Reference 13, page 158. 49. J. E. WERTZ,Chem. Rev., 55,829 (1955). 50. E. M. PURCELL,Nuouo Cimento, 6,961 (1957). 51. E. U. CONDONand G. H. SHORILEY,?Jtc Theory of Atomic Spectra, Camlnidgc University Press (1935). 52. H. S~~kazu and S. F~~~ARA, J. Chem. Phys., 34, 1501 (1961). 53. J. I. KAPLAN and S. MEIBOOM, Phys. Rev., 106,499 (1957). . 54. B. E. LASAREVand L. V. S~~~JBNIKOV, Physik. Z. Sowjetunion, 11,445(1937). 55. R. K. WANGSNESS and F. BLOCH,Phys. Rev., 89,728 (1953). 56. Reference 13, page 23. 57. H. C. Tom, Phys. Rev., 76, 1059 (1949). 58. E. L. I-Lum, Phys. Reu., 80, 580 (1950). 59. B. A. JACOBSOHN and R. K. WANGSNESS,Phys. Rev., 73,942 (1948). 60. R. GABILLARD, Compt. Rend,, 232,155l (1951); 233,39 (1951). 61. R. GALIILLARD, Phys. Rev., 85, 694 (1952). 62. R. GABILLARD,Rev. Sci. (Paris) 90,307 (1952). 63. C. A. REILLY,J. Chem. Phys., 25, 604 (1956). 64. B. V. ROLLIN,Nature, 158, 669 (1946); Rep. Prog. Phys., 12, 22 (1949). 65. Reference 13, page 46. 66. F. BLXH, W. W. HANSENand M. PACKARD, Phys. Rev., 69,127 (1946). 67. VARIAN-pm, NMR and EPR S’ctroscopy, Pergamon Press, Oxford (1960). 68. G. SURYAN,Proc. I&an. Acad. Sci., A 33, 107 (1951). 69. E. E. SALP~, Proc. Phys. Sot., A 63, 337 (1950). 70. I. SOLOMON, Compt. Rend., 248,92 (1959). 71. I. I. RABI, N. F. Rmme~ and J. SCHWINGER, Rev. Mod. Phys., 26, 167 (1954). 72. T. P. DA.Sand A. K. &HA, Phys. Rev., 93, 749 (1954). 73. H. Y. CARRand E. M. PURCELL,Phys. Rev., 94, 630 (1954). 74. T. P. DM and S. K. ROY, Phys. Rev., 98, 525 (1955). 75. E. T. JAYNES,Phys. Rev., 98, 1099 (1955). 76. A. L. Bmnr, Phys. Rev., 98, 1105 (1955). 77. E. JENSEN.Actu Pofytechnica Stand., No. 7 (1960). 78. P. GRIVET,La RCsonance Paramagn&ique Nuclbaire, Centre National de la Recherchc Scientifique, Paris (1955).
58 79. 80. 81. 82. 83. 84. 85.
HIGH
RESOLUTION
NMR
SPECTROSCOPY
S. MEIBOOM and D. GILL, Z&Y.Scf. Instruments, 29,688 (1958). J. G. Pow and D. CUTLER,Nutum, 184.1123 (1959). K. LIJ~~~~YNSKI and J. G. POWLES,J. Scf. Zmtrummts, 36, 57 (1959). M. SUSON,A. TZALMONA and A. LOEWENSTEIN, J. Scf. Znstruments,40, 133 (1963). J. G. POWLESand J. H. STRANGE, Dfscuss. Faaby Sot., 34.30 (1963. A. L. VANGEETand D. N. Hum, Anal. Chem. 37,979 (1965). A. L. VAN GEETand D. N. Hum, And. Chem. 37,983 (1965).
CHAPTER
THE ORIGIN AND
3
OF CHEMICAL
SPIN-SPIN
SHIFTS
COUPLING
INTRODUCTION 3.1 THE CHEMICAL Sm AN ASSEMBLY of magnetic nuclei each with spin I in a magnetic field, H, will
absorb energy at a frequency given by (3.1) When the nuclei are part of a rigid lattice, the resonance line is broadened over a range of field strengths and has a width dependent on the relaxation times of the system (see Section 2.7). The centre of the line is still given by equation (3.1). The presence of rapid, random molecular motion averages. the direct dipolar coupling practically to zero, thus in the liquid or gas phase, resonance should take place over a very narrow range of field strengths; the resonance line will still have a finite width owing to the inhomogeneity of the applied magnetic field. An early application of nuclear magnetic resonance was the comparison of the nuclear moments of different isotopic species by comparing their resonance frequencies in the same applied field. The most accurate way of making such a comparison is to use liquid samales since the relative positions of the sharp resonance lines can be measured with great accuracy; however, it was discovered that the value obtained depended upon the chemical environment of the nuclei. The phenomenon was observed first by Knight(l) for metals and metal salts, and later by Proctor and Yucz) for l*N compounds, and Dickinson(3) for 19F compounds. It soon became apparent that the phenomenon was a general one for all nuclei, and it was given the name “the chemical shift”. The chemical shift is observed whenever two or more nuclei of the same isotopic species have a diierent environment, a separate resonance absorption usually being observed for each distinct group with an intensity proportional to the number of nuclei in the group. Nuclei may be magnetically different because either they are in chemically distinct groups or they have a different spatial environment. An example of the iirst type of non-equivalence is the fluorine resonance spectrum of perfluoroisopropyl iodide which shows two resonance lines with intensities in the ratio 6 : 1 corresponding to the two groups of fluorine atoms - CF3 and > CF (Fig. 3.1). The other kind of non-equivalence is 59
60
HIGH
RESOLUTION
NMR SPECTROSCOPY
illustrated by the two hydrogen nuclei in the amide group, which are chemically equivalent but magnetically distinct, since the hydrogen labelled (1) /O R-c\N/H
(l)
has a different spatial environment to H (2). In this example magnetic non-equivalence exists only if rapid reorientation about the C-N bond is absent (see Section 9.5.2). The symmetry of the molecule is of the utmost importance in deciding how many resonance peaks are expected; for example,
(C F,:,CF
k-72.9 Fro. 3.1 The lgF
I
ppm-
resonance spectrum of neat pefluoroisopropyl iodide at 56.4 MC set-’
the lgF resonance in the molecule BFI consists of one absorption band, whereas the lgF spectrum of CIF, shows two peaks with an intensity ratio of 2 : 1. The molecule BF3 has DSI, symmetry whereas CIFB does not. F F\B/F
D3,,
symmetry, all B-F
F-Cl-F
bonds equivalent
The chemical shift has a linear dependence on the applied field strength and increases as the field increases. The shift arises from the diamagnetic shielding of the electrons surrounding the nucleus. In an applied field HO, the electrons precess about the field direction with their Larmor frequency eH/2Mc and give rise to circulating electric currents which produce an induced magnetic field at the nucleus opposing Ho. In equation (3.1) His the magnitude of the field strength at the nucleus and differs from Ho, the applied field, by an amount H’, that is H = Ho - H’ (3.2) H = (1 -cr)Ho or (3.3)
THE
ORIGIN
OF
CHEMICAL
SHIFTS
AND
SPIN-SPIN
COUPLING
61
where 0 is a dimensionless number known as the shielding coefficient or constant. If two nuclei of the same isotopic species in environments i and j have shielding coefficients ol and a, at the same .value of N,, , then the chemical shift difference 6,, 1s 6ij = u, - a,. (3.4) The magnitude of &, for a particular isotopic species depends upon the mass number of the isotope; in hydrogen resonance the range of shifts is of the order of 10 parts per million (ppm), whereas in fluorine resonance the shifts extend over about 600 ppm. The measurement of absolute chemical shifts is not possible since it would mean comparing the resonance frequency of an isotopic species in a particular molecular. environment with that of the same isotope stripped of all its electrons. For most applications of chemical interest it is necessary to know only the shifts measured relative to some reference compound (see Section 7.2). In hydrogen resonance the most often used reference compounds are water, benzene and tetramethylsilane Si(CH&, and the chemical shift is quoted as the dimensionless number, b = (H, - H,J H;’
x lo6 ppm
(3.5)
where H, is the resonance field for the nucleus in that particular environment and HR is the resonance field for the reference compound. In some compounds the chemicalshift is dependent on temperature, and in general this indicates that an averaging process of some sort is taking place, for example exchange of nuclei between positions with different shielding coefficients, or hindered motion about bonds. A detailed treatment of these effects is given in Chapter 9. 3.2
SPIN-SPIN
CtXJPLING
On examining spectra under conditions of high resolution it is often found that the chemically shifted peaks are themselves composed of several lines. This added multiplicity was first noted by Proctor and Y~‘~)for the “‘Sb resonance in NaSbF,. Careful purification of this compound ensured that only one form of antimony was present, nevertheless, seven equally spaced lines were observed with the intensity ratios 1: 6 : 15 : 20 : 15 : 6 : 1 (5 lines were reported by Proctor and Yuc4), but Gutowsky and McCaW) point out two extra lines in their spectrum). A systematic study of multiplet splitting by Gutowsky and MC&~(~) showed that fine structure may arise in any molecule containing two or more nuclei which resonate at different field strengths, that is, the nuclei are either of different isotopic species, or are chemically shifted. The number of lines in the spectrum for. nucleus A in a molecule AX,, where A and X are non-equivalent?, is given by 2nlx + 1 where Ix is the spin quantum number of X, and the relative intensitiesaregiven by thenthbinomialcoefficients.Themultipletlinesareequally spaced and the magnitude of the splitting is known as the coupling or spin-spin t see p. 283 for full discussion of notation.
62
HIGH
RESOLUTION
NMR
SPECTROSCOPY
interaction constant and is represented by the symbol JAx; it is usual for JAx to be quoted in the units cycles per second. Two simple examples of spectra containing multiplet peaks will serve to illustrate the above rules. The hydrogen resonance spectrum of hydrogen gas is a single line, since the two nuclei in a hydrogen molecule are magnetically equivalent. However, gaseous hydrogen deuteride has a hydrogen resonance spectrum consisting of three equally spaced, equally intense lines arising from coupling with the deuterium nucleus (for deuterium I = I). The deuterium resonance spectrum is a symmetrical doublet, and the coupling constanP) in both the hydrogen and the deuterium spectrum is 43.0 cycles sec- I. It is important to note that if the coup-
CF3 groups
KF,), CFI CF
!I-’ FIG. 3.2 The lgF
group
i-J-+J-+J-+-J+++
resonance spectrum at 56.4 MCsecml of neat perfhoroisopropyl iodide: J = 12.5 h O-3 cycles see-l
ling constant had been measured in terms of field strength, then the splittings in each spectrum would not be equal, but would be in the ratio y&n, where yH and yn are the magnetogyric ratios of the hydrogen and deuterium nuclei respectively. The spectrum of perlluoroisopropyl iodide under high resolution is shown in Fig. 3.2. The CFa peak is a doublet arising from coupling of the fluorine nuclei of the CFs group with the fluorine nucleus in the CFI group; the CF peak is split into a septet by interaction with the two CF, groups. One coupling constant can be obtained from the spectrum: J CP- CF, = 12.5 + 0.3 cycles set-‘. Before discussing a mechanism for spin-spin coupling it will be instructive to consider the properties of the interaction. (a) Coupling is not observed between groups of equivalent nuclei. Equivalence here means not just an equivalence brought about by symmetry of the molecule, but that the nuclei must also have equal coupling with all nuclei in other
THE
ORIGIN
OF
CHEMICAL
SHIFTS
AND
SPIN-SPIN
COUPLING
63
groups. As an example, the four hydrogen atoms in the planar molecule ethylene are equivalent and one line only is observed in its spectrum, but the two hydrogen atoms in l,l-difluoroethylene”’ F’ (1) F (2)
although related by symmetry are magnetically non-equivalent as they couple to different extents with the fluorine atoms labelled (1) and (2). (b) The coupling constant is independent of the strength of the applied magnetic field”‘-ll). Table 3.1 shows the values of JpF in HPF6 over a range of field strengths. TABLE 3.1 JPF IN THE MOLECULE ~~~~~~~~~~~~ OFTHEAPPLED FIELD&(~)
Field (gauss)
VALUES
Jpp (cycles set-1)
(c) The magnitude of the coupling constant between two nuclei generally increases with the atomic number of both nuclei. In hydrogen resonance, typical coupling constants JHHlare of the order of 10 cycles set-‘, whereas in ?gF, spectra Jm may be as high as 300 cycles sec.-‘. Coupling constants for Merent nuclei are best compared by reference to the hydrogen nucleus, that is by multiplying J by Y&. (d) In general, the magnitude of the coupling constant JAx decreases as the number of bonds separating A and X increases, and, although exceptions to this rule are known @e&ion 11.3), it is usually the case that JAx is undetectable if A and X are more than 5 bonds removed. (e) The coupling constant is independent of temperature.? A mechanism for the coupling giving rise to multiplet splittings has been suggested” 3, in which the coupling proceeds viathe bonding electrons. Themechanism is illustrated best by a molecule like HD. In both the hydrogen and deuterium atoms the single electron will tend to align itself so that its magnetic moment vector lies parallel to the nuclear moment vector& and in the HD molecule in addition to the nucleus-electron interaction, the two electrons in the covalent bond have their spins, and hence their magnetic moment vectors, aligned anti-parallel. Thus the two nuclear magnetic moment vectors in the HD molecule tend to he anti-parallel. It is important to note that although this t But see reference 120. $ The spin angular momentum vectors tend to lie anti-parallel, but since the magnetogyric ratios of the electron and proton (or deuteron) have opposite signs, the magnetic moments are parallel.
64
HIGH
RESOLUTION
NMR
SPECTROSCOPY
coupling mechanism suggests how the orientation of one nucleus affects the orientation of the other, it cannot explain the existence of more than one energy level. If perfect pairing of the electrons takes place then there can be only one relative nuclear configuration, i.e. with the nuclear spins anti-parallel. For there to be more than one energy level, that is more than one single spectral line, there must be excited electronic states of the molecule in which other relative orientations of the nuclei exist. On the above rigid mechanism this means that there must be states in which the electrons are paired with spins parallel. This simple coupling mechanism may be described by a term h.Zt, pr. y, in the interaction Hamiltonian, where Jr, is a constant independent of temperature and having the dimensions of frequency. The number and relative intensities of the multiplets depends on the relative spin configurations of the interacting groups; as an example consider the molecule CHJCHIX where X takes no part in’the coupling. The hydrogen nucleus has Z = +, therefore the possible spin states are those with m, the magnetic quantum number, equal to _++. Let us label the state m = + + as OLand m = - + as /I. The possible configurations of the hydrogen nuclear spins in the CH, group are total spin 1
6. a &
2
‘%ocB Y/5& i P cY3c
3
+; _+f +J ++
zpp
-$
B&B L BBa
-4
4
-+ -5
BBP
The methylene hydrogen nuclei see four energy states of the CH3 group corresponding to the four values of the total spin. The two states with total spin + 3 are each three times as numerous as those with total spin &, hence a quartet is observed for the CHI resonance with relative band intensities of 1 : 3 : 3 : 1. The spin states of the two hydrogen nuclei in the CH2 group are: total spin 1
CtLy fl
2
KB Pa 1
3
BP
0 0 -1
Thus the CHJ resonance is split into 3 lines with relative intensities 1 : 2 : 1. The analysis of spectra by means of the simple rules given so far applies only to systems which have a chemical shift large compared with the coupling constant. When two groups of nuclei are separated by a chemical shift which is of
THE
the same spectrum above. A quantum
ORIGIN
OF
CHBMlCAL
SHIFTS
AND
SPIN-SPIN
COUPLING
65
order of magnitude as the coupling constant between them, then the does not obey the rules for multiplicity and relative intensity given more complex pattern is obtained which requires a more detailed, mechanical analysis (see Chapter 8).
3.2.1 The E#iect of Time-averaging Processes on Spin-Spin Coupling In some molecules spin-spin coupling is not observed even though magnetically non-equivalent nuclei are present, for example, ethylene dichloride C1CH2CHZCl shows a single peak for hydrogen resonance even though the isotopes 35C1 and 37C1each have a spin of 3. The reason for this apparent anomaly is that any process which affects the lifetime of a spin state may cause the spin-spin coupling to vanish. In the case of ethylene dichloride the lifetime of the spin states of the chlorine nuclei are reduced to a very small value because of rapid spin-lattice relaxation arising from the nuclear quadrupole moment of the 35C1 and 37C1 nuclei (see Section 2.5.2). In Chapter 9 it is shown that the spin-spin coupling constant cannot be detected if the lifetime of the spin states is less than (2~ v,)-’ , where Y,,,,is the frequency separation of the spin states m and n. A similar effect is observed for other systems of time-averaging, that is chemical exchange processes and hindered rotation (see Sections 9.1 and 9.5). CHEMICAL
SHIFTS
.
3.3 BULK DLUAGNJXIC SHIELDING The internal chemical shift arises from the magnetic screening of the nuclei by the electrons ; however, when comparing internal chemical shifts with an external reference compound the shielding -of the nuclei by the surrounding medium must also be taken into account. Consider a molecule in a diamagnetic medium and enclose the molecule with a sphere small in dimension compared with the size of the sample but large compared with the molecular dimensions. The effective field experienced by a nucleus in the molecule will depend upon
(a) the external field He, (b) the diamagnetic
susceptibility
in the medium outside the sphere, and
(c) the shielding effect of the molecular electron and intermolecular effects. The medium outside the sphere can be regarded as continuous and with a volume diamagnetic susceptibility 2; therefore the effective field at the molecule is given by(*2),
where OLis a shape factor which is zero for a sphere and 27cfor a cylinder of length large compared with its width. Most high resolution measurements are Has. 3
66
HIGH
RESOLUTION
NMR
SPECTROSCOPY
taken on samples in cylindrical tubes so that the diamagnetic susceptibility of the medium gives a contribution of (2n/3)~ to the shielding constant u.
2nx %bs
=
cr,ol +3.
If the chemical shift is referred to an external reference compound, chemical shift B is given by H- HR 6=a-a,= HR
(3.7) then the
where u is the shielding constant of the molecule and CR that of the reference compound. The observed chemical shift bobr and the chemical shift corrected for the difference in diamagnetic susceptibility 8,,, are related by (3.8) If an internal reference compound is used then xR and x will be equal and it will not be necessary to apply the correction of equation (3.8). 3.4 CHEMICALSHIELDING IN ATOMS The magnetic shielding of a nucleus by a spherical electron distribution was first calculated by Lamb (la). The problem is to calculate the effective field at the nucleus and we may do this by first considering the current induced in a spherically symmetric electron distribution by an applied magnetic field ; the field at the nucleus then differs from the value of the applied field by the field arising from the induced current. Consider a nucleus in a magnetic field H with a vector potential A defined (15) by 9 H = curlA. (3.9) A is not d&ed uniquely by equation (3.9) since we may add an arbitrary vector to A without changing the value of H. This may be seen by substituting for A in equation (3.9) with a vector potential defined with respect to a coordinate system with a different origin. Changing the origin of A is equivalent to adding a quantity grad pl to A, where p, is any scalar function of the appropriate dimension. curl A = curl(A + grad v) = curl A + curl.grad c~ and curlgrad v is zero for any value of v. The invariance of any physical quantity consequent on the existence of a vector potential with the choice of origin of the coordinate system is known
THE
ORIGIN
OF
CHEMICAL
SHIFTS
AND
SPIN-SPIN
COUPLING
67
as gauge invariance. It is convenient to choose the nucleus as the origin of the coordinate system. The vector potential at a position r in a magnetic field is,
A = f-H x r. Now according to the Larmor Theorem(16) an electron in a magnetic fieldprecesses about the direction of the field with a uniform angular velocity given by eH
a=-
(3.10)
2Mc
where e = electron charge in e.s.u. ; M = mass of the electron ; and c is the velocity of light. The precession of the electrons is equivalent to a current flowing in the opposite direction and the current density vector, I, at r may be written j=
”
=
x r@(r)
eH x r e(r) 2Mc
=-eAe(r>
(3.11)
MC
where e(r) is the charge density at r. The current flowing in the electron distribution will give rise to a magnetic field with a vector potential A’. A’ at r’ is given by the Bid-Savat JQzw(~~) (3.12) the integration being over all r. From (3.11) and (3.12),
(3.13) For a spherically symmetric electron distribution,
and the corresponding tion is
magnetic field at the nucleus (i.e. r’ = 0) in the z-direcH’
= -
eH
3Mc2
s
e(r)
r
r>O 3”
dt
’
68
HIGH RESOLUTION
NMR
SPECTROSCOPY
The integral is the electrostatic potential V(0) produced by the electron distribution at the nucleus, hence the shielding coefficient for S state atoms is a=
--=
H’
- e V(0)
H
3McZ
(3.16)
*
The electrostatic
potential for a nucleus with charge Z may be expressed as
- Ze
, rl
therefore ZeZ
/l\ (3.17)
cr = S’\II//.
‘$ \>
1 is the quantum mechanical average value? of - where rl is the distance r1
of the ith electron from the nucleus. Equation (3.16) may also be expressed in terms of the fine structure constant u = e”/fi c, and the Bohr radius aH = - h2/M c2, a=
- u2 aHY(0)
(3.18)
3e
or, substituting(16) for V(0) (3.19)
where y is a normalised eigenfunction. An alternative derivation of (3.16) not using vector methods has been given by Dickinson(“). 3.5 MAGNETIC SHIELDINGIN MOLECULES.PERTURRATIONTHEORY The Lamb equation (3.16) cannot be applied to a nucleus in a polyatomic molecule since in the presence of two or more attracting centres the electron distribution no longer has spherical symmetry. The first full treatment of the calculation of the shielding coefficient of a nucleus in a molecule was given by Ramey(lSn 19) using second-order perturbation theory. The theory applies only to polyatomic molecules which, in the absence of an applied magnetic field, have no resultant electron spin or electron orbital angular momentum. The nuclei are considered as stationary attracting centres, on account of their large mass, and the shielding factor is calculated for a particular orientation of the molecule in the magnetic field. The vector potential at the kth electron arising from the external field H and the dipole moment of the nucleus f~ is
be =+Hxr,+~-
fH
x Bk.
(3.20)
THE
ORIGIN
OF CHEMICAL
SHIFTS
AND
SPIN-SPIN
COUPLING
69
where r, is the distance of the ti electron from the nucleus, and &, is an arbitrary vector arising from the gauge invariance. The Hamiltonian of the .kth electron in a field with vector potential A and scalar potential V is
where P, is the generaked momentum(20). Substituting for P, with the operator - i h pk, and for A, from equation (3.20),
and this may be written as r
= x* + E(l) + X(2’
(3.23)
where
-efi = 2Mci and
rn&
=
[rk - &81 x
vk
-
A?* is simply the Hamiltonian in the absence of a magnetic field, and X(l) and dF2) may be regarded as small perturbations on dP. To obtain the shielding coefficient oAfor the orientation A we need to obtain the interaction energy WA and then to select those terms which are of first order in p - H, since w, = _CTA(l.H.
(3.24)
W, may be obtained by a perturbation treatment taken to second order. In the general case aAis a second-rank tensor and is given by
where d is the unit dyadic. 0 A refers to the electronic ground state of the molecule in the orientation iz, and n to the excited state. The fist term in
70
HIGH
RESOLUTION
NMR
SPECTROSCOPY
equation (3.25) gives a positive contribution to the shielding coefficient and is known as the diamagnetic term, whilst the second term gives a negative contribution to 0, and is known as theparamagnetic term by analogy with Van Vleck’s equation for the diamagnetic susceptibility of a molecule(21). The calculation of d from (3.25) (after averaging over all values of A) is not practicable as it would require a knowledge of the excited state wavefunctions and these are not in general known. In particular cases it may be possible to simplify equation (3.25) so that only known quantities are involved. 3.5. I Molecules Symmetrical About the z-Axis’ls’ In molecules which are cylindrically symmetric about the internuclear axis it is assumed that the nuclear dipole lies in the same direction as the magnetic field, so that the vector potential of the P electron is now (3.26)
Ak
taken with the origin of the coordinate system at the nucleus. The solution of the secular equation for W, now gives a, =
Here, aAis a scalar (the z-component 0 mzk
=
r
eh
2Mci
of aA i.e. azr), and
>(
a
xkay*-yk
a
-!a&
(3.28)
i.e. the z-component of the magnetic moment operator. Equation (3.27) may be derived directly from equation (3.25) by setting Rkc= 0 (a gauge transformation), and taking only the a,, component of the tensor ul. In a liquid or gas the molecules may assume all orientations with respect to the field and hence equation (3.27) must be averaged over all A; in such an average x, y, z must be equivalent and if avArepresents the averaging process, then
thus,
THE
ORIGIN
OF
CHEMICAL
SHIFTS
AND
SPIN-SPIN
COUPLING
71
Note. C’ means that the state 01 is omitted in the summation. The%st term in equation (3.29) is simply the Lamb equation for the shielding coefficient in an atom (i.e. a spherical electron distribution); the second term arises because the electron distribution is not spherical, and because it gives a negative contribution to (I it is called the paramagnetic term. 3.5.2 Average Energy Approximations It is still not possible in general to calculate shielding coefficients from equation (3.29) since the second term requires a knowledge of the excited state wavefunctions. The calculation may be simplified by taking an average over the excited states of the molecule, as was done by Van Vleck and Frank(22) in a similar calculation on the diamagnetic susceptibility of molecular hydrogen. Using
where
mk
=i&k
+ jm,l,+km,k
where i, j and k are unit vectors along the x, y and z axes. An averaged value of (E, - EO) is defined by the equation(75)
c -l Eo) +WI w n+o@n
=-&0l~‘I0).
(3.30)
d E cannot be obtained from equation (3.30) without a knowledge of the excited state wavefunctions, but in practice an estimate is made of the magnitude of LIE. Substituting equation (3.30) into (3.29) gives
Although equation (3.31) contains only ground state wavefunctions it involves the second derivatives and is very sensitive to any error in the wavefunctions chosen. Wick(23), using a similar expression to equation (3.31) in a calculation on the rotational magnetic moment of molecular hydrogen, found a difference by a factor 8 for two different ground state wavefunctions. Snyder and Parr(76) have applied equation (3.29) to the case of atomic hydrogen in order to investigate the relative magnitudes of the diamagnetic and paramagnetic terms when the origin of the vector is varied. They also divided the paramagnetic term into tid + ode where tid and C? are the contributions of the discrete and continuous excited energy states. A complete set of eigenfunctions is available for the hydrogen atom so that all the terms of the ground and excited states may be calculated individually. When the origin of the vector potential is taken l-4 a0 from the nucleus the diamagnetic and paramagnetic terms are comparable in magnitude, and calculations of the individual terms in the paramagnetic term showed that the continuum excited states contribute more than the discrete excited states. Of the discrete energy states contribution the first excited state provides about 70 per cent of the total. It can be concluded from
72
HIGH
RESOLUTION
these calculations that in estimating the excited states, and in particular, account the calculation of c would It is possible to simplify equation shipP)
NMR
SPECTROSCOPY
a value of A E it is important to include all if only the first excited state is taken into be about 25 per cent in error. (3.31) even more by means of the relation(3.32)
oa = I I> *
Aa-& \
-Eo)
$En
(3.33)
so that the shielding coefficient is independent of even the average excitation energy d ,?Czs*26). If both sides of equation (3.32) are multiplied by
and both sides of equation (3.33) multiplied by
then added together and summed over all excited states, the left-hand side of
the equation becomes <=ollmzk;T
-
lOA>,
so that
(3.34)
~~l~l.~~}~~~(y,l”n)].
If (E, - E,,) is replaced by an average energy A E, and if each of the integrals of the type
,/Oi \
I
$-e-&
I
tzA>
and
THE
ORlGlN
OF CHEMICAL
SHIFTS
AND
SPIN-SPIN
COUPLING
73
expanded by parts, equation (3.34) simplifies to
and combination tations gives
of equations (3.31) and (3.35) after averaging over all orien-
3.5.3 Linear Molecules For linear molecules in 1x states the summation in equation (3.29) can be related to the spin-rotational magnetic interaction constant, which has been determined experimentally for some molecules. Equation (3.29) may be expressed in a form more suitable for a comparison with Wick’s theory(27) of rotational magnetic moments. Since &/ri: is a Hermitian operator,
=
$-s (rk x inlo&> dt = f
KAOA
where the current density vector
Also
where J&kis the z-component of the angular momentum Operator 4; pLois the Bohr magneton. Substituting equations (3.39) and (3.37) into equation (3.28) and averaging over iz gives
x WomLnor + Lod%,o,l. HIS.
3.s
(3.40)
74
HIGH
RESOLUTION
NMR
SPECTROSCOPY
Now, if F; represents the summation in equation (3.40) then for a linear molecule we may write FA = FL + 4 (3.41) where FL and Fu represent the values of FAwhen the molecule has its internuclear axis perpendicular and parallel to the applied field. If 8 is the angle between the z-axis and the internuclear axis then av, F, = av,(sin28, FL + cosz8,Fll) =$FL+
IF 3 II*
(3.42)
A linear molecule is cylindrically symmetrical about the internuclear axis hence F,, is zero, and rn$ has non-zero matrix elements only when n 1 = Ol; T thus equation (3.40) for a linear molecule becomes
x (Ho,/,A L
AO_,_ +
Loin A Hnro,_> .
(3.43)
If a linear molecule rotates with angular velocity, m=-
Jh (3.44)
I
where J is the rotational quantum number and I is the-moment of inertia, then the experimentally observed magnetic field at nucleus r, H,, is’27. 28)
z, is the charge on a nucleus i other than the rth nucleus; al is the distance of the ith from the rth nucleus. Substituting equation (3.45) into (3.43) and introducing a, for the Bohr radius, ar for the fine structure constant’2g’ and pN for the nuclear magneton, the result is
M,, is the mass of the proton. It might be considered a weakness of equation (3.46) that it depends on second-order perturbation theory in the derivation of both equation (3.43) and equation (3.45), but in fact equation (3.46) may be obtained by a direct method not using second-order theory. 3.5.4 Molecules Containing Heavy Atoms The main difficulty in applying equation (3.25) lies paramagnetic term, and even in the simplest molecule 20 per cent of the shielding coefficient. As the number cule increases so the magnitude of the paramagnetic
in the calculation of the (hydrogen) it constitutes of electrons in the moleterm becomes large and
THE
ORIGIN
OF CHEMICAL
SHIFTS
AND
SPIN-SPIN
COUPLING
75
comparable in magnitude to the diamagnetic term. In a large molecule the electrons in inner orbitals can play little part in shielding the nucleus and this would be reflectedinprincipleby an equality of the paramagnetic and diamagnetic terms in equation (3.25) for these electrons, but in practice approximatewavefunctions would be used and the two terms might be appreciably different thus creating alarge error in the shielding coefficient. The division of equation (3.25) into two terms is a result of the choice of a particular origin for the coordinate system; changing the origin would alter the magnitude of the two terms, but the sum of the two would be unaltered. This is strictly true only when a complete set of excited state wavefunctions is used, or if the wavefunctions are gauge invarianPO’. Ramsey (lg) has used the invariance of 0 under a gauge transformation in an attempt to calculate the shielding coefficient for large molecules. In a large polyatomic molecule we may divide all the electrons in the kth nucleus into those which are tightly bound to the kth nucleus, N,, and those which cannot be considered bound to one single nucleus in the molecule, N,. The N, electrons can be regarded as giving rise to a spherically symmetric potential about the kth nucleus so that the orbital angular momentum operator -(2M c/e) m,, about the kth nucleus has zero non-diagonal matrix elements. If k is one of N, then R,, is chosen to be R,, the radius vector of the kth electron to the kth nucleus R,, remains arbitrary for the N, electrons, and U, becomes
(3.47) Equation (3.47) may be reduced to the same form as equation (3.29) or (3.40) if the symmetry of the molecule is appropriate. McConnelP) has approached the same problem of dividing the shielding coefficient of a nucleus N in a large polyatomic molecule into the contributions from electrons strongly bound, and those remote from N. G is a group of electrons remote from the nucleus N, that is, the ground and excited state wavefunctions of the electrons in G have zero amplitude within one bond length of N. The origin of the vector potential is taken as being in the group G, i.e. rL! =r,+R 0
mk
=
(3.48)
m, -
where rk and m, are radius and angular momentum vectors of electron k referred to origin OG, and R is a radius vector which points from ONto 0,; 3a*
76
HIGH
RESOLUTION
NMR
SPECTROSCOPY
p, is the linear momentum of k. IfR is large, that is G is at long range from N, then us, the group shielding tensor, may be expressed as 3xG*R*R R=
1
N,, is Avogadro’s Number, and xG the molar magnetic susceptibility
xG= x
(3.50)
*
tensor is
-fNo(-,&$)(OGl~I-rk*rk - rk*rk~OG)+&~(&)
(OG Cm,
I InG)(nG k
cq
]k
cmk OG\ OG c 1OG\+ / (nG Ik 1 /( I*mklnG>. (3.51)
The origin of bg may be regarded as the field at N arising from an induced dipole p at G given by P =E&.$.
(3.52) 0
For liquids and gases, equation (3.50) must be averaged over all orientations of the molecule, which for a group G with axial symmetry gives
A XG 44 = 3RJN
- 3c0s2ejay.
Ax0 is the anisotropy of the magnetic susceptibility
(3.53)
of group G, that is
and 8 is the angle between R and the symmetry axis of G. Since equation (3.53) is strictly true only when the distance of the group from the nucleus is intiitely greater than the group’s dimensions, a specific value of R is implied. In real cases the group origin 0, is probably best taken at its electric& centre. An interesting consequence of the variation of the shielding coefficient with orientation is that it provides a mechanism for spin-lattice relaxation (Section 2.5). The magnetic field at a nucleus in a molecule undergoing random motion will have time-dependent components owing to the anisotropy of u, hence Fourier components with a frequency near that of the Larmor frequency will induce transitions between the Zeeman levels. For an axially symmetric molecule application of the Bloembergen, Pound and Purcell theory of spin-lattice relaxation (see Section 2.5)(3” to this situation gives a contribution to the relaxation time of 15(1 + 4n2 v2 r:> (3.54) (=hlJsotroPY = &2(d a)2 2 v re where v is the resonant frequency, Aa is the difference between the shielding coefficient along the molecular axis and perpendicular to it, and T, is the molecular correlation time.
THE ORIGIN OF CHEMICAL
SHIFTS AND
SPIN-SPIN COUPLING
77
3.6 VARIATION THEORY APPLIED TO THE CALCULATION OF MODULAR
SHIELDING CONSTANTS
The dif&ulty inherent in calculating the shielding coefficient from equations derived by perturbation theory (e.g. equation (3.25)) is that a knowledge of the excited states of the molecule is required. A different approach which does not involve an explicit knowledge of the excited states is the Variation Method(2o). If&’ is the Hamiltonian of the system then the appropriate eigenfunction is the normalised function tp which causes the energy to be a minimum, that is (3.55) must be a minimum. A function of the form
may be substituted into equation (3.55); the qn are known functions and the coefficients C,, are determined by minimising equation (3.55). Several calculati~nW-~~)ofthe shielding coefficient have been made with different forms for the functioarp, but the most successful have been th~se(~~-*~*‘~) based on a form for the trial function first proposed by! Tillieu and Guyt41)in a calculation of magnetic susceptibility, and whose validity in shielding calculations is discussed by O’Reilly”g’. The most general treatment has been that of StephexP7) who used as the trial function (3.57) I=yo(l+H’g+P.f) where tpo is the unperturbed wavefunction (i.e. in the absence of an applied magnetic field), and f and g are unknown functions. In a tensor notation, Y =
(3.58)
YOU+ K&T, + PA
(the repeated Greek suffix indicates a summation, i.e. H, g, represents H,g, + H, g, + H. g.). Substituting equation (3.58) into (3.55) and selecting those terms in h HP gives u* = -
e2
2Mc2
(
0
IXr
(r: 4#-
rke4@/r3
0 I>
78
HIGH
RESOLUTION
and using the Hermitian simplifies to
character
NMR
SPECTROSCOPY
of the operator
m, the equation (3.59)
(3.60) f and g are obtained by minimising equation (3.60) with respect to variations in these functions. If the electrons are not localised around the nucleus concerned, then the factor l/r: in equation (3.60) may be replaced by an average value of l/R”. It may be shown(37)that in this case f.: = 2gJR3 and the shielding coefficient is given by (3.61) where la5 is the magnetic susceptibility tensorC41). The variational method again divides the shielding coefficient into a positive contribution (diamagnetic), and a negative (paramagnetic) part, but all the integrals in equation (3.60) are over the unperturbed wavefunction yo. The polarisation of a@ into two terms is a consequence of the choice of the origin for the coordinate system used to define the vector potential A,, and since be0 is invariant under a gauge transformation it should be possible to transform practically all the paramagnetic term into the diamagnetic term(42’ by an appropriate choice of origin. Equation (3.59) was obtained with the origin of the coordinate system at the nucleus. The SchrBdinger equation, HV =Ey
(3.62)
is invariant under a gauge transformation only when y is the exact eigenfunction; if y is expanded as an incomplete set, as in the variational method, then equation (3.62) is no longer gauge invariant. If the vector potential 4, is referred to a new origin such that (3.63) rk = R + r;i (the prime indicates the new coordinate system), then to keep equation (3.62) invariant we must change the functions f and g to g = g’ - &T f = f’ + constant
(R + rk) + constant (3.65)
and by substituting in equation (3.55) it is found that the best origin to use is at the charge centroid of the molecule(37).
THE ORIGIN
OF CHEMICAL
SHIFTS
3.7 THE INDUCEDCURRENTMODEL
AND
FOR MAGIC
SPIN-SPIN
COUPLING
79
SHIELDMG IN MoLECULPS
An alternative to both the perturbation and variation methods of obtaining the shielding coefficient has been devised by Pople(43*4*) and consists in obtaining the expectation values of the currenr density operator j. For a continuous distribution of charges e, moving with velocities v1 the current density J (r) is da fined as
j
(4 = F ef vf.
(3.66)
The induced magnetic field H’ at a point r from the charge is given by the BiotSavat law” s,
Note. It is usual in electromagnetism to use the symbol B for the indnced field in equation (3.67), and to retain H as the applied field; in nuclear resonance it has become standard practice to use the symbol H for both applied and induced fields. The quantum mechanical value of j(r) is de&red as (y I j(r) I lp) where j(r) is the operator(45), r,) +:d(r - rk+k
+
:&)I (3.68)
where 6(r - r,) is the Dirac 6-function(46) (zero for all values of r except r, when it tends to infinity in such a way that +j&r - rk)dr = 1). The expectation value of j(r) could be foid?by means of first-order perturbation theory, but this method has the disadvantage that the form of the result depends upon the choice of the origin for the coordinate system, that is on the gauge of 4. If the origin is selected as being at one nucleus this leads to difflculties in interpretation at other nuclei. It is possible to surmount this difficulty by using modified molecular orbital functions. The ground and excited states of a molecule may be represented by functions yg, VT . .. p! where each molecular orbital is a combination of atomic orbitals qE,, (S refers to the atom and p to the atomic orbitals in that atom). Pople(43)modifk tbe functions rpi to a set y. which are combinations of functions x,,,, where
A is the vector potential at atom s. The wavefunctions for the electrons in the presence of the magnetic field are now (3.70)
80
HIGH
RESOLUTION
NMR
SPECTROSCOPY
and the coefficients C, are to be found by a variational method. The expectation value of j(r) is given by
I3 W
Iy> =
(yo
+
jaJo
Cn ya 1i (d 1y0
+
jnFo Cn Y+
=C~-~~~Y~I~~~~IYO>+~~~~~I~~~~~~~~~.
(3.71) The two parts of equation (3.71) again may be called the diamagnetic and the paramagnetic terms. The diamagnetic term j(r)- may be shown to be given by C eJoo’ 16 - r,) x
j(r)dia = &-
(3.72)
HI
in which eioo) the electron density at position r on atom s is (yoI?(r
- dlwo).
A physical interpretation of j(r)d’8 is that it represents the sum of the effects of -or precession of the electrons in each atom with a frequency of eH/ 2M c. Evaluation of the paramagnetic term j(r)pam requires a knowledge of the coefficients C, and the two matrix elements, (y. Ij(r) 1yJ and (y,, Ij(r) 1yo). The final result is(43)
in which the coefficients &$
are derived from
eon(r) = C eZ$, 9%(r) FJr) SWII and are real numerical coefficients. The current densities given by equations (3.72) and (3.73) can now be used to calculate the magnetic shielding constant. The secondary magnetic field at the nucleus in an atom as a result of the induced current density j(r) is given by the Biot-Savat Law equation (3.67) in the form j(r)
H’
(r - rs)3 dr
therefore, from equation (3.72) Q(“) dt.
(3.74)
If the applied field is taken as the z-axis and the nucleus is chosen as the origin of the coordinate system then equation (3.74) simplifies to the familiar Lamb term, (H,)dh
=
-@.
H
=
(s)Hf[ (X2 f Y2’]$‘o’dr.(3.75)
THE
ORIGIN
OF CHEMICAL
SHIFTS
AND
SPIN-SPIN
COUPLING
81
The contribution from induced paramagnetic currents in one atom to the secondary field at the nucleus of another atom is most readily calculated by considering the effective point dipole on the first atom: PJ-
=-
’
C s
(r-rr,)xjydr
j.” represents the paramagnetic current density on atom s. For an applied field in the z-direction the shielding coefficient at a point a distance R from the atom s is given approximately by 6”
= - (2&=
- ,ur
- /~r)/3
R3 H.
(3.77)
If the atomic orbitals q.,, are spherically symmetrical then the coefficients C, are zero, therefore PF will be zero and there will be no paramagnetic contribution to the shielding coefficient. 3.8 INTBIUTOWCCWRENTS So far we have considered only induced currents localised on each atom in the molecule, but in some cases, notably aromatic ring systems, electrons are free to move in an orbital in the molecule not associated with any one atom. The idea of circulating interatomic currents was first used by Pauling(47) to explain the anisotropy of the diamagnetic susceptibility of aromatic molecules, and has since been applied to the calculation of the difference in shielding coefficients between a hydrogen atom in an aromatic molecule and one in an ethylenic system. There should be no internal chemical shift between these two cases except for the presence of interatomic currents in the aromatic molecules. The internal shift may be estimated (48)by considering the x electrons in the aromatic molecule to be free to move in a circular path in the plane of, andpasaing through, the ring carbon atoms. When a magnetic fieldHis applied in a plane perpendicular to the plane of the aromatic ring the 1c electrons precess about the field direction with the Larmor frequency w = cH/2M c, and the current flowing is i = ~perelectron. There are six z-ekctrons, therefore the total interatomic current flowing 3e2H = = I. Assuming that the current moves in a circular path with a 2nMc radius qua1 to the C--C bond length, then the current is equivalent to a magnetic dipole at the centre of the ring, ‘and of magnitude 3 e2 Ha2 P
=
2Mct
(3.78)
82
HIGH
RESOLUTION
NMR
SPECTROSCOPY
The dipole is parallel to and opposed to the applied field, thus the induced field H’ opposes the applied field at the centre of the ring, but reinforces it at the positions of the hydrogen atoms as shown in the diagram. ,_---I+--., \
/
\
:
:
‘\
,,;I/ -\ \ \
/ i
‘\._+ _-f.+____-/ I
The magnitude of the induced field at the position of the hydrogen atoms is 3e2 Ha2
(3.79)
+ b)3
2Mc2(a
in which b is the C-H bond length. When H’ is averaged over all orientations of the molecule with respect to the applied field,
GO,,. =
e2 Ha2 2Mc2(a
(3.80)
+ b)3
and the chemical shift between the hydrogen atoms in benzene and those in ethylene should be 6
H’
e’ a2
(3.81)
z-z
H
2McZ(u + b)3’
Two improvements may be made to the above theory whilst still treating the induced currents according to the laws of classical electrodynamics. Instead of using the point dipole approximation the effective field caused by the induced current may be calculated directly (49*so).Moreover, equation (3.81) was derived for an induced current flowing in the plane of the ring, whereas the z-electron distribution is best simulated by line currents in two closed paths at equal distances above and below the plane of the molecule and retaining the same symmetry axis as the molecule. In terms of cylindrical coordinates e and z, expressed in units of a and with the centre of the current circle as origin, the component of the secondary field in the plane normal direction (the z-axis) is H;
=
-
(1 +
#
-
e2
-
2’)
+
[(l - e2)2 + z”]
where K and E are complete elliptic integrals of the first and second kind respectively with modulus, k=
4e
112
THE
ORIGIN
OF CHEMICAL
SHIFTS
AND
SPIN-SPIN
3e2H
For n electrons in the circle the current is 2n~c all molecular orientations gives 6 = - ( 6Z;;20)[(l
+ e)’ + z’]-‘“[K
COUPLING
83
ncos0 *- 6 and averaging over
+ it; 1 ;;;;;;,
.E].
(3.83)
Quantum mechanical methods of calculating the ring current shift have been developed by Pople(sl), by McWeeney(s2), and by Hall et aZ.(101*102)Both Pople and McWeeney extend the method of London(ro3) for obtaining the molecularorbitals of n-electrons in a conjugated hydrocarbon from the case of a uniform external magnetic field to that of a non-uniform field. The London theory is based on the simple L.C.A.0 theory of Hiicke1(104J05), that is the molecular orbital pi is written as the sum of atomic orbitals ~?r, Wr = T C*%
(3.84)
and the coefficients C, are found by solving the secular equation (Z,s - S,, E> c, = 0
(3.85)
in which z,,
= s c? xqsdr
(3.86)
$o;:G
(3.87)
Sr, = and &’ is the one-electron
Hamiltonian X=
operator,
+Mp:+ v.
(3.88)
In the Htickel approximation sss has a constant value for each carbon atom and is known as the Coulomb integral, LX.When atoms r and s are neighbours &,s is known as the Exchange integral and has a constant value of B; when r and s are not neighbours %‘,, is zero. S,,, the overlap integral, is unity when t = s and zero otherwise. With these approximations the Hamiltonian matrix has diagonal elements IXand off-diagonal elements #?or zero, and the secular determinant reduces to a polynomial in x, where x is the dimensionless eignevalue a- E -. B In the presence of a magnetic field H, with vector potential A, the one-electron Hamiltonian becomes .Ztp=
& (p+c> zA2+Y
and the best combination of atomic orbitals now depends upon the choice of the gauge of A. London overcame this difficulty by introducing gauge invariant atomic orbitals xI, which are defined as
84
HIGH
RESOLUTION
NMR SPECTROSCOPY
In the above equation A, is the vector potential at atom s, and r is the position vector of the electron referred to an arbitrary origin. The molecularorbitals Ye are now written as
and the equations (3.85) to (3.87) are modified by replacing qJ by x, where ap propriate and using the Hamiltonian given by equation (3.89). The new secular equation was shown by London to be Im exp(kJ
- x&l
= 0
(3.92)
where ‘yrsis unity if r and s are neighbours and zero otherwise, and %
=eH.S,,s 2cl
in which Sors is the signed area of the triangle formed by the origin and the atoms r and s. On expanding this secular determinant and retaining terms to second order one obtains (3.93) P(x) = C Q,,(x) WI mi P(x) is the unperturbed secular equation, and Q&x) is a polynomial in x such around a closed circuit of that Q&4 = Q,d x ). w1 is the sum of the terms CO,~ atoms, that is
The solutions of equation (3.93) to second order in o1 are x(p) = $1 + c @CO* CO, if where x$” is the unperturbed xv is given by
eigenvalue for the p molecular orbital,
(3.95) and (3.96)
in which P’ (4
E
-&p(x).
Evaluation of xi? and the Q);gives the total change in energy for each energy level on applying a magnetic field H. Pople shows that the current flowing in each ring i can be obtained from the values of .# by the equation (3.97)
where the summation p is over the occupied molecular orbitals.
THE
ORIGIN
OF CHEMICAL
SHIFTS AND
SPIN-SPIN
COUPLING
85
The calculation of the individual ring current in a polycyclic aromatic molecule requires the following: (a) The solution of the unperturbed
secular equation for 2-j’).
(b) The expansion of the London secular determinant and re-arrangement in the form of equation (3.93), and hence identification of P(X) and &(x). (c) Evaluation of P’(x@, Q1,(x$) and hence of xl,? Applying the theory to benzene gives 8n2 b e2 ZbenMe = (3.98) He S, 9112c where S, is the area of the six-membered ring. Combining equations (3.97) and (3.98) enables ring currents in polycychc compounds to be obtained relative to the ring current in benzene. The calculation of the chemical shift arising from the ring currents is best obtained from equation (3.83) (see Section 4.5). 3.9
THE INFLUENCEOFANAPPL~EDELECTRICFIELDONSH~ELDINGCONSTANTS
An atom with a spherically symmetric electron distribution in an applied field has induced currents which are wholly diamagnetic, that is, there is no obstacle to the free circulation of induced currents. If a constraint is applied to the electron distribution, for example, as in a molecule with several attracting centres, then a paramagnetic contribution to the shielding factor arises. A similar situation exists if the constraint is an applied electric field(54e81). Consider an atom in a tied orientation in applied electric and magnetic fields, then the Hamiltonian for the system may be written aP4)
(3.99)
z=Lx~o+~01+acrJ2+ax-‘lo
(In the double suffix notation the first suffix represents the order in E and the second the order in H) i2 e2
zoo=
-2Mv27
~01=-
e H*L 2Mc
x02
=
Xl0
=
(3.100) 2 t2
-
(H
* r)2]
eE-r
where L is the orbital angular momentum operator, L=ihrx
(3.101)
*.
The wavefunction is expanded in a similar way, Y=Yoo+4uol
+tvcl2
+1y10.
(3.102)
86
HIGH
RESOLOTION
NMR
SPECTROSCOPY
We need solve for only the components of electric field E is perpendicular or parallel components are aA and au, then since d is between E and H is CT= oL sin’8 + q
the shielding coefficient when the to the magnetic field, H; if these a tensor its value at some angle 0 0098.
(3.103)
The calculation for u proceeds by obtaining the expectation value of the current density vector j, -e2(H j = 2McN where
x r).)y*y
(3.104)
N =
y* y dt. s The induced field H’ at the nucleus is obtained from the Biot-Savat Law, (equation (3.61)), and may be divided into two parts H’ = H; + H;
(3.105)
corresponding to the two terms @equation contribution to the shielding coefficient, &
e2
= -
2Mc=
(3.104). Hi leads to a diamagnetic
s
e sin2 e --ddz r
(3.106)
where e, the electron density in the absence of a magnetic field, is
e
=
540+ ho0 y
10
+
2wooy2o
+
y:o
+
(Y:, + 2~00 ylo & ***)dt
s
***
’
(3.107)
Substituting equation (3.107) into equation (3.106) and carrying the expansion as far as E2 gives
LT;- =_
2 3Mc2a [
I_--*
641 a4E2 80
ez
1 1
(3.108) (3.109)
A similar calculation for Hi and hence for crPPraleads to O;/araIO
pm = I
(3.110) -233
a3E2
144
MC=
(3.111)
and combining equations (3.108) to (3.11 l), e2 ‘II = u ‘-
3Mc2 a e2
3Mc”a
439
1 --- 40
a4E2
e2
193 a4E2
1 -157’
1 1
(3.112). (3.113)
THE
ORIGIN
OF
CHEMICAL
SHIFTS
AND
SPIN-SPIN
COUPLING
87
It is interesting to note that (r is independent of the sign of E, and that the value of u is reduced for all directions of E relative to H, but that the reduction is greatest when H and E are perpendicular. A uniform electric field distorts an atom in an S state giving a reduction in the shielding coefficient of magnitude proportional to E2, for the symmetry of the atom requires that a change in the direction of E should not affect u. A particular nucleus in a molecule may not be at a centre of inversion anda may be proportional to E as well as E2. When E is an applied field then its average value at a particular nucleus will be zero if the molecules are in a liquid or gaseous state, but if E arises from a polar group within the molecule then E may not be zero. The shielding tensor uolais expanded as a power series in Efs5),
For a molecule symmetrical in x then a$ is zero, so that for an atom a$ = 0; in an X-H bond u,$ = u+ = 0 and a(‘) = a$ where the X-H bond is taken in the z-direction. For a liquid or gas Us is averaged over all orientations. 1 +--u~~&E,E~ 2
+ USE;,
+ .*a .
(3.115)
Some of the terms in equation (3.115) can be obtained by a comparison with equations (3.112) and (3.113); for in an atom CT+ is zero and the two independent tensor components are 1 uIl = a,, = u” + +;,
E2 + .*a
(3.116)
1 t2) EZ + . . . . CT1= a*, = CJO+ - a,,,, 2
(3.117)
Comparing these two equations with (3.112) and (3.113) it is easily seen that e2 UO
(2) uzzxx
(2)
= Ga;
=
q
uzzzz = q =
-
- 386 45
-
(2)
=--
-439
a3
60
Mc2
I
a3 Mc2
(3.118)
I
u% can be written as
where 6, is the substitution tensor (0 ifor 9 p; equals 1 ifor = #?).Buckingham(55) has obtained the components of uti for an X-H bond. In an axially symmetric bond there can be no component of uti perpendicular to the X-H direction
88
HIGH
RESOLUTION
NMR
SPECTROSCOPY
which is proportional to the first power of E, i.e. u$ two independent CG tensor components. oil’ = oi;;
= usi = 0 and there are
(1) = a,, = u;;;.
a:"
From equations (3.114) and (3.119) (3.120)
in which
(J(l)
1
= -
3( @
= f
05” + 2u:“)
.
(3.121)
(o(H2)+ 2uY’)
I Buclcingham has shown that u+ can be related to u(z) by considering the case of a hydrogen atom at a distance R from a point charge ;2.In a uniform electric field E the total electric force F at the hydrogen atom is F = (E,, E,, E, +1/R2) and substituting %
=
F into equation (3.119) and using the relations (3.118) gives
ez8
a’ --
3Mc2a
An effect proportional shows that
193 a3F2 45
-f&
Mc2
227
+ -
a3
-F, 360 .Mc2
FB.
(3.122)
to F2 will involve terms in E, and Ef, and Buckingham 1 (2). 1 (1) tJ:” 3 --&‘. ?I = Toll ’
(3.123)
Thus expression (3.120) becomes 881a3
e2
881 a3
A
E,’ + . . . . (3.124) 216Mc2 Substituting the values for the constants in equation (3.124) and giving 1 a value lo-lo e .s.u . and R = lO-8 cm , u=
3Mc2a
-
108Mc2 .R3
Ez -
u = 2 x 10-S - 2 x lo-12E s -
lo-‘* E,‘.
(3.125)
Forming an X-H bond gives an increase in charge between the two nuclei and an electric field applied along the bond direction will draw this excess charge away from the hydrogen nucleus, thus decreasing its shielding coefficient. The magnitude of the decrease is given by equation (3.125). When E, arises from an external field then its average value in a liquid or gas is zero at any particular nucleus, but is not zero if E, is an internal field arising from the presence in the molecule of a polar group. A polar molecule in solution polar&s the surrounding medium and the nuclei in the molecule experience a reactionfield, R, whose value is independent of the random motion of
THE
ORIGIN
OF CHEMICAL
SHIFTS
AND
SPIN-SPIN
89
COUPLING
the molecules; if the molecule is symmetrical the resultant mean field is in the direction of the dipole moment of the molecule. Bu~kingham(~“) has used a value for R obtained by 0nsageP6); the solute molecule is considered to be a sphere of radius r with a point dipole P at the centre, and the solvent is considered as a continuous medium of dielectric constant e R=
2(& - 1)(?22 - 1) 3(2e i- n2)ar
”
n is the refractive index of the pure solute, and LYis the polarisability tP - 1 . r3. If n2 is taken to be 2.5 then sphere and is equal to n2 + 2
(E-11)
R=
P
(2E + 25) * b- *
(3.126) of the
(3.127)
Equation (3.127) is probably a good approximation for the reaction field in a molecule like i3C X Y _ where the ‘“C shielding coefficient will be dependent on (e - 1)/(2e + ;-5>*ii solvents of different dielectric constant. Hydrogen atoms, however, are usually on the outer part of molecules and the Onsager model is inappropriate; an alternative has been suggested by Diehl and FreemarP’) in which the polarisable dipole is at the centre of a non-spherical cavity. The reaction field is
1)
R
LJ
(3.128)
a, b and c are the axes of the cavity ellipsoid, and 5. is a shape function,
L
=;abc
s
dl
(u2 -t j1)3’2(b2 + jl)l12 (c2 + A)1’2 ’
(3.129)
0
If the molecule has spherical symmetry, a b c = r3, E, = 3, and equation (3.128) reduces to equation (3.126). In some cases a reaction field may exist even when the molecule concerned has no electric dipole moment. The molecules in question are those which contain highly polar groups, but are symmetrical so that the total dipole moment is zero; however, polarisation of the solvent molecules near the highly polar groups can lead to an induced quadrupole moment in the solute molecule(5s). The reaction field gradient set up by the induced quadrupole moment 0 based on an Onsager model iP*) R’
=
6@-
I)8
(3e + 2) rs ’
(3.130)
90
HIGH RESOLUTION
NMR
SPECTROSCOPY
For a molecule with two opposed dipolar groups of moment P separated by a distance d, (3.131) e-2pd. The reaction field R’ is illustrated by the two examples p-dinitrobenzene acetyleneCs5). (See Figure 3.3.)
and
--
FIG. 3.3 Reaction fields arising from induced electric quadrupo1e-s in p-diitrobenzene and in acetylene. Buckingham(s ‘)
3.10 THE EFFECT OF MOLECULAR INTERACTIONSON SHIELDING CONSTANTS Itis shown in Section 3.3 that the observed screening constant for a nucleus in a medium of volume susceptibility x, differs from the value of 0 in an isolated molecule, i.e. 2z (3.132) =o+3 Xv
c
if the sample is cylindrical. But equation (3.132) accounts only for long range magnetic interactions between molecules, and in many systems short range intermola cular interactions cause considerable deviations from the behaviour predicted by equation (3.132) ~3~.84*8s). The short range forces which must be taken into account are those arising from (a) the anisotropy of neighbouring molecules (b) Van der Waals or dispersion forces (c) permanent dipole or quadrupole interactions.The most complete treatment of the problem has been that of Stephe!P) who has found the effect on the observed @ielding constant of effective magnetic and electric fields arising from the surrounding molecules. If H,, is the applied magnetic field then from equation (3.1) the effective field at molecule i as a result of long range interactionP’ is H = HO(l - AX”)
(3.133)
THE
ORIGIN
OF CHEMICAL
SHIFTS
AND
SPIN-SPIN
COUPLING
91
where A is a constant which depends on the shape of the sample. The total effective magnetic field at the molecule i will be I-&f,
=
H + G”’
(3.134)
where Go) is the total magnetic field at molecule i produced by all the surrouading molecules. The shielding constant is obtained from the interaction energy W of a nucleus, dipole moment y,, with the total effective magnetic and electric fields at molecule i. The energy W, of the ith molecule in conf@ration 1 expanded in powers of the effective field is, in a tensor notatioxP6), w, = W(A) - ,u:*‘(H, + Gk”) + @Ha - fxs(
H, + c’,“) (I$ + G;‘)
- q+,,JH,
+ G:‘) ,u;’ - &+,,(Ha + G:‘) &’
+ G;‘) ,u;’ F:” F(d[)a**
,
(3.135)
where &+ and v,+, describe the effect of an electric field F on the shielding constant, o&q,of the isolated molecule. F(*) is the electric field at molecule i arising from all other molecules. The shielding constant co) of a nucleus in the ith molecule is given by (3.136) The bar implies an average over all directions of e, a unit vector in the direction of H,, for & = #*) = 0; and the angular brackets an average over all orientations A. m(l) is the dipole moment of molecule i and is obtained by differentiating W, with respect to Ho, thus m’i’ o f?, = {Q
+ $; (H, + @;I) + [$
- a$! p;’
,g, I$”
Now, assuming that the molecules behave like point dipoles then d,” = -j F.$’ e$
where
aG"'
o
thus
aH0
my)
(3.138)
= rbS (r:, &, - 3 rllcr r*,&
=-
C E$’ xpeee
(3.139)
f 0(x2)
J*’
and Substituting equation (3.140) into (3.137) and (3.136) gives to first order, after averaging over all directions of e, p
s
A xv + a<” -I- f
-
f
pava {F:” I;‘,“) (3.141)
HIGH
92
RESOLUTION
NMR SPECTROSCOPY
which u(‘) = 3 0:. Equation (3.141) will now be applied to the three media of (a) an imperfect gas (b) a pure liquid (c) a solution.
in
(a) An imperfect gas. The shielding constant is expanded in inverse powers of the molar volume(*6) V, C”’ = cb” + +. so that
1’1” + . . .
(3.142)
(0 CO = A x, + o(I)
(3.143)
c:” =“lizic(o m
(3.144)
and - Cb”) V,.
For spherical and tetrahedral molecules xns = x 6,, and from equation (3.141) (3.145) C”’ = A xv + a”’ + 0 (x2) and since Oh*) is zero for an isotropic medium then in this case co’ and a’” are related by Dickinson’s equation’**). For anisotropic molecules,
so that if the molecules are axially symmetric, with two components xfl and XL, then C:” = A x y,~oo lim
(+ - cos’eJ))
of x$, (3.147)
vm
6, is the angle between the molecule j and the line joining nucleus ,P with molecule j. Stephen (86) has calculated x7’ for molecules with permanent dipole or quadrupole moments, and found in both cases that the value of is negligibly small (AI IO-lo) at normal pressures. (b) A pure liquid. In an isotropic liquid, ;cop= x&4; and
vlr22
=
vllJ3
etc. =
L& = 0; tpb
plllll =
932222 =
Qj3333 =
To
Substituting in equation (3.141) gives
A 1”’ = c”’ - A xv - a(” = (cpa+
29~~) (F*>_
(3.148)
A C(*) is the difference between the observed shielding constant and the shielding constant corrected according to equation (3.132). It is found to be negligibly small for methaneCe5’.
THE
ORIGIN
OF
CHEMICAL
SHIFTS
AND
SPIN-SPIN
COUPLING
93
Anisotropic molecules which are axially symmetric, with the axis labelled 3 on the molecular axis, have
E 333 =
4)1111
=
?l ’
t";
5113
v2122; 91111
+
=
t223
=
tL;
t =
t" + =
%A
%122
=
9)2211;
VI133
4)2233
vll22
+
q3311;
v3 =2yl133
+
93333
and
- (2ej:i’ (F:“2)
+ #
(3.149
The first term in equation (3.149) will be the dominant one when the value of A 31”)is large ; for example, Stephen ‘86)has estimated that for benzene this term will be M + 0.47 x 10-6. The second and third terms are dominant when hydrogen bonds are formed since in this case both E (F) and v (F2> are of the order of lO-‘j. For polar liquids not forming hydrogen bonds the term in N molecules of type b. Both molecules are assumed to be axially symmetrical and non-hydrogen bonded so that the shielding constant of each molecule is given by equation (3.149) with the last term neglected. Then C(a) = A@%?’ + (1 - /3)X:‘) + u?) + #6L, + (1 - j?)r, (3.150) and
c’“’ = A{/?%:’ + (1 - g)xf’)
+ db’ + ,5&,,, + (1 - flL,jb
(3.151)
where g and $’ are the respective volume magnetic susceptibilities of molecules a and b, and L&I = A#’ L ob
=
d#’
L bo
=
L bb
=
<&!(a - COSV,.)) - P’ (F’$‘(u’))
(3.152)
(r(;b”
(3.153)
(3
-
~s28b))
-
Fe’
d%,"'(rg(+
-
cos28J)
-
E(b) (Fp(u))
dx$b’
-
ct-,s26bV)>
<&(+
-
&“’
(p;‘(b))
.
(3.154) (3.155)
F:‘(b) is the electric field at a molecule of species u arising from molecules of type b; these fields are taken to be additive and proportional to concentration. The difference in the shielding constants of molecules a and b is AC.6
=
o!~)
-
db)
+
B(L,, -Lbo)+tl
-B)(Lsb-Lbb)
(3.156)
94
HIGH
RESOLUTION
NMR
SPECTROSCOPY
4
and is concentration dependent unless L,. = Lba and Lai, = LLL, and in general this is true only when the molecules a and b are similar in shape and electronic structure, such as methylene chloride and methylene bromide(83). If the solvent is isotropic then L,,b = 0 and measurement of the shielding coefficient c(“) at infinite dilution gives the value for an isolated molecule after correcting for the long range shielding of the solvent. When the solvent is strongly anisotropic the last term in the equations (3.152) to (3.155) may be neglected. At inf?inite dilution (p + 0) equation (3.150) becomes C(a) = &zb’ + a(‘) + AXcb)(r;,“(+ - cos*&,))
(3.157)
and Ax’“’ = AXcb’(r;;(+
- cos*8,)).
(3.158)
Buckingham et ul.(*‘) have applied equation (3.158) to the cases of benzene and carbon disulphide. Both these molecules have a largediamagnetic anistropy, but one is disc shaped and the other rod shaped so that the result of the averaging process, denoted by the angular brackets ( ), will be different in each case. In benzene, collisions of type (a) are more common than those of type (b)
0
_- _ __@__
__e--_o__
(4
@I
so that 6b i 0” and (3.159) For carbon disulphide, collisions of type (a) predominate in which the angle between the molecular axis and the solute molecule is 90”
0
_e--___
0
__
so that the dominant term in the averaging process has 19~- 90” and (3.160) Comparing equations (3.159) and (3.160) shows that for a disc shaped solvent molecule the shift should be to high field and for rod shaped molecules it should be to low field. For methane in benzene and carbon disulphide AC is found to be + 1.3 and - 0.5 ppm respectively@“.
THE
ORIGIN
OF CHEMICAL
SHIFTS
AND
SPIN-SPIN
COUPLING
95
3.10.1 The Effect of the Van der Waals or Dispersion Interactions on khielding Constants In polar molecules the effective electric field at a molecule is the “reaction field ” resulting from the permanent electric dipole moment of the molecule(s5* 57) (see Section 3.9). The average value of the effective electric field in non-polar molecules (F) is zero, but the mean square value (Fz) is not zero. (F2) arises because the mean square value of the electric dipole moment of neighbouring molecules is not zero, and the magnitude of (F’) may be calculated via the free energy, W, arising from the interaction between the time varying dipole moments(**),
w= - f
(m’>
g
(3.161)
where (m2) is the average square of the oscillating dipole moment of the free molecule, and (2n2 - 1) .’ (3.162) g = (2n2 + 2) a3 ’ n is the refractive index of the medium and a is the radius of the molecule. In a solution the partial molar free energy of the solute becomePg)
-$p (y1 ‘+‘p2>
w=
where the subscripts 1 and 2 refer to the solvent and solute respectively, and Y is a mean absorption frequency. Now, the potential energy of a non-polar molecule in a static electric field k is(go) w= where (x2 is the static polarisability
- ;a2
(F2)
(3.164)
of the solute. Thus (F2) is given by
(F2) = t[+][$--].
(3.165)
<& so that
=fhv,a,
(3.166)
+g 1= 1. vi +
(3.167)
y2
For a non-polar isotropic liquid, Stephen (*6)has shown that we may neglect all the terms in equation (3.141) except the term in (F2>, hence AC”’ = pl
(3.168)
HIGH
96
NMR SPECTROSCOPY
RESOLUTION
and thus from (3.167) .
A 2”’
v may be calculated from either the approximationCg’) y=-
I h
where Z is the ionisation potential, or through the relationship molar diamagnetic susceptibility x and <ml>, <m2>
=
_
(2$-C>
between the
(3.171)
x
0
where M = mass of the electron; c = velocity of light; and No is Avogadro’s number. From equations (3.170) and (3.166) y=
-
(3.172)
The radius of the molecules must be known in order to calculate g from equation (3.163) and this may be found from the molar volume since 4 VIII= - nNo a3. 3
(3.173)
TABLE3.2 VALUEJOF PARAMETERS USEDIN THE CALCULATIONOF CKEMKAL SHIF-I+~)
Substance
1, 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 2D.
Ethylether Acetone Neopentane n-Hwrane Cyclohexane Cyclopentane Triethylamine S&l.+ CHzClz Chloroform ccl4 CHBI CH2Br2 CHBr, CHJZ CFjCdHl 1 n-Heptane
Methane
Ethane cp*
Molar dhl&gXlCtiC susceptibility x x 106 (cm3 mole-l) 55-l 33-7 63.0 74.6 68.13 59-18 81.4 115.4 46.6 59.3 66.6 57-2 65-5 82-6 93.1 126 85-24 17.4 26.8 31
Ionisation potential Z
Refractive ill&X
(ev)
nD
9-53 9.69 IO-37 10.17 9.88 10.51 7.50 10 Il.35 11.42 11.47 9.54 10.48 10.40 9-3 16 10.06 12.99 11.65 17.8
l-35272 (15”) 1.35609 (25”) l-35130 (0”) 1.37226 (25’) 1.42354 (25”) 140363 (25”) ldO101 (20”) l-50799 (250) 1.42456 (20”) 144858 (15’) 1.45759 (25“) 1.53152 (20’) l-54180 (20”) 160053 (15O) l-7411 (20”) 1.2815 (20“) 1.3852 (25’)
optical polarisability x x KY’ W) 8.76 6-42 10.07 11.85 10.87 9.17 13.38 13.72 6.34 8.32 IO-24 7-65 8-20 11.81 12-89 13.6 13.7 255 4.39 2.89
Molecular radius cubed a3 x W4 @m3)
37.54
15.1 21.8 17.8
THE
ORIGIN
OF CHEMICAL
SHIFTS
AND
SPIN-SPIN
COUPLING
97
If equations (3.167) and (3.168) are correct then A c(*) should be proportional to (F2) for a non-polar molecule, and the constant of proportionality v should be characteristic of the nuclear species. Howard et &.(**) have tested these conclusions by comparing the gas-to-solution shifts of methane and cyclopentane in the solvents listed in Table 3.2 with the value of (F2> in these solvents calculated with values of Y obtained from equations (3.170) and (3.172) (see Figs. 3.4 and 3.5). The line on Figs. 3.4 and 3.5 was obtained with a value of 9 = 0.74 x lo-l2 cm2 e.s.u.-1 ppm calculated for the hydrogen atom by
l 14
.
013 I2
IO-“
FIG.3.4 Corrected experimental gas-to-solution chemical shifti veraus
Marshall and Pople(54). Figures 3.4 and 3.5 show that q is approximately a constant for methane and cyclopentane, but that the value of q~is considerably larger than that for the hydrogen atom. Although equation (3.169) predicts the correct trend for A co) with in the case of CF, in different solvents; in this case the experimental points do lie on approximate straight lines passing through the origin. The theoretical value of v is not known for fluorine, but the experimental evidence suggests that it is about 20 times greater than for the hydrogen atom(88). EFts. 4
.
98
HIGH
RESOLUTION
NMR
SPECTROSCOPY
d c(i) may also be related to the partial molar volume or the density of a gas through the Lorentz-Lorent? equation(a** go)
8
PP + 2
8nNoa,
= 2n2 + 1 *
3V,
1 ‘Z
(3.174)
thus, d C”’ = (jI
(n’ + 2) . 2nNo h 0~~ . L . ~1~2 (2n2 + 1) A41 a; VI + Y2 * @
(3.175)
FIO. 3.5 Corrected experimental gas-to-solution chemical shifts versus (F2> for methane (filled circles) and cyclopentane (open circles) in various solvents (see Table 3.2). (F’) obtained from diamagnetic susceptibilities. Howard, L,inder and Emerson(*s)
where e is the density of the medium and A4 the molecular weight. Equation (3.175) is in agreement with the observation of Gordon and Daileytg2) that there is a linear relationship between Ax’” and ,g for non-polar gases and liquids. The dispersion interaction can also account for Evans’ observation(g3) that the value of A c for gaseous perfluoromethane is proportional to the pressure since equation (3.175) may be re-written as (3.176)
THE ORIGIN
OF CHEMICAL
SHIFTS
AND
SPIN-SPIN
COUPLING
99
Marshall and Pople (g5)have made a quantum mechanical calculation of the effect of the Van der Waals interaction on the shielding constant of a hydrogen atom separated from a neighbouring hydrogen atom by distances large compared with the equilibrium separation in a hydrogen molecule. They conclude that Van der Waals forces tend to reduce the value of the shielding constant, and calculate that the reduction in u is about lo-’ for two hydrogen atoms separated by 4 4 (approximately twice the Van der Waals radius in hydrogen).
20
40
60 I@
CFz>
FIO.3.6 Correctedexperimeatalgas-to-solution&emicai shifts vomus for carbon tetrahoride in various aolvcnts (see Tabk 3.2). Open circles and short dashed line are from diamagnetic susceptibilities. Howard, Linder and Emorson
3.11 THEErwncr OF ISOTOPIC SUBSTITUTION ON
SHJELDING CONSTANTS
It has been observed that the shielding constant of a nucleus is increased when a neighbouring nucleus is replaced by a heavier isotope; thus Wimmett(6) observed an increase in u of the order of lo-’ for the hydrogen atom in the molecule HD compared with HL, and Gutowsky’g6) has observed changes in u of a similar magnitude in the compounds CHD,COCHJ, acetone - ds and CHsCOCHs. In fluorine resonance spectra Tierstg7*gs) has noticed changes in u as large as 0.60 f 0.05 x 1O-6in the CFsD group of a-CSF,D compared with n-&F,H. The isotope shift probably arises because of the different amplitudes of the vibrations in the ground state of the molecules concemed(gg). The shielding constant of a nucleus will depend on the dimensions of the molecule, so that the observed value of u is an average over the motions of the nuclei in the ground vibrational state. In a diatomic molecule the averaging of u may be carried out 4*
100
HIGH
RESOLUTION
NMR
SPECTROSCOPY
if the variation of cs with internuclear distance R is known, and if the nuclear wavefunction y for the ground vibrational state is known (assuming that all molecules are in the ground state). Expressing a(R)as a polynomial in R of the form(‘OO),
a(R)= a(R,)+ a'(R,)(R - R,)+ Lu"(R,)(R 2 - R,j2+ .** (3.177) where R, is the equilibrium internuclear distance, then since 00 a obs = y2a(R)d R
(3.178)
0
we may write, a ohs
=
a(R,) + (R - R,)a’ (R,)+ ((R - R,)Z)a"(R,) + - - * (3.179)
The brackets ( is
) denote a quantum mechanical average over the motion, that
(R-R,) =SmW2(R-Re)dR
(3.180)
0
((R
-
KY)
=
L2CR
-
R,)‘dR.
(3.181)
0 The function y is determined by the nature of the potential field Y(R)in which the nuclei move. If the motion is harmonic then terms such as (R - R,)are zero, whilst ((R - &)2) is non-zero for both harmonic and anharmonic motions. MarshaWoo) chose for V(R)a function with both a harmonic and an anharmonic term, F'(R) =f(R - R,)2- g(R - R,)3 (3.182) giving for y (to fust order in g) Y = y. + g
p2
/&-l/4
f-514
+ $5Y3)
(3.183)
where p is the reduced mass, and y. , yl,and y3 are the ground, lirst and third excited states for the corresponding harmonic oscillator (g=O). Substituting equation (3.183) into (3.180) and (3.181) gives (R
-
R,)
= ;ghy-'/2f-J/2 L
,&p2f-l/2 a,,,, = a(R,) + fg
fi p-1/2 f-3’2 a’(R,) + f+-1/2
f-'/201f(R,). (3.184)
The values of g and f may be obtained from the vibrational spectrum of the molecule, and the constants a(R.), a'(R,) and a"(R,) may be determined if the
THE
ORIGIN
OF CHEMICAL
SHIFTS
AND
SPIN-SPIN
COUPLING
101
variation of Q with interatomic distance is known. For hydrogen a calculation by Marshall and PopleCg5*loo) gives o’(R,) = - Il.30 x 10-6 c”(R,) = Il.06 x 1O-6 and substituting shifts,
values of g and f into equation - o(H,)
o(DJ
(3.184) gives the isotopic
= 055 x IO-’
- a(HD) = 0.30 x lo-’
a(D,)
which agree well with the observed orders of magnitude. The above calculation contains the assumption that all the molecules are in the ground vibrational state, which is a good approximation at room temperature; however, as the temperature increases, the population of excited vibrational states increases, and this should be reflected in a temperature dependence of the shielding constant (“I) . If the shielding constant for the ith vibrational tate is b(l) and that for the ground state is a(O), then do) + 7 g, a”) exp( - h q/k T> (3.185)
Uoba = 1 +
c
a
exp(-
h vi/k
0
1
where g, is the degeneracy of the vibrational mode of frequency vi. When vi > 200 cm-‘, and T - 3OO”i( equation (3.185) may be written as %bs = uw) + Cgrda”)exp 1
(3.186)
where da”) = uo) - u(O). The chemical shift v. 6 from some reference compound, whose shielding constant is independent of temperature, is given by vo
6 = vo d(O)
where dI = v. cV) - v. tY”). The variation of v. S with temperature
d(vo8)
-
dT
=
(3.187)
is
rgi4(s)exp(%)
(3.188)
so that the effect of each vibrational mode is additive. Petrakis and Sederholm(77) have in fact observed a temperature dependence of v. 6 for several gaseous compounds, with values of d (v. S)/d T of the order of - OXlO cycles set-1 deg-’ . 3.12 TEMP~TURE DEPENDENCE OF SFIIELDING CONSTANTS If the chemical shift is measured relative to an external reference signal then it will be both temperature and concentration dependent since both the diamagnetic and electric susceptibilities are dependent on both temperature
102
HIGH
RESOLUTION
NMR SPECTROSCOPY
and concentration. When an internal reference signal is used then in most cases the observed chemical shifts do not depend on temperature. However, some examples of temperature dependent shifts have been observed(77*g4); the small temperature variation arising from vibrational excited states has already been discussed (Section 3.11). The equations for the shielding coefficient discussed so far have all been for a single electronic ground state n. However, it is possible that other electronic states are occupied in each of which (I is different; as the temperature is varied the distribution between the states varies and the average value is altered. If for states n = 0,l . . . g, the quantity exp (- E./kT) is appreciable, whilst for other states it is negligible then the weighted mean average of u for the case when the molecule is symmetric about the z-axis and when all orientations of the molecule with respect to the field are equally probable is given by(“) 0 = 01 + 62 + Qj (3.189) where 01 =
and
C = [iTP(-$$)]‘-
A particularly simple case would occur if only one low lying excited state were possible apart from the ground state. The shielding coefficient would then be approximately the term a, in equation (3.189), that is
The majority of cases when the shielding coefficient is temperature dependent are a result of some form of molecular association and are not described simply by equations (3.190) to (3.193) since these equations refer to a single molecule. The temperature dependence of the shielding coefficient when the molecule
THE ORIGIN OF CHEMICALSHIFTS AND SPIN-SPIN COUPLING 103 takes part in molecular association depends upon the association of many molecules, and the various theorems for the equilibrium of the association must be used (see Chapter 9). SPIN-SPIN COUPLING 3.13 THEORY OFSPIN-SPINCOUPLING We have already discussed a mechanism for spin-spin coupling in which the interaction proceeds via the electrons (Section 3.2); however, other coupling mechanisms may also play a part and in general all magnetic interactions must be taken into account.(**i) The interaction energy A?& for two nuclei N and N’ in a molecule with a specific orientation Ais given by(5g) E NN'
=
hIN-
Jme ’ IN’ + hJNN. IN - 1~‘.
(3.194)
JNN*is a tensor of the second rank whose trace is zero. For frequent intermolecular collisions JNppaverages to zero, so that (ENN,LL
=
11JNN*&
*IN*
(3.195)
where INis the nuclear angular momentum in units of It. If a Hamiltonian for the system is constructed which represents all magnetic interactions in the molecule, then J?&~,can be obtained by selecting those parts of the Hamiltonian which depend upon both IN and INI. It is -convenient to write the Hamiltonian as *=z,+X*+*Si-pb
(3.196)
and we will discuss each part of equation (3.196) in turn. 3F’, is simply the Hamiltonian used in the magnetic shielding calculations [Section 3.5 equation (3.22)] with the addition of electron-electron interaction terms, i.e.
For simplicity a specific choice of gauge has been made. The terms flu, X Ls, Zss and flsm are respectively electron orbital-orbital, orbital-spin, spin-spin, and spin-external field; these terms may be neglected for singlet molecular states. In the absence of an external magnetic field only two terms in equation (3.197) need to be considered.
’ (3.198)
104
HIGH
RESOLUTION
NMR
SPECTROSCOPY
d is the unit dyadic.
tiZ and ZJ together describe the nucleus-electron-electron-nucleus interaction discussed in Section 3.2. The mutual interaction energy of a nuclear magnetic dipole pN with an electron dipole pLI,is Eh
=
-J+
3@N ’ rkN) hk ’ r&d
’ pk
PN
Substituting
.
(3.200)
dN
rkN
y h IN for pN and 28 h S, for & gives the Hamiltonian 3 (Sk sZ
=
28ficyN
* rkd
(IN
’ sN>
5 rkN
kN
as
(3.201)
Another term Z’S is needed in the Hamiltonian since equations (3.200) and (3.201) do not apply for electrons in S-states, since if the electron density at the nucleus is finite then as rk, -+ 0 the energy becomes infmite. A term &‘3 was first introduced by Fermi (60) for s-states to account for the hyperfke structure interaction, and the form given here has been discussed by Ramsey(2g* 5g). Imagine the nucleus enclosed by a sphere of radius R, where R is large compared with the size of the nucleus, but small compared with the atomic radius. The value of R is such that we may regard the wavefunction for the electron to have a constant value, y. , over the volume of the sphere, that is the electron density within the sphere is uniform. e(0) = (0 IQ&l
0)
where e(O) is the electron density within the sphere, and I!&) d-function. If m(0)is the magnetisation within the sphere then m(0) = -e(O)‘2~&
(3.202) is a Dirac (3.203)
and the induced field at the nucleus is
B(O)=!+(O) Since the
nUdeUS
=
16x --j-do)~sk.
(3.204)
has a magnetic moment y Ir IN, then the interaction energy is (0) E kN
16n/? =jd”)yNASkelN
(3.205)
and the Hamiltonian is (3.206)
THE ORIGIN
OF CHEMICAL
SHIFTS
AND
SPIN-SPIN
COUPLING
105
The last term in the Hamiltonian is simply the internuclear dipole-dipole interaction which gives rise to the broad lines observed in solids
S4
averages to zero when all molecular
orientations
are equally probable.
CALCULATION OF SPIN-SPIN COUPLING CONSTANTS
3.14
3.14. I
Application of Perturbation Theory calculation(5g) gives a second order energy correction,
A perturbation c
ENNI
=
-&
tEa
;
Eel
(0
icfln>
IsI
0)
(3.208)
NIl.3’
that comparing equation (3.194) with equation (3.208) gives for &@ and
so
Jwt
J NN’ =
NN’
=
-
-
:,
.;o
E i &I
(0 1dl
n>- (n I 810) n) (n 1910)
(3.209) -
JNN*6.
(3.210
For the case of all molecular orientations being equally probable Jml is zero and the coupling constant is given by equation (3.209). Ramsey’5g) has given expressions for both JNN*and JNN*for all tems of the Hamiltonian; however, since in almost all practical systems the coupling is observed only when all molecular orientations are equally probable, equations for JNN* only will be considered here. For this case cross terms between .?P,, 2, and S’s may be neglected and JNw can be written as JNN*= Jy&, + Ji2&, + JZNNV+ JsNNs. Each term in equation (3.211) is obtained by substituting Hamiltonian operator into equation (3.209),
(3.211) theappropriate
(3.212)
- (n
t
IW,,)
s, I 0) *
(3.215)
The operators &and I are those parts of the Hamiltonian that are bilinear in IN and I,..
ERS. 4.p
106
HIGH
RESOLUTION
NMR
SPECTROSCOPY
In most molecules, JaNNa makes by fdr the largest contribution to JNN,, and JINNe the smallest. Evaluation of the summations in equations (3.212) to (3.215) is in practice impossible since it requires a knowledge of the excited state wavefunctions. In the majority of calculations with these equations the average energy approximation has been applied, that is the summation over the excited states is replaced by 1 E. - &
=& 1 (0 I L%=I 0)
(3.216)
where A E is an average energy separation and is defined by equation (3.216). To solve for A E from equation (3.216) would involve the problem that we are trying to avoid, namely the evaluation of the summation. In practice a value for A E is estimated. The application of the average energy approximation to spin-spin coupling theory has been discussed by McLachlaP), AlexandeP) and KarpluP3), and other methods of evaluating the summation have been discussed by the last two authors. 3.14.2 Molecular Orbital Approach to the Calculation of Spin-Spin Coupling Constants After an average energy approximation has been applied to equations (3.212) to (3.215) the integrals are over the ground-state wavefunction for the molecule, and this function may be approximated to by either a molecular-orbital or a valence-bond modeP4). In a general molecular orbital treatment by McConnelP5) the function p is an anti-symmetrised product of molecular orbital and spin functions (3.217)
There are M orthonormal molecular orbitals yl, ty2 . . . yM which contain N electrons, two in each orbital, so that N = 2M. ‘?‘eo= ‘f’t(1) Y I (2) Y’z(3) Y2 (4) r~rs= d)B(2)d3)8(4).
. . . PM
. .a(N
-
W
-
l)B(N).
1) ‘PM (IV)
(3.218) (3.219)
P is the permutation operato?’ and represents the N! permutations of the electrons among the orbitals; any one permutation involves either an even or odd number of interchanges of the electrons in pairs and if P is even (- I)p is positive, if odd (- l)P is negative. The molecular orbitals ya are formed from a linear combination of atomic orbitals p?;,
THE
ORIGIN
OF CHEMlCAL
SHIFTS
AND
SPIN-SPIN
COUPLING
107
v; is an atomic orbital centred on nucleus p and contributing to y,_. Configuration interaction is neglected. By substitution of equation (3.217) into equations (3.212) to (3.215) the expressions for JNNnmay be reduced to’65)
J
in which
J 3NN’
=
cos2e,,
-
=
c&N
* irN#)’
c28 h)2 YN YN’ &
(3.224)
If the L.C.A.O. form for y= of equation (3.220) is introduced into equations (3.221) to (3.224) a number of atomic integrals appear which give appreciable contributions to J&I when N and N’ are bonded together, but which are diG fkult to evaluate. For coupling over two or more bonds these integrals are negligible and the equations for JNN, are considerably simplified. A further simplification results if the atomic orbitals are only S, p or sp hybrid orbitals; thus substituting for vS by equation (3.220), considering only long range (LR) coupling and retaining only the largest one and two electron integral~(~“* 67). bb’
-108
HIGH
RESOLUTION
NMR
SPECTROSCOPY
(An error in the equation for JZNNein McConnell’s paper has been corrected by Williams and Gutowsl@j’).)
case,,.= J:‘&(LR)
r
St4’ rNN'
rsyrsN,
= -
(3.226)
x I(3 cos2 6,, - 1) r$
r$ I &
Wq&@>>I-
(3.227)
In equation (3.211) only p atomic orbitals are present and 0 is the angle between thep orbital and rNNl.For the two electron terms 0 is equal to the angle between the p orbital axes. Finally, the contact term J3 NNlis gz
J JNN&R)
=
$
(2ph)2YNYN&
2M.M
+ K
)
c %P
aNaaxfiaN'aaN'j
x (vii (4 I6(rsN)I Pi (4) k4.1~ (0 I6(rrN,)I c& 0 >)I.
(3.228)
Equations (3.225) to (3.228) predict that the coupling constant will always be positive, a result that is at variance with experiment. The constant sign of JNNaarises because a set of real, orthogonal, single electron orbitals have been used containing a pair of electrons in each orbitaP). Negative coupling constants could be obtained if a molecular orbital function were constructed with configurational interaction@*). Another objection to equations (3.225) to (3.228) is that they do not predict a reduction in JNN*as N and N’ are separated by an increasing number of bonds(6g).
THE
ORIGIN
OF CHEMICAL
SHIFTS
AND
SPIN-SPIN
COUPLING
109
3.14.3 Valence Bond Theory and the Evaluation of .TNN. The inherent difficulties met in applying L.C.A.O.-M.O. wavefunctions to the evaluation of JNNnby Ramsey’s equations are mostly absent when the wavefunction is constructed by the valence bond method. The wavefunction y. is expressed as a linear combination of functions representing independent valence bond structures. (3.229) Yo = CC,%. Each y+ is a combination of 2n singly occupied orbitals a, b, c . . .2n, with a spin function 0~or p associated with each of the 2n orbitals. The value of j goes from 1 to (2n)!/[n!(n + 1) .] 1 , corresponding to the number of independent singlet wavefunctions which can be constructed from 2n orbitals. The independent valence bond structures (known as a canonical set) are readily constructed with the aid of Rumer diagrams(‘O). If the 2n orbitals are arranged in a ring and bonded orbitals joined by a line drawn within the circle, then canonical structures are those in which no lines intersecP). yJ must be antisymmetric in the electron coordinates thus(‘l),
3%=
(&)F’-l)‘R[ [(2n;!ll,2]; (- op
x P a(1) p(l) b(2)&(2) . . . a(2n)oc(2n)
(3.230)
a(l) is one of the 2n orbitals, and it is conventional to associate the spin functionB to a( 1). P is the permutation operator(66), and R represents the operation of interchanging the spin functions LTand b for each pair of bonded orbitals; (- l)R is even for an even number of interchanges, and odd for an odd number. Substitution of equation (3.229) into Ramsey’s equation for JJNNn, after first making an average energy approximation, and using the Dirac identity(46) P;* = f
(1 + 45, *S,.)
(3.231)
gives(72)
(3.232) where P& represents the interchange of the spins of electrons k and k’. The integrals in equation (3.232) are of two types, (3.233) (3.234) Substituting for yJ in I1 and I2 by equation (3.230), then if the 2n orbitals a, b... are orthogonal, and neglecting the contributions to the electron den-
110
HIGH
RESOLUTION
NMR
SPECTROSCOPY
sity at nuclei N and N’ of all electrons other than those centred on N and N’, then the integrals reduce to (reference 66, p. 239) II=
+ (
>
&(-
l)R+R’G41)B(l)
b(2)&(2)
- **Ii&J
*&.I~ll
x b(l) PO) W) 42) ’ - *I> I2=
+
>=!>.
(
(3.235)
uR’-R’ (Ml) @(I) b(2)&(2) * * .I Ioh,)
*Wk+J *Pk I
XW>B(1>42>~(2) *- .I>.
(3.236)
The integral I1 is in the form of a Coulomb integral (a b c 1JP 1 a b c) except that the Hamiltonian operator has been replaced by 8(r& c?&,). Similarly 1, is in the form of an exchange integral (a b c 12’1 b a c) since the operator P& has the same effect on the spin orthogonal@ of the integral as the permutation Pw. Both 1, and IZ may be evaluated with the aid of Pauling superposition diagrams (‘l). If after superposition of the two canonical structures yI and p1 there are ill closed polygons formed, then the coefficient of the coulomb , and therefore I, =
~(0)
(& )
C%N (0)(?JN’(O)
(3.237)
and ~~~(0) being the electron
densities at the nuclei N and N’. The coefficient of the exchange integral is fil (PNN*) where f,,(PNNe) equals - 3 ( 2”4ll > if N and N’ are in different polygons, + 1 if in the same polygon and separated by an odd number of bonds, and - 2 if they are in the same polygon but separated by an even number of bonds”“. The integral I2 is therefore fit
I, = [
When equations obtain
VNN’) p-h
1
VN
to> (0):. P)N’
(3.238)
(3.237) and (3.238) are substituted into equation (3.232) we
x [l + fidPNN')I*
If “perfect pairing” takes place only one valence bond structure considered and the value of J3 NN’for bonded nuclei N and N’ is”“) J 3NN'
=~(~)(~~YNYN.~N(O)PN'(O)
(3.239) need be
(3.240)
and zero for all other pairs of nuclei. Since the contact term provides almost all the coupling between hydrogen nuclei the existence of a non-zero coupling
THE
ORIGIN
OF CHEMICAL
SHIFTS
AND
SPIN-SPIN
COUPiING
111
constant between two hydrogen nuclei which are not directly bonded implies that canonical structures other than that for perfect pairing contribute to the wavefunction(72’. 3.14.4 Calculation of JNN. by the Variational Method As already shown in the shielding coefficient calculations (Section 3.6) the summation over excited states of the molecule can be avoided by means of the variational approach to solving the secular equation(73*80’. In Stephen’s(73’ calculation, the contributions to JNNa by the direct coupling of nuclear spins (.7’$‘&#+ J\‘h.)areobtained separately from the contributions (Jz NNn+ J3 NNn), a different variational function being used in each case. The Hamiltonian .V1 in equation (3.234) may be written in the form .W1 = INn .@;I + I,., &%$;I+ I& INSie&Z’ where the Greek suffixes indicate a tensor notation, (3.199)
(3.241)
and from (3.198) and (3.242)
&y’
(3.243)
3:
(3.244) The wavefunction p is in the form y = yo(1 + INcrfLN’+. I,*,f:N”>
(3.245)
and the unknown functions fLN’ and f p” are to be found by minimising the energy, (3.246) E NN’ = hJaB INo I,., . Jd is a symmetric coupling tensor and is given by J$’ = (Ol~‘lO) (grad&f p’* - grade f p” + grad, f p’ grad&?” +
(fp”
-f
y)Syi
0).
I
O>
(3.247)
Minimising equation (3.247) with respect to variations in f p’ and f p” gives these two functions as solutions of the equations, %N+O
+
&
PO
vff P’ + &
grad& ?l’ograd, f p’
a
= 8
(qq
112
.HIGH
RESOLUTION
NMR
SPECTROSCOPY
and a similar equation in IV. Equation (3.247) may be simplified since mkN is Hermitian and fLN) and fF" are purely imaginary functions, and the new equation as derived by Stephen(73) is
The electron-nucleus contributions to Jd can be found in a similar fashion. The Hamiltonian is (S2 + LX?,) (cf. equations (3.196), (3.201) and (3.206)) and is written (S2 + S3) = INa.%‘:’ + ZNta&&=r)
_
za
YN
c
3 @k
+ ZNa.%‘i6’ + ZNra9:‘.
’ IkN)
rky
k
-
6N
..
Sk,
(3.250) (3.251)
rkN
SP 0 is the same as AC4’ I in N’ (3.252) ti=” is the same as Z’i6’ in N’. To determine (JZNNO+ JsNN,) = J$‘, t =
PO
h
+
%(I,,
the wavefunction
hN’
+
&‘a
y is written as (3.253)
dN’))I
CJJ# is an antisymmetric spin function and v, a symmetric triplet spin function. p” and pN’ are unknown functions which are to be determined by the variational principle. With this wavefunction .I$’ can be written as
f$’ =
(JZNW
ii2
= 4M
+
JJNN,)
(grad,p;N’* grad, py” + grad&“‘*
0 (
grad, piN’)+ 0
II k
I>
+
+
(y8
‘ps
I (jly?)
+
dy’>
py
+
;
(y$
p’:
( (Gty
+
2:))
py*
+
(3f$i’
+
+ complex conjugate. The functions piN’ and pLN” are found from
- li2 FYO
c vzPP’
- $;
(xy)
+
fly’)
+
Y/y))
pp’
py
I q.$ yo)
1 cps yo)
(3.254)
1yo gradk y. - gradkpim = 0
(3.255)
k
and a similar equation in piN”. For rapid, random motion of the molecules av(J+& = +(Jxx + J,.,, + J,,).
THE ORIGIN OF CHEMICAL SHIFTS AND SPIN-SPIN 3.15 THE COUPLING CONSTANT
COUPLING 113
AND THE ANISOTROPYOF THE SFHRLDLNG CGRFFICIRNT
The relative orientation of nuclear spins by direct dipolar interaction of the nuclei is destroyed by random motion only when the shielding coefficients of the two nuclei are isotropic; when the shielding coefficients are anisotropic there is a small but significant contribution to the coupling constants. For inter-hydrogen nuclear coupling the dipolar contribution is very small, but is ‘more important when larger nuclei are considered. An approximate relationship between the coupling constant J1 NNeand the anisotropy of the shielding coefficient can be obtained by considering the nucleus N’ as a magnetic probe investigating the magnetic polarisation produced by N(74). The Hamiltonian X1 (see equation (3.198)) is, neglecting the electron-eleo tron terms, 1 St?, = p+ 2+ v. k2M >
c (
.%‘, describes the motion of the electrons in a magnetic field with vector potential &. If & arises from the nuclear moment y h IN, then
that is, the nuclear dipole gives rise to an induced electron orbital current which may be represented by an inctuced orbital magnetic moment &+ The total magnetic moment produced by the induced current is related to the shielding coefficient of the nucleus in an external field. Consider the energy of the system in the combined field of the nuclear dipole and an external field. The vector potential 4, is now &=&+ih; (3.258) where & is the vector potential arising from the nuclear dipole, and g that arising from the external field. Carrying out a second-order perturbation calculation for the energy, the cross terms in & and Ai’ give the interaction energy between the external field and the induced orbital currents, but this same cross term also gives the interaction energy between the nuclear dipole and the currents induced by the external field, which, of course, is simply what is measured by the shielding coefficient. If H, is the external field then the field at the nucleus is (H, - u,&J, where us,, is the shielding tensor. The field at the nucleus arising from induced orbital currents is - o,&, thus Interaction
energy = y h I, u4 H, =
therefore
-t-&b
*I3
&,& = - y h $,#I,,
(3.259)
where lu,, is the total magnetic moment produced by all the electrons in the molecule, but if interatomic currents are negligible then equation (3.259) can
114
HIGH
RESOLUTION
NMR SPECTROSCOPY
be regarded as arising from the electrons localised on nucleus N. The contributions to the orbital magnetic moment from circulating currents on atoms other than that containing the nucleus N is estimated by, p, = y A {3(I * R,) Rt - R:I} R;’ xr.
(3.260)
x, is the magnetic susceptibility of an atom at a distance R, from N. The coupling energy between two nuclei N and N’ via electron orbital polarisationis equal to twice the interaction energy of the magnetic moment of N’ with the currents induced by N. The currents induced by N may be divided into(74) (a) Currents on the atom with N as nucleus (b) Currents on the atom with N’ as nucleus (c) Currents on all other atoms. Currents (a) give a magnetic moment of - y zia,, Ip at N, and the magnetic field at N’ arising from this induced dipole is y fi e,@G@,, I,,, where .Q is the dipole interaction tensor, E,B = {R2 4,
- 3R,Rp) R-5
(3.261)
R is the position vector of N’ with respect to N.
The unscreened field at N’ arising from the dipole y h I, at N is simply - y A t, I,, so that the contribution from induced orbital currents of type (b) is y h cap a&, I,,, where u$ is the shielding coefficient of N’. The contribution to the field at N’ arising from currents on other atoms can be obtained from equation (3.260). If RI and R; are the position vectors of a third nucleus relative to N and N’ respectively, then the moment induced in atom i by the currents on N is - y A &iauI, xr and the field arising from this at N’ is y h xi &a@E& 1,. The interaction energy arising from currents (a), (b) and (c) is of the form(74) E%) = h Jab I, Ia where
*is a summation over all other atoms’ 1. (5: If all molecular orientations are equally possible,
and
, &a8 Eo# = R-5 (R2 a,, - 3R,R,a,,)
is non-zero only if a,, is anisotropic. If each nucleus has an axially symmetric environment then a,, has two principal values ull and o,, corresponding to the field parallel or perpendicular to the symmetry As. A0 =u\i -ul.
THE ORIGIN
OF CHEMICAL
The value of Jco*) NN’ is now 2 J(_b) NN’ = - 5 YNYN 0
SHIFTS
Jl2{R-3[Ll
- 3 C+& Ri3(&)-3
AND SPIN-SPIN
115
COUPLING
u(1 - 3 COSV) + d d (1 - 3 COSW)]
(1 - 3 COS20J}
where 8 is the angle between the symmetry axis of the shielding tensor and the internuclear line and 8# is the angle subtended at atom i by N and N’. EFFECTS
IN PARAMAGNETIC 3.16
&WTACT
MATERIALS
Sm
For some paramagnetic complexes, in which diamagnetic ligand molecules are bonded to a transition metal, the NMR spectrum of the nuclei in the ligand molecules sometimes show unusually large chemical sh.ifWo6 - log*l 19) (see Section 10.36). For example, the hydrogen resonance spectrum of an aqueous solution of the complex ion [(H20)4 Ni(NH2CH2CH2NH2)]2+ consists of two groups of chemically shifted peaks separated by N 300 ppm(lo9). These large chemical shifts are thought to arise because of a slight transfer of unpaired electron spin from the paramagnetic ion to the ligand molecule; the unpaired spin is then transmitted to the other nuclei in the l&and molecule by-the contact mechanism. A qualitative description of. the distribution of spin density in the ligand is given in Section 10.36’where it is seen that, in the nickel ethylene diamine complex, the net spin polarisation of the electrons in the N-H bond tends to be anti-parallel to the applied field and produce a shielding field at the hydrogen nuclei so that their resonance shifts to higher applied fields. There is a positive spin density in the C-H bonds, hence these hydrogen nuclei resonate at lower applied fields. The magnitude of the contact shift d H, at the ith nucleus in the ligand is given by(106.107) AH, _=
H
_a‘~Ms+ YN
1)
6kT
(3.265)
in which g is the g-value of the paramagnetic species of spin S and a1 is the hyperfine interaction constant. It can be seen from equation (3.265) that the direction of the contact shift gives the sign of al. The condition for observation of a contact shift is that
in which Tz is the electron spin relaxation time, and T, is the characteristic exchange time for the paramagnetic species. The value of uI calculated from equation (3.265) can be used to calculate the pn electron spin density at the appropriate nucleus. In the case of hydrocarbons the value of a, obtained from hydrogen contact shifts has been shown
116
HIGH RESOLUTION NMR SPECTROSCOPY
to be related to the pn electron density ec on the attached carbon atom by the relationship(116) (3.266) an = Qec. For the 6-H fragment, Q has been found to be - 22.5 gauss(ll’), and for the 6-CH, fragment the value of Q is + 27 gauss(“*). Phillips et al.(lo8) have shown that when the nuclear species whose contact shift is being measured can take part in n-electron conjugation, then equation (3.266) no longer describes correctly the relationship between a, and the electron spin densities. They suggest, for example, that in the case of lgF contact shifts in fluorocarbons the value of aF is related to both ec and & by the equation aF
=
Q&c
+
(3.267)
Q&F
where Q& is an-ucontribution to ar similar to that in hydrocarbons, and Q& is the contribution arising from z-u polarisation of the 1s and 2s electrons of fluorine by the spin density centred on a 2p x fluorine orbital. Phillips er al. suggest that eF is determined by the extent of double bonding in the C-F bond so that equation (3.267) may be written as
wherepc, is the double bond order of the C-F constant.
bond and A is a proportionality
3.17 THE OVWIAUSRREFFECT It has been shown already that in solutions of paramagnetic compounds the nuclear and electron spin systems are not independent of one another, and as a consequence the spin-lattice relaxation time of the nuclear spins in such a system is very short, and the resonance lines are too broad for observation. Overhauser(“O) has shown that another consequence of this electron-nucleus interaction is that if transitions between the electron spin energy levels are excited by radiation at the appropriate Larmor frequency, then transitions are also induced in the nuclear spin system. If the electron-nucleus interaction is the dominant spin exchange process for the nucleus, the result is an increased polarisation of the nuclear spins between the possible energy levels, and hence an enhancement of the nuclear resonance absorption signal. The nature of the electron-nucleus coupling and the mechanisms which produce the nuclear spin polarisation in an Overhauser experiment have been discussed in detail by Abragam” II), and also by Anderson(112). The interdependence of the nuclear and electron spin systems can be expressed by the phenomenological equation’“‘* 112)
((I,) - IO) =
(1- $-)e(&)
-
SO)
(3.268)
where (Zz> arid (S,) are the expectation values of the z-components of the nuclear and electron spin momentum operators, and lo and So are their values in
THE ORIGIN
OF CHEMICAL
SHIFTS
AND
SPIN-SPIN
COUPLING
117
the absence of any irradiation of the electron spin system. Tl and T,, are the relaxation times of the nuclear spins in the presence and absence of the paramagnetic material, and e is a factor which depends on the nature of the electronnucleus coupling. The coupling may be of two kinds: a dipole-dipole coupling for which e = + 3, or a scalar coupling, described by the Fermi contact term, for which e = - 1. If the electron spin system is saturated by applying a strong microwave field at the electron Larmor frequency, then (S,) is reduced to zero and equation (3.268) becomes, ((Z,)
- Z,) =
1 - $(
es, 10)
hence the nuclear polarisation is increased by the factor
which can have a value of up to approximately 1000. For most solutions the enhancement obtained by an Overhauser experiment is of the order of 100(112). Richards and White” 13-1 l 6, have investigated the Overhauser effect in solutions containing small amounts of free radicals, and besides showing that it is possible to achieve signal enhancements of the order of 100, they also show that it may be possible to obtain inter-nuclear distances between interacting molecules from the exact behaviour of the nuclear resonance signal. .
REFERENCES 1. W. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.
14. 15. 16. 17. 18. 19.
20. 21. 22.
D. KNIGHT, Phys. Rev., 76, 1259 (1949). W. G. Pmmo~ and F. C. Yu, Phys. Rev., 77, 717 (1950). W. C. DICKINSON,Phys. Rev., 77,736 (1950). W. G. PROCTOR and F. C. Yu, Phys. Rea., 81,20 (1951). H. S. GUTOWSKYand D. W. MCCAL.L,Phys. Rec., 82,748 (1951). Phys. Reu., 91,476 (1953). T. F. Wmm-r, H. M. MCCONNELL,A. D. MCLEAN and C. A. REULY, J. Chem. Phys., 23,1152 (1955). W. E. QUINN and R. M. BROWN, J. Chem. Phys., 21, 1605 (1953). H. S. GUTOWSKY,D. W. MCCALL and C. P. SLICHTEX,J. Chem. Phys., 21,279 (1953). D. P. Roux and G. J. BJZNE, J. Chem. Phys., 26,968 (1957). J. M. ROCARD,Archives des Sciences, 10, 209 (1957). D. P. Roux, Archives des Sciences, 10,217 (1957). N. F. RAMSEYand E. M. PURCELL,Phys. Reo., 85, 143 (1952). W. E. LAMB, Phys. Rev., 60, 817 (1941). L. D. LANDAUand E. M. Lmcmrz, Chssicai Theory of Fields, 2nd cd., Pergamon Press, Oxford (1959). E. H~LLERMS and S. SKAVLEM,Pbys. Reu., 79, 117 (1950). W. C. DICKINSON,Phys. Reu., 80, 563 (1951). N. F. RAMSEY,Phys. Rev., 78,699 (1950). N. F. RAMSEY,Phys. Rev., 86,243 (1952). H. MAR~ENAU and G. M. MURPHY, The Mathematics of Physics and Chemistry, Van Nostrand, New York (1943). J. H. VAN VLECK,Electric and Magnetic Susceptibilities, Oxford University Press (1932). J. H. VAN VLECK and A. FRANK, Proc. Nat. Acad. Sci., 15, 539 (1929).
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NMR
SPECTROSCOPY
23. G. C. WICK, Z. Phys., 85, 25 (1933). 24. H. EYRINO, J. WALTER and G. KIMBALL, Quatzfum Chemistry, Wiley, New York (1944). p. 111.
25. K. ITO, 1. Amer. Chem. Sot., 80, 3502 (1958). 26. K. Iro, Private communication (1959). 27. G. C. WICK, Phys. Rev., 73, 51 (1948). 28. H. BROOKS, Phys. Rev., 59,925 (1941). 29. N. F. RAMSEY,Molecular Beams, Clarendon Press, Oxford (1956). 30. H. F. HAMXA, Mol. Phys., 1,203 (1958). 31. H. M. MCCONNELL, J. Chem. Phys., 27,226 (1957). 32. H. M. MCCONNELL, Ann. Rev. Phys. Chem., 8, 105 (1957). . 33. E. ISHIGIJRO and S. KOIDA, Phys. Rev., 94,350 (1954). 34. J. F. HO~NIG and J. 0. HIRXHFELDER, J. Chem. Phys., 23,474 (1955). 35. B. R. MCGARVEY, J. Chem. Phys., 27,68 (1957). 36. T. P. DAS and R. BERSOHN,Phys. Rev., 104,849 (1956). 37. M. J. STEPHEN,Proc. Roy. Sot., A243, 264 (1957). 38. T. P. Dm and T. GHOSE, J. Chem. Phys., 31,42 (1959). 39. T. P. DAS and R. BERSOHN,Phys. Rev., 115,897(1959). 40. M. F~XMAN,J. Chem. Phys., 35,679 (1961). 41. J. TILLUXIand J. Guv, J. Chem. Phys., 24, 1117 (1956). 42. R. ~HN, Ann. Rev. Phys. Chem., 11,369(1960). 43. J. A. POProc. Roy. Sot., A239, 541 (1957). 44. J. A. POPI& Proc. Roy. Sot., A239,550 (1957). 45. L. D. LANDAU and E. M.LUXHITZ, Non-RelativisticQuantum Mechanics, Peqamon
Press, Oxford (1962). 46. P. A. M. DIRAC, Quanfum Mechanics, Oxford University Press (1935). 47. L. PAIJLING,J. Chem. Phys., 4, 673 (1936). 48. J. A. POPLE, J. Chem. Phys., 24, 1111 (1956). 49. J. S. WAUGH and R. W. FESSENDEN, J. Amer. Chem. Sot., 79, 846 (1957); (see J. Amer. Chem. Sot., 80, 6697 (1958) for a correction). 50. C. E. JOHNSONand F. A. BOVEY, J. Chem. Phys., 29, 1012 (1958). 51. J. A. POPLEZ, Mol. Phys.. 1, 175 (1958). 52. R. MCWEENEY, Mol. Phys., 1,311 (1958). 53. C. A. COULSONand H. C. LONG~JET-HICXXN~, Proc. Roy. Sot., A191,39 (1947). 54.T. W. MARSHALL aad J. A. POPLE, Mol. Phys., 1, 199 (1958). 55. A. D. BUCKINGU, Con. J. Chem., 38,300 (1960). 56. L. ONSAGER,J. Amer. Chem. Sot., 58, 1486 (1936). 57. P. Draru. and R. FREEMAN,Mol. Phys., 4, 39 (1961). 58. A. D. BWKINGHW J. Chem. Phys., 30. 1580 (1959). 59. N. F. RAMSEY,Phys. Rev., 91, 303 (1953). 60. E. Fe~hsl, Z. Phys., 60, 320 (1930). 61. A. D. M&A-, J. Chem. Phys., 32, 1263 (1960). 62. S. ALEXANDER,J. Chem. Phys., 34, 106 (1961). 63. M. KARPLus, Rev. Mod. Phys., 32,455 (1960). 64. C. A. (%JE4oN, Valence, Oxford University Press (1952). 65. H. M. MCCONNBLL, J. Chem. Phys., 24,460 (1956). 66. L. PNLING and E. B. W&SON, Xntroductionto Quantum Mechanics, McGraw-Hill. New York (1935). J. Chem. Phys., 30, 717 (1959). 67. H. S. G~TOWSKY and G. A. W-, 68. H. M. MCCONNELL, J. Chem. Phys., 30,126 (1959). 69. E. AIHARA, J. Chem. Phys., 26, 1347 (1957). 70. G. RUMER, Giittinger Nuchr., 377 (1932). 71. L. PA~~.~NG,J. Chem. Phys., 1,280 (1933). 72. M. KARpLus and D. H. AND-N, J. Chem. Phys., 30, 6 (1959). 73. M. J. STEPHEN,Proc. Roy. Sot., A243, 274 (1957). 74. J. A. POPLE, Mol. Phys., 1, 216 (1958).
THE
ORlGlN
75. N. F. Mom
OF CHEMICAL
SHIFTS
AND
SPIN-SPIN
COUPLING
119
and I. N. SNEDWN, Wave Mechanics and its Applications, Clarerrdon Press, Oxford (1948), p. 94. 76. L. C. SNYDERand R. G. PARR,J. Chem. Phys., 34,837 (1961). 77. L. PEnurcIs and C. H. SEDERHOLM, J. Chem. Phys., 35, 1174 (1961). 78. S. K. Grroarrand S. K. Sm, J. Chem. Phys., 36,737 (1962). 79. D. E. O'REJLLY, J. Chem. Phys., 36, 855 (1962). 80. D. E. O’REILJ+Y, J. Chem. Phys., 36,274 (1962). 81. H. F. -, Niiouo Cimenro, 11,395(1959). 82. W. C. DXKXNsoN, PhyS. Ret?., 81,717(1951). 83. A. A. --BY and R. E. GLICK,J. Chem. Phys., 26,1647 (1957). 84. A. A. --BY and R. E. GLICK,J. Chem. Phys., 26,1651 (1957). 85. W. G. SCHNEIDER, H. J. BERNSTEM and J. A. POPLE,J. Chem. Phys., 28,601 (1958). 86. M. J. STEPHEN,Mol. Phys., 1,223(1958). 87. A. D. BUCKINORAM, T. -atrdW.G.S cxNarom J. Gem. Phys., 3% 1227 (1960). 88. B. B. How-, B. -ER and M. T. -N, J. Chem. Phys., 36,485 (1962). 89. B. LENDER,J. Chem. Phys., 35,371 (1960). 90. C. J. F. B&-rTheory of Electric Potkrisation, Ekvicr, New York (1952). 91. F. Low Z. Phys., 63,245 (1930). 92. S. Goauo~ and B. P. DAILEY,J. Chem. Phys., 34, 1084 (1961). 93. D. F. EVANS,J. Chem. Sot., 877 (1960). 94. R. FRFPMAN,G. R. MURRAY and R. E. RICHARDS,Proc. Roy. Sot., A242, 455 (1957). 95. T. W. MARSHALLand J. A. POP& Mol. Phys., 3, 339 (1960). 96. H. S. Gurowsrcp, J. Chem. Phys., 31, 1683 (1959). 97.0. V. D. Traps, J. Amer. Gem. kc., 79,5585 (1957). 98. G. V. D. TIERS,J. Gem. Phys., 29,963 (1958). 99. N. F. RAMSEY, Phys. Rev., 87, 1075 (1952). 100. T. W. w MoI. Phys., 4, 61 (l%l). 101. G. G. HAL.Land A. HAWSSON, Proc. Roy. Sot., A268,328 (1962). 102. G. G. HALL, A. and L. M. JA~KMAN,Tetrahedron., 19,.suppl. 2,101. (1963). 103. F. LONDON, J. Phys. RaaYum, 8,397 (1937). 104. E. HU-, Z. Phys., 70,204 (1931); 76, 628 (1932). 105. A. SW Molecular Orbital Theory For Organic Chemists, John W&y, New York (1961). 106. H. M. MCCONNELL and D. B. C-, J. Chem. Phys., 29,107 (1958). 107. H. M. MCCONNELL and C. H. HOLM,J. Chem..Phys., 27.314 (1957). 108. D. R. EATON, A. D. JOSEY,W. D. Parryps and R. E. BENSON,Mol. Phys., 5,407 (1962). 109. R S. Mnpra~ and L. Parr, Discuss. Far+ Sot., 34,88 (1962). 110. A. W. Gvastx~uaaa, Phys. Rev., 91,476 (1953). 111. A. ABRAQAM,Principles of Nuclear Magnetism, Oxford (1961). p. 333. 112 W. A. A~~artaoN, NMR and EPR Spcctroswpy, Per8amoh Press (1960). p. 268. 113. R E. RICHARDSand J. W. Wrrrra, Dtkxss. Far&y Sot., 34,% (1962). 114. R. E. Rrand J. W. WHITE,Proc. Roy. Sot. A269,287 (1%2). 115. R. E. Rxand J. W. Writ=. Proc. Roy. Sot., A 269, 301 (1%2). 116. H. M. MCCONNELL,J. Chem. Phys., 24, 632 (1956). 117. s. I. w-, T. R. TUTIZEand E. DEEIOER, J. Pbys. Chem., 61.28 (1957). 118. A. D. MCLXHLAN, Mol. Phys., 1, 233 (1958). 119. D. R. EATON and W. D. PHIUPS, A&mces in Magnetic Resonance, 1, 103 (1965); edited by J. S. WAUOH,Academic Press, New York. 120. J. G. Pow and J. H. STRANOE,Discuss. Faraday Sot., 34,30 (1962). 121. M. Band D. M. GRANT,Advances in Magnetic Resonance, 1,149 (1965); edited by J, S. WAUGH, Academic Press, New York.
CHAPTER
4
THE CALCULATION OF SHIELDING CONSTANTS OF NUCLEI IN MOLECULES THE shielding constant of a nucleus in a molecule depends upon the form of the electronic distribution and can be calculated provided that the form of the electronic ground state wavefunction for the molecule is known. In practice a satisfactory calculation, either by a perturbational or variational method, is possible only for the hydrogen nucleus in simple molecules. For hydrogen nuclei in more complex systems, and for other nuclei even in simple molecules attempts have been made to express the shielding constant for a particular nucleus as a sum of contributions from the different atoms and bonds in the molecule. In this way the variations in the values of the shielding constants in related series of molecules can be predicted, and related to other molecular properties, such as the dipole moment, the electronegativity of a substituent group, and the magnetic anisotropy of the bonds. In this chapter the success of calculations of shielding constants of nuclei in molecules will be reviewed; some calculations of shielding constants in atoms have been reported but will not be discussed here”, 2. 3). 4.1 THE HYDROGENMOLECULE Almost all methods of calculating shielding constants (a) have been applied to the hydrogen molecule. In many cases the calculation of an2 has been used as a test of the validity of both the calculational procedure and the form of the wavefunctions before proceeding to calculations of c for more complex molecules. The hydrogen molecular system provides the best case not only because of its electronic simplicity, but also because a semi-empirical value of a,, can be obtained and compared with the calculated values. This arises because, as Ramsey has shown, for linear molecules the paramagnetic term uDsramay be obtained from the spin-rotational magnetic interaction constant (see Section 3.53). For a linear diatomic molecule equation (3.46) becomes e2
O=G\O
/
(4.1)
where R is the internuclear distance and ,u’ is the reduced mass of the molecule. The quantity @H,/MJ has been obtained experimentally from molecular beam measurementsC4) and is 13.66 f O-20 gauss, and by substituting for the other molecular constants of the hydrogen molecule the second term in equa120
THE
CALCULATION
OF SHIELDING
CONSTANTS
IN
MOLECULES
121
tion (4.1) gives the valuet5) of e = - 056 + 0.01 x 10-5. This value of OF is for a rigid molecule and, as a further refinement, NeweW) has found the value of e averaged over the zero-point vibrations of the molecule to be - 055 x 10-5. The first term in equation (4.1), the diamagnetic term e, can be calculated from a knowledge of the ground state wavefunction of the electrons. Newell used a wavefunction of Nordsieck”) and obtained e = 3.21 x 10-5, hence the total value of on, is %
= (3.21 - O-55) x 1O-5 = 2.66 x 10-5.
This value of uH1is the best available and can be used to obtain semi-empirical values of shielding constants for hydrogen nuclei in other molecules if their chemical shifts relative to molecular hydrogen are known. In most calculations of shielding constants, the result is for the rigid molecule at the equilibrium internuclear separation, and hence should be compared with the analogous value for hydrogen of 2.68 x 10-5.(5**) The main difficulty in calculating on, from the equation developed by Ramsey (see Section 3.5) lies in the complexity of the paramagnetic term c. To evaluate a”? exactly requires the evaluation of a summation over all the excited states of the molecule, including continuum states. This is a virtually impossible task and hence the term G can be obtained only approximately. One method of calculating e introduces an average energy A E into equation (3.29) to give the equation (3.31) which involves only ground state wavefunctions. However, this simply transfers the difSculty to that of estimating A E since the only way of evaluating A E is by carrying out the summation in equation (3.29). Snyder and Parr(g) have emphasised the dangers encountered in the estimation of A E, particularly those arising from the neglect of the continuum states. The relative magnitudes of the two terms e and c depend on the origin of the coordinate system in which the vector potential A is defined, that is on the gauge of A. When the origin for A is taken at the nucleus, the semi-empirical calculations of Ramsey show that c is - 0.56 x 10-5; however, it is possible to choose the gauge of A so as to minimise the value of G. The total value of oHz is gauge invariant only when the wavefunction is expanded in terms of a complete set, and this condition is not fulfilled when oH1is calculated by secondorder perturbation theory. Hameka(‘O) has suggested that this difficulty can be overcome by using gauge invariant atomic orbitals (G.I.A.O.). If A,, is the value of the vector potential (see Section 3.5) at the nuclear position R, of the atomic orbital QI,,,then the G.I.A.O. t is given by
and the molecular orbital y,,, is formed from the linear combination atomic orbitals xn; that is
of the’
122
HIGH
RESOLUTION
NMR
SPECTROSCOPY
The calculation of uHs with G.I.A.O. functions based on the functions of Coulson(“), Wang(r2) and RoserP3) are(‘O) UHa
x 105
I
Function ~~
Although Hameka’s G.I.A.O. method still requires the use of an estimated average energy, it reduces the effect of an inaccurate estimate of this quantity by reducing the value of G. Ito (r4) has calculated cnl by means of equation (3.36), which requires neither an average energy nor a summation over excited states. Equation (3.36) with the CoulsorPr) molecular-orbital function for v. gives e = 3.22 x 10-5, c = -055 x lO-5, that is crH1= 2.67 x lO-S which is in almost exact agreement with the value found by Ramseyc5). Unfortnnately, the application of equation (3.36) is limited to molecules which are either highly symmetrical or where the constituent atoms have electron clouds which overlap either to a large extent, or very little(r5). Variational calculations of uHzhave the advantage over the calculations using the perturbation method in that a summation over the excited states of the molecule is not required. The variation method needs only a function which can be adjusted to give the minimum energy value, and the majority of calculations of shielding coefficients, both for the hydrogen molecule and for other molecules, employ a variational method. The best results for the hydrogen molecule have been obtained with a variational function similar to that proposed by Tillieu and Guy(16) for calculations of diamagnetic susceptibilities. In its simplest form the function is Y =Yo(l +g*H) (4.2) TABLE4.1 VARLUTONALCALCIJLATION(~‘)OF 0x2 WITH DIFFERENT FOR= OF ‘po @.”
‘PO
V.B. V.B. V.B. V.B. V.B. M.O. M.O. M.O. M.O.
Heitler-London Waw Weinbaum Rosen Hiichfelder-Linnet Coulson (unshielded) Coulson (shielded) Wallis (configuration interaction) Wallis (limited ionic)
059
u x 105
3.03 2.73 2.77 2.82 2.85 2.47 2.85
2.45 3.40 346 346 3.51 3.01 3.54
- 0.75 - 0.80 -090 - 0.84 - 0.57 - 0.82
244 2.46 247 2.45 2.53 2.27 2.55
241
3.49 3.55
- 0.78 - 0.98
2.53 2.33
268
-
where g is a variational parameter whose value is determined by minimising the totalenergy of the system. For the hydrogen molecule there aremany different forms of y. which have been designed to minimise the dissociation energy of
THE
CALCULATION
OF SHIELDING
CONSTANTS
123
IN MOLECULES
the molecule; Das and Bersohn (“‘have used a number of valence bond andmolecular orbital functions with the variational function given by equation (4.2) to calculate oHI. Their results, shown-in Table 4.1, indicate that the best M.O. and V.B. functions give almost identical results (compare the results from the Hirschfelder-Linnet function with those from the Coulson shielded function). The Hirschfelder-Linnet function is(’ a) 5
= a(l) 6(2)[1 + a Z2 G*(l) %(2) + YJl)Y*(2))
+ mw
+ B Z2G(l)
%ml
Cl+ ~~2{~*(2hu) + Y,4(2)Ya(l)) + B Z2 r*(2)%(1)]
+ Y [c(l) a(2) + b(l) W2)l.
(4.3)
a(1) and b(1) are 1s atomic functions with effective nuclear charge Z centred on nuclei A and B. The Rosen (13) function is derived from equation (4.3) by setting OL= y = 0; 1y = /? = 0 gives the Weinbaum(lg) function; putting u = fi = y = 0 gives the Wang functiorP2), and if Z = 1 also, the result is the Heitler-London function(20). The molecular orbital functions may similarly be derived from the Wallis function(21), fp - = u*(l) P(2) + P(1) aA(2) + P(l) uB(2) + uB(l) P(2) N + A[&(l) aB(2) + P(1) P(2) + aB(1) ti(2) + P(1) P(2)]
(4.4)
where tP and bB are 1s atomic functions with effective nuclear charges 2, and 2, centred on nuclei A and B. With I = 1 the function is the Wallio open shell function, that is with configuration interaction; if rZis chosen so as to minimise the binding energy, the function is known as the Wallis limited ionic function. Coulson’P) shielded molecular orbital function results by putting 1~ 1 and z, = Z,, whereas Coulson’s unshielded function has 2, = 2, = 1 P 1. The variational function in equation (4.2) can be made more flexible by adding a term proportional to cc, the nuclear moment w = ~o(l + g.H
+f.~).
(4.5)
Applying equation (4.5) to a calculation of cHI, Stephen(22) (see Section 3.6) obtained slightly higher results than those of Das and BersohrP7), although only one result of Stephen, that obtained with Weinbaum’s function, is strictly comparable (see Table 4.2). It is not certain that the added complexity of the calculation is just&d by the improvement in the results. TABLE 4.2 Vam.cnoN& Function Valence bond Molecular Weinbaum
orbital
CALcu~nON
I
2 1-O l-193 1-o l-193 l-193
OF an2 BY &-mmdz2’
I
ux105 2-41 2.69 2.39 264 291
124
HIGH
RESOLUTION
NMR
SPECTROSCOPY
Zis again the effective nuclear charge, and the value of 1.193 is the value obtained by minimising the binding energy of the molecule. Both the valence bond and the molecular orbital function give a value of a,,which is in good agreement with that of Ramsey (2.68 x 1O-5)(s). It is interesting to note the success of Stephen’s variational procedure when used with a simple molecular orbital function. The function is one suggested by FixmaiP3’ and applied in the calculation of a for the hydrogen molecule and some simple hydrides. The function has the form
Ye) = wYJ;w + (1 - YMiw11’2 where am is an atomic orbital centred on the adjacent atom and v&k) is a hydrogen atomic orbital. For the hydrogen molecule itself the function is w(1) = 2-1’2 [w:(l) +
wm
(4.6)
where (4.7) and 2 is the effective nuclear charge, and r,, is the distance of the electron (1) from the 4 nucleus. 2 is a variational parameter and is determined to be 1.13 by minimising the total energy; with this value of 2 the shielding constant for the hydrogen molecnle is 2.74 x 1O-5. The good agreement shown between the shielding constant obtained with this simple [L.C.(A.0.)2]1 I2 function and that found by Ramsey illustrates that the form of the variational function is a more important factor in calculating o than the form of the ground state wavefunction. This is amply illustrated by comparing Fixman’s calculated value of o = 2.74 x lO-5 with that obtained by Ishiguro and Koida, o = 2.756 x 10-5*(24) using the 13-term James-Coolidge wavefunction(25). 4.2 SIMPLE HYDRIDES 4.2. I Halogen Hydrides The series of diatomic molecules HX, where X is a halogen, is the next step in complexity from molecular hydrogen. One might expect that as the electronegativity of the X atom increases, the shielding of the hydrogen nucleus would decrease owing to the progressive decrease in electron density associated with it. Thus it would be expected that the hydrogen nucleus in the molecule HF would be less shielded than that in HCI, which in turn would be less shielded than HBr etc. In fact, this is‘what is found experimentally(26), the hydrogen resonance in H F being at lower applied magnetic fields than in HI. Semi-empirical values of the shielding constants of the hydrogen nucleus in the molecule HX can be obtained by adding the chemical shift measured from molecular hydrogen to Ramsey’s value of aHz. The chemical shifts of the molecules HX relative to
THE
CALCULATlON
OF SHIELDING
CONSTANTS
1N MOLECULES
125
molecular hydrogen have not been measured directly, but the shifts of the gaseous samples relative to gaseous methane have been measured and are(26) 6
hi.
5
- 41, H CH4
X
F
Cl
Br
I
6 x 105
- 0.25
OGt
044
1.33
These values may be regarded as those for the isolated molecules, but shifts measured in the liquid phase are considerably influenced by intermolecular interactions, particularly in HF where a liquid-gas chemical shift difference of 6.65 ppm has been observed(26). The absolute shielding values may now be obtained by comparison with the value for methane of 3.04 x 10m5,to give X
F
Cl
Br
I
unx x 105
2.79
3.08
348
4.37
The three reported calculations of shielding constants for the halogen hydrides have all assumed a close relationship betweenon x and the charge distribution in the molecule. The calculations of Hamekac2’) and Fixmar~(~~)have been the most successful and both include the dipole moment of the molecule as an integral part of their wavefunctions. Hamekat2’) used the perturbation method of Ramsey (see Section 3.5), hence uHX is split into the two opposing terms.0~ and a=, whose relative magnitude is determined by the choice in origin of the vector potential A. With the gauge invariant atomic orbit& discussed in the calculation of oH1, Pm and G are invariant under a gauge transformation and the origin of the coordinate system may be chosen differently in each calculation. Thus the hydrogen nucleus is used as the origin for those electrons centred on the hydrogen nucleus and also for the bonding electrons, whereas the halogen nucleus serves as the origin for the electrons centred on the halogen nucleus. Hamekac2’) assumes a close relationship between a, x and the part of the electronic charge centred on the hydrogen nucleus, and also on the overlap charge between this nucleus and the halogen nucleus. The wavefunction is written as a product of one-electron functions, each of which is either an atomic orbital or a linear combination of atomic orbitals. The wavefunction also contains a variational parameter A which is adjusted to fit the experimental value of the electric dipole moment of the molecule. If the halogen atomic orbitals are sp3 tetrahedral hybrids, the ground state wavefunction for the HF molecule is VIZ% 1+(1)1+(2) t~F(4) I~F@) t2~(6) bFt7) ~JF@) ~F(g)~F@) where and
& = (1 + 2&d, r4F
=
aF(2r,)
+
+ Af;)-1’2 (s” + & f4F) bF(2PrF).
is the charge transfer parameter, a variational parameter whose magnitude is determined from the dipole moment pm. AF is the overlap integral between
jlF
126
HIGH
RESOLUTION
NMR
SPECTROSCOPY
is a sp3 hybridised atomic orbital, and aF and bF are hybridiand t4F, flF sation constants, which in this case are + and +1/3 respectively. The calculation of a= with equation (3.31) requires a value of A E and in this case only d E for the first excited state was used. This approximation may be considerably in error, but is not so important since the magnitude of a= is small compared with ad&. Table 4.3 shows the magnitudes of O$$, a&? and uHx calculated by Hamekat2’). 8,
TABLE 4.3 CALCULATED VALUES OF SHIELDINGCONSTNTIS FOR THE HALDGEN HYDRDJS””
An early application variation function(28),
of variation theory to the calculation of ai.,, used a Y=“Yo+B~zwo
where OLand p are variational parameters and S?‘: is the Hamiltonian,
xz=
~Cm:,. 2Mc k
In contrast with Hameka’s treatment, the halogen atomic orbitals were considered to be unhybridised with an effective 2 = 1.19. The calculation was performed for a purely covalent molecule and a purely ionic one, and the shielding constant determined from the ionic character of the bond by assuming a linear relationship between the two (2g). The results (shown in Table 4.4) are in reasonable agreement with experimental values except for the HF molecule. TABLE~.~ VARIATIONALCALCULATIONOF SHIELDINGCONSTANTSIN HX MOLECULFS(~~) X F Cl Br I
Covalent
Ionic
Percentage Ionic character
4.40 4.82 4.94 4.86
0.39 0.39 O-81 0.76
85 40 31 21
ax
105
o-99 3-05 3.66 4.00
The large discrepancy here may be caused by an inaccurate estimate of ionic character, but may also result from the choice of an inadequate variational function(30).
THE
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IN MOLECULES
127
Better results were obtained using the variational function of Stephen. Fixman(23) has used Stephen’s approach with a wavefunction similar to that employed in the calculation of aHI. YHX(M = [r V:(k) + (1 - r)
Ysw’2
and the total wavefunction is a product of the k functions ; y is determined from the electric dipole moment of the molecule. The calculation has been performed for hybridised and unhybridised halogen atomic orbitals and also with effective nuclear charges on the hydrogen nucleus of 1-Oand 1.2. Further, the calculations were made with two choices for the dipole moments of HF and HI. For H F the value ‘of 2GlD (Debyes) was used in order to compare the result with that of Hameka, as well as a value of 1.74 D(‘r). For HI the values used were 0*38D(32) and - 0*38D(29). The results are shown in Table 4.5. TABLE4.5 VARIATIONAL C,+~XJLATION
OF C7m BY hCMd23’
Hybridised halogen
Unhybridiscd
The results in Table 4.5 show Srstly (column 3) that the method compares well with that of Hameka, and secondly that there is little support for a hybridised halogen atom (column 4) except in the case of iodine. An estimate of the percentage s character of the H-X bond can be made by comparing the results of columns 4 and 6 with the experimental results. Thus the following values are obtained. Percentage s character 6 (/.J = 1.74) HF HCl
5
HBr
8
HI
25 (ru = 0.38)
4.2.2. Group VI HyaYides The experimental values of the hydrogen nuclear shielding constants in the molecules H2X, where X is oxygen, sulphur or selenium again follow the expected correlation with the electronegativity of X, thus the hydrogen nucleus in water is less shielded than that in hydrogen selenide. The experimental values are obtained from the chemical shifts of gaseous samples measured from a methane gas reference(26), except in the case of H2Se where only a measurement
128
HIGH
RESOLUTlON
NMR
SPECTROSCOPY
the liquid sample is avaiiable (33’. The association shifts (see Section 9.4) in these compounds vary from 4.58 ppm for water, 150 ppm for H,S to an estimated value of 1.0 ppm for H$e. Although the series H#,H$, H$e shows the expected trend with electronegativity it is noticeable that the differences in dHzx are much smaller than in the halogen hydride series. Figure 4.1 shows a plot of CHx and bH2x against the eleetronegativity difference of H-X. The much greater change in flHx with elecon
t 0.4
1.2
0.8
Electronegativity
1.6
difference
,3
(fi -Xl
FIG. 4.1 The chemical shift of the hydrogen nucleus in H2X and HX molecules versus the ektronegativity @atiag(34’) difference between H and X
tronegativity than in oHIX is most readily interpreted in terms of induced local currents and a discussion of this will be deferred unti1 Section 4.3. TABLE4.6 EXPERIMENTALSHIELDXNGCONSTANTS FOR GROUP VI HYXXUD~ Hz0
u x 10’ ]
2.98
H,Se (liquid)
H2S
1
3.03
/
3.38
The only complete calculations for uHzx are those of FixmarP) using the simple molecular orbital function discussed earlier. In this case yx is a hybrid orbital of the form yx = (1 + AZ)-I’Z(s - J?&) where pz is centred on X and points away from the hydrogen nucleus. The calculation is divided into the contributions to uulx of electrons centred on
THE
CALCULATION
OF SHIELDING
CONSTANTS
IN
MOLECULES
129
H and on X, and of non-local&d electrons. The latter contribution arises from the n-electrons in the non-bonding p-orbital on X perpendicular to the plane of the molecule, and is designated a,. The results of the calculation are shown in Table 4.7 and are in reasonable agreement with experimental values, except in the case of H&z. Although part of the discrepancy can be attributed to the effect of hydrogen-bonding in the liquid, most of it is real. TABLE4.7 CALCULATED VALUESOF u X 0 S Se
FOR
MOLBCUUSH2 X
t7H x 105
ux x 10s
a* x 105
u x 105
1.99 2-32 2-36
O-78 O-52 o-45
0.02 O-08 O-08
2.79 2-92 289
A calculation by Das and Ghose(30) of the shielding constant of hydrogen nuclei in water gave results much lower than the experimental value (1.35 x 1O-s). In contrast to the results of Fixman, they find that a, is 029 x 1O-s, but Fixman’s rather low value for u, is supported by the evidence that the shift between ethylene (with “looser” n-electrons) and benzene is accountable by assuming that it arises from the small (or zero) value of a, in ethylene, and a large value in benzene. It is interesting to note that McConnell’s calculation for shielding by electrons in groups Jar removed from a nucleus when applied to these n-electrons gives un = 0.04 x 1O-5-in good agreement with the value found by Fixman even though equation (3.53) is of doubtful validity for electrons only one bond removed from the nucleus. 4.2.3 Simple Hydrocarbons In both the hydrides of group VI and group VII the shielding constants could be correlated with the electronegativity of the X atom, and one would expect the same to be true of simple hydrocarbons. The hydrogen atoms in the molecules CHI , C2Hd and CzHz show increasing acidic properties suggesting that the electron-withdrawing power of the carbon atom increases from methane to acetylene. The hydrogen shielding constants, however, do not decrease from methane to acetylene; in fact the ‘H resonance signal of acetylene lies between the signals for methane and ethylene. The experimental values of u for gaseous samples are(26) %I, = 3.04 x 10-S;
UczH, = 252 x 10-S;
uclHs = 290 x 10-S.
Both Stephen(22) and Fixman(23) have calculated the shielding constants of the hydrogen nuclei in these molecules by the same variation method. In both calculations, the shielding constant for a hydrogen nucleus in an “isolated” C-H bond is calculated, and then the contribution from the rest of the mole cule estimated. Stephen used an L.C.A.O. molecular orbital function
HIGH RESOLUTION
130
NMR SPECTROSCOPY
where yc is a carbon hybrid-orbital, that is, sp3, sp2 or sp for the three molecules CH,, , C2H4 and C2H2. il is a parameter which allows for the difference in electronegativity between C and H atoms. The effective nuclear charge on the hydrogen nucleus is taken as 1.0. The contribution to u from the electrons not centred on the C-H bond was calculated from equation (3.53) with diamagnetic susceptibilities calculated by Tillieu (3s). The results for the three molecules are(72) @CH,
=
2.65 x lo-“;
uczu, = 2.61 x lo-‘;
uc’CzHz = 2.74 x lo-”
which predict the C2H4 and C2H2 resonances in the correct order, but not that of the CH,. The main error in the calculation lies probably in the estimate of the contribution from the electrons not centred on the C-H bond, as equation (3.53) is valid only for electrons at large distances from the nucleus. Fixman(23) calculates uC+ withy as 05 (equivalent to 1= 1) but with 2 = 1.2. It is assumed that the contributions from the rest of the molecule are negligible in methane and ethylene, but that in acetylene the contribution from the arelectrons is equivalent to that from a pair of nonbonded px and p,. orbitals and is equal to 0.52 x 1O-s. The results are = 3.00 x 10-s; uc2uz = 3*53x 10-5. ocn. = 2.97 x 10-s; a,,, In comparing these with the experimental values, Fixman(23) adjusts the value of y to make the agreement exact. The values of y required are
Y
CH,
C2I-b
C2H2
0.473
0.613
0.662
which are in the order expected from relative acidities. 4.3 THE CALCULATION OF SHIELDING CONSTANTS BY THE INDUCED CURRENT MODEL The calculatiotis of shielding constants described so far have treated the molecular system as a whole, but for more complex systems this type of calculation
becomes virtually impossible, and attempts have been made to divide the shielding constant into contributions from electrons localised on atoms and in chemical bonds. One method of doing this is the induced current model for calculating u, in which the shielding constant cA of a nucleus A in a molecule is regarded as arising from induced electronic currents in the atoms and bonds comprising the molecule. The shielding constant 0, is written as
The term flu arises from induced diamagnetic currents on the atom A, and its magnitude depends upon the electron density around nucleus A, and hence upon the electronegativity of groups attached to A. UE is the contribution to CA of induced paramagnetic currents on A, and arises as a consequence of the mixing of the ground and excited electronic states by the applied magnetic
THE
CALCULATION
OF SHIELDING
CONSTANTS
IN MOLECULES
131
field. Pople(36) has shown that an excited state contributes to the local paramagnetic current on an atom only if it corresponds to the transfer of an electron between p or d-orbitals, hence G is zero when the electrons localised on A are in pure s-states. a*,# is the contribution to uA arising from local induced currents on the atoms other than A in the molecule, and may be positive or negative depending on whether the induced currents on the atom B are diamagnetic or paramagnetic in character. The magnitude of o,,B depends solely on the nature of B and if the atom, or bond, B is at a distance RAB from A, which is large compared with the radius of A, it is given by aAB =
3R: N Kl - 3 cos2U An 0
%xX+ (1 - 3 COS~6,)%,~+ (1 - 3 COs~ez)%*z] (4.9)
where No is Avogadro’s number; xXx, xrr and %=Iare the three principal axes of the magnetic susceptibility tensor x of the atom or bond B; 8,, 13,,0, are the angles between the principal axes and the AB internuclear vector. If&% 8 %,r * Xst, and the y or x axis can be chosen such that RAB lies in the zy (or zx) plane, then equation (4.9) becomes QAB =
3N01R~ (2A%1- A%i-
A%l3ws2ez)
where AxI = %tr - xrY and Ax2 = xzz - x;*. If the group B is axially symmetric, then xXx = ty * xzz, and equation (4.9) becomes the equation (3.53) of Section (3.5.4), GAB= 3N01RL AXB (1 - 3 cos’k)
in which
(4.11)
Axe = &- xi and 8, is the angle between the anisotropy axis (the z-axis) and the internuclear vector R,,. The equations (4.9) to (4.11) show that uABis zero unless the magnetic susceptibility tensor xe is anisotropic, and for this reason oAB is sometimes referred to as the neighbour anisotropy effect. The last term in equation (4.8) arises from induced currents involving electrons which are not localised on any one atom or bond in the molecule, and is particularly important in aromatic hydrocarbons (see Section 4.5). Equation (4.8) is a good approximation to the value of o, only when the electrons localised on the atom A do not have any appreciable overlap with those on atom B (see Section 3.5.4), and hence is not strictly valid for calculating the shielding constant of small molecules such as the simple hydrides discussed in Sections 4.2.1 to 4.2.3. However, applying equation (4.8) to these hydrides to estimate the relative magnitudes of the local contributions to a,, does help in interpreting the observed variations in the values of the shielding constants, even though the magnitude of a, can be obtained more accurately by the direct methods. When A is a hydrogen nucleus the magnitude of a= is small compared with eU, because the p character of the hydrogen bonding-orbital is negligibly small. The term e” in these hydrides is zero, hence the shielding 6*
132
HIGH
RESOLUTION
NMR
SPECTROSCOPY
uH is the sum of the terms &A and aAn. If uz % aAB for a related series of molecules, then a, should be closely related to the electronegativity of the atom B, but when eAa is comparable in magnitude to flti the variation of err in the series will be more complex. For the halogen hydrides HX it has already been noted that a, decreases as the electronegativity of X increases, and it is tempting to conclude that u$$ is the dominant term in equation (4.8) for these molecules. That this is not the case is apparent when it is noted that the shielding constant of the hydrogen nucleus in the molecule HF is almost the same as that in water, and also that the shielding constants eH in the isoelectronic series CHI, NH3 , OH1, FH are practically identical”@.
constant
u x 10s
CH*
NH3
OH2
FH
3.04
3.05
2.98
2.78
The magnitudes of ci., for the other hydrogen halides are all greater than that for methane, whereas they would be expected to be smaller on the grounds of electronegativity. This anomaly can be explained by the magnitude and sign of o,.,&&. The diamagnetic susceptibility tensor x.x is axially symmetric about the direction of the H-X bond, hence the magnitude of u,x is given by equation (4; I 1). When the applied field lies along the H-X bond direction there is no hindrance to induced electron currents about this direction and xl1 is large and negative. When the field lies perpendicular to the H-X bond direction the electrons on X cannot circulate freely and xL is small. Thus Ax in equation (4.11) is negative, and since the angle ox is zero, the value of uHx is positive (the same sign as 03 and its magnitude increases as the size of the atom X increases. In the hydrogen halides both’ c&&and uxx increase as the electronegativity of X decreases, and this accounts for the high values of a, in this series. In the isoelectronic series CH4, NH3 , OHz , FH only FH is a linear moIecule with a large value of A xx and a maximum value of (l-3 cos2 0). The other molecules in the series are non-linear and give smaller values of GA,. Pople has estimated the magnitudes of bAB in these molecules by the point dipole approximation to be c3’) CH* NH3 OH, FH CAB x lo5
0’~
0.03
0.28
030
GABincreases along the series whereas fl= should decrease, thus the total change in u, is small. Other linear molecules will have large positive values of GAB, for example, acetylene. When the applied field lies along the C-C bond direction there is less hindrance to electron circulations on the carbon atoms than when the field is at right angles to this direction, hence d x is negative. The two other molecules in the series CH*, CzH4, C2H2 are non-linear and thus have smaller values ofeA,,. Pople(37) has calculated that in acetylene aABis of the order of 1.0 x 10q5 and is large enough to make the shielding constant Us greater than uH for ethylene.
THE
CALCULATION
OF SHIELDING
CONSTANTS
IN MOLECULES
133
4.4 SHIEI.JXNGCONSTANTSANDTHEMAGNE~CANISOTR~PYOFCHEM~CALBONDS The divisionof the shielding constant into local contributions has more significance for larger molecules than for the simple hydrides. The equations (4.8) to (4.1 I) are particularly useful when in the same molecule two or more hydrogen nuclei have the same immediate environment, and hence have identical but differ in their relative orientation to some distant values of a& and s, group B in the molecule. The chemical shift between these nuclei can be calculated if the principal components of the magnetic susceptibility of group B, and the geometry of the molecule, are known, or conversely, the anisotropy of the magnetic susceptibility may be calculated provided that the chemical shift can be measured. Some examples of such calculations are given in Sections 4.4.1 to 4.4.3.
4.4.1 The Internal Chemical Shift in Amides The two hydrogen nuclei labelled 1 and 2 in the amide
are magnetically non-equivalent and at low temperatures give rise to two chemically shifted resonance peaks. At higher temperatures hindered rotation about the C-N bond with a frequency greater than the chemical shift difference causes the two peaks to coalesce (see section 9.5.2). ‘Now, the electronic environment immediately surrounding the hydrogen nuclei 1 and 2 in the two N-H bonds must be ahnost identical, so that the chemical shift d1_z must arise because the internuclear vectors Ror.r(1Jand Rou,,, and the corresponding RRH vectors are different in both magnitude and direction. The difference in the shielding constants Acrlz for the two hydrogen nuclei is given by the difference between the two terms a,,B in equation (4.8), where B refers to the electrons localised in the C = 0 and C-R bonds. For formamide, R is H and the C-H bond is axially symmetric, thus grrcrr is given by equation (4.11); the C= 0 bond is not axially symmetric and aH,c_o is given by equation (4.10). Acll is given by(jg) 1 (24x1 - A& - AXI 3 COS28~~) , + ~XCH(1 - 3 co&?,,) ArJl2 = R3Wl R& 3% 4
-Ax1 (24x1 Ax2 -[
1
3 COS28~2) + Axcr.l(l - 3 cosV&
Rt2
R3HZ
11
(4.12) whereA%, and Ax2refer to the C=O bond. The values ofAx and Ax2 are not known, hence equation (4.12) cannot be used to calculate Aai2. Narasimhan and Rogers(3g) have used equation (4.12) to calculate values of Ax1 and Ax2 by equating Acl2 to the observed chemical shift difference. They assume that the three principal components of x for the C = 0 bond do not change on going
134
HIGH
RESOLUTION
NMR SPECTROSCOPY
from one amide to another, nor upon N-methyl substitution, and therefore they use equation (4.12), together with a similar equation for the shift between the two methyl groups in N, Ndimethyl formamide, and the average value of the susceptibility
in order to calculate xXx, xYYand xzz for the C=O bond. There are two main difficulties in such a calculation. First, it is not possible to decide on the sign of the chemical shift difference Ao,~. Secondly, the values of Rol and R,,2 depend upon the location of the induced dipole along the C = 0 bond. The dipole is probably best located at the charge centroid of the bond, which depends on such properties as ionic character and hybridisation of the bond, and these are not known for C = 0 bonds in amides. Narasimhan and Rogers calculated the principal susceptibilities of the C = 0 bond for the two cases of do, negative and positive, and in each case the calculations were also carried out with the induced dipole at various positions along the C = 0 bond, starting at O-81 A from carbon and then moving it towards the oxygen atom. The calculated magnetic susceptibilities were then applied to the calculation of do in acetamide and dimethylacetamide, and it was found that da was not affected by the position of the dipole provided that the same choice of dipole position was made throughout. No serious error occurs even by locating the dipole on the oxygen atom. The calculated values of Au for acetamide and dimethylacetamide are shown in Table4.8. The two values of d xC_Hare those calculated by Tiieu and GuyQ6) with two different wavefunctions. Tau4.8 CALCULATEDINTERNAL Sm-rs IN ACETAMIDE ANDD IMETHYLACETAMIDE WITH ILITW OBTAINED FROM Fo RMAMIDEAND DIMETHYLFO-~~~) MACBIETIC SUSCEFTD Au x 10’ Ax=_~ x lo6 Ao(Acetamide
+ 0.26 + 0.49
Dimethylacctamide
+@26 fo.49
ve)
- 1.63 - 1.78 1.43 -1.53
Au ohs. Au(+ ve) + 3.10 + 3.24 + 2.31 + 2.41
f 221
In comparing da calculated with da observed, it is important to note that the measured shifts are for pure liquids and do not represent the values for the isolated molecules. Hydrogen bond shifts would be expected in these molecules, particularly in formamide and acetamide which show evidence of strong hydrogen bonding in the solid state c40). The calculation also assumes that the whole of the observed chemical shift arises from the neighbour anisotropy effect, whereas part of the shift in these polar molecules will arise from the electric field effect discussed by Buckingham (see Section 3.9).
THE CALCULATION
OF SHIELDING
CONSTANTS
IN MOLECULES
135
4.4.2 The Calculation of Internal Chemical Shifts in Saturated Hy&ocarbons and the Magnetic Anisotropy of the C-C mtd C-H Bona3 Thesimplest saturated hydrocarbon molecule to show more than one resonance peak is propane, CHJCH2CHJ. The hydrogen resonance spectrum is that of an A2Bs system with a chemical shift between the two groups of hydrogen nuclei of 6 = 044ppm(Aa = 4.4 x 10-3. Now, for eachhydrogennucleus in propane the term (a”H;I+ G) in equation (48) depends on the charge density in the C-H bonds and can be calculated from a simple molecular orbital of the form Vc-Ii = N(% + 2 0 assuming all the C-H bonds to have the same degree of hybridisation. j! is a parameter which depends on the charge distribution in the C-H bond, and is assumed constant for each bond in propane as the molecule has a zero dipole moment. Thus uHu = (e + c) should be almost the same for each hydrogen nucleus in propane, and the observed chemical shift difference can be considered to arise from the difference in the terma,, for the methyl and methylene hydrogen nuclei(42). The group B refers to electrons locahsed in the C-C and C-H bonds, and since both these bonds are axially symmetric the two terms a,, c_c and a,, c_H can be calculated from equation (4.11). From the dimensions of the molecule, and taking into account the rapid re-orientation of the methyl groups, Narasimhan and Rogers(42) found that the differencf in the shielding constants of the methyl and methylene hydrogen nuclei is Aa I 0106A~=-=
- 0.124Af-H
(4.13)
in which Aa = cc,, - cc&. Substituting for Ax=-= and AxCeH values calculated by Tillieu(43) of Axcwc = 1.21 x 10m6,andAfmH = 0.24or 1.50 x 10m6, gives for Au, do = 0.98 x lo-’ (A xc-” = 0.24 x 10-6) = -053
x lo-’
(Ax=-~ = 150 x 10-6).
Comparing these values of do with the observed chemical shift of 4.4 x lo-’ suggests that the term Aa HHis not zero, and in fact provides the major contribution to Au. A value for AcrHHin propane can be estimated either from the electronegativity - chemical shift correlation suggested for CHsCHzX compounds(44*45), or from the electronic charge distribution in propane as calculated bySandorfy(73). Both methods give similar values forda,,; the 8rst method gives 3.17 x IO-’ and the second 3.38 x lo-‘. With AaHH = 3.38 x lo-’ and AaH, = 0.98 x IO-‘, the calculated difference in shielding constants, Au, = 4.36 x lo-‘, in excellent agreement with the observed shift. If it is assumed that the observed chemical shift between the methyl and methylene hydrogen nuclei in propane can be attributed entirely to the effect of the magnetic anisotropy of the C-C and C-H bonds, then the value of Axe -c calculated from equation (4.13) is (4.3 to 5.6) x 10m6cm3 mole-‘.
136
HIGH
RESOLUTION
NMR SPECTROSCOPY
This value of dxc-c
is much larger than that calculated by Tillieu(43), but it is supported by other evidence, such as the chemical shift between the axial and equatorial hydrogen nuclei in cyclohexane (74). At room temperature the hydrogen spectrum of cyclohexane shows a single peak, but on lowering the temperature the spectrum(7s) is that of an AB system with a chemical shift difference of 4.6 x lo-‘. The chemical shift is that between hydrogen nuclei in the axial and equatorial positions in the chair form of the molecule
< The charge distribution and degree of hybridisation should be the same for all the C-H bonds, and the observed shift can be attributed to the anisotropy of the C-C and C-H bonds. The bonds, C1-C6, Cl-C2 and the equatorial C--H bonds at carbon atoms C2 and Cs have the same relative orientation to the hydrogen nuclei H, and H,, and thus do not contribute to the difference in shielding constants do,,, . Neglecting the effects of electrons more than there bonds removed from H,, and H, means that the principal contributions to do,, come from the magnetic anisotropy of the bonds C&&, CS-C6, and the axial C-H bonds at Cz and Cs. With rc_c = 1.54 A, rc_ H = I.10 A, assuming ‘tetrahedral angles, and locating the induced dipoles at the midpoint of the C-C bonds, and 0.77 A from the carbon atom in the C-H bonds, gives for do,,, da,,
= 01073 x @lx=-c - Llxc-H).
Equating da, with the observed shift in cyclohexane of 4.6 x lo-’ means that must be 4.2 x lO-‘j cm3 mole-‘. A similar calculation for (Ax~-~ - Af-H) the isobutane molecule(74) gives 4.15 x 1O-6 cm3 mole-’ and both these values are in agreement with that calculated from the internal chemical shift in propane. Musher(46) has calculated Ax c -c from the chemical shift between axial and equatorial hydrogen nuclei in a number of derivatives of cyclohexanol and decalol. He found A f x to be 5.0 + 0.6 x lO-‘j cm3 mole-‘, in excellent agreement with the values found in cyclohexane, propane and isobutane. However, in these molecules containing polar groups it is possible that the chemical shift will be strongly influenced by the electric field effect(76* “). 4.4.3 Chemical Shifts in Alkyd-X Compoun& and the Magnetic Anisotropy of the C-X
Bond
The change in the shielding constants of the hydrogen nuclei in the molecules CH3X and CH3CHzX as X changes depends on changes in both aHH and $,, c x. is linearly dependent on the electronegativity of X, whereas A&,, cx A~HH depends upon the magnetic anisotropy of the C-X bond, and upon the geometry of the molecule. It is possible to obtain an estimate of the relative magni-
THE CALCULATION
OF SHIELDING
CONSTANTS
IN MOLECULES
137
tudes of eHHand a,, cx for both the methyl and methylene hydrogen nuclei h CH3X and CHjCHIX by following the change in the chemical shift with the electronegativity of X. Table 10.2 shows the hydrogen chemical shifts relative to methane gas (internal reference) of both CHs- and CHz-groups in a number of gaseous alkyl compounds(45), and in Figs. 10.2 and 10.4 the shifts are plotted against the electronegativities co’)of X. These figures show that a linear correlation of the chemical shift with electronegativity of X is found when X is H, C, N, 0 or F, but that when X is Cl, Br, I or S there is a deviation from the straight line. The magnitude of the deviations for the hydrogen nuclei in both methyl and methylene groups in the series CHIX and CH&HzX increases in the order Cl < Br < I, S, and are all negative. Both the magnitude and sign of do,, cx can be explained in terms of induced currents in the electrons localised in the C-X bond(45). Consider the molecule CH&H?X shown in Fig. 4.2, where X is a large atom. When the applied field, Ho, is along the
(b)
(0)
FIG. 4.2
Inducedmagneticfields
in CHaCHtX.
Spie#ckeand S&t&&(4’)
C-X bond direction there is free diamagnetic circulation of electrons about the bond, hence a large induced dipole opposing Ho. The induced magnetic fields at the methyl and methylene hydrogen nuclei are both in the same direr tion as H,,. When Ho is perpendicular to the C-X bond direction the electrons cannot circulate freely, and this can be represented by placing an induced dipole in the same direction as Ho, producing induced fields at the hydrogen nuclei which oppose Ho. Thus both for H,-, parallel and H,, perpendicular to the C-X bond direction there is a reduction in the shielding constant for both ocand p hydrogen nuclei. The magnitude of uH,cx is given by equation (4.11) and has been calculated by McConnelS3*)for the case X = halogen. He assumed Ax to be 20 per cent of the total susceptibility of the halogen negative ions, and locating the induced dipole on the X atom gives for au, =the values, + 01 x 10m6,f O-3x 10s6, f 0.3 x 10B6,f 0.4 x 1O-6for X = F, Cl, Br, I. Figure 10.2suggests values of eH,cx considerably greater than these; for example ou, cr is estimated from the graph to be - l-5 ppm(45).The error may arise as a result of locating the induced dipole on the halogen atom. An even more serious discrepancy between experimental and HRS
6s
138
HIGH
RESOLUTION
NMR
SPECTROSCOPY
calculated values of the shielding constant is apparent on examining the 13C chemical shifts in these compounds (45): the results are plotted against the electronegativity of X in Figs. 12. I5 and 12.16. Again the halogens show large deviations from the straight line graph, and this time the shielding constants of the cPC nucleus in CHJX and CH&H2X compounds are larger than expected; that is, cc,cx is positive, whilst for the /?J3C nucleus in CHJCHzX it is negative. This is consistent with the induced dipole model illustrated in Fig. 4.2, since the induced field at the a-carbon nucleus is in the opposite direction to that at either the p-carbon, or any of the hydrogen nuclei. If the deviations from the straight line graph (Fig. 12.15) are considered to arise only from magnetic anisotropic effects of the C-X bond then, in CHJ, the value of o,, ,-r would be 54 ppm. The /?-carbon nucleus in CH3CH2X and the hydrogen nuclei in CH3X are in almost the same position relative to the X atom, hence the magnitude of on, cx and oc, cx for these nuclei should be identical since they depend solely on the magnetic anisotropy of the C-X bond and the values of Bx and R in equation (4.11). In fact, for CH&H,I and CHjI the two values are respectively @-carbon of CH&H,I) UC,Cl = 18 ppm %, CI
=
1.5 ppm
(hydrogen in CHJ.)
suggesting that factors other than magnetic anisotropy are important in determining the chemical shifts in these molecules. The large unexplained contributions to the 13C shielding in compounds containing C-X bonds may arise from the effect on the shielding constant of permanent and induced electric dipole moments in the molecules. Buckingham has shown that the shielding constant of a nucleus in an X-Y bond is influenced by the field component E, along the bond and by EZ, the square of the total electric field at the nucleus, thus 0, = -AE,
- BE=.
(4.13a)
In addition to the electric field arising from permanent dipole moments in the molecule there is also a contribution to E2 from the Van der Waals interaction between atoms (see Sections 3.9 and 3.10). Time dependent dipole moments in an atom give rise to a non-zero value of the square of the electric field (E2) whose magnitude at a distance r is given approximately by
where a is the electron polarisability of the atom and I is the first ionisation potential. The values of A and B depend upon the nature of the atom and its environment. For hydrogen in hydrocarbons A is of the order of + 2 x lo-l2 and B is around + 1 x 10-18. For other nuclei A and B can be considerably greater, and whereas Bis always found to be positive, A may be of either sign. Values of A and B for i3C have not yet been determined, but values of 13C solvent shifts suggest that the electric field effect is an important factor in determining the magnitude of ‘“C shielding constants(92).
THE
CALCULATION
OF SHIELDING
CONSTANTS
IN
MOLECULES
139
4.4.4 The lgF Chemical Shtjlt Diflerence in Perfruorocyciohexane The chemical shift difference between axial and equatorial hydrogen nuclei in cyclohexane is 0.46 ppm and can be explained in terms of the diamagnetic anisotropy of the C-C and C-H bonds. But the chemical shift da, in perfluorocyclohexane between lgF nuclei is f 18.2 ppm and this cannot be attributed entirely to the diamagnetic anisotropy of C-F and C-C bonds; in fact the contribution to Au, from this source can hardly be much greater than 0.46 ppm. Emsley (g3)has shown that this large value of da,,, can be explained in terms of the values of E, , E2 and (E2) at the axial and equatorial lgF nuclei which arise from the polar C-F bonds and the polarisabilities of the electrons in the C-F and C-C bonds. The values of E,, the electric field component in the direction of the C-F bond, and E2, the square of the total field at the lgF nucleus arising from permanentbond dipole moments,were calculatedassuming that each C-F bond has a dipole moment of - 1.4 D located at the centre of the bond. Perfluorocyclohexane was assumed to exist entirely in the chair form and the bond lengths were assumed to be C-F
= 1.32 A
C-C
= 15s
A
and all angles were assumed tetrahedral. (E2) was assumed to arise from the electrons in the C-F and C-C bonds, and in calculating its value from equa4 tion (4.14) the following values of oLand I were used (**), UC-F
= 0.683 x lo-‘* cm3
ac+ = 058 x 1O-24 cm3
I c_F = 28.5 x lo-l2 ergs Ic_c = 18.03 x SO-l2 ergs The value of r was taken as the distance between the lgF nucleus and the midpoint of the bond concerned. The values of A E,, A E2 and d <E2>obtained are A& = E, (axial) - E, (equatorial) = -0.0484 x lo6 e.s.u.
AE2 = E2 (axial) - E2 (equatorial) = 0.0933 x 1012e.s.u.
.
A(E2) = (E”) (axial) - (E2) (equatorial) = O-3033x 1012e.s.u. Substituting in equation (4.13 a) gives
do;, = (-0.0484
x 106) A - (O-4266 x 1012) B.
(4.15)
Experiments on the pressure dependence of the chemical shift of 1gF in fluorine containing compounds suggest that A is negative and of the order of - 10 x 10-12, and that B is positive and of the order of (l&40) x 10-l*. If A and B are given the values found for CHFJ oftg4)- 10 x IO-l2 and + 15 x lo-l8 respectively, then equation (4.15) gives Au, = - 7.88 ppm which leaves a shift of - 10.3 ppm unaccounted for. If it is assumed that the whole of the ob6&*
140
HIGH
RESOLUTION
NMR
SPECTROSCOPY
served shift of - 18.2 ppm can be attributed to the diamagnetic anisotropy and the electric fields present then with a value of - 10 x lO-12 for A, equation (4.13a) gives for B the value of + 44.9 x lo-I*, which is not an unreasonable value. 4.5 CHEMICALSHIFTSOF HYDROGEN
Nucm
IN AROMATIC MOLECULES
The main feature of the shielding constant of a hydrogen nucleus in benzene is that it is considerably different from the shielding constant of ethylenic hydrogen nuclei, the ‘H resonance signal in benzene being 15 ppm to low field of that in ethylene. The contribution to the shielding constant oH from electron circulations localised on the hydrogen atom depends upon the state of hybridisation of the C-H bond and therefore should be the same in both cases. The effect on a, of the anisotropy of the magnetic susceptibility of the C-C and C-H bonds will be different in the two molecules, but this contribution to d Q~is small and can be largely eliminated by comparing benzene ring hydrogen nuclei with the ethylenic hydrogen nuclei in cyclohexadiene- 1 : 3 where the chemical shift difference is 1.48 ppm. This chemical shift difference has been interpreted as arising from the delocalised n-electrons of the benzene ring which, in the presence of a magnetic field along the hexagonal axis, are free to move in a closed loop (see Section 3.8). In benzene, the induced current gives rise to a magnetic field which opposes the applied field inside the current loop and enhances the applied field outside the loop. This is illustrated by the chemical shifts of the methylene groups in the 1&polymethylene benzenes I
I
I
I
In the lower members of this series, the CHI groups are held in positions close to the centre of the aromatic ring where the induced ring currents cause increased shielding; thus, for 1,4-polymethylene benzenes where n = 10 and 12, the central CH2 groups resonate 6.50 ppm and 6.30 ppm respectively to the high field side of benzene, whilst in 1,2_hexamethylene benzene, the CH2 groups farthest from the benzene ring resonate at 5.8 ppm to high field of benzene, a value characteristic of the methylene groups in saturated cyclic hydrocarbons. The c&HZ groups in the 1,4_polymethylene benzenes resonate in the same region as the CHI group in ethyl benzene which suggests that the induced ring currents deshield both groups to the same extent. The magnitude of the ring current shift in benzene can be calculated with the aid of a purely classical model (see Section 3.8) where the n-electrons are considered to move in two circular loops above and below the plane of the carbon atoms, with a radius equal to the C-C bond length (1.39 A) and separated by 1.28 A. Using this value for the current ring separation and equation (3.83), Johnson and Bovey(4*) have calculated the ring current chemical shift contribution for a hydrogen nucleus at any position relative to the benzene ring; their results are shown in Appendix B.
THE
CALCULATION
OF
SHIELDING
CONSTANTS
IN
MOLECULES
141
4.5. I Polycyclic Aromatic Hydrocarbons In the fist attempt to calculate the effect of ring currents in polycyclic aromatic hydrocarbons, it was -assumed that the current flowing in each hexagon is equal to that in benzene(4g). The effect ofthis current was then found by means of Pople’s point dipole calculation (see Section 3.8). Because the spectra of these compounds were obtained from neat liquid samples, the observed chemical shifts are not those of the isolated molecules and hence they will be considerably Muenced by intermolecular interactions. Some of the compounds were re-examined by Jonathan, Gordon and Dailey’so’as dilute solutions in Ccl4 or CS2 and the ring current shifts relative to benzene obtained. The same authors have calculated the ring current shifts in these compounds by lirst calculating the current intensity in each hexagon by means of the molecular orbital calculation of Pople (see Section 3.8) and then using this current intensity in conjunction with Johnson and Bovey’s tables (Appendix B) to obtain the ring current shift of a particular hydrogen nucleus. Table 4.9 gives the current intensities found from the molecular orbital calculation and shows that the assumption of equal currents in each hexagon is a poor one when the number of rings increases. The experimental and calculated ring current shifts are shown in Table 4.10; wherever possible the results are those of Jonathan et uZ.(~O)otherwise they are from the earlier work of Bernstein, Schneider and Pople(4Q). It can. be seen from Table 4.10 that the calculated shifts for the polycyclic aromatic molecules are in the cotiect order, but that most are too large by about O-5 ppm. It is interesting to note that agreement between the calculated and experimental shifts is best for the positions where it is known that steric factors are important; thus in the 4,5 positions in phenanthrene osl,, - o”&, is only - 0.06 ppm. In this molecule the steric interaction between the hydrogen atoms in positions 4 and 5 reduces the shift because of a strong Van der Waals interaction and also because the molecule is not planar. Reidt5’) has calculated that the shift arising from the Van der Waals interaction in phenanthrene is of the order of 0.5 ppm. Differences between theory and experiment are the largest for hydrogen nuclei in rings having two or more benzene rings attached to them, and also for nuclei which are not sterically hindered; thus 9,lO positions in anthracene have u_lC - u,,~ of + 0.79 ppm, and the hydrogen nuclei in coronene havea,,,, - Q,, = + 1.17 ppm. One method of improving the agreement between experiment and theory would be to alter the separation of the current loops, p, in benzene. Johnson and Bovey(4*) varied the value of p to obtain exact agreement of the observed and calculated shifts between the benzene hydrogen nuclei and the ethylenic hydrogen nuclei in cyclohexadiene- 1 : 3. Jonathan et aZ.(50)haveinvestigated an alternative procedure of adjusting p so as to obtain exact agreement between benzene and naphthalene. Figure 4.3 shows the ring current shift for benzene and for the LXand /I hydrogen nuclei in napht+alene as functions of the ring current separation ; it is seen that in order to obtain agreement for the LXhydrogen nucleus the value of p required is 2.09 A and with this value the cal-
142
HIGH
RESOLUTION
NMR
SPECTROSCOPY
TABLE~.~ CALCULATED RATIO OF Ruw CURRENT INTENSITIESCOMPARED WITH BSNZNE(~O’
-u
/CA &
current IllteLlsity z
A
1mo
A
l-093
A B
1.085 l-280
A
B
1.329 0.964
A B
1,133 0.975
A B
1.111 o-747
A B
o-979
A B
l-460 1.038
I,AU
ro A A
fB
0
O-247
'J
THE
CALCULATION
OF
SHIELDING
CONSTANTS
IN
143
MOLECULES
T~~m4.10 Rma CURRENTSmm IN Po~~cycr~c AROMA~CH~DMXAREONRELATMiTo BENZENE
a”(expG
Molecule
1
03 ti
10
B&ma
Ppm
O-54
o-19
O-85 0.42
1 2 9,lO
O-64 o-12 l-04
l-14 0.55 1.83
50
1 2 3
0.72 0.89 o-79
l-12 l-55 l-40
50
O-44 144 @16
50
0:3
O-99 1.38 0.61 O-56 O-99
i-29 O-34
l-34 091
0.82 O-14 O-33
1.03 O-50 O-84
1 2
2
a”@w
Ppm
50
1
63 0 3
3 2 r 6 7
9.10 4, 5 3, 6 2. 7 1. 8
50
50
I
423 /J
l-57
2.74
50
144
HIGH
RESOLUTION
NMR
SPECTROSCOPY
Table 4.10 (continued) Molecule
13 12 11
10
5
4
1
0 ci
6
pm
3 4 5 6 7 8 9 10 11 12 13 I 14
6 2 3 4 5 2
fJJ=(expt.)
HY~=w=
0.57
O-57 I.95 157 0.57 1.57
I
1.20
I
o-37
1 , 2 3 4 *
220 0.47
z,
oc1 (cak.)
mm o-93 l-03 @84 0.35 0.37 o-91 1.82 1.34 o-79 O-36 0.43 1.45 l-35 1.45 0.42 O-36 0.80 O-98 261 O-90 O-40 ON 0.95 o-99
Reference 49
49
49
culated shift for benzene is only 0.5 ppm. Similarly, agreement between c& and ccalc for the @hydrogen nucleus requires p = 2.78 A, and this would reduce u,~,, for benzene to almost zero. It is possible that the large values of a,,, - u~,,~~ mean that the observed shifts between the hydrogen nuclei in polycyclic aromatic compounds and benzene are not entirely caused by differences in ring current shifts. Thus to obtain an experimental value for the ring current shift in naphthalene would require a comparison with the shifts in the 1,2-dihydronaphthalene ovl”
a HZ
Jonathan et aZ.‘50)have shown that only about 0.05 ppm can be attributed to the different “neighbour-anisotropy” shifts in fused aromatic systems. An improved calculation of ring currents has been made by Hall(s2-54) using a self-consistent field (S.C.F.) molecular oFbita1 theory for aromatic systems, instead of the Htickel model adopted by Pople and McWeeney. The S.C.F. theory gives better results for alternant hydrocarbons, but still leaves a large
THE
CALCULATION
OF SHIELDING
CONSTANTS
IN
MOLECULES
145
difierence between ual, and a,,. This is seen in Table 4.11 which shows the results of calculations on naphthalene by the Htickel method and by the S.C.F. method(s2).
Cur&
loop
seporotion
m benzene
tdii
units
Fm. 4.3 Them&al ring current shifts of csand fi hydrogen meld of naphthakm with respect to bermme and of absolute ring current shiik of bmmnc. Jonathan, Gordon and Dailoycso)
TABLE~.~~R~NGC~RREN~SHIF~~N~RELA~ BY ‘I’kiEHUCKELANDS.C.F. ~BLWZENEC ALCULA-KD METHODS
Hydroeen 1 2
Hiickel
S.C.F.
0.78 @44
0.72 O-37
Expt.““’ O-54 0.19
4.5.2 Heterocyclic Aromatic Molecules All the molecules discussed so far in which ring currents are important have been altemant hydrocarbons where the n-electron charge densities at the carbon atoms are equal. The introduction of a hetero-atom into the ring introduces a polarity in the carbon u-bonds and the n-electron charge densities at the carbon atoms are no longer equal to one another. In these circumstances the S.C.F.
HIGH
146
RESOLUTION
NMR
SPECTROSCOPY
calculations of Hall et uZ,(~~)give very different results from the Hiickel theory, which tends to overestimate the ring currents. Table 4.12 compares the Htickel and S.C.F. calculations of ring currents in the four molecules 0
P ObO
Q
I (Xi
Q La‘0
LHa
I
b
II
L Iv
JII
En this
study, the rings were assumed to be regular polygons with sides equal to 140 A, the C-H bond was taken to be 1.09 A and the resonance integrals were considered to be equal to the value for benzene. Since the values of the coulomb, OL,integrals involving the 2p n atomic orbitals of the nitrogen and Tmm4.12
CALCULATED
Parameters t
_-
0.19 0.49 0.19 0.19 0.49 0.79 0.19 0.49 0.79
Comu-rxo~s HETJsocYcLFs (I-IV)
@nnw
CURRENT
I
0)
-I
q-r1.02 1.02 1.02 1.52 I.52 1.52 2.02 2.02 2.02 1
-r
Rrm
T
00
TO BBNZENE) IN m
(III)
1Hiickel 1 S.C.F. 1Hiickel
S.C.F.
Hiickel
S.C.F.
Hiickel
0.33 0*49(‘s’ 0*62”3’ O-24 0.37”” O&11,
0.53 0.65 0.73 O-44 0.56 0.64
0.18 0.37
O-82
0.56
0.83
074
O-42
0.75
0.34
0.16 0.28’12’ 0.38”“’
0.36 0.48 @56
0.65
0.31
0.67
O-26
0.52 0.64 0.71 0.42 0.55 0.63 0.34 0.47 0.55
1
1 S.C.F. .
044
.0.52 0.14 0.29 0.41 o-11 0.23 0.33
= (aNOX- d/8 t &N(O)
oxygen atoms are not known, a range of values was taken (& and B,, in Table 4.12). The available experimental data of such compounds(54) suggest the following orders of the ring currents relative to benzene, I > II III > IV III > I which is consistent with the S.C.F. calculations but not with those of the Hiickel type. 4.5.3 Azulene The simple classical calculations of ring currents and the molecular orbital theories of Pople, of McWeeney and of Hall have also been applied to the nonaltemant hydrocarbon, azulene. Experimental evidence shows that the hydro-
THE
CALCULATION
OF SHIELDING
CONSTANTS
IN MOLECULES
147
gen nuclei in the five-membered ring resonate at lower field values than those in the seven-membered ring and that the hydrogen nucleus at position 2 in the five-membered ring resonates at a lower field than those at positions 1 and 3. However, all the calculations predict hydrogens 1 and 3 to be at a lower field than 2, (cf. the results of Hall and HardissorW) shown in Table 4.13).
TABLE~.~~
RING
CURRENT
%im~
IN AzumNERELA~To
BENZENE’Hydrogen IlUClCUS
2 193 498 597 6
Ring current shift dative calcUlatcd
1.37 1.23 l-32 1.18 1.16
to benzene @pm) Observed(79’
1.04 1.37 1.61 O-82 l-11
Although the large discrepancies between theory and experiment for axulene can be ascribed in part to the assumption of equal hybridisation of the carbon atoms in the five- and seven-membered ringst5*), they arise mainly from the presence of different charge densities of the n-electrons on the carbon atomsCs3) (see Section 4.5.5). 4.5.4 Porphyrins Porphyrin molecules (Fig. 4.4) have a large cyclic conjugated path and it has been shown that some of their hydrogen nuclei can experience very large ring current shifts. The peripheral hydrogen nuclei HI, Hz, H4 and Hb show a shift of 2.7 ppm to the low field of benzene, whereas the N-H hydrogen nuclei are + 3.89 ppm to the high field side of tetramethylsilane~s5~.Comparison of the shifts of N-H hydrogen nuclei in porphyrins with that of pyrrole suggests a ring current shift of 1l*Oppmto high field of benxene(56).The ring current model of 18 electrons circulating in a loop of 3.3 A radius gives approximately twice the observed chemical shifts and furthermore does not account for the different shifts of M.W and B-hydrogen nuclei. Abraham(s6) has calculated the ring current shifts by an extension of PaulingWs7) theory for the diamagnetic anisotropy of aromatic molecules (see Section 3.8). In this theory, the induced e.m.f. in any loop is proportional to the area of the loop enclosed, and the resistance of the loop is proportional to the number of bonds. Regarding the porphyrin ring as consisting of four pyrrole rings and a large inner ring, the e.m.f.‘s are calculated to beCs6), pyrrole e.m.f. = KT = R(3il - 2iz) inner ring e.m.f. = KS = R[8(il + iJ + SiJ
148
HIGH
RESOLUTION
NMR
SPECTROSCOPY
and for benzene, e.m.f. = KS, = R(6i,) where T, S, and S, are the areas of the rings and K is aconstant. The C-C and C-N bonds are assumed to have the same resistance. From these equations and the dimensions of the porphyrin and benzene molecules the currents il and iz are . 11 = l-91 iB; i2 = 0.89 iR
where iBis the ring current in benzene.
FIG. 4.4 The porphyrin skeleton (drawn to scale). Abrahamcs6)
The effective magnetic field at the hydrogen nuclei has been calculated in the following way. The shifts of the peripheral hydrogen nuclei H1, H,, Hd, H, were calculated by the induced dipole model of Pople, taking into account the currents in the pyrrole rings and in the four hexagons (area S/4, current il + iJ. For the N-H hydrogen nuclei, the shifts arising from the ring currents in the pyrrole rings were calculated by the dipole approximation, but as the N-H groups lie inside the current loop of the large inner ring (area S) the shift arising from this ring was calculated by the method of Waugh and Fessenden, making use of Johnson and Bovey’s tables (see Appendix B). The ring S was considered to have a ring current divided between two loops as in benzene, with area 78r2 = S giving r = 2.81 A. At a point p inside the ring and in the plane of the molecule the field was obtained by looking up the value ofp/r in Johnson and Bovey’s table for benzene shifts and then correcting for the dilferent sizes and currents of the benzene and porphyrin loops. The results of these calculations are shown in Table 4.14. It can be seen that the magnitude of the calculated shifts are about 50 per cent too large, and to some extent this reflects the many approximations in the calculation, and the limitations of the classical model(s6).
THE
CALCULATION
TABLE~.~~ RING
cuRRENT
OF
SHJELDJNG
~~NIIuBUTIONS
CONSTANTS
TO
THE
IN
P~RPENRIN
‘H
MOLECULES CHEMICAL
WA:
L?wF-Is(~~’
VPpm)
230 5.24 6.70 8.16
233 020 0.09 045
3.25 3.25 715 7.15
0.83 0.83 0.08 0.08
Contributions 346 from bexa- 450 gons 643 7.04
1.88 0.85 0.30 0.21
2.50 5.05 5.05 6.70
497 0.60 0.60 0.26
Coaributions from pyrrole rings
149
216 3.25 3.25 406
Contribution from large ring
281 0.83 0.83 0.43
3.15 574 7.55 8.95
048 048 0.06 0-M
4.15 5.32 7.10 7.85
207 0.39 O-39 0.19
- 21.2
Total shift
5.91
8.25
- 16.3
3.15
4-31
Calculated shift x 213
3.9
5.5
- IO.9
21
29
ObMWd
4.2
5.5
-11.0
1.8
2.9
shifts
1
4.5.5 The Chemical Shift and Chat-g2Densities in Aromatic Molecules The introduction of a substituent group into a benzene ring gives rise to a change in the shielding constants of the hydrogen nuclei at the ortho, meta and para positions relative to the value in benzene. Electron withdrawing groups reduce the shielding constants, that is the ortho, meta and para hydrogen nuclei resonate at lower applied fields than in benzene, whereas electron releasing substituents shift the resonances to higher fields. This has been interpreted as arising from changes in the n-electron charge densities d e at the carbon atoms (see Section 10.12.7), and it has been suggested that the shift produced, 6, is directly proportional to A e, i.e.@l* 82*U) 6 = kde.
(4.16)
Not all the observed shift can be attributed to changes in e since there may also be large contributions from the magnetic anisotropy of the substituent, from permanent and induced electric moments, from intermolecular association and from solvent effects. Equation (4.16) may also be applied to aromatic ions, and to non-altemant aromatic molecules and in this case the observed value of 6 must be corrected for the different ring current shifts in these molecules. A value for k in equation (4.16) may be obtained empirically by comparing the observed shift relative to benzene of some molecule whose charge density
HIGH
150
RESOLUTION
NMR SPECTROSCOPY
is known.
The most reliable value obtained in this way is that of Schaefer and SchneideP3) who compared the shifts in the symmetrical molecules C5H;, CsHs, C,H$, giving a value fork of + 10.7 ppm (electrons)-’ with a probable error of If: 1 ppm (electrons)-‘. This value of k can n*ow be used to evaluate experimental values of g and Table 4.15 shows the results obtained for some representative aromatic molecules. Substituents with large magnetic anisoCOMPARISON OFOBSERVED AND CALCULATED ELECTRON DENS-
TABLE 4.15
IN SOMEARO-
MATICMOLECULES
w
‘osition
(Pm)
&0,, * @pm)
@ohs
ecak. tt
Reference
Aniline
N&i 0
0
m
In
P
P
+ 0.75 + 0.20 + 0.62
1,071 1.019 1.059
+ 0.42 +0*04 i 0.37
1.039 1.004 1.035
l-081 0.997 1.064
(85)
0.988 1.059 0.954 1.011 0.969 0.998
(86)
0.952 1.004 0.981
(87)
Anisole OCHZ.
0 m P
Azulene 2
2
6
193 498 597 6 9,lO
3 0 4 5
+ -
0.583 0.067 0.945 0.28 0.17
+ 0.29 + 0.55 + 0.52 + 0.17 + 0.12
- 0.29 + + -
o-49 0.43 0.45 0.05
0.97 I.05 0.96 1.04 (::;
Pyridine
; Y
- 1.31 + 0.155 -0.26
1.014 0.976
t 8’ is the correction for the differencein ring current shifts between the molecule and bzene. 4 &0,=6+S tt e talc. is obtained from the molecular orbital method.
THE
CALCULATION
OF SHIELDING
CONSTANTS
IN
MOLECULES
151
tropies have been avoided and where relevant a correction, P, has been applied for differences in ring current shifts. Schaefer and Schneider(83) have calculated 9 for other non-alternant aromatic molecules, and they &d that equation (4.16) withk = 10.7 ppm (electrons)-’ gives good agreement between observed and calculated n-electron charge densities. They do not find any evidence to support the suggestion that equation (4.16) should include a term proportional to e*. Schug and Deck(84) have obtained the chemical shifts relative to benzene of the hydrogen nuclei in a number of hydroxy benxenes. After allowing for the effects of the permanent dipole moments, and extrapolating the shifts to a medium of unit dielectric constant, they find that the observed shifts can be fitted to an equation 6 = kl el + kz e2 + k3 e3 where ,eI, e2 and e3 are the n-electron charge densities at the bonded carbon, on the two adjacent carbon atoms, and on any oxygen ortho substituent atoms respectively. The values of the constants kl, k2 and k3 were obtained by calculating the n-electron charge density distributions by means of simple Hiickel molecular orbital (H.M.O.) theory. They found that the values of the proportionality constants were markedly dependent on the values of the Coulomb and exchange integrals employed in the H.M.O. calculations, but they suggest that reasonable values are 7.61, l-26 and 0 ppm (electrons)-’ respectively for kl, k2 and k3, 4.6
CALCULATIONS
OF hamx~
&rms
OF
FLUOI&E NUCLEIIN M~LE~~TXS’
The observed range of chemical shifts for the fluorine nucleus in molecules is much larger than the range of hydrogen shifts; for example, the l 9F resonance in the molecule UFs is 955 ppm to the low field of that in HFt5’). These large chemical shifts arise because of the importance in Ramsey’s equation (equation 3.29) of the paramagnetic terms for the electrons local&d on the fluorine nucleus. Applying equation (4.8) to the case where A is a fluorine nucleus, then, in contrast with hydrogen nuclear shielding, G is the dominant term. The importance of u= arises from the presence of valency pelectrons in the fluorine atom, which on molecule formation can depart considerably from spherical symmetry (see Section 3.7). The paramagnetic term will be negligible in the spherical fluoride ion, so that in this case ti; is zero, whereas in the fluorine molecule F2, G is large and negative, with the result that the resonance of F2 gas is 630 ppm to low field of that in liquid hydrogen fluoride(33). Between these two extremes the chemical shift of the fluorine nucleus in binary fluorides is almost linearly dependent on the electronegativity of the atom bonded to the fluorine (see Chapter 11). The exceptionally large values of UF in some molecules, such as UF, , arise because of the presence of low-lying electronically excited states with the appropriate symmetry to give non-zero matrix elements of the operator n&k in equation (3.29)‘59’. Saika and Slichter(60) have calculated the chemical shift difference between the fluorine nucleus in the molecule F2 and in the ion F- by assuming that the observed shift can be equated to a= in the fluorine molecule. Thus the cal-
152
HIGH
RESOLUTION
NMR
SPECTROSCOPY
culation reduces to one of evaluating the matrix elements of the operator mzr between the ground and excited states. The calculation is simplified further by including only p electrons in the summation in equation (3.31) for u:, and the difference in shielding is given byt60) (4.17) where (l/r3)2, is the average value of l/r3 for 2p electrons and A E is an average excitation energy. The ground state of the fluorine molecule is a 1x; state, while the lowest excited state with the appropriate symmetry to give a nonzero matrix element of the operator mzk is a In, state, arising from the transfer of an electron to an unoccupied 2p orbital. The value of the excitation energy for the transition ‘c: to ‘n~iis known to be 4.3 eP1) and should be similar to that for a ‘xi to ‘n, transition. Using this value of A E and a value of (1/r3>21, of 8.89/a: where an is the Bohr radiuP2) gives a shielding difference OF60)
Aa = -20
x 10-4.
The experimental value of - 6.3 x 1O-4 between F2 and HF can be accounted for in part by allowing some covalent character in the H-F bond. Using an ionic character of 0.43 and an average excitation energy of 7~7eV for HF, Saika and SlichtePO)calculate the chemical shift between F2 and HF to be - 14 x lo-‘, in better agreement with experiment but still too large. The remaining discrepancy probably reflects the many approximations made, and in particular the neglect of higher excited states. Karplus and Das(63) haveextendedthe above treatment of lgF chemical shift calculations, and by using a molecular orbital function in Ramsey’s perturbation expression for a= they have been able to relate the change in shielding of the fiuorine nucleus in similar molecules to changes in ionic character, bond order, and degreeof hybridisation of the bond involving the fluorine atom in question. The molecular orbital function for the ground state of the molecule is written as a single determinant of the form YO = ~(2~)!I-“21~~(l)8(l)~~(2)~(2)y,(3)8(3)
. . . %(2n)a(2n)l
(4.18)
where each of the doubly-filled molecular orbitals y,(k) is formed from atomic orbitals v,(k) by the L.C.A.O. method
substituting equations (4.18) and (4.19) into the approximate expression for the shielding constant involving an average energy approximation (equation (3.31)) gives f? &la = _ cp /q(k) rk2m-rk.rk 3 2Mc2 vll ““\ ’ rk
THE
CALCULATION
OF
SHIELDING
CONSTANTS
IN
MOLECULES
153
The summations in equations (4.20) and (4.21) are over the atomic orbitals occupied in the ground state. The p,,, are elements of the charge-bond order matrix. When applying equation (4.21) to a fluorine atom in a molecule only the atomic orbitals centred on the fluorine atom need be considered. Moreover, since s-orbitals give a zero contribution to a,,-, the q, in equation (4.21) are simply the 2p-orbitals (pX,9)*and vz. The integrals may now be readily evaluated since operating on one of the p-orbitals q, with a component of mk gives either zero or transforms CJ+into another p-orbital, i.e. mXk&V
= 0
1
(4.22)
When equations (4.21) and (4.22) are averaged over all orientations of the mole cule then 47
= 00 UPXX + PYY + PZA
-
HP,,
PYY + PYY Pzz
+ HPXY PYZ + Pm PZY + Pm Pxz>>
wherep,,,
are orbital populations,
PYY, andp,,
+ PII PXJ
-
(4.23)
and (4.24)
This is equivalent to the expression obtained by Saika and SlichtePO) for the chemical shift between covalent and ionic fluorine. By specialising the argument even more to a fluorine atom directly bonded to only one other atom, Karplus and Dasor5r were able to show that G
e&l + 3P 1s + (ex+ ey)U+ s - 14 - exeJ)
= ao{[l - S - Z + Hex +
(4.25)
where Z is the ionic character of the bond, s the degree of sp hybridisation, and ex and eYthe double bond characters of the bond
ex =2-p,,
and
e,=2-p,.
When s, Z, eX, and eYare small the terms in equation (4.25) which involve products of these terms may be neglected, hence o?
=a,[1
-s-
Z+
Hex+ e33.
(4.26)
154
HIGH
RESOLUTION
NMR
SPECTROSCOPY
It might be thought a weakness of equation (4.26) that it was derived from an equation for d defined in terms of an average energy approximation; however, Karplus and Das have shown that equation (4.26) may be derived from the full expression for 6. 4.6.1 Chemical Shifts of Fluorobenzenes The most useful applications of equation (4.26) are in calculating the difference between the shielding constants of fluorine nuclei in related molecules, since having calculated a0 for one member of a series one can assume that it has the same value for the others. Karplus and DaP3) have successfully applied equation (4.26) to predict lgF chemical shifts for multifluorobenzenes. When the fluorine atom is part of an aromatic system having its ring in the yz plane, then eYis zero and oga = 00 (2 - PLZ.PXJ and, without any restrictions on the magnitude of Z, s or eX, aFFm= a0 (1 - s - Z + Z s (e/2) + e/2 (s + 1))
(4.27)
where e has been written for g,. The chemical shifts of the multi&rorobenzenes relative to the reference compound monofluorobenzene, can therefore be calculated from dap = tiy - &JWara and since de will be negligible then A@$’ can be equated to the observed chemical shift, (Hi? - &) A4y = (4.28) HR ’ Substituting AZ=Z,-I,; As=s,-sc; Ae=pR-eec into equation (4.26) gives fWY
= co{-
dZ + d@/2 - ds + (ZR - dZ + JR - ds)de/2
+ [(2+ + @R)/21d Z +
[@R/2)
+
IRj
d s
-
ill
d 81).
(4.29)
The degree of sp hybridisation in fluorine compounds is small (s x 0.05) and thus Ascan be neglected. Assuming that a0 remains constant, and neglecting terms quadratic in A Z and il ,o, we get AfJg
= aO{- AZ(1 - eR/2 - sR> + d e/2(1 + ZR + 3,)).
(4.30)
The value of u. was calculated by applying equation (4.30) to the two molecules F, and monofluorobenzene. For the fluorine molecule Z and Q are zero and s has been calculated(64) to be 0.02. For monofluorobenzene, Z was estimated to be 0.75 from the ionic character versus electronegativity relationships proposed by Dailey and TownesP*) and by Gordy(29); s was taken to be 0*05’65’, and the value of e = O-126 was calculated by means of the Hiickel molecular orbital
THE
CALCULATION
OF SHIELDING
CONSTANTS
IN
MOLECULES
theory. With these values of I, e and s, and the experimental of - 543 ppm(33), a0 was calculated to be, CO =
-0.863
155
chemical shift
x 1O-3
which on substituting into equation (4.30) gives for A-, de?“” = (0*765dl-
0.777de).
(4.3 1)
From equation (4.28) it is seen that a positive value of da means that the resonance in the compound is at lower applied fields than that in the reference, monofluorobenxene; thus a positive value of dl shifts the resonance to low fields (IR > Ic), and a positive value of A e(eR > ec) shifts the resonance to high fields. In order that equation (4.31) may be applied to the calculation of da for the multifiuorobenzenes, it is necessary to calculate values of d Zand A e. Values of A e can be obtained by means of the Htickel theory and estimates of d Z can be found by applying equation (4.31) to the three difluorobenxenes. From the experimental values of da and the calculated values of A e it is possible to obtain values of Al; the results of these calculations are summarised in Table 4.16, where A a(e) has been written for 0.777 de. TABLE~.~~ CALCULA TEDVALUESOF AZ FOR DIFLUOROBENZENB Compound
ortho meta
pore
de oao93 oaoO3 0~0115
I
Au@) x lo5
Au(cxpt.) x 10’
-@72 - 0.02 - 0.89
- 2.@33’ + @uT6a - 0.65I6”’
.
AZ -
o*o%l6
_ -to9042 + OaI31
The values of AZ for the m- and p-difluorobenzenes compare well with d I values for dichloro- and dibromobenzenes obtained from quadrupole coupling constantrP3); however, the large negative value ofd I for the ortho compound seems unreasonable and suggests that there is an added effect when two fluorines are adjacent to one another and for which no account has been taken in deriving equation (4.31). If equation (4.31) is applied to multifluorobenxenes,with calculated values of A g, and d I values estimated by assuming that a fluorine neighbour in a mera position contributes + O-0042, and one in a para position + 00031, then the calculated values of da are in close agreement with experiment. However, fluorine nuclei with o-fluorine neighbours show an anomalous shift of magnitude 1.67 x 1O-5 to high field. Table 4.17 summarises the calculated values of Au for a number of multifluorobenzenes in which the fluorine nucleus has one ortho-fluorine neighbour, and compares these values with those obtained experimentally. The shielding constant differences da in Table 4.17 are between the fluorine nucleus in the position “in bold type” and the fluorine nucleus in monofluorobenxene. The ortho effect appears to be constant in multifluorobenzenes, except that an unusually high value of - 1.88 x 1O-5 was found for ortho-difluorobenzene.
156
HIGH
RESOLUTION
NMR
SPECTROSCOPY
The calculated values of do for the fluorine nuclei in some other multifluorobenzenes(63) are shown in Table 4.18. The values of Au for fluorine nuclei with o&o-fluorine neighbours contain an empirical “o&o effect” contribution of - 1.67 x 10-5. TABLE4 17 CALCULATED VAWB
Au(e) x 10”
AZ
Au(Z) x 10”
x 105
00X8 0.0204 0~0200
- 0.68 - 1.59 - 1.55
0*0@42 0*0031 oaO73
+ @32 + 0.24 + 0.56
- 0.36 - 1.35 - 0.99
TABLE 4.18
Compound
Aa(caIc)
AQ
Compound
1, a4 I,54 kZ4.5
OF A d FOR SOME ortho SUBSTITUTEDM~LTELIJOROBEN~ENW
Aa(obs)@” x 105
- 2.04 - 3.03 - 2.65
Au ortho x 105 - 1.68 - 1.68 - 166
CALCIJLATEDVALUES OF .4 u FOR SOMEMULTIFLUOROBENZENES~~~)
Ae
Ade)
x105
AZ
A NJ
x
Au(calc)
Au(obs)
103
x 103
x 10s
1,395 1,394
0.0006 0.0103
- 0.04 -0.80
0*0084 a0073
+ 0.64 + 0.56
+ 0.66 - 0.24
+ 0.53’33’ - 0.25’33’
1,3,4,5 I,53 1,z 395 1, 2, 394 1,2,3,4.5
0*0109 0.0117 0.0109 OalO 0.0204
- 0.85 -0.91 - 0.85 - 1.63 - 1.59
0.0115 0.0042 00084 oa73 0~0115
-!- 0.88 + 0.32 + 0.64 -i- 0.56 i- 0.88
+ 0.03 -223 -1.88 -274 -238
_ 2.55’67’
The ortho effect is too large to be attributed to neighbour anisotropy, or to polarisation of the electrons by neighbouring fluorine atoms. One possible explanation(63) is that the large size ofthe fluorine atom compared with hydrogen distorts the molecule so that it is no longer planar, and thus reduces the amount of double bond character. This would mean a large positive d,o of magnitude 0,023. However, this steric effect is not supported by the observed shifts in chlorofluorobenzenes. In these compounds there is a pronounced shift to lower fields (N 21.7 ppm) when the fluorine has an orrho-chlorine neighbour, whereas the increased bulk of the chlorine group would be expected to increase dq still further. The decrease in the shielding constant of fluorine produced by an orrhochlorine atom can be explained in terms of the internal electric field produced by the polar C-F and C-Cl bonds?, and also by the Van der Waals interaction (see Section 3.10.1)(g7). The work of Karplus and Das on the shielding constants of lgF nuclei has been extended by Prosser and Goodman (go). Instead of restricting contributions to the lgF shielding constant to electrons localised on the atom in question the latter authors have developed a theory which gives the shift in terms of the n-electron distribution throughout the molecule. In fluoro-aromatic compounds t
This effect seems to be important in a wide variety of molecules(gg).
THE
CALCULATION
OF SHIELDING
CONSTANTS
IN
MOLECULES
157
the theory predicts appreciable contributions to the shielding constant from n-electron charge densities on the bonded carbon atom, q C,, on the fluorine atom, q F., and in the C-F bond p(C, F,). They show that these quantities are related to the chemical shift between two lgF nuclei by the relationship 6 = 488dE-‘[11.96q(Fz)
+ 3*9dp(C,F,)
+ 0.1 dq(C,)].
Equation (4.32) has been applied to a number of p - X-C&F by Taft and co-workers(gl). 4.7 CHEMICALSHIFT
CALCULATIONS
FOR hkW.ELLANEO~S
(4.32) compounds
NUCLEI
Only a very few calculations of shielding constants have been carried out for nuclei other than hydrogen and fluorine. The Saika and Slichter calculation has been applied both to phosphorus ~3. 6g) and to cobalt(70) nuclei; for both nuclei the dominant contribution to the shielding constant arises from the paramagnetic term in Ramsey’s equation. A completely different type of calculation has been carried out for aluminium shielding constants in which the variational method was used. More recently, Pople and Karplus have presented a theory of carbon chemical ~hifts(~~~~*).Buckingham and Schneidefig6) have predicted lggHg, 205Tl and 207Pb chemical shifts using a simple atom-in-a-molecule model. REFERENCES 1. W. C. DICKINSON, Phys. Reu., 80, 563 (1950). 2. F .T. ORMANLJand F. A. MATSEN,J. Chem. Phys., 30,368 (1959). 3. R. E. GIXK, J. Phys. Chem., 65, 1871 (1961). 4. J. M. B. KEWOO, I. I. bB1, N. F. RAMSEYand J. R. Z~aiuus, Phys. Rev., 57. 677 (1940). 5. N. F. RAMSEY, Phys, Rev., 78, 699 (1950). 6. G. F.Nmv~u, Phys. Reu., 80,476 (1950). 7. A. No RDS~ Phys. k., 57,556 (A) (1940). 8. W. biNDERSoN, Phys. Rev., 76, 1460 (1949). 9. L. C. SNVDER and R G. PARR, J. Chem Phys., 34,837 (1961). 10. H. F. I-~MEKA, Rev. Mod. Phys., 34, 87 (1962). 11. C. A. ~ULSON, Trans. Far&y Sot., 33, 1479 (1937). 12. S. WANG, Phys. Reu., 31, 579 (1928). 13. N. ROSEN,Phys. Rev., 38,20QQ (1931). 14. K. I?o, J. Amer. Chem. SOL, 80, 3502 (1958). 15. Y. KUXUTAand K. ITO, 1. Amer. Chem. Sot., 82,2% (1960). 16. J. TILLIEUand J. GUY, J. C/tern. Phys., 24, 1117 (1956). 17. T. P. DAS and R. BERSOHN,Phys. Reo., 115,897(1959). 18. J. 0. HIRS~~~L.DER and W. LINNET,J. Chem. Phys., 18,130 (1950). 19. S. WEINBAUM, J. Gem. Phys., 1.593(1933). 20.Y. SUGIURA, 2. Phys., 45,484 (1927). 21. R. F. WALLIS,J. Chem. Phys., 23, 1256 (1955). U.-M. J. STEPHEN, Proc. Roy. Sot., A243, 264 (1957). 23. M. FIXMAN,J. Chem. Phys., 35,679 (1961). 24. E. ISHIGURO and S. KOIDA, Phys. Rev., 94,350 (1954). 25. H. M. JAMESand A. S. WuDciE, J. Chem. Phys., 1,825(1933). 26. W. G. S CHNEIDER, H. J. B~~~srerrr and J. A. POPL.E,J. Chem. Phys., 2& 601(1958) 27. H. F. HAMJXA,Mol. Phys., 2, 64 (1959).
158 28. 29. 30. 31.
HIGH
RESOLUTION
NMR
SPECTROSCOPY
B. R. MCGARVEY, J. Chem. Phys., 27.68 (1957). W. GORDY,J. Chem. Phys., 19,792 (1951); 22, 1470 (1954). T. P. DASand T. GHOSE, J. Chem. Phys., 31,42 (1959). M. R. BAKER,C. H. ANDERSON, J. PINKERTON and N. F. RAMSEY, Bull. Amer. Phys. Sot., 6, 19 (1961). 32. C. T. ZAHN,Phys. Rev., 24, 400 (1924). 33. H. S. GUTOWSKY and C. J. HOFFMAN, J. Chem. Phys., 19,1259 (1951). 34. L. PAVLW~,Nature offhe Chemical Bond, 2nd ed., Oxford University Press (1950), p. 59. 35. J. TILLIEU,Thesis, University of Paris (1956). 36. J. A. POPLE,Proc. Roy. Sot., A239, 541 (1957). 37. J. A. POPLE,Proc. Roy. Sot., A239, 550 (1957). 38. H. M. MCCONNELL, 1. Chem. Phys., 27,226 (1957). 39. P T. NARA~~N and M. T. ROOERS,J. Phys. Chem., 63, 1388 (1959). 40. B POSTand J. LADW, Acta Cryst., 7, 559 (1954). 41. D. R. Wun74~~, M. SAUNDERS, L. ONSAGER and H. E. DUBB, J. Chem. Phys., 32, 67 (1960). 42. P. T. N ARASMHAN and M. T. ROGERS,J. Chem. Phys., 31, 1302 (1959). 43. J. TILLIEU,Ann. Phys., 2,471, 631 (1957). 44. B. P. DAILEYand J. N. SHOOLERY,J. Amer. Chem. Sot., 77, 3977 (1955). 45. H. SPIESECKEand W. G. SCHNEIDER, J. Chem. Phys., 35, 722 (1961). 46. J. I. MUSHER,J. Chem. Phys., 35, 1159 (1961). 47. J; S. WAUGHand R. W. FESSENDEN, J. Amer. Chem. Sot., 79, 846 (1957). 48. C. E. JOHNSON and F. A. BOVEY,J. Chem. Phys., 29, 1012 (1958). 49. H. J. BERNSTEIN, W. G. SCHNEIDER and J. A. POPLE,Proc. Roy. Sot., A236, 515 (1956). 50. N. JONATHAN, S. GORDON and B. P. DAILN, J. Chem. Phys., 36,2443 (1962). 51. C. REID, J. Mol. Spect., 1, 18 (1957). 52. G. G. HALL and A. HARDISSON, Proc. Roy. Sot., A268,328 (1962). 53. G. G. HALL, A. HARDISSON and L. M. JACKMAN, Discuss. Farad. Sot., 34, 15 (1962). 54. G. G. HALL, A, HARDI~~~N and L. M. JACK-~, Tetrahedron, 19, suppl. 2, 101 (1963). 55. E. D. BECKW and R. B. BRADLEY,J. Chem. Phys., 31, 1413 (1959). Mol. Phys., 4, 145 (1961). 56. R. J. ABRAHAM, 57. J. N. : HOOLERY and H. E. WEAVER,Ann. Rev. Phys. Chem., 6,433 (1955). 58. B. P. Dand C. H. Tow, J. Chem. Phys., 23, 118 (1955). 59. E. PITCHER,A. D. BUCKM~HAM and F. G. A. STONE,J. Chem. Phys., 36, 124 (1962). 60. A. SAIKAand C. P. SLKHTER, J. Chem. Phys., 22,26 (1954). 61. G. HERDERG, Molecular Spectra and Molecular Structure, Vol. 1,2nd ed., Van Nostrand, New York (1950). Phys. Rev., 84, 244 (1951). 62. R. STEB 63. M. KARPLUSand T. P. DAS,J. Chem. Phys., 34, 1683 (1962). 64. B. J. mNSlL, Rev. Mod. Phys., 32,245 (1960). 65. H. S. GIJTOWSKY, D. H. ANDERSON and P. J. FRANK.J. Chem. Phys., 32, 196 (1960). 66. R. W. TAFI; S. EHRENSON, I. C. Lawns and R. E. GLICK, J. Amer. Chem. Sot., 81, 5352 (1959). 67. N. BODEN,J. W. EM~LEY,J. FEENEYand L. H. S~TCLIFFE,unpublisheddata. 68. N. MULLER,P. C. LAIJTERBUR and J. GOLDENSON, J. Amer. Chem. Sot., 78, 3557 (1956). 69. J. R. PARKS,J. Amer. Chem. Sot., 79, 757 (1957). 70. J. S. GRIFFITHand L. E. ORGEL,Trans. Far&y Sot., 53,601 (1957). 71. D. E. O’REILLY,J. Chem. Phys., 32, 1007 (1960). 72. J. G. POWLES,Rept. Prog. Phys., 22, 433 (1959). 73. C. SANDORPY, Gun. J. Chem., 33,1337 (1955). 74. A. G. MORITZand N. SHEPPARD, Mol. Phys., 5,361 (1962). 75. F. R. JENSEN, D. S. NOYCZ, C. H. SEDERHOLMand A. J. BERLIN,J. Amer. Chem. Sot. 82, 1256 (1960). 76. J. I. MUSHER,J. Chem. Phys., 37, 192 (1962). 77. J. L Musxs~, J. Chem. Phys., 37, 34 (1962). 78. M. KARPLUSand J. A. POPLE, J. Chem. Phys., 38,2803 (1963).
THE
CALCULATION
OF SHIELDING
CONSTANTS
IN
MOLECULES
159
79. H. Sprrsx~~ andW.G.S CHNEIDER, Tetrahedron Letters, 468 (1961). 80. C. A. COULSON, Valence,Oxford University Press, (1952). p. 248. 81. G. Fm, R. E:CARTER,A. MCLACHLANand J. H. Rrcxi~~~s, J. Amer. Chem. SOL, 82, 5846 (1960). 82. I. C. SMITEand W. G. SCHNEIDER, Can. 1. Chem., 39, 1158 (1961). 83. T.Sand W. G. SCHNEIDW,Con. J. Chem., 41,966 (1963). 84. J. C. SWG and J. C. DECK, J. Chem. Phys., 37,2618 (1962). 85. H. BABA and S. SUZUKI,J. Chem. Phys., 32,1706 (1960). 86. R. D. BROWNand M. L. HEFFERNAN, Aurtrdian J. Gem., 13,38 (1960). 87. R D. BROWNand M. L. HEFFERNAN, Australian J. Chem., 12, 554 (1959). 88. H. H. LANDoLTand R. BCIRNSTFXN, Zahlenwerte und Functionen Springer-Vcrla8, Berlin, I. Band, I. TeiI (1950) and III. .Teil (1951). 89. R. J. W. LEF~RE, Jozirn. andl’roc. Roy. Sot. N. S. W. 95, 1 (1961). 90. F. m and L. GOODMAN,J. Chem. Phys., 38,374 (1963). 91. R. W. TAFT, F. PROSSER,L. GOODMANand G. T. DAVIS, J. Chem. Phys., 38, 380 (1963). 92. W. G. mrz~., Private communication. 93. J. W. m, unpublished results. 94. L. PsIIuKEl and H. J. BERW’IBN,J. Chem. Phys., 38, 1562 (1963). 95. G. G. HALL, Private communioation. Discuw. Faraday Sot., 34,147 (1962). 96. A. D. BUCEINWUMand W. G. SCIPIEIDER, 97. N. B~DEN, J. W. EMS=, J. EENEY and L. I-I. SUTCLIPPE, Mol. Phys., 8,133 (1964). 98. J. A. P~PLE,Mol. Phys., 7,301 (1964) 99. J. Fm, L. H. Strrfznz’~ and S. M. WUKER, Mol. Phys., 11,117,129, 137,145 (1966).
CHAPTER
5
THEORETICAL CALCULATIONS OF SPIN-SPIN COUPLING CONSTANTS OF SIMPLE MOLECULES THEREhave been many applications to real molecular systems of the equations for spin-spin coupling constants developed by Ramsey(‘), McConnelP and by Karplus(3*J). Most of the published work is on the calculation of interhydrogen coupling constants, largely because of the simplicity of the molecular wavefunctions, but also as a result of the large body of experimental values available. The calculations have revealed two interesting properties of spin-spin coup ling constants. Firstly, the valence bond approach developed by Karplus and his collaborators has shown that a non-zero coupling constant between nonbonded hydrogen atoms is a sensitive indication of departures from “perfect pairing” structures. Secondly, both the valence bond and the molecular orbital calculations suggest that the values of J HHin some molecules depend upon the inter-bond angles and thus the magnitude of JHH can sometimes give the configuration of a molecule. 5.1 HYDROGEN DEUTERIDE, HD Experiment has shown that spin-spin coupling is not observed between nuclei which are magnetically equivalent and a quantum mechanical justification for this observation is presented in detail in Section 8.12. The spectrum of the hydrogen molecule consists therefore of a single sharp line; but it is important to stress that this does not mean that JHH in the hydrogen molecule is zero. It is in fact possible to obtain an experimental estimateof themagnitude of the coupling constant in the hydrogen molecule by measuring the coupling constant JHD in the molecule HD since the ratio of the two coupling constants JHD/JHHis equal to the ratio of the magnetogyric ratios of the two nuclei Y&J” (= 0-I 54). There is a small correction to this conversion factor arising from the different amplitudes in the ground vibrational states of the two molecules(112), but this can be safely ignored at this stage since the effect is to reduce JHH by only a few tenths of one cycle set-‘. The hydrogen resonance spectrum of HD consists of three lines of equal intensity arising from coupling of the hydrogen nucleus with the deuterium nucleus of spin 1. The coupling constant measured from this spectrum(5~6’ is JHD = 43.0 f 0.5 cycles set-‘. 160
C A L C U L A T I O N S OF S P I N - S P I N
C O U P L I N G CONSTANTS
161
In calculating coupling constants, RamseyC~>considered four types of interaction and he wrote the coupling constant as a sum of four terms (see Section 3.14), that is, JHD = 1(1 ) + 1(2) ",rid "lUD + J2nD + Js.D. (5.1) By far the largest term for bonded nuclei is the contact term, in this case JSHD. The magnitude of JSMD can be obtained from equation (3.215). Assuming that the orbital and spin part of the ground state wavefunction are separable, then after applying the average energy approximation, equation (3.215) becomes / 4 "~['16~zfl h~ 2 .land = -- ~ 3 - - - ~ a J ~ ~ ynyD(V'.ol~(r~.)" t~(r2D)[ ~V~o) × (v,= IS~ • s , I v,+)
(5.2)
where ~o~oand ~oes are the orbital and spin parts of the electronic wavefunction. A Ea is an average energy defined by equation 0.216) with o~Fa in the place o f . ~ . The electron spin terms in (5.2) are readily evaluated since (~,~IS~.S~lw=)
ffi
,~
{s ~ - s~ - s~,} v,~
=
-~-.
Substitution of this result into equation (5.2) gives =
(5.3)
in which [~O,o[~s2D is the measure of the probability of finding electron 1 on H and electron 2 on D. Evaluating equation (5.3)with V,o as the James-Coolidge function gives(1) 55.8 JH D ----~ (cycles sec-~). (5.4)
,dE~
A E3 in the above expression is in Rydberg units. At present it is not possible to evaluate AE3 from equation (3.215); however, one can place limits on its possible values c1~.James, Coolidge and Present tT~ calculated the lowest triplet state of hydrogen to be 0.67 Rydbergs above the ground state for an internuclear separation of 1.4 ao (ao is the Bohr radius), whereas the energy required to doubly ionise the molecule with this internuclear spacing is 3.77 Rydbergs. For the analogous mean excitation energy A E used in the theories of the electronic contributions to the rotational and spin-rotational magnetic interactions, Brooks c8) has selected values of 1-9 and 1-1 Rydbergs respectively. Ramseytl~ assumed that the term J3sD accounts for 40 cycles sec -1 of the experimental value of 43.0 cycles sec -1, and thus from equation (5.4) the value of AE3 is 1-4 Rydbergs which is consistent with the limiting values. Ramsey estimated that J2nD is about 3cycles sec -I and that (J~Ro a) + J ~ez)) is no greater than 0"5 cycles see -1. mrs.
e
162
HIGH RESOLUTION NMR SPECTROSCOPY
A variational calculation by Stephen (°) (see Section 3.14.4) gives the magnitudes of (JtX)nD + J~a~m~)and (J2aD + J3aD) as 1"46 and 47'68 cycles see -~ respectively when a valence bond function is used, and 2.08 and 47-49 cycles see -~ on using a molecular orbital function. Although these values differ from those o f Ramsey they confirm that (J3aD and J:aD) contributes the major part o f J n o. Ishiguro(t o)has improved on Stephen's calculation by using a more flexible variational function, and also by allowing for the zero point vibrations in the molecule. The calculation was carried out for values of the internuclear separation of 1.3 ao, 1.4 ao and 1.5 ao and then JnD was averaged over the zero point vibration. This gives (t°) j¢x) - 0-254 cycles see -1 IHD j¢2) tin> -- + 0"354 cycles see -~ J2nD = + 0.202 cycles see -~
JznD = + 36"837 cycles see -~ and a total of Jr~D = 37"138 cycles see -~, which compares well with the experimental value of 43.0 cycles sec-z, particularly as the calculation does not invoke an average energy approximation. The main difference between the results of Ishiguro's calculation and that o f Ramsey is in the magnitude of (J~I~nD + J ~ D + J2nQ. Ramsey assumed that these would c o n t r i b u t e - , 3 cycles sec -1 to JrtD, whereas Ishiguro's calculation suggests that they total ,-~ 1 cycles see -1, which" leaves J3 n o as 42-0 cycles sec-~. With this value of J3 HD equation (5.4) •gives R = 1.35 Rydbergs. An important consequence of J3.D being the largest term in equation (5.1) is that the coupling constant is positive in sign, that is, the preferred nuclear orientation is the one with the spins aligned anti-parallel. Furthermore, since J3NN' would be expected to be the dominant coupling term for other directly bonded nuclei the coupling constants between them should also be positive. However, Tiers (3s) has found that this is not true in at least one case, for in the molecule CHFC12 he has found that J,~ca and J~3cv are of opposite sign (see Section 5.3.3). Buckingham and Lovering (x1)have suggested a method for o btaining the absolute signs of coupling constants but as yet the method has only been applied to the J aortho u value in p-nitrotoluene where the sign was shown to be absolutely positive (~°°). This result is in agreement with the sign predicted by an indirect method based on the assumption that &3c-a is positive.t The method of Buckingham and Lovering consists of noting the effect on a spectrum of applying a strong electric field, E. F o r example, they show that for a system A X the effective magnetic field at nucleus A when the applied electric and magnetic fields, E and H, are parallel to one another is
(
(HA)E = H[1 -- ¢r -- (~rll -- a~) Z] -- 2#x m x \4/~A/~X
3z)
R-~'.
(5.5)
f A recent calculation by Musher(I~6) predicts that Jr31.1D is negative, and a MO calculation by Pople and Santryn17) gives a positive value for l~ac,H, but a negative value for/t3cF.
CALCULATIONS
OF S P I N - S P I N
COUPLING
CONSTANTS
163
where a u and u± are the components of the shielding constant of nucleus A parallel to and perpendicular to the molecular axis. R is the internuclear distance, and Z is given by 2E2 [ p2] Z -- 45k--'-'~ (all - 0~±) + ~ (5.6) in which 0ql and o~± are the components of the static electric polaxisability parallel and perpendicular to the molecular axis and # is the molecular dipole moment. For R = 2 × 10 -8 cm, PA = #x = 10 -~3 erg gauss -~ and Z = 10-" then
12p^gxZ R3
= 2-3 cycles sec -~, which subtracts from the unperturbed
splitting when J is positive and adds to it when J is negative. 5.2 METHANE
The H - H coupling in methane cannot be observed directly since the four hydrogen nuclei are magnetically equivalent; however, an experimental value of J.. has been obtained by examining the IH spectrum of the molecule CH3D, and itis found that J.. ---- 12.4 _+ 0.6 cyclessec-I. It can be inferredfrom other experiments that the absolute sign is negativem7 (see Section 5.3.3).The firstattempt to calculateJ.. in methane was made by McConnelF 2)using the general theoreticalformulation of nuclear spin-spin couFling constants in terms of the molecular orbital approximation. McConnell came to the conclusion that the coupling constant between hydrogen nuclei which are not directly bonded is dominated by the Fermi contact term, so that J.., ~ J3..'. Applying equation (3.228) to hydrogen nuclei gives J , xn, = h-1 (2/~ h) 2 72. ~
L-~-\-~--] ~ n ' 92H(0)9~,(0)
(5.7)
where ~ . . , is similar to the quantity used by Coulson and Longnet-Higgins (~3) to define the bond order between pairs of carbon nuclei directly bonded to one another in aromatic molecules, and 9H (0) is the electron density at the nucleus. AE, the average excitation energy was assumed to be l0 eV, a value which is unlikely t o be in error by a factor of more than five. With this value of A E, and an effective hydrogen nuclear charge of l'00, equation (5.7) gives J H H ' '~
2OO ~ 2 H, 2~
(cycles sec-0.
(5.8)
An approximate L.C.A.O.M.O calculation by Coulson (~4) for the methane molecule gave I~,n,l --0.118, and substitution of this value into equation (5.8) gives Jmv ffi 0.44 cycles sec-t compared with the experimental value of --12,4 + 0.6 cycles sec-L If equation (5.8)correctly describes the magnitude of J . n , then a more appropriate value for ~,n, would be 0"62, which seems very improbable. 6"
164
HIGH RESOLUTION
NMR SPECTROSCOPY
Besides giving a low value for the magnitude of JnH' the molecular orbital treatment of McConnell predicts that the magnitude of Jx n, remains constant as the number of bonds separating H and H' increases, and also that all H - H coupling constants should be positive in sign. The valence bond approach to the calculation of coupling constants as developed by Karplus and Anderson (s) avoids these particular difficulties, and was thought to be more suitable for the calculation of coupling constants in that it deals directly with the electron spins in molecules and avoids the need for accurate wavefunctions. Unfortunately, it gives the wrong sign for J ~ . Application of equation (3.240) to inter-hydrogen coupling gives ) E(eV)/103-! ¢t ( JHH ,= 1"395:": j~lC'j
I
[l "l-2fit(pH H,)]
(5.9)
where the cs and c~ refer to a complete canonical set of valence bond structures The application of equation (5.9) proceeds in two stages; firstly,the ground state wavefunction ~'o is determined as a linear combination of the canonical valence bond structures, and secondly the values of cs are substituted into equation (5.9) together with the values of the other coefficients (see Section 3.14.3). Methane was treated as an eight electron, eight orbital problem, in which the four hydrogen electrons are in I s orbitals (b, d, f, h), and the four carbon electrons are in sp s hybridised orbitals (a, c, e, g) directed towards the hydrogen atoms. Fourteen canonical valence bond structures result under the restriction that each of the eight orbitals is singly occupied. The total wavefunction is formed by taking linear combinations of these canonical structures which form a basis for the irreducible representation of the tetrahedral symmetry group (see Section 8.9). Karplus and Anderson (a) used the symmetrised functions derived for methane from a non-canonical set by Eyring, Frost and Turkevich C15). The AI ground state for methane is then found to be a combination of three functions ~'o ffi 1.08 Wx + 0.001 W2 - 0"028 Y3 (5.10)
where ~p~ is the "perfect pairing" structure
y~ = (a b)(c d)(ef)(g h) and the 1P2 and ~3 refer to non-perfect pairing structures.
~P2 = (a h) (b g) (c f) (d e) + (a d) (b c) (el) ( f g) + (af ) (b e) (e h) (d g) ~o3 ffi (a b)(c h)(d g ) ( f e) + (af)(b e)(c d)(h g) + (a h)(b g)(c d)(f e) + (ab)(ed)(eh)(gf) + (ad)(bc)(ef)(hg) + (ab)(cf)(de)(hg) where the parentheses indicate pairs of orbitals bonded together. A non-zero JHn, arises because of the presence of ~2 and ~P3, the contribution from tpl being zero for non-bonded nuclei. The ground state function given by equation (5.10) cannot be used directly with equation (5.9) because ~/'ois not derived from a canonical set. However, Karplus and Anderson were able to use equation (5.10) by expanding each of the non-canonical structures in terms of the corn-
CALCULATIONS
plete canonical
OF SPIN-SPIN
COUPLING
165
CONSTANTS
set, and they obtained for JHHt the value of -
113
A E(eV)
cycles
With A E as 9-O eV, this gives JHw. = + 12.5 cycles set-‘, which does not compare very well with the experimental value of - 12.4 f 0.6 cycles set-‘. Hiroike(16) has developed the valence bond method for calculating spinspin coupling constants without recourse to the assumption that the atomic orbitals are orthogonal. The ground state wavefunction is expressed as a linear combination of independent canonical structures vJ, that is, set-I.
t = COY0 +ccJY,
(5.11)
J*o
where y. is the canonical structure in which all bonds are chemical bonds. Substitution of equation (5.11) into Ramsey’s equation for JsNNr (equation 3.215) gives
The non “perfect pairing” structures may be divided further into those involving the HH’ bond (groupj), and those which do not (group k), and described respectively by the functions and coefficients vJ, c, , yk, ck; equation (5.11) may now be written as II (5.13) + Tc,y, + cckyks . P = cOyO k>n The HH’ bonded structures are by far the most important (5.12) since T2 wJ c W,) 6 (rmaJ St * % (
terms in equation
I I, m
where T is the order of the overlap integrals between atomic orbitals associated with nuclei which are not chemically bonded to one another, and T 5 10-r. Thus the principal terms in equation (5.12) are
The neglect of terms in yk introduces an error of about 20 per cent when the number of bonds does not exceed ten. Equation (5.14) was evaluated for J HHnin methane adopting a A E value of 10 eV, and using the exchange integrals evaluated by Van Vleck(17’. The value of JHH, obtained”@ was + 11.9 cycles se&; if a A E value of 9 eV is chosen, corresponding to that used by Karplus and AndersorP, then a value of + 13.2 cycles set-’ is obtained. Further discussion of the limitations of the valence bond approach is given in Section 5.3.2.
166
H I G H R E S O L U T I O N NMR S P E C T R O S C O P Y
5.3 E ~
A~'D ETHYUm'~
The H - H coupling constants of interest in these molecules are those between hydrogen nuclei on the same carbon atom (geminal hydrogen nuclei) and between hydrogen nuclei on adjacent carbon atoms (vicinal hydrogen nuclei). Theoretical calculations of these coupling constants made so far are general H
H
in that they consider the fragments \ C - C /
/H and C \ H . Both the valence bond
and the molecular orbital treatments of the problem predict that there is a correlation between the vicinal H-H coupling constants and the dihedral angle between the C - - H bonds. Experimental data for unsymmetrically substituted hydrocarbons are in good agreement with the theoretical relationship. The experimental Jn H' values of most theoretical significance are those of the parent molecules, ethane and ethylene. However, the coupling constants are not obtainable directly because the hydrogen nuclei are magnetically equivalent (see Section 8.12): the required magnetic non-equivalence can be introduced by either deuterium or carbon-13 substitution.
5.3.1 Vicinal H-H Coupling Constants One of the main contributions to the calculation of vicinal coupling constant has been made by Karplus c4' 18~ who extended the valence bond approach. The magnitude of JH n' is againdominated by the contact term J3 n n', and it is assumed that the other terms in equation (3.21 I) can be neglected. If the wavefunction is expressed in the form given in equation (5.13), so that v2ois the perfect pairing structure, ~pj the non-perfect pairing structures involving an H - - H ' bond, and W~ the non-perfect pairing structures without an H - - H ' bond, then only cross terms between ~o and ~j have to be included in the calculation, provided that co >> c~, ck. Terms between v2o and ~h make no contribution to the coupling constant. Among the valence bond structures ~pj, there are some having two " b r o k e n " bonds, three " b r o k e n " bonds and so on. Structures having two broken bonds, and occasionally those with three, can be neglected since their contribution to the ground state function ~pis small. Furthermore, because the calculation of JHH' using equation (5.9) is concerned with evaluation of the coefficients co and cj appearing in equation (5.13) and not with the total energy, the electrons and orbitals associated with the same valence bonds in all of the selected structures can be neglected. The fragment I'Z\C--C'./H' in ethane and ethylene was considered as a six electron, six orbital problem of the form
A\
/B /co,
i
C~
Cbz
where A and B are I s hydrogen atomic orbitals, and Col and Cb~ are directed sp carbon hybrid orbitals. The perfect pairing structure is thus written
~0 = (A, C°t)(C°2, C~z)(C~, B)
(5.15)
CALCULATIONS
OF SPIN-SPIN
COUPLING
CONSTANTS
167
and the two structures with A B bonds are
~ot -- (A, B)(Co,, Csl)(Co2, Cb2)
(5.16)
~2 = (A, B)(C°,, C.2)(C~, Cb2).
(5.17)
There are five canonical structures for a six electron system, the remaining two being ~3 = (A, Cb~)(C°~, C.2)(C~,, B) (5.18)
(5.z9)
~o, = (A, C°,)(Co2, B)(Cbt, Cb2). These canonical structures can be represented schematically as
HA
He
HA"
*,
*o HA~
-He
*a
H8
~3
O,,
Superposition of these canonical forms gives the coefficients ij,~ andfj,~ appearing in equation (5.9) (see Section 3.14.3). The coefficients Co and cj were determined by solving the secular equation ctg> (s~',j - ~,j E) cj = 0
(5.20)
where •~Ijis the integral f ~ot ~ ~j d3, and Stj is the overlap integral f ~ ~oj d3. Karplus c+> included only nearest neighbour atomic integrals in equation (5.20), neglecting integrals other than those of the form [A, C°l], [A, Ca,], [Col, Co2], [CoI, Cb2], [C,1, C~1] and [C,2, Cb2] where IX, Y] = f x ( 1 ) Y(2) .,~'Y(1) X(2) d 3. The exchange integrals [A, Col] and [A, Co2] are dependent on the angle A Co Cb, (called 0) and the other exchange integrals: the integrals involving carbon orbitals only, are dependent not only on 0 but also on 9 the dihedral angle between the A Ca Cb plane and the Ca C~ B plane. For ethane<*), the carbon orbitals were taken to be tetrahedral ap s hybrids, and the carbon-carbon distance to be 1-543 A. The coefficients Co and cj were calculated for dihedral angles, 9~, between 0 ° and 180° at 30° intervals, and the results are shown in Table 5.1. The last column in Table 5.1 shows the values of d ~ , calculated by means of equation (5.9) in which it is assumed that the
168
HIGH
RESOLUTION
NMR
SPECTROSCOPY
values o f 9~H(0) are those o f a free hydrogen 1 s function. A E is taken as 9"0 eV. The calculated J e w values were found to fit approximately a c o s 2 ~ function jvlc
uw = 8"5 cos2tp -- 0"28 (cycles see - l ) = 9"5 cos2~0 -- 0-28 (cycles see -~)
0 ° ~_ ~0 =< 90 ° 90 ° ~ ~0 ~ 180 °
The asymmetry o f the curve about 9~ = 90 ° was attributed to approximations involved in calculating the exchange integrals [C,~, Cbl]. T~aBt~ 5.1 VICINAL H - H Spnq--SPrN CotreLr~G CONSTANTS FOR VARIOUS D n - n ~ a ~ IN ETHA~'~ Dihedral angle
3~ 60 ° 90 ° 12~ 150 ° 180 °
Co
Ct
C2
1"013 1"015 1 "020 1"022 1"019 1"014 1"011
0.0010 0.0012 0.0016 0.0017 0"0015 0.0011 0"0009
0.0168 0.0120 0.0026 0.0016 0.0037 0.0140 0"0191
Ca, C4 -
ANGLES
J m r ( c y c l e s se¢- 1,
0"0217 0"0217 0'0217 0"0217 0-0217 0"0217 0"0216
8"2 6-0 1"7 - 0"28 2"2 6-9 9'2
The ethane molecule exists in the staggered form (see Section 9.5.5), so that pairs of hydroge n nuclei are associated with dihedral angles o f either ~o = 180° (trans), or ~ = 60 ° (gauche), and there are two coupling constants, J , , ~ and Jg~k,. However, the rate of reorientation between the three positions o f minimum energy is sufficiently rapid at room temperature for the spectrum to show only an averaged coupling constant <J). In ethane itself <0r) = ~ (2Jg~che ÷ l t , ~ ) and the value of ( J ) predicted by the valence bond calculation is ( J ) =-~(9"2 + 2 x 1.7) = 4-2 cycles sec -1. This is in poor agreement with the experimental value ( J = 8.0 cycles sec -t) measured (2°) in the IH resonance spectrum of Hz 13 C--CH3. The separate values of Jt,~, and J~,che have not been found for ethane, but it has been possible to obtain the values o f the two coupling constants in some substituted ethanes. The magnitude o f the weighted mean coupling constant ( J ) in substituted ethanes depends upon the relative populations o f the three rotational isomers, which in turn depend upon both the temperature and the solvent (see Section 9.5.5). The separate values o f J t , w and J~,ch~ can be found by following the variation o f ( d ) with either the temperature ( 2 1 ) , or the solvent ( 2 2 ) . It is found that Jt,,~ is always greater than Jga,ch, in agreement with the calculated values shown in Table 5.1, but the experimental values are considerably larger than the calculated ones. Table 5.2 shows the experimental values of J~,,,~ and Joa~h~ for a number of substituted ethanes, and it can be seen that ~,°ns " Jan"
10-18 cycles see -~, whereas
r gH*H~" h " ~
'~'
1 - 3 " 5 c y c l e s s e c -1.
CALCULATIONS OF S P I N - S P I N COUPLING CONSTANTS
169
The same pattern of [3,m.I > IJe,=¢~.[ is found in some ring compounds which contain the ethane-like fragment. HCCH'; thus in some acetylated sugars L e m i e u x et al. tza) f o u n d that IJ,,-.=l was 5-8 cycles sec - I a n d that ]30,,~,1 varied over the range 2-4 cycles sec -~. TAeL~ 5.2 VICtuAL H - H COUPUNO CONSTANTSIN Su'l~nruTl~ ] [ ~
J~"
Compound :HCI~CHCI~ .'HCI2CHF2 ~HCI2CHFCI ~ICH2CH2CI ~2CHCH~CI )rC'H~CH2Br /CH=--CH~x~
J~"
(cyc~ sec-~ )
(cycles sec-1)
+ 2"01 -4- 0"08
÷ 16"08 ± 0"8
+ 2"01 -4- 0.09 +1.63 ± 0"15 - 2"62 -4- O'65 1"2 -4- 1.5 3"0 -4- 1.5 3-4 -4- 1.0 2.5 -4- 1.5 2.7 4- 0.2
+ 10"25 -4- 0"4
[ Refmence
I
21 21 21 22
+ 16"5 -4- 0"3 18"0 4- 4"0
10"5:1:3"0 15"3 -4- 1-0 16"5 -4- 2-5 9.4 -4- 0"2
It would be unwise to carry too far the comparison between the vicinal coupling constants in substituted ethanes and the values calculated by Karplus for the H ~ - - H fragment, since the calculation makes no allowance for the effect of different substituent groups, nor does it allow for deviations from the tetrahedral value of the H---C--C angle. In calculating J . w for ethylene, the carbon sp 2 hybrid orbitals were assumed to form angles of 120° with one another, and the carbon-carbon bond length was taken to be 1-353 .~ An exactly analogous calculation to that for ethane can be made, by writing the ground state wavefunction as a linear combination of the five canonical structures; however, such a calculation completely neglects #-oz interactions, and gives only the #-bond contribution to the coupling constant. The coefficients c# are found by solving the secular equation and are given in Table 5.3 together with the calculated values for dihedral angles of = 0 ° ( c b ) and ~ ffi 180° (trans) (4" is). TABLE5.3 a-BOND C o r c r ~ t c n o m TOTI~ H - H SPn~-Spm COUPLINGCOmTA~rS m E T H Y t ~ cycles sec-1 0°(cis) 180° (tram)
I
1.001 0.995
[
0.0004 0.0002
I
0.0145 0.0252
I-0.0080 -- 0.0080
]
6.1 11.9
The inclusion of #-~ interaction leads to a contributions to J . a ' via the ~electrons in the H - - - C = C - - H fragment. Karplus (2.) has shown that the magnitude of this contribution is J.n'oo = 2"1 x 10 - i s . a .A~ 'as, 6a
(5.21)
170
H I G H R E S O L U T I O N NMR S P E C T R O S C O P Y
where as and an, are hyperfine interaction constants for the radical fragment H--C--C, and A~z is the energy difference between the singlet and triplet n-electron states. For ethyleneauandA~ were taken to be - 6 5 x 10e cycles sec-1 and 6.0 eV respectively, and substitution into equation (5.21) gives Ja H,(~) -- + 1.5 cycles sec -1. The total calculated coupling constants for ethylene assuming that Jan, "" J3un' (c.f. equation (3.211)) are 13.4 cycles sec -~ for J ~ , and 7.6 cycles see -l for J ~ , . The values observed for ethylene are cz°) 19-0 and 11-7 cycles see -t respectively for J~r~ and J ~ r ; both these values are some 50 per cent larger than the calculated values, but the observed ratio of Jt,o~/Jct~ of 1"62 is in good agreement with the calculated ratio of 1.76. Conroy c25~ has used the molecular orbital approximation to calculate the contact term J3an, for the vicinal hydrogen atoms in the H - - C - - C ~ H ' fragment. The results of this calculation are essentially in agreement with those obtained from the valence bond method, giving the same dependence on the dihedral angle, except that for ethane the valence bond calculation predicts a negative value for J3aa, when ~0 - 90 °, whereas the molecular orbital calculation for this angle gives a zero value for J3 n n,, and also predicts a positive coupling constant at all other dihedral angles. Karplus C~t') has pointed out the dangers of attempting structural analyses based on the dihedral angle dependence of vicinal coupling constants. There may be effects from substituents of greater electronegativity than hydrogen, from CCH bond angle changes and from bond length changes.
5.3.2 Geminal H-H Spin-Spin Coupling Constants in CH2 Groups The valence bond approximation has been applied to the calculation of geminal H - H coupling constants in the HCH fragment t26~, by a calculation similar to that already described for the vicinal coupling constants in the H - - C - - C - - H fragment. Unfortunately, the method produces values for J ~ in poor agreement with the experimental data. The molecule XYCH2 was regarded as a four electron, four orbital problem and the effect of substituents was neglected. A four orbitalproblem involves only two linearly independent valence bond structures,
vl = (c~, H0(C2, H 0 / ~2
(Ct, C2)(Ht, H2)
/
(5.22)
which may be represented schematically as c
where tpl is the perfect pairing structure. The carbon hybrid orbitals C~ have the form Ck = 1 - cos0) -~/2 [C2pok + i(cos0) 1/2 C25] (5.23) k=l,2
CALCULATIONS
OF S P I N - S P I N
COUPLING
CONSTANTS
171
where 0 is the H C H bond angle, and C2p.k is a 2 p o-bond directed along the C - - H ~ bond. The wavefunction for the system is #oven by ~0o = ct ~
+ c2 V'2
where c~ and c2 are obtained by solving the secular equation (equation (5.20)). The exchange energy matrix elements .,~:~ and the overlap matrix elements S~: in equation (5.20) are given by •,X~'xx = 2K(Ct, Hx) - K(C2, HI) - ½K(C~, C2) - ½K(H~, H2); •,X~"22 -- K(C1, C2) + K ( H I ,
H2)
-
K(Cx, HI)
-
K(C
2, Hi);
•gg'~2 = K(Ct, H~) + ½K(CI, C2) - ½K(HI, H2) - 2K(C2, Ht);
Sl¢ = ½
S22 = 1 (5.24)
where the terms K(C1, H1) etc. represent exchange integrals K(CI, HI) -- f C1(1) Hx(2).,WH1 (1)CI (2) dT.
(5.25)
Substituting for the C~ with equation (5.23) enables each of the four exchange integrals K(CI, Hx), K(C2, H1), K(C~, C2) and K(HI, H2) to be expressed as functions of the angle0. Gutowsky, Karplus and Grant c2e~solved the secular equation for values of 0 at 5 ° intervals between 100° and 130°, and the resultant values of C1 and C2 are shown in Table 5.4. The contact term J , x w can now be evaluated from eqtzation (5.9), provided that the average excitation energy is known. If the A E value is taken as the average for the three triplet states, J3 n H0 has the values shown in column four of Table 5.4. T ~ I . ~ 5.4 CALCULATED G ~ m ~ H - H SPIN-SPIN COUPLING CONffFANT$ AS A FUNCTION OF ~ H C H BOND ANO~S ¢2e~ H C H Angle
(degrees) 100 105 110 115 I20 [25 130
C,
C2
0"968 0-979 0"987 0.992 0"997 1.000 1 "003
0"062 0"040 0"025 0"015 0"0067 - 0.0001 - 0"0059
~E~
.~r3Hl1,
(cyclessec-I)
8"23
32"3 19"7 12"1 6"91 3"05
8"60
8"77 8"99
9"05 9"09 9"14
~0 -
2"62
The terms J~t~,, dCx2)m~, and J 2 s w will make negligible contributions to J s w , hence the dependence of the geminal coupling constant on the angle 0 should be the same as that of J3 s w. Some values of J ~ in substituted methanes CH3X are shown in Table 5.5 c2~-'9). If, as seems probable, the absolute signs of all these values are negative then there is poor agreement between theory and experiment. The variation in J ~ probably reflects the direct effect of the substituent rather than alterations in the bond angle alone. 6at*
172
HIGH RESOLUTION
NMR SPECTROSCOPY
Barfield and Grant t37) have pointed out that the valence bond calculations are extremely sensitive to one of the exchange integrals, K(C2, H~), between a carbon hybrid orbital and a non-bonded hydrogen orbital. This exchange TABLE5.5 GEMINALH - H COtrPLING CONSTANTSIN SoaSTrrtrrm~ MaTHAN~S (Obtained by deuterium substitution) Compound
J * ~ (cycles sec -x) H'H
CH,
Reference 12 27 28 29 28 27 27 27 27 27
12'4 13"0 14"9 15"5 13"2 10"8 9"6 10"8 10"2 9"6
CHaCCla CHaCOCHa CHaC6Hs CHaNOz CHaOH CHaF CHaCI CHaBr CHaI
integral is determined from the differences of crude empirical exchange integrals o f the form described by Van Vleck t17). Barfield and Grant estimate that an error o f only ten per cent in the values of these integrals could give an error as large, as + 30 cycles see -1 in the geminal coupling constant. Thus valence bond calculations of such coupling constants are not feasible until a better method o f estimating K(Cz, H 0 is available. Barfieldand Granttlla) have made a semi-empirical calculation of the contribution to the geminal coupling constant by an adjacent z~-electron bond. The contribution was estimated to be - 1"5 cycles see -~ compared with the experimentally observed value o f about I 1"91 cycles see -~ for each .~-electron bond.
5.3.3 Relative Signs of Geminal and Vicinal H - H Coupling Constants The valence bond calculation predicts that for 0 < 125 ° the vicinal andgeminal coupling constants have the same sign, but in nearly all cases measured ~a and J ~ are in fact of opposite sign. The measured values o f J ~ and jge,, jv~c HH for some substituted ethanes are shown in Table 5.6. The bond angles in these compounds are not known but they are unlikely to be greater than 125". TABLE 5.6 GEMINAL AND VICTNALH - H COUPt~G CONSTANTS IN SOME SUI~TITUTED ETHANES j g.m
Compound CHaCHz0(S0zCHzCH3) CHaCH20 [CH(CHa)0CH±CHa] C H z - CHCH2CI \0 7 CHBr(C00H)CH2Br
m-x
(cycles s ~ -x ) -T 10"45 q: 9"30 :F 11"7 ¥
9"9
j ~c
~
(cycles soz-I)
Reference
±4"0
30 30 31
± 6"6 ± 4"6, -4- 10"9
32, 108
± 7"12 _ 7-03
CALCULATIONS
OF S P I N - S P I N
COUPLING
CONSTANTS
173
The substituent groups will change the values of J ~ " and J ~ from those in ethane, but they are not expected to change their signs. J ~ ' and J~,,~ have been measured for one molecule for which the bond angles are known with certainty. The molecule is 2,2-metacyclophane, whose structure has been determined by an X-ray analysis of the solid compound (33). The two methylene groups are locked in the virtually symmetrical, staggered positions shown in Figure 5.1, and the H C H angles have the tetrahedral value. Gutowsky and Juan (3") have carried out a detailed analysis of the A A' B B' type of spectnJm for the - - C H 2 - - C H 2 - - bridges in the molecule. They have shown that J ~ and r" E~U R have opposite signs, the values being J ~TTon$ . = +_ 12"3 cycles sec -~ j.o,~h~ ""
{ + 3"2 cycles sec-~ =
_+4.0cycles sec -~
J ~ . = ~ 12"0 cycles sec-L
~
(
J.
2o.._ B
~>t.
L~.
\
,'
(
B
Fio. 5.1 The molecular skeleton of 2,2-metacyclophane.The hydrogen atoms in the -C'HeCH~- groups are labelled A and B. Gutowsky and Juan(s4)
The opposing signs of the coupling constants in this molecule cannot be attributed to angular distortion or internal motional effects. It is probable that the geminal coupling constants in all substituted methanes and ethanes have the same sign. This is apparent (27' 35) from a consideration of the values of J ~ ' shown in Tables 5.5 and 5.6. The magnitudes of J ~ in the fragments C--CH2---O and C--CH2--Br are similar to those in CH3OH and CH3Br respectively, hence it can be assumed that the effect on J ~ ' of replacing a hydrogen with a carbon atom is small and will not change the sign of J ~ . Thus the J ~ ' values in all these compounds can reasonably be assumed to have the same sign. Furthermore, the gradual change in IJ ~ l in the series CH3F, CH3OH, CHaNO2, CH3CCI3, CH4 suggests that its sign is the same in all these molecules. Of the two possibilities, J ~ positive and J ~ negative, or J ~ negative and J ' ~ positive, the latter has the more convincing theoretical justification. McConnelP e) has shown that applying the Dirac vector coupling model to the problem of spin-spin coupling leads to the conclusion that J ~ is negative in
174
HIGH RESOLUTION NMR SPECTROSCOPY
methane, and also that JHH' will alternate in sign, being negative if the twohydrogen nuclei are separated by an even number of bonds and positive otherwise. A negative J ~ in methane means that the vaience bond calculation is in error. If the mechanism of the coupling between two directly bonded nuclei is dominated by the Fermi contact term then the sign would be expected to be positive. Hence, it has been suggested° ' ' 3s. 39) that the absolute signs of J ~ and r°~, "HH could be deduced by determining their signs relative to that of J,3cH between directly bonded nuclei in the same molecule. The double resonance method of determining the relative signs of coupling constants has been used by Lauterbur and Kurland ¢4°)to showthat J13c~ and J ~ are of the same sign in the molecules 1,2-dichloroethy!ene and 1,2-dichloroethane; Anet c4~)has used the same method of sign determination to ~how that J,3c~ and -~rorg"'(and hence J~,~) are of opposite sign in the molecule CH,DOH. This appears to be reasonable proof that . / ~ is positive in ethanes and ethylenes. However, this assignment depends upon the absolute sign of J~cR being positive, and there is some evidence that this is not necessarily correct.t The argument for a positive coupling constant between directly bonded nuclei means that J~cn and J,3c~ should always have the same sign, but Tiers °s) has found that these coupling constants are of opposite sign in the molecule CFHCl2 where the values of the coupling constants are J13c~ = ±220"0 ± 0.I cycles sec -1 Jt3cv = ~=293-8 ± 0.2~cycles s e c - 1 JHF = ~ 53"65 ± 0"16 cycles sec -~. The available experimental evidence points to a negative geminal coupling constant, J ~ , in substituted ethanes which suggests that the predicted variation of J ~ w i t h the HCH angle, 0, is incorrect. But J~'~ may still be angular dependent, changing in sign for some value of theHCH bond angle, since in ethylene (2°) and some substituted ethylenes(42-4") J ~ and r,t, -ma are of the same sign, suggesting a positive geminalcoupling constant. 5.4 COUPLING CONSTANTS BETWEEN HYDROGEN NUCLEI SEI"ARATi~ BY
F o u r a-BONDS Spin-spin coupling through four a-bonds has been observed in a number of molecules, but as yet no detailed theoretical predictions of their magnitude have been made. Karplus (24) has suggested that the magnitude of the spin coupling through four a-bonds should be £ 0.5 cycles sec -t, whereas the majority of the observed values lie in the range 0,3 to 1.5 cycles sec-t (see Table 5.7). There is some evidence that these long range coupling constants are strongly stereospeciflc, thus, in methacrolein dimer <47) H~O
LI
lr
~/CHa
)\CliO
I:
cI%/j~/\H It t Sc¢ footnote on page 162.
It
CALCULATIONS OF SPIN-SPIN COUPLING CONSTANTS
175
the aldehydic hydrogen nucleus couples with only one of the hydrogen nuclei on the/~-carbon a t o m (J -- 1.3 cycles sec-:). Similarly in 1,2-dibromoisobuty. rate CH2Br Br /C~ CI-I., COOCH3 TABLE5.7 COUPLINGCONSTANTSBETWEENHYDROGL~NUCt~ SLn'XXXTED BY FOURO-BOm>S Molecule
./m~(cycles sec - 1 )
meso-2,3.Dibromobutane (4s) me$o-2,3-Diphenylbutane(as) d/-2,3.Diphenylbutane (4s) ds-2,2,4,5-Tetramethyldioxolane ('e) 1,2-Dibromoisobutyrate (4~) Methacrolein dimer(4~)
0-26t -0"29 -0.20 -
+ 0-55
+ 0.75 +1.3
1" Negative signs are assigned relative to a positive vicinal coupling constant. the C-methyl hydrogen nuclei couple to only one of the methylene nuclei (4" ( J = 0.75 cy0es see-l). Both these results suggest that the coupling constant depends on ~, the dihedral angle between the two C - - H b o n d directions, but there is conflicting evidence about the exact f o r m o f the angular dependence. Anet ('s) found that in the six-membered ring compounds, diastereo isomers o f camphane-2,3-diols, the H - - C - - C , - - C - - H coupling constants are in the range 1 to 5 cycles sec -1 when the five atoms are approximately planar, that is e = 180 °, and zero otherwise. An exceptionally large value for a four-bond coupling constant J ^ , -- 7 cycles sec -~ was found in the molecule exo-5-chloro-bicyclo[2.1.1 ]-hexane-exo-6-t-butylcarboxamide, H
{A) H
_CL
H
ONHC (CH3) z
H
-
~(s)
in which ~ ~- 180 ° for the two bonds C - - H ( A ) and C---H(B). In the isomer exo-5-chloro-endo-6-amide in which ~ ~ 60 ° the value o f JAB is approximately zero (49). This suggests that the four-bond coupling c o n s t a n t has a m a x i m u m value for 9~ ffi 180 °, as is the case for J ~ . However, Bothner-By and N a a r Colin (45) found that in 2,3-disubstituted butanes the magnitude o f the coupling constant for the fragment C H 3 m C - - C - - H is almost zero when f = 180 ° and approximately 0.30 cycles sec -1 when ~ = 60 °.
176
HIGH RESOLUTION
NMR SPECTROSCOPY
Only in the case o f some 2,3-disubstituted butanes ('s) have the relative signs o f J ~ and the four bond coupling constant, J a n , been measured, where they are found to be o f opposite sign, in agreement with the Dirac vector coupling model. 5.5 LONG RANGE H - H COUPLrNG CONSTANTS IN UNSATURATED COMPOUNDS There are m a n y known examples of long range coupling constants between hydrogen nuclei separated by four or more bonds when one or more of these bonds are unsaturated (see Tables 5.8, 5.9 and 10.44). TABLE 5.8 LoNo KANGE H - H COUPLING CONSTANTS IN OLEFI/~ICMOLECm.~
XH nuclei separated by four bonds
jc~ ma (cyclessea-t)
CH;:CH.CHs CH2:-CH.CH2CH3 CH2: CH.CH2C2Hs C6HsCH:CH.CHs CHs.CH: CH.CH3 CHCI: CCICH2CI CHCI=CH.CH2CI CH2= CH.CH2NHz
jtr,~ rm (cyclessec-t) --1"3
-- 1"7
1"7 -1"5 - - 1"8 - 1"8 ~0"4 --0"5 --1"8 --1"6 1"2 1"3 1"6 -
(CHO).CH=CH.CH3 (COOH)CH-CH.CH3 CHsCH= C(CH3)COOH (COOH)CH= C.(CHs)COOH C~HsCI'I=CH.CH3
CH3CH=C(CHDCOOCH3
--
-- 1"2
0"8 -0.9 -1.7 1"4 1"6 1"4 1"4
1"2
CHsCH=C=CHCI
1"3
--5"8
Reference 50 50 50 51 52 53 53 54 55 56 57 58 59 57 61, 62
XH nuclei separated by five bonds CHsCH- C(CHs).CHO CHsCH- C(CHs).COOH CHsCH-C(CHs).COOCHa CHsCH- CH.CHa CHsCBr= C(CHD.COOCH3 C~Hs. C(CHs)- C(CHa). COOCH3 (CHs)2C=C=CH2 ( ~ s ) 2 C l
e l
+ 1"0 1"2 1"2 1"2 1"6 1"0
1"5 1"5 1"2
3"03 4- 0.06 2"14 4- 0-08 +2"4 4- 0"1
~
CHaCH--C- CHCI
60 57 57 52 55 58 61 61 61, 62
The experimental results have been reviewed by Hoffman and Gronowitz (ss), who suggest that the long range coupling in unsaturated molecules arises almost entirely f r o m hypereonjugation between the n-electron system and the electrons in the C - - H a-bond. Karplus cz4) has applied the valence b o n d approximation to calculate the n-electron contribution to long range coupling constants via a--~ eontiguration interaction. H e showed that .Inn',, can be expressed as Jaw,, = 2"1 x 1 0 - ' s ~. {an(T) T
aw(T)}IAE~(T ).
(5.26)
CALCULATIONS OF S P I N - S P I N COUPLING CONSTANTS
177
ax (T) and a., (T) are hyperfine interaction constants in cycles sec -x, and A E. (T) is the triplet state excitation energy for the ~r-electron, expressed in electron volts, a.(T) is a measure of the extent to which the spins of the hydrogen nucleus and an unpaired electron in a carbon 2p~ orbital are coupled together in the radical fragment T. The values of a. (T) for some radical fragments measured from their ESR spectra are shown in Table 5.10. In Table 5.10, ~v is the angle between the H - - C - - C plane and C ~orbital axis. In some molecules, the coupling of the spins of the two nuclei H and H' TAm.E 5.9 LoNG RANaE H - H COtWL~O CONSTAm~ n~ Ac~ow.n~c Co~r~ov-~s 1H nuclei separated by four bonds
Coupling constant
(cycles f~-l) 2"9-3 "6 2.6 2'7 2"8
CH-CCHs CH~CCH2CI CH--CCH2Br CI'I,,C~2I CI'I - C C H 2 0 H CH-CCH2CHs
2"4 --2"4
63-65 66 66 66 55 77
zH nuclei separated by five bonds +2"6 2-7
C[-][sC~=CCH,CHs CHsC,-CCH~, CHsCmCCH20H CHsC~CCH2CI HC=CC-CH
2"5 2"2:1:0"2
78 67 55 55 61
1"274- 0"05
61
1"3 -I-0-I 1"1 1"0 4- 0"2
61 55 61
0"4
61
2"4
~H nuclei separated by six bonds CHsC--CC--CH aH nuclei sepuratcd by seven bonds
CI'I3C--CC"CCH~ CI']sCmCCICCH20H CICH=C-CCmCCH2CI 1H nucJei separated by nine bonds
CHsCmCC-CC-CCH~OH
TAm~5.10 H
~
b r r e ~ c n o N ComTAtCm aE (T)
Radical fragment
H-~ (.V') I-I-~ (,V,') H - C - C (sp and sp 2) CH3-C-C
(cycles ~c -~) - 65 x 10 e - 9 5 x 10e + 150cos2~i× 106 +75 x 10e
178
H I G H R E S O L U T I O N NMR S P E C T R O S C O P Y
by ~--~ interaction may proceed by more than one path; for example, in allene (2"), CH2=-C=CH2, there are two ways in which the nuclei can correlate their spins, as illustrated in Fig. 5.2, and each path mustbe included in the summation in equation (5.26). Other molecules such as butatriene have only one path for ~-zc interaction. In applying equation (5.26) to calculate J..,,, for some ethylenic and acetylenic molecular fragments, Karplus (2") took the values of d E,, to be 6.0 eV for ethylenic, and 8.0 eV for acetylenic fragments; the results shown in Table 5.11 are in good agreement with the observed long range coupling (see Tables 5.8 and 5.9). TABL~ 5.11 CALCULATEDVALUESOFJX~n FORSO~,~ UNSATURATED HYDROCARBONMOLECULAR FRAO"
~ r r s (I¢~a~L~ (2")) Molecular fragment
J m r n cycles sec -~
H-C=C-C-H' H-C-C-C-H' H-C=C=C-H' H-C-C=C-C'H' H-C-C~C-C-H' H-C=C=C=C-H'
1"7 -3"7 -6"7 + 2"0 +2"9 + 7"8
- -
Where the sign of the coup/ing constant has been measured (relative to a positive vicinal coupling constant) it is found to follow the expected pattern of negative values for nuclei separated by an even number of bonds, and positive values when they are separated by an odd number of bonds. The valence bond theory also predicts that for both c/s and trans configurations of the molecular fragments H - - C = C - - C - - H and H - - C - - C - - C - - C - - H substitution of a methyl group for a hydrogen atom changes the sign of the long range coupling constant and the magnitude remains approximately the same. Differences between the longrange coupling constants in molecules containing these fragments probably arise from some residual a-bond coupling. Because of the angular dependence of a. for the radical fragment H--C--(~, if a ring structure or some other form of constraint causes the planes containing the HmC--(~-- and C = C - - H fragments to be in a fixed orientation relative to one another, then J a . will be a maximum when the two planes are mutually perpendicular (~ = 0°), and zero when they are coplanar (¢ = 90°). This can be used to explain the abnormally large value of d a , = 2"5 cycles sec -t found between the methyl and methylene hydrogen nuclei in the molecule 5-methyl-4-thiolen-2o n e (55). H
H
C--C--H CHa--C C=O ~S//~
CALCULATIONS OF SPIN-SPIN COUPLING CONSTANTS
]79
One striking feature of long range coupling constants in unsaturated hydrocarbons is that the magnitude of the coupling is not strongly attenuated as the number of bonds between the two hydrogen nuclei increases. Thus, the coupling constant between a hydrogen nucleus in the fragments H - - C = C - - or H---C-= C--, and some other hydrogen nucleus, H', in the molecule, has almost the same magnitude as Ju H. between a nucleus in either of the fragments C H s - - C = C or CH3--C-- C and the same hydrogen H'. This is consistent with the mechanism of ~-~ interaction proposed by Karplus (24), which predicts that methyl substitution should not change the magnitude of the above coupling constants, but -65x I0 s
+150 xlOS
+ 150 x I0 ~ (o)
+ l S b x I0 s
+ 150 x I06
(b)
Fzo. 5.2 Hydrogennuclear coupling in (a) allenc and Co)butatriene. The arrows repr'-~-nt the alim~ment of electron spins; (--) indicates the =-bond pairing of electrons; and (---) the ~-:~perturbation interaction of electrons. Kaxplus(a4) should change the sign. This arises because of the relative magnitudes and signs of the hyperfine interaction constant, a . , for the radical fragments H - - C = C and H - - - C - C on the one hand, and CHs----C=.C and CHs---C= C on the other (see Table 5.10). For a freely rotating methyl group the values o f a s are similar in magnitude but opposite in sign. Hoffman and Gronowitz <ss) have suggested that methyl substitution could be used to test whether the principal contribution to a particular long range coupling constant in a molecule is determined by ~-~ interaction. I t has been suggested by Holmes and Kivelson (6s) that the ~--~ electron interaction mechanism for coupling nuclear spins is responsible for the long range coupling observed between the methyl hydrogen nuclei in acetone and acetoxime, (CH3)2 C = N O H , even though the direct through-bond path between the nuclei does not involve an unsaturated bond. In both these molecules the hydrogen resonance spectrum consists of a single sharp line from the six equivalent methyl hydrogen nuclei. On each side of this single line there are weak
180
HIGH
RESOLUTION
NMR
SPECTROSCOPY
taCH satellite peaks at + 63.6 cycles scc-t comprising a weak 1 : 3 : 3 : 1 quartet of lines arising from the methyl nuclei in the 2.2 per cent of acetone molecules containing one laCHa group. In the case of acetone the separation of the quartet components is 0.54 + 0.05 cycles sec -t. The contribution to the methyl-methyl coupling constant from a or-bond mechanism is probably less than 0.2 cycles sec -t since the t3CH satellite lines in similar compounds not containing unsaturatedlinkages, (such as (CHa)2CHOH) are single sharp lines(68)2 The methyl coupling in acetone is thought to arise from coupling of the two groups of methyl hydrogen nuclei with the same zr-electron in the C = O bond. A similar approach to that used by Karplus C2.) gives JM.,,~in acetone as - 2 . 1 x 10-1s22(a,)2 J..,~ =
A E. (C = O)
(5.27)
where aH is the hyperfine interaction constant for the radical fragment CH3--C, 22) is the ionic character of the C = O bond, and A E= (C=O) is the excitation energy for the triplet state (CH3)2C---O. Comparing equation (5.27) with equation (5.26) it is apparent that the methyl-methyl coupling constants in the molecules acetone and 2-butene, CH3CH=CHCH3, should be related to one another by (1 -
JH, (acetone) = -- 22/i E.~(C= C) . dnn (2-butene). E~(C =
o)
The value of the long range coupling constant dHn' in2-butene (57) is 1"2 cycles sec -t, thus withdE~ (C=C) = 6.0 eV a n d d E t (C=O) = 6.3 eV, the value of (1 - 22)iscalculatedc~8)to be 0.53, in reasonable agreement with the values given by Paulingt~9) of 0.42 to 0-52. The indirect ~r.electron contribution to the coupling constant present in the acetone molecule should also affect the magnitude of the coupling between hydrogen nuclei in other molecules containing unsaturated bonds. Holmes and Kivelsonc~a) have estimated that the geminalHH coupling constant in ethylene and vinyl derivatives contains a contribution of - 1.5 cycles sec -t from the indirect =-eiectron mechanism. It is also possible that the unusually largo long range H u H coupling constants found in some unsaturated aldehydes(j*) arise in part from indirect as well as direct ~¢-eiectron effects. 5.6 AROMATICHYDROCARBONS The values of the coupling constants between the ring hydrogen nuclei in substituted benzenes have been found to lie in the ranges J ~ * = 6 to 9 cycles see-t, j ~ a = 1 to 3 cycles sec -1 and dp~a = 0 to 1 cycles sec -1. The absolute sign of d ~ ° is positive in p-nitrotoluene (t12). Where the relative signs of the coupling constants have been determined it is found that J ~ ° and -l=°*a -tr~ are of the same sign. Both g-bond and ~r-bond coupling mechanisms are important in determining the magnitudes of the coupling constants between ring hydrogen nuclei in aromatic hydrocarbons, hence JHH"may be written as J.., = J..,(a) + J..,(~).
(5.28)
CALCULATIONS
OF S P I N - S P I N
COUPLING
CONSTANTS
181
j ~ o (a) is probably comparable in magnitude to the calculated value of J ~ ( a ) in ethylene of 6.1 cycles scc -l. The magnitude of JaW(a) decreases rapidly as the number of bonds separating the two nuclei increases and if one assumes that the G-bond contributions are comparable with those in saturated hydrocarbons, then J ~ ' ( a ) is probably not more than 1 cycle sec -1, and J~m~'~(a) is almost certainly negligibly small. The values of dn,,~ for aromatic hydrocarbons have been calculated by McConnellCTO. 71~using the valence bondmethod and including a-n configuration interaction. The calculation invokes the aid of electron-nucleus hyperfine interaction constants, and the expression for dn,,,~ is similar in form to that developed by Karplus for JM,',~ in non-aromatic molecules. MeConnell c7t~ showed that JHM'~ is given by J..,. =
-
(g/~Q)ZANN,
~
(5.29)
where Q is the isotropie hypertine interaction constant in gauss for the aromatic free radical,/~ is the Bohr magneton and g the electronic g-factor. The value of (2/O A E h)) (g • Q)2 is approximately unity, thus JHl'l'n '~ - - A N N "
(5.30)
ANN, is a spin correlation matrix which describes the correlation between the n-electron spins on the two carbon atoms N and N ' directly bonded to the hydrogen nuclei H and H'. ANN, has, the form
where ~o is the ground state wavefunction, written in the valence bond method as a sum of valence bond canonical structures ~0~, ~o = ~ cl ~'l. l
McConnell C71~showed that ANN' is given by --
Ct Cj ~Jij T O
C1cjTtj)
•
(5.32)
T u and ~tj are determined from the Rumer-Pauling superposition diagrams 1 (see Section 3.14.3) Tts = ~ where n is the number of bonds in the valence bond structure, and ltj is the number of islands in the ~0~,j superposition diagram. ~tj refers to the carbon nuclei N and N' and is zero when they are in different islands of the superposition diagram. When N and N' are in the same island, ~h: has the value of + 1 if they are separated by an odd number of bonds, and a value of - 1 if they are an even number of bonds apart. Thus for even alternant hydrocarbons ~Tu is positive when N and N' are separated by an odd number of bonds and negative when they are separated by an even number of bonds. If the coefficients c~ are all of the same sign then the value o f J n a , n will alternate in sign, being positive when the number of bonds separating H
182
HIGH
RESOLUTION
NMR SPECTROSCOPY
and H' is an odd number, and negative when it is an even number. The c4's are assumed to have the same sign in valence bond calculations on even alternant molecules, but there is no fundamental reason why this should always be true o f these molecules. McConnelF ~ ) has applied equation (5.32) to calculate the values o f ANN' in benzene and naphthalene; his results are summarised in Table 5.12. TAataS 5.12 CALCULATEDVALUESOV T~m SPIN C O l LATIONFUNCTION ANN S IN BENZENE AND NAPHTHALENE (McCo~,mLL(7~)) N, N I 1
©: ,$
$
1
5
4
z~NN'
1,2 1,3 1, 4
+ 0"45 -0"15 +0"15
I, 2 2, 3 1, 3 1,4 1,5 1,6 1,7
+ 0"58
1,8
2,6 2,7
+ 0"47 - 0"21 + 0"23
+O.28 -0-11 +0"I1 + 0"02 -0"02 + 0"02 --0"02 + 0"02 -- 0.02"
I" Calculated using only principal valence bond structures. # Calculated using principal structures plus three excited Dewar structures. The calculated values of/HH' (#) for the ortho and recta hydrogen nuclear coupling constants are much smaller than the observed range o f coupling con= stants indicating that the main coupling mechanism in both cases is via #-bonds. Further evidence for this is the inconsistency of the signs predicted compared with those observed. The #-bond mechanism probably dominates the coupling between hydrogen nuclei in positions para to one another; the calculated value of JHW(#) of approximately + 0"23 cycles sec -~ is in good agreement with the value o f J ~ = 4- 0"2 cycles sec -~ observed in the spectrum of 3, 5, 6-trichlorobenzene (v2), In this connection it should be noted that spin-spin coupling con= stants o f substituted benzenes are scarcely affected by the nature of the substi= tuent (variations o f about I cycle sec -~ are most common(43)). McConnell's method o f calculating the #=bond contribution to inter-nuclear coupling can be applied to nuclei other than hydrogen if the necessary hyper= fine splitting data is available from electron magnetic resonance spectroscopy. F o r nuclei other than hydrogen, the anisotropic as well as the isotropic part o f the hyperfine splitting is required. The observed isotropic splitting for the ~3C nucleus at the ~ position in the naphthalene negative ion corresponds to a Q value (~3) of 39 gauss, very similar in magnitude to hydrogen nuclear splitting
CALCULATIONS OF S P I N - S P I N COUPLING CONSTANTS
183
(O ~ 30 gauss). By contrast, the anisotropic hyperfine splittings for 13C in this radical could be about ten times as large as 39 gauss, because the presence of coniigurational interaction is necessary to give the isotropic but not the anisotropic hyperfme splitting. The net result is that ~ 3 C ~ 3 C coupling constants originating from ~r-electron interaction could be a hundred times greater than the hydrogen coupling constants given in Table 5.12. 5.7 AMMOSL~ The magnitude of the inter-hydrogen nuclear spin coupling in the ammonia molecule has been estimated by Saika (74) using the molecular orbital method developed by McConnell (2) (see Section 3.14.2). In McConnell's treatment only the coupling across three or more bonds was considered; in the present case, where only two bonds separate the nuclei, some of the integrals neglected by McConnell have to be included. The coupling arises mainly from the contact term, J3 n,,, and for the case of intermediate coupling (I.R.) Saika obtained the expression 7.754x 1 0 4 1 4 ( ~ ) 33n.,(I.R.) =
AE(eV)
la.. a~ ((cp.(s)16(rm)[qo.(s)))
× ( ~ ~., a.~ar~tj(~(s)16(r,N)l~.(s)))]
(5.33)
where the summation ~ extends over the orbitals of the two hydrogen atoms N~H
a n d the nitrogen atom. With a self-consistent field molecular orbital function derived by Duncan
,t E(eV)
"
The selection of a Value for A E presents the usual difficulties: Saika chose a value of 8-3 eV, and, therefore, J3 . . , ffi 8"3 cycles sec -1. The only other appreciable contribution to the coupling was found to be ~m~,, and adopting the same S.C.F.M.O. function this was found to have the value of 1"2 cycles sec -1, giving a total calculated coupling constant of 9.5 cycles see- 1. A direct experimental determination of the value of for ammonia is not possible became of the magnetic equivalence of the hydrogen nuclei, and the best comparison that can be made at present is with J . - c . , - 5 cycles sec -~ in RNHCH~ (74). A value for could be obtained indirectly by measuring J . v in a deuterated ammonia molecule, but it would be necessary to examine anhydrous XSNH2D rather than X'NH2D, since quadrupolar line broadening effects associated with the quadrupole moment of the I"N nucleus (spin number I = 1) would probably mask the small splitting of .., 1.5 cycles see -1 from H - D coupling.
Jnn'
Jnn'
5.8 ALIPHATIC AND AROMATIC FLUOROCARBONS
The experimental data on coupling constants involving 19F nuclei show that the magnitude of the coupling can be much larger than is the case for hydrogen, even allowing for the 7F/F. ratio; for example, the value of J ~ " in perfluoro-
184
HIGH RESOLUTION
NMR
SPECTROSCOPY
cyclohexane ¢75)is 284 + 1 cycles see -I whereas J ~ in cyclohexane is 12 cycles sec -x. Moreover, the magnitudes and signs of JFF and ,In, do not always show the same trends with changes in molecular structure. There are two aspects of F - - F spin coupling which are particularly striking. Firstly, the magnitude of J ~ in saturated fluorocarbons is often the greater when four bonds rather than three bonds separate the two interacting fluorine nuclei (see Table 11.3). Secondly, the relative signs of the fluorine coupling constants in fluorocarbons do not always follow the pattern found for hydrogen coupling constants in hydrocarbons and predicted by the vector coupling model. Thus, Evans ~76) has found that the sign of J ~ " is opposite to those of J ~ and j~m in the compounds CFz=.CFX, in contrast to the analogous hydrocarbons C H z = C H X in which Jma ' ~ " and Ym~ a , always have the same relative sign. And, as noted earlier, Tiers <3s~has found that Jt,cn and J~cF in the molecule CFHCI= are of opposite sign. If the coupling between directly bonded nuclei operates predominantly by the nucleus--electron--electron-nucleus mechanism, then the vector coupling model predicts a positive coupling constant. Therefore, it is probable that the x3C~F coupling is strongly influenced by some mechanism other than the Fermi contact mechanism. It has been suggested that electron-orbital and orbital-orbital interactions become more effective nuclear spin coupling mechanisms when the atoms concerned possessp or datomie orbitals.PopleCVg)hasestimated thecontribution to ~9F couplingconstants made by the electron-orbital interaction. He obtained an expression for J ~ , in terms of the anisotropies of the shielding tensors of the two nuclei N and N' (see Section 3.15). J ~ . = - (2/3h) 7N 7N. h = {R-3[Aa( 1 -- 3 cos=0) + Aa'(1 - 3 cosS0')] -- 3 ~ * Z, R/3(R',)-3( 1 - 3 cos=0,)}
(5.34)
where R is the distance between the two nuclei N and N', and Rt and R; are the distances from N and N' to a third nucleus; 0 and 0' are the respective angles between the symmetry axis of the shielding tensors and the line joining N and N'; 0t is the angle subtended at nucleus i by N and N'; Z~ is the magnetic susceptibility of atom i and is usually negative because most atoms are diamagnetic. If each nucleus has an axially symmetric environment then the anisotropies of the shielding tensors, Aa, may be written as A ~r = crll - a±. The final term in equation (5.34) represents interactions via a third nucleus and an upper limit on its possible magnitude may be placed by considering the extreme case of two hydrogen nuclei (Ys = ?N, = 2'68 x 104) at distances of 10-s cm on opposite sides of a carbon or a nitrogen atom (Zt "~ 1 0 - = 9 ) such that 0t is 180°. In this extreme case the contribution to J~, is 4.8 cycles sec -1 and therefore it may be supposed that in most molecules the coupling via a third nucleus is negligible. The first term in equation (5.34) depends upon the shielding anisotropies Aa, and in Chapter 4 it is seen that the shielding constants for
C A L C U L A T I O N S OF S P I N - S P I N
C O U P L I N G CONSTANTS
185
hydrogen nuclei extend over a much smaller range than those for other nuclei, and thus the magnitude o f Jm~,is much smaller than for other nuclei. Pople (79) applied equation (5.34) to the case o f F - - F and F - - H coupling. H e estimated that A~ for a fluorine nucleus in a C - - F or H - - F bond is approximately + 10 -3, and by neglecting the anisotropy o f the hydrogen nucleus one can write
(5.35)
Jr=~ = 75"2 R ~ ( 3 cos20 - 1) x 10 -24 cycles sec -1
J~,=~ = 70"8 R;3v [(3 cos=0 - 1) + (3 cos20 ' - 1)] x 10 -24 cycles sec -1. (5.36) Numerical values o f J ~ and J ~ were computed from equations (5.35) and (5.36) with the following molecular constants: H - - F and C - - H distances -- 1 A, C - - F and C = C (aliphatic) distances = 1.35 A, C = C (aromatic) distances - 1-4 A. Valence angles o f 109.5 ° and 120° were assumed for tetrahedral and trigonal carbon atoms respectively. Table 5.13 contains the results o f these calculations for a variety o f molecules. TAex~5.13 CALCULA3"~DOltSlTAL CoNTRIBUTIONSAND O]~RV~ TOTALCOUPLrN(3COwSTAN'I~ FOR SOME FLUORINE CONTAINING MOI..F.CUL~ ('/9)
Compound
J~n~(calc)eyclessec -~
HF
Monofluoroethylenes Monofluorobenzenes
gem trans cis
ortho meta para
Jm~(obs)cyclessec-x
Reference
+ 150
615
80,
+13.3
40--81
81,'82, 83
+ 12.9
+ 72-81 + 12-52 =1=1-20 =1=6-10 -*-6--8 ~:2
84, 85, 86 S4, 85, S6 84, 85, 86 87, 88 87, 88 87, 88
+ 3"6 + 0.0
0.0 + 1"0 + 1"1
jorb FF (calc) cycles sec-:
l,l-Difluoroethylenes 1,2-Difluoroethylenes o-Difluorobenzenes m-Difluorobenzcnes ~-Difluorobonzenes
trans c/s
.I
J w (obs) cycles sec-I
+ 13"3
150-270
83, 89, 90
+ 13.8
27--87 115--124 33-58 20 2--4 12-15
84 84 84 87 87 87
+5"2 - 1-8 - 1.7 + 1.6 + 1"7
Equations (5.35) and (5.36) can lead to contributions o f either sign, but for most molecules positive values result. Comparison o f the calculated values o f J ~ and J ~ with the observed coupling constants shows clearly that although the orbital contribution is appreciable it is too small to account for the total interaction.
186
HIGH
RESOLUTION
NMR
SPECTROSCOPY
5.8. I Fluoroethylenes The above results suggest that the F-F and F-H coupling constants, although containing significant contributions from Jo’*, are dominated by the Fermi contact term JsNN,. Karplus (4) has applied his valence bond method to predicting the values of JjFp and J3 HF in fluorinated ethylenes. He assumed that fluorine substitution does not change the coefficient c, of the structures in the ground state wavefunction p. , so that J3 FF and J3 NF can be calculated directly from equation (3.240) using the obtained for ethylene provided that the values of a*(O) and d E are known. For a normalised fluorine bonding hybrid orbital F, written F = a Fzs + b F,, (5.37)
YJ
cJ
where F,, and F,, are 2s and 2pa orbitals on fluorine atoms, and a and b are suitable coefficients, then ~~(0) is given by a* F&(O). Rather than calculate the value of a, Karplus estimated the value of e(O) to be approximately 2~” (0) from the value of the electron-nucleus hype&e splitting constant in a fluorine containing aromatic radical. Also, Williams and Gutowsky(gl) have found that a consistent interpretation of the coupling constants in some fluorobenzenes requires that f&O) N 39&O). Karplus used both these values, together with d E = 8-OeV (twice the C-F bond dissociation energy) for the calculation of J 3FFv, and A E = 8.5 eV (the sum of the C-F and C-H bond dissociation energies) for the calculation of J3 Hp The values obtained are shown in Table 5.14 together with some observed values of JFF and JHp The close agreement between ‘the magnitudes of JsHF and JsPF and the observed values suggests that the contact interaction is the dominant feature of coupling in fluoroethylenes. However, the calculated contribution for the contact term is positive for both cis and tram F-F coupling whereas in some substituted fluoroethylenes Jg and J&- are of opposite sign (76). The wide range of JFF values in fluoroolefins shows that the substituent has a large influence on the spin coupling, but it would not be expected to change the sign of JJFF, hence it is probable that there is a large negative contribution from some other source. TABLE 5.14 CALCULATEDVALUES OF JaH~ AND JspF AND OBSERVE-D VALUES OF JHp AND JFp FORFLUORINATEDETHYLENES J3HP D*=
o” (cis) 180”(rrms)
n(O)-?afl(o)
13 25
(cyda
w-9
J
opm-*(o)
19 38
(A
J3FF
(OW a-9
I-20 12-52
m
eN=
(0) - 2-$Q (0)
28 53
=-9 (pp (0) -3w0,
62 120
Jq$g!?,) (w= 33-58 115-124
5.8.2 Aromatic Fluorocarbons In an early application of molecular orbital theory to F-H and F-F coupling in fluorobenzenes, Gutowsky and Williams (gl) chose to neglect configurational interaction”‘) and it is likely, therefore, that their results are considerably in error. However, their calculation does show that terms in equation (3.211) other
CALCULATIONS
OF SPIN-SPIN
COUPLING
CONSTANTS
187
than the contact term J3 NN' are important in fluorocarbons andhence the results will be given, bearing in mind the above reservations. The four terms contributing to dry and JFr, were calculated from equations (3.225), (3.226), (3.227) and (3.228) (see Section 3.14.2). Each term may be divided into the contributions from nucleus-electron-nucleus interactions (one-electron terms), and nucleus-electron-electron-nucleus interactions (twoelectron terms); J~2~NN,and J2NN, have both one-electron and two-electron terms, J~N~, has only one-electron terms, and J3N~' only two-electron terms. In calculating the one-electron terms for H ~ F coupling the p-orbitals were assumed to be completely localised, a value of ,4 E - l0 eV was used for all terms, and H - - F distances were taken as 2.61 A (ortho), 4"50 A (meta) and 5"20 A (para), based on a C - - F bond length of 1"35 A. The one-electron terms are shown in Table 5-15; the total contributions from these terms are all negative and comparable in magnitude with the observed coupling constants. T~d~t~ 5.15 THE CONTRmtmON OF ONE-Et~CrRON TO H - - F COUPLING CONSTAN'I~ IN FLUOROBElqZENIZS
(GUToWSk'Y AND W
ILLI
,MS (91)) meta
para
+0"3 --3"9 + 1"0
+0"2
J2HF
+ 1"7 -- 17"8 -- 1"0
Total (cyclessec -1)
--17"1
-2"6
-
ortho j(l) IHF
J2nv
-
2"6
+ 1"6 1-4
The two-electron terms are all zero for localised p-orbitals but when some form of delocalisation is introduced the two.electron terms are obtained as functions of the H m F bond order, ~THF, M
' ~ . r = 2 ~ a~, aF.
(5.38)
where a ~ and aw are coefficients of the H and F atomic orbitals, and the summation is over all occupied orbitals ~ (see equation (3.220)). The two-electron terms are negligible except for J3 HF, the contact term, for which Gutowsky a n d Williamst91~ obtained J3Hr = + 1"9 x I(PUHF(S).
(5.39)
The symbol ~THv(s)refers to the s-bond order between H and F, and thus the summation in equation (5.38) is taken over s-orbitals only. Now the observed values of JHr in fluorobenzenests7~ are JHF (cycles sec-O
ortho
meta
para
4- 8"6
+ 7"0
:]: 2-2
and if it is assumed that both ~Hv fo-so and J ~ ° are positive, and J ~ " negative, then J3HF has values of + 25, + 10, and + 1 cycles sec -1 for ortho, meta and
188
HIGH
RESOLUTION
NMR
SPECTROSCOPY
para nuclei respectively; hence 7/He(s) must be given the values o f 12 x 10-* (ortho), 5.3 x 10 -a (meta), and 0.5 x 10-" (para). I f configuration interaction were included in these calculations then these conclusions might need modifying. The calculation o f F m F coupling constants is very similar to that for Jar. The one-electron terms were obtained with the aid of F m F distances o f 2-74 A (ortho), 4-75 A (meta) and 5"48 A (para), and A E w a s again taken to be l0 eV; the results are shown in Table 5.16. TABLE5.16 Ti-m Cot,rrv.mtrrtoN ov ON~-m.~crRoN TERMS TO J'FFt IN FLUOROBENZENE8(91)
ortho
meta
para
j ( lI )F F , j~2) IFF' J2FF'
~ 2"7 -- 28"8 -- 1"7
+0"5 -6"4 + 1"6
+0"3 -4"4 +1"7
Total (cyclessee-t)
- 27.8
-4"3
-
2.4
All the two-electron terms are important in F - - F coupling, and may be expressed in terms of F - - F ' bond orders. The various two-electron terms are j~2) ~vF,= + 1"11 x lO'%lvv,(p~r)rlvv,(p:OcosO (5.40) where ~lvv,(PU) and ~/vv,(pTr) are the bond orders o f p a and pzr bonds as calculated from equation (3.220), and 0 is the angle between the two C - - F bond directions. The J2vv, contribution is Jzvv, = +2.08 x 10a(3cosa0 ' -
1)~2r(p)
(5.41)
in which 0' is the angle between the axes o f the p-orbitals on F and F'. Equation (5.41) may be divided into contributions from p :r and p a orbitals; the p ~r contribution is 2 d2vv,(p~z) = +4"2 x 103 ~FF'(P=) (5.42) and is the same for ortho, meta andpara F - - F ' coupling because of the properties o f aromatic z~-orbitals; the pcr contributions are J2vv,(pa) = - 5.2 x 102 ~7~F,(po)
ortho and meta
Jzvv,(Pa) = + 4"2 x 103 r/~v,(pcr) para.
(5.43a) (5.43b)
Finally, the contact term is given by
J a w , = + 1.67 x 106 r/2v,(S)
(5.44)
Gutowsky and Williams (gL) showed that ~Tvv,(s)can be related to the s-bond order in aromatic hydrocarbons by the equation r/~ F' (s)
z = avz,~Trm,(S )
(5.45)
C A L C U L A T I O N S OF S P I N - S P I N
COUPLING CONSTANTS
189
where at2, is the amount of s-character in the C - - F bond. The contact term was evaluated for at2, ffi 0.03 and the results are shown together with the other two-electron terms in Table 5.17. TABLE 5.17 THE CONTRmtrnON OF TWO-ELF,CTRON ~ TO JFF' IN F L U O R O 3 E N ~ (91)
para (2) ]IFF" J3FF'
+44 +7 -20 +57
l'otal(cyclessec-x)
+88
12w, (p~) ]2FF" (po')
-2.2
-22
+7
+7 +12 +3
-5
+ 15 -5
0
The total calculated values of Jr F, are shown in Table 5.18 together with some experimental values. TAI~.I3 5.18 CALCULATEDAND OBSERVEDVALUESOF JFFt IN FLUOROm~NZEN1eS
j ortho 'Observed(s~) (cycles sec-1)
20.5
meta I para 3"
I
13-2
The calculated and observed values can be brought into closer agreement by increasing ~rF'(P or) and decreasing ~/Fv, (p ~z); however, the neglect of configurational interaction and the uncertainty in the signs of the observed coupling constants renders the results of doubtful validity. 5.8.3 Fluoroalkyls
There have been no detailed calculations of F - - F or F - - H coupling constants for saturated fluoro-organic compounds. An estimate of the magnitude of the contact term J3NN' for J ~ and J ~ in fluoroethanes can be made in a similar manner to that used by Karplus for fluoroethylenes by assuming that the replacement of H by F in ethane leaves unchanged the coefficients c: of the valence bond structures tpj. Also, the value of ~F(0) is assumed to lie between the limits 2~0u(0) and 3~, (0), while the value ofA E is taken to be 8.0 eV for F - - F coupring, and 8.5 eV for F - - H coupling. With these values it is found that j ~ c and J ~ are related to J ~ by the equations = =
~ yo,c 3"2~
or
~
or
IF~c zv~c r = 4"6 .,m~ T*'tc = 10"2 .T~tC -,v ,..
(~F(o) = 2 ~ . ( 0 ) )
(~(0)=3~(0)) (~t~(0)-- 2 q~(0)) (~F(0) 3 ~(0))
.~
.f
(5.46) (5.47)
190
H I G H R E S O L U T I O N NMR S P E C T R O S C O P Y
so that the contact term contributions to the gauche and trans coupling constants are estimated to be j~,cs, = 3.6 to 5-4 cycles sec -t J~'~"* = 19.3 to 29.4 cycles see -x j~,~,ch, = 7.8 to 17.3 cycles see-: J~'~' = 42.3 to 93.8 cycles see -x. The separate values of ~rg*'¢s"and J ~ ' in fluoroethanes (C2H,F6_,) have not yet been determined experimentally, but the average of ro~¢ -m, over the three rotational forms, (J~,~?, has been found to be 25-2 cycles sec -t in CH3CHzF (Tz), and 15.5 cycles see -~ in CF3CH2F (9-a. 94), and in both cases (J~¢> and J ~ , have the same sign. In CH3CHF2 the value (x°9) of<J~t,¢) is 20.9 + 0"I cycles see -x. For HF coupling the average value of the contact term over the three rotational forms is ( J ~ ) = 8"8 to 13-4 cycles see -t which, even taking the upper limit of 13.4 cycles see -z, is only one half of the observed value (cf. ro~c in ethane, Section 5.3.1). Gutowsky et al. (2~) have found the separate values of Ja~h~ and Jr,,,~ for both F - - H and F - - F coupling in some substituted fluoroalkanes by following the temperature dependence of the averaged coupling constant.l" Their results (see Table 5.19) show that, although the substituent has a large effect on the coupling IJ,,~l is always larger than IJ~,~¢h,I and may sometimes be of opposite sign. TABLE5.19 H - F AND F - F COUPLINGCONSTANTSIN SOME SUBSTITUTEDETHANES(21) Compound
C'FC12CHCI2 CHC12CHFCI CF2CICFC12
Ysa,~, (cycles see-1)
I Jtram (cycles see-1)
H - F Coupling ± 1-00 -4- 0.02 I ± 18.2 ± 0.08 ± 13.2 ± 0.4 q: 2-8 ± 0.4 ± 37-3 5:0.3 F - F Coupling ~: 21.17 ~ 0-13 I ±40.045:0.13
J~,m (cycles see=x)
±49.1 ± 0.2
The occurrence of opposite signs for j~,c,e and l ~ ' could account for the very small values of ( J ~ observed in some saturated fluoro-organic compounds (see Table 5.20). The observation that the F - F coupling constants, J,b, between interacting nuclei separated by three bonds is often much smaller in magnitude than that, J,c, between nuclei separated by four bonds in many saturated fluoro-aliphatic compounds (see Table 5.20) led Sederholm and Petrakis (97) to suggest that the coupling of tgF nuclei by bonding electrons is negligible, and that the dominant contributions arise via electrons not involved in bonding. They suggest that this "through space coupling" is dependent on the F - - F distance, and is almost t N©wmark and Se,derholm (11s) have shown that the values of Jg.,c~, and Jt,o,j obtained by this method may be considerably in error.
CALCULATIONS OF S P I N - S P I N COUPLING CONSTANTS
191
zero for F - - F = 2"73 A, the normal separation of vicinal fluorine nuclei in a ---CF2---CF2-- fragment. The relatively large values of Jrv, over four bonds (J~ in Table 5.20) can be explained since in some configurations of the molecule the interacting fluorine nuclei are closer than 2.73 A, and hence can give rise TABLE5.20 F - F Cotrpxa~G Com'rxrcrs ~ FLUORINATEDAL~HATICCOMPOUNI~
Compound C~aCF~CF~I CF3CF2CF2Cl CF3CFICF2CI CF3CF2CF2H
CF3CF2CF~NF~ CFaCF2CF2COOH CFaCF~CFCll
(cycles sec-x) 0.8 11"8
(cyclessec-1)
(cyclessec-1)
Reference
4.7 1-6 18"8, 18"1
9"4 9"0 9"4, 11"8
95 96 95 97 97 97 97
4-0, 4.5 <1 <1 <1
1
_-<1 14-6
8"6
9"9 10"8
to a larger through space coupling. Similarly, the large values of J ~ for saturated fluorine compounds can be accounted for by the small value of the F n F distance of geminal fluorines (approximately 2.17 A). However, Petrakis and Sederholm in their correlation of internuclear distances with F - - F eotipling constants assumed the signs of the latter to be the same. In fact, an investiga-. tion of the relative signs of F m F coupling constants by Evans showed that for all the compounds studied the relative signs are j~m + , j~c :1:, j ~ a 5:. In view of this inconsistency in the correlation, the importance of the spatial mechanism may have been overstressed.(12°~The fact that the values of J ~ " ~ and J ~ " are different in magnitude indicates that the contact term doesplay an ira° portant part in determining the magnitude of vicinal F n F coupling constants, but a "through space" mechanism may also be important (~~9). The large values of J r r between fluorine nuclei separated by four or more a-bonds probably arise from the "'through space" mechanism, since the contact term would be reduced by a factor of approximately ten for each additional a-bond, and hence would be ~ 2 cycles sec-L Thus in the molecule (CF3)2NCF2CFBrCFa where the coupling (99) over six bonds, Jr-c-N-c-c-c-r is 1"7 cycles see-~, the geometry of the molecule allows of configurations in which the two fluorine nuclei are spatially close together. 5.9 aaC--H COUPLING CONSTANTS Although IaC--H coupling constants can be obtained either from the a3C or the ~H resonance spectrum of a molecule, the latter method is normally adopted since it gives the more accurate values, because of the relative ease of examining the hydrogen nucleus. Measurements of J c . for directly bonded nuclei have revealed a very close dependence of the magnitude of this coupling constant on the s-character of the C m H bond (zT. lo3) (see Section 12.2.11).
192
HIGH
RESOLUTION
NMR
SPECTROSCOPY
Thus the values of Jc H for methane, ethylene and acetylene are methane ¢e7) ethylene (2o) acetylene¢2°) J c . (cycles sec -1)
125
156-4 + 0"3
248.7 +_ 0.5
These can be shown to obey the relationship c67) 2
JCH = J o a n
(5.48)
where ~ is the s-character of the C - - H hybrid orbital described by the molecular orbital function ~Pca = a H S + bH p (5.49) Jo is a constant and from the data given for methane, ethylene and acetylene its value is (2°) methane ethylene acetylene JcH 500 495 497 cycles sec -1 The above values of Jo show that changes in Jcu in going from the sp 3 hybrid bonds of methane to the s p hybrids in acetylene can be attributed entirely to the change in s-character; the polarity of the C m H bond does not seem to be important. This behaviour is explicable if it is assumed that Jcn is dominated by the Fermi contact term JSCH, that is ((i7" I01) JCH ~ J 3 c . =
YcYH(~'°IE ~(r'c)" ~ ( r J " ) ' S ' ' S J I~° \ i k,j
3hAE
(5.50) Gutowsky and Juan (t°~) have treated the case of methane and substituted methanes C H X Y Z in detail, using the valence bond method. For directly bonded nuclei, the contribution of non-perfect pairing structures to the coupling is unimportant, hence for C H X Y Z the wavefunction ~Pois taken to be the separated electron pair function ~Po = (8Dr/2 ~ ( - 1)PP[~Pox (1, 2)~Pb, (3, 4)~P,= (5, 6) ~Pah(7, 8)]
(5.51)
p
in which ~vr(i, j) --/~r(i, j) {[~ (i) fl(j) - fl (i) oc (j)] / ~/2"} where # , ( i , j ) is the valence bond function with inclusion of ionic terms, p. (1, 2) ---~/x [~o.(1) ~ox(2) + %(2) ~ox(1) + 2.~.(1) %(2) + 2~ qo~(1)qo~(2)] where %, ~ , ~oc and q0, are carbon atomic orbitals and ~o~, %, q% and ~ are atomic orbitals on the atoms X, Y, Z and H respectively. ~/is the normalisation coefficient, and 2, and ~ are coefficients of the ionic terms. When equation (5.51) is substituted into equation (5.50), and the term S~-Sj is replaced by the Dirac identity 1
s~. sj = ¥ (2 P~, - 1)
CALCULATIONS
OF
SPIN-SPIN
COUPLING
CONSTANTS
193
the equation for Jcu can be reduced to jc.=
YcY" ( 1 6 z t f l h ) 2 h.4----E a . ~/2@dlt~(rtc)lga)(9.13(ra0[9.).
(5.52)
9d is assumed to be a sp a hybrid orbital and for the C - - H bond is given by 9~
=
a.
2
s +
(1 - -
a~)t/2p o
(5.53)
a 2 is the s-character of the C m H bond, and the p o% are linear combinations of the p~, py and p= orbitals. Substituting for 9a in equation (5.52), and taking 9 . to be a ls atomic orbital gives J c , = hA'----'E ~
'7Za~1~"(0)121~c(0)12
(5.54)
which may be written as
/ A'72~ 2 Jc. = ~-Z-E--} a- cycles s e c - 1 .
(5.55)
By comparing this with equation (5.48) 3"o = A • ' / A E
(5.56)
A is simply a collection of constants, and A E is expected to be constant for a C - - H bond in a related series of molecules, ~/depends uponboth an and the ionic character of the C - - H bond, that is, upon the values of ~ and 2.. However, Gutowsky and Juan (a°l) have shown that the changeint/2 with changes in these quantities is negligibly small, hence Jo can be regarded as a constant. The value of a~ for a substituted methane can be calculated from equation (5.48), if the value of Jcn is known, but the reverse process of calculating J c . in C H X Y Z from known values of a~ is not of practical importance since the effect of substituents on the hybridisation of the C - - H bond is not accurately known. It has been observed that the effects of substituents on Jcn in methanes are additive "°2), that is J c . ( C H X Y Z ) = J(cx' + Jtc~Y'+ Jto~z)
(5.57)
and Gutowsky and Juan (ten have shown that this additivity rule is consistent with equation (5.48). To do this they introduce a parameter Zlx which measures the affinity of a substituent X (compared with that of a hydrogen atom) for carbon 2a electrons, d x is positive when the s-affinity of X is less than that of hydrogen. They also assume that only one carbon 25 orbital is used in forming the four hybrid bonds, so that a 2 + aXx + av2 + az2 ---- 1.
(5.58)
If the effect of the substituent is distributed equally among the bonds, then for
CH3X a2.(CHaX) = aXn(CH,) + ¼Ax ]rR$.
7
(5.59)
194
HIGH RESOLUTION
NMR SPECTROSCOPY
and for C H X Y Z a~(CI--IXYZ) -- a~(CH,,) + ¼A x + ¼Ay + ¼A z.
(5.60)
Eliminating Ax, As, and Az from equation (5.60) gives a~(CHXYZ) = as2 (CH3X) + a2n (CH3Y) + at2 (CH3Z) - 2a~ (CH,,) (5.61) and re-arranging the above equation gives a2n ( C H X Y Z ) = [a~(CH3X)
-
2
2
2
2
~" an
+ [a~(CH3Y) - - T a s
(CH,)] (CH,)]
2 + [a~(CH3Z) - -~- a2n (CH,)]
(5.62)
which, together with equation (5.48), gives JCH ( C H X Y Z )
= J~cx' + JCc~ + J~c~
which is the relationship found empirically by Malinowski (x°2) (see Section 12.2.13). From equations (5.48) and (5.61), Ax is given by Ax = (4/Jo) [Jcn (CH4) - dcn (CH3X)]
(5.63)
and the values of Ax for a number of substituents are shown in Table 5.21. The magnitude of Ax increases as the eleetronegativity of X increases which means that the s-character of the C - - X bond is greater for the more electronegative elements. This can be interpreted in the following way cx°*). The sp s bonding orbitals in methane are arranged in space so as to give the minimum charge repulsion between the electrons, the bonding electrons being in pairs in the tetrahedrally disposed orbitals. If hydrogen is replaced by a more electronegative atom X, then the pairs of electrons in the C - - X bonding orbital are centred further from the carbon atom than they were in the C - - H bond, consequently charge repulsion is diminished and the bonding orbitals can move closer together. This can be interpreted in terms of unequal hybridisation in the C - - X and C - - H bonds and a consequent increase in the s-character of the C - - H bonds. There is not, however, a direct correlation between a~ and thebond angles in substituted methanes (~°a' xo3, ,o4> The dependence Of Jcn on s-character given by equation (5.48), and the additivity rule (equation (5.57)) for substituents also apply to substituted ethylenes and acetylenes(l°n. The olefine, CH2 = 13CI-IX is regarded as a substituted methane having as substituents X and =CHz; the s-character of the *3C--H bond is 2 H 2 -- 13CHX) = ~1 [ 1 + A =cna + Ax] as(C 1 = as2 (CH2 = CH2) + "z-Ax
(5.64) (5.65)
CALCULATIONS OF SPIN-SPIN COUPLING CONSTANTS
195
w h e r e / I x has the same meaning as before, and ,4-cH~ is equated to ,4c,~ ( = + 0.008). Values o f a~ for some ethylenic compounds are included in Table 5.22, where observed Jca values are compared with values o f JcM calculated from equation (5.48). There is no simple means o f allowing for the substituent effect on the hybridisation in these compounds for which ~ is listed as sp2 or sp. Ax OBTAINED FROM VALUES OF J u c l t IN C H 3 X COMx~OtmDS(GUTOWS~:YANDJu~'~x°t))
TABLE 5.21 StmS'Trrtr~T P ~ a t
CH3X
lc~ (cycles × ~-
/Ix
I)
CH3X
Ax m
d2(CH3)6 ~i(CH3)4 CHs)sSiCN H'I3C(CH3)3 :H, ~'IsCOCHs ,"I-IsCHs ,'I'13CH(CH3)
113 118 122 124 125 126 126
0"226 0"236 0"244 0"248 0"250 0"252 0"252
126 126 127 128 128 130 131 131
0"252 0"252 0"254 0"256 0"256 0"266 0"262 0"262
0.096 - 0"056 - 0.024 - 0.008 0.000 + 0.008 + O.008 -
I
Br H'IsCeHs H'IsCHO H-IsCH2Br ~'HsCH2CI H'IsCOOH H'IsN(CH3)2
+ 0.008 + 0"008 + 0"016 + 0-024 + 0.024 + 0.040 + 0-048 + 0-048
CHsCH2I CHsNHCH3 CHsC~CH CHsNH2 CHsCCI3 CHsCN (CHs)2NCHC CHsSH (CHs),S CH~SOCHs CHsOH CI'I30CsHs (CHsO)~CO CHsF CHsCI CHsI CHaBr
132 132 132 133 134 136 138 138 138 138 141 143 147 149 150 151 152
0"264 0.264 0"264 0"266 0"268 0"272 0-276 0-276 0.276 0-276 0-282 0"286 0"294 0"298 0-300 0"302 0"304
+ O'O56 + 0.056 +O.056 +O.O64 + 0.072 +0.088 + 0-104 + 0"104 + 0-104 +0-104 + 0-128 + O.144 +0.176 +0-192 + O-2OO + 0-2O8 + 0-216
The observed values o f J c . for substituted ethylenes lie consistently 5 to 10 cycles sec -x below the values calculated from equation (5.48). This probably arises because o f a contribution to J c . from o--~r interactions (see Section 5.6). The term Jca (~r) can be estimated from the equation J~0z)--2"1
x 10 -xs a . . ac
(5.66)
where a . and ac are isotropic hypertine interaction constants for the radical fragment H---~ (sp2). With a . ffi - 65 x 10e cycles sec -1, ,4 Es -- 6 eV, and ac =, + 40.7 gauss, equation (5.66) gives Jc~(~r) -- - 2 . 6 cycles sec -x, which is o f the correct order o f magnitude and sign (l°x). Muller and Pritchard (x°s) have suggested that Jcs should be directly related to both the bond distance r e , and the inter-orbital angles since both these quantities depend upon the state o f hybridisation o f the carbon atom. They derive an empirical equation relating J c . to r e , (see equation (12.3)), which predicts bond lengths in fairly good agreement with the experimentally observed values (see Section 12.2.11). 7*
196
H I G H R E S O L U T I O N NMR S P E C T R O S C O P Y
The bond angles in substituted methanes m a y also be calculated from the observed JcM values, but the agreement with experimentally obtained bond angles is poor (see Table 5.23). The bond angles in such compounds as CH2C12 cannot be reconciled with any state ofhybridisation based upon a combination ofs and p atomic orbitals, which m a y explain the above discrepancies (t°". TABLE 5.22 VALUES OF JcH A N D a 2 FOR SOME UNSATURATED COMPOUNDS
J c u (observed)
Compound
cycles see-*
Naphthalene( t o ~) [~D72a¢(
TM
)
Mesitylene (ring H)(*°*) 2,3,5,6-Tetrachloronitrobenzene(**o) 2,4,6-TribromophenoF *xo) 2,4,6-Trichlorophenol( t i o) 2,4,6.Trinitrophenolo lO) 2,6-Dibromo-4-nitrophenol(*xo) 1,3,5-Trichlorobenzene(xxo) 1,3,5-Tribrornobenzene(,, o) 1,3,5-Trinitrobenzene(t ~o) ~-Dichlorobenzene(*1o) ~,4,6-Trimethylpyridine( t l o) 7-Dimethoxybenzene(11o) ',CH3)~C = C = ~3CH2 o o , ) ~yclohexene(t o,) =I-I2= CHz(*o*) ~"'I-.ICI~ 13 C H 2 (cis~.~.)( 101 ) ~'-'I'-IC ! =ffi13 C H 2 (tra/./2)( I 0 t ) ~CI2 = C-'I..I2(I o x)
=H2: laCHCl( *o*) :is-CHCI=CHCI
(*o*)
T a n s - CHCI= CHCI(* o l) = C I ~ = CI-ICl( " o ' ) ~H3Cmet 1 3 C H ( l O 1) [.-[-- C ~ C t cis
157 159 160 175"0 176'9 172"5 171"3 175"8 174"1 174"3 179"5 169"0
$p2
158"5 161.1 166 170 157 160 161 166 195 198"5 199"I 201"2
sp 2 sp 2 sp 2
C__--C- H (101)
= 500 ~ cycles sec-J
sp 2 sp 2 sp 2 sp 2 sp a sp z sp 2 sp 2 sp 2 sp 2 sp 2
sp 2
0"336 0"341 0"341 0.349 0"402 0-408 0"408 0.416
248 251 259"4
~6HsC~*3CH (t°*)
JcM (calculated)
2 aH
168 170"5 170"5 174 201 208 208 213
sp sp sp
refers to CI and H across the double bond.
TABLE 5.23 "IwrER-ORarrAU" AZqOt.ES CALCULA'rED FROM a~2 AND a~ for HALOtUmTF,AN~ A N D THEIR COMPARISON WITH OBSERVED B O N D ANGLES (tot) HCX
Difference Calculated
Observed
:I
102"2 101"8 102'6
106.9 107"3 108"0
F
103.0
108.5
Br
HC){ in CI'I2X2(°)
in C H a X (o)
X
4"7 5"5 5"4 5-5
Difference Calculated
Observed
100.5 99-7 99.7 98.5
114.7 112 111.8 108.3
14"2 12.3 12"1 9"8
CALCULATIONS OF S P I N - S P I N COUPLING CONSTANTS
197
A number of ~3 C ~ H coupfing constants between nuclei separated by two bonds have been measured, and these also appear to depend upon the state of hybridisation of the ~3C atom c~°s~ (see Table 12.34). An estimate has been made of the magnitude of the Fermi contact term for interacting nuclei of this type by the method used for calculating F - - F coupling in fluorinated ethane and ethylene (see Sections 5.8.3 and 5.8.1). Although such a calculation gives the correct order of magnitude for s p a and s p 3 hybridisation it predicts positive coupling constants, whereas the signs of Ji3c-ca determined so far have been found to be negative c~°~ (relative to a positive J:3ca), except for the anomalously large value of + 49"3 + 0-3 cycles sec -~ found for acetylene ez°). When the ~3C and ~ H nuclei are separated by more than two bonds the Fermi contact term is no longer the only large term in equation (3.211), and the magnitude of the coupling constant does not depend solely upon the s-character of the bonds o°e' ~o7~. As with F - - F coupling constants there are examples where the coupling over three bonds (J~3c_c_ca) is less than that over four bonds (JJ at-c-c-ca) within the same molecule. REFERENCES 1. N. F. RAMSE~,Phys. Rev., 91,303 (1953). 2. H. M. M C ~ O N ~ t ~ J. Chem. Phys., 24, 460 (1956). 3. M. KxgPLus and D. H. ANDL'nSON, J. Chem. Phys., 30, 6 (1959). 4. M. KxP.~Lus, J. Chem. Phys., 30, I1 (1959). 5. H. Y. C ~ x and E. M. PURCELL, Phys. Rev., 88, 415 (1952). 6. T. F. WIMMEr'r, Phys. Rev., 91, 476 (1953). 7. H. M. JAMES, A. S. COOLIDGEand R. D. IhtEs~rr, J. Chem. Phys., 4, 193 (1936). 8. H. BROOKS, Phys. Rev., 59, 925 (1941); Phys. Rev., 60, 168 (1941). 9. M. J. S ~ , Prec. Roy. Soc.,A243, 274 (1957). 10. E. Ismotmo, Phys. Rev., 111, 203 (1958). 11. A. D. B u ~ i ~ and E. G. LOVEmNG, Trans. Faraday See., 58, 2077 (1962). 12. M. KXnPLUS, D. H. ANDERSON,T. C. F~RAR and H. S. GUTOWSKY,J. Chem. Phys., 27, 597 (1957). 13. C. A. COULSONand H. C. LO~UFr-I-hGGINS, Prec. Roy. Sac., A193, 447 (1948). 14. C. A. COULSON, Trans. Faraday See., 28, 433 (1932). 15. H. EYmNG, A. A. FROST and J. ~ C H , J. Chem. Phys., 1, 777 (1933). 16. E. Hmoll~., Prog. Theor. Phys., 21, 943 (1959); J. Phys. Soc. Japan, 15, 270 (1960). 17. J. H. VAN Vt~'K, J. Chem. Phys., 1, 219 (1933); 2, 20 (1934). 18. M. KA~LUS, J. Phys. Chem., 64, 1793 (1960). 19. L. PAUL~G and E. B. WILSON, Introduction to Quantum Mechanics, McGraw-Hill, New York (1935). 20. R. M. LYNDEN-BELLand N. SHSPt'XnD, Prec. Roy. See., A269, 385 (1962). 21. H. S. GUTOWSgY,G. G. BFJ~VOP,D and P. E. McMxHoN, J. Chem. Phys., 36, 3353 (1962). 22. N. S~.PpXXD and J. J. T ~ Prec. Roy. Sac., A252, 506 (1959). 23. R. U. ~ u x , R. K. KuI2~O and R. Y. Morn, J. Amer. Chem. See., 80, 2237 (1958). 24. M. KTa~Lus, J. Chem. Phys., 33, 1842 (1960). 25. H. CONROY,Advances in Organic Chemistry, Vol. II, Interscience, New York (1960). 26. H. S. GtrrowsKv, M. K ~ L u s and D. M. GRANT, J. Chem. Phys., 31, 1278 (1959). 27. N. SH~pPXp.Dand H. J. BEnNSTI~, J. Chem. Phys., 37, 3012 (1962). 28. M. BAv,E~J) and D. M. GRANT,J. Amer. Chem. See., 83, 4726 (1961). 29. G. V. D. TIEns, J. Chem. Phys., 29, 963 (1958), 30. F. K^PLAN and J. D. ROBERTS,J. Amer. Chem. See., 83, 4666 (1961). 31. C. A. REILLYand J. D. SWALEN,J. Chem. Phys., 35, 1522 (1961).
198
HIGH RESOLUTION
NMR SPECTROSCOPY
32. K. A. McLAUCHL~ and D. H. Wmr~e~, Proc. Chem. Soe., 144 (1962). 33. C. J. BROW~, J. Chem. $oc., 3278 (1953). 34. H. S. GUTOWSKYand C. JUAN, Discuss. Faraday Soc., 34, 52 (1962). 35. C. N. BANWeLLand N. SHL~PAZD,Discuss. Faraday Soc., 34, 115 (1962). 36. H. M. MCCOtC~qh'LL,J. Chem. Phys., 23, 2454 (1955). 37. M. B A ~ L D and D. M. GRANT, Private communication. 38. G. V. D. T i r e , J. Amer. Chem. $oc., 84, 3972 (1962). 39. M. KA~LUS, J. Amer. Chem. Soc., 84, 2458 (1962). 40. P. C. L A ~ u I t and R. J. KUP,L A ~ , J. Amer. Chem. Soc., 84, 3405 (1962). 41. F. A. L. Ate'r, J. Amer. Chem. Soc., 84, 3767 (1962). 42. T. SCH~FeR, Can. J. Chem., 40, I (1962). 43. P. F. Cox, J. Amer. Chem. Sot., 85, 380 (1963). 44. W. BItUI~3EL,TH. A~r.L and F. I~U~B~RO, Z. Elektrochem., 64, 1121 (1960). 45. A. A. BOTHeR-BY and C. NA~-CoLr~, J. Amer. Chem. Soc., 84, 743 (1962). 46. F. A. L. A ~ T , J. Amer. Chem. Soc., 84, 747 (1962). 47. D. 1t. DAvls, R. P. LUTZ and J. D. ROSeRTS, J. Amer. Chem. Soc., 83, 246 (1961). 48. F. A. L. A ~ T , Can. J. Chem., 39, 789 (1961). 49. J. MSI~'WALDand A. LeWLS, J. Amer. Chem. Soc., 83, 2769 (1961). 50. A. A. B ~ - B Y and C. NA~-CoLnq, J. Amer. Chem. Soc., 83, 231 (1961). 51. R. W. FE.SSENDL~and J. S. WAUGH,J. Chem. Phys., 30, 944 (1959). 52. A. A. BOTHeR-BY, Private communication. 53. F. S. MORTXM~, J. Mol. Spect., 3, 335 (1959). 54. S. ~ R , J. Chem. Phys., 32, 1700 (1960). 55. R. A. H o F ~ and S. GRoNowrrz, Arkiv. Kemi, 16, 471 (1960). 56. S. FUJXWARA,H. SmMIZU, Y. ARATA and S. ~ o R I , Bull. Chem. Sac. Japan, 33, 428 (1960). 57. R. R. FRASER,Can. J. Chem., 38, 549 (1960). 58. L. M. JACF~MANand R. H. WXLEY,J. Chem. $oc., 2886 (1960). 59. J. A. POPLE, W. G. SCHN~DER and H. J. B~P~ST~, High Resolution Nuclear ~fagnetic Resonance, McGraw-Hill, New York (1959). 60. R. A. HO~MAN and S. GRoNowrrz, Acta Chem. Scand., 13, 1477 (1959). 61. E. I. S~,Del~ and J. D. ROSERTS,J. Amer. Chem. Soc., 84, 1582 (1962). 62. S. L. MANATr and D. D. ~ , J. Amer. Chem. $oc., 84, 1579 (1962). 63. J. N. SHOOL~tY, L. F. JOHNSONand W. A. A~DEP.SON,J. Mol. Spect., 5, 110 (1960). 64. E. B, Wmp~..e, J. H. GOLDsrr.L~and W. E. STEWART,J. Amer. Chem. Sac., 81, 4761 (1959). 65. W. R. VAU¢3~,'~and R. C. TAYLOR,J. Chem. Phys., 31, 1425 (1959). 66. E. B. WH~PLE, J. H. GOLDSTEn~, L. M ~ E L L , G. S. REDDY and G . R . McCLu~, J. Amer. Chem. Soc., 81, 1321 (1959). 67. N. MULLet and D. E. P z r r ~ , J. Chem, Phys., 31, 768 (1959). 68. J. R. HOLM~ and D. K2VF2ZON,J. Amer. Chem. Sac., 83, 2959 (1961). 69. L. PAULINO,The Nature of the Chemical Bond, 3rd ed., Cornell University Press, Ithaca, New York (1960). 70. H. M. M c C o t ~ . L , J. MoL Spect., 1, 11 (1957). 71. H. M. McCoNN~.L, J. Chem. Phys., 30, 126 (1959). 72. C. N. BA.'qWELL, A. D. C o ~ , N. S~Pp^go and J. J. TuR~a~g, Proc. Chem~ Soc., 266 (1959). 73. T. R. TUTrLe and S. I. We~SSMA~,J. Chem. Phys., 25, 189 (1956). 74. A. SAu¢~, Physica, 2~, 51 (1959). 75. G. V. D. TIERs, Proc. Chem. Soc., 389 (1960). 76. D. F. EVANS,Moi. Phys., ~, 183 (1962). 77. P. T. N ~ and M. T. ROGERS, J. Chem. Phys., 33, 727 (1960). 78. B. BP~LON, Private communication in reference 55. 79. J. A. POPL~, Mol. Phys., 1, 216 (1958). 80. I. SOLOMONand N. BLOEMBERG~,J. Chem. Phys., 2~, 261 (1956). 81. L. H. MEYERand H. S. Gu'rows~Y, J. Phys. Chem., 57, 481 (1953). 82. J. F~ENEYand L. H. Su'rcLrrre, Trans. Faraday Soc., 56, 1559 (1960).
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J. LEE and L. H. SuTCI.IF~, Trans. Faraday Soc., $$, 880 (1959). H. M. McCONN~, C. A. Ren~y and A.D. McLEAN, J. Chem. Phys., 24, 479 (1956). C. N. B~,'qw-E~ and N. S ~ r ~ P ~ , Proc. Roy. Soc., A263, 136 (1961). N. BOD~, J. W. E~L~Y, J. F ~ Y and L. H. S u ' r ~ unpublished work. H. S. GuTowsKY, C. H. HOL~, A. S ~ and G. A. WrLu~MS, J. Amer. Chem. $oc., 79, 4596 (1957). 88. M. KnvKn~ S. MATSUOKA,S. HATrO~Jand K. SEND^, J. Phys. Soc. Japan, 14, 684 (1959). 89. S. L. MANATr and D. D. ~ , J. Amer. Chem. Sot., 84, 1305 (1962). 90. L. M. ~ and C. H. S~'D~OI~, J. Chem. Phys., 33, 1583 (1960). 91. H. S. GUTOWSKYand G. A. Wn.J.~a~, .7. Chem. Phys., 30, 717 (1959). 92. J. D. BALD~SC~V~t~ and S. L. STAFFORD,J. Amer. Chem. Soc., 83, 4473 (1961). 93. S. L. lVtXNATr and D. D. ELt~MA~, .7. Chem. Phys., 36, 1945 (1962). 94. S. L. MANATr and D. D. ~ , .7. MoL Speet., 7, 307 (1961). 95. N. BOD~N, J. W. E ~ Y , J. F~"NEY and L. H. SUTCt.W~ unpublished data. 96. E. Prrc~at, A. D. BuCgJlqo~M and F. G. A. STo~, .7. Chem. Phys., 36, 124 (1962). 97. L. P~ra~a~ and C. H. S~J~et~IOLM,.7. Chem. Phys., 35, 1243 (1961). 98. D. F. EVANS,Discuss. Faraday Soc., 34, 139, (1962). 99. M. T. Roo~.s, Private commuuication. 100. A. D. BucKn~ox~M and K. A. McLAucle..AN, Proc. Chem. Soc., 144 (1963). 101. C. JUAN and H. S. GuTowslcY, J. Chem. Phys., 37, 2198 (1962). 102. E. M ~ m o w s ~ , .7. Amer. Chem. Sot., 83, 4479 (1961). 103. N. MULL~ and D. E. ~ , J. Chem. Phys., 31, 1471 (1959). 104. N. M t n 2 ~ J. Chem. Phys., 36, 359 (1962). 105. G. J. ~ A T S O S , J. D. GRAHAM and F. M. VANE, J. Phys. Chem., 65, 1657 (1961). 106. G, J. KARABATSOS, J. Amer. Chem. Soe., 83, 1230 (1961). 107. G. J. ~ T s o s , J. D. G ~ and F. M. VA~e, J. Amer. Chem. Sot., 84, 37 (1962) 108. R. F p . J ~ . ~ , K. A. MCI.,AUCm.AN, J. I. M U s ~ and K. G. R. PACm.Y~R, Mol. Phys. 5, 321 (1962). 109. J. D. B A I , D F ~ and G. W. FLYNN, J. Chem. Phys., 37, 2907 (1962). 110. H. M. HUTrON, W. F. REYNOLDSand T. ~ Can. J. Chem., 40, 1758 (1962). 111. C. C. COSTAnq, J. Chem. Phys., 29, 864 (1958). 112. H. S. GUTOWSlCY,J. Chem. Phys., 31, 1683 (1959). 113. M. B ~ and D. M. GRANT, .7. Amer. Chem. Sot., 85, 1899 (1963). 114. M. K~PLUS, J. Amer. Chem. Sot., 85, 2870 (1963). 115. A. B. F. DUNCAN, J. Amer. Chem. Sot., 77, 2107 (1955). 116. J. I. M u s t r ~ Private communication. 117. J. A. PoPI.E and D. P. SANTRY,MoL Phys., 8, 1 (1964). 118. R. A. NeW~qK and C.H. S ~ o t ~ , J. Chem. Phys., 39, 3131 (1963). 119. S. NG and C.H. SI~D~OLM, .7. Chem. Phys., 40, 2090 (1964). 120. N. BODEN, J. F ~ Y and L. H. StrrcLm~., J. Chem. Soc., 3482 (1965).
CHAPTER
NMR
6
SPECTROMETERS
AND
THEIR
ACCESSORIES THIS chapter describes high resolution nuclear magnetic resonance spectrometers, their operation and the equipment commonly used in conjunction with them. The necessary theory is developed wherever it seems appropriate. 6.1 BASICREQUIREMENTS Although several types of high resolution N M R spectrometer are available commercially, they all possess the following components: (i) a magnet capable of producing a very homogeneous field (usually in the range 1000-25,000 gauss) ; (ii) a sample holder (probe) which also contains the radiofrequency receiver coil; (iii) a sweep unit to allow the main magnetic field to be varied slightly in a controlled manner; (iv) a radiofrequency oscillator operating in the range 4-100 Mc sec-~; (v) a radiofrequency receiver and amplifier; (vi) an oscilloscope and/or a pen recorder for permanent presentation of the spectrum.
Radiofr
Requl(:le<: gower
suoi)~y
__2_
1
Mmn Field Coil
q
Correction and slow sweep coil
FIG. 6.1 The single coil nuclear magnetic resonance spectrometer 200
NMR SPECTROMETERS
AND THEIR
ACCESSORIES
201
Block diagrams of both single and double coil (nuclear induction) nuclear magnetic resonance spectrometers are given in Figs. 6.1 and 6.2. The difference between these two types of instrument has already bccn described in Section 2.11. Spectrometer types may be further subdivided into those having a permanent magnet and those having an electromagnet. The major components of a spectrometer will now be dealt with in detail.
I
I Moia f~llld
Correction and slow sweep coil ~ Modulotion c o;I -..,,.~
co. I I
~)oer I I
I
[
FIG. 6.2 The double coil (nuclear induction) nuclear magnetic resonance spectrometer 6.2 THE MAGNET In Figs. 6.1 and 6.2 the spectrometer is shown to have a large electromagnet to produce the steady magnetic field; however, several commercial instruments use a permanent magnet. The advantage of the electromagnet is that it enables the field strength to be set anywhere within the range of about 1000 to 25,000 gauss, thus permitting all magnetic nuclei to be studied at more than one frequency. This feature is extremely useful in the interpretation of complex spectra (see Section 8.19.1). The disadvantages of the electromagnet lie in the high cost of the electronic control equipment and the need for a certain degree of operator skill. The permanent magnet has clearly not got the flexibility of operation, and the upper limit of field strength is probably about 15,000 gauss, but it scores in the ease whereby resonance can be located and maintained because of the long term stability of the field strength. This property is particularly useful when analytical problems are being undertaken. However, electromagnets give a similar performance when run continuously under thermostatted conditions or when a sensing probe is used as for example in the Varian A60 spectrometer (see Section 6.2.1). The uniformity of field attainable by either type of magnet is comparable, both being capable of giving homogeneity of the order of 3 parts in l09. In this connection, it is interesting to note that the phenomenon ItRS.
71
202
HIGH RESOLUTION NMR SPECTROSCOPY
of nuclear magnetic resonance provides the only means of obtaining such an estimate. The upper limits of the field strengths quoted above arise out of the need for field homogeneity over a volume of 0.1 cm 3 and this limits the separation of the pole faces to not less than 3 cm. Also, the volume over which a field is homogeneous is dependent upon the ratio of the pole face diameter to the gap width, the recognised factor (1) being about l0 to I. The required homogeneity calls for pole faces which are parallel, free from machining marks and are almost optically flat(2). The pole cap material must be metallurgically uniform. As a rule, pole pieces are chromium plated to reduce the effects of corrosion. The field distribution is dependent to some extent on the magnetic history of the magnet: at the end of this section will be described the method of conditioning the magnet adopted by most users of electromagnets to produce a uniform field. Homogeneity can be improved by attaching small coils to the pole faces°): currents passed through these various shim coils enable a number of shlrnrniug patterns to be generated quickly. Intense fields in the neighbourhood of 15 kilogauss having field gradients of up to 200 mgauss cm -1 can have inhomogeneities of 1 in l0 s over the sample volume reduced to 1 part in l0 s with the aid of electric shim coils. Anderson (~)has discussed the design of these coils. Most magnets have a closed-yoke design which provides a good magnetic circuit and, what is more important, good mechanical stability. The degree of mechanical stability required may be illustrated by the fact that finger pressure on a 3 ton, 12 inch electromagnet can cause a nuclear magnetic resonance signal to be displac¢¢l appreciably. The magnetic forces tending to pull the pole pieces together amount to about 7 tons for a 12-inch magnet operating at a field of 14,000 gauss(2); in the double-yoke magnet the pole pieces will move but still remain parallel. The cross-sectional area of the yoke should be sufficiently greater than that of the pole faces so that the leakage flux can be carried without saturation being approached. The yoke material must be magnetically soft but its composition is not critical. The pole piece material is very important, requiring the properties of uniform density and high saturation value. Permanent magnets are often constructed from Alnico V or Ticonal either in the form of a cylinder or as sectors of a cylinder held together in a strong tube. A permanent magnet has a negative temperature coefficient of about 1 part in 5000 per °C therefore it is essential to provide adequate thermostatting (s' ~) Electromagnets operated continuously at peak performance require thermostatting of the room (to within + 2°C) and of the magnet cooling water c7) (to within + 0.1°C). All magnets usually have thermal insulation covering the yoke. 6.2.1 Magnetic Field Stabilisation The permanent magnet gives rise to fewer problems since the operator has only to control external influences by means of careful thermostatting, lagging, and placing of magnetic shields. The stability of an electromagnet, however, depends upon the stability of the current that passes through its coils: the upper limit to the field strength of about 25,000 gauss is imposed by the prob-
NMR SPECTROMETERS
AND THEIR
ACCESSORIES
203
lems of cooling the magnet sufficiently at high currents, and by the difficulty of providing the necessary electronics to control these currents. Higher fields can be attained by using superconducting solenoids operating in liquid helium. ~8) Control is usually achieved by passing the magnet current through a small resistance and then comparing the resultant voltage with a reference voltage from a battery. The difference between the voltages is amplified and the error voltage is then fed back by application to the grids of the valves through which the magnet current is passed. Although simple in principle, this feedback system is compficated by phase shifts leading to oscillation, and by noise from components. A direct current amplifier is required, which takes advantage of the superiority of an alternating current amplifier usually by means of a "chopper". All the sections of a magnet current supply must be fed by highly regulated voltages. High resolution N M R spectrometers incorporating either a permanent magnet or an electromagnet employ correction coils c9) (fiuxstabilisers)wound round the pole pieces. These improve the field stability from about 1 part in l0 s to better than 1 part in 108. The coils are used to detect changes in total flux across the gap and the voltages induced in them are applied to a galvanometer which acts as an integrator, the current flowing through it causing a proportional deflection. The deflection can be detected photoelectrically ff the galvanometer has a lamp and suspension mirror. On deflection, the difference in currents from two photocells placed side by side can be amplified and applied to a set of compensator coils. The flux produced by these cots diminishes the original change of flux and also restores the galvanometer deflection to zero. Normally the pick up and compensator coils are wound on the same former. The method is most useful for controlling rapid variations, and transients from the mains supply. From Figs. 6.1 and 6.2, it will be seen that the means of varying the main field in a controlled manner is linked to the flux stabiliser. A field sweep is produced by applying a small steady voltage to the galvanometer thus causing a deflection in the same way as does a change in the main magnetic field. The flux stabiliser attempts to restore balance by feeding current to the compensator coils hence the main field is changed continuously by a minute amount as long as a voltage is applied directly to the galvanometer. An approach to the long term stability problem is to control the magnetic field itself thus correcting the drift to less than 1 part in l0 s. A nuclear magnetic resonance signal from a control probe or internal reference can be used to generate an error signal which is then injected into the input of the amplifier of the magnet power supply ¢1°) or into the flux stabiliser (11). By utilising a control probe, Baker and Burd (12) built one of the first spectrometers having frequency scales printed on the recorder chart paper and having a predetermined reference line. In first instruments the spectra are obtained in terms of frequency variation rather than the more usual main magnetic field variation. Primas (13) has devised an ingenious method of field stabilisation which requires only one radiofrequency probe. To the sample is added an internal reference compound having a signal well away from the sample (tetramethylsilane for ~H resonance spectra). The dispersion mode signal of the reference compound is fed as an 7 a*
204
HIGH
RESOLUTION
NMR
SPECTROSCOPY
error signal to the flux stabiliser; this in turn corrects the main magnetic field by maintaining exact resonance of the reference signal. A radiofrequency oscillator excites the reference nuclei and detection is accomplished by means of a radiofrequency lock-in detector. An audiofrequency sideband of the radiofrequency is used to examine the spectrum of the sample. The spectrum is recorded by sweeping the audiofrequency; separation of the spectrum from the reference signal is achieved with the aid of a second radiofrequency lock-in detector circuit. The principle has been adopted by Freeman and Whiffen(14) to enable them to carry out frequency-swept spin decoupling experiments (see Sections 6.8 and 8.19.5). They kept magnetic field variations within a range of 0.25 mgauss for a main field of about 14,000 gauss. The Varian A60 NMR spectrometer utilises a sideband control loop "5) to provide field stability sufficient to enable spectra to be recorded directly on precalibrated recorder chart paper (see Section 6.9).
6.2.2 Cycling Procedurefor Electromagnets The most homogeneous portion of a magnetic field shifts slightly from day to day and in the case of an electromagnet restoration is possible by a cycling procedure if adjustment of the shim coil settings is inadequate. Cycling the magnet consists of increasing the magnetic field to a value somewhat greater than that required to observe resonance and allowing it to remain at this value for a short time. The effect of cycling on the field shape will now be discussed. On increasing the magnetic, field H r r o m zero to a given value the field shape will be as shown--referred to as "peaked"
I" \ \
J
,
H
~z
/
/
By adding a small increment to the given value for a few minutes the field shape then becomes "dished" as shown ~ory
I
I
J
l
,~z
\ )
/ J
NMR SPECTROMETERS
AND THEIR ACCESSORIES
205
After finding the optimum increment and time of application for a given operating field (and for a particular magnet) the homogeneity of a 12 inch magnet can be good for about 2 inches in either the x or the y direction. Cycling will not improve fiat or "dished" fields--obtained immediately by increasing the field to too high a value from zero. Several cycling operations may be required to get the best field shape. If a field develops too large a "dish" then a temporary complete reversal of the magnet current is necessary. A "peaked" field gives a resonance signal as shown in Fig. 6.3(A): only the centre of the receiver
H
H
FI~. 6.3 Signalshapes obtained from the cutting of the receivercoil by contours from "peaked" (A) and "dished" (B) fields coil is at the highest and most homogeneous magnetic field so that nuclei within this region of the coil resonate first, followed by those in the parts of the sample above and below the coil thus spreading out the trailing edge of the absorption line. In a "dished" field the nuclei both above and below the coil resonate first, followed by the nuclei near the receiver coil thus causing the signal to end rapidly and often giving rise to considerable "ringing" (see Section 2.10). A perfectly flat field at the receiver coil gives a symmetrical line shape. To achieve this condition it is usual to have the main magnetic field slightly "dished" to the extent of a few milligauss per inch. The field shape tends to become more "dished" with time but such small changes can be compensated for either by distorting the magnet yoke very slightly or by adjusting the shim coil currents. The inhomogeneity of the applied magnetic field normally determines the signal line width because line broadening caused by spin-spin and spin-lattice relaxation is usually negligible. This fact enables the width of a suitable line,
206
HIGH
RESOLUTION
NMR
SPECTROSCOPY
(for example the hydrogen resonance of the C H O group in oxygen-free acetaldehyde) measured at half itsheight to be used as a measure of fieldhomogeneity (spectrometer resolution). In practice,the half-height width of a multiplet component from a spinning sample of acetaldehyde should be better than 0"4 cycles sec-x if reallyhigh resolution work is being undertaken.
6.2.3 Sample Spinning Further improvement of the homogeneity of the magnetic field experienced by the nuclei may be brought about by mechanical motion of the sample( 16' 17). The hydrogen resonance signal from water is thus reduced in half-height width by a factor of 17 and the band height is increased by a factor of 7. One of the simplest means of providing motion is to spin the sample at a speed greater than 20 cycles sec -x about the y-axis. An example of the increase in resolution obtained in this way is shown in Fig. 6.4. The undesirable situation arising from spinning at too low a rate will be dealt with in Section 6.7. Good mechanical construction of the turbine and the selection of uniform glass sample tubes is essential, otherwise spinning will increase the noise level. In order to avoid hay= ing moving metal parts in the vicinity of the radiofrequency receiver coil, it is
H
ID
/•
(A)
(B)
(c) I
I
)
[
I
[
Fie. 6.4 The XH spectrum of the ethyl group of diethylamine at 40 M¢ sec-x. (A) without rapid sample spinning, (13)with rapid sample spinning, (C) with slow sample spinning. Each division on the scale represents 20 cycles s¢~-x
NMR SPECTROMETERS AND THEIR ACCESSORIES
207
customary to employ a plastic turbine driven by compressed air to give speeds up to 20,000 rev min-I: usually spinning rates of 30 to 40 rev sec -1 are used. Sample spinning is not effective when the resonance line has a large natural line width: ItJs difficult but not impossible to spin sample tubes having diameters as large as 15 mm when occasion demands. Bloch c1~ has given both a qualitative and a quantitative explanation of the effect of spinning. It is assumed that the main magnetic field varies over the sample by an amount ,4 H and that a molecule within the sample experiences the whole of this A H in a time interval t. Hence the more rapid the motion, the more a magnetic nucleus in the molecule behaves as if it were exposed all the time to the average field ( H + A HI2). The effect becomes appreciable when
t ___
(6.1)
where 3, is the magnetogyric ratio of the magnetic nucleus. Generally, only a partial narrowing is observed since the averaging process only occurs for those regions of the field through which a given molecule is carried. Using the phenomenological equations of nuclear induction, Bloch" s~ found an effective field inhomogeneity A Heft given by
:t
=
<(,t
o.
(6.2)
After a mean time 0 the motion within the substance exposes a given molecule to a randomly different field within the sample. ((/1 H) 2) is the mean square deviation of the magnetic field within the sample from its average value. Irrespective of the manner in which the external field varies over the sample, a Lorentzian line is observed with an effective spin-spin relaxation time (T2),ff given by (T2)eff---- [IIT2 + 3,(/IH),f¢]-I (6.3)
where T2 isthe spin--spinrelaxationtime correspondingto naturallinebroadening.On the one hand, Williams and Gutowsky ~29~have pointed out thatrandom processes are about twice as effective as coherent processes (spinning of a cylindrical sample) in reducing local field effects. They calculated that the effective spinning frequency ,: is given by ,, ~_ 5 ~/~7/I H/~z2 .
(6.4)
On the other hand, Kaplan <2°) has shown that periodic rotation of the sample is generally more effective in producing line narrowing. 6.3 Tim S ~ u ~
Hox~
By the sample holder or probe is meant the assembly which carries the air turbine for sample spinning, the receiver coil, the linear sweep coils and the pre-amplifier. The sample holder is mounted on a manipulator (which may be calibrated) to enable the best position in the main magnetic field to be sought. The probe body is fabricated from non-magnetic material, usually alnminlum. The body has to be rigid and has to provide accurate location of the receiver
208
H I G H R E S O L U T I O N NMR S P E C T R O S C O P Y
coil. Probes containing a combined transmitter and receiver coil have to be adjusted electrically, while the double (crossed) coil probe may be adjusted either mechanically or electrically. The two types of probe are sufficiently different to warrant separate discussion.
6.3.1 The Crossed Coil Probe This detection system was employed by Bloch, Hansen and Packard (21) in the first successful nuclear induction experiment and has since been used in instrumentsbuilt for high resolutionwork (see forexample reference 22). A commercial version of it is shown in Fig. 6.5. The probe assembly shown is mounted on a Transmitter input Nylo n s p nner~ _ _
Recurrent sweep ~
A
I: I~samp )e ~
tube(Store o.o.)
,//
Fine paddies
~ . ~ . ~ r ~ ~ ~ ~ - - T o
receiver
\ ~eceivercoils on removable insert \One transmitter coil Fro. 6.5 A sample holder for the crossed coil method of detection. By courtesy of Vad~n Associates
manipulator to enable the best position in the main magnetic field to be found. In this type of equipment, any lack of orthogonality between the two coils will cause the transmitter coil to couple with the receiver coil and hence a voltage (the leakage voltage V,) will be induced in the latter. Electrostatic coupling is reduced to a minimum by placing a Faraday shield between the two coils. Typical experimental conditions are such that V~ ~ O × 103 volts where (90 ° - 0) is the angle between the receiver and transmitter coils. Since a nuclear induction signal (Vo) is of the order of I millivolt, in order to make VdVo l, 0 will have to be only about l0 -6 radians. Such a fine adjustment is not necessary because (i) only a reduction of the leakage signal by a factor of 104 is required, since some leakage is desirable, and (ii) this reduction can be accomplished without disturbing the fine adjustment of the geometry of the coils. This required adjustment is made by a very small rotation of the slightly skew mounted receiver coil within the fixed transmitter coil. In order to control the leakage flux without moving the receiver coil, Bloch, Hansen and Packard (21) mounted a semicircular sheet of metal (called a "paddle") at the end of the transmitter coil. The paddle was mounted on a spindle lying ~
NMK SPECTROMETERS AND THEIR ACCESSORIES
209
along the axis of the transmitter coil thus giving control of the radiofrequency flux as shown in Fig. 6.6. In the experimental arrangement shown in Fig. 6.5, two types of paddle give a variable degree of inductive and capacitive coupfing between the crossed coils. Adjustment of these paddles gives a controlled finite Poddte
',,.
C,
','
.
",
,
,
,',It
Poddle
FIG. 6.6 The effect of a paddle on the fines of force of the radiofrequcncy field produced by the transmitter coil
i
O
/' i t
~} /
",
j[i
FIG. 6.7 The magnet3sationvector Mo and its component in a system of rot~ tmg coordinates
leakage voltage enabling the receiver detector to operate at an efficient level, and also allowing the desired mode of the complex NMR signal to be selected. These points will now be explained. Using a system of rotating coordinates (see Section 2.12.3) the x y plane is rotated about the z-axis as shown in Fig. 6.7. If the main field/4"o is at a value far below that required to satisfy the resonance condition, the application of a fixed value of the radiofrequency field at
210
H I G H R E S O L U T I O N N M R SPI~CTROSCOPY
low amplitude leaves the magnetisation vector Mo coincident with the z-axis. As the value of Ho is increased slowly, the magnetisation vector will move away from the z-axis and start to precess and will therefore have projection components in the x y plane. When the resonance condition is reached, the component M,,, will have its maximum value and will then die away to zero as Ho is increased further. The magnetic field represented by the vector M~.y. induces a voltage B
~'\\ Mx%,"
--//
t,H~ x"
y"
~o
_J
0
~
&Ho
FIo. 6.8 A. The x' y' planeviewedfrom above showingthe locus of the Mx, ~, vector. The heavyline showsits positionat resonanceand the dotted linesshow its position away from resonance. B. The magnitude of the M~, andM~, vectors as a function of the main field strength Ho V-mode siqnal
O-rood, I signal I
V-mode leokoqe
r,o.
elver
Coil
"
I1
Fio. 6.9 Suppression of the U-mode (M~,) component of the magnetisation vector by adding in-phase leakageto the V-mode(M,,) component in the receiver coil which can be amplified, demodulated and recorded graphically or presented on an oscilloscope. From Fig. 6.8 B it may be seen that only the component My, is of interest ff the usual absorption line is required. The component M=, gives the dispersion mode (U-mode) which produces a spectrum more complex in appearance than the absorption mode (V-mode). The M~, component is suppressed by introducing leakage in phase with the x'-axis into the receiver coil from the transmitter coil. The leakage voltage serves as a reference for the absorption signal and suppresses the undesired mode(23). Figure 6.9 illustrates diagrammatically the relationship between the various voltages involved. Changes in the length of the signal vector are proportional to cos~0. The V-mode paddle is a small cylinder of copper attached to the end of a plastic rod. The U-mode paddle is similar except that the cylinder is replaced by a ring of resistive material such as gold leaf. The equivalent cir-
211
NMR SPECTROMETERS AND THEIR ACCESSORIES
cults are a shorted coil with little series resistance for the V-mode paddle and a shorted coil with a high distributed resistance for the U-mode paddle. Any particular paddle adjustment affects both modes due to the inability to reduce the inductance or the resistance to zero but, however, the V-mode is relatively pure in comparison with the U-mode paddle. Usually it is necessary to make a compensating adjustment only if U-mode leakage is introduced. The magnitude of the effect is diminished as the amplitude of the radiofrequency field is decreased.
The optimum leakage voltage. The amount of leakage required depends upon the signal strength: low leakage values are employed for weak signals and vice versa. Large leakage to signal ratios give rise to instabilities (due to changes in the radiofrequency power level) appreciable in comparison with the signal. Small leakage to signal ratios will clearly cause distortion of the absorption line. The recognised compromise between these two limits is a leakage to signal ratio of 5 to 1. The need for this compromise has an important bearing on the usefulness of absorption band areas t24). If an optimum leakage to signal ratio is not adopted errors can be introduced into the integrated absorption signal and these arc particularly important in the application of NMR to quantitative analysis. From the vector diagram in Fig. 6.9, the observed integrated absorption signal is proportional to t24~ (6.5)
f(R - L)dt -- f ( l L -F V +jU[ - L ) d t
where L is the leakage voltage, V the absorption mode signal, U the dispersion mode signal and R the resultant signal. The required integrated signal is
f Vdt
(6.6)
Williams c2.~showed that for a damped oscillator shape function, and in the absenee of saturation, the ratio of the actual integrated signal to the required one, expressed as a function of R --- V,_./L, is OD
(2R + observed integrated voltage required integrated voltage
lg
R2)qt/~
]
--cO o0
l~dt -
(6.7)
R ~ ~-7-7--7i-,2dt
,}l+t
This ratio, which represents the ratio of the observed band area to the true band area, is unity only when [ R ] ~ I. When R is positive the ratio is greater than unity and when R is negative the ratio is less than unity. In the latter case, R cannot be less than - 1 without serious distortion o f the signal. The calculated error in the observed integrated voltage is plotted in Fig. 6.10 and thus it is
212
HIGH
RESOLUTION
NMR
SPECTROSCOPY
possible to correct the voltages of absorption signals recorded under conditions of insufficient leakage. When the widths of the bands in a spectrum are determined by the inhomogeneity of the main field they will all have the same value, and the band heights are unaffected by the leakage to a first approximation. Spectrometers employing a phase-sensitive detector (25) following the last stage of radiofrequency amplification in the receiver avoid these errors in the observed signal intensity because the dispersion mode contribution to the absorption signal is eliminated(15): the instrument is then usually operated with minimum leakage. 6O
-~
40
b
-40 -I
,
0
i
~
2
3
~laxirnum siqnol voltage L.eckaqe vottoqe
FIG. 6.10 The error introduced into the integrated signal voltage due to insufficient leakage voltage. Williams(2'*)
Temperature control of the specimen. With the probe arrangement shown in Fig. 6.5 the sample experiences the ambient temperature. Occasionally it is necessary to obtain the spectrum of a substance at other temperatures, for example, (i) Improved spectra may be obtained at elevated temperatures when (a) the viscosity of the sample is reduced significantly thereby decreasing line broadening (b) solids are melted to low viscosity liquids (c) the solubility is increased appreciably. (il) Measurements must be made at several controlled temperatures for the determination of the heights of potential barriers in interconverting molecules, of the enthalpies of equilibria and of chemical reactions (see Chapter 9). The simplest and most versatile temperature device for the Varian crossed coil probe is that developed by Brownstein (26~ (see Fig. 6.11) from an earlier design of Shoolery and-Roberts C2~). Any design has to overcome the difficulties imposed by the small space available, the need to spin the sample tube and the necessity of providing insulation to avoid changing the temperature of the probe
NMR SPECTROMETERS AND THEIR ACCESSORIES Receiver .~insert coil
71
Cold or hot gos
•.,~-- 3 cm - - - ~
vessel
tl IIII c°°*i°' ~]~l~p,~ Nylon
FIG. 6.11 The Brownstein(2e) thermostatted assembly for the VarianHR-60 probe
Hostoflon somple tube (slotionory)
J[-
~,~[]mm
R ~ . . ~ GlOsS shield for
£q F~-/-/~II|I| ~ ~t
~///~IBIBIB
,,oo_.., co,
~ly///L_Co.de.sed
somple
.o..s ,o, ~ _ ~ / / / A I I I B I ~ bkoge~[~///./jl~ [ ~ II[~.,,~,._Tronsmitter ,r--,., "°~-'-~.. V.~/~ll IB "~S~ oo, o,,,
~
R
e
c
e
i
v
e
r
cotl
FIG. 6.12 Device for handling liquefied gases by cooling the sample zone. Hamer, Lee~ and Bentley(~9)
213
214
H I G H R E S O L U T I O N NMR S P E C T R O S C O P Y
itself too drastically. The sample may be heated by passing a stream of air through a transfer dewar which contains a spiral nichrome heating wire (not shown). Low temperatures may be reached by using a stream of cold nitrogen from a liquid nitrogen boiler. In Fig. 6.11 the receiver coil may be seen to be surrounded by a cylindrical unsilvered vacuum jacket open at both ends, but sealed by nylon plugs when in position. Silvering of the dewar is best avoided because of the severe attenuation of the radiofrequency power. The only iraTurbine stotor~
N ~ex,t .,=~-------_.-.~-- ' "O"rinq.
ompressed air"
.~ ~
N
~
inlet
B 2~
joi~t
L
i
,
)
,
cm
P.T.E E:
beefing FIe. 6.13 Dewar insert to enable spectra to be obtained in the range - 150 to + 50°C. The hatched portions in the upper part of the diagram represent nylon
portant modification which needs to be made to the original Varian probe is the fitting of an aluminlum extension to the aluminium spinner starer. Furthermore, the whole assembly is so unobtrusive as to allow normal operation of the spectrometer at room temperature without impairing its performance (giving a resolution of better than 1 part in 10s). The thermostatted assembly has been found to provide control over the range - 150 to + 150°C in the authors' laboratory, the temperature being measured by means o f a thermocouple located about 5 mm from the receiver coil. For accurate work it is necessary to apply a correction for the-difference in temperature between the sample and the thermocouple; corrections (for a given rate of flow of gas) can be found directly by locating a second thermocouple within a spinning liquid sample
NMR S P E C T R O M E T E R S AND THEIR ACCESSORIES
215
at the centre of the receiver coil. Another, more complicated, version of Shoolery and Roberts' insert dewar is that designed by Franconi and Fraenkel (2". Compounds which are gases at room temperature may be liquefied by cooling the sample zone in the range of - 140 to + 20°C with a stream of cold nitrogen without damaging the probe ¢29~.This was carried out by enclosing the entire probe containing dry air in a Perspex box (see Fig. 6.I2). Because the sample tubes were left connected to the gas handling system, sample spinning was not
/ 7
1 8 / 9 B$
'
'
B 29
,. Tungsten wire
......
P.T.F. E. turbine
i" Soml~
tube
. Tube ground / " to a point I
=
I
I
I
I
I
Iocm
Fxo. 6.14 Temperature control device designed by Schnmder, Bvrnstein and Pople(so)
possible and the resolution was then about I part in l0 T. However, a simpler system has been developed in the authors' laboratory (shown in Fig. 6.13). Sample tube spinning is preserved, the temperature range is - 150 to + 50°C and a given temperature may be held to within + I°C. It provides considerable thermal isolation of the receiver coil from the cooled region. The O-ring is necessary to prevent the nylon dewar head from sinking too far into the B.29 socket on cooling. Theleakage controls need to be adjusted when the temperature is changed, because small thermal movements affect the cross-coil coupling. Some sensitivity is lost when the device is used since a 15 mm diameter receiver coil has to be used thereby reducing the filling factor (see Section 2.11). The apparatus is particularly useful when highly unstable compounds have to be
216
H I G H R E S O L U T I O N NMR S P E C T R O S C O P Y
transferred from say a vacuum line to the spectrometer with a minimum of warming up. Another thermostatting assembly which also requires a 15 mm insert is that described by Schneider, Bernstein and Pople c3°) (see Fig. 6.14). Its construction requires highly-skilled glass-blowing techniques and the alignment of the receiver coil is easily disturbed in operation because of the rigid nature of the apparatus. Also, introduction of samples is difficult. The main advantage of the assembly is that it can provide thermostatting in the range - 150 to + 300°C. The version constructed in the authors' laboratory has a turbine head fabricated from
FiG. 6.15 The sample holder of the nuclear induction spectrometer manufactured by Triib, T~uber and Co., A. G., Ziirich polytetrafluoroethylene in the shape of a straight sided spiral reamer. Close temperature control is achieved by fitting a nichrome wire spiral of 50 ohm resistance into the central tube of the horizontal part of the apparatus: potentials up to 100 volts provide the necessary fine heating control. A copper--eonstantan thermocouple is located about 5 mm from the sample coil and the leads pass through the nitrogen outlet tube. All the descriptions of apparatus given so far in this section refer to the Varian Associates HR-60 probe. Another double coil probe is that manufactured by Triib, Tiiuber and Co.,A. G., Ziirich~3.t' 32), (shown in Fig. 6,15). The receiver coil E is wound on a tube 4 m min diameter, inside which the 3 mm diameter glass sample tube is situated. The liquid sample is introduced via a capillary
NMR SPECTROMETERS
AND THEIR
ACCESSORIES
217
of 0"1 mm diameter into a spherical cavity of 1 to 2"5 mm diameter. At each end of the sample tube sapphire bearings are located on points of beryllium bronze enabling the tube to be spun at speeds as great as 10,000 rev rain -1 by means of the air turbine T. The four transmitter coils S are wound on a Perspex former R so arranged that the radiofrequency field is approximately at right angles to the axis of the receiver coil. Paddles are not used to adjust the leakage in this instrument; instead simple movement of the Perspex ring provides leakage compensation. Final precise adjustment is made electronically in the combined cascade pre-amplifier and leakage compensator (a2). Since the leakage is practically balanced out, the resonance signals due to absorption and dispersion are obtained by means of a phase-sensitive detector. Auxiliary apparatus is available for varying the temperature of the sample. 6.3.2 The Single Coil Probe In the first s uccessfulnuclear magnetic resonance experiment, Purcell, Torrey and Pound (33) used the single coil method of detection in conjunction with a bridge circuit t-'3). The bridge network balances out the very large transmitter C' ........ ~
C,"
I !f ~~- - - . . . . . . . . . . . . . . . . . Sample coil
F, - I F - - ~ - - - ~ ' ,~i ..~,,
71- % To ©' transmitter
r I
L.
¢.
To receiver I~ 'R
I Haft woveleclgth of cable
? F Dummy
C2
It
FIo. 6.16 A radiofrequencybridge whichincludesa tuned circuit for the sample and a dummy tuned circuit for balancing purposes. Bloemberl~zb Purcell and Pound(2S) signal and allows the small absorption or dispersion signal to appear as an out of balance electromotive force across the bridge. Bridge circuits were used extensively in the early work on nuclear magnetic resonance; they have been discussed in detail by Andrew (3.). In high resolution work, the twin-T bridge has been adopted for spectrometers built by research workers (3s-~9) and by manufacturers. The twin-T bridge has the advantage that it gives good stability
218
HIGH RESOLUTION NMR SPECTROSCOPY
of bridge balance which is essential if a mixture of U- and F-mode signals is to be avoided. The unwanted dispersion signal can be suppressed by taking advantage of the fact that the absorption and dispersion signals differ in phase by 90 °. A slight out of balance of the transmitter signal (leakage) is necessary for the reasons given in Section 6.3. I. A diagram similar to that shown in Fig. 6.9 could be drawn in which a little of the transmitter signal (F-mode leakage) supplements the absorption signal (also F-mode) thus swamping the unwanted dispersion signal (U-mode). A quantitative interpretation can be made (4°) using the bridge circuit of Bloembergen, Purcell and Pound (23) as an example (see Fig. 6.16). The bridge used by these workers consists of two almost identical tuned LC circuits in parallel, connected to the same radiofrequency oscillator by means of small coupling capacitors and properly terminated coaxial cables. An extra electrical half wavelength of cable is inserted in one circuit and the outputs are connected to a point D and then to a receiver circuit. The half wavelength of cable serves the purpose of reversing the phase of the voltage output of the sample circuit at D compared with the phase of the dummy circuit, hence the voltages are subtracted./:or two identical circuits balanced for the radiofrequency used, the only voltage reaching the receiver would be that induced by the magnetic nuclei. It was shown in Section 2.11 that the potential difference across the sample coil induced by the nuclei is given by A F ffi - ~ 8 r * A H I co(-- Z' since t + Z" cos co t).
(6.8)
This may be related to the characteristics of the tuned circuit and the radiofrequency voltage, Vo, applied to it. On applying the definitions Vo = Vo° cosco t and
(6.9)
1°Re = V°
(6.1O)
Lo I ° = 2 H t A
(6.11)
to equation (6.8) it becomes Re ( - g ' since t + Z" cosco t) V°o.
/I V = - ~, 4~r c o l e
(6.12)
The term coLe~Re may be re-written as Q, the quality factor of the circuit. The term in the real part of the susceptibility Z', is out of phase with Vo hence equation (6.12) can be written in the explicit imaginary form / I V = - ~ 4~zQ(Z" + iz' ) Vo.
(6.13)
The voltage induced by the nuclei can be seen to be a mixture of the nuclear dispersion and absorption modes. By controlling the nature of the unbalance of the bridge, the two effects can be separated and observed individually. Taking first the situation of amplitude unbalance where the output voltages at D are exactly in phase but have different amplitudes, the voltage v at D is then the difference between the voltage Vt from the dummy circuit and the eel-
NMR SPECTROMETERS AND THEIR ACCESSORIES
I
219
rage V from the sample circuit. The latter is the voltage Vo applied to the sample plus the A Y from the nuclei, hence
vfVh 1 -- V (Vo + ~ F ) Vl° cosco t
--
[Vo° cosco t -- ~ 4 : ¢ Q ( -
Z'
sinco t
+
X" cosco t) Vo°]
(6.14)
(I~i -- Vo° + $ 4~zQx" Vo°) coso~ t - ~ 4 = Q z ' Vo° sino~ t 0 ~ (V10 - l~o + ~ 4 ~ Q z H V;)cos¢o t = Vl - Vo(1 - ~ 4=Qx")
If the residual balance is appreciable in that Vt° - Vo° is several times larger than ~4n Q Z' Vo°, then the term x' sinco t is responsible mainly for the shift in phase of v but its amplitude is left unchanged. As Ho is swept through the resonance condition, the receiver detects ~", the absorption (or V-) mode of the resonance. For phase unbalance with amplitude balance, VI ffi Vo° cosco t + V°o sinco t
(6.15)
which gives v = V°o sinco t + ~ 4 ~ Q ( - Z' sin.) t + g" cosco t) Vo° I
= (V°o - ~ 4=Qz' Vo°) sinoJ t + ~ 4=Qz" v~o cosa) t i
(6.16)
"" (V°o - ~ 4 ~ Q z ' Vo°) sino) t The xeason for dropping the Z" term is the same as that for dropping the ~' term in equation (6.14). For this situation the dispersion (or U-) mode will be detected by the phase unbalance of the radiofrequency bridge. E F P'-" [ i B/ C/
i .
•
~---~F U Top view
Side view
A -"--"
~D
B, / C,/
Flo. 6.17 Sample holder of a single coil nuclear magnetic resonance spectrc~ meter. Leane, Richards and Schaefer ~39)
220
H I G H R E S O L U T I O N NMR S P E C T R O S C O P Y
If the bridge is balanced too closely, a mixture of Z' and Z" will result despite the presence of pure phase or amplitude unbalance. More rigorous analyses of bridge balance have been made (33) in which the undesirable intermediate line shapes can be described. In practice, the bridge is balanced by having two manipulator
o0,o,,,o,eo , ,leoo
~
~
,~ir s,Jppiy tO turbine
FIG. 6.IS(a). Side view of the probe. Hertz and Spalthoff (al)
Pole cop
Coils
Shim
=
-~'FIll ~ ~ ~ - - - _ 7 - - A i r
[
Centering rinqs
Shim
Pole cop
(b) F[O. 6.18(b).
Top view of the probe. Hertz
and Spaithoff ('~j)
NMR SPECTROMETERS AND THEIR ACCESSORIES
221
adjustments to control the phase and amplitude independently as in the Anderson bridge. The phase of the output voltages from the tuned circuits can be adjusted by means of the main tuning condensers C3. The amplitudes of the voltages can be adjusted with the coupling condensers C1 a n d / o r (=2. Since the coupling condensers are effectively in parallel with (73 via the small impee Heati COi
,uency :ouple
FIo. 6.18(c). Modification of the probe for examining samples at temperatures above that of the room. H~'tz and SpalthoflX4z) dance matching resistor Rx then a change in Ca affects the phase as well as the amplitude balance. This is avoided by using a differential condenser combination CI and C4 so that capacitance removed from Cl is added to C4. In this manner, the phase control C3 and the amplitude control CI--C4 are essentially independent thereby allowing the bridge to be balanced readily. The tuning elements are usually mounted on the magnet close to the probe containing the radiofrequency coil. As in the case of the crossed coil system, it is essential to make the probe from a material like alumin/ttm. A typical sample holder for a single coil spectrometer is shown in Fig. 6.17. The single radiofrequency coil is wound on a 6 mm diameter thin-walled glass tube, and
222
H I G H R E S O L U T I O N NMR S P E C T R O S C O P Y
the leads are brought upwards to a short coaxial cable connected to the bridge contained in a cubical copper box set into the top of the magnet. The coil is screened, and protected from contamination, by the aluminium outer tube D. The aluminium ends B and B' of this tube have inserted into them PTFE bushes, C and C', to allow the sample to spin freely. The sample tube is about 12 cm long and 4.5 mm in diameter and is mounted in a plastic turbine. Air Slandard somple tube
e coil
3cket
ports
FIo. 6.19 Section of the A.E.L R.S.2 variable temperature probe in position on the manipulator. By courtesy of Associated Electrical Industries Ltd.
for spinning purposes enters from the jets F after being brought into thermal equilibrium with the permanent magnet. Hertz and Spalthoff cat ) have also described a single coil high resolution N M R spectrometer. Their probe is shown in Figs. 6.18(a), Co) and (c). Another version is that of Kowaleski and Hoffman ('2). A commercial version of the single coil probe is that of the A.E.I.R.S.2. spectrometer, shown in Fig. 6.19. This design allows the ready fitting of a dewar assembly (shown in Fig. 6.19) containing a heating coil which enables the sample to be heated up to 250°C. The probe block has incorporated in it current shims for final adjustment of magnetic field homogeneity.
NMR SPECTROMETERS AND THEIR ACCESSORIES
223
6.4 MAGIs~'nc Fna.n S w E ~ As in optical spectrometers, a means of scanning the spectral region has to be provided. The conventional method of scanning by varying the frequency (or wavelength) can be used (12) but in the nuclear magnetic resonance experiment one has a choice between varying the radiofrequency and keeping the steady magnetic field fixed, or varying the steady field and keeping the radiofrequency fixed. Most spectrometers have available two types of magnetic field sweep, a recurrent sweep and a slow sweep. Both are linear in the sense that the rate of change of the sweep field is constant with time. In the former case, the output of a saw-tooth generator is amplified and then fed to two small Helmholtz coils usually mounted on the outside of the probe body (see Fig. 6.5) and having their axis in the same direction as that of the main magnetic field.
HI
~0 . . . .
Mz
/VVVV Fro. 6.20 Magnetic field scanning with a recurrent sweep
The purpose of the recurrent sweep is to allow oscillographic presentation of the signal to be made by feeding the output of the sweep amplifier to the X plates of the oscilloscope. The effect of the recurrent sweep is shown in Fig. 6.20. If the steady field Hz lies sufficiently close to the resonant field value Ho, the signal may be displayed oseillographically: the resonant condition is passed through on the return as well as on the forward sweep but this is so rapid as to be unnoticed. For the conditions shown in Fig. 6.20 the signal would appear at the high field side of the oscilloscope trace. Because the saw-tooth output provides a field sweep through a balanced cathode follower, the average value of the total msgnetic field is the same with or without the sweep. The sweeps on the oscilloscope and the field are automatically synchronised and no phase shifter is needed. By varying the time base and the output of the saw-tooth generator one can control both the rate (from several seconds to several minutes) of the recurrent magnetic sweep and its amplitude (from a milligauss to several gauss) (43). This linear sweep is brought into play when searching for the resonant field of a particular nucleus and when making adjustments to the spectrometer such as tuning the probe for a particular line shape, or when locating the probe in the magnetic centre of the main magnetic field. When the sweep rates greatly exceed the relaxation time ofthe nucleus under examination, the absorption line will be distorted. In particular, large amounts of ringing (see Section 2.10) will be seen at the traillng edge of the absorption line. Since
224
HIGH RESOLUTION
NMR SPECTROSCOPY
ringing can be a good indication of field homogeneity it is an asset when adjusting the shim coil voltages. The slow sweep unit is provided to enable the signal to be recorded, by means of a pen recorder, under slow passage conditions in order to minimise line distortion. The slow sweep can be introduced simply by applying a small d.c. voltage to the galvanometer in the flux stabiliser (see Section 6.2.1). Thus the change in the static magnetic field is brought about by an increase in current through one of the correction coils.The magnitude of the d.c. voltage determines the rate at which the small applied field is varied. Normally, the rate is continuously variable from zero to about 500 milligauss min-L The method cannot be employed for fast rates of scanning because the high inductance of the correction coils, (because of their being mounted on the pole caps) restricts their rate of response. Field or frequency instability can give rise to broadened lines but it should be remembered that minimum distortion of the signal occurs when the resonance is swept through in a time long compared with the relaxation times T~ and 7'2, that is, the sweep time is much greater than the inverse of the line width. Unless this condition is met, the broadening of the natural line may exceed the field inhomogeneity broadening. Some calculated line shapes resulting from various sweep rates are shown in Fig. 2.7. A line 0.2 cycles sec -t in width would require a sweep rate of about 4 cycles see -~ min -~ to avoid appreciable broadening. This is so slow as to require some kind of field-frequency controller(~4~ which is usually part of a commercial spectrometer. The lines recorded in such an experiment are followed by very little "ringing". In order to eliminate wiggles the rate of change of the main field d H / d t must satisfy the requirement (45) d H / d t <= (4 17~1T2)-'.
(6.17)
A lower limit is imposed on the sweep rate by the slow drift of the static magnetic field. Another parameter affecting the choice of sweep rate is saturation (see Section 2.8): this is of particular importance when intensities of bands are to be measured, as will now be discussed. Starting with the equation (developed by Bloembergen, Purcell and Pound (23)) for the excess of population between the energies of two spin states d n / d t = (no - n)/T1 - 2n W
(6.18)
where n is the excess population, no is the equilibrium value in the absence of radiofrequency radiation and W is the transition probability, Williams(24) made several simplifying assumptions: as a first approximation to the solution of equation (6.18)only the cases for which 2n W ~ (no - n)/T1 were considered. This situation can arise when T1 is large and passage through resonance is sufficiently rapid, so that dnl/dt ~ -2nl
W.
(6.19)
NMR SPECTROMETERS AND THEIR ACCESSORIES
225
A better approximation is obtained by substituting for dn/dt in equation (6.18) the approximate value given by equation (6.19): n2 ~
no 2 WTt nt + 1 + 2WT~ 1 + 2WT:
(6.20)
where n2 and nl are the populations of the upper and lower energy levels respeetively. Assuming that the signal shape is that corresponding to a damped oscillator, that the field sweep is linear and that the centre of resonance is at t = 0 on the time scale of the field sweep, then
r" tt~r, [ 2 W = 1 + T~(A to)2 = ZT;'
(6.21)
[1 + (T2 r t)2] -'
where Z is the saturation parameter equal to y2 ././12T1 T2 and r is the sweep rate given by r = d (,4 to)/d t = y d H/d t (6.22) Williams t2+~ obtained the following solution for equation (6.19)
(tan-' T2rt + ~t]2)]. nl N no exp [-Y2H------~ r
(6.23)
The power absorbed, P, is P ~ 2n2 W/~ 1,
(6.24)
and the total energy absorbed is
A ~ h • f 2n~ W d t
7 H12
r
+f! •
1
Z exp
.~ ~ h , n n - ~ + v dH/dtL~/1 + Z -®
(6.25) r~r2 2
(tan-I x + ~12)
dH/dt + Z + x2)(1 + x 2)
dx
(6.26) The integral in the square brackets was evaluated by means of an electronic digital computer for a range of values of Z and
-t-t~
[
dH/dt
d H [ -1 7'T~ dt [
over the integration limits (6.27)
where T] includes a contribution from the main field in homogeneity. Figure 6.21 is a plot of the entire expression in the square brackets of equation (6.26) as a function of Z and 7'H~ . When the latter term is large, it may be seen
dH/dt
~
8
226
HIGH
RESOLUTION
NMR SPECTROSCOPY
that the slow passage result is obtained; when it is small, the saturation parameter Z has only a negligible effect. An NMR spectrometer usually measures the voltage induced in the sample coil by the rotating magnetisation, and the total energy absorbed is related to the V-mode magnetisation by the equation ¢D
A = Hi ~o f v d t.
(6.28)
--(30
The time integral of the signal voltage from the spectrometer is proportional to A/H1. There is a transient oscillatory contribution to v during non-adiabatic rapid passage giving rise to wiggles but, from the results of detailed treatments T
2 2
HI T~ T 2 0
i.O
i
I
m~
I
i,~:~
0.6
i~:
0.4
4,
0-3
'
[
I 2
O0 P
OH
T H J~-i-
Fro. 6.21 Absorptionof radiofrequencyenergyas a functionof saturation and the rate of field sweep. Williams(24)
of non-adiabatic passage (45, 4~>, Williams considers that the integral of the oscillatory term is either zero or negligibly small when Z ~ 1. In the above equations account has not been taken of the contribution to 1"2 from inhomogeneity of the main magnetic field. Line broadening is dependent on 2"2 (see Section 2.7) whatever the mechanisms contributing to 2"2(47). The area of an absorption band broadened by field inhomogeneity is independent of the degree of inhomogeneity. There is a practical point here, namely that in order to obtain an undistorted signal, the receiver and recorder circuits must have a minimum bandwidth, which can pass all the important frequency components of the signal. This bandwidth depends upon the observed width of the absorption band. The time constant requirements of the receiver-recorder system (see Section 6.5) for low distortion are given by the approximate relationship dH 3-1 > yT~' d--t-"
(6.29)
NMR SPECTROMETERS AND THEIR ACCESSORIES
227
6.5 THE RADIOFREQUENCYOSCILLATORAND REGEIVER The upper limit of about 53,000 gauss for the main magnetic field sets the highest radiofrequency required at about 220 Mc sec -1 (for the 1H and 19F nuclei): the lower limit is about 3 Mc sec -1 (for I"N nuclei). All nuclei could be studied with the latter frequency but the sensitivity relationship(4s) given below dictates that the highest value of the static field should normally be used. S =
C N H 2 7'31(I+ I)
(6.30)
7
(o)
{c)
FIG. 6.22 The hydrogenresonancespectrumof the CH2 group in ethyl alcohol at (a) 12Mcsec-I (b) 30Mcsec -t (c) 40Mcsec -I. By courtesy of Varian Associates
S is the signal strength, N is the number of nuclei under consideration, H is the applied field strength and T is the absolute temperature at which the experiment is being performed. The analysis of spectra is aided considerably by the use of more than one irradiating radiofrequency for a given nucleus (see Section 8.19.5). Since the magnitude of the chemical shift is directly proportional to the strength of the static magnetic field whereas spin-spin inter8*
228
H I G H R E S O L U T I O N NMR S P E C T R O S C O P Y
action is independent of it, we have a means of distinguishing between them by obtaining the spectrum at more than one field strength. It is common to encounter molecules in which the chemical shift is of the same order of magnitude as the spin-spin coupling constant. When this occurs a complex spectrum is obtained and it can be difficult to assign completely the fine structure in such a spectrum. Some of the multiplet splittings are inversely proportional to the main field strength. Figure 6.21 shows how the appearance of a given spectrum containing all the above features is affected by the change of radiofrequency (and main field). It is often necessary to examine complex systems at more than one radiofrequency to confirm a spectral analysis. H (a)
f
!b)
Fie. 6.23 The hydrogen resonance spectrum of 5 ~-stigmast-22-en-3-one in
CDCI3 at (a) 40 Mc sec-1 (b) 60 Mc sec-1. Slomp(s°) The improved sensitivity and dispersion afforded by working at as high a field as possible is particularly useful for large molecules having low solubility or available in insufficient quantity. Steroids fall into the latter category(49); they are normally studied by dissolving from about 5 to 50 mg of substance in about 0.5 ml of CDC13. Figure 6.23 is a comparison of the spectra (5°) of 5/~stigmast-22-en-3-one at 40 and at 60 Mc sec -~. The stability requirement of the radiofrequency oscillator is clearly similar to that of the main magnetic field, namely 1 part in l0 s to 109 per minute. The general method of producing frequencies in the range required for nuclear magnetic resonance is to multiply the frequency of a quartz crystal. Some transmitters give different frequencies by controlling the extent of multiplication of the output from a crystal-controlled master oscillator. Others operate by the substitution of different quartz crystals. Some oscillators have their crystals installed in a small oven regulated to within about :t: 0"01°C. In addition to
NMR SPECTROMETERS AND THEIR ACCESSORIES
229
frequency stability the transmitter must have a constant level, usually accomplished by incorporating automatic gain control into the circuit. A means of controlling the radiofrequency power is an essential part of the circuit. So far in this chapter it has been tacitly assumed that a high degree of stabilisation of the radiofrequency oscillator and the magnetic field is necessary. However, since one is only interested in the strict proportionality o f the frequency and the field strength there is no need to stabitise them separately. Baker and Burd c12~(see Section 6.2.1) have controlled the oscillator by the u s e o f a nuclear magnetic ~resonance probe situated in the field, but a more general method which greatly reduces the requirements for time and temperature stabilisation o f the magnet has been devised by Pound and Freeman ¢51). In their method, a constant wave oscillator is phase-locked to a simple super-regenerative oscillator producing a coherent signal at the magnetic resonance frequency of a sample in its coil. This method enables high resolution N M R spectra to be obtained with simpler equipment.
(o}
Ib)
FIo. 6.24 The IH resonance spectrum at 60 Mc sec"1 of 1 per cent ethyl benzene in carbon tetrachloride (a) one conventional spectrometer sweep (b) the computer-average of a hundred sweeps. A filter bandwidth of 0.4 cycles sec-1 was used in both cases. By courtesy of Varian Associates~s~
230
HIGH RESOLUTION NMR SPECTROSCOPY
Enhancement of sensitivity. The effective sensitivity of a spectrometer can be improved by a factor of about fifty by time-averaging signals over a long period C95). One way of achieving this is to feed the spectrometer output into a computer of average transients (" CAT")c52' 53). The technique requires that the spectrum be scanned many times in a synchronous fashion. The repetitive experiments then synchronise the averaging process with the result that the desired signal is reinforced because all additions of a positive signal are in phase. Noise signals will add out of phase and thus be averaged out. The noise adds as a root mean square thus giving a signal-to-noise ratio improvement by a factor of l/n, where n is the number of experiments performed. Figure 6.24 shows the improvement which results from a hundred repetitions: trace (b) is the integrated computer output fed into the chart recorder of the spectrometer. A computer having 400 channels only allows about 80 cycles see -a (at 60 Me see -a) to be scanned at a time. A computer having the desirable number of channels (about 4000) required to give a typical scan of 500 cycles see -t costs as much as the spectrometer itself. The full benefits of the method are only realised if the spectrometer has a field-frequency control loop. The main advantage of the method is that it is possible to follow the increase of sensitivity during measurement and to stop the repetitions when the sensitivity seems to be sufficient. A disadvantage of the method is that the resolution is diminished by the finite number of points taken into the calculation. An alternative and essentially equivalent means t54) of increasing the spectrometer sensitivity is to increase the sweep time of the spectrometer to several hours. Again, a field-frequency control loop is essential, and furthermore, it must be of the type using a single probe. (The stability of a spectrometer having two spatially separated probes--one for providing the spectrum and one for the stabilisation--is only satisfactory if the field between the probes is constant.) Ernst and Primas c55~ have achieved a long-term stability of 0.0003 ppm over several days for a spectrometer operating at 25 Me see -1, hence it is feasible to record long-term spectra overnight. The main advantage of the method is that it is simpler than the computer method. 6.5.1 Radiofrequency Power and its Measurement From equation (6.28) it is evident that the application of a large radiofrequency magnetic field, H t , produces a large signal strength. One must bear in mind, however, that saturation (see Section 2.8) and radiation broadening will set in as the magnitude of Ht increases, resulting in a decrease of signal strength: to avoid saturation the condition HI
NMR SPECTROMETERS
AND THEIR
231
ACCESSORIES i
tration of free radical used. High concentrations cannot be used because of the onset of paramagnetic line broadening. Saturation can have a profound effect on the appearance of a spectrum (see Fig. 6.25). Because of the uncertainty of knowing whether saturation is occurring in the whole or part of a spectrum, it is advisable to measure and to report the optimum value of H1 used when publishing spectral data.
(o)
(b)
I0 cycles $ec-~
-I
Flo. 6.25 The 1H resonance spectrum of 2-bromo-5-ch]orothiophene at 30 Mc sec=1 obtained at a sweep rate of 5 cycles sec-Z n ~ - t (a) ~diofrequency of low
power applied (no saturation) Co) z~diofz~uency power increased by a factor of 3.4 accompa~ed by a co~esponding decrease in receiver gain (c) a further
increase in radiofrequencypower by a factor of 3"4 with a reduction in receiver gain. In spectrum (c) the appearance of a new band in the centre is attributable to a double-quantumtransition(57). Anderson(Se) A reliable method of measuring the strength of the radiofrequency field, HI, has been described by Anderson (59). In this method a weak radiofrequency field is required in addition to the usual radiofrequency field//1. Another requirement is that this latter radiofrequency field must be sufficiently strong to saturate partially the nuclei under examination. The weak field can be obtained by modulating the main magnetic field at a frequency large compared with the line width, by means of an audiofrequency oscillator capable of giving a small amplitude. The index of modulation must be kept small so that the sideband signals (see Section 6.7) do not perturb the system appreciably. Under these conditions the measurement of the sideband signals is very difficult because the
232
HIGH
RESOLUTION
NMR SPECTROSCOPY
signals are situated on the sharply inclined sides of the main resonance line. The latter signal can be suppressed by using an audio phase-sensitive (lock-in) amplifier to detect the signals: a suitable circuit is shown in Fig. 6.26. With this experimental arrangement, it is possible to measure H1 values greater than 3 milligauss with a precision better than _+ 5 per cent. When making measurements, the radiofrequency detector is adjusted to select the usual absorption mode (V-mode) and the audio phase sensitive amplifier is adjusted to be in phase with the weak audio field. Under these conditions, the sideband signals from the output of the phase sensitive amplifier will be in the absorption mode but one of the sideband signals will be inverted. The main resonance signal will be suppressed almost completely. The spacings of the sideband signals are Audio modulator o Output {to recorder)
c
Signal input ! (from spectrometer) o
_I° 5"F T
o O.OO2/.~F
_•
To sweep coits
_-~Re ference input "(from oudio OSOIIntor)
FIG. 6.26 An audio modulation circuit. Varian Associates then measured at different audiofrequencies. The resonance conditions for the sideband signals are given by the equation (sg> 2 = ( y H - to)2 + 72H12 ton,
(6.31)
where oJ= is the angular frequency of the audio modulation. Providing that this relationship will hold at the two slightly different values of the main magnetic field strength, H, produced by the modulation at an audiofrequency. In order to measure HI, it is only necessary to measure the distance d between the sideband signals, keeping the sweep rate constant for all measurements. A plot of d 2 against to~ gives a straight line having an intercept at
~,, > 7H1,
(T HO ~.
Stehling and Williams (6°) have described a method for determining H~ which does not require a lock-in amplifier. 6.5.2. The R e c e i v e r
The signal produced at the receiver coil is extremely weak and requires a good deal of amplification. The ultimate amplification is determined by noise in the circuits and that generated mechanically by the spinning of the sample
NMR S P E C T R O M E T E R S AND T H E I R ACCESSORIES
233
tube. An effective method of reducing electrical noise is achieved by mounting the first stage of amplification (the pre-amplifier) as near to the receiver coil as possible. This pre-amplifier is followed by a main amplifier tuned to the transmitter frequency and having a gain of about 105. The signal-to-noise ratio in N M R spectrometers has been discussed in detail by Bloembergen, Purcell and Pound c23) who derived the following expression for the absorption mode setting
r'o =
kTBFT~
4SkT
where V~/I/,, is the final signal to noise ratio after rectification and Y is a factor of the order of unity dependent upon several factors including the detector law. ~ is the falling factor of the receiver coil, N is the number of resonating nuclei, V, is approximately equal to the volume of the receiver coil and Q is its quality factor. B is the effective bandwidth of the amplifier and F is its noise factor t~x). Most of these factors affecting signal to noise ratio have already been discussed. The need for an amplifier of low noise factor and narrow bandwidth is obvious from an examination of equation (6.32). B is of the order of the time constant of the output circuit of the amplifier hence it can only be reduced at the expense of reeording speed. A convenient time constant of about 1 see, prior to the display or recording stage, is effective for the suppression of background noise in the form of very short-term fluctuations. The receiver stage of a spectrometer is usually fitted with a means of varying the time constant in the range 0".01 to 10 see. When long time constants are selected, it is essential to employ a sweep rate slow enough to give an undistorted absorption signal. Williams c24) has pointed out that filtering at the output of a diode detector (used in most spectrometers) with too great a time constant will distort the signal, leading to an error as great as 50 per cent in the observed band area. The observed area is too large when the V-mode voltage and the leakage voltage are of the same sign and too small when they are of opposite sign. Williams t2"~ obtained the following expression for the time constant ~ of the filter for a damped oscillator band shape by assuming that the changing time con. stant is appreciably less than T-2yT~2~
(1 + q 2 ) ( ~ - q 2 + R )
(6.33)
where q2 =
[R 2 + 16(1+R)] 1 / 2 - 2 - R 6
(6.34)
and R --- V ~ x / L (Vm.z is the maximum signal voltage and L is the leakage
voltage). For large and positive values of R we get / d H \ -1 T ~ ~ , 7"* --~--)
(6.35)
which shows the dependence of the observed line width on the rate of magnetic field scan. 8~
234
IIIGH RESOLUTION
NMR SPECTROSCOPY
The output of the pre-amplifier is usually fed to several stages of radiofrequency amplification located in the main amplifier. A separate oscillator within this amplifier generates a frequency which is mixed with the amplified radiofrequency signal. Then follows one or more stages of intermediate frequency amplification and finally the signal is rectified. An integrating falter generally follows the rectifier in this high gain superheterodyne receiver. It is possible to control the gain of the receiver at the mixing stage. The rectified output from the receiver can be injected either on to the Y plates of an oscilloscope or into a d.c. amplifier preceding a pen recorder having a response time of about 0"1 sec. 6.6 INT~srfY MEASUREMENTS AND QUANTITATIVE ANALYSIS Most of the instrumental criteria required for accurate band area determinations have been dealt with in Sections 6.2, 6.4 and 6.5 according to Williams' treatment(24). The band area of an absorption band is directly proportional to the number of nuclei absorbing at that particular resonance frequency (equation (6.32)). By measuring relative intensities of two or more absorption bands one can estimate the relative numbers of nuclei in chemical groupings. Errors can be introduced by measuring line heights since they are not necessarily proportional to band areas because of the variation of line shape. The proportionality relation contrasts with the familiar Beer-Lambert law for the absorption of electromagnetic radiation. The difference arises because less than about i per cent of the total radiation is absorbed in the nuclear magnetic resonance experiment and therefore the radiation intensity can be regarded as constant. The direct proportionality of the band intensity to the number of absorbing nuclei and the use of experimental arrangements which balance out a large fraction of the applied radiofrequeney power make it more convenient to express the extent of absorption in arbitrary units rather than as a percentage of absorption or transmission. Absolute intensity measurements are very difficult to make owing to the dependence of the signal strength on the instrumental factors Q, ~, Vc, B and F (equation (6.32)) most of which are difficult to estimate precisely. The problem is not insuperable, because absorption band intensities can be measured with respect to one another in the same sample or referred to a standard. For this latter purpose, the entire spectrometer must have adequate stability, linearity and reproducibility; measurements have to be made at the same radiofrequency. In quantitative analytical applications, the spectrometer has to be operated under standard conditions using a calibration curve obtained from standard samples. It is necessary to standardise the conditions because of the dependence of the signal voltage V on T1, 7"2 and H1
NyZI(I + 1) TzHt V~ 1 + (eJo -- co)z Tzz + y2 H z T1 Tz"
(6.36)
For maximum precision, area ratios should be determined at different values of the radiofrequency power, H1, and extrapolated to zero H1 values. The determination of relative areas within a spectrum is a valuable aid in the eluci-
NMR SPECTROMETERSAND THEIR ACCESSORIES
235
dation of organic structures. Occasionally, however, nuclei of the same species in a single compound have sufficiently different values of ]'1 and 2"2 to give discernible effects. Clearly, the viscosity of the sample should be kept low so that 7'1/1"2 approaches unity. The dependence of band intensities on H1 can be reduced by working with dispersion rather than absorption signals (4°). The nuclear dispersion signal is proportional to Xo COo
-
1 + (coo - co)= T~ + y2 H~ r l 7"2
(6.37)
H
41ram
, "~-~" ~
~
+4.81+5-47
+ 18-89 ppm, from externOI 8 5 % H3PO,t
FIG. 6.27 The ~ap resonance spectrum at 40.5 Mc sec-1 of a mixture of sodium tripolyphosphate and sodium pyrophosphat¢ in water. The inset traces with millimetre
markings are integrator outputs. Varian Associates(64)
The maximum of this function occurs along the frequency scale at coo - co -
(I +
,2H
rITD
(6.38)
when it gives a signal of value
Z0COOH172 2(1 + ),=H~TI T2) I/="
(6.39)
This value does not decrease for large values of H I , as does the absorption signal, but increases monotonically. In estimating intensities by the dispersion method, the peak height from the first maximum to the first minimum of the recorded signal is measured both for the reference and the unknown substance under the same operating conditions. The measurement of band areas can be avoided by the use of electronic integrating circuits applied to the output stage (62, 6a, 96) without interfering with
8a*
236
HIGH RESOLUTION NMR SPECTROSCOPY
the normal recording arrangements. Integrators are available commercially and normally form an essential part of a spectrometer supplied with a field/frequency control loop. To avoid errors the spectrometer must be operated in the field sweep mode. An example of an electronically integrated spectrum is shown in Fig. 6.27 which depicts the 3, p resonance spectrum of a slightly supersaturated aqueous solution of an impure sample of sodium tripolyphosphate. The doublet centred at 4-81 ppm and the triplet at 18.89 ppm can be assigned to the tripolyphosphate ion 0
0
0
]5-
i
i
J
l
O--P--O--P--O--P--O|
l
0
l
0
L
0
l
]
The remaining band at 5.47 ppm is due to the presence of the pyrophosphate ion
°i°T° ? ?1' The integrator tracings clearly show (i) that one phosphorus nucleus is in a chemical environment different from that of the other two in the tripolyphosphate ion (ii) that the pyrophosphate impurity is present at a concentration of 21 mole per cent. 6.7 CALmKATIONOF SPECTRABY THE SmEBAND METHOD In Section 6.2.3. it was seen that under slow spinning conditions spectra show extra bands. The spacing of these bands is related to the frequency of spinning. Obviously, choice of a suitable frequency will give a less confused spectrum and if this frequency be known the spacing of the bands in the spectrum can be determined(65). A more practical method of applying a known frequency is to modulate either the irradiating radiofrequency or, as is more usual, the main magnetic field with an audiofrequency applied to the recurrent sweep coils. For high quality work an ordinary audiofrequency signal generator can be used if it can provide a given frequency which is constant and reproducible to better than 0"I per cent. The frequency can be read from a dial setting on a precision oscillator or measured with a frequency counter when a slightly poorer oscillator is used. Frequencies are normally required in the range 50 to 10,000 cycles sec -I. In Fig. 6.28 is shown the effect of applying a sinusoidal audiofrequency signal of I00 cycles sec -t at different power levels. The intensity of the first pair of sidebands is proportional to the Bessel function Jt (x) where x is A o)/o), for the case of frequency modulation or 7A H/o),, for the case of field modulation, o)m being the modulation frequency. Field
I~MR SPECTROMETERS AND THEIR ACCESSORIES
237
modulation can become difficult when ~m is greater than 1000 cycles sec -~ while frequency modulation can introduce serious errors by shifting the radiofrequency. Alexakos and Cornwell (9~)have described simple circuit modifications which can be made to a commercial radiofrequency transmitter in order to give pure amplitude modulation. This technique pro~des sidehands of reasonably constant intensity over the range 200 to 40,000 cycles sec-;.
| I00 cycles se~l I00 cycles sec"~
lO0
j
(b) ioo
cycles sec-~ cycles sec': __~
(o)
, "Jr~
j
I00
t
I00
cycles sec-In cycles sec-I L .JL.
:/~
v4__..
(c)
i=Io. 6.28 The hydrogen resonance spectrum of cyclohexane at 40 Mc sec-I when modulated at 100 cycles sec-z. The audiofrequency power p r ~ y increases from (a) to (c). The c~tral line is the single band of cyclohexane ,D
An early theoretical treatment of the modulation of microwave absorption signals has been given by Karplus (6~). Other workers have extended the treatment to N M R spectroscopy( 1 9 , 5 9 , 6 7 - 7 1 , 9 7 ) .The approach selected for discussion here is that of Williams and Gutowsky(19). The Hamiltouian for a system in which the static field H is perturbed by the local magnetic field HI (t) produced by audiofrequency modulation is o~" = o~"(°) + ~?'")
(6.40)
N
where
~lr(o) = ~
_ y h~r,: [H + H(t)]
(6.41)
_ 27 hItx Hx cos ~ t.
(6.42)
f:l N
and
,~,,(1) : ~ 1:1
238
HIGH RESOLUTION NMR SPECTROSCOPY
For a particular nucleus in a given state m o.(~0, t) = - (2~r) -1/2 exp(im 90) e x p ( - im 9, H t) exp [-- im 9'
f H~(t') dr'] dt
-® (6.43) Taking the case when I = ½ and for an initial state m = + ½ ~ m = - ½, Williams and Gutowsky calculated the time dependence of the coefficient a+ ½(t) of the eigenfunction by application of first-order time dependent perturbation theory, r 2~
a÷½(t)=
T- i
f Jf~,~l~)gct~b_t +½ dt.d~ --aO
(6.44)
0
!
t
= ig,Hx fcoso~texp(ig,Ht)exp[ig,f H~(t')dt']dt -oo
(6.45)
--oo t
= i9, H1/2 f
[exp(i oJ t) + e x p ( - ioJ t)] exp(ig,
-®
Ht)
exp[ig, f H,(t') dt'] dt - ~o (6.46)
where H~(t) is chosen to be a cosine waveform having an angular frequency oJ,, tlaen H, (t) = H ° cos(o~, t + 9) where 9 is the phase angle. (The effects of other types of waveform estimated by substituting the Fourier transform of the waveform). tution of equation (6.47) into (6.46) gives
audio(6.47) can be Substi-
t
a+~(t)
i7 Hx/2 f
[exp(i ~o t) + e x p ( - ice t)] exp(ig,
Ht)
-- o0
i9, H o
x exp/ sin(o~, / + 9 ) / d t . L m, J Bessel functions of the first kind were introduced by the relation c72~ exp(i z sin0) =
~
Jo(z)exp(in 0).
(6.48)
(6.49)
I1~ --00
Substitution of this relation into equation (6.48) followed by integration yields a+~(t)=
7Hz2 ,=-~o ~ exp(inq~)J~(~,){ exp[i(9,H+nc°'+C°)t]'lg,tt+n°~a+~° exp[i(9,H + n~o, - oJ) t] - 1~
+
~--h :~ ~--~, - ~
J
(6.50)
The second term in the braces corresponds to an absorption having a transition probability given by 72H2
~ exp[i(n' - n)~]J~(k)Jo,(k) a**(t) a÷~(t) = 4 ,,=-® ,,,=-oo {exp[-i(TH+no~,-~o) t ] - l.}{exp[i(~,H+n'o:,-co) t ] - 1 } × r ~ + ~-o,-, - g ~-~-+ -~;o,--,-'-d (6.51)
NMR SPECTROMETERS
AND THEIR
ACCESSORIES
239
where k = yH°/co°. The diagonal terms n -- n' in the double sum give upon rearrangement 72H~ ~ j2(k ) 4 sin 2 [½(yH + n co° - co) t] 4
.=_®
(~H+
rico, - co)2
(6.52)
One of these terms can be large and absorption can occur when
co = 7H
+ n co°
(6.53)
where n = 0, ~ 1 • • ". This equation is identical with equation (6.31) when the insiLmificant term containing H~ in the latter is dropped. The condition can be fulfilled by adj ustment of either the mean field H or the radiofrequency co. In the customary NMR experiment, the latter is kept constant (co = coo) so that the fundamental absorption band appears at a field H = cool7, there being sidebands of this spaced by co,/7'- From Fig. 6.28 it may be seen that the intensifies of sidebands are dependent upon the amplitude H ° of the audiofrequency modulation. The intensity of absorption is proportional to the transition probability per unit time and was evaluated by Williams and Gutowsky(.9) for a small region of magnetic field about the value 7 H = (coo - n co,)/y,
giving
I[ (co°- "co*q ~2
= [a*a/t].,,
(6.54)
_In=n"
= y" H~ -
2
./~(k) e ( r ~
(6.55)
where o 0 ' H ) is a line shape function describing the density of spin states in the absence of the audiofrequeney modulating field. Multiplication of equation (6.55) by 0(3,//) gives the total absorption of energy of the sample assuming that conditions of saturation are avoided. Excluding the effects of cross terms, equation (6.55) gives J .2 (yH~0 [co,) as the intensities of the resonance components at 7 H -- coo - n o~°. Similar results are obtained if, instead of magnetic field audio-modulation, the radiofrequeney source is frequency(6e) or amplitude modulated(73). Using either of these latter methods the receiver has to be tuned for each sideband. Care needs to be taken in applying the method to the measurement of small splittings because the band pass of the receiver section has to be less than the applied audiofrequency. The audio oscillation can be applied in two ways (i) the oscillator can be adjusted until a sideband from the reference compound coincides with an absorption line in the sample or (ii) any convenient audiofrequency can be applied and then the spacings of the absorption bands measured by interpolation. Method (i) eliminates errors introduced by nonlinearity of the slow sweep but it is not very convenient to use. However, for accurate band separation measurements it is the only satisfactory method (see Section 7.4.2.). Sideband techniques are not only used for spectral calibration but also find application in optimising the performance of NMR spectrometer magnetsO*).
~-40
H I G H R E S O L U T I O N NMR S P E C T R O S C O P Y
They are also very useful in double irradiation experiments as may be seen in the next section. 6.8 DOUBLEIRRADIATION Another valuable auxiliary technique in NMR spectroscopy also involves irradiation with a frequency other than that responsible f o r / / 1 . Whereas the previous section dealt with irradiation only at audiofrequcucies, the method referred to as double irradiation or double resonance may employ either an audiofrequcncy(Sg, vs. v~) or a second radiofrequencycTv-sl). The main application of nuclear magnetic double resonance is to the analysis of spectra, including the determination of the relative signs of coupling constants. Details are given in Section 8.19.S. In Chapter 8 it is shown that when two non-equivalent magnetic nuclei, A and X, are coupled together by indirect spin-spin interaction, the resonance spectrum of each nucleus is split into a multiplet structure. If one of these nuclei jumps rapidly between the various spin states corresponding to the multiplet lines of the other nucleus, then only a single average resonance absorption frequency will be observed at the centre of the original multiplet. Spin multiplets in the X resonance absorption are observed when the nucleus A spends a longer time in a given spin state than the time corresponding to the reciprocal of the frequency separation of the multiplets(S2): sidebands appear in the intermediate stages. Partial or complete spin decoupling can occur naturally if one of the nuclei possesses an electric quadrupole moment (see Section 2.S.2). Here the lifetime of such a nucleus in a particular spin state is limited by rapid spin-lattice relaxation. When only partial spin decoupling takes place it may be completed artificially by irradiating the nucleus having the nuclear quadrupole moment with the appropriate resonance radiofrequency. In the case of nitrogen coupled to hydrogen nuclei, the hydrogen resonance absorption bands can be made considerably sharper by irradiating with the nitrogen resonance frequency. The example given in Fig. 6.29 shows the 1H resonance spectrum of formamide both with and without double irradiation (93). In this example, the spin decoupling is complicated (but provides more information) by the e]imination of quadrupolar broadening caused by the 14N nuclei. Pople (s3) has discussed this effect in detail (see Section 9.1.3). Spectrum (a) in Fig. 5.29 is rather featureless but much fine structure is revealed when the ~4N nuclei are irradiated (spectrum (b)). Apart from aiding naturally occurring spin decoupling one can also promote it entirely without the necessity of one of the nuclei concerned possessing a quadrupole moment. In applications where different magnetic nuclei are being dealt with, the multiplet of one nucleus can be collapsed by irradiating the other interacting nucleus with the appropriate resonance frequency. Used in this way, the double resonance technique is a valuable aid for the analysis of complex spectra, for example those of the boron hydrides(s4) (see Sections 8.15.5 and 12.1.3). In these experiments the transmitter coil is double tuned for both the radiofrequencies to be employed. The decoupling radiofrequency is provided by an oscillator developing several watts of power and being tunable
NMR SPECTROMETERS
AND THEIR
ACCESSORIES
241
over a frequency range greater than the range of chemical shifts known to occur for the irradiated nucleus. The frequency selected by this fine tuning has to be constant to about I part in 105. Audiofrequency modulation is commonly used to provide fine adjustment of the second radiofrequency~Ss). By means of a refined technique, Anderson (sg) demonstrated the decoupling of chemically non-equivalent hydrogen nuclei. He used two crystal controlled radiofrequency oscillators having a controlled frequency difference. A similar
H
(o)
JAB
HA
Hc
JACj (b)
_t, i
JAB
i
~6
Fto. 6.29 The hydrogen resonance spectrum of formamide (CHO-NHa) at 40 Mc sec-I (a) without spin decoupling(b) with simultaneousirradiation of the lSN nuclei by means of an applied radiofrequencyof 2.8904 Mc sec-x. By courtesyof Varian Associates
result was acb/eved with much simpler instrumentation by Itoh and Sato (75) and by Kaiser¢76)who devised sideband methods. The decoupling of close resonance signals is clearly of great value in simplifying the hydrogen resonance spectra of complex organic molecules. In the method of Itoh and Sato c7s), an audiofrequency, oJa, of low amplitude is applied while keeping the radiofrequency responsible for / / I sufficiently intense to saturate the main resonance signal. By choosing coa/2~ to equal the difference between two absorption bands A and B in the spectrnm~ one of the sideband peaks of the A resonance will then appear at the B resonance and vice versa, As long as the sideband peak is not saturated it will appear on the
242
H I G H R E S O L U T I O N NMR S P E C T R O S C O P Y
broadened line of the saturated main resonance. When HI is large enough, the spin-spin coupling between A and B will be inoperative and multiplets existing previously will merge into a single absorption band. The simplicity of the method is attractive but the resonance line under examination is superimposed upon the broadened peak from which it is being decoupled. This difficulty is avoided in Kaiser's method (76) by using a little more equipment: here the unwanted signal is eliminated with the aid of a lock-in detector. The method is
"
M
WA
Fxo. 6.30 The production of a double resonance spectrum of acetaldehyde. Kaiser t76)
illustrated schematically in Fig. 6.30 using the hydrogen resonance spectrum of acetaldehyde as an example. Figure 6.30A shows the spectrum obtained under ordinary conditions with the spectrometer in V-mode operation with a weak resonant radiofrequency field. When this field is increased (accompanied by a decrease in receiver gain) until the condition 7H1 > n J is satisfied (see the Bloom and Shoolery treatment (81) given later in this section) then the multiplet structure disappears and the fundamental spectrum has the appearance of Fig. 6.30B. This procedure enables the value of Hx needed to saturate the fundamental spectrum to be determined. After this adjustment has been made, a very small audiofrequency field is applied to the sweep coils, and the resulting first order side bands are observed with a phase-sensitive detector locked to the audiofrequency. The frequency coa and its amplitude Ha are adjusted to the
NMR SPECTROMETERS AND THEIR ACCESSORIES
243
magnitudes calculated from the following relationships derived from Anderson's treatment 's')
~,H. < CO.l~372
(6.56)
COo2 = (~ COo10-6)2 + 7'2Hf
(6.57)
where 6 is the chemical shiftin p p m between the two groups of non-equivalent hydrogen nuclei whose spins are to be decoupled. The two sideband spectra
FzG. 6.31A: the I H resonance spectrum of the CH2 group in propionaldehyde, with markers of 10 cycles sec -~ separation. B: the same spectrum but now the aldehydic hydrogen nucleus has been irradiated strongly. C: spectrum calculated withJ/~ = 0.087 andnocoupling~fthealdchydichydrogennucleus. Freeman (as)
(Fig. 6.30C and 6.30D) can be observed in the output of the phase-sensitive detector. Providing that the reference voltage to the phase-sensitive detector has been adjusted to be in phase with the audiofrequency field, the fundamental spectrum will be suppressed by the phase-sensitive detector and the side band spectra will be in the V-mode display. A slow sweep superimposed upon the static main magnetic field will then result in a spectrum of the type shown in Fig. 6.30E in which the fundamental bands are de.coupled. One set of experimental conditions which will produce the acetaldehyde spectrum shown are COo = 2~z x 56"4 Mc sec -I, CO. = 2~ x 427 cycles sec -1 and HI ~, 10 milligauss. Freeman (st) has described a circuit for this method of double irradiation. Figure 6.31 shows the spectra he obtained for propionaldehyde with and without spin decoupling: the important feature of this experiment is that tedious
244
HIGH
RESOLUTION
NMR SPECTROSCOPY
conventional analysis of the spectrum is eliminated. Another useful circuit for this type of work is that of Elleman and Manatt(*‘). Commercial units are available from Nuclear Magnetic Resonance Specialities, Inc., 145, Greensburg Road, New Kensington, Philadelphia. A drawback to the field modulation methods described above is that they produce a symmetrical pair of sidebands. The unwanted sideband does not interfere with the experiment if the chemical shifts in the spectrum are large; however, in cases where the chemical shift difference between the two groups of nuclei to be decoupled is small, say 50 cycles set-‘, it may happen that an unwanted sideband overlaps another part of the spectrum which then becomes superimposed on the part being recorded. Turner@*) has overcome the difficulty by using a single sideband suppressed carrier modulation of the spectrometer fixed radiofrequency. The latter frequency provides both H, and the single sideband Hz. Another way of eliminating the unwanted sideband employs two modulation frequencies(r4); one sideband, having a high modulation index, provides the decoupling field Hz at a frequency Ye, the other sideband, of low modulation index, provides the field HI at frequency rl. The advantage of the method is that although 1v1 - v2 1 may be only 50 cycles se&, both v1 and v2 are separated from the centre band by a much larger frequency so that the centre band and any unwanted sidebands fall well away from the recorded spectrum. In all sideband decoupling experiments there are two possible methods of traversing the spectrum. The field sweep method keeps (vl - v2) a constant and traverses the spectrum by varying the applied magnetic field H,, . It has the advantage of experimental simplicity but has the disadvantage that the whole of the uncoupled spectrum cannot be recorded in one experiment. For example, in an APX type spectrum (see Section 8.2 for definitions) the effect on the P and X spectra by decoupling nucleus A can be observed only by performing two experiments, one in which (vr - IQ = (vs - YJ, and the other in which (% - YJ = (VP- vJ. In the frequency sweep method, the applied field H,, and decoupling field at frequency y2 are maintained constant and the spectrum is recorded by varying vl. The frequency sweep experiment is simple to carry out with spectrometers having a field/frequeficy control loop (see Section 6.2.1). Bloom and Shooleryc81) have made a mathematical analysis of the effects of radiofrequency fields on nuclear spin coupling, and they were able to predict the various spectra obtained, intermediate between unperturbed multiplet splitting and collapsed multiplets, as the second radiofrequency was tuned and changed in amplitude. The procedure they adopted was to make the Hamiltonian stationary by transforming to a system of rotating coordinates (see Section 2.12.3). On applying a steady magnetic field Ho in the z direction to a system of two nucIei with the magnetogyric ratios y1 and yz (an AX spin system), then their resonance frequencies will have the values y1 H,, and y3 H,, . The magnetogyric ratios used are those observed experimentally which thus takeinto account the effects of chemical shifts. The Hamiltonian for the system is
NMR S P E C T R O M E T E R S AND T H E I R A C C E S S O R I E S
245
where J is the coupling constant in radians sec- ~. Under the usual experimental conditions, only the expectation values for spins in the z direction are effective in the spin-spin coupling. Thus, the allowed transitions between energy levels occur at ~ I H o + m2J and ~'2Ho + m l J where m - - I , I - 1 . . . 1. Hence each nuclear resonance spectrum is split into a field-independent multiplet. Performing now the double irradiation experiment, the first nucleus is investigated by application of a weak radiofrequency field H1 having a frequency of7'1 Ho. The second nucleus is subjected i
I
1 ~o I
g
.
:/ ,
---
L .......
1
~---.-__ ~
i
I~G. 6.32 The forces acting On 12 at different orientations of 11. In this example, /1 ffi/2 ~ ½. Bloom and Shoolery~sx)
to a strong radiofrequency field H~ having a frequency near72Ho. After transference of the system to rotating coordinates, terms including//2, Ho and Ix" I2 are time independent, while Hx can be regarded as a perturbation inducing transitions between otherwise well-defined energy levels. H2 and the spinspin interaction term have only a small effect on the first nucleus and therefore spectrum frequencies calculated in the rotating frame can be transferred back to the laboratory frame by adding to them the factor oJ2/2~. The Hamiltonian is described in the rotating frame by ~o, _- - h [ I ~ - k ( y x H o - w2) + I2" k(~2Ho - oJ2) + Ix "i(y~H2) 4" 12 " i ( ~ ' 2 H 2 ) "1- J-I 1 " I2].
(6.59)
Whereas the first nucleus is quantised only in the z direction the second nucleus is quantised in the direction of its effective field which is at an angle 0 to the z
246
HIGH RESOLUTION
NMR SPECTROSCOPY
direction. 0 may be determined by considering JI~ • 12 in terms of an equivalent magnetic field acting at the nucleus. From Fig. 6.32 we obtain a(mt) = [(y2Ho + J m l - o~2)2 + 7,2H2]÷
(6.60)
cos0(mi) = (y2Ho + J mt - oJ2)/a(mj)
(6.61)
sin0(mx) = Y2 H2/a(ml)
(6.62)
t h e energy levels in the rotating frame are W(mt,m2) = h[ml(ytHo
- -
co2) + m2(y2Ho - 0)2 + mlJ)cosO(mx)
+ m2 7'2//2 sin0(mt)].
(6.63)
For the case o f l t = I2 = ½, the initial states may be defined W(mt = + ½) and the final states W(mt = -½). Only two of the four possible transitions are allowed for the unperturbed situation in w h i c h / / 2 = 0. In the present case, all four transitions may be observed because the angle 0 changes during the transition (see Fig. 6.32) therefore the initial and final states of m2 may not be orthogonal. The transition probability is given by P = cos 2 (9/2)
(6.64)
where qo is the arigle between the initial and final states. The line intensities depend not only upon P but also upon the population distribution amongthe energy levels which is non-Boltzmann because o f / / 2 . Since the thermal relaxation process is not present in the type of spin coupling considered here, the population distribution can be calculated by assuming that the two types of nuclei are non-interacting. Hence the total spin population for each value of mt is governed by the Boltzmann distribution, but at each value of mt the population difference between adjacent levels of m2 is proportional to 3t/.. Here Mz is the slow passage population difference> I, it is a good approximation to use M. = Mo cos20, where Mo is the Boltzmann distribution value. In the general case for spin values greater than ½ there may be as many as 211 (212 + 1)2 lines in the multiplet when H2 is applied. This is because transitions starting from different mt levels may no longer be superimposed since equation (6.63) is not a linear function of m~. Further lines may appear due to the breaking down of the selection rule, d ml = __. 1, caused by the second radiofrequency field. The following detailed calculations made by Bloom and Shoolery ts~) apply only to two nuclei of spin value ½. The four possible transition energies (in the laboratory frame) are A W, = W(½, ½) -
W ( - ½, ½) + h ,o~
A w~ = W(½, - ½ ) -
AWo=
W(½,~) -
w(-½,
w(-½,-~)
,t w~ = W(½, - ½) -
- ½ ) + ha,2
+ h,o~
w ( - ½, ~) + h o,2.
(6.65) (6.66) (6.67) (6.68)
NMR
SPECTROMETERS
AND
THEIR
ACCESSORIES
247
The normal experimental procedure is to adjust the second radiofrequency until 02 = y2H,,, w1 is fixed at y1 H,, and the spectrum is obtained by means of the field sweep d H. Substitution of (Ho + AH) into equation (6.63) and solving for transitions at A W = h o1 in equations (6.65) to (6.68), spectral lines will appear at the positions AH=0
(6.69)
and The intensity of the line at A H = 0 is given by the relative transition probability P, from 4y:Hf P, = Pb = (6.71) (J2 + 4~:.H:) * To find the intensities of the other lines requires the application of equation (6.64). When y1 > y 2, increase of the amplitude of H2 causes the original doublet to spread out and become weak while a new line appears at y1 Ho at the expense of the doublet. When y2 H2 % J only a single line appears due to transitions CIand b, and these are twice as intense as each of the original doublet lines. When yl < y2, the line structure collapses as the radiofrequency power of H2 is increased giving only a single line for the condition 4y2 H2 # J. The theoretical expectations were verified. by Bloom and ShooleryCsl)‘by obtaining the fluorine-19 resonance spectrum of aqueous Na2POjF at 3OW MC Se471 while simultaneously irradiating the sample with the phosphorus-31 resonance frequency of 1291 MC set- l. Freeman and Whiffeno‘g) have applied the above theory to the AX spin system, where A and X are both hydrogen nuclei. They calculated both the field swept and frequency swept spectra of dichloroacetaldehyde and obtained excellent agreement with the corresponding observed ‘H spectra. Baldeschwieler and Ra.ndaW5) have presented detailed quantum mechanical analyses of the double resonance behaviour of AX, A3X and AB spin systems: the frequency of the strong oscillatory field, 02, was considered to be close to the resonance frequency of the X or the B nucleus, while the transitions of the A nucleus are promoted by Hl . In another type of double resonance experiment, namely that involving the Overhauser effect(“0 go*gl), the signal intensity is greatly enhanced by double irradiation (see Section 3.17). Here, the hyperfine coupling between electronic and nuclear spins is removed by irradiating at the electron resonance frequency. For the effect to occur it is necessary that the principal means of thermal relaxation is via spin-spin coupling. In the situation under discussion, however, the thermal relaxation process for each nucleus is mainly through direct interaction with the lattice, therefore the enhancement of polarisation predicted by Overhauser does not occur. Tomita (g2)has developed a general theory of magnetic double resonance which covers all types of coupling.
248
HIGH RESOLUTION NMR SPECTROSCOPY
6.9 COMMERCIALNMR SPECTROMm'~'RS The specifications of some of the high resolution instruments currently available are listed below in alphabetical order. It is not practicable to give a price list but as a rough guide the cheapest spectrometer costs about £10,000 while a more flexible instrument having a comprehensive range of accessories can cost more than £30,000. These high performance spectrometers incorporate most of the refinements described earlier in this chapter.
AE1 RS2 Spectrometer (Associated Electrical Industries Limited, Instrument Division, Scientific Apparatus Department, Trafford Park, Manchester 17. England).
Fro. 6.33 RS 2 nuclear magnetic resonance spectrometer. All the electronic components are housedin the one console.By courtesyof AssociatedElectrical Industries Ltd.
The equipment (shown in Fig. 6.33)has an electromagnet capable of providing a homogeneous field from 1000 to 18,000 gauss with guaranteed resolution of 1 in 108. The detection system is of the single coil type. Radiofrequency units can be supplied for any frequency: six units can be built into the console. Many accessories are available including apparatus for controlling the temperature of the sample and units for the observation of wide line spectra of solids. Spectra can be recorded on a 2 in. or a 10 in. chart recorder.
JNM-3H-60 Spectrometer (Japan Electron Optics Laboratory Company Limited, Mitsubishi Naka 28th Building, Marunouchi, Chiyoda-ku, Tokyo). The instrument (see Fig. 6.34) is of the same type as the RS2 above and has roughly the same range of accessories. A spin decoupler is also offered. The
NMR SPECTROMETERS AND THEIR ACCESSORIES
FIo. 6.34 JNM-3H-60 nuclear magnetic resonance spectrometer. By courtesy of Japan Electron Optics Laboratory Co. Ltd.
F£G. 6.35 .INM-C-60 nuclear magnetic resonance spectrometer. By courtesy of Japan Electron Optics Laboratory Co. Ltd.
249
250
HIGH RESOLUTION NMR SPECTROSCOPY
maximum field strength attainable with this instrument is 15,000 gauss but a spectrometer is produced (the JNM-4H-100) which has a maximum field of 24,000 gauss. Figure 6.35 shows a version of the JNM-3 H-60 spectrometer which is supplied with the two radiofrequencies 60 and 56.448 Mc sec -t. The spectrometer is designed for routine operation, having pre-calibrated chart paper and a built-in integrator.
PerMn-Elmer NMR Spectrometer (Perkin-Elmer Limited, Beaconsfield, Buckinghamshire, England).
FIG. 6.36 The Perkin-Elmer high resolution NMR spectrometer. The permanent magnet is housed in the thcrmostatted cabinet shown on the right. By courtesy of Perkin-EImer Ltd. Unlike the spectrometers listed above, this instrument (see Fig. 6.36) has a permanent magnet (of field strength 14,092 gauss). The performance, however, is similar to those having electromagnets. Because of the stability of the permanent magnet pre-calibrated chart paper (9½ in. x 16 in.) can be used. The single coil detection system is employed. The radiofrequency units available permit hydrogen, fluorine, phosphorus and boron resonances to be studied. Provision is made for integration of the spectrometer output.
Triib Ttiuber KIS 2 Spectrometer (Triib TAuber and Company, A.G., Ztirich 37, Amperestral3e 3, Switzerland). The spectrometer shown in Fig. 6.37 has a 5 ton electromagnet giving a maximum field of 21,000 gauss (1H resonance at 90 Mc sec -t) with a guaranteed
NMR SPECTROMETERS AND THEIR ACCESSORIES
251
resolution of 5 x 10 -9. A series of probe heads allows radiofrequencies down to 5 Mc sec-1 to be used. The magnetic field is held steady with the aid of a control loop comprising a IH stabiliser unit for the frequencies 30, 60 and 90 Mc sec -~. Thus automatic calibration of spectra can be made at these frequencies. Spot frequencies of 5, 10, 15, 20, 30, 60, 80 and 90 Mc sec -~ can be selected by means of push buttons. A simultaneous pen recording of the spectrum and its integrated form can be made.
Fro. 6.37 KIS2 nuclear magnetic resonance spectrometer. By courtesy of Trilb Ttiuber and Co., A.G. Among the accessories offered are wide line equipment for the examination of solids and an attachment for high resolution studies at high temperatures. The Trfib Tiuber Company also manufactures a high performance but less costly N M R spectrometer (type KIS I/L) in which the electromagnet is replaced by a permanent magnet of field strength 6000 gauss. This spectrometer also has a i H resonance stabiliser and a built-in integrator. Additional equipment may be bought for wide line N M R studies in solids and for variation of the sample temperature in the range - 160° and + 150°C.
Varian A-60 Spectrometer (Varian Associates, Pale Alto, California, U.S.A.). This is a sophisticated routine instrument designed for recording and integrating IH resonance spectra at 60 Mc sec-:, Extensive use of transistors in all
252
H I G H R E S O L U T I O N NMR S P E C T R O S C O P Y
the circuitry, including the magnet power supply, has resulted in a compact design as may be seen in Fig. 6.38. A field-frequency control loop introduced via a second reference probe keeps the maximum rate of change of the field homogeneity at I cycle sec -~ hr -z. The resolution claimed is 1 part in 10s. The spectrum is recorded or integrated on the pre-calibrated chart paper (11 in. x 26 in.) of a flat-bed recorder. The main accessories offered with this apparatus are a sample temperature control unit ( - 6 0 ° to + 200°C), a microc¢ll having a capacity of 25/~1, a spin-decoupler, and a time-averaging computer (C.A.T.).
FIG. 6.38 A-60 nuclear magnetic resonance spectrometer. By courtesy of Varian Associates
Varian HR- I O0 Spectrometer The instrument allows hydrogen resonance spectra to be obtained at 100 Mc sec -1. A general view is shown in Fig. 6.39. The electromagnet provides a maximum field of 23,500 gauss hence a 94 Mc sec -1 frequency unit is required for 9F resonance: any radiofrequency can be supplied. Two other versions of this crossed coil spectrometer are available, the HR-60 and the DP-00. Both operate at maximum field of 14,100 gauss. The DP-60 spectrometer can be used for both high resolution and broad line studies, while the HR-00 finds its greatest use in variable-temperature high-resolution work. The DP-60 may be further adapted to electron magnetic resonance spectroscopy or to wide-line, variablefrequency NMR specfroscopy by adding the necessary components for conversion. More recent versions of these spectrometers have a field-frequency control locking system (HA-60 and HA-100).
NMR SPECTROMETERS AND THEIR ACCESSORIES
253
Included in the wide range of accessories are (i) an integrator and base line stabiliser (ii) a variable temperature probe accessory (ill) a microcell holding 25/~1 of sample (iv) a spin-decoupler and (v) a time-averaging computer.
FIG. 6.39 HR-100 nuclear magnetic resonance spe~rometer~ The compommt shown on the extreme right regulates the temperat/tre of the water flowing through the magnet c~oling coils. By courtesy of Varian Asso~ates
REFERENCES
1. MI E. RosE, Phys. Rev., 53, 715 (1938). 2. A. L. BLOOMand M. A. PAC~tJ~, Se~ne~, 122, 738 (1955). 3. M. J. E. GOLAY, R~. Sei. Instruments, 29, 313 (1958). 4. W. A. AND~aSO~, Rev. Sd. Instruments, 32, 241 (1961). 5. B. A. EVANSand R. E. RlcF~,Itm, J. Sci. Instrumems, 37, 353 (1960). 6. A. P. McCANN, F. SmTa, J. A. S. SMitH and J. D. ~ m ~ s , J. Sci. Instruments, 39, 349 (1962). 7. G. SLo~, Rev. Sci. Instruments, 30, 1024 (1959). 8. F. Blra~K, Rev. Sci. Instruments, 37, 342 (1962). 9. H. PRn~tAsand H. H. G ~ , Rev. 8¢i. Instruments, 28, 510 (1957). 10. M. E. PACF,~P~D,Rev. S d . / n s : ~ s , 19, 435 (1948). 11. R. B. W - J x ~ , I.S.A. Proc., 4, 163 (1958). 12. E. B. BAK~ and L. W. BURD, Rev. Sci. Instruments, 28, 313 (1957). 13. I-L PKIto~, 5th European Congress on Molecular Spectroscopy, Amsterdam (1961). 14. R. FR.~F~7.~and D. H. WmFE~, Mol. Phys., 4, 321 (1961). 15. W. A. ANDERSON,Rev. Sci. Instruments, 33, 1160 (1962). 16. F. BLoc~ Phys. Rev., 94, 496 (1954). 17. W. A. ANDERSONand J. T. ARNOLD,Phys. Rev., 94, 497 (1954). 18. F. BLOC~, Phys. Rev., 70, 460 (1946). 19. G. A. WrLLT~tS and H. S. GtrrowsKy, Phys. Rev.,104, 278 (1956). 20. J. I. K~LAN, J. Chem. Phys., 27, 1426 (1957).
254
HIGH RESOLUTION NMR SPECTROSCOPY
21. F. BLOCH,W. W. HANSENand M. E. PACKARD,Phys. Rev., 70, 474 (1946). 22. J. T. ARNOLD,Phys. Rev., 102, 136 (1956). 23. N. BLO~ERG~, E. M. PURCELLand R. V. POUND, Phys. Rev., 73, 679 (1948). 24. R. B. WILLIAMS,Annals N.Y. Acad. Sci., 70, 890 (1958). 25. VA~ANASSOCIATF.SSTAFF,N M R and EPR Spectroscopy, Pergamon Press, Oxford (1960), p. 90. 26. S. BROW~STEr~,Can. J. Chem., 37, 1119 (1959). 27. J. N. S~OOLERYand J. D. RoBE~, Rev. Sci. Instruments, 28, 61 (1957). 28. C. FP.ANCONXand G. F~t~o~L, Rev. Sci. Instruments, 31, 657 (1960). 29. A. N. HAMER,J. LEECEand P. G. BENTLEy,U.K.A.E.A. Technical Note, IGR-TN/CA-1048 (1959). 30. W. G. SCHNEIDER,H. J. BV.m~STVZNand J. A. POPLE,J. Chem. Phys., 28, 601 (1958). 31. H. PmMASand H. H. GthCrHAm3, Chimia, 11, 130 (1957). 32. H. PR~tASand H. H. GONTHARD,Helv. Phys. Acta, 30, 315 (1957). 33. E. M. PURCELL,H. C. T o ~ Y and R. V. POUND,Phys. Rev., 69, 37 (1946). 34. E. R. A z ~ , Nuclear Magnetic Resonance, Cambridge University Press (1956), p. 40. 35. W. N. ~ , Proc. Inst. Radio Engineers, 28, 23 (1940). 36. P. Gmvrr, M. SOUTrfand R. G ~ , Physica, 17, 420 (1951). 37. H. S. Gtrrowsg~,, L. H. MeYea and R. E. McCLu~, Rev. Sci. Instruments, 24, 644 (1953). 38. C. E. WAm~o, R. H. SPeN~R and R. L. CUSTER,Rev. Sci. Instruments, 23, 497 (1952). 39. J. B. L E ~ , R. E. R~CHAnDSand T. P. SCHAE~, J. ScL Instruments, 36, 230 (1959). 40. H. S. Gu'rowsKY, Analytical Applications of Nuclear Magnetic Resonance in Physical Methods of Chemical Analysis, Vol. 13, Ed. W. G. Berl, Academic Press, New York (1956). 41. H. G. I-IXRTZand W. SPALTHOFF,Z. Elektrochem., 63, 1096 (1959). 42. V. J. KOWALEWSKXand R. A. HOVFMAN,Nuc. Instruments and Methods, 6, 357 (1960). 43. R. BRADFORD,C. CLAYand E. STruCK,Phys. Rev., 84, 157 (1951). 44. R. FR~temN, Private.communication. 45. B. A. JAcomol-~ and R. K. WAN~SNV.SS,Phys. Rev., "/3, 942 (1948). 46. E. E. SAI.P~XR, Proc. Phys. Soc., A63, 337 (1950). 47. A. M. PORTm,Phys. Rev., 91, 1071 (1953). 48. E. H. ROGERS,NMR-EPR Workshop Notes, Varian Associates. 49. J. N. StlOOLERYand M. T. ROGERS,J. Amer. Chem. Soc., 80, 5121 (1959). 50. G. SLOMP,Private communication. 51. R. V. POUNDand R. FREEMAN, Rev. ScL Instruments, 31, 103 (1960). 52. O. JAP.DErZKY,Private communication. 53. L. C. ALLI~ and L. F. Jormso~, J. Amer. Chem. Soc., 85, 2668 (1963). 54. R. ERNSTand H. PR~AS, Private communication. 55. R. ERNSTand H. PRIMAS,Discuss. Faraday Soc., 34, 43 (1962). 56. L. M. JACKMAN,Applications o f NMR Spectroscopy in Organic Chemistry, Pergamon Press, London (1959). 57. S. WILK~G, Z. Phys., 157, 401 (1959). 58. W. A. ANDERSON,Phys. Rev., 104, 850 (1956). 59. W. A. A~D~RSON,Phys. Rev., 102, 151 (1956): see also reference 25, p. 164. 60. F. C. STEm.~G and R. B. W I L ~ , Private communication. 61. L. A. MoxoN, Recent Advances in Radio Receivers, Cambridge University Press (1949). 62. G. CAMPONOVO,B. MARUGGand L. WEGMANN,Archives des Sciences, 11, 203 (1958). 63. Reference 25, p. 185. 64. VARDaq ASSOCIATES, N M R at Work, No. 84. 65. J. T. ARNOLDand M. E. PACKARD,J. Chem. Phys., 19, 1608 (1951). 66. R. KAS~LUS,Phys. Rev., 73, 1027 (1948). 67. K. HALnACH,Heir. Phys. Acta, 29, 37 (1956). 68. J. H. Btrao~s and R. M. BROW~, Rev. ScL Instruments, 23, 334 (1952). 69. B. SMALL~R,E. YASArrLSand I~. L. ANDERSON,Phys. Rev., 81, 896 (1951). 70. E. YAsArns and B. SMALLER,Phys. Rev., 82, 750 (1951). 71. H. PPaMAS,Helv. Phys. Acta, 31, 17 (1958).
NMR SPECTROMETERS AND THEIR ACCESSORIES
255
72. P. M. MoRsE and H. F~SHBACH,Methods of Theoretical Physics, McGraw-Hill, New York (1953), pp. 619, 1322. 73. H. S. GLrrOWSKYand C. J. HOVFMAN,J. Chem. Phys., 19, 1259 (1951). 74. R. ~ Rev. Sci. Instruments, 33, 495 (1962). 75. J. PrOH and S. SATO,J. Phys. Soc. Japan, 14, 851 (1959). 76. R. KAISER,Rev. Sci. Instruments, 31, 963 (1960). 77. F. Bt~CH, Phys. Rev., 93, 944 (1954). 78. V. ROYD~, Phys. Rev., 96, 543 (1954). 79. J. BRC6SELand F. BrrTE~ Phys. Rev., 86, 308 (1952). 80. T. 1~ CAxv~ and C. P. SucI-rrER, Phys. Rev., 92, 212 (1953). 81. A. L. BLOOMand J. N. SHOOLVmY,Phys. Rev., 97, 1261 (1955). 82. H. S. GwrowsKY, D. W. McCALL and C. P. S L I ~ J. Chem. Phys., 21, 279 (1953). 83. J. A. PoPtm, Mol. Phys., 1, 168 (1958). 84. J. N. SHooI.~Y, Discuss. Faraday Sac., 19, 215 (1955). 85. J. D. ~ D c ~ w z n ~ . ) ~ and E. W. RANDALL,Chem. Rev., 63, 81 (1963). 86. R. F ~ , Mol. Phys., 3, 435 (1960). 87. D. D. l~vi)w~¢ and S. L. M,~A'rr, J. Chem. Phys., 36, 2346 (1962). 88. D. W. T u m ¢ ~ J. Chem. Soc., 847 (1962). 89. R. ~ and D. H. Wl-m,er.a%Proc. Phys. Soc., 79, 794 (1962). 90. A. W. OvmtuAus~, Phys. Rev., 92, 411 (1953). 91. J. KOmPa~GA,Phys. Rev., 94, 1388 (1954). 92. K. Tos~rA, Prog. Theor. Phys., 20, 743 (1958). 93. L. H. P,~LI~, J. D. RAY and R. A. 0oo, J. Mol. Spect., 2, 66 (1958). 94. L. G. At~XAKOS and C. D. C o m ~ x ~ , Rev. Sci. Instrmnents, 34, 790 (1963). 95. M. P. ~ and G. W. B A R ~ , Rev. Sci. Instruments, 34, 754 (1963). 96. P. J. p~)nJm~ and W. D. COOWV,Anal. Chem., 36, 1713 (1964). 97. O. HAWORTH and R.E. R I c I - I ~ s , Progress in Nuclear Magnetic Resonance Spectroscopy, 1) 1 (1966); edited by J. W; EMSt~sy, J. F~NEY and L. H. S t r r ~ Pergamon Press, Oxford.
CHAPTER 7
EXPERIMENTAL
PROCEDURES
IN CHAPTrm 6 is given a detailed description of the equipment used in high resolution NMR studies. However, no mention has so far been made of the overall experimental procedure to be followed in order to obtain the nuclear magnetic resonance spectrum of a sample. This chapter deals systematically with details of the preparation of the sample in a suitable form right through to the recording of the spectrum. A full analysis of the spectrum normally entails evaluation of chemical shifts and coupling constants while a more extensive investigation might require relative intensities of bands and their line widths to be determined. 7.1 PREPARATION OF THE SAMPLE It is usually preferable to examine samples as dilute solutions in an inert solvent such as carbon tetrachioride or cyclohexane, where the intermolecular effects on nuclear shielding will be negligible. The spectral parameters measured under these conditions can be used more reliably for characterisation purposes a n d they will have greater theoretical significance than those obtained from the pure liquid. The pure liquid at room temperature, however, is not to be shunned entirely because it is, at worst, a reproducible standard state. The sample must be a mobile liquid or gas, otherwise incomplete averaging of the direct dipoledipole interaction leads to broad spectral lines and the highest resolution will not be attained. To observe the high resolution spectrum of a solid, the sample must be melted or dissolved to form a mobile liquid. It is important that the liquid is not turbid since the presence of solid particles in suspension can give rise to broad spectral lines from the auisotropic averaging of the field inliomogeneities on spinning the sample. Polycrystalline solids can be made to give fairly narrow lines by spinning the sample about an axis making an angle of 54° 44' with the main magnetic field~1-*~. The rate of rotation should be large compared with the static line width. Using this method the alp resonance spectrum of solid phosphorus pentachloride has been found to consist of two lines 377 ppm apart corresponding to the environments of PC1+ and PC1;. 7.1.1 Gases The spectrum of a gas at normal temperatures and pressures is that of an isolated molecule, that is, all the intermolecular effects on the chemical shifts are negligible (see Section 3.10). The chemical shift obtained from a gaseous sample depends to some extent on the temperature and pressuremfor hydrogen the 256
EXPERIMENTAL
PROCEDURES
257
effect is very small ¢5). Evans te~ has noticed that the fluorine resonance in gaseous carbon tetrafluoride is dependent on pressure, a shift to low fields of 1"15 + 0.15 ppm atm -1 being observed on increasing the pressure. Nevertheless, this is only small compared with the chemical shifts usually observed for fluorine-containing compounds.
7.1.2 Liquids Viscous liquids must be diluted or the sample temperature raised: even dilute solutions of high molecular weight compounds may be highly viscous; for example, a 10 per cent by weight solution of polymethylmethacrylate in chloroform is too viscous to give a good high resolution spectrum unless the temperature of the sample is increased to 90°C ¢7). Chemical shifts obtained from the spectra of liquid samples may be strongly concentration and temperature dependent owing to strong intermolecular interactions (see Sections 3.10 and 10.39). Therefore, in order to obtain a spectrum representative of the isolated molecule it is necessary to use a dilute solution of the compound in an inert solvent. The ideal solvent for nuclear resonance work is one which does not interact with the solute in any way and which is magnetically well-behaved, that is, it is magnetically and electrically isotropie. It is also essential that the solvent does not have a spectrum overlapping that of the solute: in 1H resonance spectral studies deuterated solvents can be useful in this respect. For example, CDCI3 is used extensively as a solvent for organic compounds of high molecular weight. Other useful solvents for hydrogen resonance studies are the inert isotropic compounds carbon tetrachloride, tetramethylsilane and eyclohexane. For fluorine resonance, a convenient solvent is fluorotrichloromethane (CFC13) which, although neither inert nor isotropic, gives solvent shifts small compared with the magnitude of fluorine chemical shiftsc6"s). It is, of course, desirable that the sample be soluble in one of the inert solvents but often this is not the case and a less ideal solvent must be used. The effect of some common solvents on polarisable solutes will now be discussed. It should be remembered that the use of these solvents precludes comparison of chemical shifts from one solvent to another.
Water N M R spectral data obtained from aqueous solutions must be treated with extreme caution. Not only is water a strongly hydrogen bonded liquid but i also may enter into an exchange reaction with H or OH groups present in the solute. Exchange can affect the chemical shifts and may also bring about collapse of spin multiplets in the spectrum. The 1H resonance spectrum of pure ethanol shows a 1 : 2 : 1 triplet for the hydroxyl hydrogen nucleus but in aqueous solution this multiplet becomes a single line whose chemical shift is strongly concentration dependent. The presence of an exchanging group in such a situation may be detected simply by dissolving the sample in D 2 0 and observing the subsequent diminution of the appropriate signal as chemical exchange takes place. HRS.
9
258
HIGH RESOLUTION
NMR S P E C T R O S C O P Y
Aromatic solvents Although aromatic compounds have only weak complexing properties, benzene and other aromatic molecules often produce large changes in the chemical shifts of polarisable solutes due to the large diamagnetic anisotropy associated with the induced ring currents (see Section 3.8). If the formation of a weak complex gives rise to a preferred solute-solvent orientation then a shift in the resonance position occurs relative to that of the non-complexed solute. The shift may be either to high or to low fields depending upon the nature of the complex. Thus the hydrogen resonance signal of chloroform moves by 1"5 ppm to high fields on going from pure chloroform to a 5 per cent solution of chloroform in benzene(9). The upfield shift takes place because the preferred orientation of the complex is as shown below
CI3C--H........ The chloroform hydrogen nucleus thus experiences an applied field which is reduced in magnitude by the induced field arising from diamagnetic ring currents in the benzene molecule. Molecules having hydrogen atoms with some degree of charge polarisation will form complexes of this type. Table 7. I shows the tH chemical shift differences of some representative molecules when dissolved in benzene and in an inert solvent. TABLE 7.1 SOLVENT Sm.vrs IN BENZENE
Solute Chloroform Propargyl chloride (acetylenic hydrogen)
(B)H,, H(A) Vinyl bromide ~C=C (C)H 'Br NO.o H ] ~H(A) p-Methyl~ ~J'~ nitrobenzene ~ J~ H~~ ' H(B) CHs
Solvent shifts~" + 1"35 +0"31 + 0"86(A), + 0"88 (B), + 0"75(C)
+ 0.S7(A), + 0.9603)
t The quoted shifts are measured as the difference in ppm betw~'n the chemical shift of the solute extrapolated to infinite dilution in benzene and that of the solute extrapolated to infinite dilution in an inert solvent.
Substituted benzenes have similar effects on the chemical shifts of solutes, that is, the hydrogen resonance signal is displaced to high fields. However, if the substituent itself has strong complexing properties then a low field shift may be observed. For example, in the case of propargyl chloride (re>, CICH2C = C H . changing the solvent from cyclohexane to nitrobenzene causes a low
EXPERIMENTAL PROCEDURES
259
field shift of 0.71 ppm for the acetylenic nucleus. This is due to complex formation between the nitro group and the propargyl chloride molecule.
Pyridine. This has been found to be a useful solvent for NMR investigations of polar steroids c~). Because of the existence of specific interactions between pyridine and the steroids, the large anisotropic ring current of the pyridine int~uences the shielding of the hydrogen nuclei of the steroids to different extents. The effect can be put to good use in unravelling spectra containing superimposed absorption bands. For example, the 1H resonance spectrum of 4,4,17~trimethyl- 17/~-hydroxy-5-androsten-3-onedissolved in deuterochioroform shows only two bands attributable to methyl hydrogen nuclei: a solution of the compound in pyridine reveals separate absorption bands for each of the five methyl groups. The coupling constants are unaffected by the solvent. Fortunately, the spectrum of pyridine itself is well removed from the region of interest in the steroid spectrum. Trifluoroacetic acid. The nuclear magnetic resonance spectra of amino-adds and peptides can be conveniently measured using trifluoroacetic acid ¢x2~ as solvent rather than water ¢~s. ~4~. Tritluoroaeetic acid is an excellent solvent for these compounds and also has the advantage that, unlike water, it will dissolve sufficient tetramethylsilane for use as an internal reference. Porphyrins exs~ have been examined in both trifluoroacetic and deuterotrifluoroacetie acids--advantage is taken here of both the solvent and strong a d d characteristics. The solvent has the disadvantage of bringing about a certain amount of line broadening. Liquid sulphur dioxide. This is a useful solvent for many organic compounds. Furthermore, it has the useful property of being aprotic. Sulphur dioxide may be kept at the liquid state at room temperature with the aid of the screw-top sample tubes described in Section 7.3.6. Acetone and nitromethane. These and other donor molecules all form specific complexes with molecules having electron acceptor properties; in all cases the resonance signal of the solute undergoes a shift to low fields, the magnitude of which depends on the strength of the solute-solvent complex. 7.1.3 Impurities in the Sample Large amounts of impurities in a sample may obscure a region of the spectrum but in general as much as 5 per cent of a diamagnetic impurity can be tolerated. There are types of interacting impurity which take part in chemical exchange with the sample and need to be removed. For example, small amounts of water may have a large effect by promoting a proton exchange reaction of the type R O H + HsO ~-- R O H + I-~O
Traces of acids and bases can aggravate the situation by catalysing the exchange process. Exchange reactions are discussed fully in Chapter 9. 9"
260
H I G H R E S O L U T I O N NMR S P E C T R O S C O P Y
It is necessary to exclude small amounts of paramagnetic materials from the system since they can affect markedly the nuclear spin lattice relaxation times (see Section 2.5.1). Because the Bohr magneton is approximately 103 larger than the nuclear magneton, mole fractions as low as 10 -6 of a paramagnetic impurity can lower Tt considerably. Large m o u n t s of paramagnetic impurities can reduce Tt to such a low value that the line widths broaden sufficiently to obscure the hyperfine splitting. However, the presence of traces of paramagnetic compounds can be an advantage when the value of 7"1 is large, as for example in atp and 13C resonances--paramagnetic species are often introduced deliberately(16) in order to allow the use of higher values of radiofrequency power before the limit mposed by saturation is reached. Dissolved oxygen is always present in samples which have not been degassed but despite its paramagnctism its effect on the spectrum is quite smallcry). If there is any doubt about the effectof oxygen then samples should be degassed in a vacuum system in order that the natural narrow line widths can be observed. 7.2 REFERENCE COMPOUNDS It is not possible to measure absolute values of shielding constants (see Section 3.I) and therefore all chemical shifts are measured from the absorption band of an appropriate nucleus in a pure standard reference compound. The standard compound may be either dissolved in the sample (internalreferencing) or placed in a separate container (externalreferencing). Additionally, the reference compound often serves to provide a locking signal.
7.2.1 External Referencing Relative shifts measured in this way must be corrected for differences in the bulk diamagnetic susceptibility between the sample and reference compound (see Section 3.3), a procedure which presents some difficulties. For cylindrical samples the corrected chemical shift is given by
where A Z = Z, - Z,, the difference in bulk volume susceptibilities of the reference and the sample. It should be noted that Bothner-By and Glick eta' tg~ have found for some systems empirical correction factors some 20 to 30 per cent greater than the theoretical one [(2,-r/3) A ;~]. If both sample and reference compounds were contained in spherical sample tubes then no correction for differences in diamagnetic susceptibilities would be necessary; unfortunately good spherical containers are not easily made. Thus in many cases the method of external referencing is unsuitable because the form of the correction factor may be in doubt and because the accuracy of the corrected shift will rely upon the measured values of the bulk diamagnetic susceptibilities. When external referencing is unavoidable then it is most conveniently carried out with the aid of precision coaxial tubing. Usually, the reference is contained in the central capillary and the sample in the surrounding annulus. Zimmerman and Foster ¢2°~
EXPERIMENTAL P R O C E D U R E S
261
have examined the practicability of such a method and they found that if the sample is magnetically and electrically isotropic then accurate and reproducible results can be obtained. External referencing has the advantage of not contaminating the sample and is often used in N M R studies of nuclei other than hydrogen where susceptibility corrections can frequently be regarded as negligible. In order to evaluate A :~ use can be made of the tables given in Appendix C. Susceptibilities can be measured with the aid of a high resolution N M R spectrometer in the method devised by Frei and Bernstein (21). Figure 7.1 shows the experimental arrangement used in this method: two small reference containers, one spherical and the other cylindrical, are positioned inside a conventional
\
/ Nylon s ~ c r
I
jO'SmmO.D. ~ - 2.5ram QD. sphere
FIo. 7.1 5 mm o.d. sample tube containing a cylindrical and a spherical referencetube. Frci and Bornstein(21) 5 mm o.d. spinning sample tube with the sphere at the centre of the receiver coil. The N M R spectrum of the reference compound contained in the two inner vessels consists of two chemically shifted sharp absorption bands, one from the contents of each container. The separation between the two signals is linearly dependent upon the volume susceptibility of the sample contained in the conventional sample tube and is given by the expression (R) -
--
[go,
- g, h] [z,(R)
- z,(s)]
(7.2)
where ~ is the chemical shift in ppm, ~, is the volume susceptibility in cgs units and g is the geometrical constant which is dependent on the shape of the interface between the sample and the reference, but is independent of the shape of the outer tube. For an ideal case [gcr~ - g, ph] would be expected to be 2~/3 but in practice this constant will need to be determined by calibration with liquids of known volume susceptibility. Some of the volume susceptibilities
262
HIGH
RESOLUTION
NMR
SPECTROSCOPY
obtained from this method are given in Table 7.2. The accuracy is comparable with that of classical methods. TABLE 7 . 2 VOLUME SUSCEPTIBILITIES DEO,~SSF.D LIQUIDS (21)
Si(CHD,. CI-L,CN CsI"Ito CS2
SiCl4 CHaI
OF SOME
-0.543 x 10-6 e.g.s, units - 0.532 - 0.628 - 0.693 - 0-755 - 0.938
Li, Seruggs and Becker(2z) and Douglass and Fratiello (.3) have also reported a method for determining diamagnetic susceptibilities with the aid of a NMR. spectrometer. They used a simpler sample cell than the one described above. In their method, first the spectrum of a sample contained in the central compartment of a precision coaxial cell is recorded, the external reference benzene being contained in the annulus. Second, the same assembly is used for making the susceptibility measurements simply by stopping the spinning, when the benzene then displays two absorption bands whose separation is a linear function of the volume susceptibility of the liquid in the inner cell. The separation (n cycles see -t) is given by the equation (43) n = 4~ % [(21 -- g2) (a/r) 2 + (X2 - Z3) (b/r) 2]
(7.3)
where % is the fixed radiofrequency, a and b are the internal and the external radii respectively of the inner glass tube, r is the mean radius of the annulus, Zt, Z2 and Zs are the volume susceptibilities of(1) the liquid in the inner tube (2) glass and (3) the annular liquid (benzene in this case). With precision bore tubing, a, b and r are constant. The linear plot of n versus the known volume susceptibilities of water, carbon tetrachloride, cyclohexane and methanol was obtained and then used to determine the susceptibilities of samples. Assuming the shape constant to be 2~t/3 the accuracy of the method is within the limits required for correction of spectral measurements, that is, _+0.004 x 106 c.g.s units corresponding to 0"5 cycles sec -t at 60 Mc sec -t. Measurements of paramagnetic susceptibilities have also been made with the aid of a high resolution NMR spectrometer(24). Even when diamagnetic susceptibility corrections are applied, chemical shifts measured from an external reference will often contain an appreciable error because of the uncertainties involved in the correction procedure. It is permissible to compare chemical shifts from an external reference for a series of closely related compounds, as for example a set of geometric isomers. Here the volume diamagnetic susceptibilities can be regarded as constant. Accurate conversion can be made from one external reference scale to another if corrections are made for the difference in diamagnetic susceptibility between the two reference compounds. Some useful approximate conversion factors for external referencing are given in Table 7.3.
EXPERIMENTAL PROCEDURES
263
The effect o f solvents on the position o f fluorine resonance lines cannot be predicted by equation (7.1) even when isotropic solvents are used c2~. 27) (3lick and Ehrenson ~2" have shown that for 1,2-dibromotetrafluoroethane the extent o f the deviation from the behaviour predicted by equation (7.1) is proportional to the polarisability o f the solvent and they suggest that intermolecular influences on the shielding o f ~9F nuclei may be eliminated by extrapolation to zero polarisability. The most commonly used external reference compound in fluorine resonance work is trifluoroacetic acid. Shifts measured relative to the line from this compound are usually quoted uncorrected for differences in diamagnetic susceptibility between reference and sample. Although deviations from equation (7.1) may be as large as 1 ppm even for isotropic solvents, they are small compared with the range of 19F chemical shifts ( ~ 200 ppm). TABX.E7.3 C O - - I O N
FACTORS FOR Ex'rI~NAL RlWl6REN~ C O M P O U N D S ¢2s)
t H Resonance ~C6H6
H20 =6e= +2"49 d~ffi~H6= 6si(c~s'4 + 7"27 ext
6~H6 __ •SH12 -t- 5"83
6 ~ a~ = 6~ c~,c°cH~ + 5"1S CeHe = ~C~C~3 + 0"02
eXg
--egt
6~I~' = o~m 2ct' + 1"94 19]:: Resonance 6Cox~ v3c°°x
"= ~,,,t'sca3F - 78"6
6cF3c°°H ~ = 6ct4Fs + 61.49 13C Resonance ~CH~COOH _- - o.~H6 e=
+
50
Footnote. The conversion factors for xH resonance were obtained from the diffe~mce in the chemical shifts (measured relative to tetramethylsilane) of various compounds in dilute solution in carbon tetrachloride. To convert 6,zt to t~mt, corrections must be made for the difference between the volume stmceptibilities of the solution and the external reference. The conversion factors given in this table are not n__~zes_~rilythose used throughout the t e x t : preference has been given to conversion factors suggested by the original authors. 7. 2.2 Internal Referencing The introduction o f about I per cent of a reference compound into the sample has the distinct advantage that corrections for differences in diamagnetic susceptibility are not required since both sample and reference are surrounded
264
H I O H R E S O L U T I O N NMR S P E C T R O S C O P Y
by the same medium. However, internal referencing contaminates the sample and for this reason it is desirable to employ a reference compound of low boiling point in order that it may be removed readily by distillation. A more serious objection to an internal reference is the possibility of specific solvent-solute interactions which would affect to differing extents the positions of the reference and the sample resonance lines. It is essential therefore that the reference be chemically inert and magnetically and electrically isotropic. Hydrogen resonance. For ~H resonance studies of non-aqueous solutions, recommended reference compounds are tetramethylsilane (TMS) and cyclohexane. Tetramethylsilane, Si(CHa)4, is a volatile liquid (b.p. = 27°C) and it is magnetically and electrically isotropic, reasonably chemically inert and it does not appear to associate with any common compounds (2a). The magnetically equivalent methyl hydrogen nuclei give a single sharp band at fields higher than almost all other compounds: this property also makes it very suitable for providing an internal locking signal. The chemical shift ~ (in parts per million) of a resonance line from tetramethylsilane is defined as
HTMS
]
(7.4)
where H, and H~n~sare the resonance field values of the sample and tetramethylsilane respectively. Defined this way most 8 values are negative: it should be noted that this definition of t~ is used throughout this monograph. Tiers ~2a) has suggested that instead of the resonance line of tetramethylsilane being taken as 6 = 0 ppm it should be assigned arbitrarily to the value of + 10 ppm thus making most chemical shifts positive. Shifts measured for dilute solutions in carbon tetrachiodde relative to tetramethylsilane at 10 ppm are known as values c2s) where • is defined as T = 10.000 + 6.
(7.5)
On this scale almost all hydrogen resonances have a positive value of I=; increasing values of z imply increase in shielding. Table 7.4 shows a comparison of r values obtained using tetramethylsilane both as an internal and an external reference compound (28). The internal chemical shifts were obtained from approximately 5 per cent solutions in carbon tetrachloride, while external shifts were determined by extrapolation to zero concentration of two or more measurements made on carbon tetrachloride solutions of different concentration. There is some evidence (29) that the resonance line of tetramethylsilane is affected by solvent interactions but the changes are small enough (about 0"025 ppm) to he neglected in most cases. In the most accurate work it is advisable to introduce a second internal reference compound and to use the constant separation of their resonance lines as an indication that both are unaffected by the surrounding medium. Cyclohexane is a good reference compound giving a single absorption band at ~ = 8.564, that is, 86.16 cycles sec -t downfield from tetramethylsilane at 60 Mc sec -t . This separation can be used to test for solvent interaction.
265
EXPERIMENTAL PROCEDURES
Tetramethylsilane is virtually insoluble in water, hence a different 1H internal reference c o m p o u n d is required for aqueous solutions. Tiers (s°) has suggested the use of the methyl resonance band in the 1H spectrum o f the sodium salt o f 2,2-dimethyl-2-silapentane-5-sulphonic acid, (CHs)sSiCH2CH2CH2SO~Na" TAI3LE7.4 COMPARmONOF T VALUESOBTAINEDWITH AS A N INTERNAL A N D ~XTERNAL R~,~.RF..NCE(2s)
Compound C6He
c6nsC2Hs (C685CH2)2 p-CeK,(CH3). CsH, (cydooctatetraene) CHsNOs CeHsOCH3 CHsOH (CeHsCH2)2 CeHsCH2CHs CeHsCHs (CH3COhO CHsI CH3COCHs CHsCOOH CHsCN Cell12
• (external refegec~e)
• (internal reference)
2"74 2.89 2.89 3-05
2"734 4- 0.003 2"888 q- 0.002 2"893 4- 0.001 3.053 4- 0"003
4.26 5"69 6"31 6.60 7.13 7"42 7"67 7.81 7.81 7.91 7.90 8.10 8.51
4.309 4- 0.004 5.720 4- 0.002 6-266 4- 0"002 6"622.4- 0-002 7"129 4- 0-001 7.382 4- 0.003 7.663 4- 0.003 7.809 -t- 0-003 7.843 4- 0-004 7.915 4- 0.003 7-930 4- 0.004 8"026 4- 0-002 8.564 4- 0-002
TABLE7.5 A COMPARISONOF SICmLnlNG VALLr~ %"(TMS As ~ A L
REFERENCE) AND T' ('DSS AS ~ A L I~d/:FERENCIE) MEASURED IN SIMILARSOLLrIIOI~O0) .fl
Compound CHsCOOH CHsOH (CHs)sC.CH20H in CHaOH (CH3)3C.CH2OH in (CHs)2SO (CH3)3C.CH2OH in CH3CONHCHs CHsCN CHsN02 (CH3)2SO
7"934 6"650 9"117 9"175 9-109 8"033 5"640 7"457
7"945 6"659 9"133 9"165 9-124 8"022 5"646 7"434
k n o w n as DSS. The salt is prepared by the addition of sodium bisulphite to atlyltrimethylsilane. The three strongly coupled methylene groups give rise to a complex multiplet which at 1 per cent concentration is indistinguishable f r o m the b a c k g r o u n d noise in the spectrum. DSS is stable in acid media and is soluble in concentrated salt solutions. The position o f the methyl resonance band is almost unaffected by the nature o f the solvent or the p H o f the solution. Table 7.5 HRS. 9 a
266
HIGH
RESOLUTION
NMR
SPECTROSCOPY
shows that a x' scale, defined relative to the DSS methyl band at 10"000 ppm, agrees with the x scale to within 0.02 ppm. Aeetonitrile and dioxane have also been shown to be suitable internal reference standards for use in *H resonance studies of aqueous solutions ° t ) . Chemical shifts measured relative to these compounds can be converted to T values using the following relationships: "t" ~ (~acetonlttlle ""~ 7"98
z = ~a~o~,°, + 6"30 where ~ is the chemical shift in ppm measured from the reference compound and it is defined as in equation (7.4). Fluorine resonance. Tiers (') has suggested that monofluorotrichloromethane, CC13F, is suitable both as a solvent and as an internal reference compound (b.p. TABLE 7.6 APPARENT FLUORINE SHIELDING VALUESis) (~o*)
Vol., % Compound
C6HsSO2F CFBra
CFzBr2
BrCF2CF2Br
C6HsCF3
CFCI2CFCI2 CFaCO2H CFaCCI3
C6HsF (CFaCC12)2 (C2Hs)aSiF n-C6HxaF
c o n c h , in
~0 'll
CCI3F
(ppm)
5 20 3 2.O 40 6O 80 2 I0 80 10 4O 80 3 I0 20 40 5 2O 5 2O 3 I0 4O I0 40 2 15 I0 4O I0 4O
-65"509 - 65-547 - - 7"384 - 7.309 - 7-231 - 7"143 - 7"052 - 6"768 - 6.763 - 6"768 + 63"394 + 63"373 + 63"332 + 63"719 + 63"651 + 63"574 + 63"385 + 67"753 + 67"762 + 76"542 + 76-552 + 82'209 + 82"220 + 82-230 + 113-150 + 113"231 + 114"086 + 114"086 + 176.24 + 176.240 + 219-017 + 219-002
Std. dev.
__0.005 ± 0"006 ±0.004 0.002 ---0.007 ± 0.003 ± 0.OO3 0"002 ±0"004 ± 0-O05 ± 0"005 ~ 0.009 ± 0.007 -4-0"003 -4-0"010 ± 0"OO5 ±0"OO9 ±0"OO6 -*-0"006 ± 0"OO5 ±0"OO6 -4-0"002 0"008 -4-0"009 ± 0.002 -4-0"003 ±0"009 ± 0.003 __+.0.02 ± 0.006 ±0.0O6 q- 0-013
EXPERIMENTAL PROCEDURES
267
= 24°C). The chemical shift of a sample relative 'to CCI3F used as internal reference is defined
(, H. -
Bo~l,v
/.
C~.6)
There are small variations of ~* values with concentration, therefore for accurate comparisons the shifts must be extrapolated to infinite dilution (they are then known as 9~values). Table 7.6 shows some ~* values of compounds at different concentrations in CClsF. Evans (~) has asserted that 19F shifts measured at infinite dilution in CC13F are appreciably different from those measured on the gaseous sample. Carbon tetrafluoride and sulphur hexafluoride have ideal characteristics for internal references but the fact that they are gases at room temperature introduces a handling problem. 7.2.3 Tetramethyl$ilane as a Reference for ttll Nuclei A method has been suggested by Jackman (s2) whereby the chemical shift of any magnetic nucleus may be obtained relative to the internal reference tetramethylsilane provided that the sample contains hydrogen coupled to that nucleus. In Section 8.19.5 it is shown how accurate values of chemical shifts can be obtained by means of double resonance experiments. Baldeschwieler(ss) has demonstrated how the the chemical shift of I"N nuclei coupled to hydrogen nuclei can be found by observing the XH resonance spectrum whilst irradiating strongly at the 14N resonance frequency. The frequency of the dccoupling field is locked to the hydrogen resonance frequency by means of a series of frequency dividers and multipliers, final adjustment being made with audiomodulation. In this way the shift difference between the I"N nucleus and the IH nucleus is known to within 1 cycle see -x at 40 Mc sec-1. If the • value of the decoupled hydrogennucleus is known then the shift of the X'N resonance from tetram~thylsilane is known. The method can be applied to any two groups of spin coupled nuclei. 7.2.4 lntereonversion of Reference Scales In the past many internal and external reference compounds havebeenadopted for use in chemical shift determinations and it is sometimes necessary to convert shifts from one reference scale to another. Such a procedure is fraught with difficulties and it is usually impossible to obtain accurate results because of the absence of necessary data for the original sample and reference compounds (for example, concentration of solution, extent of intermolecular effects, diamagnetic susceptibilities and sample shape). Conversion from one internal reference scale to another will only be accurate if both reference materials are ideal and present in small amounts and if the chemical shifts to be converted have been extrapolated to infinite dilution/n the same solvent. The converted shifts will then only apply to solutions of the sample in this solvent. It is impossible to convert accurately chemical shifts measured against an external reference to those on an internal reference scale and vice versa. 9m*
268
HIGH RESOLUTION NMR SPECTROSCOPY
7.3 SAMPLE CONTAINERS
Most spectrometers are designed to hold a cylindrical sample container capable of being spun by means of an air driven turbine. For the routine study of hydrogen and fluorine resonance spectra, glass sample tubes approximately 5 mm external and 3 mm internal diameter and about 15 cm long are used. A typical sample volume is 0-3 ml. If sample spinning is employed then it is essential that the outside of the tube is straight and that the wall thickness is uniform. In some instruments good sample spinning is assured by locating the tip of the sample tube in a nylon bush at the bottom of the probe assembly. It is important that the bottom of the sample cell be symmetrical, either rounded or conical. Although some manufacturers of commercial instruments provide standard sample tubes, it is possible to make satisfactory containers in the laboratory. Precision bore sample tubes must be used if the maximum signal-to-noise ratio is required. 7.3.1 Coaxial Sample Tubes A convenient method of external referencing is to use precision coaxial tubing, the reference being contained in the central capillary and the sample in the surrounding annulus. The effect on chemical shifts of imperfections in coaxial cells has been discussed by Zimmerman and Foster ~2°). Coaxial cells can be obtained commercially or they can be constructed in the laboratoryO *). A less satisfactory arrangement is t ° introduce a thin walled capillary tube containing the reference compound into a conventional sample tube. When the reference and sample signals are well separated, Both compounds can be placed in separate standard tubes and measurements made by interchanging the tubes during a single sweep of the magnetic field.
7.3.2 Spherical Sample Cells When discussing external referencing (see Section 7.2.1) it was noted that corrections for differences in diamagnetic susceptibilities need not be applied if both sample and reference compounds are put into spherical sample cells. Perfectly spherical glass sample holders are difficult to make and any deviations from perfect symmetrywill introducediamagnetic susceptibility shielding effects. The Triib Tfiuber N M R spectrometer is supplied with spherical sample cells. The cell devised by Frei and Bernstein ~21> (illustrated in Fig. 7.1) can be used simply as a spherical cell. Spiesecke and Schneider ~16~have designed a spherical sample cell having a large filling factor suitable for the study of t3C resonance spectra. Figure 7.2 depicts the sample container comprising two concentric thinwalled spherical bulbs. The smaller bulb has a capacity of 0"2 ml and it contains the reference compound. The outer sample bulb has a volume of 1.5 ml. The receiver coil is mounted inside the main cylindrical tube and encloses the two bulbs. Richards t3s) has improvised spherical sample holders with the aid of gelatin. A cylindrical sample cell is filled with aqueous gelatin setting at about 50°C :-,nd the sample is introduced into the gelatin by means of a syringe so that the
EXPERIMENTAL PROCEDURES
269
sample is located at the centre of the receiver coil. Finally, to ensure that the globule of sample is spherical, the tube is rotated until the gelatin sets. The main drawback of the method is that the compound under examination must be immiscible with water. Air
I ,bnteol holes
4 0 m*l ~ungste wire
Inner sphere for reference liquid
sphere mpte
Bokelit beoring
F~. 7.2 Spherical sample tube having a large filling factor. Spi~l~k¢ and
7.3.3 Sample Cells with Large Filling Factors In the study of nuclei other than those of hydrogen and fluorine, it is often necessary to improve the sensitivity by increasing the size of the sample within the receiver coil. This is achieved by increasing the diameter of the sample cell and using a correspondingly larger receiver coil. Because it is difficult to spin large sample tubes it is customary to make the measurements on stationary samples and this results in loss of resolution. However, a design of a large sample cell which can be spun is shown in Fig. 7.2. When a large sample is spun under conditions of veryhigh radiofrequencyamplification, avarying coupling between the transmitter and receiver is introduced of sufficient magnitude to overload the detector. The problem can be solved by using a phase sensitive detector capable of rejecting signals having the spinner frequency.
270
H I G H R E S O L U T I O N NMR S P E C T R O S C O P Y
An obvious way of improving the filling factor is to use sample tubes having very thin walls, or to mount the receiver coil inside the sample tube. Also, the signal-to-noise ratio can be improved by increasing the number of turns of wire in the receiver coil. Any attempt to improve the sensitivity by increasing either the sample or the coil size will result in a loss of resolution since the main magnetic field homogeneity has then to be maintained over a larger volume element.
7.3.4 Containers:for Small Volumes Although only about a 5 mm length of sample is in the region of the receiver coil, in many spectrometers it is necessary to have a depth of liquid of approximately 2 cm in the sample cell before the receiver coil volume is satisfactorily 80
rl
fl
-AL t ~Id r: 0"0 7 3'!""
"
( snug fit in tube) T
~c~s"
-N\~0.020"
Fro. 7.3 Nylon plugs used to reduce the volume of a conventional sample tube. Shoolery(s~)
Me
'~
H
Me 0 I~..~/C--Me
~
Ho
6-Me iC_I 9
I '
8"0
7-0
6"0
5"0
':,.0
3"0
1
fll
=
,6.M, I
2"0
I'0
ppm
FIe. 7.4 The I H resonance spectrum at 60 Mc sec -z of 6/3, 16~-dimethyl-6 c~hydroxyprogesterone in 25/el of CDCI3 containing tetramethylsilane (located at 0 ppm). Shoolery (3~)
0
EXPERIMENTAL
PROCEDURES
271
occupied. Part of the long length required is due to the formation of a vortex during spinning. W h e n only small quantities of the sample are available it is possible to reduce the volume required by fillingthe dead space at the bottom of the tube either by inserting a nylon or P T F E plug, or by constructing the sample tube so that this portion is made of solid glass. In this way a signal can be obtained with as littleas 5 m m length of liquid.The vortex can be eliminated by trapping the sample between two plugs.The resultingsmall cylindricalsample gives rise to a loss in resolution since the nuclei near the ends of the sample are in a less homogeneous field,owing to the discontinuity in the diamagnetic susceptibility.Shoolery (36)has overcome the difficultyby enclosing the sample between two nylon plugs forming a hemispherical cavity (see Fig. 7.3).Theupper plug has a fine hole bored down itslongitudinal axis for fillingpurposes. The cavity occupies only one third of the volume (25 pl) enclosed by the receiver coil but the most efficientuse is made of a fixed quantity of sample. Figure 7.4 shows the IH resonance spectrum at 60 Mc sec -1 of 6/~, 160~-dimethyl-6o~hydroxyprogesterone obtained using a 560 ~tg sample dissolved in deuterochloroform. If more sample is available the plugs can be spaced a little farther apart.
7.3.5 Sample Cells .for Use in Temperature Studies It is often desirable to observe N M R spectra at different temperatures: several thermostatted probe assemblies are described in Section 6.3.1. If the sample is a solution then it is advisable to have as little dead space above it as possible because changing the temperature will change the concentration. It is also advisable to have as small a volume as possible since the longer the samplethe greater the temperature gradient along it. Petrakis and Sederholm(ST), in a study of the temperature variation of chemical shifts of gaseous molecules, restricted the volume by sealing the tube just above the receiver coil while retaining the upper part of the tube for spinning purposes. When working with a gas or a volatile liquid an increase in temperature can lead to a large increase in pressure. Normal 5 mm o.d. borosilicate glass tubing with 1 mm wall thickness will withstand pressures up to 60 arm (e). When high pressures are expected the sealed end of the tube can be made the weakest part to act as a safety valve.
7. 3.6 Sealing of Sample Tubes Temporary seals can be made with moulded polythene caps. Such seals are useful when one is working with expensive precision bore sample tubes intended for repeated use. A more elaborate temporary seal is shown in Fig. 7.5. This consists of three precision turned pieces of an inert plastic such as PTFE. The sample tube needs to be modified by flanging its open end. The assembly gives a good seal in that it will hold liquid sulphur dioxide almost indefinitely at room temperature. If care is taken in the construction, spinning can be perfect.
272
H I G H R E S O L U T I O N NMR S P E C T R O S C O P Y
It is very dangerous to attempt to fill and permanently seal a sample tube simply by immersing an open-ended tube in liquid nitrogen, because oxygen condensed in the tube can cause a violent explosion on warming to room temperature.
~ - \~PTFE
Flanged 5ram O.D. glass sample tube
FIG. 7.5 Polytetrafluoroethylene screw cap for temporary sealing of sample tubes
7.4 RECORDINGTHE SPECTRUM In this section a discussion is presented of the factors which need to be considered if the best resolution of an NMR spectrometer is required.
7.4.1 Homogeneity of the Main ~lagnetic Field The first priority is to obtain a homogeneous magnetic field over the sample volume. The shape of the magnetic field contours in the region of the sample depends upon the geometry of the magnet, particularly the parallelism of the pole caps, and also upon its immediate magnetic history. The shape of the magnetic field may be changed by alteration of the alignment of the pole faces,
EXPERIMENTAL PROCEDURES
273
and in the case of an electromagnet, by magnetic field "cycling" (see Section 6.2.2). Some magnets are also equipped with homogeneity coils (see Section 6.2.1), which change the field shape when controlled currents are passed through them. A typical procedure for obtaining a homogeneous field for an electromagnet is as follows. A sample is chosen which gives a strong sharp single line, and the main magnetic field increased until the resonance signal is observed at the centre of the oscilloscope trace. By moving the sample horizontally across the pole faces it is possible to plot the contours of the magnetic field by observing the direction in which the signal moves on the oscilloscope screen. This experiment is repeated for the vertical axis. The desired field shape is one giving field homogeneity over the sample volume, that is, fiat in both the vertical and horizontal directions. The sample itself and the probe assembly distort the field shape and it is found that an apparently slightly dished field gives the best spectra. Successive cycling can change the field shape from domed to dished but the reverse is not true (see Section 6.2.2). If a magnetic field is dish-shaped then it may be changed to a domed field by reversing the direction of the magnetic field for a few minutes in the case of an electromagnet, or by altering the alignment of the pole caps of a permanent magnet. The final adjustments to the field shape are made with the homogeneity coils if available, the homogeneity being checked first by observing the decay pattern of the transient signals following a sharp resonance line and finally by recor
274
H I G H R E S O L U T I O N NMR S P E C T R O S C O P Y
Having obtained a good spectrum from the test compound, the next step is to record the best possible spectrum from the sample. Changing the sample in the probe may alter the homogeneity of the field slightly and necessitate adjustment of the homogeneity coil controls for optimum resolution. Finally, the spectrum is recorded with the radiofrequency power well below the saturation level but high enough to give a good signal-to-noise ratio (see Section 6.5.1), and with the magnetic field sweep rate slow enough to reveal all the resolvable peaks. The sweep must be fast enough (about 0-2 cycles sec -2 in the absence of a field/ frequency control loop) to avoid spurious bands coming from instability of the magnetic field. At slow sweep rates the signal to noise ratio can be increased by reducing the band width of the receiver, care being taken to avoid distorting H
S;C-I
J
Fro. 7.6 The tgF resonance at 56.4 Me sec -1 of the band (40.6 ppm to high fields of external CF3COOH) due to the - C F 2 - group in liquid CFsCF2CF2I
the spectrum by using too narrow a band width. When very weak signals are to be recorded it may be necessary to increase the value of the radiofrequency power level beyond that sufficient to cause saturation under slow passage conditions. The effects of saturation may be minimised by increasing both the magnetic field sweep rate and the receiver band width. Under these conditions it may be possible to record otherwise undetectable signals but the resolving power of the instrument is decreased. In such cases, in addition to using a higher sweep rate it would be advantageous to record the dispersion mode of the signal since this does not saturate as readily as the absorption mode (see Section 2.9).
7.4.2 Calibration oj Spectra Most nuclear magnetic resonance spectrometers produce a graphical record in which the abscissa is proportional to the applied magnetic field. When the main magnetic field is stabilised by means of the N M R signal from a reference compound (see Section 6.2.1) or if a permanent magnet is used it is possible to
EXPERIMENTAL PROCEDURES
275
use pre-calibrated chart paper and hence to measure chemical shifts and coupling constants of simple spectra directly from the chart (see Fig. 1.2). But in some spectrometers the change of magnetic field produced by the slow sweep unit is neither accurately linear nor exactly reproducible for any particular setting of the controls, thus it is necessary to calibrate each individual spectrum as outlined in Section 6.7. Several calibration procedures are given below.
Superimposition o.f bands. For accurate measurements of the separation between two signals A and B, the sideband of A must be arranged to coincide exactly with the B signal. In favourable cases exact superposition of signal and sideband can be achieved by observing (on an oscilloscope or with a recorder having rapid response) the transient pattern generated when coincidence is approached, the beat pattern disappearing at complete coincidence. The first step in the procedure is to ensure that band B gives a good sequence of "wiggl e s " b y employing a low sweep rate and a low receiver time constant. A sideband of A produced by means of an audio oscillator is now located close to B and thi.~ results in a modulation of the "wiggle" envelope giving a beat pattern. By altering the frequency of the oscillator until the "'wiggle" pattern is again that of a single band, indicating that the sideband of A and signal B are perfectly coincident, an accurate value for the separation of ,4 and B can be read from the audio oscillator dial, Using the pen recorder to inspect the superimposed bands allows band separations to be measured with an accuracy of about 4- 0"! per cent because the method is independent of variations in the sweep rate. Interpolation method. When band positions in a complicated spectrum have to be measured, the superimposition method can be difficult to apply and resort is often made to the inferior interpolation method of calibration. By means of the audio oscillator, sidebands of known separation can be introduced (see Section 6.7)into the spectrum and the line separations calculated by interpolation between convenient sidebands. The most serious source of error arises in the non-linearity of the non-recurrent sweep: such errors can be minimised by using numerous sidebands and interpolating between small frequency intervals. It is often useful to generate sidebands from the signal of the referenee compound. The method is accurate to _ 0.5 per cent. Sidebands can be generated either by frequency modulating the radiofrequency output or by producing an oscillating magnetic field at the sample. The latter method consists of connecting an audio oscillator to the Helmholtz coils norreally used for the recurrent sweep. When band separations of more than about 2000 cycles sec -I have to be measured it is customary to employ the frequency modulation procedure. This can lead to errors due to shifting the frequency of the crystal controlled radiofrequency unit. Such errors can be miniraised by keeping the audiofrequency oscillator in circuit throughout the recording of the spectrum and its sidebands. The method of superimposing a sinusoidally varying weak magnetic field upon the main magnetic field is not subject to this limitation. However, both methods are likely to introduce errors as may be seen from the following discussion. Let us examine the resonance con-
276
H I G H R E S O L U T I O N NMR S P E C T R O S C O P Y
dition in relation to modulation techniques. Rearranging equation (6.31) in Section 6.5.1 we obtain for the positions of the sidebands H =
+
-
(7.7)
H,')*.
In describing the calibration techniques given above it was assumed that the /-/1 term (the radiofrequency power level) is unimportant. That is, that the sidebands appear at intervals of oJ=/7 rather than the real intervals (~o~/72 - H~)~. Taking a value for Hx of one milligauss and an audiofrequency of 25.0 cycles sec =1 (oJ, = 157 radians see-l), the sidebands would appear at 24.7 cycles sec -1 at either side of the main signal in a hydrogen resonance spectrum. The error present in any given situation can be estimated by finding the value of HI (see Section 6.5.1). Wiggle beat method. A rather special type of band separation measurement, applicable to separations of less than 15 cycles sec -1, is based on the occurrence of wiggle beats. The advantage of the method is that it can also be used to measure the very small band splittings which are unresolved because of shortterm field instability. The wiggle pattern from a spin multiplet containing n equally spaced fines (of separation J) whose intensities are in the ratio of the nth binomial coefficients, is given by V(t) = 2 "-1 Vo e x p ( - t/T2)cos[½ F R t 2 - (n -
1) ~r Yt] cos"-a(• J t) (7.8)
where V(t) is the amplitude at time t, and R = d H o / d t . T h e spacing between modulation maxima is 1/J see; the distance between the signal and the first maximum depends on 7"2 as well as on J. ReiHy(zg) has used this method to obtain the coupling Jcr~-cF2 ( = 1.48 cycles see -x) in CF3CF2COOH which has a t 9F spectrum containing unresolved multiple~. Figure 7.7 (a) shows the wigg-
(al
I
Cb) I
] sec
I
Cc) Fro. 7.7 (a) Shnpleringing pattern from the 19F resonance of CF3COOH (b) complex ringing pattern from the CF3 group in CF3CF2COOH (c) complex ringing pattern from the CF~ group in CF3CF2COOH. Reilly(39)
EXPERIMENTAL PROCEDURES
277
les from the single 19F resonance band in CFsCOOH, while Figs. 7.7 (b) and (c) show the respective wiggle beat patterns from the CF3 and CF2 bands in CFa CF=COOH.
Turner (4°) has extended the above analysis to include an arbitrary series of signal intensities Vo, V,, V2 ... V, which are passed at times t -- 0, h , tz ... t,. The intensity of the decay pattern is now given by V(t) = e x p ( - t/Tz) [P cos½ a t 2 + Q sin½ a t z]
where
P = Vo + V, c o s ( a t t ,
(7.9)
- ½atZ,) + ... V, c o s ( a t t = - ½at2.) + ...
Q = V, sin(a t t, - ½a t~) + ... Vo sin(a t t~ - ½a t.a) -6 ...
and a is the rate of change of the Larmor angular velocity in units of sec -2. The beat method is more reliable than the usual slow sweep pins sidehand means of measuring small band splittings. De Kowalewski and Kowalewski(4.~ have compared the two methods in determining the small long range coupling constants between the aldehydic and a ring hydrogen nucleus five bonds distant in some ortho monosubstituted benzaldehydes. Their findings are listed in Table 7.7. TABLE7.7 Co~LnqO CONSTANTS(CY¢"~SF.C-*) rt~ ortho.sums,,it~**a, Bm~ZALDmaYD~St'tn Compound
Direct method
Beatsn~hod
o-OCHs
0-67
0.7?
o-OC2Hs o.NI-I2 o-C] o-OH o-NO~
0.64 0.60
0.80
0.51 0.48
0.31
0"75 0.72 0.61 0.40
7.4.3 Simplification oJ Spectra
Chemical shifts can be determined direcdy from recorded data only when the various groups of magnetic nuclei are weakly coupled. Otherwise, considerable effort may be expended in analysing a spectrum completely to derive values for the chemical shifts and coupling constants (this is explained in Chapter 8). However, spectra can be simplified occasionally by choosing a suitable solvent or by raising the main magnetic field strength to as high a value as possible (see Section 8.19.1). As spectrometers working at higher and higher fields become available more classes of compounds will give simple spectra. Double irradiation can be useful in deter~nining the chemical shift between two coupled groups of nuclei when the bands from one of the groups are obscured by another part of the spectrum (`2) (see Section 8.19.5).
278
HIGH RESOLUTfON NMR SPECTROSCOPY
7.4.4 Reporting oJ Spectra At present there is no accepted standard method of presenting high resolution N M R spectra and the data derived from them. However, for a spectrum to be regarded as complete it should be accompanied by the following data (i) an indication of the direction of field (or frequency) sweep (ii) a scale, calibrated for preference in the units Of the raw data (iii) the frequency of the fixed radiofrequency unit. It is also useful to note on the spectrum the proposed molecular structure of the sample. Additional information such as the solvent used, the solution conditions, the radiofrequeney field strength and the sweep rate are valuable. When reporting chemical shift data it is preferable to use the dimensionless unitsppm which render the data independent of the main magnetic field strength. Coupling constants are best quoted in cycles sec -1. It is only permissible to quote the origin of a spin multiplet in dimensionless units. A clear indication of whether a detailed spectral analysis has been carried out enables one to assess the usefulness of a particular set of results. REFERENCES 1. E. R. ANDREW,A. BI~DSURY and R. G. EADF.S,Nature, 183, 1802 (1959). 2. E. R. A,'qDREW,A. BRADBtmY, R. G. EAD~S and G. J. JEh'KS, Nature, 188, 1096 (1960). 3. S. CLOUOHand K. W. GRAY, Prec. Phys. Soc., 79, 457 (1962); 80, 1382 (1962). 4. E. R. ANDREWand G. J. JEt,~zs, Prec. Phys. Soc., 80, 663 (.1962). 5. L. ~ and C. H. SL~SgHOLM,J. Chem. Phys., 35, 1174 (1961). 6. D. F. EvANs, J. Chem. Soc., 877 (1960). 7. G. V. D. Tings and F. A. BovEY, J. Polymer Sci., 44, 173 (1960). 8. G. V. D. TIERSand (3. FmIPOVmH,J. Phys. Chem., 63, 761 (1959). 9. W. G. S ~ E R and L. W. RJ~ms, Can. J. Chem., 35, 251 (1957). 10. J. V. HAa'rONand R. E. RmHAaDS, Trans. Faraday See., 57, 28 (1961). 11. G. SLOMPand F. MAcKI~LLAR,J. Amer. Chem. Soc., 82, 999 (1960). 12. F. A. BOVEyand G. V. D. TIERS,J. Amer. Chem. Soc., 81, 2870 (1959). 13. M. TAKEDAand O. JARDETZKY,J. Chem. Phys., 26, 1346 (1957). 14. O. J A g D ~ Y and C. D. JAgD~TZKY,J. Biol. Chem., 233, 383 (1958). 15. R. J. ABRAHAM,A. H. JACKSONand G. W. KENNER,J. Chem. Soc., 3468 (1961). 16. H. SPmSECrd~and W. G. SCmqE~D~R,J. Chem. Phys., 35, 722 (1961). 17. D. F. EVANS, Chem. andlnd., 526 (1958). 18. A. A. BOTHeR-BY and R. E. GLICK,J. Chem. Phys., 25, 362 (1956); 26, 1647 (1957). 19. A. A. ~ - B Y , J. 1Viol. Spect., 5, 52 (1960). 20. J. R. ZIMMERMAIqand M. R. FOSTER,J. Phys. Chem., 61, 282 (1957). 21. H. J. B~saam~ and K. Fgr~, J. Chem. Phys., 37, 1891 (1962). 22. N. C. Lx, 1LL. SCRUC~Sand E. D. BECgER,J. Amer. Chem. See., 84, 4650 (1962). 23. C. A. R ~ t . y , H. M. McCo~qELL and R. C. MWSENHEUe~R,Phys. Rev., 98, 264 A (1955). 24. D. F. EVANS,J. Chem. See., 2003 (1959). 25. S. B R o ~ , Chem. Rev., 59, 463 (1959). 26. D. F. EvANs, Prec. Chem. See., 115 (1958). 27. R. E. GLIC~ and S. J. Ettm~soN, J. Phys. Chem., 62, 1599 (1958). 28. G. V. D. Tm~, J. Phys. Chem., 62, 1151 (1958). 29. E. D. B s c ~ J. Phys. Chem., 63, 1379 (1959). 30. G. V. D. Tlva~sand A. KOWALSKY,Private communication. 31. R. A. Y. Jo~s, A. R. KATarrzKy, J. N. MURR£LLand N. S~PPAgO, J. Chem. See.. 2576 (1962).
EXPERIMENTAL PROCEDURES 32. J. A. Gt.~s~l.., L. M. JACr~.,~ and D. W. T u R ~ Proc. Chem. Soc., 426 (1961). 33. J. D. B ~ . ~ ~ ' ~ , J. Chem. Phys., 36, 152 (1962). 34. M. G. M o ~ , G. PAu~'rr and M. E. Hoeas, J. Phys. Chem., 60, 1594 (1956). 35. P. HIOIiAM and R. E. RJC:~IARDS,Proc. Chem. Soc., 128 (1959). 36. J. N. SItOOLERY,Discuss. Faraday Soc., 34, 104 (1962). 37. L. I>L~RAKISand C. H. SEDERHOLM,J. Chem. Phys., 35, 1243 (1961). 38. J. F~NEY and L. H. SUTct.n~ Trans. Faraday $oc., 56, 1559 (1960). 39. C. A. l ~ n J y, J. Chem. Phys., 25, 604 (1956). 40. J. J. TURNU, Mol. Phys., 3, 417 (1960). 41. D. G. DE K o w ~ . e w s ~ and V. J. K o w , ~ v s r ~ , J. Chem. Phys., 37, 1009 (1962). 42. S. L. MANATr and D. D. F J J ~ , J. Amer. Chem. Soc., 83, 4095 (1961). 43. D. C. DOUGLASSand A. FRATr~IO, J. Chem. Phys., 39, 3163 (1963).
279
CHAPTER
8
THE ANALYSIS OF H I G H R E S O L U T I O N S P E C T R A 8. I INTRODUCTION
WhEN an NMR spectrum consists of more than a single absorption band the spectrum can be expressed in terms of two parameters, namely chemical shifts (1,o6) and coupling constants (J). The purpose of this section is to indicate how numerical values of these parameters can be extracted from an experimentally observed NMR resonance spectrum. Most of the spectra previously
rH Resononce
~gF Resononce
I
~,j-D Fro. 8.1 Diagrammatic N M R spectra of C H s F - t h e tgF resonance giving 2(~) + I = 4 lines and the t H resonance giving 2({) + 1 = 2 lines
considered have consisted of well-separated chemically shifted absorption bands possessing multiplet fine structure due to spin-spin interaction between the differently shielded nuclei (this is always true for nuclei of different species). Under such conditions, the number of multiplet splittings on any absorption band is given by the simple (2/1- + 1) rule and the intensities of the components are distributed symmetrically about the centre of each multiplet (see Chapter I). This type of spectrum is referred to as first order, and it is possible to obtain chemical shift and coupling constant parameters directly from such a spectrum. Thus methyl fluoride, CH3F, should have a X~F resonance spectrum which is composed of a symmetrical quartet due to the fluorine nucleus coupling with the CHa group of total spin, IT = 3/2: the 1H resonance spectrum consists of a symmetrical doublet due to coupling with the single fluorine atom (see Fig. 8.1). 280
T H E A N A L Y S I S OF H I G H R E S O L U T I O N
SPECTRA
281
The necessary condition for the observation of a first order spectrum (i.e. % ~ ~, J) is sometimes satisfied by interacting nuclei of the same species. In the case of 2-bromothiophene, the chemical shift differences between the three non-equivalent hydrogen nuclei at 60.00 Mc sec -1 are much greater than the J values involved and the ZH resonance spectrum (shown in Fig. 8.2) of this molecule closely approximates to a first order spectrum. Here, the three differently shielded nuclei give rise to three separate absorption bands each of which is split into four components of roughly equal intensity due to spin-spin A
X
I]
f"
I
,
H~.S )St
I
I i I
FIG. 8.2 The ~H resonance s p e c t r u m o f 2 - b r o m o t h i o p h e n e at 60 M e sec -~. Cohen and McI.~ueld~O)
interaction with the other two non-equivalent nuclei (the four components being a doublet of doublets). The three coupling constants involved can be obtained directly from the spectrum by subtraction of the appropriate bands as indicated in Fig. 8.2 and the chemical shift differences between the nuclei can be obtained by ass-mlng that the true resonance position for each nucleus is at the centre of its symmetrical multiplet. An analysis of this kind will not yield the relative signs of the coupling constants and this information can only be obtained from a first order spectrum by resorting to further experiments (such as spin-decoupling(~), see Section 8.19.5). When the magnitudes of the chemical shift differences are comparable with the couphng constants, the simple rules for analysing the spectra break down.
282
HIGH RESOLUTION NMR SPECTROSCOPY
Such a state of affairs is found in the 1H resonance spectrum of the hydrogen atoms in glyeidonitrile~z),
ri\,
a
shown in Fig. 8.3. There are thirteen absorption bands in this spectrum as opposed to the twelve bands predicted by first order theory for a three spin system of nuclei of spin value ½. The components of the multiplets are of unequal intensity and the chemical shift differences cannot be obtained by assuming that the true resonance frequency for a particular nucleus is at the centre of its multiplet absorption. Chemical shift differences and spin coupling constants can only be obtained from such a spectrum by subjecting it to a full H
M~\,C__c/CN H ,~/',,Oz/
'H ,:
il
~k
5
cycles
sec-i~
i*~ i
iI
,
j'
i
i~' 1 I
Fio. 8.3 The ~H resonance spectrum at 40 Mc sec-t of glycidonitrile. Reilly
andSwalen(2)
quantum mechanical analysis. This requires evaluating the nuclear energy levels and stationary state wavefunctions appropriate to a system of nonequivalent interacting nuclei in a static magnetic field in the absence of the applied radiofrequency field. By considering certain selection rules it is possible to decide which energy levels are involved in transitions when the radiofrequency field is applied at the resonance frequency. The transition energies (which correspond to the measured absorption bands in the spectrum) can then be obtained by subtracting such energy levels. Calculations of the transition probabilities of the various transitions lead to values of their predicted relative intensities. By assigning the transition energies to the absorption bands in the spectrum (paying due regard to the calculated intensities obtained from estimated values of the parameters involved), it is possible to construct a theoretical spectrum which agrees with the observed spectrum and from which numerical values of chemical shifts and coupling constants are obtainable. By this procedure one can also usually ascertain the relative signs of the coupling constants.
T H E A N A L Y S I S OF H I G H R E S O L U T I O N S P E C T R A
283
The quantum mechanical principles and the evaluation of theoretical spectra form the main part of this chapter. Monographs dealing exclusively with the analysis of N M R spectra have been written (3' 4). 8.2 NOTATION Berustein, Pople and Schneider (s) have introduced a convenient notation for referring to systems of magnetic nuclei occurring in molecules and it is intended to use this notation (with slight modification) in the present text. Non.equivalent nuclei of the same species with resonance absorption bands separated by chemical shift differences of similar magnitude to the coupling constants involved, are denoted bythe letters A, B, C, D etc: such a group is called a basic group of nuclei. Other non-eqnivalent nuclei in the same molecule separated from this basic group by large chemical shifts, but which are separated from each other by chemical shifts of magnitudes similar to the coupling constants involved, are denoted by X, Y, and Z (these need not necessarily be nuclei of a different species from A, B and C). A further set of non -equivalent nuclei separated from the other two sets of nuclei by large chemical shifts, but separated from each other by chemical shifts comparable in magnitude to the coupling constants between them, are referred to as P, Q, R . . . . Thus vinyl fluoride CH2 = CHF, is referred to as an ABCX system. The three vinyl hydrogen nuclei have absorption bands separated by chemical shifts which are not large compared to the coupling constants, at normal values of applied magnetic fields, and are therefore classified as ABC nuclei. The fluorine absorption bands are separated from those oF, the hydrogen nuclei by a chemical shift large compared with the J,v coupling constants and thus the fluorine nucleus is classified as X. Nuclei which are magnetically equivalent are referred to by the same symbol and the number of nuclei in a molecule having the same symbol is indicated by means of a suffix on the symbol. Hence, 1,3-dibromo 2-nitrobenzene H NO= is said to form an AB2 system. Ethyl chloride, CHsCH2CI, constitutes an A2B 3 system on this nomenclature. Nuclei which show chemical shift equivalence but are not magnetically equivalent are referred to by the same letter but are distinguished from each other by means of primes. An ABB' system describes a molecule in which the two B nuclei have the same chemical shift but couple to different extents with the third nucleus A (6). In the molecule 1,1-difluoroethylene, CF2=CH~, there are two pairs of symmetrically equivalent nuclei which are not magnetically equivalent. This is a consequence of the two hydrogen nuclei coupling with each individual fluorine nucleus to a different extent (J~, ~ J~vw). Hence, this molecule is classified as an AA'XX' system. The notation of Bernstein, Pople and Schneider does not distinguish between magnetic equivalence and chemical
284
HIGH RESOLUTION NMR SPECTROSCOPY
shift equivalence, therefore it has been necessary to introduce the modification outlined above to clarify this point. Some authors tx'7~ refer to an AA'BB' system as an A 2* B2s system but this convention will not be used in this text. Table 8.1 lists a series of molecules together with their classification in the terminology to be used in this text. TABLE8. l CLASSIFICATIONSOF SOMETYIUCALMOLECt~ Compound H Clf~tiF Clt~Cl
Classification
AX
c1 CH2ffiCCIBr CH2ffiCFCI CHzffiCHBr CH2FCl CHzF CHzffiC=CHCI CF_,ffiCHz H
AB ABX ABC A2X A3 X AB2 AA'XX' AA'BB'
H (~taCH2)PFz
AaB2PX2
ir
o
8.2.1 Equivalent Nuclei Nuclei in molecular systems can exhibit various kinds of equivalence and it is imperative that the meaning of each be clearly defined at this stage: (a) Chemical shift equivalence. Nuclei which have the same chemical shift are said to show chemical shift equivalence. (b) Symmetrical equivalence. Nuclei which interchange their positions in a molecule upon the application of a symmetry operation are symmetrically equivalent: they are of necessity chemical shift equivalent. (c) Magnetic equivalence. Nuclei which have the same chemical shift and which couple to the same extent with all other magnetic nuclei in the molecule are referred to as magnetically equivalent. Hence, if we have a molecule containing a group of nuclei G containing several nuclei G1, G2, Ga ... G~, then these are magnetically equivalent if (i) they all have the same chemical shift: ~ai -- ~a~ -- ~a3 . . . . = ~ , (ii) each nucleus in this group couples to the same extent with each nonequivalent nucleus in the molecule: Jaw'~ -- J ~ ; ffi Jaw'~ . . . . Ja,~j where G~ is some nucleus j in another set of nuclei G'. Whilst it is not necessary for symmetrically equivalent magnetic nuclei to be magnetically equivalent, symmetrical equivalence often does result in magnetic equivalence. Thus, although 1,1-difluoroethylene,
THE A N A L Y S I S OF H I G H R E S O L U T I O N S P E C T R A
285
Hi ~Ft has two pairs of symmetrically equivalent nuclei neither pair is magnetically equivalent: this is because the symmetrically equivalent nuclei are not coupled to the same extent with each non-equivalent nucleus in the molecule e.g. because J.xFx • Jtl2V*, Hx and H2 are magnetically non-equivalent and likewise for the two fluorine nuclei. On the other hand, difluoromethane, CH2F~, contains two pairs of symmetrically equivalent nuclei which are also magnetically equivalent: the two nuclei in each pair have the same chemical shift and both nuclei in any one pair couple with each non-equivalent nucleus in the molecule to the same extent. The same is true for the non-planar molecule 1,1-difluoroallene,
F?C=CfC4H F/
\H
where the symmetrically equivalent nuclei are also magnetically equivalent. Both molecules give spectra of the type A2X2. The requirements for magnetic equivalence can sometimes be satisfied accidentally in a group of nuclei which are not symmetrically equivalent. Magnetic equivalence of this kind cannot be predicted by examination of the symmetry of the molecule but can be detected only by experimental observation. A case of accidental magnetic equivalence is provided by ethylene monothiocarbohate(7)t. Symmetrically equi'valent magnetic nuclei in substituted aromatic compounds are frequently ~nagnetically non-equivalent. Mono-substituted and o- and p-disubstituted benzene derivatives all possess two pairs of symmetrically equivalent nuclei which are magnetically non-equivalent. In the molecules S
S
S
S = Substituent H~
H4
S'
H1 and H2 are magnetically non-equivalent since Jmns ~ J.2as and likewise for Hs and ]Lt. Coupling constants between magnetically non-equivalent nuclei can feature in the spectrum. Whenever the term "equivalent nuclei" is used without further qualification in the present text it is implied that such nuclei are magnetically equivalent.
8.2.2 EquivalenceResultingfrom Con/ormationalMotion Rapid rotation of a group of magnetic nuclei about a molecular axis of rotation can sometimes result in the group of nuclei becoming effectively magnetically equivalent. The methyl and methylene hydrogen atoms in an ethyl 1" The spectrum can be explained also in terms of a "deceptivelysimplespectrum" from magneticallynon-equivalentnuclei.
286
HIGH RESOLUTION
NMR SPECTROSCOPY
derivative form two groups of magnetically equivalent nuclei as a result of internal rotation (see Section 9.5.5). If the methyl groups were held in a fixed position with respect to the methylene group as shown X
Hf~H2 H4~V//'--~H5 H3
then
~1 =~2 :#~3 d3,
=
dl, ~ d2,
and the methyl hydrogen atom would therefore be non-equivalent. At room temperatures, the molecule undergoes rapid internal rotation about the carboncarbon bond, and the averaging process associated with the rotation results in the methyl hydrogen atoms and those of the methylene group becoming two groups of magnetically equivalent nuclei. The spectrum of the fixed rotamer is of the type AA'BB'C while that of the rapidly interconverting molecule is of the type A2B,. If the nuclei of a group in a molecule are rendered magnetically equivalent by rapid internal rotation it is possible that by lowering the temperature of the sample, free rotation will cease (the rotational isomeric forms will no longer be equally populated) and the previously magnetically equivalent nuclei will lose their equivalence (see Section 9.5.5). Rapid interconversion of ring compounds can also result in nuclei which are non-equivalent in the fixed conformational isomer becoming magnetically equivalent (see for example Section 9.7 in which is described the low temperature investigation of perfluorocyclohexaneCS)). An important property of magnetically equivalent nuclei is that the coupling constants between the nuclei in such a set have no effect on the observed N M R spectrum and therefore the coupling constants cannot be measured. Hence, in the molecule SFsCI, which has the tetragonal bipyramidal structure
.... t-CL the four equatorial fluorine atoms at the corners of a square are magnetically equivalent and hence the coupling constants JF2 F3 and JF2 F4 are not observed in the 19F resonance spectrum of this molecule. However, coupling constants
T H E A N A L Y S I S OF H I G H R E S O L U T I O N
SPECTRA
287
between symmetricallyequivalent nuclei which are not magnetically equivalent are observed in nuclear magnetic resonance. For example, in 1,1-d/fluoroethylene, CF2-~-CH2, values of dam and.lvF are obtainable from both the IH and the 19F resonance spectra. In favourable circumstances, suitable isotopic substitution into a molecule can cause a group of previously magnetically equivalent nuclei to become non-equivalent with the result that coupling constants between the nuclei are observed. By enriching a sample of ethylene with lsC atoms many of the resulting molecules have the structure H, /
-
\m
and the four hydrogen atoms (which are magnetically equivalent in ordinary ethylene) form two pairs of magnetically non-equivalent nuclei, and from the i H resonance spectrum it is possible to obtain the coupling constants ./12, ./,s and -/1,- In other molecules containing groups of magnetically equivalent nuclei on adjacent carbon atoms, for example CICH2CH2CI or dioxane, examination of the resonance of the hydrogen nuclei attached to the 13C present in natural abundance (1"1 per cent) gives the desired coupling constants (9). 8.3 QUANTUMMECHANICALFORMALISM It h~s been shown in Chapter 2 that a collection of magnetic nuclei in a magnetic field has a discrete energy spectrum, and transitions between energy levels can be stimulated by applying a secondary magnetic field oscillating with the appropriate frequency (see Section 2.5). The general problem in constructing any theoretical spectrum is that of obtaining expressions for the energy levels of the system and then of computing the energies and probabilities of allowed transitions. No more than an outline of the general method can be given here, and the reader without prior knowledge of quantum theory is recommended to consult referencesl0 to 13. Experiments aimed at obtaining values of the chemical shift and spin-spin coupling constants are carried out with the aid of a stimulating magnetic field //1 of intensity small enough to avoid saturating the system. In this case/'I1 does not change the energies of the stationary states of the system, but simply induces transitions between them. The nuclear system can be regarded therefore as being in an external magnetic field whose intensity does not change with time, and the energy levels are found by solving the time-independent SchrSdinger equation (Chapter 2 of reference 10)
in which ~lff is the Hamiltonian operator and ~ is the stationary state wavefunction. Before discussing the form that ~ and ~ take in the particular case of nuclear resonance it will be useful to look at some of their general properties. It is always possible to find the form of the Hamiltonian but the statefunction is not usually known, except of course that it is a function which satisfies equation (8.1). However, an arbitrary statefunction ~ may always be expanded
288
H I G H R E S O L U T I O N NMR S P E C T R O S C O P Y
as a linear combination of the eigenfunctions q~, of some physical quantity f
i.e. ffi E c. ~.
(8.2)
n
where the ~,. satisfy the equation
f~o. = f, ~.
(8.3)
f~ being an eigenvalue o f f . The summation in equation (8.2) extends over all possible values of n, and the set of functions ~o, is known as a complete set (p. 8 of reference 10). The coefficients C, satisfy the relationship IC.[ = = 1
(8.4)
n
that is, the series is normalised. Further, the functions ~o. are orthogonal,
f,~,,, ~.* dr = ,~,..
(8.5)
(d~ symbolises integration over all the possible values of the coordinates). (Sin is the Kr6necker delta: (~,, = 1 i f m = n, zero otherwise. A set of functions satisfying equations (8.4) and (8.5) is known as an orthonorrnal set. If the orthonormal set (8.2) is substituted into equation (8.1) we have X" Z C. ~0. = E E C . ~ ..
(8.6)
Multiplying both sides of (8.6) by ~v* a/~d integrating over all space
f ~*~ F; C. ~. dT = E f ~*. Z C. ~. d~.
(8.7)
The integrals on the L.H.S. of (8.7) will be denoted by the symbol X',.. i.e.
x ' . . = f ~ * x " q~. d'r.
(8.8)
The integrals on the R.H.S. of (8.7) are zero unless m = n because of the orthonormality of ~o.. Equation (8.7) may be written as
Z a~?',.. C. = E ~. C. Or,. n
(8.9)
n
thus (x'.,. - ~..~)Z
c . = 0.
(8.10)
n
Equation (8.10) gives a set of homogeneous linear equations in C., and the necessary and sufficient condition for non-trivial solutions is that the determinant of the coefficients should be zero " t ) , i.e. IX'..
- &..El
= 0
(8.11)
Equation (8.11) is a polynomial in E of degree n and is known as the secular equation. The integrals X',,, may be arranged as a matrix which can be regarded as representing the Hamiltonian operator (Chapter 9 of reference 11). Similarly, the matrix formed from the coefficients C. represents the statefunction ~ .
T H E A N A L Y S I S OF H I G H R E S O L U T I O N S P E C T R A
289
The matrix formed from the elements .,~m, obeys the normal rules of matrix algebra. The elements o~?'=, are not unique since the wavefunction W could be expanded in terms o f another complete orthonormal set q,~ so that ~'O
J'
~ e . ffi f ~..,~ ~, dT.
(s.12)
However, the eigenvaiues obtained from the secular equation are unique even though the matrix elements are not. If the functions ~, are in fact eigenfunctions of the Hamiltonian, that is, ~. ffi ~ ~. (8.13) then the matrix representing.~ has all elements ~ , , zero except ~¢o , that is, the matrix is diagonal. This follows from equations (8.8) and (8.12),
but since ~, is an eigenfunction of .~,
~ , . -- f~,_~ ~, dT ffi F ~ f ~ * ~0. dT = E, ~ ,
(8.14)
so that the diagonal elements are the eigenvalues of ~ . The operators in quantum mechanics which correspond to measurable physical quantifies must have only real eigenvalues, so that if () is an operator with real eigenvaiues g, then
fv,*
6wd~ =
fv, 6*w* d~.
(8.1~
6 * is the operator which satisfies the equation 6* V'* ffi g* V'*.
(8.10
If ()' is the transposed operator (), defined as
then for operators with real eigenvalues 6 . ffi 6,.
(8.18)
That is, the transpose of the operator is equal to its complex conjugate. Such an operator is said to be Hermitian. The matrix elements of a Hermitian operator, such as Jff, obey the relationship
which for operators with only real matrix elements means that ~°m, ffi ~ m .
(8.19)
Thus the matrix representing the Hamiltonian operator is symmetrical about the leading diagonal. m~s. 10
290
HIGH
RESOLUTION
NMR
SPECTROSCOPY
8.4 FACTORmINGTim SECULAREQUATION The secular equation (8.11) is a polynomial in g of the order n, so that the exact solution of the equation would become impossible for n > 3, unless the order of the polynomial can bo reduced by factorisation. Factorisation may be achieved by a suitable choice of the functions 9,- If the functions 9n are eigenfunctions of an operatorfwhich commutes with the Hamiltonian, then ( M ' f - f X ' ) = 0.
(8.20)
Consider two functions 9m and 9n, both eigenfunctions of f with eigenvaluesfm and fn respectively, then
f g* fg, dr
= f g*M'f~9, dz = f~j'9*ov:9,dr.
(8.21)
f 9 * f g 9,, d*.
(8.22)
Since ~ andfeommute,
f 9:,.~taf9nd*
=
Now f is an operator corresponding to some measurable quantity and is therefore Hermitian, thus (fgm)* = (f= 9m)* -- fm 9* (8.23) and
(fg~,)* = 9*f.
(8.24)
The R.H.S. of equation (8.22) is seen to be f 9~*f~'~g'9, dz = f m f 9* vYa9. dr
(8.25)
and, equating (8.25) with (8.20
f . f g * x ' 9 , d, = f . f g = a ' 9 ,
d,.
(8.26)
But in general fn ~ f , , , so that f9~* ~ 9, d~ = O, unless 9m and 9n belong to the same eigenvalue off. This means that the matrix ~ = ~ is considerably simplified and the secular determinant contains many zero elements. If the set of functions 9~ can be grouped into sets 9t belonging to the eigenvalue f . then the secular determinant is factorised into sub-determinants of order i 2 and the secular equation is factorised into polynomials of order i,
8.5 THE HAMILTONIAN The form of the Hamiltonian operator for a collection of nuclei in a magnetic field is (I 3) o~ = + ~ y , h H,I=, + ~ Z d u I , .Ij (8.27) i
i <./
(Ht is in the negative z direction) where Ii is the spin-angular momentum operator for the i th nucleus with components (Ixt, Iy~, I=0. ~'~ is the magnetogyric ratio of the i th nucleus, and du is the coupling constant between nuclei i and j. The eigenvalues belonging to o*e'have the units of energy and as the equation stands it has the units ergs. It is more convenient to work in units of linear
T H E A N A L Y S I S OF H I G H R E S O L U T I O N
SPECTRA
291
frequency, so that J u is then in cycles sec °1. The change of units is effected by dividing through by h leaving
1 ~" = + ~ - ~ 7, H,/~, + ~ ~ :,j It" Ij
(8.28)
t<J
with J u in cycles sec-L Ht is the resonance field for the i th nucleus so that ~r, = tto(1 - at)
(8.29)
where Ho is the applied magnetic field and at is the screening coefficient for the i th nucleus (see Chapter 3). The chemical shift ~u between two nuclei i a n d j is dimensionless and is simply ~u = at - aj. (8.30)
7tilt is equal to cot, the resonance frequency in angular units, and if v~ is the linear frequency (cot/2~t) then M' - + ~ ~'t I~t + ~ J u l i t
l<j
"Is.
(8.31)
8.6 SPIN ANGULAR MOMENTUM The Hamiltonian, equation (8.31), consists of a combination of spin angular momentum operators so that to apply the Hamiltonian to any set of functions requires a knowledge of the properties of the spin.angular momentum. In the parallel case of the orbital angular momentum of a particle the properties of the quantum-mechanical operators are built up with the aid of the correspondence principle, which states that for high quantum numbers the classical theory applies. However, for spin angular momentum there is no classical limiting case. The spin angular momentum of a particle with spin I is simply 81 so that in the limit h - * O, the spin angular momentum disappears. This difficulty is resolved by assuming that there is a close analogy between orbital and spin angular momentum such that the same commutation rules apply in each case. Thus the commutation rules for the components of spin angular momentum are (reference 10 p. 78) xx¢ - z,1. = iI~ |
I,I: I,I, --. iI: [ f I,I:, -- I~I, iI, ]
(8.32)
ffi
and, the operator 12 ffi I - I commutes with each component I., I . I,, [12, Ifl ffi o.
(8.33)
Now it is a general theorem of quantum mechanics (reference 10 p. 14) that when operators in a set commute with one another then a set of functions exists which are simultaneously eigenfunctions of each member of the operator set. From equations (8.32) and (8.33) it is seen that 12 and any one of its components, but not all of them, forms a commuting set. Commuting operators have the physical significance of representing physical quantities which can be meal0*
292
HIGH
RESOLUTION
NMR
SPECTROSCOPY
sured simultaneously, so that for spin angular momentum we can have knowledge only of the eigenvalues of 12 and 1, say. If ~x. m is an eigenfunction of 12 and 1:, then
I2~U,.. = Iff + 1) ~ , . ,
(8.34)
l,~tz1.= ---- mgJ x.m
(8.35)
I, the nuclear spin quantum number, may have integral or half integral values depending on the nucleus, and m may take the values I, I - 1, I - 2 . . . - I + 2, - I + 1, - 1 including zero. For a particular value of I there are 21 + 1 values of m. The effect of operating on 'P~.... with the operators Ix and Iy is found via the two operators,
I ÷ = I,, + il~, I i-
=
,r~ -
ii,
(8.36)
!
the so-called "raising" and "lowering" operators (reference 10 p. 91). Iq-~uz.m -- [(I ~ m) (I +_ m + 1)1t/2 ~,.~,zt (8.37) 1 From equation (8.39), Ix = ½(1 ÷ + I-) and Iy = ~ (1" - 1-). Almost all the nuclear resonance spectra recorded have been concerned with nuclei with spin 1 = ½, and for this case the quantum number m may have only the values + ½, so that the system is either in the state -
gJl/2, +~/2 - o ~ I. ~1/2. -t/2 fl /
or
(8.38)
The effect of operating on 0~ and/3 with the angular momentum operators follows from equations (8.34), (8.35), (8.36) and (8.37). I2o~ = ¼o~; I2fl = ¼fl (8.39)
¢~, = ½ifl;
eft = -½i~ t[
/.~,=½~,;
/~flf-½fl
/
I" o~ = 0;
X*/3 = ,~
[.
I-~=fl;
I-/3 = 0
/
(8.40) (8.41)
The discussion so far has been concerned with a sin#c, isolated nucleus of s p i n / , but the.equations can easily be generalised to apply to a collection of nuclei. The operators I, I~ and 12 for a collection of n nuclei are
(8.42)
I =ZIj n
(8.43)
L- = Y & R
12 = Z I J n
+ 2ZEI,.Ij. l<j
Similar expressions can be written for Ix, ly, I + and I-.
(8.44)
THE ANALYSIS
OF H I G H
RESOLUTION
293
SPECTRA
The spin functions for the individual nuclei are assumed to be independent of one another so that the spin function for n nuclei with individual spins o, or # can be written as --- o~(1) ~ ( 2 ) . . . ~(n). (8.45) For I ---- ½, there are 2" product functions. It is important to note thatalthough the product functions are eigenfunctions of 1. they are not in general eigenfunctions of 12. The operators 12 and I: given by equations (8.42) and (8.43) obey the same commutation rules as the operators for the individual nuclei (equation (8.33)), so that a set of functions does exist which are simultaneous eigenfunctions of 12 and 1:. Such a set may be formed from suitable linear combinations of the product functions. The eigenvalues of total I: are m --- ~ m~ (8.46) R
and m may have the values
-z,.
+
that is [2 (.~ 1,) +1] values. But, for spin ½nuclei there are 2" product functions, L
/
J
which is greater than the possible number of values of m. The number of produet functions with the same. value of m is n!
m),
""
As an example of product spin functions let us consider a collection of four nuclei each with spin 1 -- ½. There are 24 - 16 product functions with values o f m of 2, l, 0, - l, - 2. Equation (8.47) shows that only one function belongs to the m value + 2, that is the product with all spins o~, m =
-I- 2 ,
~0ca~x.
The labels l, 2 . . . n will be assumed from the order in which the product is written. For m ffi + 1 equation (8.47) gives 4 functions, which are
m = 0 has 6 functions,
a pp,
pap, pa p,
app
and since the number of product functions belonging to m is equal to the number with - m we can write down the 16 functions in Table 8.2. Of the 16 functions in Table 8.2, only ua and u ~ are already eigenfunctions of 12.
294
HIGH RESOLUTION
NMR S P E C T R O S C O P Y
The operator 12 has eigenvalues lr(Ir + 1), where lr is the total spin quantum number (-- ~ 1,), and which for n nuclei of spin ½, can take the values 0 n
n
2'
2
for
n odd
n
(8.48)
I, ~ - - 2 , " "
_1 2
for
neven
TABLE 8.2 PRODUCT FUNCTIONS FOR Fo~mNUCLEIEACHWITHSPIN]"=
Function
m - ~
ml
Abbreviation
Function
m
=- ~ m i
n
Abbreviation
n
/11 //2
3~3~ ~#33 3~33 3#=3
M3
cctt~
0 0 0 -1 -1 -1 -1 -2
//4 //5 //6 //7 //8
3333
U9 UlO Ul l U12 //15
U14 UlS //16
The number of spin functions which are eigenfunctions of I z with a given value of Ir is n ! ( n - 2k + 1) (8.49) gm= (n-k+ l) tk! n
where
Iv = - 2 i 0,1""
and
k = 0, 1 " '
n2
k
for
(n
n even
1.__.._~) for
2
n odd.
Figure 8.4 shows (y) the values of g#v for values of n from 1 to 10. The eigenfunctions of 12 can be obtained as linear combinations of the product spin functions by solving the equation, (c.f. equation (8.10)), Y.(t~. - 2. ~..) C, = 0
(8.50)
n
where C, are the coefficients in the expansion of the eigenfunction of 12 as a linear combination of product spin functions, i.e. ~v,., = Z C. ~0,
(8.51)
II
in which the q~, are product spin functions for n nuclei. As an example of the construction of eigenfunctions of I z consider the case for two nuclei of spin ½.
THE ANALYSIS OF HIGH RESOLUTION
SPECTRA
295
The product functions are listed in Table 8.3. The wavefunctions ~IT . . can be expressed in the form (8.51) i.e. ~/ITOin ---~ C l Ul JI- C 2 u 2 -I- C 3 u s --I- C 4 u 4.
and the coefficients are obtained by first solving the secular equation
l I2mm -- 2~ ~mn I ffi 0
(8.52)
5
4
3
t IT 2
I
½
n
Flo. 8.4 Total spin quantum numbers and their statistical weights for n equivalent spin ½ nuclei. Corlo¢7) TABUB8.3 PRODUCTFUNC~Om t'ORTWO Nucuu EACHwrm I -- ½ Ftmction
I m [ Abbreviation
~p
0
u~
~ pp
0 - I
u3 u,
for the eigenvalues ~,. At first sight it would seem that equation (8.52) gives a polynomial in 2 of order 4, however, in Section 8.4 it was shown that the secular determinant contains non-zero elements only between functions belong-
296
HIGH
RESOLUTION
NMR SPECTROSCOPY
ing (in this case) to the same cigenvalu¢ of 1z. The secular determinant divides into two 1 x 1 determinants and one 2 x 2. : 2
::I~
-- 2
0
0
0
I ~22 -
2
o
zh
zh-2
o
o
o
0
1223
ii
0
-- 0 .
o izh-2i
It follows immediately that ut and u, are eigenfunctions of P , with the eigen-
values l~t and I~4. The integralsI,,,are readily evaluated e.g. P ul = 02 + I~) ul + 2(L. I2) u,.
(8.53)
Note that the operator Ij operates only on the jth nuclear spin function. (12 + I2)o~o~ = ¼c¢o~ + ¼o~x = ~xc¢.
(8.54)
To evaluate the second term on the right of equation (8.53) the product It • 12 is expanded into its components, (I1" I2)o~ = (1~,1~2 + [,,I,, + I~,1,~)o~o~
(8.55)
and from equations (8.40)
(11 "12)o~c~ = ¼ ~ .
(8.56)
Substitutingequations (8.56)and (8.54)into equation (8.53)gives I z ~x~x = 2 a n
so that l~x = 2 and since l~t = IT(IT + 1) then 1T = 1. A similar calculation for l~,t gives again 1T = 1. Operating on u2 with 12 does not generate only the function u2, because 12 u2 = (I 2 + I22)afl + 2(I1 • I2)~fl
(8.57)
and 3 3 (c¢ t) + ~ (o~t) = -~- (o~t )
(I2 + I~) o~fl =
1
1
(~,¢) + _~ (8 ~) + 4" (8 o,)
thus
I2
U2 "~ / / 3 "
(8.58)
similarly, The 2 x 2 determinant is
1 2 / / 3 = u2 "~" //3.
(8.59)
And
//2 =
(l - 4) l
t [= o (1 - 4) I
(8.6O)
which on expansion yields a quadratic in 2, --24+22
=0
(8.61)
T H E A N A L Y S I S OF H I G H R E S O L U T I O N
with roots
,~=0
SPECTRA
297
or 2
so that
Ir=O
or
1.
The coefficients C2 and Cs may be obtained by substituting the values of 2 into the equation (8.52) (1 - ;.) c2 + c3 = o I C2 + ( 1 - , l )
Cs = 0
}
(8.62)
and remembering that the normalisation of the wavofunctions demands that Cl + Cl = I .
(8.63)
Substituting ). = 0 into equation (8.62) gives C2
=
-
Ca
so that, from equation (8.63), l Ca = I
and the function ~o.o is
-~
-- C3 =
(~ fl - fl o~).
For 2 = 2,
l" C: =
thus,
~,,o
C3
I = -~-
=--~ (~,P + P~,).
The eigenfunctions of 12 for two nuclei are shown in Table 8.4 TABL~ 8.4 S~LTA~BOUS E I o ~ W u N c n o m Or I ~ AND I s n Function
m
oux
/T
l
1
-~-(~p + p~)
o
~2(~p - pc,)
o
8.7 BAsic PRODUCTI~'NCTIONS The wavefunctions of the Hamiltonian a~' =
Ii'~8.
lOa
+2,,I,,+ i
XZJuI,.la t<J
n -- 2
298
HIGH
RESOLUTION
NMR
SPECTROSCOPY
can be expressed as a linear combination of a closed set of functions ~0,,; the product spin functions form such a set, with the added advantage that they are eigenfunctions of an Hermitian operator 1, which commutes with ~ ' . The matrix elements J~'m, constructed from product spin functions will be non-zero only when the functions q0m and ~, belong to the same eigenvalue of I,, thus the secular equation factorises into (2Ix + 1) equations of lower order. For n = 2 there will be two 1 × 1 determinants, and one 2 x 2; for n = 3 there are two 1 x 1, and two 3 x 3 determinants. In general the order of the sub-determinants is given by the nth binomial coefficients. Even with the factorisation achieved by means of product spin functions the secular equation will still involve equations of order ~ 3 for small values of n. 8.8 BASICSYMMETRYFUNCTIONS So far it has not been necessary to discuss the environment of the interacting magnetic nuclei: it has been sufficient that the nuclei have coupled together. But the existence of a finite coupling constant between two nuclei only occurs if they belong to the same molecule, and it will be shown that the symmetry of that molecule is an important factor in the analysis of spectra (14, 15), and may lead to added factorisation of the secular equation (t~). Consider a molecule belonging to a symmetry transformation group containing g elements R (reference 10, Chapter 12). Each element R transforms some single-valued function of the coordinates ~01into a different function ~k, and the g elements generate g functions ~k, not all of which, however, are linearly independent. In general, a set of linearly independent functions ~ are obtained such that k = l, 2 . . . f with)' _~ g, and the functions are transformed only into linear combinations of themselves by the g elements in the group. An element R transforms the function ~ into a linear combination of the form $
=
(8.64) J=l.
where the coefficients Rjk form a matrix representing the operator R. The q~k may be chosen to be orthogonal and n0rmalised so that
_- f
R
dr.
(8.65)
The collection of the matrices representing each element of the group is called a representation of the group and the functions ~k are known as the basis of the representation. The number of functions f is known as the dimension of the group. A linear transformation of the coordinates transforms the functions 9~ into some other set ~f, and a matrix representing R may be constructed with the ~I as base functions, and of the same dimension as before. The two representations are said to be equivalent. Symmetry operators fall into the class of unitary operators, and such operators have matrix elements Rjk and Rl~ related by the equation E R~'~R,k = ~1. (8.66) k
THE ANALYSIS
OF HIGH
RESOLUTION
SPECTRA
299
and it may be shown that a linear transformation of a unitary matrix leaves the trace unchanged (reference 10 p. 330, Trace R ffi ~ R~ ffi ~ . ~ k
(8.67)
k
where RJk is the matrix formed from the elements, ffi
R
(8.6g)
The trace of a matrix representing a symmetry operator is known as the character Z(R) of the operator in the ~ representation. After performing a linear transformation on the base functions 9~ it may be possible to divide theffunctions ~ into sub-setsf~ ,f2 . . . . f~ such that the functions in each sub-set are transformed only into combinations of themselves by the group operators; that is, the functionsf~ ,f2 • • .f~ themselves form a basis for representations of the group, but of smaller dimension. In such a case the ~vk representation is said to be reducible; conversely if decomposed into representations of smaller dimension by a linear transformation then the representation is said to be irreducible. The importance of irreducible representations in quantum mechanics is a consequence of the invariance of SchrSdinger's equation to symmetry transformations. Multiplying both sides of the equation ~f'lo = Elo by an element R of the group, gives RoW ~ -- R E w .
(8.69)
The Hamiltonian operator oW must be unchanged by any symmetry transformation so that R and oW commute, thus (8.69) may be written as oWR~p =
ERtp
which shows that the operator R transforms the function ~pinto another function (R~0) which is an eigenfunction of oW belonging to the same eigenvalue as tp. Apart from accidental degeneracy, this can only mean that the function R~0 is a linear combination of the function tp so that the ~0's form the basis for an irreducible representation of the symmetry transformation group. If the matrix elements of the Hamiltonian are constructed from functions 9t which form the basis for an irreducible representation of the symmetry group of the molecule, then it can be shown that (p. 344 of reference 9) the elements oWjt are non-zero only when 9j and 9~ belong to the same irreducible representation. The secular equation (8.11) factorises into a number of equations of lower order depending on the number of irreducible representations (or on the number of classes) in the group. 8.9 CONSTRUCTIONOF BASIC SYM~rKY FUNCTXONS
To achieve the maximum factorisation of the secular equation it is necessary that the functions ~., which form the basis for the matrix representation of the Hamiltonian, should be eigenfunctions of the total z-component of angular 10 ~*
300
H I G H R E S O L U T I O N NMR S P E C T R O S C O P Y
momentum, and also should form the basis for an irreducible representation of the symmetry transformation group of the molecule. The product spin functions ~n form the basis for a reducible representation and they may be transformed into a set wn-(~)belonging to the ath irreducible representation via the equation, (p. 336 of reference 10)
~c~ = y Z~-~(R), R ~
(8.70)
R
where Z('°(R) is the character of the operator R in the o~th irreducible representation, and may be found from character tables of the symmetry point groups (p. 338 of reference 10). The number of times that the o~thirreducible representation is contained in the reducible representation is given by
,7c~, = ! E z c~'(R) z (R) g~
(8.71)
in which g is the order of the group and z ( R ) is the character of the operator R in the reducible representation. As an example of the construction of basic symmetry functions from product spin functions we will work through the details of the calculation for a system of four nuclei at the corners of a square. Strictly speaking, the point group is D4h, but since a reflection in the plane of the nuclei does not change the spin system in any way, the group may be regarded as D4. The elements of D~ are illustrated in Fig. 8.5. The permutations of the spin coordinates brought about by the elements are: Permutation E, the identity operator, 1234 A, a 2-fold axis of rotation perpendicular to the plane, 3412 B, a clockwise 4-fold axis of rotation perpendicular to the plane, 4123 C, an anti-clockwise 4-fold axis of rotation perpendicular to the plane, 2341 D, a 2-fold axis of rotation passing through nuclei 2 and 4, 3214 F, a 2-fold axis of rotation passing through nuclei 1 and 3, 1432 G, a 2-fold axis of rotation in the plane, 4321 H, a 2-fold axis of rotation in the plane, 2143 The first step in the calculation is to obtain the characters of the elements in the product representation. The effect of operating on the 16 functions (Table 8.2) is shown in Table 8.5. The characters are obtained by forming the products u~' R u~ and summing over allj and k, remembering the orthonormality of the ~o
The characters are: z ( E ) = 16;
z(A) =4
Z(B) = z ( C ) = 2 Z ( n ) = Z(F) = 8
Z(G) = z ( H ) = 4.
THE ANALYSIS
OF H I G H R E S O L U T I O N
301
SPECTRA
D , contains five irreducible representations which are denoted by the letters
At, A~, Bt, B2, E and the character table is given below (Table 8.6). TABLE 8.5 Tm~ Eev~cT oF ELm~l~rrs oF D4 ON PRODUCT FUNCTIONS FOR FOUR N u c t . n
/'/1 U2 /'/3 U4 U5 U6 U7 us u9 /~10 ffl I
H12 UI3 /'/14 U15 U16
E
A
B
C
Ul /42
Ul U4 U5 U2 U3 U9 U?
Ul 115
H] U~
U2 113
U4
//3 U4 US
ue ~7
us u9 "
Uxo Ull
Ulo u6 us
HI I //9 UlO U6 H? HI5 U12 UI3 H14 U16
U9
~11 /dl 3
U13 Ul& UI5 UI6
U15 U12 UI3 MI6
/dt$ U12 Ul 6
II3 U2 H5 U8 U7 U6 UI0 U9 Ul 1 UI2 UI5 U14 //13 UI6
It3 Ulo U7 119 U8
ue Ul 1 U14
U13 UI2 U15 //16
TABLE 8.6 CHARAg-IJ~t TABLE FOR ~ E
A 1 1 1 1
A1
A2
E
-2
B
C
1 1 -1 -1
1 1
0
I,, il 1 ,I/3 U2 H5
u,, U6 Ul 1 Ulo
U9 Ull U7 /4.1.3 U12 UI5 /'/14
UI6
D 4 POINT GROUP D
F 1
-1 -1
H 1
1 1 -I
0
0
C
-I 1
-1 -1
c; Ul I15 it,; 113 U2 119 Ul 1 U8 U6 UI0 U7 UI5 U14 UI3 U12 UI6
U4
U4
//8
us
Ul
/41 U2 U5
U5 //2
111o 1111 1~6
[.,
r
0
c,,c2
FIG. 8.5 Elements of the D4 point group
-1
1 1 0
302
HIGH RESOLUTION NMR
SPECTROSCOPY
At, A2, Bt and B2 are all of dimension one, whilst E is of dimension two. The number of times that each irreducible representation occurs in the product representation can be found by applying equation (8.71), ~(Az)={(16+4+2+2+8
+8+4+4)=6
~](Aa) = {r(16 + 4 + 2 + 2 - 8 - 8 - 4 - 4) = 0 ~7(Bz) = ~ r ( 1 6 + 4 - 2 - 2 +
8 + 8-4-4)--3
~7(B,)=~r(16+4-2-2-8-8+4+4)=
1
r/(E) = ~ ( 3 2 - 8 ) = 3 . The product representation contains Az six times, Bz three times, B2 once and Ethree times. This result may be checked since i f f (~) is the dimension of~, then W(00f c~') -- H ( 2 I j + 1) (8.72) c,
J
which in this case gives (1 x 6 ) + ( 1
x 3 ) + ( 1 x 1 ) + ( 2 x 3)-- 16
and is equal to I I ( 2 1 s + 1) when I = ½ for a l l j and when four nuclei are involved, s The combinations of the product functions which form the basis for the irreducible representations can be found with the aid of the operator (see equation (8.70)). ~?(o0 = ~ g c~')(R) R. (8.73) R
For the representation At, ~}(At) is ~/(Az)=E+A
+ B+ C+D+F+
G+H
and applying ~/(A t) to the sixteen functions in Table 8.2 gives (neglecting numerical factors)
~(Al) ut
=
ul
~/(At) u2 = u 2 + u 3 + u 4 + U s *7(At) u3 = u z + u 3 + u ,
+us
v(At) u 9
---- U6 + Us + U9 + Uto
,7(A~) Rio ~(A~) UZl
= U6 + US + Uo + Uzo = U7 -4- 1111
~/(At) u, = u 2 + u 3 + u , + u s
~(At)
*7(At) us = u 2 + u 3 + u , + u s
~ u t 2 + Uta + u z 4 + Uls
uz2 ---- u t 2 + u l a + u t 4 + uzs
TI(At) U6
= U6 + Us + U9 + Uto
,7(at) Uta ~(AI) Ut4.
~ ( A t ) uv
= u-/ + u t t
r/(A~) UjS
= Ut2
~(Az) us = u 6 + u s + u g + u t o
~(A~) lit6
= U16
= U12 + U13 + ~t4- + U15 +UZ3 + Ut4 + UZS
THE
ANALYSIS
OF
HIGH
RESOLUTION
SPECTRA
303
From the above it is apparent that the six independent functions belonging to the representation At are
u2 + u3 + u4 + us
.1 U?
+Ull
U6
+Us
I
+ U9 + Uto
U12 + U x 3
+ Ul.+
[
Uls
UI6
The operator ~(Bt)
/
is
~(Bt) = E + A - B - C + D + F - G - H and operating on the uj gives the non-zero functions, u2
+u a--u3
u3
+ Us--U2--
-- us
U7
-- Ull
u,
UI2 + u14 - - u13 - - u15 U15
t
+ uts --ux2
-- u l a .
Thus the three independent Bt functions are U2 + u , U7
-- u 3 - - u S
/
J
--Ull
U12 + U 1 4 - -
UtS - - U15
A similar calculation carried through with the operator, tl (B2) -- E + A - B - C - D - F + G + H I/6 +
gives the B2 functions as
U9 - - U 8 - - U t o .
The operator for the E representation is =
2E
-
2A
and this gives the non-zero functions, Ua - - U 4
U9
~
~6
UlO - - U 8
U12 ~ U14
u4
-- u2
U$
~U 3
U13 ~ U 1 5
u6
-- u9
~14~
U8
~
UlO
~12
~15 - - ~13
304
H I G H R E S O L U T I O N NMR S P E C T R O S C O P Y
so that the six independent functions are U2
~
U,~
u3
~
u5
u6
~
u9
/,4 8
--
UIO
U12 --
Ul, ~
U13
U15
--
(the E representation is two dimensional and occurs three times). The sixteen basic symmetry functions for the point group D, are shown in Table 8.7 (suitably normalised). 8. I 0 CALCULATION OF THE MATRIX ELEMENTS OF THE HAMILTONIAN
We have at our disposal all the information necessary for calculating the matrix elements of the Hamiltonian, but some labour can be saved by studying the relationships between the matrix elements. First, consider a basis for the Hamiltonian more general than the product functions but still eigenfunctions of I=. An operator R is defined (t :) which has the effect of reversing the spins of all the nuclei so that Rg(mt) = 9 ( - m3. (8.74) TABLE 8.7 BASIC SYMMETRY FUNCTIONS FOR FOUR NUCLEI BELONGING TO T ~ POINT GROUP D , Symmetrised spin function
D4 Symmetry
m = 2..
Iz/
J
2
al
~ ( ~ 8 # + =#8= + 8t~o~ + 8==~ 2-~(~,8=# + P=8=) ½(=888 + 8=88 + 8/~8 + 8/~8=) PP88
At At A1 Al At
½(===8 + ~ = = - ~ 8 = - 8a=a)
BI
1
Bt Bt
0
~(~=P~ + 8P== - =88= - ~==~ 2-~'(~x= 8 -- oq~==)
2-~(~flc, - / ~ ) 2-~(~,88= - 8 ~ ¢ ) 2-÷q~=SP -/~/~P=)
1
0 0 --1
-2
-1
B2
0
E E E E E E
1 1
0 0 -1 -1
where ~0(mr) is the eigenfunction of 1= with the eigenvalue ml. R commutes with two components of the spin angular momentum, JR,/j and anti-commutes with/.-.
= JR,/,] = 0
(8.75)
THE A N A L Y S I S OF H I G H R E S O L U T I O N S P E C T R A
305
The Hamiltonian may be written as
= ~ + ~
(8.70
A ffi ~ ~,z,,
(8.77)
where g
§ = ~ ~ J,~l,. i~. !
(8.78)
<j
The operator ,~ contributes only to the diagonal elements of the Hamiltonian matrix, whereas ]3 contributes to both diagonal and off-diagonal e]ements. The diagonal elements of the Hamiltonian may be written
~ ~ = (~k (m) I ~ I ~ (m))
= (~(m)I~-I ~(m)) + (~j(m)I§l~(m)) = ak~ + b~.
(8.79)
The off-diagonal elements ~Fqk are
~ . ~ = (~,(m) I ~ l ~ ( m ) )
= (~q (m) I fi I q~k(m)) = bqk.
(8.80)
HOW the operator R commutes with B and anti-commutes with A.
Rfi-~R=0 RA + ~R = o thus,
}
~--R§R
(8.81)
= --RAR
Substituting the relationships (8.81) into equations (8.79) and (8.80) gives
(~pk(- m) I-~1 ~ ( -
m)) = - a ~ + b~k
(8.82)
(~p,(-- m) I ~ 1 ~ ( -
m)) = bek
(8.83)
so that when the matrix elements for the states with 1, = m are known, the matrix elements with 1, = - m are readily obtained from the above equations.
When m = 0 the rules must be applied with caution. If the operator R simply permutes the ~t(0) then the diagonal elements are given by equations (8.79) and (8.82), but if R does not permute the ft(0) then a.~ is zero and the diagonal elements are simply btt. In general, a mixture of these two possibilities is found for m -- O. The base functions are almost always product spin functions (sometimes symrnetrised), and it is found convenient in this case to express the Hamiltonian as
Jla = ~ v, I=, i
+
~., ~., d,j I=, /=j i
<J
+
½ ~ ~., du (l; J'; , <J
+
I;I;).
(8.84)
306
H I G H R E S O L U T I O N NMR SPECTROSCOPY
From equations (8.35) and (8.37) the diagonal matrix elements of ~ are (~0~(m) I ~ 1 ~(m)) = ~ ,, m, + ~ ~., J,j m, mj |
(8.85)
~ <J
and the off-diagonal elements are (¢p(...I/, mr+ 1 . . . I j , m j - 1 . . . . m ) I ~ 1
~(...
I,, m i . . . .
I j , mj . . . m))
= ½JtJ [(/, - m,) (lj + mj)(Ii + m, + 1)(Ij - mj + 1)]~.
(8.86
When all the nuclei have spin I = ½ the equations (8.85) and (8.86) simplify to
(~0~t~ 1 ~ ) = ½Y.E J,J uu (8.88) where t <j S~ is + 1 for spin oc and - 1 for spin/ff Ttj is + 1 if spins of nuclei i and j are parallel - 1 if they are anti-parallel Uu is + 1 if the functions ~0~and ~k differ only by the interchange of spins i and j, and is zero otherwise. As examples of the above rules let us calculate some of the matrix elements between the product spin functions o~a~ff and ocoqffa.The diagonal elements are (from equation (8.87)) - J~, - J~,)] + Y~, - y~,)]
and the off-diagonal element (from equation (8.88)) is
(~HI ~l~fl~)
= (~1-~1~/~)
= + ½J~,
8.11 T~NSITmN I~rrm~SlTW.S Solving the secular equation (8,11) gives the energy levels of the system, and substitution of these values into equation (8.10) gives the coefficients C. and hence the dgenfunctions ~p. are known. The probability of transitions between the levels yj and ~Pkinduced by a magnetic field H, rotating in the x y plane has been discussed in Section 2.2 and is given by or
e j _ ~ = (7, H1) 2 I(~j II=,1 ~=)12
(8.89)
P j ~ = (Yi H1): [(%lI, i[ VD[ ~
(8.90)
since the x and y directions are equivalent. In many examples where the simple rules for analysing spectra cannot be applied, that is, other than first order spectra, the magnetogyric ratio 7t is the same for all nuclei. The equations for Pj_~ can also be written in terms of the matrix elements of I-, so that Pj~k ~ I(%1/-I ~h)[2
(8.91)
T H E A N A L Y S I S OF H I G H R E S O L U T I O N S P E C T R A
307
and since the effect of operating on a function with I- is to reduce the value of m by unity, then because the functions ~0kare orthonormal the matrix elements in equations (8.89), (8.90) and (8.91) are zero unless tpj and tpk differ only by - 1 in the value of m, thus transitions take place only between energy levels with Am---- --1. In Section 8.9 it was shown that eigenfunctions ~ must form the basis for an irreducible representation of the symmetry transformation group, and since the operators I=, I, and 1- are unaffected by symmetry transformations it follows that the matrix elements in equations (8.89) to (8.91), and hence the transition probabilities, are zero unless ~ and ~p~belong to the same irreducible representation. Also transitions only take place between the energy levels of nuclei of the same species. 8.12 EQUIVALENTNUCLEI The four hydrogen nuclei in a molecule of methane C H , all have the same shielding coefficient, and a calculated coupling constant between any pair of hydrogen nuclei o f - 12.4 cycles sec-1 ; however, only a single peak occurs in the N M R spectrum of methane. That the single absorption band in methane is not necessarily the result of zero coupling between the hydrogen atoms, but is an example of a general phenomenon is shown by the following theorem c~3>. . The Hamiltonlan ' '~=
E ",I=, + E E 1
I
Ju I, " IJ = ~ + ~)
<J
commutes with lz but not with IL Evaluating the commutator [ ~ , 12] gives°>, [ ~ , i ~] = ~ ~ 2i(,, - ,j)CX., 6J - I,, x.j) t
(8.92)
<J
so that [3if, 12] -- 0 when ~ -- ~j for all i and j, and in this case the Hamiltonian becomes .)Eo = , I, + ~ E J u I , . I:. (8.93) t <J
The operators ~ o , I, and B (defined by equation (8.78)) form a commuting set with simultaneous eigenfunctions ,p~ (eigenvalues E,, m, bn respectively). Suppose that Wn-~ W,' is an allowed transition, ~ o ~ . = (m~, + bn) V', -- En~0n ,,to ~o.' = (m' • + b,,) Vn' -- E., ~0.,
and
E . , - E , -- (m' -- m ) )
+ b,, -- b , .
(8.94) (8.9~ (8.96)
It can be shown that both the transition energy and probability are independent of the coupling term ]3. The transition probability P,_,,. is, (*)
e,-,,, -- I(v,.I exp[i(§ + ~ HI/~) t] I~o,)12
(8.97)
308
H I G H R E S O L U T I O N NMR S P E C T R O S C O P Y
and the matrix element in equation (8.97) may be written as
(~0.,Iexp[i(B + Y Ht Ix) t] ]~.) = (exp(- iBt)~., Iexp(iyHt Ix t) I~,.)
= exp(ib., 0(v'., Iexp(i~,~tXx 01~D. (8.98) The squared modulus of the term on the R.H.S. of equation (8.98)is independent of b..,thatis,the transitionprobabilityisindependent of the coupling term §. Now, B and Ix commute, and so do the exponentialoperators exp (ifit)and exp (iy Htlxt),so that the matrix element in equation (8.97)may be writtenas iv,., I exp [i(B + y H1 Ix) t] 1%) = (~/,,, 1 exp (i;, r r, z~ t) I exp (i t3 t v/.)) = exp(ibn t)(Vn, [ exp(iy n : Ix/) I ~,.). (8.99) From equations (8.98) and (8.99),
exp(i b., t) (V.' I exp (i r HI Ix t) I W) = exp (i b. t) (~p., I exp (i r H, Ix t) I V.) (8.100) But ff ~n,'~ ~pn, is an allowed transition 0Pn, I exp(i y Ht lxt)[ ~.) cannot be zero, so that b~, = b~ (8.101) and substituting equation (8.101) into equation (8.96) gives
E , , - E. = ( m ' - m ) v '11.
Thus the transition energy is independent of the coupling term and a single resonance peak is observed. The coupling term can be dropped from the Hamiltonian for a group of magnetically equivalent nuclei (this should be done at the beginning of the setting up of the energy levels of a system), so that o¢:o = v / . .
(8.102)
The above theorem is perfectly general and does not invoke symmetry arguments nor does it depend on perturbation theory. Equivalence of chemical shift is a necessary but not sufficient criterion for magnetic equivalence when two or more groups of nuclei couple together. If the nuclei can be divided into sets G such that the nuclei within each set have equal shielding coefficients, and there is no observed coupling between nuclei in the set, then the Hamiltonian is = E ~'o I~6 + E E J o , G j l o , "IG3.
(8.103)
Gf G 1 G
6;
J~,G~ is the coupling constant between the tth nucleus in set G with t h e j t h nucleus in set G'. The condition that the operator I~ commutes with the Hamiltonian is the necessary condition for an equivalent group. The operator I~ is defined as I~ = E I2c,, + 2 E E In," I~, (8.104) Gi
G|'~G 2
T H E A N A L Y S I S OF H I G H R E S O L U T I O N S P E C T R A
309
and the commutator [I~, . ~ is (is)
[.,¢', I~=] =
2i~ Z ~.(Jo,~
-
J~,a~)[l~,~(l~, I~',= - Io,= Iai,)
G~ G | < G j
+ zo,,
1oh - zo, i ;3 (8.105)
Therefore, I~ and ~ commute if
Jo,o~
=
(8.106)
Jo~o'k
for all i a n d j and for all groups G. That is, the nudei within the group G should all couple to the same extent with all other nuclei in other groups. The condition for magnetically equivalent nuclei may be stated as being first that they have the same chemical shift and second that they couple to the same extent with all other nuclei (see Section 8.2.1). When a collection of nuclei can be divided into sets of equivalent nuclei it is advantageous to construct the Hamiltonlan matrix elements from functions which are eigenfunetions of I~ as well as 1=; this is a direct consequence of I~ commuting with ~ . The matrix elements ( Jl l win now be zero unl a new selection rule for allowed transitions appears since I~ commutes with 1- = ~ IN. The selection rules are, therefore a
Am f ~ Amo = - 1
o
AIo=O
]
l"
(8.107)
Summary oJ rules.for analysing spectra. I. The Hamiltonian is, (i
G < G'
where G and G' refer to groups of equivalent nuclei so that there is only one coupling constant between G and G'. 2. The Hamiltonian matrix is constructed with linear combinations of product spin functions which are simultaneous eigenfunctions of 1: and I~. 3. The basic functions should form an irreducible representation of the symmetry transformation of the nuclear assembly. 4. The matrix elements (~0j I.,~'l ¢~) are non-zero only when ~0j and ck (a) belong to the same eigenvalue of 1= (b) belong to the same eigenvalue of I~ (c) belong to the same irreducible representation.
310
HIGH RESOLUTION NMR SPECTROSCOPY
5. Transition intensifies are proportional to
I(~ojII-I wk)l 2 where ~ and ~k are eigenfunctions of the Hamiltonian. 6. The selection rules for transitions between the levels of ~pj and ~Pk are: (a) A m = -- 1 (b) ,tI~ = 0 (c) y j and W~ must belong to the same irreducible representation. 8.13* THE ANALYSISOF SPECTRA CHARACTERISEDBY ONE COUPLING CONSTANT The NMRspectrum of a molecule containing two different groups of magneticaUy equivalent nuclei depends on only one coupling constant JAB, and one chemical shift difference tSXe. Apart from a sealing factor, the spectrum is dependent upon only a single parameter, the ratio d/vo6 (the ratio of the coupling constant and the chemical shift difference between the two different types of nucleus). It is relatively easy to compute the spectrum expected for various values of d/vo ~ and to compare them with the experimentally observed spectrum. F r o m this both J and ~ can be found. This section will deal with common values of n in ABn systems in turn, and then with common values of p and n in ApBn systems. 8..13.1 The A B System The simplest N M R spectrum featuring both a chemical shift and a spin-spin coupling constant arises from a molecule containing only two non-equivalent magnetic nuclei. If the two nuclei are separated in their spectrum by a small chemical shift of the same order of magnitude as the coupling constant, then they are said to constitute an A B system. Several workers have discussed the detailed analysis of the N M R spectra of such molecules. GeneralJeatures oJ an A B spectrum. The N M R spectrum obtained for an A X system consists of a pair of doublets, the components of which are of equal intensity as shown in Fig. 8.6. The separations between the components of each doublet are identical and are equal to the spin coupling constant, JAX, between the two nuclei. An accurate value for the chemical shift difference between two such nuclei is obtained by assuming the resonance frequency of each nucleus to be at the centre of its doublet. In an A B system, in which J/vo6 ~ I, it is necessary to apply the results of the full quantum mechanical treatment of the problem to obtain the spectral parameters and to predict the relative intensities of the transitions. Spectra similar to that shown in Fig. 8.7 are observed: the components of the doublet are no longer equal in intensity nor can the chemical shift difference between the nuclei be measured directly from the spectrum. The following sections deal with the analysis of the more important spin systems; less common systems are tabulated in Appendix I. All magnetic nuclei referred to have ! = 1/2.
THE
ANALYSIS
OF HIGH
RESOLUTION
SPECTRA
311
Analysis of the AB system. The four basic product functions, ~u, ~3, 3 °~ and 33 are used as a basis for obtaining the stationary wavefunctions for the molecule: they are referred to as u l , u2, us and u. respectively. It will be seen presently that
r
A I ~, I I
\
I\ I \ I I
\
l~o. 8.6 Theoretical A X type spectrum
H i • Hydrogen
5
Hydrogen 4
J Band number
I
3
I0 cycles seC"
FIG. 8.7 Example (>fan AB type spectrum. Ring aH ~'sonance at 40 Mc sec-z of ]/quid 2-methylthio-3-thiopbenethiol. Hoffman and Gronowitz(22)
the only off-diagonal elements for this system are Has -- Hs2 -- ½Y: thus the mixing of basic product functions will occur only between u2 and us (that is,
up and pu). The Hamiltonian operator for this system is
ar = ~ H o [ 0 - o ~ ) Z,. + (l - a , ) I,.] + J^, I^. Z,. + ½Z^B(G I; + E, G).
312
HIGH RESOLUTION NMR SPECTROSCOPY
The secular equation for an AB system can be written Hlx--E
H~2
H~3
Hx~.
I
H2x
H22 - E
/-/23
H2.
//31
/'/32
H33 - E
H34.
H,~
H4z
H,~n
H,,,, - E
I
= 0
(8.1o8)
where H11, H,2 etc., are the matrix elements, which can be obtained by applying the simple rules outlined in Section 8.10, in conjunction with the basic product functions for the system. The values of E are the eigenvalues or stationary state energy levels set up when an appropriate molecule is subjected to a magnetic field: E values can be obtained by solving the above equation. If the matrix were completely diagonal then the eigenvalues would be given by the diagonal matrix elements themselves. H l I represents the matrix element (ul I~1 u 0 or (~oq~lo, x), and H=t represents (o~t[,~[o~o0 etc. Hence the secular equation could have been written more fully as
((~,~I.~I~)-E) (~tl~l~,~)
(~l~l~t)
(~l.~lt~)
((~,tI~I~D-E)
(~l.~ltt)
(~tl-~lt~)
(o,t l g e l t t )
•o !
(ttl~
~,)
(triPlet)
(ttl~lfl~)
((fltl~ltt)- E)I (8.109)
The non-zero matrix elements for this system are 1
Y (HA+H,,)+
H,, = (o,fllglo, t) ---7l [ ~r ( ~ * - ~0 - + J ] ~q~ = (t~,l -YtlS:,) = ~- ~
( - HA + ~n) -- 7
H. = (ttl~ltD
(-H.-HB)+TJ
=7
H=3 = ( ~ t l ~ l tic,) =
1 --J 2
1 H32 = ( H ~ I ~ I
~)
= -= J L
THE A N A L Y S I S OF H I G H R E S O L U T I O N S P E C T R A
313
and substituting these into equation (8.109) we obtain the equation .~_.~
1 j_El 0
0
0
0
[ 4"4~"(HA -- H.) - -~-1 J - El ~1- J
0 •0.
0
2
0
E
o
o~
( - H ^ - ~/.) + ~ J - E (S.ll0)
HA and H . are the magnetic fields experienced by nucleus A and by nucleus B: HA = Ho(1 - aA) and HB ffi Ho(l -- a.) where aA and am are the dimensionless screening constants. If we denote the coefficients by as~ in the set of linear equations which the above determinant represents, then the following equations
areobtained
l
]al
(HA + H . ) + - - J - E t
4
( R A -- X . ) = 1_ J _
at, -
Ja,,
4
.
[.~_(_ H^ "+liB) (--HA-
1
-- ~ J -
HB) + -
4
= o
(S.lll)
= o
(8.i12)
E.] a3 ~ - ~I J a , k =
J-
a,,,
0
(8.113)
ffi 0.
(8.114)
The coefficients have the following values a11~0,
a22#0,
a2s = - Q a s 3 ,
as3#0,
a,,#0
as2 = Q a ~ 2
with all other values of aj~ = 0. A further restriction on the coefficients is that they obey the normalising condition J
For example, ~ t = 1 and ~ t + a~t = 1 etc. The matrix (equation (8.110)) can be factorised into two 1 x 1 matrices and one 2 x 2 matrix. Solutions obtained from the 1 x 1 matrices are
r ( - H ^ - H.) + ~ J . 4
(8.116)
314
HIGH RESOLUTION
SPECTROSCOPY
NMR
The 2 x 2 matrix is H22 -- E
/'/23 E
H32
~3
= 0
-
which can be evaluated (H~2 - E)(//33 - / r )
- #~s H~ = 0
and this expands to E 2 -- E(H22
q- H 3 3 ) at-
H22/-/33 - - H 2 3 H 3 2 = 0 .
Solution of this quadratic equation gives the two roots /r2 = ½ [ ( ~ 6" + y2)~/2 _ ½~q E~ = - ½ [ ( , ~ 6 2
where
~o=7Ho/2~
and thus
and
6=(re-(rA
H22 = ½(~o 6 - ½J)
We have
(H22
+ J ' ) + ½Jq
-
and
//33 = - ½(~o 6 + ½J).
E2) a22 - / - / 2 3 a32 = 0
(Hz2 - E2) a2z - Has Qaz2 = 0 and
(8.117) (8.118)
(/-/33 -
E 2 ) aa2 -
H32
(8.119)
a22 ----0
(H33 - - E2) Qa22 - H32 a22 = 0
(8.120)
Rearranging equations (8.119) and (8.120) gives Q2 = H33 -E
2
V0 6 - - ½ J -1- ½ J - ~0 ~ - ½ J "+" ½ J -
(1Jo 2 65 -+- j2)1./2 ( ~ 62 "+" j 2 ) 1 / 2 "
Multiplying above and below by - [~0 ~ + (%262 + j2)]1/2 gives Q2 = j2/[~, ° 6 + ( ~ 62 + j2)1/212
Q = J/[~o 6 + ( ~ 6" + j2)~/2]. Stationary state wave~unctions ./or the AB system. The stationary state wavefunctions will be denoted by ~1, ~P2, W3 and W4. It has been seen already that the wavefunctions ~1 and W4 associated with the energy states E1 and E4 are equivalent to the basic product functions ux and u~. The two remaining wavefunctions, ~2 and ~P3, can be obtained as the linear combinations of the basic product functions uz and u3 which would diagonalise the matrix. The coefficients of the basic product functions in these linear combinations are the coefficients of the linear equations obtained from the matrix (equations (8.112) and (8.113)): thus ~/)2 = a22 u2 + Qa22 ua ~/)3 =
- - Q a 3 a u2 q- a33 u a .
T H E A N A L Y S I S OF H I G H
RESOLUTION
315
SPECTRA
Because of the normalising condition a~ + Q~ a~ ffi1 a]3Q2+a]3ffi 1 hence a22 = (1 + Q2)-1/2 a3s ffi (1 + Q2)_1/2. Therefore Q2)-x/2 Q2)-x/2 u2 + Q u~ ~2 = ( 1 + u2 + Q ( 1 + u3 = (I + Q2)X/2 ~p~ = -- Q(1 + Q~)-II2 u~ + (1 + Q~)-x/2 u~ :
(s.12D (s.122)
u 3 ~ Q U2
(1 + Q2p/2 •
The four normalised functions are u2 + Q u3 (1 + Q~)~/2 U3 ~
Q U2
(I + Q2)I/2 ~v4 = u4.
IntensitiesoJ bands. The intensity of a transition between statesp and q is proportional to the square of the transition moment P P = OP, lMal W,) ' where ~0p and ~ are the stationary state wavefunctions and M~ is the x-cornponent of the magnetic moment operator
Mx = ~ ~ I
lz(i) -----~. ~, Ix (i).
zy[
I
The transition moment can be expanded in terms of the matrix elements of single products according to the rules given in Section 8.11. It is now possible to calculate the transition moments for the allowed transitions 7 P~-,., ffi P1-,.2 ffi ~ ' n (V= I/.lVl) ffi 7
2n
2n
([u2(1 + Q2)-l/= + Qua(1 + Q=)-a/=] I l x l u~)
([o~(I + Q2)-I/2 + Q p ~ ( I +
Q~)-I/~]Ixz I ~ )
= v ([~/~0 + Q2)-I/2 + QO~(I + Q2)-In] II.(~) + I=(B) I ~ ) 2~
ffi 7 [(I + Q2)-I/= + Q(I + Q2)-I/=] 4~ y(1 + Q) 4~(I + Q2)ll~ •
316
HIGH
RESOLUTION
NMR SPECTROSCOPY
The intensity is proportional to the square of the transition moment
Similarly,
p2+2__e2..~__
Y2( t + Q ) 2 16~2(1 + Q2)
P~-~
y2(I _ Q)2 16~z(1 + Q2)
.9
= P2-,a =
Since only relative intensities are required, these are given by L 2 . ~ = L,_~2 --
(I + Q)2 1 +Q2 (I -
La~
= Zl ~ a =
1
Q)2
+Q2
1
F r o m the normalising condition are obtained the equations
a2s Q2 + a]n __ 1. These equations will be satisfied if we choose for coefficients a22 = cos0
and
Qa22 = sin0.
F r o m equations (8.121) and (8.122) cos0 = (1 + QD -l/z sin0 ffi Q(1 + Q2)-x/2. 1 - Q2 cos20 = 2 cos20 - 1, cos20 = - I+Q2" Defining a positive quantity C
Since
1
c = -g
+" +
j2)1/2
(8.123)
then it can be shown that 1 C co820 = 2 % ~
(8.124)
1
and
C sin20 = - - J. 2
The stationary state wavefunctions may be written now as ~0t =
ut
~'2 -- cos0 u2 + sin0 u3 ~v3 = - sin0 u2 + cos0 u3
(8.125)
THE ANALYSIS OF HIGH RESOLUTION SPECTRA
317
Using these wavefunctions the transition intensities can be expressed in terms of cos20 and sin20 instead of explicitly as given above: the trigonometric expressions are more convenient. For transition ~03-~ ~4, the relative intensity is given by e~..~ -- [ - sinO(~/~) + cosO(floO I Iz(.4) + / = ( B ) IPPP y2 7z = 16=2 ( - s i n 0 + cos0) 2 = 167r~ (1 - 2 sing cos0) that is,
L~_~ ffi L,.~3 : (1 - sin20).
Similarly, L 2 - . , -- L1-~2 ffi (1 + sin20). Returning to the stationary state energy levels, equations (8.115-8.118), it is seen that substitution of the C parameter brings about some simplification of form
E,
=
E2
+
(HA
1(
+-~-
1 j)
2c
-T
H,)
- T
2c + 7 1
l[~
(H.+H.)_
,
l ~
z'=-TT=
T
Table 8.8 gives the wavefunctions and stationary state energy levels tor the A B system. T~us 8.8 EN~ROYLma~LSANDWAV]mONCnONSFOg ANAB S ~ Wavcfunction
¥3
[
Total spin IT
¢XO~
1
cos0(a~) + sin0~0,)
0
-sin0(ap) + cos0(~c,)
0 -1
~ T
level
( H , + H~) + T J
2
(H,l + Ha) - T J
1
C cos20 = ~ ,,06 1
C sin 2O ffi.-ff J
The transition energies can now be obtained by subtracting energy levels between which transitions are allowed, that is those having A IT ---- + 1.
318
HIGH RESOLUTION
Band Number
Transition
1
3~1
2
4--,2
3
2~1
4
4-+3
I f these
NMR SPECTROSCOPY
Transition Energy
transition energies are referred to the mean chemical shift,
1 [ - ~ - (HA + HB)], they become
T
1
1
c +--2.r
2
C --~J
1
3
1 -C+--J 2
4
-C
1 ---J. 2
A summary of the transition energies and relative intensities is given in Table 8.9. TABLE8.9 TaAr~'HO~ E~r~RcIESANDR~LATrVEIm'ENsrrmsfOR ~ Band number
Transition
Transition energy (cycles see-t ) c+~J c-
½.r
-c+½J -c--!,J
AB sYsrr.M
Relative intensity 11+ 1+ 1-
sin20 sin20 sin20 sin20
Figure 8.8 illustrates in schematic form a typical A B type spectrum with its important features labelled: nucleus A is assumed to be more shielded than nucleus B. Although the high field components of the high fielddoublet are referred to as A lines, the transitions from which they originate are mixed, in that they involve some change in the spin of nucleus B. It is always possible to measure J
THE ANALYSIS
OF HIGH
RESOLUTION
SPECTRA
319
directly from an AB spectrum, providing the absorption bands are resolved. Since C : 1 (.12 + r~ a2) ~12 then
r~ ¢~2
= 4C2
-- J2
= (2c + ~)(2c
- J~
The chemical shift difference ro ~ between the A and B nuclei can be calculated from the separation of the outer lines (1-4) and the separation of the inner lines (2-3) as follows ro ~ = [(rl - r4)(r2 - rs)] 1/2 (8.126) H
It
B
A
4-;-2 -]-"
Transition .3 -,'-Ii_
V
Bond
number
I
~
2"-.-I 2C-J
"
=j
"I
4-'-3_[
['-
2
3
o
-I 4
FIG. 8.8 Schematic AB type spec3rum
Also the intensity ratio of the central to the outer bands is (2C + f )
(~1
(2C-])
(~2-"D
-
r,) (8.127)
The bands shown in Fig. 8.7 can be assigned readily to their corresponding transition energies as listed below: Band number 1
2 3 4
Transition
Band energy
energy
(cycles sec -~)
c+½J
c-½] -c+½] -c-½.r
20.7 15"2 5-5 0"0
ReJatiw intcm~ty
1-0 2"1 2"1 1"0
Note that spectral line positions are often measured in the low field direction since this is the direction of increasing energy. Suitable subtractions of bands gives the values, J = 5.5 cycles sec -i and ro ~ = 14.2 cycles sec -i for liquid 2-methyl-3-thiophenethiol(22). The general appearance of an A B type spectrum depends only on the ratio J/ro ~ as can be seen from Fig. 8.9 where the ratio has been varied by varying
320
HIGH
RESOLUTION
NMR
SPECTROSCOPY
J for a fixed value of % ~. When the coupling constant J is zero, two absorption bands of equal intensity are observed. F o r very small values ofJ(J/% ~ ffi 0.02), two doublets are observed, each doublet having components of equal intensity; this type of spectrum is described as A X . As the coupling constant is increased with respect to the chemical shift difference, two asymmetric doublets characteristic of an A B system are observed. The central components increase in
J
00
II
8 0'6
I
0.4
I
[
tl
3. ?
!
r
O.G
Cycles seC
FIG. 8.9 Theoretical AB type spectra for two nuclei of spin ½ having ~o~ = l0 cycles sec- t intensity at the expense of the outer components as J/% ~ increases: at very large values of this ratio, the central bands coalesce and the outer bands disappear. F r o m examination of an A B spectrum alone it is impossible to decide (i) the sign of J and (ii) which of the two nuclei is the more shielded.
8.13.2 The Three Spin System, AB2 This system consists of two magnetically equivalent nuclei coupled with a third non-equivalent nucleus of the same species. Although many molecules have symmetry properties such that their magnetic nuclei constitute a system
321
THE ANALYSIS OF HIGH RESOLUTION SPECTRA
of this type, very few examples of AB2 type spectra have been publishe& A good example of an AB~ system is provided by a symmetrically substituted 1,2,3-benzene derivative (6) such as
H
H
R
R'
where R and R' are non-interacting atoms or groups. Monosubstituted arlene derivatives in which the substituent is non-interacting are also classified as AB2 as for example 1-chloroallene (17)
~\c__c__c/~ c~/ \H where the substituents on the two terminal carbon atoms lie in two p l a n e s perp ~ d i ~ ] l a r to each other. An AB~ system can be described completely by two resonance frequencies, OJA and o~, and by one coupling constant JAB- There is no need to consider the coupling constant between the two equivalent B nuclei, JB, s., since this will have no effect on the spectrum (see Section 8.12). The complete Hamiltonian for this system is afD -- + {a~A(I.A) + ~O,(I~. + I~.) + J^e(I~. + Xln~)I.A + J...B.I.~.I~.
+ ½ JaB [(1~. + I;,) 1~ + (I;, + I;,) I~] + ½ Se, B.(I;,'I;. + I~ I~0}. 'The terms involving are.m, the coupling constant between the two magnetically equivalent nuclei, must be removed from the Hamiltonian at this stage and hence the Hamiltonian can now be written
= + {,,.,,,.(x,,O + ,,,.,,,(.,",~,++r~) + .5,.,(J"~,+ T:,,.)X,A+½.r,.B [(~ + +
(I~"
r~)~
+ I;.)I~]}
or
a¢ = + {~ X~(I.A) + ~ ~,(X...+ Z.,) + S~,(Z.,~ + Z~) X.~ + ½S^, [(I;.
+ ~;)z;~ + (~;, +
I;)I~]}.
There are eight basic product functions for th~ system and they are given in Table 8.10. TABLe 8.10 THe BAsic PRODUCTFUNCTIONSFOR AN ABa SYSteM Igl
1'12
I/3
U4
~-~
lg6
U7
/311
o,o,+, o,po, po,o, +,o,p po,p o,pp ppo, ppp
Since the AB2 system possesses(sg) certain symmetry properties it should be possible to construct a set of symmetrised spin functions based on the basic product functions, which reflect the symmetry of the system. The use of symmetrised spin functions results in the system being generally more easily maniHiss.
ll
3~2
HIGH RESOLUTION
NMR SPECTROSCOPY
pulated (for example, further factorisationof the Hamiltonian is possible when symmetrised spin functions are employed). A n A B z system has (72 symmetry since the three nuclei concerned can be represented as being situatedat the corners of an isoscelestriangle as shown below. A
8
8
A reflection about a plane bisecting the angle B,~B and perpendicular to the plane of the triangle will result in the two B nuclei interchanging their positions. Certain of the basic product functions for this system already remain invariant under this symmetry operation: thus o~x~x, flo~x, ocflfl, and tilt can be regarded as symmetrised spin functions for the AB2 system. The four remaining basic product functions change under this reflection operation,
In Section 8.9 we have seen that symmetrised spin functions can be obtained from suitable linear combinations of the basic product functions. For two nuclei, the appropriate set of symmetrised functions is ~ , ( ~ + 3 o , ) / 4 2 , (o,3-30,)/42
and 88
(1/~/2 is a normalising factor).
Each of these can be combined with either of the two spin functions for A (a or fl), the eight resulting direct products representing the symmetrised spin functions of the system. These symmetrised spin functions are designated by Yl, Y2, ~3 etc. and they are given in Table 8.11. The notation for a particular spin function has the generalformArAX,A Bxsz~B where IA and IB correspond to the total spin of the A and B nuclei respectively and I~A and I ~ are the respective z-components of the angular momenta for these nuclei. On this nomenclature, the spin function ~xaa would be represented by A½.t BI.,. It is also usual to denote to which irreducible representation of the point group the symmetrised spin function belongs: only transitions between energy levels which have functions belonging to the same irreducible representation are allowed. In the AB2 system the symmetry properties of the functions are not destroyed by the perturbing effects of the spin coupling and thus one can use the selection rule that there will be no transitions between states of different symmetry. This is not the case for the AB system where the symmetry properties are destroyed by the coupling. The next step is to calculate the diagonal and the off-diagonal elements for this system: the diagonal elements are given in Table 8.11 (Js,s~ is neglected in
-~23
THE ANALYSIS OF H I G H R E S O L U T I O N SPECTRA
calculating the matrix elements). The only off-diagonal elements observed are those between ~ and ~ and between ~, and ~07 and they are both equal to
J/V2. =
= J&/2.
All other off-diagonal elements are zero. TABLE 8.11 BAsic FUNCTIONSAND DIACONALE
Basic
~
-
Notation m = l ,
~
~O~ TH~ AB~ S ~
D ~ o n a l matrix ~tst
Symmetry
7
1
p2 o,O,~+ ~o,)l~'2
.4
.4½.½B,.o
½
s:ma.
p~/~
a
~½._½B,.,
½
Sym.
p, ~,(o,a- ~o,)/y2
B
a½.½ ~o.o
½
Antisym.
7H^
ps ~(o,/z-/~o,)/~2
a
~½._½ao.o
-½
Antisym.
-" 7 HA
p, ~(o,~+ ~o,)/y2
A
x½._½.B,.o
-½
Sym.
Ps ~ ;
X
A ½._½ B, _,
_a
Sym.
t
~-~(-u,, + 2he) - T s 4,~
4,~
--~'HA
,t•
~,
1
--~ (-- HA -- 2He) + "~ J
Where J - Jxs.
The secular equation for the system may then be represented in the matrix formulation as follows (//11 - E )
//1, (/'/., - E )
//~3
•
//Is
=0
//el
• (Has -E)
On substituting in the matrix elements it is seen that the 8 x 8 matrix factorises into four 1 x 1 matrices and two 2 × 2 matrices: these equations can be solved 11"
324
H I G H R E S O L U T I O N NMR S P E C T R O S C O P Y
to give the stationary state energy levels for the AB2 system as shown in Table 8.12.
TAmS 8.12 WAVeX~NCTIOm AND STATIONARYSTATBENERGY L m Wavefunctions
El = ~
~2 cos0+o~(o,3 + 3~)/2~ + sin0+3o,o, ~s
AB2 S~rrt~t
Stationary state energy levels
o,o,o,
~t
FoR ~
- sinO+o,(o,~ + 3a)/2~ + cos0+/~o,o~
7
1
(ttA + 2Ha) + T s
7Ha E2 Es =
4~
7Ha 4~
1 4 J + C÷
1
4 J-C+
YHA
~a,t
o,(~'t~ - ~o`)/2÷
E.=
~5
3(o,/~ - 300/2÷
Es
4~
~e cos0_ o'33 + sinO_ 3(o`P + 3a)/2~
E6
7Ha 4~
I 4 J+C_
~h - sin0_ o`33 + cos0_ 3(o,3 +/~o,)/2~
E7=
7HB 4:z
1 4 J-C_
~s
~' 1 Es = --~ (-- HA-- 2HB) + T J
4~z
?HA
The positive quantities C÷ and C_ a n d the angles O+ and O_ (having values between 0 a n d ~z) are defined by 7Ho 1 1 C+ cos 2 0÷ = ~ (~e - ~.,) + - - J = Vo 0 + - - J 2~z 4 4 C÷ sin20÷ = .1/2~ 7Ho
C_ cos20_ = ~ ( ~ a 2.'~
1
where Hence,
(8.129) 1
- ~s0 - - - J = Vo 0 - - - J 4 4
C- sin20_ = .1/2 ~-
(8.128)
(8.130) (8.131)
0 = ~B - CrA.
C+ = ½[(Vo 0) 2 + Vo 0 J + ]j2]½
(8.132)
C_ = ½[(Vo 0) 2 - Vo 0 J + ~_j2]~r
(8.133)
We are n o w in a position to calculate expressions for the allowed transition energies and their relative intensities for this system, and a list o f these is given in Table 8.13. W h e n a transition is described as ' A ' in origin it means that in the limit when J --, 0, only the spin o f the A nucleus will be changed in the transition; f o r example, v/3 --* ~01 becomes u3 ~ ut or 3o, --, ~ , ~ . One o f the transitions involves
THE A N A L Y S I S OF H I G H R E S O L U T I O N S P E C T R A
325
the simultaneous change of spin of all three nuclei, and this is referred to as a combination (or multiple) transition. Thus the combination transition ~Pe ~ ~0s corresponds to the simultaneous change of spin of the three nuclei, ~8~ - - , / ~ . Combination bands become forbidden in either of the two limiting cases when either J --* 0 or when *o (~ "* 0; when they are observed for intermediate values of J and Vo (5 their intensities are always low. TAm.~ 8.13 "IMANSmONENI~OmS AND P,.ELA'HV~~
Band number
Transition
Origint
Transition energy
Y3 - 1ol
A
3 "~-~(~s + hr.) + ~-.I+ c÷
A Y4
A
~ s -- Y6
A
Y6 -- I02
B
Ys -
5 6 7
B
1O7-- 1O3
B
YHB 2n
volt ' r ~ AB2 S~rEM Relative intensity
[2½sinO+- cosO+]a [2½ sin(O+ - 0_) cosO_]2
+c÷+c_
+cosO+
~,H^ 2n
1
[2½sinO_+ cos0_]z
2~ + c ÷ - c _ ~ 3 T g (n~ + ~a)+'~ . r - c ÷ y ltB 2=
c÷ + c_
[21 cos(O. - 0_) +cos#+ sin 8_] 2 [2t c~sO++ sino+J2 [2' cos(O+ - 0_) -
8
los -- 1O7
9
I" Assuming fib > O'A or
B
Comb.
y 3 4~ (HA+ ~ ) - - T J - C_ y/'/B 2= - C ÷ - C _
sinO+cosO_]2
[2~ cosO_- sinO_]2 [2'~ ~dn(O+ - 0_)
+ sin 0+ sin0_]2
Ha >/'/B.
As for other A B , systems, the general appearance of the spectrum depends only on the ratio J/~o ~. Figure 8.11 illustrates the changes which occur in an AB~ spectrum as theJ/~o¢~ ratio varies. When J/~o ~ is large and 1,0d has a smaU but finite value, a symmetrical triplet is observed (see Fig. 8.10) and the separations between components of the triplet are given by ~ o d. In only one of the spectra illustrated does the combination band 9 appear (see Fig. 8.10) and its intensity can be seen to be low. A closer examination of the theoretical AB2 spectrum for J/~o ~ •ffi 0"5 reveals the following interesting features: (i) F r o m the transition energies given in Table 8.13 it may be seen that band 3 in Fig. 8.10 gives the true chemical shift for the A nucleus and contains no contribution from spin--spin coupling. (ii) The true chemical shift of the B nucleus is given by the mean position of bands 5 and 7.
326
H I G H R E S O L U T I O N NMR S P E C T R O S C O P Y
It is possible to say which of the two types of nuclei in an AB2 system is the more shielded from examination of its resonance spectrum but it is impossible to find the sign of the coupling constant in this way. If the absorption bands in the experimentally observed spectrum are well resolved, then the analysis is achieved simply by assigning the absorption bands of the spectrum to the transition energies given in Table 8.13 and then finding the chemical shift difference between the A and B nuclei by direct measurement
d
i...%8-(c+-C
-
)..~ •
I..-~,o8 +(C+-CJ I I
2
'
3
1
_--;
I I
4
~ =0.5
It
FZG. 8.10 AB2 type spectra having .//~oG = 0-0 and 0.5
of the separation between band 3 and the mean of bands 5 and 7. Values of (7+ and C= can also be obtained from the spectrum by suitable subtractions of the absorption bands, and knowing these values and )'o ~ one can obtain the J value for the system using equations (8.132) and (8.133). An alternative method of assigning the bands in an AB2 spectrum has been used by Bernstein, Pople and Schneider c2°). They have drawn up a table of the transition energies in an AB2 system for various values of J/% ~ (see Appendix D) using a fixed value for ~o 6 of unity in each case. The problem of assigning the observed spectrum to the transition energies is one of finding the 3/),o 6 which most closely reproduces the observed spectrum in general appearance, and then adjusting the J/% (3 ratio so that an exact fit of the theoretical and ob= served spectra is achieved. In order to do this, one must first scale down the separations in the observed spectrum to correspond to a % ~ value of unity: it is relatively easy to derive a value of ~o 8 from the observed spectrum and
THE ANALYSIS
OF HIGH
RESOLUTION
SPECTRA
327
employ it in the scaling-down process. Bernstein, Pople and Schneider (=°) have analysed the 1H resonance spectrum of the ring hydrogen atoms in 2,6lutidine (see Fig. 8.12): neither the hydrogen nuclei in the methyl groups nor the nitrogen nucleus interacts with ring hydrogen nuclei and consequently they can be excluded from the analysis. Because of the symmetry of the molecule the hydrogen atoms in the 3 and 5 positions are magnetically equivalent. The three
3"042
1-333
~
0 6 "59 I] l,,
o.o,
II,
0 • I00
~000
_4LO
i
J -20
0
i
20
1
I 40
Cycles sec t
FIG. 8.11 AB= theoretical spectra for nuclei of spin ½ and a chemical shift difference ~o~ of 10 cyclessec-I ring hydrogen nuclei thus constitute an AB2 type system. The spectrum shown in Fig. 8.12 may be seen to resemble some of the theoretical AB= spectra depicted in Fig. 8.11. An analysis of the spectrum was performed (2°) by fitting the observed spectrum to a theoretical spectrum having J/~o ~ = 0-375. From the resulting theoretical spectrum, the above authors calculated the following numerical values for the chemical shift difference and the coupling constant between the A and the B nuclei: • o ~ = 21"9 cycles s e c - j 3" = 8"2 cycles sec -1. The A nucleus is less shielded than the B nuclei.
328
HIGH
RESOLUTION
NMR SPECTROSCOPY
Relative intensities can be obtained by using equations (8.128-8.131) to provide values for 0÷ and 0_, f r o m which the intensities can then be calculated. Agreement between the observed and calculated intensities can be taken as an indication that the assi~mment is correct. Table 8.14 gives the observed and calculated transition energies and intensities for the 1H resonance spectrum of 2,6-1utidine and the agreement between them can be seen to be fairly good. By examining the sample at a different magnetic field strength and showing that the observed spectrum is consistent with the parameters obtained from the original assignment one can confirm that the analysis is correct. Tables similar to Table 8.13 can be drawn up for all Ap B, systems, where the general appearance of the spectrum depends upon a single parameter, J/vo & Some of these tables are reproduced in Appendixes D - H . F o r systems containing H
C H~.."-.l~f,~2CH.x
4
iO c~cles sec-4
A (Hydrogen 4)
B (Hydrogens 3ond 5)
FIG. 8.12 Example of an AB2 type spectrum-the IH resonance spectrum of 2,6-1utidine obtained at 40 Mcsec -1. Bernstein, Popl¢ and Schneidert2°) TAeLB
8.14 CO~AmSON OF CALCULAX~.DAND OeS~RV~) SPEC'mA FOR 2,6-Ltrrmn~mt=°) Energy relative to line 3
Relative intensity
Line Calc 1
2 3 4 5
-- 9"4 cycles sec-t -- 3.0 0 6.5 18.3" >
19.0. 25"5 27"7 46"7
Obs
Calc
Obs
-- 9"4 cycles sec- t
0"47 0"68 1.00 1"86 2"85 I 2"52 J
0.45 0-75 0.9 1.6
--2.8
0 6"6 19"3 25"6 27"7
5"0 3"3
0"0025
THE ANALYSIS OF HIGH RESOLUT ION
SPECTRA
329
more than one coupling constant and chemical shift separation, it is impossible to provide for all possibilities in tables of this sort since there are several variables involved. It is usual in such cases to restrict calculations of the transition energies and their intensifies to those expected for a few typical examples using reasonable values of the various parameters and varying only one, and that over a limited range, at any one time.
8.13.3 The Four Spin System, AB3 A molecule containing three magnetically equivalent nuclei coupled to a fourth non-equivalent nucleus, which is separated from the other three nuclei by a chemical shift of the same order of magnitude as the coupling constant between them, is described as an AB 3 system. Both the cis and trans isomers of the fluorinated alkene, CFsCCI=CFCI constitute AB s systemsc23). Other molecules giving rise to spectra of t h i s t y p e are methylacetyleneC~'. 2s), CH-=CCH3, methyl mercaptan c7. ~6), CH3SH, and methanol czT. 28), CH3OH, where rapid rotational motion about the C--S and C - - O bonds results in the methyl hydrogen atoms becoming magnetically equivalent. It is necessary to examine methyl alcohol in acetone solution in order to suppress the chemical exchange of the hydroxyl proton, which obliterates the fine structure from / spin-spin interaction ceT). The grouping C H 3 - C H also forms an AB3 system because the three coupling constants JAB are averaged by rapid rotation about the C - - C bond at room temperatures: examples of spectra from this grouping are given on p. 97 of reference .29. As in other ABn systems, the observed N M R spectrum can be described in terms of the two resonance frequencies o~Aand OJBand the coupling constant J. Coupling constants between the three magnetically equivalent nuclei need not be considered in the analysis. The Hamiltonian operator for the AB3 system is
= [O~AIA. + O~BIB. + JI^. IB] = [~'HAXA, + 7'HBIB, + JIA, I , , + ½J(I~,l~ + I ; I ; ) ]
where and
Ie,-- I~, + I~, + I~, I~ = I~ + I Bs + "Jr- I+]S~
The nuclei in an ABs system can be considered, for symmetry arguments, to be at the four apices of a trigonal pyramid £
HRS.
11~
330
HIGH RESOLUTION NMR SPECTROSCOPY
the three equivalent B nuclei being at the corners of the basal equilateral triangle. Such an arrangement of nuclei is described by C3o symmetry. The basic symmetry functions for the system can be constructed in such a manner that they remain invariant under operations of the point group C3,. The appropriate symmetrised spin functions are given in Table 8.15 together with the irreducible representations to which they belong. Also given in Table 8.15 are the diagonal matrix elements for the AB3 system. Theonlynon-zero off-diagonal elements are (~,1 ~ 1 ~ ) = ~' (~6[.,Wl~pT) = x/3J/2 (~0xol~l~0,0 = - ½ J
The four E states are all doubly degenerate, and consequently since only one of each is considered in Table 8.15 it is necessary to double the intensities of any transitions involving the E states. The matrix corresponding to the secular equation for this system can now be constructed (16 x 16) and factorised into six(l x 1) and five (2 x 2) subdeterminants. As in all other AB~ Systems, the eigenvalues and the eigenfunctions can thus be obtained from solution of linear and quadratic equations only, as described in Section 8.13.5. The transition energies can be evaluated from the stationary state eigenvalues by subtracting those which are involved in allowed transitions, and the relative intensities of these transitions can be predicted from the eigenfunctions (p. 386 of reference 7). Table 8.16 gives the expressions for the transition energies and their relative intensities. There are sixteen transition energies in all, two of them being combination bands of rather low intensity. The positive quantities D÷, D_, Do, Do' and the angles 0+, 0_, 0o, 00' between 0 and zc have been introduced for convenience and they are defined as follows D+ cos20. = ½~o 6 + ½J D+ sin20+ = 31': J/2 Do cos20o Do sin20o D_ cos20_ D_ sin20_ D~ cos20~
= = = = =
½Vo,3 J ½-Vo~ - ½J
31;2 .1/2 .tv o 3
D~ sin20o = ½J By elimination of the angles in the above expressions one obtains the relationships for the D values in terms of Vo6 and J. D+ --" +½{(%c$) 2 + 2J(~ oc$) + 4J2} 1/2 Do = + ½ {('0 ~)2 + 4./2}1/2
D = +½ {(% ~)2 _ 2J0'o 6) + 4j2} 1/2
THE ANALYSIS
OF HIGH
~,~.,~ ,+
+
+
RESOLUTION
~.~.
_,.
i,,-q
~#
m
I
~
331
SPECTKA
'--,
I
+
!,-i
~+I
"~#
,~I
I
fl
,,~ N
,,~
,@
'
N
..,,.
i
,@
~
0 N I
z
•,T
~I
,,T
,T
',T,,T
,,T
,,T
,,T
~
~
I
I
~
°~
OW
•-I-
11 a*
+
+
+
I
I
332
HIGH
RESOLUTION
NMR
~'1
÷ 0 o
SPECTROSCOPY
~'1
-L
+
-L
~
m
~
(vl
-~- Jl-
+-L ~
"~
0
~
0
~
0
÷
I
I
++
o
c,4
=? <
7. + +
Z <
I
I
+
+
+~_=
+
~ + ~ , +
+
~
"
o
I
I "._:~
I
t
o
o ,
o
o
0
I
I
I
t
1
I
I
I
[
I
I
[
I
I
I
I A
¢ 0
-H-
THE A N A L Y S I S OF H I G H R E S O L U T I O N S P E C T R A
]B
r
,
,I
i lll ~.o
,I
6
*U2
13 14
4 5
l
3
t,
li 12
1~,14
Ill
J
5 2
"i
,I
IIi12 13,14
i 4
, 2
I 5
13112 II114
3
11 I
tl J 1
I0
~5
0.50
I 8
I 4.5 IO
0.75
2-25
8i t0.
I, II114 7111 i3 , i l
0-15
O-25
7 9 II
79
0-025
?~10
";'§l ,lO
JdJI I J lilt, I
I 6
0
8-0
23
45
J/uo8
8.0
ill,
6
~ 8
333
1.00
J
4-5
I
4.5
~ o . 8.13 AB3 typc theoretical spectra. Codo (7)
5.00
t:O
334
HIGH RESOLUTION NMR SPECTROSCOPY
The eigenfunctions ~o,, ~02,~03, etc. are linear combinations of the symmetrised spin functions ~t, ~2, ~3, etc. Bands 3 and 5 and also 8 and 13 are separated by the coupling constant J: hence, if the observed spectrum is correctly assigned to the transition energies given in Table 8.16, the J value can be obtained from the spectrum by direct subtraction of these bands. Corio has outlined a general analysis of the ApB, system for nuclei with spin numbers I = ½(see Section 8.13.6) and he has shown that whenever group A contains a single nucleus and group B contains an odd number of nuclei (such as AB, AB3, ABe etc.) then there are always two pairs of transitions in the spectrum the components of which are separated by the J value for the system.
CH~SH
JJl
~o sgz
'J~~'~'H
I I ,
3
2
s
4
FIo. 8.14 AB3 type spectrum. The 40 Mc sec - t *H resonance spectrum of liquid methyl mercaptan. The theoretical spectrum was calculated using J/~,o,~ i= 0,295 Corio (v)
Appendix E contains calculated transition energies for the AB3 system for various values of the parameter d/~o ~ (for a fixed Vo ~ value of unity) and some of these theoretical spectra are reproduced in graphical form in Fig. 8.13. It is possible to assign the bands in a spectrum of an AB3 system to a theoretical spectrum in a similar fashion to that adopted for the AB2 system: by comparing an observed spectrum with the theoretical spectra shown in Fig. 8.13 one can select a value of J/re ~ which Approximates to that for the system under consideration. This value when slightly adjusted and used in conjunction with a value of J (directlyestimated from the spectrum) can be used to predict a theoretical spectrum which agrees with the observed spectrum. Methyl mercaptan is a typical example of an AB3 system and its *H resonance spectrum has been fully analysed. Fignr¢8.14 reproduces the observed *H resonance spectrum of methyl mercaptan together with the theoretically predicted spectrum obtained by using the following values of the chemical shift difference and the coupling constant between the A and B nuclei J = 7.42 + 0.17 cycles sec-t ~'o ~ = 25"18 4-0"21 cycles sec -t
THE A N A L Y S I S OF H I G H R E S O L U T I O N S P E C T R A
335
Liquid methyl acetylene, CH3C---CH, contains two different types of hydrogen nuclei which are accidentally almost magnetically equivalent. Thus, when the 1H resonance spectrum of this molecule is examined at 40-00 Mc sec -1 a single absorption band is observed. However, gaseous methyl acetylene or a solution of the compound in various solvents give ~H resonance spectra at 40.00 Mc sec -1 which would be expected of an ABs system¢~'L Shoolery, Johnson and Anderson ¢2~) have examined the IH resonance spectrum of liquid methyl acetylene at 60-00 Mc sec -~ where the chemical shift difference between the methyl and acetylenic hydrogen nuclei becomes sufficiently large for a spectrum of the AB~ type to be observed (J/% ~ ffi 1-33, J --- 2.93 and ~o ~ -- 2.1 cycles sec-: at 60.00 Mc sec-~). That the apparent magnetic equivalence observed at lower irradiating frequencies is not due to chemical exchange had previously been established by Vaushan and Taylor ¢~°)who examined the XHresonance spectra of C H 3 Cffi - CD and C D ~ C - C H where H - - D coupling of 0.5 cycles sec -~ is observed. TABLE 8.17 THE A t . ~ w ~ Transition Number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Transition
Ips-+ ~o.,
Energy
Transition A A
2A~ "+ A~ 2A'o-* 1A;
2E~ ~ Ex 2A'--1"* IA'o
'lP11 " ~ ~O9
~'I"* lP4 A Ys "* IPe tp6 ..-*~o.,. ¥1o -*" ~°9 ¥4 ~ ~o2 ~o2 ~ ~ox Ys ~ ~os
TRXNSmONS m AN A B s SYSTEM
E-x-*
IF-~
A _ 2 "* l A ~ 1
1A'-.I~ 1A~ B
J + D+
-½~'o ~ +.D+ + Do
½J + D~
-½Vo~ + Do ÷ D_
-½J + D/,
--J+D_ --½~o~ + Do - D_
½ ~ - D~
--½~'o ~ + D÷ -- Do B B
I A'x"~"A2 2,4~+ 2Ai 2A'-.1 ~ 2A~
E-I "-~2 ~
~Pxz-* ~PlI Comb. Comb.
A-2 "* 2A~I 1 A ' , "* 2A~
J-- D+ --½~o~ D+ + Do - ½ ~ o 6 - Do + D_ - -
-½J-
D~
--J--D_
-½~o 6 - ~o - D_ -½~o 6 -- D+ -- Do
It should be noted that Bernstein, Pople and Schneider ¢5) have used a different notation for referring to the symmetrised spin functions and the eigenfunctions of any system. If the irreducible representation to which the function belongs is that of state .4 (this .4 does not refer to the A nucleus in this context) then we would represent it as .41T where the susbeript l r is the total spin associated with the function. Hence, the basic product function o~o~, which is represented by .41/2. lnB3/2.3/2 on Corio's nomenclature, is given by.42 in the terminology of Bernstein, Pople and Schneider. If the eigenfunction is different from the symmetrised spin function to which it reduces in the limit then the
336
HIGH RESOLUTION N M R
SPECTROSCOPY
eigenfunction is referred to by the primed notation of the symmetrised spin function. For example, the symmetrised spin function o~fl(flo~ - o~3)/l/2 has the notation 1 Eo: the eigenfunction which reduces to 1 Eo when the chemical shift difference is large compared with the coupling constant involved is referred to as 1E6. The number preceding the symbol for the irreducible representation is used to distinguish between functions which have the same total spin value and belong to the same irreducible representation. Table 8.17 contains the sixteen transitions for the AB3 system presented using this nomenclature. It is not intended to use this form of notation in this text.
TABLE8.18 SYMME'rRm~ SPIN PRODUCTFUNCTIONSFOR A B ,
Funotion ogOgOgogo~
3~xoaxo~ 2-~0,(~,~,3 + o~o,~ + o,3o~ + 3owa) 2-~ 3(~,,,.a + ~ , ~ , + ~po~, + pow.o,) 6%o~(o~flfl + flo~3 + o,pp,,, + ppo~ + o,po,p
+ p~p~,) e"¢p(o,o,pp + p,,o,p + o,pp,,, + ppo,o, + ~,p~p + po~3oO 2 - ~ , 0 f l 3 o ~ + pp~p + ~ p p + o~ppp) 2-~ ~(pppo, + ppo~p + po,pp + o,ppp) ~PPPP PPPPP ~2-~o,(o,o,t~p + p ~ p + o~ppo,+ ppo~,
Notation
D, s:mmaetry
A~r, ~rBu.2 A b - t B2.2 A{t,{rB2.1 At, _+ B2. t A½, ½B2,o
At AI AI At At
Ab -,I" B2,o
At
Ai. ~ B2.-1 A½, _½B2._x
A½, _~ B2.-2 Ab ½Bo,o
Xl At At At At
A½, -Jr Bo,o
At
A½, ~ Bt. t A½. -~r Bt.z
B1
A½, ½Bt.o A½, _½Bt.o A{, ~ Bt.- t Ab -{ Bt.-t
Bt Bt BI
- 2o, p,,,fl - 2 p , , . ~ )
12-½fl(o~ap,B + 3e.o~p+ ~,ppo, + p p ~ - 2o,3o,p - 2po, po,) 2-~ pO,,,,o,p - ~P~, + ~,po,o, - fl~,,,,o,) 2-÷ p(o,~o,p
-
~e, Bo,)
p(pppo,
-
ppo,p
2-*
+ [ ~
-
o,ppp)
2-t,~O,o, p3 - P ~ P - o,ppo, + ~o,o,)
2-t-~,(~pt~ - .ap,x,,)
A½.½ Bo.o A½, _½Bo.o A½, ]r BI,1 A½, _~-B1, t A½.½ Bt. o
2-t p(o,o,p,~ - p p ~ ) 2-+o~(pppo, - p~,pp) 2-+ p(pppo, - ,~e,pp)
A½, _½Bt.o A½, ½Bt.-t A~r, -½ BI,-I
2-~(o~p~
A~.,½Bi,t At-, -½ Bt.t
- pew~)
2-~ p(o~p,, - p , , ~ ) 2-+~,(a~p o,ppo,) 2-~ p(p~p - ~pp,,) 2-~o~(ppo, p - o,ppp) 2"~ p(epo, p - o~ppp) -
A½, ½Bt.o AS, - t Bt.o A~, ~ BI,_ 1
A½, _½ Bl,-t
BI B2 al E E E E E E E E E E E E
337
THE A N A L Y S I S OF H I G H RESOLUTION SPECTRA
8.13.4 The Five Spin System, AB, Systems having nuclei that fall into two magnt~ically equivalent groups containing one and four nuclei respectively axe not very common, but do occur in derivatiws of molecules with octahedral symmetry, such as SFz in which one of the substituvnts has be~n replaced by a non-interacting group ¢~), for exami~le SFsC1 ¢~=~.An AB, type spectrum can be analysed by the method of McConnell, McLean and Reilly¢~) in the same manner adopted for AB, ABz and AB~ systems: the method will be utilised below. However, in Section 8.13.5 it is shown that m u c h of the labour involved is unnecessary and the problem is best tackled by the "complex particle" method. The Hamiltonian has the form shown in equation (8.27) with/:~ = 1=~, +/=~, + / : ~ + I=~,, and the sixteen product spin functions axe shown in Table 8.2. The four nuclei in the B group may belong to either the D , or the T~ symmetry point group: we will consider the case where D, is the appropriate point group. The thirty-two symmetrised product functions are shown in Table 8.18; they h a w been obtained by combining each of the B, functions with ~ or ~. Bearing in mind the selection rules for the Hamiltonian matrix elements, Table 8.18 shows that of the functions in the A~ representation the following are already • igvnfunctions of the Hamiltonian: ~1/2,1/2B202,
A1/2,-~/2~2._2,
~1/2,1/2B0.0,
A1/2,-1/2~0,0
T,~d~L~8.19 F ~ O ~ C n O N S AND F.~O~V~Lt~S OF T ~ A B , S ~ ' r ~
Eig~nfunction
A½.½ B=.2
(1 + Qi)-½(A½._½B2,: + Q~ Aid Bz,~) ~Ps Ys
~6 ~P8
~P9 ~P~o
~P~s
lo16
(1 + Q[)-~" (QI A½,_½B2.2 - A½,½B~.I) (1 -I- Q~)-½(A½,..½B2,: + Q~ A~,t B2.o) (1 + Qi)-½(Q2 a½._½ B2.~ - ai~ B2.o)
0 + ~)-½ (a½._½~ . o + Q3 a½,½B2._,) 0 + Q|)-ffQ~A½._½ B~.o - A ~ = . _ ~ ) (~ + L~,)-½(A~.,_½B~._, + Q,A~.½ B~._~)
(1 + C~)-½(Q,A½,_½n,._1 - A~,½B~._~) A½,_½B~._~ A½,½Bo,o A½,_½Bo,o A½,½B~,I 0 + ~|)-½(A½._½B~,~ + ~sA~,½B~,o) 0 + ~)~(QsA½...~ B,,I -- ~,~ ~,.o) (1 + Q~)-½(A½,_½B~.o+ ~A½,½ ~1.-~) (1 + Q~)-½(Q~A½,_½B~,o - A½,½B~._~) a~._½ ~1,_~
Energy
.
m
½v^ + 2~s + J
A~
2~
I
½(3~B - ½.I)+ R~ ½ ( 3 v B - ½ J ) - R~ ½(~B-½J)+ R, ½(vB - ½Y) - R2 -½(vs+½J)÷R3 --½(vB+½J)--R3
A~ A1 A1 A~ AI AI
21 ~ ½ ½ -½ -½
1 1 1 1 1 1
--½(3~'e+½J)+ R4
AI
-~
1
-½(3v,+ ½J)- R, A~
-~
1
-½0'A + 4 ~ ) + J
al
--~
1
½,A
A~
½
2
-½~^ ½v^+,,~+½J ½0,- - ½J) + R,
A, B,
-½ ~
2 3
B~ BI B~
½ ½ -½ -½
3 3 3 3 3
½(~B -- ½J)-- Rs -½0's+½J)+Re -½0'e+½J)-R6
--½vA--~B+½J
BI B~
--i
338
HIGH RESOLUTION NMR SPECTROSCOPY
r e p r e s e n t a t i o n , A t/2. t / 2 B 1 , t a n d A t / 2 . _ l / 2 B t . t are eigenfunctions of,Yl, as are also both the B2 functions A t/2, t/2Bo, o and A I/2.-t/2Bo. o. The B 2 functions are degenerate with the At functions having the same values of m and Ia. The secular determinant formed from the remaining At functions factorises into four 2 x 2 determinants corresponding to m values of i, ½, - ½ a n d - i ; solution of the resulting quadratic equations gives the eigenvalues and functions fisted in Table 8.19. Similarly, the Bx secular determinant factorises into two 2 x 2 determinants corresponding to m = ½ or - ½. The twelve functions in the E representation would seem to give rise to two 2 x 2 determinants and two 4 x 4 determinants corresponding to the m values ~-, - ~ , ½, - ½. However, the twelve E functions can be divided into six doublydegenerate functions, and further, each doubly-degenerate E function is degenerate with the B t function having the same value of m and le. Thus the g functions can be neglected and each transition in the Bt group multiplied by three for intensity purposes. The eigenfunctions and energy levels are shown inTable 8.19, while the frequencies and intensities of the allowed transitions are listed in Table 8.20. The constants Q and R in Table 8.19 are defined as: IntheBx
½[('0(}) 2 "+" 3VO(}J'4" ~ j 2 ] 1 / 2 R 2 = ½[('0 (})2 _1..Vo (}J + ~ j 2 ] 1 / 2 R a ~-
R,
½[('o (})2 _ "o (}J +
= ½[('o (})2 _ 3~'o
~j211:2
(}J + ~j2]~/2
R s = ½ [0'o ~)2 + ~'o ~ J + 9 j211/2
9 j2]1/2 R 6 = ½[(~o (})2 _ ~o (}J +
d)/J]
Qi =
¼[3 + (4R1 + 2re
Q2 --
(2 `/6)-x [1 + (4R2 + 2*'0 6)/J]
Q3 =
(2 `/6) -x [(2,0 (} + 4 R a ) / S - 1]
Q, =
:[[OR, + 2re (})/J - - 3]
,/2 ,/2
Q6 ~ m
[I + (4R s + 2re (})/s] [(4R6 + 2,o 6 ) / J - 1]
Examination of Table 8.20 shows that both Vo (}and J may be obtained directly from an A B , spectrum: first, the separation between bands l l and 12, 13 and 14, 15 and 16, and between bands 19 and 20 gives 2v o (} in each case; secondly, adding the frequencies of bands 1 and l0 (relative to "A) giVeS ~'0(} --~J, 2
THE A N A L Y S I S OF H I G H RESOLUTION SPECTRA
I
1
I
~
-
~ ~
~ +
- ~
~
~ ~
~
~
~
~ I + I
I +
339
~
~
~
~
~1 ~
,
~
,
I -
~ ~
immmm ,~+~mmmmmm,-~-~~mN.++++ ~
~
~
~
:
~
~
~
~
+++++
+++ . . _ + + + + + + + +
++++ . . . .
~ ~
~,~+I+~+~+~++++
+
+++_
•
o
°
°
I I I I I I I I I I I I I I I I I I I I I I I I I
340
H I G H R E S O L U T I O N NMR S P E C T R O S C O P Y
similarly adding the frequencies of 5 and 17 gives ~o ~ + I J, adding 7 and 18 and adding 19 and 21 gives ~o ~ - I J and % t~ + ~J respectively. Appendix F lists the frequencies and relative intensities we have computed for a range of ,1/1'o ~ ratios: in Fig. 8.15 are illustrated some typical AB4 type spectra showing the dependence on the J/~'o ~ ratio.
z, 8
,JIL, i -!o
¢
r~
LO
ill
I
I ~l.,
it
J/z, $
10
,~
fC
0,025
iO
5
0"30
5
0 " 40
,h tiJi I
-5
I
5
0"60
• I'O0
5
I 1
-io
L !
o
~;
FIG. 8.15 AB4 type theoretical spectra
O0
THE A N A L Y S I S OF H I G H R E S O L U T I O N S P E C T R A
341
8.13.5 The General Features of an A B , Spectrum The analysis of AB, systems with the aid of product spin functions becomes unwieldy when n is greater than 4. Even when n ffi 4 and all nuclei have spin ½, there are 32 product functions; when n ffi 5 there are 64 product functions. Also, in the AB4 system it was seen that the construction of symmetrised product functions does not lead to any added factorisation of the secular equation. It can be shown that with functions which are eigenfunctions of 1~ and I~ the secular equation factors into quadratic and lower order equationsC17, tg. ss); we will write such functions as A : x m A B z ~ . Restricting the discussion to spin ½ nuclei, then 1A = 3 and m^ = ~=3, thus the functions are A1/2 ± 1/2
Ble,
The secular determinant can be factored first according to the value of 1B, and the dimension of each sub-determlnant depends upon the number g1~ of spin functions with a certain value of I~. gxB is given by the equation (8.49) (see Section 8.6) he! (n, - 2ke + 1) glB = g<~/2-k~ --- (ne - k~ + 1)! ke! " (8.134) Each 1B sub-determinant can be factored further into determinants of lower dimension according to the value of m, the expectation value of I: (I~ -- I.A + ~ l.e) m ffi mA + ms.
(8.135)
For any particular value of IB there are (2Ie + 1) possible values of me, but since ]A ffi 3 there are only two possible values for mA of + 3, thus m ---- me + 3
(8.136)
and only two functions can have the same values of m; one with m •ffi m~ + 3 and the other with m ffi (me + 1) - ½ ffi me + ½. This means that the largest sub-determinant involved in an AB, system is of dimension 2 × 2. The functions corresponding to the maximum and minimum values of m for a particular IB are already eigenfunctions of the Hamiltonian; they are Ax/2.1/2 B~e, le
and
,'Ix~,. -~/2 B~e.-~e
with eigenvalues In re + ½rA + ½IeJ
(8.137)
- 1 e re - ½rA + ½1e:.
(8.138)
and The remaining eigenvalues and eigenfunctions can be obtained by solving the quadratic secular equations. The matrix elements of the Hamiltonian are (Ax/2. x/2 B1e. ,,e I °W lax~2.1/2 Bxe, me) = ms re + ½rA + 3 m e J
(,4x/,.-x/2B~e.,e÷,
(8.139)
IOWI/q/2.-x/2B~e.m~+D = (me + 1 ) r e - ½ r ^ - 3(me + 1)J
(8.140) (Ax/~, x/.B,~.~e Iowl A~/~,_x/~ B~,m..) ffi½ J[(~e - me)(Xe + me + I)]* (8.141)
HIGH R E S O L U T I O N
342
NMR
SPECTROSCOPY
and solution of the quadratic secular equation for m = ms + ½gives the eigenvalues ½[(2ms + 1),B -- ½J + R~B.ms] } (8.142) ½[(2ms + l)~'s - ½J - R,s..,s] where R~s.=~ = {[0'o ~) - ½(2ms + D-I] ~ + J2(xs - ms)(Is + ms + 1)P ~2. (8.143)
The corresponding eigenfunctions are
(I + Q~B.,~)-11~(,~,I2.-I12B~s.,s+t + Q~s.,sX~.~1, Bzs.,~) (8.144) and
(I + Q~B.~s)-~/2 (Qzs,,,,eAt/~._t/2Bxa.~.B+~ -- At/2.tl2 B,B.,,B) (8.145) where J[(IB -- ms)(I.~ + m s + I)]I/z Q x L q = (Vo~) -- ½(2ms + 1) J + Rxs..e"
(8.146)
Having obtained the energy levels and the corresponding eigenfunctions, the allowed transitions follow from the selection rules d m ffi
- 1 and
d I s ffi O
and the transition intensities arc given by the square of the matrix element of I-. The transition energies and intensities are given in Table 8.21 : the spectrum of any AB, system may be computed from this table by substitution of the appropriate values of Is and gxa. For the A B a system, the possible values of Is are 3 and ½, thus the spectrum consists of a superposition of one set of lines computed with Is = ~ and gls = I, and another set computed with Is = ½ and gIB = 2. For the A B system, the only possible value of Is is ½ so that the transition frequencies and intensities follow immediately from Table 8.21 with Is = ½ and grs = I. The system A B z is a superposition of a sub-spectrum computed with IB -- 1 and one computed with Is = 0. The A B 4 system has the same sub-spectra as AB2, that is, with Is = i and 0, and an additional sub-spectrum with Is = 2 but with gxB = 3, 2, and I respectively. The general AB~ spectrum consists of a superposition of all spectra AB~ where q has the values
q=
f
l, 3, 5 ... n for n odd 2, 4, 6 . . . n f o r n e v e n
(8.147)
and each spectrum is weighted by the factor gxB; this means that AB, spectra m a y be divided into those with n even and those with n odd: (i) /I even.
All systems with n even have a sub=spectrum with Is = 0; examination of Table 8.21 shows that for IB = 0 there is only one allowed transition and that is between the energy levels ,41/2.t/2Bo, o and AI/2. _t/~Bo. o occurring at the fro-
THE ANALYSIS OF HIGH RESOLUTION SPECTRA
8
"i
I I .
+ + i
I
+
++~
++"
b
I1
b
t~
l
I
~. ~-" .~ ~ - ~ , . . •- ~ . - . ~ _L,~_~ +
I
i
+--i~i--
,,-"
-. ,. t . - ~ ...,,
~.~
.
t
~ ~'~
ill I
I~1~ ~ + ~
~
I
~.~ ~ , - ~ / ~ ~- ~ + ~--
.~.
~..l~
m
~+ + + J+
~
+ Ii
~
I1
+
~4~
344
HIGH RESOLUTION NMR SPECTROSCOPY
quency VAwith unit intensity. This absorption line is observed to have a narrower natural line width than the widths of fines corresponding to transitions with Is > 0, indicating that the states with Is - 0 have a longer lifetime than the others. The lifetime of a spin state depends upon the probability that a transition will be induced by the randomly fluctuating magnetic fields produced at the nucleus by the surrounding lattice, and the probability per unit time of such transitions depends upon the total spin angular m o m e n t u m / , decreasing as I increases ~4). All AB~ systems with n > 1 have the mean of the frequencies of the third and fourth B transitions corresponding to vs, hence for n even, the chemical shift difference v^ - vs (that is, Vo 6) can be measured directly from the spectrum. The coupling constant may also be obtained from the spectrum since the mean of the firstA and the first B transition frequencies relative to VA is ½(% 6) +
1(is + ½). (ii) n odd. When n is odd there must be a sub-spectrum corresponding to Is = ½; examination of Table 8.21 shows that for I = ½ the first two A (or B) transitions are separated by the coupling constant d.
8.13.6 The General Features of an ApB~ Spectrum When the A group contains either more than one nucleus or a nucleus with spin IA > ½ then the secular equations to be solved are of order ~ 3. But even in the most complex ApB, systems it is still possible to obtain values of J and 6 from the spectrum. With the general spin functions AtA.mABx~.mB, which are eigenfunctions of I 2, 12 and 1~, as the base functions, the secular determinant factors first into sub-determinants corresponding to the possible values of IA and Is, and then into sub-sub-determinants depending upon the different values of m for a given pair of 1A and Is. Consider a determinant corresponding to particular values of 1A and lB. The maximum and minimum values of m are
mmax=ls+[A m~l, = --IA -- /8 SO that the sub-determinants belonging to the values m=,~ and mmia are of dimensions 1 x 1, and the functions
AIA. IA BtB.~B and ArA. -IA Btn. -IS are eigenfunctions of the Hamiltonian. The second largest value of m is 1A + IB - 1 which can be arrived at in two ways: when mA = IA -- 1 and - - m s = Is, or when mA = 1A and mR ---/13 1. In general, the state with -
m=Ix
+ ls - - r
THE ANALYSIS OF H I G H RESOLUTION SPECTRA
345
can be arrived at in the r + 1 ways m^
mB
I^
/B--r
I^-1
ID--r+ l
IA -- r
IB
NOW the value of mA must lie between -t- IA, similarly mB must be between :i: IB, so that if 1^ _~ IB the maximum value of • is 21^. This means that the value of m common to the largest number of spin functions is mf
l s - - I^
and the highest order of equation to be solved is (21^ + 1). In systems where all nuclei have spin ½ the secular equations are of order equal to or less than (p + 1); that is, A2 B, involves a cubic equation, A3 B~ a quartic equation and so on. The spectrum of an Ap Bn system is made up of a superposition of all subspectra A, B, where • and s have the same parity (that is, even or odd) as p and n; each sub-spectrum is weighted according to the values of gZA and gIB" A nomenclature has been developedcsS, 13s) for the sub-states corresponding to the possible values of I^ and Is in which the state with I = 0, ½, l, ~, 2, ~ . . . is known as a singlet, doublet, triplet, quartet, quintet, sextet etc., (abbreviated to S, D, T, Q, Qt, Sx) respectively. Thus each state is known by the number of magnetic sub-levels in that state. For example, the A~B~ system has the substates
S^TB, T^TB, T^Ss, SAS~ and the AsBz system has the sub-states SBD^,
SBQA, TsD^,
TBQ^.
Dividing the spectrum into the contributions from sub-spectra (sometimes known as the "complex particle" method) shows that an A~B, system contains lines at the same positions as in all systems with smaller values ofp and n, providing that they have the same parity. For example, the A2B6 spectrum of propane c35~ contains the lines of the A2B2 system (see Section 8.13.8), so that if all absorption bands are resolved and can be assigned to particular transitions, then the chemical shift difference ~o ~ can be measured directly from the Sl~'trum. The types of sub-spectra contained in an ApB, spectrum depend on whether n a n d p are odd or even. Ifp is odd then the general features of an AB. spectrum appear in A~B,, hence for n even it may be possible to obtain ~o ~ and J directly
346
HIGH
NMR
RESOLUTION
SPECTROSCOPY
from the spectrum, and for n odd two lines in the spectrum have a separation equal to J. When both n and p are even there are lines at the frequencies v* and vEhaving the relative intensities (l ‘I)2* l P and T-’ n. Also, since the AzB3 system has sub-spectra in common with all systems havingp and n even, then two lines will occur separated by 2 J (see Section 813.8). The number of lines in an APB,, spectrum can be calculated(“) in a manner similar to that adopted for AB,. The number of A transitions for n and p odd or even is given in Table 8.22. The AB, spectrum contains only one type of combination transition, that is when d mA = + 1, d m, = - 2, but in the TABLE8.22 THE NUMBEROF A TRANSITIONS M AN APB. SYSTW
NA
D
II
odd
odd
;(P-t
odd
even
;
even
odd
&p
t 2)(/r + l)(n + 3)
even
even
&P
c Z)n(fl + 4) + 1
ly(n+l)(n+3)
(p + 1)z!2(n f 4) + 1
TABLE8.23 THE NUMBEROF COMBWA~ONTRANSITIONS IN AN A,,B, SYSTEM IN WHICHAm,,
p
I n
- -r, Am, = (r-l)
!
NM
odd
odd
$ (p i l)(p + 3 - 2r)(n + l)(n + 5 - 2r)
odd
even
$(p+l)(p+3-2r)n(nf6-2r)
even
odd
$p(pi-4-2r)(n
even
even
+ l)(n + 5 - 2r)
-2r)n(n+6-2r)
A,& system, mixed transitions involving changes d mA = + 2, d mB = - 3 etc., are possible. In general, a combination transition involves dm, = - r anddma = +(r - 1) (or vice versa) where r may take the values 2,3 . . . 2I,4, assuming that ZA< ZB.The number of combination transitions in which :1 nt,& = - r is given in Table 8.23: the number of transitions with dmB = - r and d mA = (r - 1) can be obtained from this table by interchangingp and 12.
T H E A N A L Y S I S OF H I G H R E S O L U T I O N S P E C T R A
347
In all ApB, systems the transition frequencies and intensities may be expressed in terms of a single parameter J/% ~. The form of the spectrum is determined solely by this parameter (apart from a scaring factor) so that a trial calculation with an approximate value of .l/% 6 enables the spectral bands to be identified with the calculated transitions. The next three sections are concerned with some important examples of ApB, systems. 8.13.7 The Four Spin System, A2X2
A system containing two different pairs of magnetically equivalent nuclei separated by a chemical shift large in comparison with the coupling constant between the non-equivalent nuclei is described as an A2X~ system. The system is included here to emphasise the nomenclature used throughout this monograph. Analysis is of the simple first order type described at the beginning of Chapter3. The spectrumcomprises two symmetricaltriplets (of intensity 1 : 2 : 1); molecules giving rise to this type of spectrum are methylene fluoride, (CH2F~) and 1,1-difluoroallene allene, (CF2 = C ffiCH2). 8.13.8 The Four Spin System, A,B,
A system containing two different pairs of magnetically equivalent nuclei separated from each other by a chemical shift of the same order of magnitude as the coupling constant between the non-equivalent nuclei is described as an A2B2 system. In this system, both the A nuclei are coupled to an equal extent with both the B nuclei, and neither the coupling constant between the A nuclei, JAA, nor that between the B nuclei, J~s, feature in the NMR spectrum. There are only a few molecules which have magnetic nuclei satisfying the above criteria: examples are provided by certain c/s-disubstituted SF6 derivativesc3" (see Section 11.16.2). Sulphur tetrafluoride~3~ (see Section 11.16.3)is a rather exceptional example of an A2B2 system: the molecule has a trigonal bipyramidal structure with one equatorial position occupied by a lone pair of electrons. Although the two magnetically non-equivalent pairs of fluorine nuclei give rise to an AzB, type spectrum at - 100°C (see Fig. 11.19), the chemical shift difference between them is large compared with the coupling constant <36~. The appearance of an A2B2 type spectrum has been attributed to the large absolute values of % ~ and J. The hydrogen resonance spectra of the molecules fl-propiolactone ~Is~ (I), ethylene monothiocarbonate~7~ (II), trans-dibromocyclopropane~39> (III) and 2,3 dihydrofuran°9> (IV) can be explained by an A2B2 analysis. It is now fairly certain that the spectra of these molecules are deceptively simple AA'BB' type spectra. Each of the molecules is cyclic and hence there is no possibility of free rotation about a C--C bond to average the cis- and trans- H - H coupling constants between the A and B nuclei. It was originally thought that since only one coupling constant could be measured from the spectra, that the coupling constants v~eT~t"and J~'~' must be accidentally equal. However, Gutowsky and Grant c'°~ have made some theoretical calculations which show that these
348
HIGH
RESOLUTION
NMR
SPECTROSCOPY
spectra can also be reconciled with an AA’BB’ system (see Section 8.16.2), provided that IJAA’ + &WI z 5IJAB - JAB’1
IJAA’- JBWIk 5IJm - JAWI
and
6
G-d=0
i
b
I
II a
CHBr-CHBr
CH-CHn
CI-Ia III
IV
In the AzBz system, both IA and IB may take the values 1 and 0 so that on the complex particle nomenclature the sub-states are S.&, S,T,, T& and T,TB. The S.& sub-state contains no magnetic sub-levels and hence does not contribute to the spectrum .The S,T, and TASBdeterminants are both of dimensions 3 x 3 but in each case the secular determinant factors into three first order equations corresponding to the m values 1, 0 and - 1. The determinant for the T,T, sub-state is of dimensions 9 x 9 but the secular equation factors into two ht order, two quadratic and one cubic equations, corresponding to the m values of 2, -2, 1, - 1 and 0 respectively. The 3 x 3 sub-matrix in the T,T, sub-state is (-~8fJ) -J 0 s = -J 0 -J (8.148) 0
(
- J (Y,, 6 + Jl )
The eigenvalues El, E2 and E3 are obtained by solving a cubic equation; the eigenfunctions corresponding to El, E2 and Es are of the form El
aIIAI,
1 B1, -1 + azIAIP
o &,
o +
a31A1,
-1
&,
~52
a12Al,
1 &,
o 4,
o +
a32A1,
-1
&,I
E3
a13AI, 1 B~,-l+a23Al,oB1,0+a33A,,-,B,,,
-1
+
a22A1,
1 I
(8.149)
I
where the coefficients ai1 are normalised eigenvectors belonging to the eigenvalues El, E2 and El. There are seven A, seven B and four combination transitions in an A2Bz spectrum: since the lines are disposed symmetrically about the frequency +(v* + ye) this forms a convenient frequency origin, so that v* = *vo 6 where
vg = -9%8 vg 6 = VA- vg
THE ANALYSIS OF H I G H R E S O L U T I O N S P E C T R A
I "~+ I
]
~
i +"~'~.+
I
I < i-
I
+
÷~
+~
~ +
~÷ +~
~+
+
,~
2
z
g~g
aa~aa
+++
++I++
z
~
z
m
[..,
o
2-
<<<
349
350
H I G H R E S O L U T I O N NMR S P E C T R O S C O P Y
Ill
J 3"04
I
,
I
L
I
5
I
12
1.33
]
34
i
5
34
I
6
I
7 8
Ir I~.
III
0.86
0"37
4~65 7
] II
0"20
1i 416 17 235
0.I0
i
-20
I
-I0
I
I
0
I0
l
20
0.00
Cycles sec"~
Fie. 8.16 A2B2 type theoretical spectra for a chemical shift difference (Vo~) of 10 cycles sec - t . A n d e r s o n (41)
THE ANALYSIS OF HIGH RESOLUTION SPECTRA
351
Only half the lines are given in Table 8.24, in which the quantities Q and R are defined,
Q = 2J/(~o ~ - R) .R = [(% ~)2 -I-
4J2] 1/=.
Examination of Table 8.24, shows that the difference in frequency between lines 2 and 7 is 2J: the frequency difference between line 4 and the reference origin is ½6. Appendix G lists the line frequencies and relative intensities for half anA2B2 spectrum for values of J[vo ~ ranging from 0 to 10.0. Some of the spectra are plotted in Fig. 8.16 to show their dependence on the J/vo ~ parameter.
8.13.9 The Five Spin System, A3B2 The analysis of this system is important because it occurs in the hydrogen resonance of the ethyl grouping. The system has been analysed by a perturbation method ~ls) and by m e a n s o f a n exact solution of the secular equation ~42' 43); only the latter method will be presented here. IA can have the values ] or ½ and la may be 1 or 0, thus using the complex particle nomenclature the sub-states are D^SB, D^TB, Q^SB and QATB. The
TABLE8.25 FACTORSOr ~,~ SECUL~a~EQUATIONS FOR~m~ SUB-SrAI--zSOFAsB2 Sub-state
m
DASB
-½
½
2
-i ½
-½ 2
½
QA SB
-½ 2 2
QATB
Order of the • secular equation
_3 2
½
-½
two 1st order two 1st order
I I
four 1st order four quadratic
four 1st order
two 1st order two quadratic two cubic
~
H I G H R E S O L U T I O N NMR S P E C T R O S C O P Y
÷÷÷
÷
~
~
c~
e~
e4
_
ol
e~
e4
e~
eq N
N
~~ + ÷ + ~ I e ~
O
~,~ ~
~,~
o~o~
THE A N A L Y S I S OF H I G H R E S O L U T I O N S P E C T R A
~
~+~+~
Itl
~~
I
~
~
~ ~ ~+ + ~
I
I
I
~ N
~ ~
~ ~ - I ~ 1 ~
Z I{
fro
~ + ~
~
{3
o
, ~
~ ~ % ~
,
~
,
~
~E~(~c~o~
~
I
353
354
HIGH RESOLUTION
NMR S P E C T R O S C O P Y
I++1 + + 1 1
++1111
I
N
N
N
e4
N
~.~
r4
~ + + + + cq I ~ + . , . , . ,
~1 ~,~ ~ ' ~ ~
N
~
~1 ~
~ ,
~,;
--
I
THE A N A L Y S I S OF H I G H R E S O L U T I O N S P E C T R A
B
A
355
Vo ~ J//vo S
J
9.0
,11,
,
, 3
9
>0~,1
,11 Ill
3 9
©,6 II , 4
IJl II j;, ,.o
HI
8,5 12,2
I ?
tO
.
,
!
i
I1
4
12,4
it~2,1
12,~1
6
2
6 I0 t2
8
I| 7 2
I
4
IL!!IIL
2"25 0"50
"13.'3
(2,8)
(~,9) (t6,tO,J2)
=l=ilh
/////1\\\\~,, I
|2,8)
f
'
|
i
•
2
2.25 ' i' O
0.75
2(5,8) 1,7,5,9J3,3 JoJ2,6
Ih i,,,
I
II FIG. 8.17 Theoretical A3Bz type spectra. Corio (7)
12"
0"15
o..
9
19,12) 6
e,~e
5,3
9113 5,3
I0
,=
9,1z
11II III~J ~l,,~°
, l tIll6 4 12= 8II 25
I0
3
(3
5-0
5-0
5-0 >10.0
356
HIGH RESOLUTION NMR SPECTROSCOPY
factors of the secular equations for each sub-state are indicated in Table 8.25. The two 3 × 3 matrices are
I'~'(-Y- 1'0 3 - J) S ~: =
+
J 0
1
+(2) l/z J
+ (2) tn J
(8.150)
1
-='(-- ~o~ - J)
and the eigenvalues of S + are E t , E2 and E3 with eigenvectors (art, a2t, a3,), (at2, a22, a32) and (at3, a23, a33) respectively. Similarly, the eigenvalues of S- are E4, Es and E~ with eigenveetors (al4, a24, a34), (ats, a2s, a35) and (ate, a26, a36), respectively. The frequencies and intensities of the allowed transitions are given in Tables 8.26 and 8.27, combined transitions are listed in Table 8.28. The quantities R , , R2, R3 and R4 are defined, RI = [0'o~ - ½j)2 + 6j2],/2 R2 = [(Vo~ + ½j)2 + 6j2],:2 R3 = [0'o~ + ½j)2 + 2j2],,,z
(8.151a)
R4 = [(~'o~ -- ½j)2 + 2j2]t,2 Q, = x/6S/O'o ~ - ½J - R1) Q2 = - ~ / 6 J / ( , o
~ + ½J + R2)
03 = x/2J/0'o ~ + ½J - R3)
(8.151b)
Q , = - , / 2 J / ( ~ o ~ - ½J + R , )
All the frequencies are relative to va and vA - ~% is equal to 1,o & The latter quantity can be obtained from the relationship ~'o0=(B4+B5)/2-A6 where B4, B5 and A6 refer to frequencies corresponding to bands within the B and A parts of the spectrum. After 7,o~ has been found, J can be calculated from any of the line spacings involving R , , R2, R3 or R4. Some typical A3B2 type spectra for various J/~'oO values are shown in Fig. 8.17 while in Appendix H are given line frequencies and relative intensities of A3 B2 spectra over a wide range of J/1,o8 values. 8.14 SPECTRA CHARACTEI~/SED BY T w o COUPLING CONSTANTS
For this situation to prevail, a molecule must contain at least three nonequivalent nuclei: such a molecule would then usually have a N M R spectrum characterised by three coupling constants; an exception will now be discussed.
THE ANALYSIS OF H I G H RESOLUTION SPECTRA
357
Certain molecules contain three magnetically non-equivalent nuclei of the same species, two of which accidentally have the same chemical shift (an ABB' spin system). Two coupfing constants are featured in the spectrum of such a molecule, namely J^B and JAB', and they are comparable in magnitude with the single chemical shift difference present, namely Vo& Richards and Schaefer (~) have examined the XH resonance spectrum of 2,SMichloronitrobenzene and they have shown that the magnetically non-equivalent HB and HB, ring hydrogen atoms have the same resonance absorption frequency, that is they are chemical shift equivalent. They have calculated the transition energies and relative intensities for an ABB' spin system. They showed that it is only possible to obtain the mean of the two coupling constants [½(JAB + J^V)] and not their explicit values from an ABB' type spectrum. Furthermore, one cannot obtain the coupling constant JBB' from the analysis even though the two B nuclei are magnetically non-equivalent. It should be noted that the ABB' system is a limiting case of an ABC spectrum where vB = Vc. 8.15 SPECTRACHAXACTEmSEDBY THREE COUPLING CONffrANTS
8.15.1 The Three Spin System, ABX An ABX system is comprised of three non-equivalent magnetic nuclei, two of which (A and B) have J^B ~ V0~Aa, and a third nucleus X separated from the other two by a chemical shift which is large compared with the coupling constants involved. The latter nucleus need not necessarily be of a species different from the other two providing that the above condition is satisfied. A molecule having an ABX type spectrum is 2-fluoro-4,6-dichlorophenol(*4). OH Cl(/~F
H~,JH Cl
This molecule possesses two non-equivalent ring hydrogen atoms and a ring fluorine atom. Several types of molecules containing three magnetic nuclei of TABLE8.29 BAsic PRODUCT FU~C'nONS AND DL*OONXLMATRIX ELEM~rrs roe T ~ A B X SYSTEM Diagonal matrix elements ~I
~,~.0~
½(~A+ ~'B+ ~ ) + ~(.TAB+ JAX + J~X) ½(~A+ ~ -- VX)+ ¼(J*B -- JAX -- "YSX) ½(~^ -- ~S + ~ + ¼(--JAm + .tAX-- JBX) ½(--~A + ~B + VX)+ ¼(J--AB -- JAX + -laX)
m, ~o
~PP P~P
½("A -- ~'a -- ~'×) + ¼ ( - - J ^ . -- -'r^x + .1.x)
q~,
PPP
~ ( - - ~^ -- ~ . + ~X) + ~ ( . f A - + J^X + JSX)
~-(-"^ + ,'s - "x) + ¼(- J^s + J^x - Jax) ½(- ~^ - vB + ~'x) + ¼(3AS -- 3 ^ x -- JSX)
358
HIGH
RESOLUTION
NMR SPECTROSCOPY
the same species have resonance spectra closely approximating to that expected o f an ABX system: vinyl ethers (*s. 4~) and mono-substituted thiophenes (4~) are of this type,
H\C=C H
H F"X CN
H/
H~'S)H
R
=
~oR alkyl group
Three resonance frequencies and three coupling constants are required to describe an ABX system as indicated by the diagram
JAB
taAAj~Bx
°Js
X ta X
The system has no symmetry properties and consequently the basic product functions (given in Table 8.29) are themselves used as a basis for the formation of the wavefunctions. The Hamiltonian operator for the system is
X" = (~'AIA. + COSI~, + a'XlX, + ,rASlA.I~, + JAxlA=&. + Jsxls, IX, + ½JAB[I~I~ + I~,I~] + ½dAx[Z~,Ix + I~I~] + 3dBx[I~Zx + I~I~]). In Table 8.29 are listed the basic product functions and the diagonal matrix elements for the ABX system. The only non-zero off-diagonal elements are given by H3a ---- Hs6 ---- 3JAB H2, = H57 = ½JAX
H~3 = H6, = 3I~x The basic product functions 9t and 9s (o~o~ and ~/~/~) are stationary state wavefunctions since they do not mix with any others. The remaining elements of the 8 x 8 matrix can be factorised into two 3 x 3 sub-matriees corresponding to IT = + 3 and IT = -- 3. For IT = + 3, the appropriate matrix is
(/7.,2 - E)
½J, x
tJAx
½J, x
(H3~ - E)
½I^,
½J~x
½JA~
(H,,-
=0.
E)
To obtain the exact solutions of this equation it would be necessary to find the roots of a cubic equation. However, since the chemical shift difference
THE ANALYSIS OF HIGH RESOLUTION SPECTRA
359
between nucleus Xand the other two nuclei is large compared with JAX and Jex there will be very little mixing between functions f2 and functions ~vs and ~v4 even though they have the same value of total spin. This is also true for the function ~v7 which does not mix with ~vs and ~ve. Hence, the ½JAx and ½Jnx off-diagonal elements can be neglected and the diagonal elements H22 and H~7 can be regarded as solutions of the s~g~nlar equation and thus are stationary state energy levels of the system. The roots of the remaining 2 × 2 matrices can be readily obtained by solving the appropriate quadratic equation; the energy levels and the mixed wavefunctions are found to be: Wavefunctions
Energy levels ½~X-- ~JAI~+ D+ ½~x -- ¼JAI -- D+
(¢.osO+)~s + (sin 0+)~,
~3
- (sin 0+) ~s + (oasO÷)~, (cosO_)~s + (sinO_)~6
~5
-+,x -+~x
-(sinO_)~s + (cosO_)~6
- ¼JA! + z)_ - t J A i - z)_
The four non.mixing basic product functions (that is, wavefunctions) and their stationary state energy levels are:
Wavefunctions
Energy levels
,p~ ~, ~
½(~A+~+~X)+¼(J~+JAX+J'X)
V2
~(1¢A "Jr- $~B -- $~X) "~ ~(JTAB -- JAX -- J B ~
~192 0t0~fl
The usually positive quantities D, and D_ and the angles 0, and 0_ (lying between 0 and ~) are defined by the following relationships D÷ cos20+
ffi½vo ~^s + ¼(YAx - Jsx)
D+ sin20+ = ½JA~ D _ cos20_ ffi½~o ~^. - ¼(JAX -- Jsx)
D_ sin20_ ffi½JA~. Hence
D+ = {[½~o 6^n + ¼(JAx -- JBx)] 2 + D _ = {[½,o ~
- ¼(J^x
~x)P
¼j~},/2 I
+ t J ~ } ~/~
J
(8.152)
½(J.x - JBx) = [(D+ + ½a^~)(D+ - ½J.~))'/~ - [(D_ + ½J^~) x (D_
½%~ ^ B
ffi ( D +~ -
-
½J.B)] '/~
¼s~)'/~
-
¼(~Ax -
SBx).
There are fifteen possible transitions between the eight energy levels and these are given in Table 8.30 together with their relative intensifies. Three of the allowed transitions are combination bands, one of which (transition number 13)
360
H I G H R E S O L U T I O N NMR S P E C T R O S C O P Y
I I++++I '+'÷'~d
I
I÷ ÷I
I I ÷+
÷ + ~ ~ ~
÷÷
l l l i + + + +
+ I I + + I I +
+ I I + + I I +
Z
44444 +
~
Z
~ +
+
~
4
~ +
+
#
~
+
+
+
+ ~+ ~
Z 0
o
8S8
0 f-
tfttttttt~tt~tt
+
THE ANALYSIS
OF H I G H
RESOLUTION
SPECTRA
361
has zero intensity; the other two are of such low intensity that they are seldom observed at the main magnetic field strengths usually employed. Examination of the transition energies in Table 8.30 reveals that a value for the J^a coupling constant can be obtained by subtracting the frequencies of any one o f t h e following four pairs of bands: J^B = 3 - 1 = 4 - 2 = 7 - 5 = 8 - 6. Similarly, subtraction of transitions 9 and 12 leads to a value for (J^x + Jinx). It is also possible to obtain values for D÷ and D_ directly from the observed spectrum by subtracting the frequencies of the following bands 2D+ = 6 - 2 = 8 - 4 2D_=5-1
=7-3.
By using these values in conjunction with equations (8.152) the three coupling constants and absorption frequencies for this system can easily be calculated. The assignment of the absorption bands in an observed spectrum to the theoretical transition energies can be made in several different ways depending upon the relative signs of the coupling constants involved. Any assignment of bands must, however, obey the same subtraction rules as the transition energies, namely, 3-1= 42= 7--5= 8-6 | 2-
1 ffi 4 -
3 = 11--9=
10-9=12-11= (lO-
11)=(8-4)-(5-
6-5=
12-
10].
(8.153)
8-7
l)
Figure 8.18 shows a typical theoretical ABX spectrum where the significant separations between the absorption bands have been labelled. The first step in assigning the AB transitions is to recognise the symmetrical quartets (2, 4, 6, 8) and (I,3, 5, 7). It is impossible to ascertain the relative sign of the J ^ s coupling constant from the analysis of an ABX spectrum without recourse to further experiments such as spin-decoupling (see Section 8.19.5). Although the positions of the resonance frequencies are not affected by the relative signs of J^x and Jex the relative intensities of the absorption bands do reflect the signs: one can decide on relative signs by comparing a set of calculated intensities with those observed experimentally. However, even this method is limited to cases satisfying the condition ~s~ JAS/~O~^S --~ 0"5. NO information on absolute signs is obtainable since the combination of + -TAxand - Jsx gives rise to a spectrum identical with that from - JAx and + Jex. The two combination transitions possible for an ABX system (transitions 14 and 15) have an appreciable intensity when 0'A -- ~e) becomes small; they appear in the X region of the spectrum (they need not always be situated outside the X lines as shown in Fig. 8.18). Lines 14 and 15 are intense when JAx is of opposite sign to Jex.
36~
H I G H R E S O L U T I O N NMIi S P E C T R O S C O P Y i~ I
I
l
2[)_
Il 2D+
I1 "'
b
JAB
~-ri
I
I l
( Tl
JA~B 7 6
B
4
2
2 21o,+o_I '~----~,
IS
t2
!0
LI I
Io~~ '~xl '[
1
I1
l
9
~4
Fie. 8.18 A theoretical ABX type spectrum showing the separate AB and X parts. The bands are numbered assuming .TAx> 3sx > 0 and ~A > ~,~. Bands (2, 4, 6, 8) and (1, 3, 5, 7) form two symmetrical quartets
H:~(~ c/C~H5
cycles sec "l
5 6
7
B
f ', 2 3
I 4
l 9
E I l 10 II 12
FIG. 8.19 The t H resonance spectrum of liquid styrene oxide at 40.00 Mc sec-l. Reilly and Swalen¢2)
THE A N A L Y S I S OF H I G H R E S O L U T I O N S P E C T R A
363
The XHresonance apectrum oJ atyrene oxide (2). This molecule has three nonequivalent ring hydrogen atoms and it approximates to an ABX system since the resonance of the hydrogen nucleus situated at the same carbon atom as the phenyl substituent is separated from the other two by a large chemical shift compared with the coupling constant involved. The 40.00 Mc sec -~ spectrum is shown in Fig. 8.19. Table 8.38 fists the measured absorption band positions for this frequency together with the transitions to which they have been assigned. Values of JAB, (JAx + JBx), D, and D_ can be obtained by the subtraction of appropriate band frequencies and then by using equations (8.152) the following values of chemical shifts and coupling constants between the three nuclei can be calculated: JAB = 5"66, JAx = 2"42, JBx = 4"10 cycles sec -1 ~'o~^ = 100"39, ~o~e = 112"25, ~o6x = 144"46 cycles sec -1. These values can be seen to give calculated transition frequencies which are in good agreement with the observed spectrum (see Table 8.38). However, the calculated relative intensifies for the system are found to differ from the observed ones to a large extent due to the fact that the molecule gives a spectrum at 40.00 Mc scc -1 which only approximates to an ABX type. Precise values of the relative intensities can be calculated by treating the molecule as an ARC system (2) (see Section 8.15.4).
Deceptively simple ABX spectra. Sometimes fewer absorption bands than predicted are observed in the spectra of molecules expected to be of the ABX type. Some of the bands coalesce and others weaken when certain relationships between the various parameters prevail. Less information is available from a simplified spectrum of this kind in that only average values of coupling constants can be determined. Abraham and Bernstein ¢4s) have drawn up a series of theoretical ABX spectra by way of illustrating the simplification that can occur in s o m e limiting cases (see Fig. 8.20). When ~:0~Aeis large compared with JAe a normal ABX type spectrum having 12 bands results. If ~o~Ae ---- 0 or ½(JAx -- Jex) = 0, then the X region of the spectrum becomes a triplet and the AB portion consists of a strong pair of doublets together with four weak absorption bands (see Figure 8.20). If *o ~Ae and the ratio ½(JAx -- Jex)/JAe are both small compared with JAn, then further simplification of the spectrum occurs: the X part of the spectrum is a triplet (the central band being an unresolvable doublet) and the AB part comprises a doublet. The latter system is essentially of the type AA'X, an example ofwhichis provided by the XH resonance spectrum of the ring hydrogen nuclei of 2-furfurol at 6{>00 Mc sec -~ (see Fig. 8.21). Although this spectrum can be explained by assuming that JAX = ,/eX ---- 1"4 cycles sec -1, it is certain that such an equality does not exist (see Section 10.16). The deceptively simple spectrum can be interpreted equally well by using more reasonable values for the coupling constants found in the complex spectrum of the substance dissolved in dioxane (-TAx = 1"89, Jex --- 0.91 and JAe --- 3-16 cycles sec-X). The separation of themultiplet components in Fig. 8.21 gives the average value of the two coupling constants, namely ½(JAx +-Tax) -- 1.4 cycles sec -1. 12a*
364
H I G H R E S O L U T I O N NMR S P E C T R O S C O P Y
In connection with deceptively simple spectra, Schaefer C5°~has discussed the relationship between the spectra obtained from the systems A2X, AA'X, ABX and APX. x
aB
,, Ill I,,
Iff I'1 ~ 0 " 9
87
ill
12 lO,tl 9
8
12 I0,11 9
8
7
7
654
32
tlrl
64
53
4,6 3,5
t
(ii)
2
I
2
[
FiG. 8.20 Theoretical A B X spectra for the relatiomhips (i)V0aAa iS large compared with J.s(ii) yogas or ~(J^x - Jax) = 0 (i~) V0~Ae -- 0 and ~(JAx -- Jsx)/JAe approaches zero. Abraham and Bcrnstein (4s>
1"40 cvcJes sac -~
--m. I
'
I
2.80
cycles
sec -~
i
J
2
' j
H×
Ha ,
H8
FIG. 8.21 The 1H resonance spectrum of the ring hydrogen atoms of 2-furfurol (25 per cent in benzene) at 60.00 Mc sec -t. Abraham and Bernstein (4e)
THE ANALYSIS OF HIGH RESOLUTION SPECTRA
365
8.15.2 The General A~BXp System
A system of three groups of magnetic nuclei gives rise to secular equations of order greater than or equal to three. However, when one of the groups (X) has a chemical shift difference from the other two which is large compared with the coupling constants, then to a first approximation there is no mixing between functions with different values of lx., and the order of the secular equation is reduced. The approximation is valid When I~o~Axl, IVo~Bxl ~ [Vo~Asl, l/Asl,
IJ^xl,
[Jsxl
(8.154)
and the Hamiltonian can be written = (~AIA. + ~81~. + ~x/x. + JAsI^ "IB + J^x]A~ "Ix. +
JBxl,,." Ix.)
(8.155) so that the group X can be regarded as a first order perturbation on the A and B groups, and the order of the secular equations is determined by the numbers of nuclei in the groups A and B. When B contains only one nucleus then only quadratic equations are involved and equations for the transition frequencies can be written in a closed form ~'9~. The operators I ] , I~, I~, (1A. + Is.) and Ix. all commute with the Hamiltonian so that the set of spin functions ~IAo mA BI/2. + 112 X l x , ra,x ,
which are simultaneous eigenfunctions of all these operators, are taken as the basis for the Hamiltonian matrix. It follows that there are non-zero matrix elements of ~o only between the functions with the same values of IA, (m,,, + mB): Ix and rex, that is, only between the two functions ~1 =="~IA,mABll2. 112 Xlx.r~
both of which have
and
9~2 ffi AzA.mA+ , B~/z.- ,/a Xzx,.~x
(8.156)
m--m^+mx+3.
The matrix dements of the Hamiltonian are readily evaluated: ( ~ t [ ' ~ l ~1) ffi m^ ~^ + ½~s 4- mx ~'x + ½ m A J ^ B 4- ½mxJsx + m^ m x J ^ x
(w2 I ~ 1 ~2) -- (m^ + 1)~^ - ½~s +
mx~x
--
½(m^ + 1)J^B
--
½mxJsx
+ m x ( m ^ + 1)JAx
(w~ I,~1Wa) = (~a I ~ l ~00 ---- ½J^s[(IA - mA)(I^ + m^ + 1)] ~/2. Solving the appropriate secular equations gives the energy levels ½[(2mA + 1)~A + 2mx Vx - ½JAs + J^x mx(2m^ + I) + RCZA.,A.~,X)]
and
(8.157)
½[(2mA 4- 1) ~^ + 2mx ~x - ½"/AS 4- JAX mx(2m^ + 1) - R(l^.mA.mX) ] where
(8.158)
R,,A.,.A.MX~ = {[re 6^s - ½(2m^ + 1)J^s + (JAx - JBx)rex] 2 2 + J,,e(I^ - mA)(I^ + m^ + 1)} ~/2
(8.159)
HIGH RESOLUTION
366
NMR SPECTROSCOPY
The eigenvalues corresponding to the two energy levels are 2
-1.
[1 + Qct.~.,,,.~.,,,x,] ['4~A.,,,A+~B~/z.-t/Z Xxx.,,,x + Q'~A.,,'A.,"X~
and IX
× A I A , mAB1/2, t/2 X'lx.lw~¢] 2 -1 Jr" ~(IA,mA.r~X)] [~(/'A, mA,mX) AIA.mA+xB1/2 .- i/2X'zx,mx
(8.160) (8.161)
-- ~'~lA, raA B l / 2 , !/2 XIx, mX]
where the constant QCzA.,,A.mx) is defined as JAe[(IA -- mA)(IA + mA + 1)]'/2 Q(IA, mA. mX)
~'0 ~AB -- ½(2mA + 1)JAe + (JAx -- JBx)mx + RcXA.,~.,,x~"
Each energy level has a degeneracy gtA g~X" (8.162) The frequencies and relative intensities of the allowed transitions can be obtained from the above equations, remembering that the selection rules for a group G transition are /Ira~ = - 1 ,
A m~, = A I~ = A I~, = O
and that the relative intensities are obtained from the matrix elements of the operator I- in the manner outlined in Section 8.11. The results are shown in Table 8.31. Note that in this table a function of the type ArA.~,AB1/2. ± l/2Xx~mx vanishes if mA exceeds q- 1A: for the function ArA.mA+XB1/Z,± 1/2Xrx.mx mA may take the values IA -- 1, IA -- 2 . . . -- 1A, -- IA -- 1. It was shown in Section 8.13.6 that it is convenient to regard an ApB. system as composed of "complex particles" ArB, each of which have fixed values of the spins IA and Is, and the number of such particles is given by the allowed values of 1A and Is. The same approach can be adopted for A,BX~ systems, so that the complete spectrum can be regarded as a superimposition of sub-spectra arising from "complex particles" with each sub-spectrum weighted with the factor gx,.g~x. For example, the system AsBX2 will contain sub-spectra arising from the complex particles,
I & SxDT
SxDS QDT
QDS DDT DDS
5 5 2 3 2 3 2 1
2 1 2
Ix 1 0 1 0 I 0
gl A gl X
THE A N A L Y S I S OF H I O H R E S O L U T I O N S P E C T R A
367
,}]~~ ='i~"~"~'~'~ =., o
~
~.~ =~ .<.<j ~ ~ ~,+
~
..
..~--
~
+ .
: ! : : 1!i
¢
368
HIGH
RESOLUTION
NMR SPECTROSCOPY
General features of the spectrum. There are two main changes in going from an A.B system to one of the type A.BXv. First, the spectrum is no longer governed by the ratio J/~o 8, where J is the coupling constant between the groups A and B. Second, three coupling constants J . 3 , YAXand ./3 x are involved whose signs may sometimes be determined without recourse to special experimental techniques such as double resonance. The effect of the X group of nuclei on the AB region of the spectrum is simply to repeat an A.B spectrum p + 1 times with the intensities of lines in one A.B pattern related to those in another by the binomial coefficients. For example, ifp = 2 each line in the A.B spectrum is split into a triplet having components with the intensity ratios 1 : 2 : 1 with a separation equal to JAX or Jn x. The X region of the spectrum is more complex as will be described below. X Transitions. The notable feature of the X region of the spectrum is that it is symmetrical about the frequency ~x w h e n . = l, but asymmetric otherwise. This may be seen by examination of Table 8.31 and by noting that for n > I, R~xA,m..mx~ and QuA.-mA-,..~x~ are changed into RuA.-,~..-J,.,x> and Qc,..-A, ~ , , respectively by changing the sign of JAB. The latter two combination transitions fall in the X region and are also asymmetrical about ~'x f o r . > 1. This asymmetry of the X region enables the relative signs of JAS, JAX and Jnx to be determined when n > 1. When mA = IA, the two X transitions have the frequencies ~x - ½ J ^ x ( 2 I ~ + 1) + ½ ( J ~
- 4~)
~x + ½.r~x(2I~ + l ) - ½ ( J . ~ - 4 ~ )
each having the relative intensity F~'x.mx-*) 2 g~Ag~x" The lines are separated by
2I^-rAx + 4 x ; unlike all other X transitions they are independent of the applied field strength. When n is even, states with IA = 0 contribute pairs of lines separated by Je x. Examination of Table 8.31 and equations (8.159) and (8.162) shows that when n ffi 1, it is possible to obtain the relative signs of JAx and Jnx. When n > 1 it is possible to determine the relative signs of JAn, JAX and Jnx from the X part of the spectrum Ca1" 52)
B Transitions. (i) n even. The states with IA = 0 give lines at ~e + Jnx mx and, since mx may take the values - Ix to Ix, there are pairs of lines symmetrical about ~s separated by Jn x and of relative intensity gu,- o" glx. If p is also even, the state with IA ----0, Ix ffi 0 gives a line at ~'n of intensity glA-o • glx-o. (ii) n odd. The pair of lines arising from the states IA -- ½have the separation JAn. There is a pair of such lines for each value of mx, thus there are 21x + 1 pairs for a given Ix and summingover Ix gives (½p + I) 2 forp even and ¼(p + 1) x (2 + 3) for p odd.
THE ANALYSIS OF HIGH RESOLUTION SPECTRA
369
A Transitions. The m e a n frequency o f the two A transitions listed in Table 8.31 is ~A + JAX mx SO that for mx -- 0 it is vA. The separation o f the two A transitions is ~ ( I A , m A . reX) - - ~ ( I A . m A - 1, reX)
which is equal to JAn f o r n odd a n d I ^ ffi ½, mA = + ½, o r f o r n even a n d IA ---- 0.
Number oftrar~itions. These can be calculated (49) in a m a n n e r similar to that a d o p t e d f o r A B , a n d ApB,; the results are summariscd in Tables 8.32, 8.33, a n d 8.34. TABLE8.32 THE NU~mEROF A TRANSITIONSIN AN A.BXp SPIN SYST~ Number of transitions cvcn
ev~[l
cv¢~
odd
odd
@vcm
odd
odd
o(÷÷,)(++,)" 1 1 y (n + I) 2 (p + l)(p + 3)
TAsLE8.33 Te~ Ntr~m~ oe B TRANSmONSIN AN AwBXp SPIN S~yrmd Number of tmmilions
¢wm
odd
.~-
+ 1
[(p + I) (p + 3)]
odd
even
~ [(n + I) (n + 3)1
odd
odd
-~- (n + 1)(n+ 3)(p+ 1)(.p+ 3)
+ 1
1
TAm~ 8.34 Tim Nv-~mm oF X TR~-~aoss IN AN AmBXp SPIN SYsteM Number of transitions
(÷ /
even
odd
-~- (p + 1)2
odd
even
P P ..~-.(-~-+ 1)(n + l)(n + 3)
odd
odd
-~- (p -i-I)2 (n -{-I) (n + 3)
1
+ 1
370
HIGH RESOLUTION
NMR
SPECTROSCOPY
TXSL~ 8.35 THE Ntr~mn o7 Co)~t~A'noN TZANSmOtZSIN AN A.BXp 8 ~ r ~
Number of transitions
P even
cven
÷(÷+,) [÷(÷+1).(÷+1)]
oven
odd
¥ 1 p
odd odd
odd
_
1)2]
1 - ~ ( p + 1) [(n2 -- 1)(/7 + 3) + 2(n + 1) (/7 + 1)]
The general AnBPqXp system. Although six coupling constants are involved the treatment of this system is a logical development of that for the A,BXp spin system. The effect of the presence of an additional group P having a resonance frequency greatly different from those of A, B or X has been discussed by Pople and Schaefer (St) and by Corio (49). 8.15.3 The Six Spin System, A3B,X When an A.Bp system is further complicated by the presence of a group X leading to the relationships [~'o~^xl,
[~o~nxI'>IJAxl,
IJ~xl,
I~O~ABI, IJAnl
then the analysis of the spectrum is essentially the same as that for an isolated A.B~ system in that secular equations of the same order are involved. The A3B2X system is important because it often occurs in the spectra of ethyl derivatives (C2Hs),X<53-55): the analysis requires the solution of cubic and lower order equations. A complete solution of the problem may be obtained in a manner similar to that adopted forthe ABnXqsystem. In many examples of A3 B2X systems, the X region of the spectrum is not recorded along with the A3B2 part because in molecules likeH g (Calls)2,Sn (C2Hs)4 and Pb (C2H5), the resonance frequencies of the metal nuclei are far removed from that of hydrogen. In such a case,analysisof the A and B regionsof the spectrum isconveniently approached by an A3B2 analysis utilising"effective chemical shifts''(s3).Assuming first that there is no coupling between the groups A and B, the energy of the system to firstorder is then E (1) = (~A + mXJAX)mA + 0'e + taxi, x)m, + m x ~x.
(8.163)
Now, introducing coupling between the groups A and B, the condition that the interaction may be regarded as first order is 10'A + mxJ^x) -- (~e + mxYex)l >~ Y^.-
(8.164)
THE A N A L Y S I S OF H I G H R E S O L U T I O N
SPECTRA
371
The quantity on the left of expression (8.164) can be regarded as an effective chemical shift % ~ z . When X has a spin value of ½ then there are two pos~ble
P{CzHs)z
(o)
(b)
-4 0
///" /
-~./~-, 2 o, / J .-.~s,x \ Cyc les se~J \
\
\
/
\ \
/ / / /
\ \
/ J
/
\ \
(c)
,,I,
-20
5 [l- _j
o
h II
12
,i ildh,,
,~ll I
-4o Cycles sec"~
-12
0
12
v~
1:10. 8.22 (a)The I H rcsonan~ ~ of phosphorus triethyl at 60Mcs~ -I, (b) the theoretical spectrum, (c) the two AsB~ parts which make up the theoretical spectrum. Narasimhan and Rogers(s4)
372
H I G H R E S O L U T I O N NMR S P E C T R O S C O P Y
values of the effective shift, namely and
•o ~
= % ~A, + ½(.TAx -- JBx)
for
mx= ½
(8.165)
~o ~
= ~o ~AB -- .~(JAx -- Jsx)
for
mx = -- ½.
(8.166)
Thus if the inequality (8.164) is satisfied there will be two first orderA3B2 spectra characterised by the two ratios JAB/VO~'~ and JAB~%~'~. If the first order approximation is not valid then a full analysis of the two A~B2 systems must be made. Use of effective chemical shifts was made in the analysis of the spectrum of phosphorus triethyl C54), P(C2Hs)3. The ~H resonance spectrum at 60 Mc sec -t is of the A3B2X type where X is 3~p of spin value ½(see Fig. 8.22). The spectrum was analysed by first picking out a pair of intense bands, one almost at the centre and the other at the high field side of the spectrum, and assigning these to the CH3 group split into a doublet by interaction with the ~ P nucleus. The doublet separation gives JAx. The high field component of the doublet was then assigned to ~,~,,that is 7'A + ½JA x, and the remainder of the bands on the high field side were fitted to an A3B z pattern by comparison with tabulated spectra. In this way all the lines on the high field side of the spectrum and some of those on the low field side were fitted to an AaB2 pattern generated from J/*'o = 0"309 (see Fig. 8.22). Subtracting all these lines from the spectrum left an AsBz pattern having J/~'o ~ = 0.731. Having separated the observed spectrum into two A3B2 patterns it is possible to obtain ~o 6' and ~o ~" directly (see Section 8.13.9) and hence JAB may be evaluated. From the two values of the effective chemical shifts and with the aid of equations (8.165) and (8.166), ro ~AB and (JAX -- JBx) may be calculated. Since JAX is obtainable directly from the spectrum, J8 x can be found: it is interesting to note that the relative signs of the two coupling constants are available and in the example dealt with they have opposite signs. The chemical shifts and coupling constants determined by N a r a s i m h a n and Rogers c54) are listed b e l o w % ~AB =
17"5 cycles sec -
~o ~
=
24"6 cycles sec -1
Vo ~
=
10.4 cycles sec -I
Jcl'I3-Clt2 ~ JAB =
7"6 cycles sec -t
1
Jcn3-P
- JAx = --+ 13"7 cycles see -l
Jcn2-P
-- Jsx = -T- 0-5 cycles see -1 .
8.15.4 The Three Spin System, ABC A molecule containing three magnetically non-equivalent nuclei of the same species, separated from each other by chemical shifts of the same order of magnitude as the coupling constants between them constitutes an ABC system. Many examples of molecules exhibiting ABC type spectra are known and they include vinyl compounds, mono-substituted thiophenes and epoxides, asymmetrically trisubstituted benzene derivatives and disubstituted pyridines, as
373
THE ANALYSIS OF HIGH RESOLUTION SPECTRA
indicated below. It is possible to extract three coupling constants, JAB, JBc and J ^ c and three resonance absorption frequencies ~A, ~B and *c f r o m a full analysis o f the spectrum of an A B C system.t One can also obtain the relative (but not absolute) signs o f all three coupling constants. Typical ABC type molecules
/-kx \ S/
CH~CHX CI-Ie- - CI'IX \0/
/-\x
or
\ S/
RR\J/N~¢ "
R'
R'
R0R
or
ROR"
R or
R,0R,,
As in the case o f the A B X system there are no symmetry considerations to be t a k e n into account, and the basic product functions are themselves used as a basis for the wavefunetions o f the system. Table 8.36 gives the basic product functions and the diagonal matrix elements for the A B C system: they are seen to have the same f o r m as those of the A B X system (Table 8.29). The matrix TABLE8.36 BASICPRODUC'rFUNCTIONSAND DIAGONALMATRIXl::.1m~l~rrs FOR ThE ABe SYs'r~ ABC
Diagonal Matrix Elements
o,~,fl
~a
½0'^ ÷ ", ÷ "c) + ~(orA, ÷ .rAc+ .r,c) ½0',. + ,,, - ,'c) + ¼(JA, -- .r^c -- .r,c) ½0'A -- ", + "C) + t(--JA, + .b.C -- JJC)
Y, tPs We ~7 ~Ps
½(-- "A + "~ + "C) + ¼(-- JA, -- G C + .reC) ½0'^ -- "' -- "C) + r~(--.rA, -- "b,C + .l,C) ½(--"A + ", -- "C) + :t(--'/A, + .IAC -- JBC) 3(-- ~A -- ~, + "C) + ¼(~rA, -- ~r,c - Jsc) ½ ( - ~ A - vB - ~c) + ¼(~rAB + JAC + ~,C)
IP2
The only non-zero off-diagonal elements are H2~ = t/67 = ½J, c
//2, =//sT = ½J^c H3, =/'/~6 = ½J^, representation of the Hamiltonian can be factorised into two 1 x 1 and two 3 x 3 sub-matrices, each o f the four sub-matrices corresponding to one of the four different values o f total spin IT = ], 3, - ½ and - J. The 3 x 3 sub-matrix for lv = + ½ is H22
½J, c
½~.c
H33 ½G-
½G,:
I,c I (8.167) H,,
I
and because the corresponding secular equation is cubic, the three roots and the three eigenvectors o f the equation cannot be expressed in simple algebraic f o r m as could be done in previous examples. However, one can usually obtain reasonably * See reference 161 for a comparative account of the various methods of analysis of this spin system.
374
HIGH RESOLUTION NMR SPECTROSCOPY
good values of the parameters involved by approximating the spectrum to an ABX system, for which explicit expressions for the eigenvalues are known. Using these values, an approximate matrix can be constructed and its roots compared with the measured eigenvalues. By repeatedly modifying the input parameters in the matrix until the roots agree exactly with the measured eigenvalues one can thenobtain accurate values of theABC chemical shifts and coupling constants from the modified matrix elements. This procedure involves the use of an electronic computer to attain the convergence of the calculated and observed eigenvalues. A major disadvantage of the process is that there is no assurance that the final result is either correct or unique. For some molecules, two or more completely different sets of parameters have been found to be consistent with the experimentally observed spectrum to within the experimental error and in other cases convergence is never achieved. These problems are particularly acute when the separations between the absorption bands are small and of similar magnitude. One must therefore be extremely careful in the application of iterative processes to spectral analysis (s6>. One way in which the ambiguity in assignment can arise is when the combination bands are of comparable intensity to the fundamental bands (4s). In such a spectrum one can interchange assignments of the combination bands and the fundamental bands without disturbing the regular subtractive features which an ABC spectrum must show (see equation (8.153) for the ABX system: the same equation holds for the ABC case). The combination bands 14 and 15 have appreciable intensity when ~'o 6As is very small and this can lead to three possible assignments of the absorption bands. The third combination band is invariably so weak in intensity that there is no danger of confusing it with a fundamental absorption band. When two nuclei are coupled strongly together, to differentiate between 'fundamental' and 'combination' transitions is really meaningless since all transitions involve the change of spin of more than one nucleus (Se). A further possibility which increases the number of sets of parameters satisfying the observed transition frequencies is that the coupling constants can have different signs. Examination of the transition frequencies alone cannot lead to predictions of the relative signs of the three coupling constants and it is necessary to inspect the intensities of the transitions to distinguish between the four possible combinations of the relative signs, namely +JAB
+JAC
+JBC
+JAB
+JAc
--JBC
+JAB
--JAC
+JBC
--J*B
+JAc
+JBc.
Absolute signs of the coupling constants cannot be obtained, since the combination of signs + + + gives the same spectrum as - - - . It will be remembered that the ABX analysis, unlike the ABC, could not provide any information on the relative sign of JAB.
375
T H E A N A L Y S I S OF H I G H R E S O L U T I O N S P E C T R A
Because of the importance of the ABC spin system in the development of the analysis of spectra it is intended to discuss this case in detail. Analysis of an ABC spectrum using a perturbed A B X system: The A B K approximation (=). The basic problem encountered in the analysis of the completely asymmetric A B C system is that of obtaining explicit algebraic expressions for the roots of the cubic equation represented by the 3 × 3 sub-matrices of the type shown in equation (8.167). A complete solution of the m a r x equation (8.167) would be readily available if we knew the unitary matrix transformation U, (where UU-* ffi 1) which would diagonalise the matrix of the spin-HAmiltonian, that is U-I~U = A where A is a diagonalised matrix. For the ABX matrix with IT = ½, the ½JAx and ½J=x off-diagonal elements can be approximated to zero due to their being very small compared with the chemical shift difference between the X and the A, B nuclei, and the appropriate transformation is 1
0 0
H~
0
0
cosO+ sinO+
0
/'/~s
sinO. cosO÷
0
½.TA, //,,
0
-
0
0
• H~2
cos0, sinO÷
0
- sin0, cos0+
°1
0 0
I!0 .
/-/~,
If the matrix for the ABC system with the value IT = ½ is now subjected to the same unitary transformation which diagon~lised the ABX matrix, the resulting matrix is non-diagonal 1
0
1.2
0
0
cos0.
sin0÷
0
- sin0.
cos0.
½J, c
j! 0
H3s
½1.,
cos0+
H•
sin0+
[½JAC ½J.e
- sin0+ cos0+[
//22 62, 62, J = I 62s
[ 6a,
H~3
0
0
//',,
Expressions for the elements of the transformed ABC matrix are g~ven in Table 8.37. A similar transformed matrix for I T = - - ½ can be constructed H~5 0 0 Hie
6s7 I | ~67 [.
6,7 a6~
H77[
I
Second-order perturbation theory is now applied to calculate the A B K energy levels and expressions for these are given in Table 8.37. For the case of
376
HIGH
RESOLUTION
NMR SPECTROSCOPY
IT = ½, this is done by considering initially the elements ~2a and ~2, to be zero and then allowing for the fact that they are not zero by treating Jn c I n . Ic and Js c I s . Ic as perturbations of the ABX type matrix. Wavefunctions for the system are obtained similarly by means o f first order perturbation theory. Reilly and Swalen (2) have extended this method to the AA'C system where they use symmetric and antisymmetric combinations of spin functions as a basis for the wavefunctions. TAnI~ 8.37 EN~g~VLEVELSANDWAVEI~JNCTIOI~FOXTHEA~K SYSTEM State
Energy /-]'11 //22
Wavefunction q01
2
-1
2 A.2~ -1 H~a + &,a
,2 -I 2 -t H[~ -- ds~ As~ 2 -1 H6e - c~s7 -467 /./77 -~ 867`467 2 -1 -~ ~ 2 A ~ l
¢.2 - es(~z3 `4;~ cos0+ - a~4`4; t, sin0+) + ¢ , ( - ~.23`4;[ sin0+ - ~;~ `4~ cos0.) ~.2sA ~ q0.2+ ~s cos0+ + q0, sin0+ ~.2,`4~, ~.2 - ~3 sin0+ + ~4 cos0+ - ~ s sin0_ + ~6 cos0_ - ~s7 `4~ ¢7 q0s(~,7/t;~ cos0_ - ~67/t;~ sin0_) - ~6 (-~s 7/I ;~7 sin0_ - 8~7 -t ;17cos0_)+ ~ ~s
where
Hxt = hA + he + hx + ~J^a + ¼(JAx + JaK) //22 = hA "/r" his - hK -k- ¼JAB -- ~(JAK "+" JBK)
H'~a
=
hx - ¼JAn + D+
H 4 4 . = h K - - ¼ J A B - - D+ .H~s = - h x -- ¼JAa + D_ H 6 6 == --hK -- ¼JAB - - D _ =
` 4 z , = H.2.2 -
¼(JAK -k YaK) --hA -- hi) - hK "{- ¼JAB + ¼(JAg + JaiL)
H77 = --hA -- he + hx + ¼JAe
Has
a.2. = ½JAx C 0 S 0 + . ½Jisx sin0+ 8.23 = ½J, x sin0+ + ½Jax cos0+ ~67 = --½JAx sin0_ + ½J, x cos0_ ~s7 = ½J.~xcos0_ + ½Jax sin0_
-
H~
A.2a = H2z - H~3 `467 = H,66 - / / 7 7 `457
Hs~
H77
& = ½n
The remaining constants have been defined elsewhere. To apply the ABK method, one initially assigns the observed transition frequencies to an ABX set of energy levels. The approximate values of the coupling constants and chemical shift separations can then be evaluated and used to obtain approximate values for the angles 0÷, O_ from which 823, ~24, ~57 and ~67 are calculated. Approximate values for A23, d2~, ds7 and .467 are given by subtracting the appropriate frequencies in the observed spectrum. It is then possible to obtain values for the various ~2 .4- t terms which are the perturbations on the original energy levels. When these corrections have been applied to the experimentally observed energy levels, the resulting energy levels are a much better approximation to a set of ABX energy levels from which more accurate values of the chemical shifts and coupling constants can be extracted. These values are then used to calculate better values of 8 and d and the new values for the 0 z d -t corrections are applied once more to the observed energy levels. Even better values of the parameters of the system can be obtained from these
377
T H E A N A L Y S I S OF H I G H R E S O L U T I O N S P E C T R A
energy levels and the process may be repeated until consistent values of the parameters emerge. An electronic computer greatly reduces the mathematical labour associated with the A B K iterative process, but ff one is not available quite reasonable results can be achieved after only one or two iterations. The relative intensifies o f the transitions are calculated from the corrected wavefunctions given in Table 8.37. A similar ABK analysis has been outlined by other workers <ss. s . Reilly and Swalen have examined the ZH resonance spectra o f a series o f monosubstituted epoxides o f general formula
H/ ~O /
\H
where R is a substituent which does not couple with the three ring hydrogen atoms. They have interpreted the observed spectra of the molecules in three separate ways by assuming that the three hydrogen nuclei form (i) an ABX TAm_~8.38 THE Om~WV ~
CAt~
Tsat~n~c~ ~ u e Q ~
OF TIIE I H RI~ONANCE b'~I~CTRUMOF S ~
~
OXIDE AT 40"000 M¢
Rm~vs sec°x
~m~ ABX, ABK AND ABC MetHoDs o~ ANAI.~LS T
Band frequencies (cyclessec-i)
•T
number 5 6 7 8 1 2 3 4 9 10 11 12
.
Obs
95.60 98"19 101"37 103"83 108"05 112"11 113"70 117"70 141-20 143"74 145"23 147"68
Calc
Calc ABK
Calc ABC
95"66 98"16 101"32 103"82 108.06 112"07 113"72 117"73 141"21 143"71 145.22 147"73
95"66 98"16 101"31 103"82 108.06 112"08 113"72 117"73 141"21 143"71 145"22 147"73
95-66 98"16 101"31 103"82 108"06 112"07 113"72 117"73 141"21 143"71 145"22 147"73
]"
Relative intensities
Cal¢ 0"13~ 0"16~" 0-33] 0"36~ 0"31~ 0"38( 0.141 0'17~ 0.28.~ 0.27: 0.23! 0-20"
0"136 [ 0"148 I 0"364 [ 0"352 I 0"364 [ 0'352 [ 0"136 [ 0.148 ] 0"250 [ 0"250 I 0"250 I 0"250 I
Calc ABC 0"130 0"157 0"341 0"372 0"321 0"393 0"117 0.171 0"298 0"264 0"231 0"206
0.133 0.153 0"339 0"376 0"320 0"395 0"118 0"168 0"298 0"264 0"231 0"207
system (ii) an A B K system (or a perturbed ABX system) and (iii) an ABC system. A comparison o f the predicted transition energies and relative intensities with the values observed in the XH resonance spectrum o f styrene oxide (see Table 8.38 and Fig. 8.19) shows that while the ABX approximation leads to reasonable values for the transition frequencies, it is completely inadequate to predict the correct relative intensities. Both the ABK and the ABC methods gave accurate values for the transition frequencies and their intensities. The latter two methods also provide the relative signs o f the three coupling constants.
378
HtGH" RESOLUTION NMR SPECTROSCOPY
The complete ABC solution. Rcilly and Swalen achieved a complete ABC analysis by using a computer to solve the matrix given in equation (8.167). They used, as trial input parameters of the matrix, elements calculated from a sot of ABK parameters (without iteration) and then they employed a computer to solve successively the matrix equation and to modify the input parameters of the matrix until convergent solutions consistent with the observed transition frequencies were obtained. The cigenvectors of the matrix elements can be used to calculate the relative intensities. From the measured transition frequencies it is possible to calculate a set of eight eigenvalues for the systemcsg) and the iterative process (involving the use of a computer) is designed to modify the trial matrix until the roots axe equal to measured eigenvalues of the system. It is then a relatively simple matter to use the matrix elements of the final modified matrix to calculate the parameters of the system (using the expressions contained in Table 8.36). It is possible to apply a similar iterative process to converge onto the measured frequencies rather than upon the extracted eigenvalues of the system. Rcilly and Swalen(2), in both the ABK and ABC methods, derived expressions for the energy levels rather than the transition energies, and then they used these to calculate the chemical shift and coupling constants for the system. By this procedure, it is possible to obtain more accurate values of the parameters, and the iterative method is easier to apply. The complete ABC analysis involves the following stages: (1) The first step is to obtain a sot of approximate matrix elements by conducting an ABK analysis (without iteration). (2) Secondly, the trial spin Hamiltonian, ~ ' , is diagonalised by a unitary transformation U, U -x ,,~' U = A .
(8.168)
(3) The eigenvectors (the columns of U) can be re-applied to the experimental energy levels measured from the spectrum, Aorta, to give H,~pt U A ~ p t U -1 -~ H , xp,
(8.169)
where H, zpt is a set of linear equations relating the various constants, which can be solved to give more accurate values of the coupling constants and the chemical shifts. The process is re-iterated until convergence to consistent values is obtained. Reilly and Swalen used only the diagonal values, (Hexpt)H, but in theory all the matrix elements can be used to solve for the coupling constants and the chemical shifts. A serious failing of all the methods of analysing ABC spectra previously discussed is that the set of chemical shifts and coupling constants produced by an iterative process is not necessarily the only one consistent with the observed spectrum. Fessenden and Waugh (s¢) have devised a method of analysing ABC spectra which gives all the possible sets of parameters consistent with the observed spectrum. To do this one uses certain simple properties of the Hamiltonian which effectively reduce the number of variable parameters in the ABC system and at the same time provide good approximations of their numerical values. The two properties used are known as (see Section 8.19.3)
379
T H E A N A L Y S I S OF H I G H R E S O L U T I O N S P E C T R A
(i) the Trace Invariance of each sub-matrix (ii) the Intensity Sum rule. The latter rule is particularly useful in that it is capable of assisting in the prediction of the relative signs of the three coupling constants without solving for the eigenvalues at all.
Trace invariance of the sub-matrix. There are eight stationary state energy levels for an ABC system and if the system could be regarded as APX in type (that is, all J/~o 6 ratios --* 0) then these eight energy levels would be given by the diagonal matrix elements of the four sub-matrices since all the off-diagonal elements would be zero. A spectrum consisting of three groups of quartets (12 bands in all) with components of equal intensity would result from three such nuclei (see Fig. 8.2), and the separations between the absorption bands would satisfy the subtraction rules previously considered in Section 8.15.1. For example, ",,
-
"~3 -
(Hi,
-
H.)
-
(H3~
-
II~,)
=
~3~ -
~,. =
JAB
where ~'4, corresponds to the frequency for the transition ~o4 -, Io, at constant field. When the energy levels of the ABC system are considered, one must take the off-diagonal elements into account by diagonalising the matrix with a unitary transformation. Such a transformation does not change the sum of the diagonal elements (the trace) which has the same value in the initial non-diagonal and the final diagonalised matrix: this is also true for the sub-matrices. This property is referred to as the trace invariance of the matrix. Hence s~
= El = - ( E s + E~ + ET) = - ( H .
+
H~
H s , = Es = -(E2 + E, + E4) = -(H22 + H s ,
+ H~7) ]
+ H,~) I (8.170)
The transition frequencies can be evaluated by subtracting the appropriate energy levels "4, - "73 -- E, - E4 - E3 + E7 -- "3~ - ~ 4 -- E, + E~ - ( E 3 + E4) = -(E, + ~ --
+ ZT) + Z, + (Z, + Z~) = (E~ - Z~)
(Z, - E,) --- ~62 - ~,, etc.
(8.171)
Hence, the subtraction rules encountered in the ABX system are found to apply also to the ABC case. However, it is no longer possible to obtain an exact value of JAB simply by subtracting the appropriate pairs of absorption bands. It is also worth noting that whilst none of the repeated spacings in an ABC spectrum gives the exact individual values of JAe and JAc + J e c , the sum of the appropriate repeated spacings is equal to [ J , , + JAc + J , c [ , due regard being paid to the signs of the coupling constants. Consider the case when all J values are greater than zero: we can define two summations of the transition frequencies, S, and $2, such that S, = ~'2, + *'3, + I'41 = 3E, -- (E2 + E3 + E4) = 3 H ~ , + H s s
Sz = ~e5 + "ee + ~e7 = - 3 H s e - H , , .
(8.172)
(8.173)
380
H I G H R E S O L U T I O N NMR S P E C T R O S C O P Y
The difference between $2 and St is a measure of the sum of the three coupling constants S , - S t = - 4 ( H s s + H i t ) = --2(JAB 4= JAC q" Jsc). (8.174) This property enables us to neglect one variable from the problem. The sum of St and $2 provides a relationship involving the three chemical shifts but this does not lead to any further simplification since the sum depends upon the origin chosen: $2 + St = 2 ( - H 8 s + H i t ) = 2(VA q- ~11 -t- ~C). The intensity s u m rule. Application of this rule often allows the correct choice of the relative signs of the three coupling constants to be made. The fifteen allowed transitions for the ABC system fall into one of the three groups
IT i---½
½
-½
It has been shown (56) that the total intensities of these transitions in each group is unaffectedby the presence of strong coupling (that is, when all Jibe ~ ~ 1 as in the ABC system). Hence, if we consider the transitions associated with ~'t2, ~t3, and ~'t4, corresponding to the i ~ ½ group, we find that the sum of the intensities g is equal to 3 when the total intensity is normalised to 12 g (1/2, 3, 4) -- 3 i.e.
(gt,--2
+gt,--3
+gt,-~) = 3
(8.175)
and for the other two groups we have g (8/5, 6, 7) = 3
(8.176)
g (5, 6, 7/2, 3, 4) = 6.
(8.177)
The intensity of the combination bands is included in the summation (8.177). It is sometimes possible to decide which of the absorption bands in the spectrum satisfy these intensity sum rules and this enables the relative signs of the three coupling constants to be obtained. W h e n this rule isapplied to the tH resonance spectrum of styrene c5~) at 40.00 M c scc -~ (see Table 8.39 for the observed and calculated transition energies and relative intensities of this molecule) for the four possible combinations of the relative signs of the coupling constants, one can predict that the three coupling constants all have the same sign. Table 8.40A gives the summations of the intensities for the three groups for each of the four sign combinations and it is seen that assignment I is the one satisfying the sum rule most closely. CasteUano and W a u g h (89' so) have used both the trace invariance and the intensity sum rule to analyse A B C type spectra. The firststep in their analysis is to use the experimental frequencies to derive all the sets of eigenvahies with which they are consistent. This involves only the use of the trace invariance property of the Hamiltonian. By applying the intensity sum rules, given in equations (8.175 - 8.177), several of the sets of cigenvalues can be eliminated at this stage. The various sets of chemical shifts and coupling constants associated with the remaining sets of cigcnvalucs are then obtained. Often these sets of parameters
THE ANALYSIS OF HIGH RESOLUTION SPECTRA
381
a r e v e r y different a n d o n l y r a r e l y a r e t h e y r e l a t e d b y a sign reversal. By this m e t h o d , all p o s s i b l e sets o f s o l u t i o n s o f t h e H a m i l t o n i a n c a n b e d i s c o v e r e d . A m a j o r d r a w b a c k o f t h e m e t h o d is t h a t m o s t o f t h e t h e o r e t i c a l b a n d s in t h e spectrum must be resolved and measured experimentally if a complete solution TAstes 8.39 O a s ~ v ~ AND~ ' r ~ D F~s~a~NC~S ANDl)¢rmsm-r~s FOR'r~XH RnONXNC~ S P ~ U M OF NEAT ~ n ' R E N E (se) AT 40 Mcsec -~ Frequency
Intensity
Transition Obs
¥7 "" tPs tPs "-" ~'s tPs "*" ~'2 ~7 "-~ ~ 4
VPs~ ~6 tP2 "" q'1 ~%-')" ~7 •
2"60 13"50 20.78 31 "78 36"87 51"23 53"05 68"68 70"60 74"15 75"96 84"41 86"05
Calc
Obs
3.00 13-74 20-74 31"47 36"67 51-07 52"87 68"81 70-60 74"00 75.80 84.74 86-52
0"28 0"53 0"94 1"60 0"02 1'51
Calc 0-51 0"64 1"12 1"70 0"02 1"19
1'86
1"65
0'75
0"70 0.48
0.48 1"28
1-10 0.78 0-86
1-30
1"05 0"81 0-82
~tandard
deviations:
0.06 cycles sec-x
0-03
Frequencies are measured in cycles sec-1 from the centre of the ring hydrogen signals. TAa~ 8.40A APPUCATXO~Oe THE I h ~ , s r r Y SuM Rtn..~ (eQUAtOrS (8"175) TO (8"177)) TO ~ ExP~v~rr~d~ SPECTRUMOF TEIE VINYL H~ROOEN ATOMS IN STYm~ AT 40.00 MC sec-x Assignment [ [I [] [V
I~'~, J]
g(1/2, 3, 4)
~(8/5, 6, 7)
g(5, 6, 7/2, 3, 4)
30.23 26.63 18.20 12.83
3-07 3-24 2.82 2.97
2.94 2.39 2.52 3.25
5.99 6.37 6-66 5-78
is t o b e a c h i e v e d . Because o f this, m o r e c o m p l i c a t e d systems t h a n A B C a r e better analysed by the more usual iterative processes provided they are applied w i t h c a u t i o n . A n o u t l i n e o f t h e d e t a i l e d a n a l y s i s o f a n A B C system u s i n g t h e m e t h o d o f C a s t e U a n o a n d W a u g h (sg, 6o) will n o w b e given.
Determination of eigenvaluesfrom experimentalfrequencies. F i r s t w e m u s t exam i n e t h e m a n n e r in w h i c h t h e n u m e r o u s sets o f p o s s i b l e eigenvalues c a n b e o b t a i n e d f r o m t h e e x p e r i m e n t a l f r e q u e n c i e s o f a n A B C s p e c t r u m . I t is c o n v e n i e n t ,
382
H I G H R E S O L U T I O N NMR S P E C T R O S C O P Y
for this purpose, to classify the 15 allowed ABC transitions as indicated in Table 8.40s. We have already seen that
S2
+ S t = 2(VA + Vn +
VC)
l
(8.178)
TABLE 8.40B A GROUPING OF TRANSITION FREQUENCIES FOR AN A B C St'IN SYSTEM¢s~'~°) A
B
C
Et ~-E2
1
2 3 4 Comb. X
Ez~-~
El is the eigenvalue, in cycles sec- I , corresponding to I r = ] and Ez, Es and E , the eigenvalues for I r - ½, etc. TABLE 8.41 BAsic PRODUCT FuNcnoNs AND MATRIX ELEMENTS FOR T~ ABC S~s-r~t ABC
/r
3¢0~X
+T
Diagonal matrix elements
3
r-/~ = ¼(3"As + J~c + Is¢)
1
~2
+T
tps
+T
t/22 = ¼(J~s
1
-
I^¢ -
Jsc)
-
~'c
/'/33 = ¼ ( - J , , - + ; ~ c - Jsc) - "s
1
H.
+T ¥s
1 2 1 2 1 2 3
= ¼ ( - JAn - JAc + 3"sc) - ~'A
B'ss ffi ¼(--JA. -- 3"AC+ Jsc) + ~'A //~6 = ¼ ( - J ^ . + 3Ac - Jsc) + v~ H77 = ¼(JAB -- JAC -- JBC) + ~C
Non-z~ro off-diagonal elements are H23 :
H 6 ~ ffi ½ J s ¢
H34 : Hs~ :
~;..
H2,
½JAC
=
Hs7
=
t Where the origin of the frequency measurements has been chosen so that ~A + ~'e + VC = 0.
383
THE A N A L Y S I S OF H I G H R E S O L U T I O N S P E C T R A
the chemical shifts being referred to the same origin as was used in measuring the frequencies of the lines. When the origin is chosen in such a manner that both sides of equation (8.178) vanish (that is, ~A + ~B + ~c •ffi 0), the diagonal matrix elements for the system are given by the simple expressions presented in Table 8.41. Castellano and Waugh have used the trace invariance property of the Hamiltonlan to show that under these conditions the following rules are upheld: (1) The sum of all the line frequencies is zero (15 bands in all). The sum of the 3 combination frequencies is zero. The sum of the 12 fundamental frequencies is zero. One can thus find the proper origin of the spectrum before any calculations of eigenvalues are attempted. (2)
, ( ~ 1 ) + , ( A , ) ffi ,(A2) + ~ ( ~ )
= -~(A.)
and similarly for columns B and C of Table 8.40 B. This property enable one to obtain the frequencies of experimental lines when one or more of them are vanlshingly weak or unresolved. A set of bands having this property is referred to as a grouping: there are 15 independent groupings for an ABC spin system and since each grouping has four possible combinations of relative signs this leads to 60 different arrangements. (3) Several pairs of frequencies are separated by the same value referred to •as "repeated spacings". • (At)
,(A=)
= ,(,4~)
-
~(A,)
=
,,(C~)
-
,,(C~)
= ,,(C=)
~'-~G) -
,,(A,)
= ,'(,'I=) -
~(A,)
ffi ,,(B~)
-
~(B~)
ffi ,,(B:)
-
,,(B,)
,,(B,)
,,(B,.)
ffi ,,(a~)
,,(B,,)
ffi ,'(co
-
,,(c:)
ffi ,,(c~)
-
,,(c,,).
-
-
-
,(C,)
-
(4) A measure of the sum of the three coupling constants is given by E1 = E8 = (]A. + ]Ac + J . c / 4 ) ffi e.
(5) Among the 64 sums of the type
where/,.i, k -- 1 . . . 4, there are only 8 values which occur only once and these form four pairs, one member of each pair differing from the other simply by areversal of sign. The 8 values are the sums of the "'repeated spacings" observedin rule 3 with the eight possible permutations of sign. (6) One pair of these non-repeating values satisfies the equation S111 ~--" --$4~1.4:
4£.
Rules l, 2, 3 and 5 can all be applied to the measured frequencies of an ABC spectrum to enable a table of frequencie s which corresponds to Table 8.40 B to be drawn up. E: can now be found by applying rule 6, and by using this value in conjunction with Table 8.40B and the corresponding table of frequencies, the
384
HIGH RESOLUTION
NMR SPECTROSCOPY
remaining eigenvalues are deduced. Numerous sets of eigenvalues can be obtained by this procedure since the table of line frequencies can usually be set up in several different ways all of which satisfy rules 1, 2, 3 and 5. To apply the rules to a measured spectrum, the first step is to select the repeated spacings required by rule 3. Any missing or unresolved lines are given suitable values to satisfy rule 3 completely. At this stage the origin is adjusted to ensure that both sides of equation (8.178) vanish. A trial grouping is thus established using rule 2 to provide the combination lines. The 64 sums, Stjr, are next computed and the trial table is permutated so that rule 6 is satisfied in turn in each of the four ways possible under rule 5. A set of eigenvalues for each of these arrangements can be calculated by using rule 6 and making a comparison of Table 8.40n with this trial table of frequencies. Further sets of dgenvalues also result when the indices of E2, E3, E4 and Es, E6, E7 are permutated. Eventually one extracts 10 independent sets of eigenvalu~ by this procedure. Cavanaughc1"3~has presented a formulation for obtaining all 10 sets of eigenvalues from one particular grouping of the frequencies as indicated in Table 8.4013. Groups A, B and C each give rise to a quartet such that any pair of quartets possesses acommon splitting. The splitting common to groups A and B is designated a, that common to groups A and C designated b and that common to B and C designated c. If the mean frequencies of the groups A, B and C are designated as va, ~ a n d v, respectively (~a + v~ + ~c = 0) we obtain (v)a~ + (v)A,
=
(v)a2 + (~)A3
-O,)Ax
=
=
2~.
(0BI + ( 0 B . = (0B2 + (~)B3 = - ( ~ ) B ~ = 2,,. (~) C~ + (,) C , = (v) C~ + (,) C3 = - (~) C~ = 2 ~
Because the arrangement of frequencies shown in Table 8.40B is not unique there are 10 independent sets of eigenvalues and these can be evaluated as follows E1 = ~ ( a + b + c), E2 = - v , + ¼ ( a - b -
c),
E3 = -~'b + ¼ ( - a + b -
c),
E, = -*'a + ¼(--a -- b + e), E5 = v~ + ¼ ( a - b -
c),
En--- vb + ¼ ( - - a + b E~ = ~. + ¼ ( - a
c),
- b + c),
E. = ¼(a + b + c).
C1)
In (1), replace a by - a.
(2)
In (I), replace b by
b.
0)
In (1), replace c by - c.
(4)
-
THE ANALYSIS OF HIGH RESOLUTION SPECTRA
In (I), replace
385
b by vb - v. - ½(b + c), c bYcb -- ~'= + ½(b 4- c), vo by ½ [ - v¢ + ½(b - c)], vb by ½ [ - v~ - ½(b - c)].
In (5), replace
a by - a .
In (1), replace
a by v¢ - v, - ½(a + c),
(5) (6)
c b y v ¢ - Vo + ½(a + c),
• o b y ½ [ - Vb + ½(a -- c)], • , by ½ [ - ~,b - ½(a - c)].
(7)
In (7), replace
b by - b .
(s)
In (1), replace
aby~=
-
,b -
½(a + b),
b by v, - ,b + ½(a + b),
• b b y ½ [ - v ° + ½(a - b)], v, by ½[-,,~
In (9), replace
c by -
-
½(a -
6)].
c.
0) (I0)
The above f'onnuladon is completely independent of the choice of the original grouping. Calculation oJ ChemicaI Shlfts and Coupling Constants. If the diagonal matrix elements of the H~miltonian are known, it is a simple procedure to calculate values for the chemical shifts and coupling constants in the ABC systenL Examination of Table 8.41 shows that the parameters of the system can be expressed as follows 2 v . = Hs5 - H , ,
2vB = H ~ - H3~ 2v c = H ~ - - H = =
./^,,
= -(H33
+ H.
÷ H~s + Hee)
JAC = --(/']22 4- H44 4" H55 4- HT-/)
JBc ffi - ( H ~ 2 + H33 + H66 + H~7) Hence, the problem has now reduced itself to one of obtaining numerical values for the diagonalmatrixelements from the experimental oigenvalues of the system. Two of the diagonal matrix elements are already known to be equal to the corresponding eigenvalues themselves, namely H11 =~:1, H.
ffi E.
386
HIGH RESOLUTION NMR SPECTROSCOPY
Castellano and Waugh invoked the trace invariance property o f the Hamiltonian to calculate the following six independent relationships for the remaining diagonal matrix elements in terms of the six other eigenvalues: H22 =
-
1 ~ {e - [(2 - x/3) T+] t/2 cos( Z - zt/3)
-
[(2 +
x/a) I'-] '/= cos(~/
-
z~/3)}
H33 = - - ~ 1- { e + [(2 -- x/a) 2"+]1/2 cos z + [(2 + x/3) 1"-] ~/2 cos~/} 1
H , , = - -~-{e - [(2 - ~/3)1+] '/2 cos(z + zq3) - [(2 + x/a)T_] '/2 x
cos(~/ + zt/3)} l
Hss = - -~-(e + [(2 + x / 3 ) T+] I/2 cos(z + zc/3) + [(2 - x/a) T-] 1/2 x
cos(~/+ #/3)} I
H66 = - -~- (e - [(2 + x/a) 7'+] t/2 cos Z - [(2 - x/3) T-] t/2 cos~} Hv7 = - 1~" {e + [(2 + x/3) T+]I/2 cos( Z - ~/3) + [(2 - x/3) T_p/2 x
cos(rt - z~/3))
where T± are defined as T± = W~:(1 + x/3)/2 + W ± ( I - x/3)/2 - 3e 2 w.
=
+
+ E,
w_ =
+
+
The angle Z = 0 - 9, and 7/ = 0 + 9, and if values of 0 and 9 can be evaluated it is possible to calculate exact values for all the diagonal matrix elements, H22 to H77. These angles are defined as follows: Let
k = (1"+/I"_) 1:2
Q± = 7 4 ~/2,(T+ I / 2 / T ~ : )
=
- k ± 3 9_
N ± = R± - e(W~: - 2 W± + 9e2)/6 R+ = E 2 E a E .
R- = EsE6E7
P+ = 4- 6 ~/2[(1 + ~/3) N~ + (1 - ~/3) N+]/T~:/2 Finally we must introduce a fourth degree polynomial ,t
ft. + ~ " = 0 tl.O
(8.179)
THE ANALYSIS OF HIGH RESOLUTION SPECTRA where
387
13o = 4[k2p+2 + k - 2 p 2_ - (k + k-l) 2] fla = 4[k2P* Q- + k-2P-Q+ +'P÷ P- + 2 (k + k-t)] f12 = k2Q 2 + k-2Q2+ + 2(P+ Q+ + P_Q_) + (k + k-l) 2 - 4 fla = Q+ Q- - 2 ( k + k - ' )
fl,=l This equation can be solved exactly using Ferrari's method; alternatively the method of successive approximations can be adopted to reach a solution. By selecting one of the four roots of equation (8.179) the angles g and 9 can be defined = -- 2 cos2 9 (Q_ - Q + ) ~ 2 + 2 ~ + / z sin 30 = 4 s i n g ( k + k -1 - ~)2 where A = 2 ( P + - P - ) + ( k / z = 2 ( k -2 + 2 k - 1 -
-2 + 2 k - 1 -
1) Q. - ( k 2 + 2 k -
1) P _ - 2 ( k 2 + 2 k -
1) Q_
1) P÷
The above method will lead to real values of chemical shifts and coupling constants only if T+ > 0, and X and ~ are real. The final stage in the analysis is the application of the intensity sum rule to assign the correct set of parameters to the molecule under investigation. For some molecules it is difficult to make a final decision about the relative signs of the coupling constants, because the calculated intensities arising from various combinations of signs are accidentally very similar. In such cases, one can sometimes use empirical data found in analogous molecules to assist in the assignment. For example, a useful linear relationship between the sum of the three ABC coupling constants in a vinyl group and the Pauling electronegativity of the substituent has been pointed out by several workers (45' 59. ~) (see Fig. 10.16) and this relationship can be used to eliminate many of the possible sets of eigenvalues obtained from an ABC analysis. The 1H resonance spectrum of tetravinylsilanc has been analysed (6~) and a study of all the sets of eigenvalues for the molecule shows that there arc ten possible values of the sum of the three coupling constants, ~ J , namely 9.5, 10.7, 18.3, 20.1, 25.7, 38-5, 45-9, 47.9, 55.3 and 84.3 cycles sec -1. Examination of Fig. 10"16 shows that, if the correlation holds, then the only possible value of ~ J consistent with the electronegativity of silicon is ~ J = 38.5 cycles sec -~. Only the sets of eigenvalues leading to coupling constants which satisfy this requirement need be considered further. In some cases, where the tH resonance spectrum is consistent with more than one acceptable set of parameters, it is possible to decide the correct set of parameters by examining the taCH satellite spectra (~.2). This procedure has been applied successfully to the analysis of the spectra of tetravinylsilane and acrylonitrile. Sudhanshu's method of solving an A B e system (62). Sudhanshu has reported an exact method of solving an ABC type system which involves the use of Saltzer's method (~3) to solve the cubic equations obtained from expansion of 13.
388
HIGH
RESOLUTION
NMR SPECTROSCOPY
each of the two 3 x 3 sub-determinants. The same author has also made estimates of the line-widths of the components in an ABC spectrum.
8.15.5 The Four Spin System, ABX2 An ABX= analysis can be used to interpret the resonance spectra of molecules containing two magnetically non-equivalent, strongly-coupled magnetic nuclei (A, B) and two other magnetically equivalent nuclei (X) separated from A and B by a chemical shift large compared with the coupling constants involved. Typical examples of ABX2 systems are the hydrogen nuclei in 2,3-dichloropropene-I (e4), 2,3-dibromopropene-I ets), cis- and trans-l,3-dichloropropene-I ¢64, 65) (see Fig. 8.23). H
•
i I 37'5 39.41 4)~3 38,4 4 0 . 4
' 46.1
I 47.9
9'5.6 96"6
CyCleS sec-'
FIG. 8.23 The tH resonance spectrum (ABX2 type) of neat 2,3-dichloropropene-1 at 40 Mc sec-s. The frequencies were measured from the external reference benzene. Mortimertes)
An ABXz spectrum features three resonance frequencies and three coupling constants, namely, ~A, re, )'X and JAa, -TAx, Jax. The relative signs of .Tax and JAx (but not JAa) can be deduced from ass ABX2 spectrum. The waveftmctions for the system are based on a set of symmetzised basic product functions and they are given together with the diagonal matrix dements of the Hamfltonlan for the system in Table 8.42. The only non-zero off-diagonal elements are
~1o I * [
~11 = ½JA.
THE A N A L Y S I S OF H I G H R E S O L U T I O N S P E C T R A TABI~ 8.42 BASICPRODUCTFUNCTIONSAND THE DIAGONALMAIRIX ~
Total spin
Function
l omao I $
389
OF TB~
Diagonal elemmts*
$2
! ½(~^ + ~s) + ~.rA,,
1
]
~
~ l f f (~# + ~ )
1
s
1 s~
½(,'. + ~ ) + ~JAS
1
0
s s s
2s~ 3s, lso
½(.,~ - ~ + 2,,z) - ~.rA,, + b ~('A + ,,s - 2,x) + *.~AS -- d
0
s
2So
½("^ -- "S) -- ¼]AS
0
s
3So
½ ( - - ' A + , s ) - ~E.~A.
0
s
4So
½ ( - ~^ - ~s + 2~x) + } ] A S -- d
0
a
J ao
½('A - "s) - ~ ' A .
0
I
½ ( - ~,~ + ,~ + 2,,x) --'}.rAS -- b
1
~, ~ ire (~; + fl~) ]
1 1 a
2ao
½ ( - ~^ + , , ) - "}JAS
-1 -1
s s
ls_~ 2s_~
½(,~ - , , s - 2,,x) - ",t-rAS ½ ( - ~^ + '.,s - 2~x) - ¼]AS + b
-1
s
3s_~
½ ( - ~'. - ~
9,. PP ~ (~P - P~,)
-1
0
~-1
~,~ PPPP
-2
$
5_ 2
! + ~'^s
1
Z, - ½(J*x - .~sx)
d = ½(lAX + '~
* J x x valuta have not been included in the diagonal elements since they have no effect on the transition eneegies.
TABLE 8.43 EXPLICIT EN~OY EX~Um~ONS AND W x v ] ~ c - n o m FOR ABX~ S~s~e~ Eigmtunction
Ys
tp7 Ys ~0~o ~olx t/~12 ~o13
cos01 Q~4+ sine1 ~s -sinOz Q~4+ cos0z Q~s cose~ ~ + sin0~ 9's sinSl qP7+ cosel ~Ps cose= ~1o + sinS: ~ x - s i n 0 = 9,~o + cos0= ~11 cos03 ~;= + sin03 ~
FAgmvalm
-
- sin0s ~o1= + cos03 ~:3
-,'x+ ¢3 -¼&m - , ~ - c ~ - ¼.rA~
390
HIGH
RESOLUTION
NMR
SPECTROSCOPY
The remaining stationary state wavefunetions are given by the non-mixed symmetrised spin functions i.e. *Pl = 9~, ~P2 = 92, ~v3 = 93, ~P6 = 96 etc. The constants used in Tables 8.43 and 8.44 are defined as follows: VO D I = - ~ - ( o ' A - - O'B) - - b =
Ct
= (D~ +
- ½ 7 ' 0 (~AB -- b
¼J~n) ':2
,ro
D,. = "7 =
(,r:, -
=
- ½ ,,o ,,
+
VO
Ds = - ~ - - ( a A - - a a ) + b = --½~'o ~AS + b
C3 =
(D]
+
¼ J~r~)'/2
TABLE 8.44 Tile ENERGIES AND RELATIVE INTENSITIES OF THE TRANSITIONS IN AN ABX SYSTEM Transition
~P16''~ ~t3
[ ~ls ~ 'eft I I 'Pl, ~" 'Ps / [ tP7 "4" 'tP3 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
~P9 ~
~Ps
/Px6 "-*/Pt2 ~ t 3 ~ ~6
[ ~'ts "-"v:'to [ ~Pt,t ~ 'P7 [ ~Pt t -" 'P2
~s ~
~t
v:t2 ~
~s
~P7 ~
Origin
~Ps
~ s ~ ~P5
Relative intensity
B B
K- C3--½JAn--d K-- C3+{JAn-d
I - ½:AdC3
B
K-
C2 - ½JAB
2 - JAslC2
i + ½J~BIC~
B
K-
Cz + }JAn
2 + JAs/C2
B B A A
KK-
Ct-½JAn+d CI + ½JAa + d
1 - ½JAS/G
K + C3 - ½JAn -- d
1 + ½J,,8/cs
K + C3 + ½J A n - d
1
A
K + C2 - ½dAn
2 + ]As~C2
A
g + C 2 + ½JAn
2 -- . 6 , d C 2
K + K + LL-
1 + ½JAn/Cl 1 - ½.&dc~ 1-N I-M
A A Comb. X
{ 7Pi6~ t/",4. }
Energy
X X Comb. Comb. X
[ 'P6 = q'3 }
X
?,013~ 'f~7
X Comb.
CI -- ½JAB + d Ct + ½JAB + d C3-Cz C2-CI
I + ½J^dCx -
½1AdCs
4
L-d
L + C3-- C2
I+N I+M I+M I+N
L+d
4
L + Cz + Ct L + Ca + C2
I-M 1-N
L-
C3 + C 2
tCz + Ct L + C 2 - Cx
K = 1~(vA + vn)
M=
L = PK
N~
D 2 D t 4- ¼J.~B
C2 Cl. Dz Ds ~_ , ± 4 J , ~2B c z c~
T H E A N A L Y S I S OF H I G H R E S O L U T I O N S P E C T R A
391
and cos 0t sin01 ffi ¼JA,/CI cos20t =½(Dl/Ct + 1) where
i-
1,2,3
~Ae --ere - ~ A since all those involving JAx and Jex can be neglected, due to these coupling constants being small compared to-%~Ax and vo~ex respectively. Hence the diagonal matrix elements for all functions other than •4, ~5, ~7, ~., f l o , ~:x, ~0:2 and ~:3 are cigenvalues of the system. The remaining eigenvalues, calculated for the mixed wavefunctions, are given in Table 8.43. Table 8.44 lists the transition energies and relative intensities for the ABX2 system: there are 28 allowed transitions in all, 12 of them being doubly degenerate. A new notation for the spin functions is used for these systems. The symbols s and a are used to indicate whether or not the symmetrised spin function is symmetrical or antisymmetrical under the operation of a reflection in the plane of symmetry of the system. The suffix gives the total lz component: thus So refers to a symmetric spin function with a value of total Iz of zero. If there are two spin functions with the same symmetry and the same lz value they are distinguished from each other by numbers preceding the symbol e.g. lso, 2so, 3so, 4so. If the symmetrised spin functions are not the stationary state wavefunctions, the mixed functions which form the stationary states are denoted by a similar notation but differ in that they are primed e.g. 3s~, la~ and 2a~ etc. For the purpose of assigning observed frequencies to theoretical transition energies, it is convenient to arbitrarily set the values of L and K to zero (the parameter L - K does not affect the form of an ABX2 spectrum). The A transitions 3 and 4 and the B transitions 9 and 10 constitute an AB type spectrum whichis independent of the two X nuclei. If observed bands can be assigned to these transitions, then possible values of K, %~Ae and JAe are obtainable. From the completely symmetrical X2 spectrum, values of L and d are found by a trial assignment of the strong lines, transitions 15 and 20. If the assigned values of *o~Ae, JAe and d are permissible, a value for b can be obtained by plotting possible values of this parameter against (C2 - C:) and against (C3 - C1). A value of b is chosen which gives the best fit of (C2 - Cx) and (C3 - CI) to the observed frequencies tentatively assigned to transitions 16, 17, 18 and 19. Finally, a theoretical spectrum, constructed using a set of calculated parameters, is compared with the observed spectrum.
8.15.6 The Four Spin System, ABC2 The di- and tri.substituted propenes previously considered ¢18. e,. es) only approximate to ABX2 systems, and to achieve precise values of the parameters of their spectra it is necessary to conduct a full ABC2 type analysis. Such an analysis provides in favourable circumstances the relative signs of all three coupling constants. There are 34 possible transitions for an ABC2 system, six of them being weak combination transitions involving the C nuclei. In the ABX2 system these six transitions have zero intensity. The spin functions and the diagonal elements for the system are the same as those given in Table 8.42 for the ABX2 system. We
392
HIGH
RESOLUTION
NMR SPECTROSCOPY
can no longer ignore off-diagonal elements involving JAx and Jax and a complete list of all the non-zero off-diagonal elements is given in Table 8.45. Explicit algebraic expressions for the eigenvalues cannot be written down and it is necessary to use an iterative process to modify trial values of the matrix elements until convergent solutions of the determinantal equation consistent with the observed eigenvalues are obtainedct3s). An electronic computer is used for this purpose ce'). T~mf~ 8.45 O ~ - D ~ O O N ~ Eia~n~wrs voR T~m A B C 2 Srn~ $Yurv~ 1
~,
I,~l~
- o 1
1 v,,,. l,,'e'l v,, = -y,z.,,,, 1
1
I J,
I 1
I 1
Mortimerc6s) has successfully analysed the tH resonance spectra of e/sand trans-2,3,-dichloropropene-1 and similar disubstituted propenes in terms of a slightly perturbed ABX2 system.
8.15.7 The Six Spin System, A3B2C The A3B2C system often occurs in ethyl derivatives, for example in ethyl mercaptan and ethyl acetylene, but the analysis is much more complicated than for the A3B2X system since the secular equation can only be factorised into equations of the sixth order and less. Naxasimhan and Rogers c66) have made a full analysis of the former system: the results of their calculation for the matrix elements of the "complex particle" sub-matrices axe shown in Table 8.46. The eigenvalues and eigenvectors are obtained from these sub-matrices by numerical computation. 8.16 SPECTRA CHA~umO~D BY FOUR COUPLING CONSTAHTS
8.16.1 The Four Spin System, AA' XX' In principle, six coupling constants charactefise this system but the impficit symmetry of the molecule makes two sets equivalent. McConnell, McLean and Reilly" ' ) have interpreted the XH and the tgF F: ~HH, in terms of an resonance spectra of 1,1-difluoroethylene, F~C--C
393
THE ANALYSIS OF HIGH RESOLUTION SPECTRA T,,m~ 8.46 Ma1"Jt~ ~
s. j 3
m~. Tm~ AsB=C Non-z~ro elen~nts
Diagonal ~ t s
QTD
D0~e + ~SAC + S A c
QTD
H . ~ : - - 2 . , , + 5,'0 6 . . -- 5,'o ~ c + ~.~.e --¼.r.c - ½JBc
/ / 1 . - H.x - 5 Y2.r.,c
H =. = -- 2,'. + ½,'o 6A.. + 5"0 ~ C + t 3 * c s ~ , : - 2 . . + ~,'o 6 . , + ½,o e~c + ½~,, +¼.rAc + ½Ac
//1_. ,= Hsl = ½ ~'3JaC H2s = H32 = ½ f 6 G e
H;= =/-/2; = H.s - S ~
+ :}3.c - ½ A c QTD
= 5 Y6JAa
//22 = - ,'. + ½,'o'~.. + 5"o ~ c + ¼-~.¢ - - ' A + ~,'06.,, + 5"0~,C -- 5 3 - . -~^c + SAc H,.-
-,'.
-
5,0 6AS -- ½"o
~c -
H , . - ~r4. - H=, Hs=- 5 Y21,,
~Ac
Hss-= - ,'. + 5,'06A~, -- 5"0 ~,C + 5 . V . . -¼ 3 . c - 53~c
H=s - Hs= -
H.
H,~ - - 5 , ' o 6A., + ½"O~C -- 5.r,.- + ¼.r.c -½3,,c
0
QTD
H.2 -- 5,'0 6A.. + ½,'0 ~ . c -- ¼.IAC S - S -- 1,'O 6A,. + 5 " 0 ~ C -- ~.r.,, -- i.r.C + 5.r,c s. - ~ , o 6 . . - 5 , o ~ c - ~ . ~ - t~AC + ½Ac s. = - 5 " o 6A. - ½"o 6 . c - ¼.I.c //66 = 5,'0 ~^- - ½,'o 6 ec - 5.I.~. + ¼.r.,c -
QSD
S,
QSD
H11 = - ' .
- 5"o 6Ae -- 5,'0 8,C -- ~J*c
H22 = - v .
+ ½"o 6AS +.½~o 6.C + ¼JAC
13,
- H,s - J.c
Itx= . - H = I -
He,-
Hse "
Y2J,,
H.. - S.l =Sae He~ - 5 YS~,c
HIs - H . I - H=e He= = 5 Y2.I,,c H=3 = H32 - H~, -S s . = 5 Y6~A. H=5 = Ss= = .I.c
½.,'.c
H]~B.
Y2~A,
= - 2 , , . + 5,'o 6A, + t ~ , c
none
H;= - H=. - 5 ] / 3 I . c
394
H I G H R E S O L U T I O N NMR S P E C T R O S C O P Y TAeLe 8.4¢~ (continued)
Is
State
0
QSD
Non-zero off-diagonal elements
Diagonal elements
/'/12 = Hzt
Htx ~ --½~O~AB -- ~o0~nC -- ¼JAC
=
JAC
/'/'22 = ½VO ~AB + "l£t'O~BC -- ¼JAC
2
1
DTD
Httffi-2v.+
2~v0~xs+½VoCSnc+-~JAB+
DTD
HI1 ~ - v ~ + JVo ~,,B + ½~'o O,c - ½.r^B ¼J^c + ½J-c H22 - v~ + ½Vo ~..,, - ½re ~Bc + ½JAB -
none
Ht2 = / ' / 2 t = ½ J , c H~3 = H3~ = ~ ¢2JAB
¼]Ac - ½J'Bc H23 ffi H3~ ffi ½ ¢2JBc
/'/'33
0
DTD
H t l = ½~'o ~....* - ~Vo 6 e c - S JAB + ¼JAc ½1Be 1/22 :=~ ½"B OAB + ½~0~BC-- ¼~^e H~3 -½~o ~ +
~o ~ec-~J.B
=½V2~c
+~J.c-½/Bc
1
DSD
//1 t = - vA + ½=o ~AB+ ½re ~Bc + ¼J^c
none
0
DSD
H1, = ½re ~AB + ½~o ~Bc -- ¼JAc
/'/12 = H2t = ½JAc
t t z z ffi - ½ V o ~As - ½Vo ~ . c - ¼J'^c
Note that the D T D and DSD states are both doubly degenerate so that the relative intensities of transitions within these states have to be multiplied by two.
A A ' X X ' analysis. I n a n A A ' X X ' system all f o u r nuclei are m a g n e t i c a l l y n o n equivalent a n d f o r m two pairs o f s y m m e t r i c a l l y e q u i v a l e n t nuclei s e p a r a t e d f r o m each o t h e r b y a c h e m i c a l shift large c o m p a r e d with the c o u p l i n g c o n s t a n t s involved. A n A A ' X X ' s p e c t r u m c a n be e x p l a i n e d in t e r m s o f t w o r e s o n a n c e frequencies ~A a n d ~'x a n d f o u r c o u p l i n g c o n s t a n t s , JAA', J x x ' , JAx a n d J ^ x o as i n d i c a t e d d i a g r a m m a t i c a l l y in Fig. 8.24. VA
'fAX
A ........ JAA'
~X
""J^x' A'
K = JAA" + J × x '
X
L = JAX -- J A x ' 'fxx"
"'X'
M = J^^. - Jxx' N = JAx + J ^ x '
FIG. 8.24 The Coupling Constants and Resonance Frequencies in an A A ' X X' system
THE ANALYSIS OF HIGH RESOLUTION SPECTRA
395
The Hamiltonian for the system is given by
a" = ~^H~(I~(A) + ~,(A')) + ~H~(~(X) + I,(X')) + ~ ( , ~ )
-I(X)
+ J ^ x . l ( A ) " I(X') + J ^ . x l ( A ' ) " I(X) + dA.x.l(A')" I(X') + JA^.I(A)" I(A') + dxx.l(X) " I(X') where
d~x - d^,x.
J^x,
--
~'^,x
due to the symmetry of the system. The appropriate symmetrised spin functions for the system and the matrix elements of the Hsmiltonian are presented in Table 8.47. The secular determinant associated with the Hamiltonian matrix factorises into two 1 × 1, five 2 × 2 and one 4 × 4 determinants. Further factorisation occurs when T~
8.47 S Y ~ ' r n ~ D Srm FUNCtiONSAND MATRIXE ~ a ~ z r s FoR AN AA' X X ' SYs'r~M
]~lnc-
X
A
tion
tX0~
2-ff~p + ~o,)
0~(X
~0~
2-t(o,~ + p~)
pa ~5
2-ff~,~ - p~) 2-~(,,a + ao,) 2-ffo, p + ~o,) ~9 Pzo P11 P12 PlS P,~,,
2-~(ap o,o, 2-½(o,~ + 2-~(o,p 2-~(,,p -
PP
2-ff,,~ - Pc0
~P 2-ff~,p + ~,,) P~
I Nota- Diagonal matrix I tion elements * I s2 lsa
2zl
~+
~A . lso --VA +. ~.-½~V •~ - ½ N 2$o " --K 340 0 4ao ls_x VB 25_1 VB $-2
2-t(,,p - ~ ) 2-+(o,p - po,) 2-ffo,~ + p~) ~P 2-ffo,p - ~o,)
2a-l
n~A
po,) p,,) ~)
-½L
-,,+½N vB--
¢X0~
~,+½N
~B
lal 2a~ lao 2ao la-I
a~,)
Off-diagonal elements
½(K + M) • A - ½(K- M) } ~13[X'[~. = - ½ ( K - M) -½(K + M) -½L -~.½(K+ M) --
½(K- M)
* For convenience,¼Khas been subtracted from all diagonal matrix elements. This makes no difference to calculated transition energies.
several of the off-diagonal elements (those involving J^ x and dx x,) are approximated to zero because they are small compared with ~'o~^x. The matrix now factorises into twelve 1 x 1 and two 2 × 2 sub-matrices. Hence, the mixing which occurs is between basic spin functions ~6 and ~7 and between functions 923 and ~ 4 only: for all other functions the stationary state wavefunctions are giwn by the symmetrised spin functions listed in Table 8.47. The mixed
HIGH RESOLUTION NMR
396
SPECTROSCOPY
wavefunctions and their corresponding energy levels can be obtained from solution of the 2 x 2 sub-determinants. AU other energy levels in the system are given by the unperturbed diagonal elements of the matrix. The frequendes of the allowed transitions, and their relative intensities, can now be calculated and explicit expressions for these are presented in Table 8.48. One needs to consider only the A transitions since the X part of the spectrum is identical to the A part. In each half of the spectrum there are 12 possible transitions (24 in all), but four of them in each case are doubly degenerate so that only 10 distinct transitions can be observed (20 in all) as shown in Fig. 8.25. X part of lhe spectrump
1,2
5,4
9
5
6
~2
I1
I0
I I
F~o. 8.25 Typical A spectrum for an A A ' X X ' system with JAX > "fAX' > 0 and Jxx' :~ ~ A ' > 0. Pople, Schndder and" Bcrnstein(~7)
TABLE 8,48 TRANSITION EN~zenm eoR AN A A ' X X ' SYSTEM FOR ~ SPECTRUM
No. I 2 3 4 5 6 7 8 9 l0 L1 L2
Transition 1 $1 -'~ s= lso "~ Yzo "~ Ys Ys -*" Ys ~9 -~tP7
¥9 "*"~6
Is_1-~ 3 eo 2L, 4s o 2s-1 2% 2a_l
- , 2al "~ -,- 2sl -~ 3s; "-* 2ao
la~
la;
2at 2 a _ 1 ~ 2,,;
}
}
Transition energy
Relative intensity
~zN
2
-{N
2
½ g + ½(K = + L~)* - ~ K + ~ ( K = + Ls)~
sin = 0s COS=O,
½K - ½(K~ + LS)~r -½K - ½(K= + L=)* ½M + ½(M2 + L=)*
COS20s
--~-M + ½(M=+ Ls)~
COS2 0a COS2 0 a
½M - ~(M 2 + L~)* -½M - ½(M2 + L=)~
Energies are m c a s u r ~ relative to ~'AK, L, M, N are defined as follows f f = JAA' + JXX' L = ,TAx -- ,TAX'
A A ' PORTION OF
M = /^A' --/XX' N = l a x + JAx' and cos20,: sin20s: 1 = K : L : ( K 2 + Lz)½ cos20,: sin20a: 1 ~ M : L : (M z + Lz)~
sin=O, sin=Oa
sina0.
397
THE ANALYSIS OF HIGH RESOLUTION SPECTRA
Examination of Table 8.48 shows that the A part of an AA'XX' spectrum will always contain: (i) two strong absorption bands (transitions 1, 2 and 3, 4) separated by N -JAx + "/AX'and centred on the resonance frequency of the A nuclei, *O~A; (ii) twO pairs of symmetrical quartets (transitions 5, 6, 7, 8 and transitions 9, 10, 11, 12). Both quartets are centred on the resonance frequency of the A nuclei, and the components of each quartet nearest the centre of the spectrum are always the most intense. Figure 8.25 shows a typical theoretical spectntm of the A part of an AA'XX' spectrum which illustrates the above features diagrammatically.
3,4 ,/~ 7,12 40
-
r 80
1,2 9,6 / ' ~ I
]It--
120
"
I--
, 200
160
Cyctes sec"
FIo. 8.26 The 19F spectrum o f 1,1-difluorocthylene at 40-00 Mc sec -1. McConneU, McLean and Reilly (ldu)
H~"~.C
"C/F
1,2 3,4
6
7
9
12
f
]
20 cycles sec -~
FIG. 8.27 The 1H spectrum of 1,1-difluorocthylene at 60.00 Mc sec-:. Roberts (4)
398
HIGH
RESOLUTION
NMR
SPECTROSCOPY
The intense transitions 1,2 and 3,4 correspond to the degenerate pairs of transitions and they contain half the intensityof the complete multiplet. McConncLl, McLean and Reilly¢~4)have measured the IH and the tgF resonanc~ spectra of 1,l-difluoroethylcne: reproductions of spectra axe shown in Figs. 8.26 and 8.27. In each case only eight of the ten theoreticallypossible absorption frequencies are observed. The tgF resonance spectrum consists of four intense bands and four weaker satellitebands. Transitions 6 and 9 and transitions 7 and 12 have been assigned to the two central bands of the four intense inner bands and the remaining two bands to the strong transitions 1,2 and 3,4. The other four transitions are assigned as shown in Fig. 8.26. This assignment leads to the following set of coupling constants for the molecule JAA' = JHH' ~' 4 cycles sec-I
Jxx' = JFF" = 37 cycles Sec -~ JA x = J ~
~ 1 cycles sec-I
JAx' = J~"* ---- 34 cycles sec -1 (more recent values (t4°) are 4'8, 36"4, + 0.7 and + 33"9 cycles see-l). At the time it was guessed that J ~ " > J ~ : it has since been confirmed that this is so (see Section 11.11.3). The resonance frequencies for the A and X nuclei arc given by the geometrical centre of the respective multiplets (for exampie the mean of subtracting transitions 1 and 3). Values o f the coupling constants are found from values of K, L, M, N which are obtained from suitable subtractions of the observed transition energies: the absolute signs of K, L, M, N cannot b6 extracted from the spectrum and hence only the relative signs of in, and J ~ are attainable from the analysis. However, Flynn and j~l~ Baldeschwieler c~4°) have shown that the relative line widths in the ~H resonance spectrum of gaseous CHzCFz can be used to identify the origin of the lines. The identification of an AA'XX' spectrum allows IK[ and [MI to be distingnished thus giving the relative signs of Jail and JF~. Similar experiments have been made on cis. and trans- C H F C H F : J a a and JFF are of opposite sign in these compounds (~s°).
Ortho-dichloroperfluorobenzene. An excellent example of an AA'XX' system is found in the molecule 1,2-dichloro-3,4,S,6-tetrafluorobenzene. Figure 8.28 shows the zgF resonance spectrum of the AA' fluorine nuclei in this molecule with the band assignments shown on the spectrum. All ten theoretically possible transitions give rise to resolvable absorption bands and analysis of the complete spectrum gives the following set of parameters for the system l a x = +_ 20.8 cycles sec -~ lAX, = -T- 1"7 cycles sec -~ [JAA" 1 =
7"8 cycles SeC-
l/xx.l ---- 19"9 cycles sec-l vo ~ = If00 cycles sec -~ (at 56.4 Mc sec -l)
T H E A N A L Y S I S OF H I G H R E S O L U T I O N S P E C T R A
399
~,4 t~
CL
Fx .
I0 cycles sec -~
8
Fio. 8.28 The low fieldpart (AA' fluorinenuclei) of the XgF resonancespectrum of neat 1,2,-dichloro-~,4,5,6.tetrafluorobenzeneat 56.4 Mc sec-x
8.16.2 The Four Spin System, AA'BB' A molecule containing four magnetically non-equivalent nuclei forming two different pairs of symmetrically equivalent nuclei separated from each other by a chemical shift which is of the same order of magnitude as the coupling constants involved is described as an AA'BB' system. Numerous examples of this system have been investigated by NMR and they include disubstituted aromatic compounds, thiopbene, furan, monosubstituted pyridines, 1,2-disubstituted ethanes and disubstituted cyclohexadienes as indicated in Fig. 8.29. In all the molecules listed in Fig. 8.29 the four nuclei are magnetically nonequivalent because
J.A'x~ , # JHA'"~" The symmetrised spin functions and the diagonal matrix elements are the same as for the A A ' X X " system and they are listed in Table 8.49. Also given in Table 8.49 are the non-zero off-diagonal elements for the system: it is seen that the only non-mixing spin-functions are s2 and s_ 2 corresponding to the functions o~0~ and ~ / ~ respectively. Thus the spin functions s~ and s_ 2 are already stationary state wavefunctions and the corresponding eigenvalues are given
400
HIGH
RESOLUTION
NMR SPECTROSCOPY
e-
,.*,fig o~
r~ Z
II
II II II
~-
II
t
Ill
II
Z
=
O
I
ilii
E
o
+
~+
+
If+
I
Z
Z
E 0
+
I÷+
I
÷I
I
J
THE A N A L Y S I S OF H I G H R E S O L U T I O N S P E C T R A
R
R
i T~,-~,O;~,,,
B
(i)
40]
(ii)
(iii)
e,,~s?e A.
OC'H,CH,e,
(v)
(vi)
(iv)
R is a n o n - m a g n e t i c s u b s t i t u e n t .
(i) 1,2-Disubstitutedbcnzenes (two identical substituents)
(ii) 1,4-Disubstitutedbenzenes (two differentsubstituents) (iii)4-Monosubstituted pyridines (iv) Furan (v) Thiophen¢ (vi) 1-Chloro-2-bromoethane
Fro. 8.29 Molecules which are typical AAtBB ' systems
TAme 8.50 EXl,L~crr E~rzeoY Ex~mesmo~ AND WAVmn~c-noNs FOR 12 I~.NJ~Oy Energy s2 1st' 2$1'
25_1 ~ $-2
1 ax ~ 2ax' I ao ~ 2o0' 1a': 2a~-x
"A+'~+½N ½('A + ~ ) -- ½(('Oaf + ~Vt)½
½('A + '~) + ½(('Oaf + JVt)½ --½('A + "B) + ½(('Oaf + ~V2)t --½('A + ~ -- ½(('Oaf + Art)½ --~A -- "~ + ½N ½('A + ~ ) -- ½ X 3 ( , . + ,~) - ½K + - ½K + - ½X--
½{('oa + M ~ + Lt} t ½((,oa + ~ 0 t + L t ) t ½ ( M t + Lt)½ ½ ( M t + Lt)½
- ½ ( , ~ + ~ - ½ K + ½{(,oa - M ) 2 + Lt}½ - ½(",, + ' ~ - ½ X - - ½{(~oa - M ) t + Lt} ½
16 A A ' B B '
Wavefunction (St) ( l S i ) CO~V -- (2S,) ~ n e (ls~) sin~ + (2s~) cose (ls_l) cose + (2s_~) sine - (lS_l) sine + (2s_~) cose (s_t) (1 a:) cosy+ + (2al) sin~o+ - (1 al) sin~+ + (2al) cosy+ (1 ao) cos0o - (200) sin0, (lao) sin0, + (2ao)cosO,, ( l a _ l ) cosy_ - (2a_1) ldnv_ ( l a _ l ) sine_ + (2a_l)cosv_
The angles e, ~p+. 0s and O, am defined by
and
oe T i m
cos2~ : sin2~ : 1 - ~o~ : N : ((,oo')2 + Nt)~ cos20, : sin20, : 1 - K : L : (K 2 + Lt)½ cos20, : sin20a : 1 -= M L : ( M t-I- Lt)½ cos2~+ : sin21p_+ : 1 : ,06 ~ M: L : ( ( ( ' o ~ ± Air)t + Lt} ½ 6 =aB -a^.
402
HIGH
RESOLUTION
NMR SPECTROSCOPY
by the diagonal matrix elements. The 16 x 16 matrix representing the Hamiltonian for the system can be factorised into two 1 x 1, five 2 x 2 and one 4 x 4 sub-matrices. Although one can calculate the roots of the five 2 x 2 secular determinants and express them in explicit algebraic form, the 4 x 4 sub-determinant cannot be solved in this way (it involves the solution of a fourth power equation). Hence, the only explicit eigenvalue expressions and wavefunctions for the A A ' B B ' system are those given in TaMe 8.50. A consequence of this is that explicit expressions for six of the allowed transitions and their relative intensities are not available (all those involving lsh, 2s~, 3sh and 4s~ wavefunctions). Table 8.51A lists the transition frequencies and relative intensities for all the other transitions. Altogether, there are 28 transitions, four of them being combination transitions of very weak intensity which are unlikely to be observed in practice and are therefore not included in Table 8.51 A. Moreover,
TABLE 8.51 A TRANSITION ENERGIES AND RELATIVE INTEN~FI1~ OF A TRANSITIONS FOR
AA'BB' SYSTV.M Transition
4 5 6 7 8 9 10 11 12
l S1 t ~ S 2 1SO' "~ l s x ' S_2 ~ l s _ t ' Is_x' --~ 2So' 3So'
Transition energies relative to ~r(~A + ~e)
Relative intensity
½N+ ]r{(Vo~)z + N2}÷ Not obtainable
1 - sin2q0
- ½ N ' + ~{(Vo~)z + NZ}+
1 + sin2~
Not obtainable
-'~ 2 s t '
2s_t' ~ 4So' [ 4So' ~ 2st' I 2s_t' -+ 3So' J 2ao' ~ 2at' 2a_l '--~ lao' la o' ~ 2a t' 2a_l '--~ 2no'
Explicit algebraic e~xprmsionsfor the~ transitions are not obtainable ½{(rod + M) z + L2}÷ + {(M z + L2)~ {{(Vo8 - M) z + LZ}~r+ ½(Mz + L2)~ ½{(%~ + M ) 2 + L2}~ - { ( M z + L2)~ ~{(POB -- 1~4)2 + Z2}~ -- ~r(M2 + L2)+
sin2 (~, - 'e+) cos2 (0, + v/_) cos2(0, - ~÷) sin2 (0, + ~_)
the A region of the spectrum will always be a mirror image of the B part and consequently only the twelve A transitions need be considered. It should be noted that there are no degenerate energy levels as were found in the A.A'XX' system. The complex relationship between the form of an A A ' B B ' spectrum and the chemical shift and coupling constant parameters leads to the analysis of a spectrum of this type being very difficult. Because of this, several rather indirect methods of analysis have been adopted. F o r example, one can construct a general theoretical spectrum for the system under consideration assuming the chemical shift difference, %~, to be large (that is an A A ' X X ' system) and then, by progressively decreasing the chemical shift difference, the modification in the line positions for each new value of vo3 can be followed. This information enables one to assign experimental bands to
THE ANALYSIS OF H I G H RESOLUTION SPECTRA
their correct transitions ff the original theoretical AA'XX' spectrum has been constructed using coupling constants not too dissimilar to those in the molecule being examined. When an assignment has been made, the trial values of the parameters are successively modified until a theoretical spectrum in good agreement with the observed spectrum is produced. That the assignment is correct can be confirmed by a series of internal checks on the bands of the spectrum. The problem of choosing approximate values of coupling constants is overcome by examining the spectra of similar bnt more simple compounds of the type being studied. One can make the valid assumption that providing no drastic changes in hybridisation occur on going from the simple molecules to those being analysed then the coupling constants will remain fairly constant. Thus in aromatic molecules the H - H spin-spin coupling constants between atoms in different ring positions are given approximately by the values in Fig. 8.30 regardless of the nature of the substituents.
J° ~ 8"0 cycles sec-I J " ~ 2"0 cycles sec -1 JO < 0"5 cycles sec -1
FIG. 8.30 The H - H Coupling Constants in an Aromatic Ring
If we consider the case of an orthodisubstitutedbenzene (shown in Fig. 8.29 (i)) then from a knowledge of H - H spin-spin coupling it is probable that J ^ s >> JAs. > 0 JBs, >> J^A' > 0. Assuming all the coupling constants to be positive, the A portion of a theoretical AA'XX' spectrum consistent with the above features can be constructed (see Fig. 8.31) and compared with a similar spectrum for an analogous AA'BB' system. We have already discussed the spectrum of the AA' nuclei in an AA'XX' system (Section 8.16.1). If JxA. = 0, then K = - M and the four pairs of transitions (5, 10), (9, 6), (7, 12) and (11, 8) become degenerate and each pair gives rise to a single absorption band: there will then be 6 bands in all in the A spectrum. When the chemical shift difference decreases to the point where the system becomes AA'BB' in type, the A part of the spectrum is modified as shown in
404
H I G H R E S O L U T I O N NMR S P E C T R O S C O P Y
Fig. 8.31 Co). The components of the A multiplet nearest the B spectrum gains in intensity at the expense of the low field part and the transitions 1,2 and 3,4 axe no longer degenerate and give rise to two pairs of resolvable absorption bands in the AA'BB' spectrum. Also, the separations between transitions (9,6) and (7,12) axe increased in the AA'BB' spectrum. It is possible for the bands of the outermost pair of transitions (11, 8) to overlap the corresponding B part of the spectrum but the more intense bands for transitions (3, 4) never cross the bands for the corresponding B transitions. In the actual analysis of a spectrum which approximates to this particular case of an AA'BB' system one can assign the intense transitions (1,2) and (3,4) fairly easily. Lines 1 and 3 can then be manipulated to find the chemical shift of the A nucleus and the parameter N = J^s + ./As. N = v, -
vz
(8.180)
{(~o ~)z + Nz}l/2 = h + ~3.
(8.181)
Lines 9 and 11 can be used to find two of the other parameters
(8.182) (8.183)
( M 2 + L 2 ) 1/2 = ~9 -- ~11 {0'0 ~ Jr M ) 2 + L2} I/2 -- '['9 "31-~"11.
X Part of S~"~'C t rIJ m
',2
3,4
(a) 5 10
,i 8
i
II
i
B ~rt 34
of s p e c t r u m =
~)
Ill
I
II
FIe. 8.31 (a) Theoretical spectrum for A A I nuclei in an A A ' X X ' system having YAx ) ,TAx' > 0 and Jxx' > JAA' > 0 (b) theoretical spectrum for A A ' nuclei in an A A ' BB' system having JAn :~ JAB* > 0 and Jan' > JAA' > O. Bernstein, Pople and Schneider (e~)
A value for K can be obtained from transitions 6 and 7 by an indirect process involving trial and error fitting. If values for K, L, M and N are known then the complete set of spectral parameters for the molecule can be found. Relative signs o f coupling constants in an A A ' B B ' system. Grant, Hirst and Gutowsky(6a) have considered in detail the effects on AA'BB' type spectraofthe relative signs of the four coupling constants. It can be shown that no informa-
THE
ANALYSIS
OF
HIGH
RESOLUTION
SPECTRA
405
tion on the signs of Land M e a n be obtained from the spectrum since an AA'BB' spectrum is independent of the relative signs of these parameters. However, the spectrum is sensitive to the relative signs of K and N, but not to those of either M or L with respect to K and N. When the spectrum is analysed to give the absolute values of K, N, L and M and the sign of K relative to N, it is possible to obtain values of the four coupling constants and also all their relative signs. It should be noted that from an AA'BB' spectral analysis one cannot differentiate between JAA'and J n , nor between JA~ and JAw. To assign these coupling constants to particular pairs of nuclei in the molecule it is necessary to compare the observed coupling constants with values obtained from independent measurements of similarly substituted compounds where the assign. ments are unambiguous. Thus, Gutowsky and co-workers(68) have been able H
I¢
AL' P0 rt
Po r'~ A
r
?,.4
1,2
I T
I r
13.710-28.2
tl 3-70-~
6-4
3.76.4 z 0.8
Cycles
10.2 15.7 ~'2
sec " l
1=Io.8.32 The IH resonance spectrum at 40 Mc sec-I of naphthalene in dioxane. Pople, Schneider and Bernstein(GsT) to show that all four H H coupling constants in thiophene have the same relative sign, that J12 and Jxs in furan have the same sign, and that r~]BUff ~ o s, r,,,,* '~ ] B H a and jv~=oin several disubstituted benzenes have the same sign. Hutton and Schaefer (t49) have calculated theoretical spectra for l,l-disubstituted cyclopropanes. Ortho-disubstituted benzenesmNaphthalene. A typical AA'BB' spectrum of the type we have been discussing is given by the ring hydrogen atoms of naphthalend ~7) HA HA,
The two aromatic rings are identical and the IH spectrum can be considered to arise Rom the four ring hydrogen atoms in either aromatic ring. There is no
406
HIGH R E S O L U T I O N
NMR
SPECTROSCOPY
coupling between hydrogen nuclei in different rings. Figure 8.32 shows the ~H resonance spectrum of naphthalene measured in dioxan¢ solution at 40.00 M c sec -x and it can be seen to be completely symmetrical about the centre of the spectrum. The assignments of the various bands to the transition frequencies are achieved by comparing the observed spectrum with the theoretical spectrum shown in Fig. 8.31 which is known to be typical for an ortho-disubsfitutedaromatic molecule. The assignments are shown in Fig. 8.32 and in Table 8.51n. Although examination of the spectrum cannot predict which of the nuclei is the more shielded, the low field part of the spectrum (the A part) has been assigned to the o~-hydrogen nuclei on the basis of empirical considerations. There are fewer observed bands than predicted for the system and it is necessary to postulate that several of the doublets in the theoretical spectrum (see Fig. 8.31 (13))have coalesced. Based on the assignments given in Fig. 8.32 for the A part of the spectrum one can calculate values for the various parameters of the system. For example N = 7,1 - ~3 = 9.95 cycles sec -~ {0'o ~)z + N:},/z = 1,, q- va --- 17-45. Thus
0'o t~) = 14"34 ( M 2 + LZ) 1/z = v9 -
Now
~'11 = 9"4
{(re t~ + M ) z + Lz} 1/z = ~'9 q- I ' l l =
11-0.
TABLE 8.5111 OBSERVED AND CALCULATED TRANSITIONENERGIES I~LATS~ I~NsrrIES FOR Trm ~H Rr~oN~rCS SPEcrRuM oF N~mTHALE~ ~7) AT 40 MC sec -~
Line
Energy relative to centre of band (cycles see-t) Calculated
11 8
4 3 12 7 6 9 2 1
10 5
0.8 1.6 3"2 3-7 1 6"1 7-5 9"9 t 10"2 13"2 13.7
15"5 15.8
Observed 0.8 3.7 6"4 8"2 10"2 13"7
Relative intensity Calculated 0-52 0.45 1.87 1.57 0.93 0.62 0.53 } 0"49 0.46 } 0"43 0"07 0"07
Observed* 1.1 2.8 0.9 0.6 1.3 1.1
* Normalised to same intensity as calculated values. T h u s M = - 6.03 a n d L = 7-21 cycles sec -1 i f it is a s s u m e d t h a t J A s > J A n ' . I f t h e r e a s o n a b l e a s s u m p t i o n is m a d e t h a t JAA' ~ 0 t h e n K = - M , a n d w i t h
THE ANALYSIS OF H I G H R E S O L U T I O N S P E C T R A
407
this information a complete set of parameters for the spectrum can be obtained, J^B =
8"6 cycles sec -~
J^B, =
1.4 cycles sec -~
JBB' =
6"0 cycles sec- ~
JAA' =
0"0 cycles sec -1
re 5Ae = 14"3 cycles sec -~ . The relativesigns of the coupling constants were not deduced from this analysis. A check on the original assignments can bc carried out by calculating(9) the position of transition 12 with the known parameters and comparing it with the assicmment for this transition. Another check of the assitmments involves calculating the complete theoretical spectrum for the A lines (12 in all) from the parameters extracted from the tentative assitmment. To do thisthe numerical solution for the roots of a fourth order secular equation must be undertaken to find transitions 2, 4, 5, 6, 7, 8. Table 8.51A gives the comparison of the observed and calculated 1H resonance spectrum of naphthalene at 40-00 M c sec-L The IH spectrum of ortho-dichlorobenzene has been analysed in a similar fashion(eV, eg) The most difficult aspect of an AA'BB' analysisis to find a suitable value for K. For many molecules its value cannot be found directly and an assumed reasonable value must be used; good agreement between bbserved and calculated spectra justifies the choice of a particular K value.
Benzofurazan. Dischler and Englert (7°) have conducted a comprehensive study of several AA'BB' type molecules. A particularly good example of such a system with Jew >> J . ^ , > 0 and J^B ~ JAB' > 0 is benzofurazan A N B~J \ B,L I 0 -A%~,~ N / The 1H resonance spectrum at 56-4 M c sec -I for benzofurATsn is shown in Fig. 8.33: all but two of the transitions (I and 2) give rise to resolvable absorption bands. The assignments of the bands are shown on the spectrum and the observed and calculated transition frequencies and relative intensities for a positive K value with (i) N > 0 and (ii)N < 0 are given in Table 8.52. The assignments made by assuming N > 0 were shown to give a better agreement with the observed spectrum, thus indicating that the coupling constants have the same sign (N.B. the relative signs of all coupling constants are available). They used an AA'BB' analysis outlined by Dischler and Maier (71) similar to that suggested by Rao and Venkateswarlu(72). It is convenient to decompose the spectrum into four groups of bands as shown in Fig. 8.33, for the purpose of indicating the significance of the various separations between the bands.
408
HIGH RESOLUTION NMR SPECTROSCOPY
Para-disubstituted benzenes. Richards and Schaefer ¢7a~ have applied the AA'BB' analysis to interpret the XH resonance spectra of several para-disubstituted benzene derivatives. They overcame the difficulty of finding a value for K by assoming that the two meta H - H coupling constants are equal and the para H-H coupling constant to be zero, that is, in the molecule R!
H~'..jHB, RH
TABLR8.52 O ~ V E D AND CALCULATEDtH SeECrRAOF BRNZOFURAZAN(7°) AT 56.4 Mc sec-1
Observed frequency
(cycl~ ,~dgC-I)
Relative int~mity (2' =
8)
Calculated assuming N > 0 correct assignment Assignment
N<0 Incorrect assja~ment Assign- Frequency Intensity (cycles merit
Frequency! (cycles Intensity SeG- t )
21"06 19"27
0.11 0-16
17"20
1.16 {
14"32 13"68 10-56 9"30 7"01 6"61 4"35 3"13
0-59 0.66 0"71 0"80 1"46 1"50 0"45 0"40
5 I0
21-18 19"27
1 2 9 6 7 12 3 4 11
17"20 17-11 14-32 13"70 10"51 9"30 7"02
8
3.04
6"60
4"35
5 10 3 4
20"89 19"27 17"20 16"61 14"32 14"54 9"67 9"30 7"02 7"08 4"39 3"40
0.09 0-12 0-58 0"62 0.55 0"55 0"87 0.88 1"42 1"37 0.45 0"50
3 x 10-4 [ Comb.
31"25
10-4
I Comb.
40.81
10~
0"08 0"12 0"58 0"58 0"55 0"64 0"81 0"88 1"42 1"54 0"45 0"37
6 7 12 1
2 I1 8 !
Comb.
30"79
Comb.
41-29
K = 7"7 M = -5-3 Nffi10"2 Lffi8"4 vo t~ ffi 22.0
YsB'= 6"5 J~,o = 1"2 JAB=9'3 J~m'=0"9 J = 0"389
10-6
K ffi 6"9 Jse' ffi 6"1 M = - 5.3 JAA' ==0"8 Nffi -- 10"2 -/'Aa= --0"9 L ffi 8"4 J A a ' f f i - - 9"3 • o ~ ffi 22-0 ,~ = 0"389
JAA' = Jss" a n d JA, a = JAa, ~" 0. Both M a r t i n a n d Dailey C74>, a n d R a o a n d V e n k a t e s w a r l u t72) have e x a m i n e d similar molecules a n d they were successful i n achieving a c o m p l e t e A A ' B B ' s o l u t i o n w i t h o u t recourse to a n y simplifying a s s u m p t i o n s . T h e f o u r energy levels for the w a v e f u n c t i o n s I s ~ , 2sb, 3sb a n d 4s~ are the f o u r roots of a f o u r t h o r d e r secular d e t e r r n l n a n t to which reference has a l r e a d y b e e n m a d e . These f o u r eigenvalues are d e n o t e d by E l , E2, Ea a n d E4 respectively for the four w a v e f u n c t i o n s a n d they are the solutions of the f o u r t h
THE A N A L Y S I S OF H I G H R E S O L U T I O N S P E C T R A
409
power equation (equation (8.184)) which results from the expansion of the corresponding 4 x 4 determinant.
(K + E) {£~ + ]v£=-- £[¼1v= + (~o ~^.)=] - ¼iv,} -
~L = {3£ = -
[~N= +
(,o ~.~)=]} =
0
(8.1s4)
The six transition energies and their relative intensities for the A transitions involving the So states can now be written down and they are given, together with the six other transitions in Table 8.53. H
"
"
1,2 "
N4 96
I
7 12
~'E
I1 8
;
i c=[(~o~)'÷N']' .---
N~-----I; G= (I~+M=) =
12 9
} i"
I
'I°
:
G
i i i
~4 I
7
il
I
8
--- - (E=+E,)
It ""2 E~
2 "-----(E,+E 21 - - ~
2~
0
I I0
20
Cycles sec -t Fxo. 8.33 A A t B B ' type spectrum. ZH resonance spectrum at 56.4 Mc sec-1
of benzofumzan (11 mole per cent m CSD. Dbchler and E n ~ x t ~7°)
\ --0
410
H I G H R E S O L U T I O N NMR S P E C T R O S C O P Y
The first step in the A A ' B B ' analysis is t o assign the intense transitions 1 a n d 3. The separation between these transitions is N = JAB + JAn' and if a n a p p r o x i m a t e value f o r N is k n o w n then this part o f the assignment is easy to m a k e ( N is always a b o u t 9 cycles see -~ for para-disubstituted benzenes). F r o m this tentative assignment, values for N a n d *,oc5 are calculated using equations (8.180) and (8.181). Inpara-disubstituted benzene molecules, the value o f M will be small a n d N and L will be almost equal (since JAA" ~ JBB' a n d JAB' ~ O) a n d this results in transitions 9 a n d 10 being very close to transition 1, and transitions 11 and 12 being close to transition 3. F o r m a n y molecules of this type, the frequencies c o r r e s p o n d i n g to transitions within these two g r o u p s c a n n o t be resolved. All the remaining six transitions 2, 4, 5, 6, 7 a n d 8 involve the roots o f equation (8.184) and values o f K and L. Examination o f Table 8.53 indicates that the sum o f the transition frequencies 5 and 8 or 6 and 7 are both TABLE 8.53 TRAuSrrloN E~mgGms AND RELATIVE INTENSITIES FOR THE A A ' PART OF AN A A'B B ~ SPIN SYSTEM(72)
No.
Transition
Energy relative to ½(VA+ vB) ½ N + ½[(%~)2 + N2]½
1 2
15o ~--~ l s l '
--½ [(% 6) 2 + N 2 ] ½ - El
3 4
s-z"*" ls.-t t ls_t'-*" 2So'
- { N + { [ ( % 6 ) 2 + N2] ½ E2 - {[(Vo ~5)2 + N2] ~"
5
330t-* • 2$t t
½[(Vo6) 2
6
25_t ~~ 4So ~
E , + ½[(Vo ~)2 + N2]½
7
4So' ~ 2st'
½[(Vo 6) 2 + N 2 ] ~ - E ,
8
2s_t' "~" 3So'
E3 + ½[(~o~)z + N2]~
1 S1 t ~
$2
+ N21½- Ea
Relative intensity 1 - sin29 [cosg(all + at,) -- sin9 x (at2 + at,)] 2 1 + sin29 [cosg(az2 + a 2 , ) + sin9 X (a21 + a24)12
[cos9(a32 + %,) + x (a31 + a3,)l 2 [cosg(a,l + a**) x (a,2 + a**)]2 [cosg(a,2 + a**) + x (a,t + a**)]2 [cosg(a3t + a3,) -
sin9 sin 9 sin9 sin9
X (a32 + a3~.)]2
9
2ao' -~ 2 a t '
10
2a_t '-+ l a o '
~[(Vo ~ + M ) 2 + L2]½ + ½ ( M 2 + L2)~r
½[(Vo~ - M) z + L2]~
sin2 (0° - V~+) cos2 (0, ~- re_)
+ ½ ( M 2 + L2)½
11
lao ' ~ 2 a t '
12
2a_l ' ~ 2ao'
+ M) 2 + L2]½ - ½ ( M 2 + g2)~
cos2 (0,, - ~'_)
½[(v 0 6 -- l~/)2 + L2]½
sin2 (0= + q,_)
½[(Vo 6
- ½ ( M 2 + L2)½
Stationary state wavefunctions v& are expressed as linear combinations of eigenftmctions 9= tpq = ~ aqm 9 where aqm are the coefficients. equal to [(Vo~)2 + N2] U2, a value which is already k n o w n . This facilitates the assignment o f two pairs o f bands in the observed spectrum to transitions 5,8 and 6,7. D e n o t i n g the experimental frequencies of these lines by vs, vs, vt, vT, we find that E3 = ___ (v5 - ~s)/2 (8.185)
E4 = + (v6 - vT)/2
(8.186)
THE ANALYSIS OF HIGH RESOLUTION SPECTRA
411
O n l y o n e o f the t w o values f o r E3 a n d f o r E4 is correct in e a c h case. By s u b s t i t u t i n g the f o u r v a l u e s o f E i n t o e q u a t i o n (8.184), f o u r e q u a t i o n s i n v o l v i n g g a n d L 2 a r e o b t a i n e d . I t is i n i t i a l l y a s s u m e d t h a t L = N (which is a l r e a d y k n o w n ) a n d f r o m this a v a l u e o f K f o r e a c h o f t h e f o u r e q u a t i o n s c a n b e c a l c u l a t e d . T h e t w o v a l u e s o f E w h i c h give p o s i t i v e values o f K close t o e a c h other, a n d o f r e a s o n a b l e m a g n i t u d e f o r t h e u s u a l meta H - H c o u p l i n g c o n s t a n t , a r e c h o s e n as the c o r r e c t r o o t s E3 a n d E 4 . I f E3 a n d E , a r e k n o w n t h e r e m a i n i n g t w o r o o t s E1 a n d E2 c a n b e d e d u c e d easily a n d t r a n s i t i o n s 2 a n d 4 w h i c h involve these e n e r g y levels c a n b e assigned. W e h a v e n o w assigned t r a n s i t i o n s 1 t o 8 a n d if t h e r e a r e o t h e r n o n - a s s i g n e d b a n d s in t h e s p e c t r u m t h e y c a n b e assigned to t r a n s i t i o n s 9 t o 12 a n d a v a l u e f o r M c a l c u l a t e d . I f a n A A ' B B ' s p e c t r u m c o n t a i n s m o r e t h a n t w o p a i r s o f b a n d s in the A s p e c t r u m satisfying t h e c o n d i t i o n s t h a t t h e i r c o m p o n e n t s a r e s e p a r a t e d b y [(%t~) 2 + N2] 1/2 o r if o n e o f t h e lines in a p a i r is n o t o b s e r v e d d u e t o its w e a k intensity, it is n o t p o s s i b l e to m a k e a n u n a m b i g u o u s a s s i g n m e n t i m m e d i a t e l y . I t is n e c e s s a r y to c a r r y o u t t r i a l a n d e r r o r c a l c u l a t i o n s t o d e c i d e u n e q u i v o c a l l y w h i c h a s s i g n m e n t is the c o r r e c t one. A l t h o u g h a t t w o stages in this analysis, k n o w n , a p p r o x i m a t e values o f t h e c o u p l i n g c o n s t a n t s a r e u s e d t o assist in t h e s o l u t i o n o f the p r o b l e m , R a o a n d V e n k a t e s w a r l u (T2) h a v e p o i n t e d o u t t h a t in t h e a b s e n c e o f such i n f o r m a t i o n t h e m e t h o d c o u l d still b e a p p l i e d b u t w o u l d involve more trial and error calculations. F i g u r e 8.34 c o n t a i n s several t h e o r e t i c a l s p e c t r a f o r the A p a r t o f a n A A ' B B ' s p e c t r u m f o r different values o f K, L, M a n d N o f t h e m a g n i t u d e e x p e c t e d in para-disubstituted b e n z e n e s (they d o n o t v a r y a g r e a t d e a l f r o m o n e m/)lecule t o a n o t h e r ) . T h e c h o s e n values o f ~ot~ a r e also t y p i c a l o f t h o s e e n c o u n t e r e d in this t y p e o f molecule. T h e s e t h e o r e t i c a l s p e c t r a c a n be very useful aids in the assignm e n t o f t h e s p e c t r a o f para-disubstituted benzenes t o t h e t r a n s i t i o n frequencies o f a n A A ' B B ' system. C o x (Is9) has also i n v e s t i g a t e d s o m e para-disubstituted benzenes. TABLE8.54 ENERGY PARAMETERS FOR THE AA'BB' SPXNSYSTEMXNpara-DISUlb,,uTl~ Bm~z~Nrs (m vm'rs or cycles sec-') o4) Energy parameter E~ where E~ ,= El + ~o 6 + ½N Chemical shift difference, *o K 3"5 4"0 4-0 4.0 4.5 5-0 5.0 5-0 5.5 6.0 6.0
8.0 7.4 8"0 8.6 8-0 7.4 8.0 8-6 8.0 7.4 8.6
N
4"0
6-0
10"0
15"0
9"0 8"6 9"0 9"4 9.0 8"6 9.0 9-4 9"0 8"6 9"4
-- 5-97 -- 5"60 --6"11 -- 6"63 -- 6.25 - - 5"88 -- 6-40 -- 6"92 -- 6-56 -- 6"20 - 7"25
- - 4.88 -4"52 -- 4-99 - 5"46 - - 5.10 -- 4"75 - 5"22 - 5-71 - 5"36 - 5"02 - 5"99
- 3-42 -3.12 -- 3"48 - 3"87 --3.56 - 3"25 - 3"64 - 4"03 - 3"72 - 3-41 - 4-21
2-41 -2"17 - 2"44 - 2"73 -248 -2"24 -- 2"53 - 2-82 - 2"57 - 2"32 - 2"92 --
30"0
oo
-
1.23
0.0
-
1.I0
0.0
-
1-22
0.0
-
1"39
0.0
-
1.24
0"0
-
1"12
0.0
-
1.26
0.0
-
1-41
0.0
-
1.27
0.0
-
1.14
0.0
-
1-44
0-0
412
H I G H R E S O L U T I O N NMR S P E C T R O S C O P Y
TABLE 8.54 (continued) Energy parameter ~
where E~ - E2 -- ¢o t~ + ½N
Chemical shift difference, % K
L
N
4"0
6"0
3.5 4"0 4"0 4"0 4"5 5"0 5.0 5"0 5.5 6-0 6"0
8.0 7-4 8.0 8-6 8.0 7.4 8.0 8"6 8"0 7"4 8"6
9"0 8"6 9-0 9"4 9-0 8"6 9"0 9"4 9"0 8"6 9"4
5"64 5"26 5'63 6.00 5"63 5-26 5"62 6.00 5"62 5.25 6.00
4'49 4"14 4.48 4"81 4"47 4"13 4"46 4"81 4"46 4"11 4"80
[
10"0
15"0
3"08 2"79 3.06 3"31 3"04 2"76 3"03 3"31 3"02 2"74 3"28
2"18 1"96 2"17 2-36 2"15 1"93 2"14 2"36 2"13 1"91 2"33
I
30.0 1"16 1-03 1.15 1.26
1.14 1"02 1"14 1.26 1.13 1.01 1.25
I oo 0"0 0"0 0"0 0"0 0"0 0"0 0"0 0"0 0"0 0"0 0"0
Energy parameter E~ where E~ = Ea + ½ [(K 2 + Lz)½ + K] Chemical shift difference, vo 8 K
L
3"5 4"0 4"0 4"0 4"5 5"0 5"0 5"0 5"5 6"0 6"0
8"0 7"4 8"0 8"6 8"0 7"4 8"0 8"6 8"0 7'4 8"6
•
9"0 8"6 9"0 9"4 9"0 8"6 9"0 9"4 9"0 8"6 9"4
4"0
6-0
19"0
1"31 1"49 1"58 1-68 1"88 2-07 2"17 2"27 2"46 2"65 2"88
1"06 1"18 1"28 1"38 1-51 1"62 1"75 1"86 1"98 2"07 2"36
0"66 0"70 0"79 0"88 0"93 0"97 1"08 1-18 1"21 1"23 1'49
I
15"0
30"0
oo
0"38 0"40 0"45 0"51 0"53 0"54 0"61 0"69 0"69 0"68 0"87
0"12 0"12 0-13 0-16 0-16 0"16 0"19 0"21 0"19 0"26
0"0 0"0 0"0 0"0 0"0 0"0 0"0 0.0 0"0 0"0 0"0
30"0
oo
0.04 - 0-05 - 0.04 0.04 - 0.05 -- 0.07 - 0.06 - 0.05
0"0 0"0 0.0 0-0 0"0 0"0 0"0
0"20
Energy parameter E~ where E~ = E4 - ~ [(K 2 + L2) ½ -- K] Chemic, a l shift difference, %
K 3"5 4"0 4"0 4"0 4"5 5"0 5"0 5"0 5.5 6"O 6.0
8"0 7"4 8"0 8"6 8"0 7"4 8"0 8"6 8-0 7"4 8.6
N
4"0
9"0 8"6 9.0 9"4 9"0 8"6 9"0 9"4 9.0 8"6 9"4
- 0"97 - 1"16 --1-11 - - 1"06 -- 1"25 -
1"69
-
1"40
-
1"35
-
1"53
-
1"69
-
1.63
6-0 - 0-67 - 0"79 - 0"77 --0"74 --0"87 - 1-17 - 0"98 - 0-95 - 1"07 - 1"17 - 1-16
I0"0
15.0
- 0"32 - 0"38 - 0"35 -0"35 -0"42 -0.56 - 0"47 - 0-46 - 0"52 -0-56 -0.57
-0.15 -0"18 -0.17 -0.17 - 0.20 - 0.27 - 0"23 -0-22 - 0.25 - 0.27 - 0.28
-
0"0
0-06
0"0
- 0"07
0"0 0"0
--
- 0-07
THE A N A L Y S I S OF H I G H R E S O L U T I O N
413
SPECTRA
Martin and Dailey (74) have simplified the analysis of the spectra of parasubstituted benzenes by computing energy parameters related to the roots of equation (8.184) ~'E',,~=,=' etc.) for typical values of the parameters K, L, N and • o a , , (they are independent of the value of M). Table 8.54 contains a list of
|,2
,oj
~,4
i
B !
/
~ It;
11/I ltd
,)
/ !~, i Js I
//q
;
i 1
A~ XX'
, I/'
/
~
/
illl ; I
'!i!..
I
1! I
i
///
•
I=:
i !
/.,
i /
i
fl /"
uo~=7-9 cycles
, i\
I ~
!i! q
/
I
vo~ = 2 0 - 7 cycles sec-~
/ t] !
!
/
/ ~J /
J//'
sec-'
J
/,7/ "'/i~ //
'
~,o~ =4-0 cycles sec-~ i
|0
I
,
5
0
5
I
!
I
,
IO cycles s ~ -~ I
FIO. 8.34 The A parts of A A ' BB' and A A ' X X'.~t~2pespe(mm having N.- 8.5 cycles sec-*,L ,- 8.0 cye/es see-* and .,%,I,,.0.5 cycles sec-*. P,ao and Veokateswarlu (72)
the compiled values. The manner in which these values can be used to assist in the analysis of an AA'BB' type spectrum is best illustrated by reference to the theoretical spectrum of a para-disubstituted benzene shown in Fig. 8.34. The quartet of bands 5, 6, 7 and 8 are usually easy to detect in the spectrum but the remaining bands often overlap each other to form two main groups of bands dominated by bands 1, 2, 3 and 4. Bands 9, 10, ll and 12 are a weak
414
HIGH
RESOLUTION
NMR SPECTROSCOPY
quartet: at least one member of this quartet always features as a satellite to each of the two strong groups. The separation between the two strong groups of lines is taken as an approximate measure of N, while the separation between the resolvable satellites in the 9 to 12 quartet is taken as L (this will only be true exactly if lines 2 and 4 coincide with lines 1 and 3 respectively and if M is equal to zero). A rough value for [(%6)2 + N21112 can be obtained from the separation between the centres of the two half spectra. Using these approximate values, values of E~ and El, can be extracted f r o m Table 8.54 by interpolation. Lines 5, 6, 7 and 8 can be assigned
8f
CL
5 cycles sec -~
I Transitions
8
3 411 12
I iJl
7
6
91 ~02
5
FIG. 8.35 The observed and theoretical 1H resonance spectra at 60 Mc sec-t of para-bromoehlorobenzenc. Martin and Dailey<74) readily in the spectrum and thus values of E3 and E4 are immediately available from Table 8.53. Values of K and L can now be derived since El = E3 + ½[(K 2 + L2) 1/2 + K] E~ = E4 - ½[(K 2 + L2) ~/2 - K]. At this stage the two derived values of L can be compared: if they agree to better than 0"2 cycles sec -1 then the original assumption, that M = 0, is correct. If the L value obtained f r o m lines 5 to 8 is considerably greater than the original value it is assumed that the observed satellites of the strong bands are 10and 11, while bands 9 and 12 are obscured beneath the bands 1 to 4. The value of L derived f r o m lines 5 to 8 is taken as the correct value: an estimate of M can be obtained f r o m the difference between the two L values. Lines 2 and 4 can now be calculated using values E ] and E [ interpolated from Table 8.54. I f lines 2
T H E A N A L Y S I S OF H I G H
RESOLUTION
415
SPECTRA
and 4 are separated significantly from lines 1 and 3 then the parameters must be further adjusted to give a better fit between observed and calculated lines until satisfactory agr~ment between the spectra is reached. Figure 8.35 shows the observed and theoretical IH spectra ofp-bromochlorobenzene and is typical of a para-disubstituted benzene having H - H coupling constants all of the same sign. When J ~ ° is negative relative to the other coupling constants, lines 6 and 7 spread apart rapidly as ~ot~is decreased leaving a wide space in the centre of the half-spectrum. Another feature of such a spectrum is that lines 5 and 5' (from the other half-spectrum) merge at a relatively high value of ~o6 ( " 10 cycles sec -1 at 60 Mc sec -1) to form a very intense central band.
1,2-Chlorobromoethane, C1CH2CH2Br (:6). At room temperature one would expect this molecule to be rapidly rotating about the single C u C bond, spending equal times in each of the three rotational isomeric states. Only two of the rotational isomeric states of this molecule are different (i) when the halogen atoms are in positions trans to each other (an AA'BB' system), (ii) when the halogen atoms are skew to each other, thus forming an ABCD system (see Section 9.5.5). When rapid internal rotation is occurring the coupling constants assume average values of the constants in the fixed isomeric states. Despite the fact that the chemical shifts of the hydrogen nuclei in each of the two CH2 groups become equally shielded due to averaging processes associated with the
H
m,
CLCHz CH2Br
[
)
IO cycles sec "t
FIo. 8.36 The XH resonance spectrum of liquid 1,2-chlorobromoethane at 60 Mc sec-1. Abraham°e)
416
H I G H R E S O L U T I O N NMR S P E C T R O S C O P Y
internal rotation, the four nuclei remain magnetically non-equivalent (that is, an AA'BB' system) since the average spin-coupling constants between one of the A hydrogen nuclei and the two B hydrogen nuclei need not necessarily be equal (dAs 4= JAw). Figure 8.36 shows the t H resonance spectrum of the molecttle and it is typical of an AA'BB' spectrum, containing at least 20 lines. If the molecule had been an A2B2 system, with JAB = YAe', then the spectrum would contain no more than 18 lines (4 of which are combination bands) as shown in Table 8.24. A detailed AA'BB' analysis of the spectrum gives the following values of chemical shift and coupling constant parameters Vo JAB -- + 12.60 cycles see -1 (at 60.00 Me see-0 N --- JAS + JAB' ---- 15"00 cycles sec -1 L = JAB -- JAB" = 4"3"07 cycles see -t M = JAA" -- Jns, = + 0"95 cycles sec -t. 8.17 SPECTRA CHARACTERKSED BY SIX COUPLING CONffFANT'3 & 1 7 . 1 The Five Spin S y s t e m , AA'BB'X
A system containing five magnetically non-equivalent nuclei which form two different pairs of symmetrically equivalent nuclei, AA' and BB' and another nucleus separated from the others by a large chemical shift compared with the coupling constant involved constitutes an AA'BB'X system. Two molecules of this type are p-fluoroaniline
CI-Is H
F
A' Fx
To completely describe an AA'BB'X spectrum, one needs to know three resonance frequencies and ten coupling constants. If we designate the molecule
F5
then the complete set of parameters for the system is 7,A, vs, Vx, J1, = J23,
J,s = J35
J25 = J15,
J24 ------J l 3
J12 and J3, It is assumed in the analysis described here that Jr2 "~ Js4. The remaining coupling constants form four pairs of equal coupling constants, due to the symmetry of the system: thus only six coupling constants will feature in the spectrum. The symmetrised spin functions and the diagonal matrix elements for the general AA'BB'C system are given in Table 8.55 together with the off-diagonal elements. For the AA'BB'X case, several of the wavefunctions do not mix and
THE A N A L Y S I S OF H I G H R E S O L U T I O N S P E C T R A T~t~.8.55 SYM~TaemD SPIN F U N ~ O ~ AND MA~UX ~ q ' r s A A ' B B ' X SPIN S ~ s = ~ ~s)
417 FOR
Wavefunction Number
Symmetry
Diagonal elements
1 ~o~=c~
(AD~/=
½Vo(4 - 2~^ - 2 a . ) + ½ % ' ( I - ax) + ½(Jx= + J13 + J~3 + J~s + J3s)
2 ~o~=¢
(AD~/=
½,,0(4 - 2~^ - 2,~,) - ½ , , o ' 0 - a x ) + ½(.r~= +
Jxs + J23 - J15 - J3s) ½,,0(2 - 2 ~ + ½ % ' 0 - ax) + ½(Jxs + ./1=)
3 ( 1 / 1 / 2 ) ( o ~ o , +~o,/b,a) 4 (1/}/2)(~===o, + ~/b,o,o,)
½,,0(2 - 2 a s ) + ½ , , o ' 0 - Ox) + ½(J35 + J l = )
(A1)1/= 6 (1/}/2)(~,~0~,/~ + ~o,o,=~ 7 (1/}/2)(=~o,,~0, + ~o,o,) 8 O/}/2) 0 ~ / ~ = + =/~/~o,o,) 9 o,=/V~o, 10/~=o,o,
(AD-I/=
½,,0(2 - 2a^) - ½ % ' 0 - ox) + ½ ( - J . + + Jx=) ½,,o(2 - 2~.) - ½,,o'(] -C~x) + ½ ( - . r 3 s + Jl=) ½ % ' ( I - crx) + ½(Jls - Yl= - J23) ½,,o'(1 - ax) + ½ ( - J13 - Y,: + Y23) ½v0(-- 20"A + 20"B) "i" ½%t(I -- ¢~f.)"I- ½(--Jl~ -J=s + . r , - .r3s + J=~) ½,'o (2~A -- 2a,0 + ½,,o'0 - ~x) + ½(-.Y13 -
J~s - J~s + J=s + JxD ½ % ( - I + ~x) + ½(.r,s - .l~= - .r23) ½,,o'(- I + ~x) + ½(-J~3 - ./'i= + ~ ) ½ , , o ( ~ , , - 2'~s) -- ½ , , o ' 0 -- ~X) + ½(.~,= - -
J13 -- J23 + J l s - J3s) ½ , o ( - 2~,, + 2 o . ) - ½ % ' 0 - ~x) + ½(./i= -
14 o~=,6,~6 z5 (1/}/2) ~,~/b,o, + ,~o,/~o,)
½ , , o ( - 2 + 2~D + ½,,o'0 - c~x) + ½(-.v~s + ./i=)
½ , 0 ( - 2 + 2,~,,) + ½%'0 - ~x) + ½ ( - ~ . s +
./1=) (AD-sI:
½,,0(-4 + 2~. + 2 ~ . ) + ½%'(1 - ~ x ) +
½(Jx=
½,,0(-2 + 2~D - ½%'0 - ~x) + ½(.r~s + .r,=)
½ % ( - 2 + 2~.) - ½,,o'0 - ~x) + ½(Y~ + ./1=) 20~
(XD-sl:
½,,o(-4 + 2+, + ~ . ) - ½,,o0 - ~x) + ½(.rx= + Yl~ + J ~ + J ~ + J ~ )
21 (1/1/2)(o=o,x.6o~ - ~x=/~o~o~) (1/}/2) 0~==o,o, - =/~o,)
(1/I/2) ( ~ o , -/~o,,0o,o,) 26 (1/}/2)(/~=o~o~- =~.6==) 27 (1/}/2) (o~8o~8~-- ,&x.8=.8)
½,,o(~ - 2~,,) + ½ % ' 0 - ~ x ) + ½ ( . ~ s - . r ~ )
½%(~ - ~ , ) - ½%'(1 - ~ x ) + ½ ( - . ~ s - J~=) ½,,o(~ - 2~.) - ½ % ' 0 - ~x) + ½ ( - . r z s - J~=) ½%'(1 - ~ x ) + ½ ( - J~= + . r ~ - J , D (B~)_I/:
(1/}/2) ~ = o , ~ -- =/~,~o,~
29 (1/}/2)(..8.80==- .~/~o;.~a)
½,,o'(- ~ + ~x) + ½(~rx~ - .v~= - .r~s) ½,,o'(- ] + ~x) + ½ ( - J~. - ~'x= + J~D
½ % ( - 2 + 2 ~ ) + ½,,o'0 - ~x) + ½(-J~= J~s) ½ , , o ( - a + 2c~.) + ½,,o'0 - ~x) + ½ ( - J ~ = -
(/;'=)_./= ~.
14
½ , , 0 ( - 2 + 2~,,) - ½,,o'(~ - ~x) + ½(.r,s - .rx=) ½ , , o ( - 2 + 2~.) - ½,,o'(I - ~x) + ½ ( J s s - - J ~ D
418
HI(~H R E S O L U T I O N
NMR SPECTROSCOPY
Off-diagomfl elements
/-/~,= (I/~2).~**, /-/2,= (1/)/2)z~s, /'/,,= ½(J=~ + .~,~), //se = ½(J:* 4" J:~), H s ? = H~. = ½'/xs, //s9 = (I/)/2) J3s, Hs. ~o = H e ~ =//9, ~o = 0, H ~ ? = H e . =' ½Jas, He. to = (1/)/2) ./is, H~. s = ]12, Hr. 9 = / ' / 7 , 1o = (1/)/2) Z , , , H . 9 = H . . io = ( 1 / ~ 2 ) -r13, H11, 12 = 0"1~t, H l l , ta = / ' / 1 t . t* "= (I/Y2) J(2a, H I t . *s = H1(2, ts = ½dis,
H11'16 : H12ele=~'I~$' H12,13 ==HI(2.1(.=(1/)/2)dt3, H'13.14.=H13.16 =H1~t..13"=0, (1/1/2) 33s, /'/v). ,~ = (11)/2) Jls, //'is ) 1~ = ½(3~, + 3"I,), 'F'~I3, I 5
/-/~. ,s = (1/)/2) .,%, H ~ . 19 = (11)/2) Zl~, //is. 19 = ½(J~3 + zl,), H~,. v, = 'l'(J~3 - J,3), B'~3. ~, = ½(.r13 - .V~3), /'/33. ~s =/'/(2,. (2s = ½.r,s, ~/~,,~ = - H ~ , . = s = -½~%, ~ r 3 ~ , ~ : O, ~g~27.2S ==809 ~=7. (29 == ~28. (29 == --~Jl$) .eI., H3t,
3o - - ~',,. , , . = ½(.;'t,, -
3o
= &)).
-'.)'&,. /'/29.-',o =
{(I,,
-
Jr,),
TABLe 8.56 STATIONARY STATE WAVEIH.,TNCTIONSFOR TH]8 A A ' B B ' X S e ~ S ~ ' r ~
(~s)
3' (1/)/2) { - s i n 0 ( p ~ + o , p ~ ) + cos0(o~,p~, + o~po~)} / 4' (I/)/2) { c o s 0 ( p ~ : + o,po~o,) + sin0(o~,p~ + o,~pao,)} j (,41),/2 5' 0/I/2) {-sin~(~p~p + po~a,~p)+ cos~o(_._,~___pp + o~p)) } 6' (1/y2) {cos~(ap~p + :o~,~p) + s i n ~ ( ~ p : + ~:~p)} (Axh/2 Z5' (Z/)/2){ c o s 0 ( ~ +/~p~,/~) + sinO(appp~ + p(xpp~)} 16' O/)/2) ( - s i n o ~ p p ~ + ~papa) + cosO(,,ppp,, + l~aPp~)} j (AO_,/~ lS' 0/)/2) {cos~(ppp~¢ + pp~pp) + sin~(po~ppp + o,pp/~p)} 19' O/)'2) { - ~ ( p p p ~ , p + ppo,pp) + cosq~(p,,ppi3+ ~,pppp)} ) (A1)_,:,_ 21' (I/Y2) {- sinvCe~(~ - o~p~,)+ c o s ~ ( ~ p ~ -o,ap~.a)} i 22' (I/)/2){cosv(po~o~ - o~p~a) + sin¥(o~,)p~- o~/~#.~)} j (~2)*I~ 23' 0/)/2) { - ~ r ( ~ P ~ P - po~#.~p)+ c o s r ( ~ p p - ~p~p)} 1 24' (I/I/2){cosv(~p~p - / ~ p ) + sinr(~.~pp - o,o,po~p)} ~ (B2h/e 29' 0/')/2){cos,~Op,e~- p,S,,,p~)+ sin,~(o,pppo~ - P,,,PPoO) 3O' (II)/2){-~o(p/~p~ - pp~/~) + co~e(=ppp~ - p~p#~)} ) (e~)-I/2 3V (x/)/2) {cosr(ppp,,p + pp~,~p) + ~r(P~,PPP - *,PPPP)} I 32' (1/)/2) {-~y(ppp~,p ppo,pp) cos~,(po,ppp ~,pppp)) ~ (~)-*/~ -
+
-
all off=diagonal elements containing J ^ x or JBx can be neglected. T h e matrix for the H a m i l t o n i a n then factorises into f o u r I × I, ten 2 x 2 a n d two 4 × 4 sub=matrices, Since mixing between spin functions is only appreciable if the difference between the corresponding diagonal elements is large enough or if the off=diagonal elements are small, then the spin functions 8 and 12 will be stationary state wavefunctions if-/23 ~" . / ~ , ./13. Table 8.56 gives the complete set of stationary state wavefunctions f o r the A A ' B B ' X system. There are 46 possible c o m b i n a t i o n transitions b u t they are all expected to be weak in intensity a n d are not considered further. O f the remaining transitions, 16 can be referred to as X transitions, 24 as A and 24 as B transitions (64 transitions in all). F o r each o f the three types of nuclei only 12 o f the transition energies can be written down in explicit form, due to the difficulties encountered in solving a 4 × 4 secular determinant. Tables 8 . 5 7 - 8 . 5 9
THE ANALYSIS OF HIGH RESOLUTION SPECTRA TABLE8.57 TRA~srno~tsAND ~ Transition 1
2-+1
2 5 ' ~ 3' 3 6'-+ 4' 4 I 1 ' ~ 7' 5 12' ~ 8' 6 13'--* 10' 7 14' ~ 9' 8 18'-* 15' 9 19"* 16' 10 20 -* 17 11 2Y-* 21' 12 2 # ~ 22' 1327"+25 14 28 "+26 15 31"+ 29' 16 32"+ 30'
419
Or NUCLEUSX IN THEAA'BB'X SPIN Svsr~,~(re) Energy
Relative intensity .
%'(1 - ax) + J~s + Ja~ ~o'(1 - a x ) + ½ ( J ~ + J ~ D - A + B %'(1 - ax) + ½(J~s + J s D + A - B
1
(cos0 cos~ + sin~ sin0)2 (cos0 cos~ + Sin~ sin0)2
• o'(1 - a x ) - ½ ( J l s + 3"35) + ,4 - B ~o'(1 -- ax) -- ½($15 + J3s) -- A + a %'(1 - ax) - (J15 + J35)
(cosO cos~ + sinq~ sinO)2 (cosO c o s f + sin~p sinO)~
• o'(1 - a x ) + ½(J1, + J3s) -- C + D
(cos7, cosy - sin7 sin~,) 2 (cos7, cosy, - sinT, sin~,) 2
%'(1 - ax) + ½ ( J . + J . ) + C - D %'(1 - ~x) %'(1 - ax) ,,o'(1 - ~,x) - ½ ( . r . + . r . ) + c - D %'(1 - ax) - ~ ( Y . + J~s) - c + D
1
1 1
(cosT,cos~, - sin~, sinv) ~ (cos~, cosy - ~ sinv) =
present the allowed transitions for the X, A and B nuclei respectively. It has been assumed that as < aA so that A nuclear transitions will be at higher fields than the B transitions. The positive quantities A, B, C, D and the angles 0, ~v,~p and 7' are defined as .,1 c o s 2 0 = ½"o(~^ -
aB) + ¼ ( ; .
- ;.),
.4 s i n 2 e = ½(J23 + ; ~ s ) ; B cos2cr = ½,,o(a^ B sin27, = ½(;.
a . ) + ¼(;15 - ; . ) ,
+ ;~3);
Ccos2v
= ½~o(a^ -
Csin2v
= ½(;2a -
~.) + ¼(J.
;~3);
D cos2~, = ½,'o(O'.,, - ae) + ¼ ( ; . D
- ;.),
- ;35),
sin27, = ½(;~s - ;23).
The I H and the 19F resonance spectra of p-fluoroaniline are reproduced in Figs. 8.37 and 8.38.
The X spectrum (/rom the XgF nucleus) o/ para-fluoroaniline. Due to the simplicity of the 19F resonance spectrum one can easily assign the bands to the X transitions given in Table 8.57. The spectrum consists of a triplet o f triplets caused b y equal coupling o f the two ortho-hydrogen nuclei with the fluorine nucleus (J35), and equal coupling of the two meta-hydrogen nuclei with the 14"
420
H I G H R E S O L U T I O N NMR S P E C T R O S C O P Y
f l u o r i n e n u c l e u s (Jxs). F r o m t h e a s s i g n m e n t s s h o w n i n F i g s . 8.37 a n d 8.38 it is found that J l S + J35 = 13"5 cycles s e c -x
Txm.~ 8.58
A -
B = + 1"8 cycles sec -1
C -
D =
T g ~ m ' r [ o m AND ~
Transitions 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
4'-,1 T"4" Y
:t: 1"8 c y c l e s sec -~ oe "rein A NUCL~ IN Tt~ A A ' B B ' X Sere SYs'reM(73~
Relative intensity
Energy {~'o(2 - aA - as) + ¼(2J~3 + 2J'23 + dxs + d3s) - A
2(1 + sin20)
{vo(2 - aA - - a . ) + ¼(--2Jr3 -- 2J23 + Jts + J 3 D - A
2(1 - sin20)
½vo(2 - aA - as) + ¼ ( - 2 J r 3 - 2.}'23 - - J r 5 - J 3 5 ) - B ½Vo(2 -- aA -- as) + ¼(2Jt3 + 2J23 - J13 - J3s) - B ½Vo(2 - aA - orB)+ ¼ ( - 2 J 1 3 + 2,}'23 + J13 + ./35) - C
2(1 2(1 + 1+ 1 -1 -1+ 1+ 11+ 1-
8'-, Y
1 0 ' - , 4' 1 1 ' - , 5' 1 2 ' - , 5' 1 3 ' - , 6' 1 5 ' - , 7' 15'-, 8' 1 6 ' - , 9' 1 7 ' - , 16' 1 8 ' - , 12' 18'-,*- 11' 1 9 ' - , 14' 20 -* 19' 6'-~ 2 25 - , 21' 26 - , 21' 27 -* 23' 28 - , 23' 2 9 ' - , 26 2 9 ' - , 25 31'-* 27 3 1 ' - , 28
Although the X approximations can be assigned J35 = 8.5 cycles
½~'o(2-- a A - - ~B) + ¼(+2J'~3-- :'/'=3 + Yt5 + J'35) -- C ½ " o ( 2 - - a A - a-) + ¼ ( - 2 J ' , 3 + 2J'23 - J ~ s - J 3 D - D ½,'0(2 - a,, - ~B) + ¼(:a',3 - 2J23 - J'~s - a'3s) - D {Vo(2 - aA - aa) + ¼(-- 2Jr3 + 23"=3+ Jts + 3"33) -- C
½~o(2 - aA - as) + ¼(2J,3 - 2.,% + .Its + ~35) - C ½1'o(2 -- aA -- a s ) + ¼(2J13 - 2J23 + J l s + J3s) -- D
½Vo(2 -- aA -- ~n) + ¼(--2d13 + 2d2_~ -- dts -- d3s) - D
t r a n s i t i o n s 4, 5, 6, a n d 7 axe n o t k n o w n explicitly, first o r d e r o f t h e m will b e v e r y n e a r l y c o r r e c t in t h i s c a s e a n d t h e y as s h o w n in F i g . 8.37. T h i s gives v a l u e s o f J t s = 5-1 a n d s e c - L B y i n s p e c t i o n it is s e e n t h a t (A -
and therefore
sin2~) sin2~p) sin2~ sin2Tp sin2F sin2F sin2¥ sin2tp sin27 sin27'
B) = (C -
A = C BfD
D) =
1"8 c y c l e s sec -~
0 = ~0 7 =
-9
w h i c h m e a n s t h a t J13 = 0. I t w a s a l s o p o s s i b l e t o d e c i d e t h a t J r 5 a n d J 3 s axe o f t h e s a m e s i g n s i n c e i f t h e y a r e n o t t h e n (A - B) -- :i: 7.0 cycles sec -x a n d t h e c a l c u l a t e d i n t e n s i t i e s in t h e X s p e c t r u m c a n b e s h o w n t o b e i n d i s a g r e e m e n t w i t h t h e o b s e r v e d values.
THE A N A L Y S I S OF HIGH RESOLUTION S P E C T R A TABI~ 8.59 TaxNSmOm AND ~
OF T ~ B NUCL~ IN TI~ A A ' B B ' X ~
Transitions 41 3 ' ~ 1 42 5 ' ~ 2 43 8 ' ~ 4 ' 44 9 ' ~ Y 45 I 1 ' ~ 6' 46 12'-* 6 ~ 47 14'-* 5' 48 15'-* 10' 49 16'-* 7' 50 16'-* 8' 51 17'-+ 15' 52 19' -* 11' 53 1 9 ' ~ 12' 54 20 -* 18' 55 18~-* 13' 56 7 ' - * 4' 5725-*22' 58 26 -* 22' 59 2 7 - * 2 4 ' 60 28 -* 24' 61 30'-* 25 62 30'-* 26 63 32'-* 28 64 32'-* 27
421 SYs'r~ vs)
Relative inten~ty
Enecgy
2(1 - sin20)
½ , 0 ( 2 - ~A - ore) + ¼ ( 2 J l s + 2J=~ + J15 + J . )
+ .4 ½Vo(2 - ~ . - am) + ¼(2.r,3 + 2J=3 - Jx~ - Jss) + B
2(1 - s i n 2 0
R
n
n
½~'o(2 -- a~ - =e) + ~ ( - 2J~3 - 2.;23 + .;,5 + J~s) + A
2(I + sin20)
½~'o(2 - ~ . - =B) + ~ ( - 2J~3 - 2J'2~ - J;5 - J3s) + a
2 ( I + sin29o)
½,0(2 - =A ½,0(2- C~A½~0(2 - =A ½,0(2 - ~ . ½~,o (2 - ~ . -½,0(2 - o,, -
~ ) + ¼ ( - 2 J . + 2J23 + J . + J35) + c ae) + ¼ ( + 2Jxs - 2J=3 + .rts + J3s) + c , or,,) + - ~ ( - 2 J x 3 + 2 J = s - J ~ s - 6 s ) + D ~ =a) + ¼ ( 2 J , 3 - 2 . ; .
- .;,s - J.)
+ D
on) + ¼(2Jxs -- 2J=s + Jx, + Jss) + C
i ]
2 . / ' . + 2./'23 + . r . + . / . ) + ci 2 . r . + 2.rz3 - J . - J3s) + D ! ½~'o(2 - ~,., - =B) + ¼ ( 2 J t ~ - 2J=s - J'x5 - .r3,) + D !
½,0(2
-
a,,) + t ( -
a . - ~,e) + t ( -
11+ 1+ 1I + 11+ 1-
sin2v sin2~o sin2y sin2y sin2~ sin2~ sin2~, sin2~,
13 14
10
I 15
9 8 [7) (4) (6) 2 16 15 (5) II
I I0
I 5
I 0
I 5
Cycles
s e c "I
3 12
I
I I0
] 15
t"'"I !lTItJ " t !j J FIG. 8.37 The 19F resonance spectrum of neat p-fluoroanlllne at 29.92Mcsec -a Richards and Schaefer¢~s)
'
422
HIGH
RESOLUTION NMR SPECTROSCOPY
The AA'BB' part oJ the specttTon of para-fluoroaniline, This spectrum simply consists o f a normal A A ' B B ' type spectrum for a para-disubstitutcd benzene derivative (see Fig. 8.35) with each c o m p o n e n t split into a doublet due to coupling with the fluorine nucleus. Knowing this, the bands can readily be assigned to the transition energies shown in Tables 8.58 and 8.59, as shown in Fig. 8.38 and analysed in a similar fashion to an ordinary A A ' B B ' system. The following parameters gave the best fit of the theoretical and experimental spectra of p-fluoroaniline A = C = 8"2 cycles see -1
J35 = 8"6 cycles see -1
B = D = 6"4 cycles see -1
dzs --- 5"1 cycles see -z
1z3 + / 1 3 = 9"0 cycles see -1
~o ~AB = 12"2 cycles sec -x
J13 ~ 0 ( < 0"3) cycles see - t
i
I I 30
I I !
31 (2_~2}(28) 2'956 e6 27 39 34 38
t 120
32 2"1 I I 4 6 35 5417 40 6033 23 64~7 2024 52 5 5
r
r I0 ,
l ~
53 51 (43) (5:0] 41 5 8 4"5 57 61 4 7 62 59 44 63 4_2 48
0
C y c l e s sec -~
5--6
FIG. 8,38 The tH resonance spectrum of the ring hydrogen nuclei of neat p-tluoroaniline at 29.92 Mc sec-1. Richards and Schaefer(Ts) F o r p-fluorotoluene cTT), the chemical shift between the hydrogen nuclei in
the ortho- and meta-positions to the fluorine nucleus is found to be very small when the pure liquid is examined at 29.92 Mc sec -1. Schaefer c77) has thus analysed the spectrum in terms of an A A ' A " A ' " X system. Although the spectrum can be satisfactorily explained by assuming that j ~ h o = J m ,.e,. . = 7"4 cycles see -1,
T H E A N A L Y S I S OF H I G H R E S O L U T I O N S P E C T R A
423
Schaefer and Schneider c75)have shown that this is not a necessary condition, since only an average value of the two JFw coupling constants can be obtained from an analysis of this type. By adding various solvents to p-fluorotoluene, the chemical shift difference between the ring hydrogen atoms can be increased and the resulting spectrum subjected to an AA'BB'X analysis cTs~. It is found that in fact J~,~' ~ J ~ " but.that j ~ o = 8.7 and j ~ r o ffi 5.8 cycles sec -x. The CH3 group also couples with the fluorine nucleus, Jcs,-F ffi 0"9 cycles sec-L An analysis of the simpler AA'PP'X spin system has also been reported c79~ 8.17.2 The Four Spin System, ABCX A molecule containing four magnetically non-equivalent nuclei, three of which are strongly coupled together (ABC), and the remaining nucleus CAr) being separated from the others by a large chemical shift compared to the coupling constants involved, constitutes an ABCX system. Banwell and Sheppard have analysed the IH and 19F N M R spectra of vinyl fluoride, CH2-- CHF, in terms of an ABCX analysis and have shown that one can deduce the relative signs of all the coupling constants in such a spectrumC,s°). There are 50 allowed transitions in an ABCX type spectrum, 18 being combination bands. For the ABCX spin system, the basic product functions are used as a basis for the formation of the wavefunctions and Table 8.60 gives a list of the basic product functions and the matrix elements for the system. The matrix for the Hamiltonian can be factorised into four 1 × 1 and four 3 × 3 sub-matrices. Since there js no mixing between states involving opposite spins of the X nucleus, one can write the Hamiltonlan in two separate halves corresponding to the ~ and p spin of the X nucleus. ABC type transitions occur between spin states in the same half of the Hamiltonian, whilst X transitions are between states in separate halves of the Hamiltonlan. Four resonance frequencies and six coupling constants are required to completely describe an ABCX spectrum, namely • ^, ~B, ~'C,~X, JAB,JAC,JBc, JAx, Jsx, Jcx. The ABC region of the spectrum (tH resonance) corresponds to two overlapping ABC type spectra each having its origin in the transitions within one half of the Hamiltonlan. Examination of the diagonal matrix elements in Table 8.60 reveals that there is a strong resemblance between each half of the Hamiltonlan and the matrix elements for a true ABC system (see Table 8.36): they differ only in the additional terms ~x, J^x, Jex, Jcx. The term ~x features in all the diagonal matrix elements of the ABCX system and thus transitions within each half of the Hamiltonlan will be independent of it. Writing • ~ -- (~^ + ½JAx); ~ ffi (~e + ½JBx), etc., the resemblance between the ABC Hamiltonian and the ABC part of the ABCX Hamiltonian becomes more obvious. The observed spectrum can be considered to be ABC in type with the effective resonance frequencies, ~ , ~ and ~ replacing ~^, ~B and ~c. The ABC type spectrum arising from transitions within the other half of the Hamiltonian can be described by similar matrix elements involving effective resonance
424
HIGH RESOLUTION NMR SPECTROSCOPY
frequencies v~ ffi (v^ - ½JAx), etc. T h e p r o c e d u r e f o r analysing a n A B C s p e c t r u m u s i n g a n iterative process c a n t h e n b e a p p l i e d to the A B C p a r t o f a n ABCX spectrum. F o r t h e X r e g i o n o f t h e s p e c t r u m , o n l y t r a n s i t i o n s b e t w e e n t w o halves o f t h e H a m i l t o n i a n n e e d b e c o n s i d e r e d . T w o o f t h e t r a n s i t i o n energies c a n b e w r i t t e n d o w n i m m e d i a t e l y since t h e e n e r g y levels i n v o l v e d a r e given b y t h e u n p e r t u r b e d d i a g o n a l m a t r i x elements, n a m e l y ,
El - E , . = (~x) + ½(JAx + J . x + Jc~)
~'.
-
E..
= O'x) -
½(JAx +
T ~ u s 8.60 P~0DUCT FUNCnONS AND MATRZX E x , ~ r s
J.x
+ Jc~J
FOR TH~ A B C X SPIN SYSl't~(s°)
Diagonal elements n
ABC
X
1
~
~
2
==p
3
=p=
o~
4
/~==
=
=Pp
=
6
~=p
=
7
p#=
=
8
ppp
o,
~l~,,cx +2
+ 1
~[AlC
HI
+~
½0,,. + .,. + .,¢ + vx) + ~ ( J . . + IAC + Juc + J~x + Jex + :cx) [ ½("" + " - "~ + "~0 I +¼(I.. - : ^ ¢ - & c + 1,,x + & x - : c x )
+½
J
½0'A - ~'B + Vc + Vx) I + ~ ( - J^" + J . c - & c + : . x I ½ ( - , ' . + ~. + ,'c + , x )
" +¼(-:.s
I'
=o~
p
2'
~
Y
~/~=
p
4'
:~
/~
6'
/~c=~
/~
7'
/~/~=
/~
s'
~pp
~
o
- ~
+J
0
+½
1
--J-
-
-2
~ . c + 1.~x - : s x - : c x ) + ~'x) : . c -- J*X + & X -- : c x ) + Vx)
t +¼(:As -- IAc - ./.c - JAx - : s x + :cx)
-j
+~
- J . c + & c - JAX + & x + Jcx)
f ½(~A--VS--~C+'X) [ + ¼ ( - - : A S - IAC + J ½ ( - ~ ' A + v . - Vc I +¼(--:A. + I . C -[ ½ ( - - V . -- vs ~- ~c
-'t"
- &x + :cx)
½ ( - ~'A - ~ - ~'c + ~'x)
+¼(:~s + :AC + :sc - :AX - .rex - Jcx) ½(vA + va + Vc - ~,x) + ¼ ( : ^ B + :AC + :sc - JAx - Jax - :cx) [ ½0'A + ~. - Vc - "x) I + ¼ ( J A . - :Ac - :Bc - : A x - :Bx + :cx) | ½(~^ - ",, + ~c ] + ¼ ( - - : A s + : . c - J s c - : , . x + : . x - .rcx) | ½ ( - ~ ' . + " - + ~'c - ,'x) " +¼(-:^. - :Ac + & c + : A x - & x - :cx) ~(vA - ~s - ~'c - ~x) I + ¼ ( - :As - :.c + :.c - :Ax + :.x + :cx) I ½ ( - ~',, + ,. - ~c - ~'x) + ¼ ( - : . . + :Ac - :.c + :AX - Jsx + .rcx)
I ½(-..-.'~+.'~-~0 I + ¼ ( : ^ s - -r.c - Jsc + : ^ x + & x - J~x) ½ ( - ~ , - vs - ~c - vx)
-~
+¼(:~a + :Ac + : . c + :Ax + :,,x + Jcx) Off-diagonal elements
H3,
= H3'," = ½JA.
H56 = / / 5 ' 6 " = ½J^B Hs7 = Hs'7' = ~J^c H~7 = H6'7 ° = ~'J~c
THE ANALYSIS OF HIGH RESOLUTION SPECTRA
425
These lines are usually the strongest in the X region of the spectrum since they do not share intensity with the combination bands. The mid-point between them gives the resonance frequency of the X nucleus, ~x. In all there are eight fundamental bands and 12 combination bands possible in the X region of the spectrum. It is necessary to observe the best fit of the measured and calculated relative intensities in both the ABC and the X region of the spectrum to deduce the relative signs of all six coupling constants. For vinyl fluoride the values of the coupling constants obtained from the ABCX analysis are given in Table 8.61. TAmE 8.61.
HA/=C\Ha •tax -: +84.7 cyclessec-1 . t , , x - +$2.4 cycles sec-1 .tcx ': +20.1 cyclessex-1
1"JACu + 12.8 cyclessec-1 J ~ - + 4-7 cyclm aec-x J a c - - 3.2 cyclessec-1
1" JAc is assun~d to be positive and the signs of the other coupling constants are quoted with respect to this.
8.17.3 The Four Spin System, ABCD A molecule containing four nuclei separated from each other by chemical shifts of the same order of magnitude as the coupling constants between them constitutes an ABCD system. There are no symmetry considerations to be taken into account for such a system, and the appropriate spin functions are the basic product functions given in Table 8.62. Table 8.62 also contains the diagonal and off-diagonal elements of the matrix representing the ABCD Hamiltonian. The matrix can be factorised into two 1 x l, two 4 x 4 and one 6 x 6 sub-matrices. A numerical solution of the resulting secular determinant can be achieved using a computer, the eigenvalues and the eigenvectors being obtained if correct values of the matrix elements are known. Initially, approximate values of the matrix elements are used and these are progressively modified until the solutions of the secular determinant are the same as the observed eigenvalues. The complete description of an ABCD system requires four resonance frequencies and six coupling constants, ~^, ~ , ~c, ~o, JAB, JAC, J~a~, Jsc, Jmo, JcD. There are 56 possible transitions between the energy levels of the system, 24 of them being combination bands of weak intensity. Clough(81) has used an ABCD analysis of this type to interpret the IH resonance spectrum of the ring hydrogen atoms in o-bromoaniline
HB Br ~
14a
426
HIGH
RESOLUTION
NMR
SPECTROSCOPY
I] II il II II II II
II II II II II II
II H II II,
il
II II ;i II il II II II ~8
C
0
m
~++
Z
~'~I
I I I~
~@++ I I ~
I
~i+
<
~ +
~+i i + ~ I + ~
~i++~
~+,
~l++i~
~+~
~+i
I ~
"0
I
I + ~
Z
~ +~
I =
÷I==
+I©= -l-+++-I-
,-+-
e
~
÷++÷ ~0
÷+++ .'--
44 I
!
÷11+ I ÷ 1 +
~4+i
=+11+
I I I ÷++
+1+1
IIII
+ + o
I
ram7 + ~ +
.= 0 0
IT II II II
TT~
THE ANALYSIS OF HIGH RESOLUTION SPECTRA
427
Approximate values of the coupling constants are well known for aromatic molecules and approximate values of the characteristic frequencies of the four nuclei can be obtained by locating the centre of gravity of the corresponding multiplets using the moment method of Anderson and McConnelF e2) (see Section 8.18.2). One must use empirical arguments to assign the ring hydrogen nuclei to their corresponding resonance frequencies. The shielding of the hydrogen nucleus in the ortho-position to the amine group in o = b r o m o ~ n e decreases upon the addition of acetone and this solvent effect can be used to obtain information concerning both the HA frequency and the coupling constants in which the HA nucleus is involved. Theoretically, the relative signs of all the coupling constants in an ABCD system can be deduced from a comparison of observed and calculated intensifies. Reilly and Swalen(s3) have fully analysed the ABCD type spectrum given by the hydrogen nuclei in glycidaldehyde O (A)H
C--H(D)
/\ -c- \ (B)H
\O /
c/ H(C)
at both 40.00 and 60.00 Mc sec-*. The analysis was achieved successfully in two ways, (i) using an ABXY and an ABKY approximation similar to ABK (see Section 8.15.4), (ii) using an exact ABCD iterative approach cl 3s) similar to that of Clough (sl). The ABXY method is essentially similar to the ABX approximation used for three spin systems: in both cases all off-diagonal elements with the exception of those equal to ½JXB are neglected and the complete Hamiltonian factorises into 1 x 1 and 2 × 2 sub-determinants which can be easily solved. At 60.00 Mc sec -1, the resonance frequency of the aldehydic hydrogen is well-separated from those of the other hydrogen nuclei (see Fig. 8.39) and although the ABXY approximation is inadequate to describe the spectrum of the molecule, the ABKY approximation gives excellent results. In the ABKY analysis all the off-diagonal elements are included, with the exception of those involving coupling with the Y nucleus, (JAr, JBv and Ji
428
H I G H R E S O L U T I O N NMR S P E C T R O S C O P Y
frequencies to theoretical transition energies: it is apparent that the problem o f finding the correct assignment of bands from the many possible sets will be even more acute than it is for ABC systems. When the components of the four multiplets overlap, the assignment of bands using an ABXY approach can be extremely difficult. In such cases the assignment can sometimes be effected by using a computer to calculate transition energies from reasonable values of T/s and ~0a/s and then comparing the calculated frequencies with the observed bands. Reasonable values of the parameters can be obtained from the spectra of similar but more simple molecules (here the coupling constants and chemical shifts found in simple ABC type epoxides can be used). AB ,,.
Y(O)
K (C)
^
// H H~
,,3,
4, ,o,6
" ~ ' I 6 " Y '~T 'T '~ V T ~:~ 2) ' 23
~., ~
T
T
zQ "=a
LTT2~
0
31 lt3L
Hc
T
--4 :--
i :'/ties sec-:
! cycles Se,=-I
ill
!
FIG. 8.39 The ZH resonance spectrum of glycidaldohyde at 60 Mc sec -1 : example of an A B K Y spin system. Rc~lly and Swalen (sa)
Govil and co-workers (s4) have also used an exact ABCD type analysis to interpret the IH resonance spectrum of coumarin. 8.18 OTHER M~THODS OF ANALYSING SPECTRA
8.18.1 Perturbation Theory The extraction of cigenvalues by solving the secular equation presents no difficulties when the equation can be factorised into quadratics. Even higher order equations can be solved by numerical methods, but the problem of extracting values of the chemical shifts and coupling constants becomes a tedious process without giving the satisfaction of an analytical solution. In such circumstances it can be more profitable to apply other methods of analysis which give a good approximation to the exact solution, without the necessity of solving a secular equation. In the perturbation method (1°) the Hamiltonian is expanded as . ~ = .~(o) + ..~(1) + . ~ ,
(8.187)
I
THE ANALYSIS
OF H I G H
RESOLUTION
SPECTRA
429
and the energy of the state with total quantum number m is E . = E'.°' + ~ 2 ' + E .'
(8.188)
where
(8.189)
G
.,~F") =
~ Jaa, I ~ . I:o,
(8.190)
G
Jaa. (I~ I~, + I~ I~,)
•~0' ffi ½ ~
(8.191)
G < G'
and G and G' refer to two groups of equivalent nuclei. The zeroth order Hamiltonian.,~'c°) refers to a system in which Joo, is zero, hence the spectrum consists of a single line for each group of nuclei with relative intensities in the ratio no : no,. The zeroth order energy E ~ ) is
E(~°) = ~ ~o ma.
(8.192)
The first order correction to the Hamiltonian ,~c1) is appropriate when Ja~, is finite but much smaller than the chemical shift, i.e.
•GG '
~o where ~ is the dimensionless chemical shift defined by (~a - ~a.)/*o. The product spin functions are eigenfunctious of (jr,co) + ~c1)), hence the Hamiltonian matrix is diagonal with energy levels, EM ffi E~°)+ E ~ ) - - ~ , ~ m G + ~ Jaa, momw. G
(8.193)
G~G'
For a transition in group G the selection rules are Amaffil;
Am~,ffi0;
AloffiO
thus the transition frequencies are = ,~ +
Y~ ;~o, too.. G~tG"
(s.194)
The number of group G transitions is given by the (2/o, + 1)values of ma,, and the relative intensities of the fines are given by the binomial coefficients of nw. All higher order corrections are included in the term o~" and may be obtained from standard perturbation theory (reference 10, chapter 6). The second order correction to the energy is
E~ ) -- E' T
I~'j,I 2
E¢no) -
E 0,
(8.195)
430
HIGH RESOLUTION
NMR SPECTROSCOPY
where ~t'Jm is the matrix element of .~" defined with respect to the zeroth order functions q0~°), which in this case are the product spin functions. The summarion in equation (8.195) is over all the functions ~0~°), corresponding to all possible values of the quantum numbers I e , m e , Ieo, m e , , and the prime against the summation sign indicates that the term j = m is omitted. The matrix elements ~*'~m are (~m~- ~),<mo"+X); m [ ~ ' l 9o~. toe';m) = ½JAG' [(/G -- me + 1)(IG + mG)(IG, -- me,)(IG, + mG, + 1)1t
(8.196)
which on substitution into equation (8.195) gives =
½X
J G.
G < G" 'IJG ~
[mG(Zo + Ze, - me.) "- - raG, (IG + Ie - m~)].
(8.197)
'PC"
The transitions in group G are now, to second order, " = % + E 3"~G°mG, + ½ E G~G"
G*G'
J~e'
lie, (IG, + 1) -- raG. (mG. + 1)
lPe~'PG"
(8.198)
+ 2mG me,]
and are dependent on the value of the total spin quantum number 1 of all groups outside G. The relative intensities of the transitions are obtained with the aid of eigenfunctions ~o. = 90~) + q0~) (8.199) where ~0~) is a first order correction to the product spin functions ~ ) . functions ~0~) are given by the equation (p. 134 of reference 10)
E'
The
(8.20o)
j
so that substituting for the matrix ~ m from equation (8.196) gives ~l)mG,m = q)rnG,m-- ~ E G
- fGe" -
'PG~'PG"
IF(me') F(m o + 1 ~fl(mo+ 1), (me'- 1); m) m
(8.201)
-- F(mo" +1) F(mo) CP(ma- 1), ( m a ' + t ) ; m]
where
F¢.,G) = [(IG + mG)([o -- mG + 1)]2.
(8.202)
The relative intensities are proportional to I(vmo - t IIg I v:~,o)l 2
(8.203)
and substituting for the functions ~p,.o gives Pin-.m, OF.([G
m G + 1)(Io + , n G ) ~ l - 2 ~ L
G=i=G
J-?o--'m°'- 1 . '
~G - -
"JIG'_[
(8.204)
THE ANALYSIS
OF HIGH
RESOLUTION
SPECTRA
431
Equations (8.198) and (8.204) show that the second order correction to the transition frequencies is important only when ~, ~
•~ -
J6o'.
8.18.2 Moments of Spectra The normal method of obtaining coupling constants and chemical shifts from a complex spectrum is by solving the secular equation with trial values of J and t$ and then comparing the calculated and observed spectra. Even in systems with only a few unknowns this method can become long and tedious and methods of simplifying the process have been sought ts~, ss-s~). Values of J and ~ may sometimes be obtained directly by measuring the moments of the spectrum ~s2~. The method is similar to that used by Van Vleck tas) in relating the second moments of the N M R absorption lines of solids to the relative positions of the nuclei. The nth moment of a spectrum is defined by <s'> =
Ajd Jk. aj,
d,k
(8 05)
where VJkis the transition frequency and A~k is the relative transition into~ty. Ajk is proportional to the area under an absorption peak, but when peaks have the same width then Ajk is proportional to the height of a peak. The intensity is also proportional to 117kl2 where 17k is the matrix element of the operator I-. Equation (8.205) may be written <S'> =
J'*
~1i~1=
.
(8.206)
ilvi)~k =l~Ik+j
(8.207)
J.k
The term II~kl 2 may be expressed as N
II~] 2
=
(Ifk)(lfk)* = (Ilk) ~ (I,a-ill
and summing over j and k gives II Jk
l2
= ~ I -Jk Iz~ + = Tr(I- I +) ffi Tr(l + I-).
(8.208)
j.k
The symbol Tr denotes the trace of the matrix, that is the sum of the diagonal elements. The transition frequency ~jk is given by (Ej - Ek) which in turn is equal to oW~ - oW,t where oW~ are the diagonal elements of the matrix representing the Hamiltonian, using eigenfunctions as basis functions, "~
= X ~r ]zr + X J,r Is. It.
(8.209)
Substituting for vjk and II~kl2 in equation (8.206) gives <S"> = ~ (~'k~ -- 3~'~)" 6k I~/Xr I +I-.
(8.210)
432
HIGH
RESOLUTION
NMR SPECTROSCOPY
Applying equation (8.210) to give the first moment ( S ) gives<ss) ( S ) = ~ ( . ~ - ae'a)Is~ l ~ / T r I + lJd¢
= Z (~ Lk
I ~ - t ~ ~gss) l ~ / T r I + I-
= Z ['~,I+]tJb';/TrI+IAk
= Tr[af',I+]I-/TrPI -
(8.21 I)
Similar calculations for the higher moments give ( S 2) = Tr [-.~, I +] [I-, ~ ] / T r I + I-
(8.212)
( S 3) = T r [ ~ , [.~, I+] ] [I-, ae']/Tr I + I-
(8.213)
(S*) = Tr [gf, [,Yg',I+]] [[I-, af'], ,Y~/Tr I + I-.
(8.214)
The diagonal sum equations for the moments have the advantage that the trace of a matrix is independent of the matrix base functions and may be evaluated on any convenient representation without solving a secular equation. As an example of the evaluation of a trace, consider the denominator in equations (8.211) to (8.214) Tr I + I- = Tr
(~ (Iz. + i ly,)) (~ (Ix, - i ly,)) (8.215)
In equation (8.215) only the terms with r -- s contribute to the diagonal elements, h e n c e ~ 2 Tr P / - - - [Tr (" x, + /,2)1 = 2 T r ~ I z, (8.216) r
since the directions x, y and z are equivalent. When all the nuclei have the same spin quantum number, equation (8.216) simplifies to Tr I + I- = 2 N T r I 2 .
(8.217)
The diagonal sum in equation (8.217) may be conveniently evaluated on the product spin function representation, in which I~, is a diagonalmatrix which can be expressed as a product of a matrix involving the coordinates of nucleus r alone, and an identity matrix of dimension (21 + 1)N -t. Tr 15, over the coordinates of nucleus r alone is +I
Tr P,, = Y. m, =
t ( t + 1)(2I + 1)"
--l
(8.218)
hence
2N Trl÷l-
-
3
I(I+
I)(21+
I) •.
(8.219)
THE A N A L Y S I S OF H I G H R E S O L U T I O N S P E C T R A
433
The commutators in the numerators of equations (8.211) to (8.214) can be evaluated with the aid of the equations (8.32) and (8.36). The results are [~r, I÷1 = ~ ~, I,+
(8.220)
g-, .e3 = Y ,,, I ;
(8.~.1)
f
[~', [~', ~+]]
=
[[z-,~j,~
= E ~,~ I,,- + E : , ( , ,
Z ,2, I, + + E a, (,, - ,.) i, + i,. P
-
,,)z;~..
(8.2u) (8223)
r~8
Evaluating the diagonal sums and substituting in equation (8.211) gives the first moment as <S> = N -~ ]~ ,,. (8.224) r
It is convenient to set the first moment equal to zero, that is the axis about which the first moment is taken passes through the "centre of gravity" of the spectrum. Frequencies measured from this arbitrary zero will be denoted as 3 ~,, and the moments as (A S~). In this notation the first four moments are <~S> ffi N-I ~ A , , = O
(8.225)
<~ s2> ffi N-~ Z (A ~,)2
(8.~0
ffi N -1 ~ ( a , , ) '
(8.~7)
r
----N - ' [ Z ( Z 1 , , ) 4 + ½ I(I + 1)(21 + 1) Z J ~ ( A , , -tr
A,.), 1 (8.228) J
In applying equations (8.225) to (8.228) the spectrum is divided into groups such that within a group (,, - ,,)
a limitation imposed to avoid taking the difference between two large quantities.Withln each group there will be sub-groups of equivalent nuclei A,B, C . . . A group containing two sub-groups A and B. The equations for the moments are, (assnmlqg all nuclei to have spin ½) n...Av, + nBA~B ffi 0
(8.2,~)
nA(LI~A)2 + n B ( ~ B ) 2 = ~r
(S.~O)
n^(,~,AP + nB(~vBp = ~V <,~SS>
(8.~)
nA(ACA)" "t" nn(A,B)a "t- ½n^ n ~ J ~ ( ~ 0 6) 2 --- N
(8.~2)
434
HIGH RESOLUTION
NMR SPECTROSCOPY
in which vo~ ffi ~A - v., and N = nA + ha. Solving for alas and Vo ~ gives N2 (~o ~)2 _ _ _
(/1 S 2)
(8.233)
nA nB n3A + n~
2 J2s = N ~
nAns
So that values o f [JAsl and [vo ~1 may be obtained from the second and fourth moments of the spectrum provided that thelines can be assigned unambiguously to A or B transitions; an A transition is one in which A m, A = - 1, A m,8 = 0. Mixed transitions are excluded from the calculation.
A group containing three sub-groups A, B and C. In a system ABC there are three chemical shifts and three coupling constants to be obtained. Equations (8.225) to (8.227) give three simultaneous equations for the chemical shifts. nA A vA + ns A vs + ne Avc = 0
(8.235)
nA(A*,A)2 + nB(AvB) 2 + nc(A,,c) 2 = N ( A S ~)
(8.236)
nA(AVA)3 + ns(Avs) 3 + nc(A~,c) 3 = N ( A S 3 ) .
(8.237)
But the three coupling constants dxe, JAc, JBe can be obtained only from the values of (A S*), (~i S 5) and (A Se). The calculation of accurate values for the higher moments is virtually impossible because the value of (A S T becomes strongly influenced by very weak lines which go undetected. The equations (8.235) to (8.237) suffice to calculate the three chemical shifts, but as the equations (8.236) and (8.237) are of the second and third degree, an analytical solution is impossible, and the shifts are best obtained by a graphical method as follows. Let Arc X~
and R= then
(__~_)
A vB (A S 2)
((A S3>)i
~t [(nB + x nc) 2 + nA(ns + x 2 nc)]
R =
[n#A(ns + x 3 nc) - (na + x nc)S] t
(8.238)
Curves of R against x for particular values of n^,ns andnc can be constructed (s2) from equation (8.238) and possible values of x obtained from experimental values of R. Figure 8.40 shows a plot of R against x for nA = ns ----n o Having obtained a value for x, the chemical shifts are given by the equations (n~ + x n c ) 2 + ' n A(nB + x 2nc) VnA
Avs = n](n 8 + x~nc ) _(nB + xnc) 3 [. ~'A-S-~ .j" / I r A ~-
( na + x nc ) A v B
(8.239) (8.240)
?1A
A )'c = x 1)'B.
(8.241)
T H E A N A L Y S I S OF H I G H R E S O L U T I O N S P E C T R A
435
The moment analysis can be a useful method of obtaining chemical shifts and coupling constants, but in practice the measurement of moments presents some difficulties from the following causes¢59~: (a) The measurement of intensities is complicated when peaks overlap. (b) Non-linearity of the base line of the spectrum has a large effect on the higher moments. (c) The higher moments are strongly dependent on weak lines with larger values of A ~j and such lines may even go undetected. 1.6
o.8
x
o
- 0.8
-i-¢o.711
I -0"4
I
i
0
I
0"4
I 0.71
R
Fro. 8.40 Chemicalshifts of a spin system containing three chemical groups of nuclei having nA'= ns ----nc. Andersonand McConnellcsa)
8.18.3 The Direct Method o/ Calculating Spectrat All the methods described so far for the calculation of NMR spectra first require calculation of the energy levels of the system, and then the transition frequencies are obtained as the differences between these energy levels. Bearing in mind the appropriate selection rules, the transition intensities are then obtained from the matrix elements of the Ix operator. However, it is possible to obtain the transition frequencies directly without recourse to solving the secular equation for the eigenvalues of the Hamilton/an. In the direct method the transition energies are obtained as the solutions of an eigenvalue problem involving ~peroperators (sg).
Superoperators. An operator 0 is defined in such a way that a function 10is transformed by it into some other function ~p',
O r - - V'. t In this section all operators will be denoted by bold capitals, and all superops~tors by bold lower case letters.
436
HIGH RESOLUTION NMR SPECTROSCOPY
A superoperator o is similarly defined by its effect on an operator X, that is oX -- X' for all X. In discussing the eigenvaluo problems of operators it was foumi useful to define orthogonality, and normality of functions, and similarly when discussing eigenvalue problems of superoperators, it is useful to define similar properties for operators. First, the scalar product of two operators A and B is defined as (A, B) - Tr(A +, B) (8.242) the so-called trace metric. A* is the adjoint of A. Normalised operators are such that (A, A) = l, and for orthogonal operators (A.B) = 0
for A , B
The adjoint a ÷ of the superoperator a is defined by
(x, aY) = (a+X, Y) for all X and Y. a is hermitian if a = a ÷, and unitary if a a ÷ = 1 = a ÷ a The derivation superoperator. A superoperator of particular interest in the calculation of N-MR spectra is the derivation superoperator a v belonging to the operator A and defined by
a D (X) = [A,X]
(8.243)
that is, it forms the commutator of X with A. If A is an hermitian operator then a D is also hermitian. Consider the derivation superoperator I~ of the Hamiltonian ~ . h D X = [~Y, X] (8.244) for all X. The set of operators X~ which satisfy the equation h D (X~) = o~, X~
(8.245)
are termed the eigenoperators of h v, and the ~ are the corresponding eigenvalues. Since both ~ and h D are hermitian the eigenvalues o~ are real, and the eigenoperators X, form a complete orthonormal set. If ~ is of dimension n (n ffi 2s forNparticles each of spin½) then h D is of dimension n 2, hence there are n 2 e i g e n o p e r a t o r s X~.
In order to find the eigenvalues of h D an operator Sjk is defined such that and
Ssk ~v~ ffi ~ [ Sj~ t0~ = 0 u n l e s s l = k !
(8.246)
where ~,j and ~Pi,are eigenfunctions of the Hamiltonian belonging to the eigenvalues Ej and Ek.
THE A N A L Y S I S OF H I G H R E S O L U T I O N S P E C T R A
Now
437
h a (S~) ~, = [ a ' , S~] ~ = a ' S ~ ~ , - S~ a ' ~ = X~
- St, E,~o,
= E~r
- E,~r
= (Er - E A ) %
(£~ - £~) $# ~ok
(8.247)
hence Srk is an eigenoperator of ha with the eigenvalue (£r -- E~). If the Hamiltonian is n dimensional, i.e.j, k = 1, 2 . . . n, then there are n2 linearly independent values of Sj~, and hence all the eigenvalues of h a are found. The eigenvalues of the Hamiltonian derivation superoperator I/a are therefore the transition frequencies observed in the NMR experiment. The intensities of the transitions are obtained from the matrix elements (Xo~II-[ ~t), which may be expressed in terms of the eigenoperators of hv in the following way: !
-- Tr(I-, SjA) = Tr [(I+)* , Sr~]
= (I÷, Sr~)
and since the S~ are the eigenoperators of h D, A~ ffi ¼ I(I+, x . ) l '
(8.248)
where .4= is the intensity of the transition a.
Matrix representation of superoperators. The representation of a superoperator by a matrix is exactly analogous to the matrix representation of operators, thus the derivation superoperator ha may be represented by the matrix h a formed from elements, h ~ - (O r, h D 02) (8.249) where the n 2 operators Oj form a complete orthonormal set. If the operators Oj are in fact the eigenoperators of h D then the matrix h D is an n 2 x n 2 diagonal matrix with elements equal to the transition frequencies. When the operators Oj are not the eigenoperators of h D, then the matrix contains non-zero off-diagonal elements and the eigenvalues to, are obtained by solving the secular equation of the matrix [cf. equation (8.11)]. The eigenoperators X~ are expanded in terms of the orthonormal set O j, n2
x . = y~ c . r o r J=l
(8.250)
where the expansion coefficients Co~, and the eigenvalues to~ are obtained from the secular equation,
438
HIGH
RESOLUTION
NMR
SPECTROSCOPY
(h~ - oj) C~, = 0.
(8.251)
The order of the secular equation may be reduced by choosing the operators Oj to be eigenoperators of derivative superoperators belonging to operators which commute with the Hamiltonian. This arises because if an operator A commutes with o~, its superoperator a D commutes with h D. Application of the direct method to the AB spin system. The first problem is to choose a suitable basis for the matrix representation of h D, and since the operator I= commutes with X the operators Oj are chosen to be eigenoperators of the superoperator i~ i~ (X) = [L, X]. (8.252) From a similar argument to that leading to equation (8.247) the eigenvalues of i~ are simply A m where m is the eigenvalue of the I, operator, i.e. i °, Oj, -- A m Oj~ (8.253) and for the AB case the possible values of A m are + 2, ± 1 and 0. Only transitions corresponding to A m = ± 1 are of interest here, and Primas and BanwelF sg) show that the four orthonormal operators corresponding to A m = + 1 are
01 ffi I1+ (I2= + 3) ] 0 : = I~ (Ix z + 3) [ 03 =I +(I2=-3)["
(8.254)
/
0,=1~(I1~ 3) With this basis the matrix elements h~ are h~, = (Oj, hDO,) = Tr {Of [3r, O,]} j, k = 1, 2, 3, 4 and the commutators are h ° ( O 0 ffi [ ~ , O d
ffi (7,~ + ½ a 3 0 , - ½YO~
hD(O2) ffi [or, O2] = (v: + ½ J ) O , - ½J O1 h°(O3) = [or, 03] = ( ~ - ½J)O3 + ½ J O , h D ( 0 4 . ) = [o~, 04.1 = (~'2 - - ½ J ) O , + ½JO3. The matrix h D is therefore hD=
/,i +½Jo-½s
(-½Y ,'2 + # J ............... "
,_
0 ~J
\
½s
)
(8.255)
½J vz - ½J and the eigenvalues follow directly by solving the two quadratic secular equations. oJ,=½(~l+~:+R+J) o,2 -- ½(~1 + ~2 - R + J) oJ3=½(vl+7',+R-J) oo, =
½(~i
+
~2 -
R -
J)
THE ANALYSIS OI: HIGH RESOLUTION SPECTRA
439
where • --- [(,o~) 2 + j 2 ] t and vo~ = ~ - v2 The eigenoperators are found to be X~ = O~ cos ½0 + 0 2 sin X2 = - O1 sin ,I~ + O2 cos ½0
X3 ffi a s cos ½0 - 0 4 sin ½0 X4 ffi 03 sin ½0 + 0 4 cos ½0 and from these eigenoperators and equation (8.248) the intensifies of the four lines are given by A, = ¼ ICi ~, x ~ l 2 and since I + -- I~+ + I2÷ = O1 + O2 -- a s + O~ .4, ffi ¼ I((O~ + o 2 - os - o , ) , x.)12
and since the Oj are orthonormal, then AI ffi ~(I - sin0) ,~2 = ¼(1 + sine)
As ---¼(1 - sinO)
A, ffi ~(1 + sinO). The correlation function. Instead of calculating the frequency and intensity of each llne, Primas (se) has shown that the form of the whole spectrum can be expressed in terms of a single function, known as a correlation function. The experimentally observed spectrum can be written as
s(o = Y, s. (,
-
,O
where S~ is the shape function of the line centred on v~. The intensity of the line is defined by the zeroth moment of S~(~) oo
A.
= _f s.
(v -
~,,,) dr
0
and v~ is defined as the frequency for which the first moment is zero, oo
O. 0
In the ideal spectrum, each line would be infinitely sharp and S=(~) could be represented by a delta function ~ (~ - ~=) (zero unless • = ~=), thus s ~ (~) = Y . A . ~ ( , - ~.).
440
H I G H R E S O L U T I O N NMR S P E C T R O S C O P Y
Now, A~ is given by a , = I(I,, X3I ~ = (I~, X9* (I,, X,) thus S,,0') = ~ (Ix, X,)*(I,, X~) ~0' - v,)
(8.256)
and it may be shown that the R.H.S. of equation (8.256) is equal to S,,(~) -- (Ix, 8(v - hD) Ix).
(8.257)
The last step in deriving equation (8.257) is an example of a general theorem(78) which states that if M and N are any operators, hDan hermitian superoperator with eigenoperators X~ and eigenvalues %, andf(x) is any function of x, then (M,fOaD) N) = ~ (X~, M)* (X,,, N)f(~,x).
(8.258)
Equation (8.257) is an expression for the form of an ideal spectrum; a correlation function for a real spectrum is obtained by taking the Fourier transform of the equation (8.257) and normalising to give a new function,
K(t) = j'S,, (v) exp (iv t) dv --oO
(S,, (v) dv
(8.259)
--o0
where t is an arbitrary variable. The integral j Sta (v) d v is simply the sum of all the line intensities, hence K(t) = E a, exp(iv~t)/E A,. (8.260) O~
to,
The lines at % and - % have the same intensity, hence equation (8.260) may be simplified by writing A~ exp (i v, t) + A~ exp ( - i v, t) = 2A~ cosv~ t
(8.261)
K ( t ) = S', A, cos(v, t ) / X A~.
(8.262)
to give la
Now applying equation (8.260) to equation (8.261) gives A~ exp (i v~ t) = ~ (X=, I.~)*(X~, Ix) exp (i v~ t) = (Ix, exp (i hr' t) Ix) and f o r t = 0 ,
~A,=(I~,I~) ,.x
hence
g ( t ) = - -1~ (I~, exp (ih D t) Ix)
(8.263)
and Banwell and Primas show that this may be written as I Tr (I~ exp(i ~'t)I~ exp ( - i~rrrr~t)) N
K ( t ) -- ~
(8.264)
THE A N A L Y S I S OF H I G H R E S O L U T I O N S P E C T R A
where
441
N ffi ~ A. ffi (I=, I=) ffi Tr(I=)=.
Equation (8.264) may also be written in terms of the I ÷ and I- operators,
K(t) ffi Tr(I-exp (i ~ ' t ) I ÷ e x p ( - i Ar t))/Tr(I- I+).
(8.265)
Evaluation of the correlation function. Both equations (8.264) and (8.265) involve a term of the type exp ( / ~ ' t ) 0 exp ( - ioTlrt) where 0 is either Ix or I ÷. It can be shown that this is cqual to (89)
(8.266)
exp(i ~ ' t ) 0 e x p ( - i sTt) ffi e x p ( - h Dt) 0 and expanding the exponential involving !~ gives
"=® (it)~ D. e x p ( - - i l ~ t ) Offi.ffi~o n! ( h ) 0 (i t) 2 (i 0 3 = 0 + i tta~t', O] + --:7-. r~',t,lr, o1] + - : -~. ! - r- ~- [ ~ [ ~ ,
(8.267) o ] ]- -] + ...
that is, equation (8.267) gives the successive commutators of ~" with O, which are new, linearly independent operators. In the case of I= or I* the process will break off after a finite number of steps.
Application of the correlation function to a system of nuclei each with the same chemical shift. In Section 8.12 it was shown that there is no observed spinspin coupling between nuclei which are magnetically equivalent. The same result can be proved very neatly with the aid of equation (8.264) without actually evaluating the correlation function. Consider a system with a Hamiltonian ~'1, and another whose Hamiltonian may be written as *~'1 + ~'~, when [~',, o~'2] ffi [~'2, I,] -- 0.
(8.268)
The correlation function of the first spectrum is K(t), ffi ~I- Tr(ixexp(t(2t, xt)I=exp(_i~t, l t ) )
(8.269)
and that of the second is
K(t)2 ffi 1----Tr[I= exp(iCArx + ~',) t)I= exp ( - iCAY, + ,at2) t)]. N
(8.270)
Now because the condition (8.268) holds csg), exp(i(3f'l + ~'2) t) = exp(i At1 t) exp(t ~1~=t) ffi exp(i ~'2 t) exp(i ~'1 t) (8.271) and e x p ( - i ~'z t)I= e x p ( - i ,T"2 t) --- I, (8.272)
442
H I G H RESOLUTION NMR SPECTROSCOPY
hence substituting equations (8.271) and (8.272) into equation (8.270) gives 1 Tr(Ixexp (io~'x t) Ix e x p ( - io~'x t)) thus K(t)2 = K(t)t. If two systems can be described by the same correlation function they must have identical spectra. For a system of nuclei all with the same chemical shift the Hamiltonian may be written as
where
~
ffi ~ I ~ , J
and
X z = ~. 3,: It'I:
(8.273)
t<./
and both ,~r I and X~ obey equations (8.268), hence the coupling term does not affect the correlation function and hence the spectrum does not show spin-spin coupling. 8.19 ANALYTICAL AIDS
In the foregoing sections it has been demonstrated that molecules containing as few as three magnetic nuclei can give complex spectra. Thelabour of extracting coupling constants and chemical shifts from such spectra can be considerable. Short cuts can often be taken by the special experimental techniques described below. 8.19.1 Magnetic Field or Solvent Variation
A glance at diagrams of the type exemplified by Fig. 8.9 will show how increasing the chemical shift difference between magnetic nuclei in a molecule can transform a strongly coupled spin system into a weakly coupled one--in the present instance from an AB to an AX type. The chemical shift difference can be manipulated by changing the static magnetic field or, in favourable situations, by changing the solvent. Simplification of the spectrum usually results by increasing the chemical shift difference. Decreasing the chemical shift difference is often utilised to check relative signs of coupling constants: a spectrum may be analysed as obtained from an irradiating frequency of say 60 Mc sec -t then a theoretical 40 Mc sec -1 can be computed from all the possible sign combinations of the coupling constants. Finally, comparison of computed spectra with that determined experimentally at 40 Mc sec -1 can show which sign combination is correct. An example of the method is to be found in the analysis of the XH spectrum of ethyl mercaptan ~6). Because the resonance frequency of a nucleus in a molecule is affected by the diamagnetic susceptibility of the medium, the chemical shift referred to an external reference is concentration dependent (see Section 3.3). Tn the absence of anisotropic effects, the diamagnetic susceptibility of the medium affects all the nuclei in a molecule equally so that the internal chemical shifts should be independent of the surrounding medium. Often internal chemical shifts are solvent
THE A N A L Y S I S OF H I G H R E S O L U T I O N S P E C T R A
~43
dependent. This may be a result of (i) a specific interaction between the solvent and solute molecules (ii) self-association of the solute, (iii) non-zero averaging of the anisotropy in the diamagnetic susceptibility of the solvent molecules (iv) H / Br
vc-~s' >>0 I
\H c
~c - vB<< 0
(e)
I,
0
r
I0
;
20
r
~o
40
50
60
Cycles sec-~
FIG.8.41 Th¢ IH resonance spectrum at 60 Mc sec-; of vinyl bromide at various conc~tratious in benzene. Approx/mate volume ratios of vinyl bromide to benzene are (a) 1:1 Co) 2:3 (¢) 1: 2 (d) 1: 3 (e) 1:20. Schaefer and Schneider(91) polarisation of the solute by the solvent (see Section 10.39). The largest interactions usually originate from hydrogen bonding (see S c i o n 9.4), thus the effect often occurs with alcohols, alkynes, alkenes and aromatic molecules. The solvents most effective in taking part in solute-solvent interactions are
444
H I G H R E S O L U T I O N NMR S P E C T R O S C O P Y
those with strong electron donor properties, for example, acetone and benzene (see Section 10.39). Solvent shifts can be as large as 2 ppm in hydrogen resonance spectra(9°>. Pairs of solvents which give large changes in internal chemical shifts are polar and non-polar, isotropic and anisotropic. The spin-spin coupLing constants are not dependent on concentration, except when either the sol= vent-solute interaction changes the equilibrium concentrations of a mixture of rotamers, or when an exchange of atoms or groups is occurring. In the latter case the atom or group may be completely decoupled from the other nuclei in the molecule (an example is found in the spectrum of aqueous ethanol). H m CHCL 3 reference NOz
::©:'o, H,=
(i) H2 H5
H4 ,H6
.
C6H 6 r-'~
IIi
klf
,,,) Hz I
J
H~ "' ')
;i
AA H~
I
I
I
cycles sec"~
Fro. 8.42 The 60 Mc sec-z ~H reso~ace spectrum of 5 mole per cent solution of m-d~tzobem~ne, referred to external chloroform, {n (i) acetone, (fi) benzene.
Schaefer and Schneider(93) When the chemical shift differences in a solute molecule are affected by the solvent, then changing the solute concentration or changing the solvent alters the form of the spectrum as outlined above. Vinyl bromide is a good example of a molecule having a spectrum affected markedly by solvent variation. The spectrum of neat vinyl bromide (91) at 60 Mc sec -I is of the ABC type (see Fig. 10.14). In solution in benzene a specific association takes place between the olefinic hydrogen atoms and the # electron system of the benzene ring(92). Changing the concentration of vinyl bromide dissolved in benzene changes the relative chemical shifts of the B and C nuclei such that ~B - vc varies over the range =I=4 cycles sec -~, including zero (see Fig. 8.41) when the spin system is converted from an ABC to an ABB' type. Meta=dinitrobenzene is an example of a spin system which can be changed from being strongly coupled to being almost first order (93), Figure 8.42 shows the spectra of 5 mole per cent solutions ofm=dinitrobenzenein acetone and in benzene solvents.
T H E A N A L Y S I S OF H I G H R E S O L U T I O N S P E C T R A
445
In the solution with acetone, hydrogen bonding between the hydrogen atoms in the benzene ring and the carbonyl group takes place to differing extents: however, the chemical shifts between the absorption bands are not large and a complex spectrum is obtained. In benzene solution, the association between the ring hydrogen atoms and the ~ electrons of the benzene shifts the resonance signals to high fields by large differing amounts so that the spectrum becomes almost first order. 8.19.2 lsotopic Substitution There are numerous examples, some of which are given in the earlier part of this chapter, of relatively simple molecules which have extremely complex spectra because the coupling constants and chemical shifts have the same order of magnitude. To extract values of these parameters from the spectrum may involve solving secular equations of order equal to or greater than three. But it is sometimes possible to obtain all the necessary information on the shifts and coupling constants by examining the spectra of molecules where one or more of the interacting nuclei in the original molecule have been replaced by different isotopic species. Replacement of the C nucleus in an ABC spin system by a different isotope changes the system to either ABX or AB depending upon whether or not the isotope has a magnetic moment. In the former case, the new coupling constants JAx and Jsx are related to JAc and Jac by the ratio of the gyromagnetic ratios of the two isotopes, that is
a'Ax = (~,xh, c.) .1At. The isotope X may also possess a quadrupole moment, in which case the coupling JAx and JBx may vanish. Isotopic replacement of C by X does not change appreciably the electronic configuration of the molecule and has only a very small effect on the chemical shift difference ~o ~AB and on the coupling constant Ja~. Thus isotopic substitution may have the effect of removing an interacting nucleus from the analysis without changing the other magnetic interactions. In the instance where the C nucleus is hydrogen and the X deuterium, the ratio 7x/7c is 0"1535 so that the new coupling constants Jax and Jex may have become negligibly small. Most of the examples of spectrum simplification by isotopic substitution involve the exchange of deuterium for hydrogenCgS~;an early example is to be found in the NMR investigation of fluorobenzenec99~. The 19F resonance spectrum of this compound (see Fig. 8.43) at 30 Mc sex -I is the X region of an AA'BB'CX spin system hence a complete analysis would be somewhat difficult. However, the three coupling constants J~u,~° J ~ ° and J ~ ° can be obtained readily by examination of the spectra of the two molecules F F tt I 2,3,5,6-D,cfluorobenzene
D II 2,4,6-D3-fluorobe~ene
Molecule I has a single hydrogen atom in the para position relative to the fluorine atom hence the 19F and IH spectra should both consist of a doublet,
446
HIGH
RESOLUTION
NMR SPECTROSCOPY
assuming that coupling with deuterium is negligible. The XH spectrum (see Fig. 8.44(a)) shows a single intense line surrounded by a number o f unresolved weaker bands arising from weak coupling with the deuterium nuclei. It is obvious from Fig. 8.44(a) that J~x,° must be small, and this was confirmed by spin decoupling the deuterium nuclei by irradiating strongly at 4.6053 Mc sec -t whilst observing the t H spectrum at 30 Mc sec-L The decoupled spectrum (see Fig. 8.44(b)) shows a dominant single Line with a width at half height of 1 cycles sec -1, hence j ~ a must be < 0.5 cycles sec-L H
T
°15"Z
11
'
;
0
~5.2
Cycles sec-I
FIo. 8.43 The 19F resonance spectrum of fluorobenzene at 30 Mcsec-j. Bak, Shoolery and Williams (99)
(b)
Cycles sec -t
Fxo. 8.44 (a) 1 H resonance spectrum of 2,3,5,6-D4-fluorobenzene at 30 Mc sec-1 (b) as in (a) but strong irradiation of the deuterium nuclei at 4.6053 Mc sec-t is present. Bak, Shoolery and Williams(99)
THE
ANALYSIS
OF
HIGH
RESOLUTION
SPECTRA
447
Molecule II has a : H spectrum consisting of a broad doublet (see Fig. 8.45 (a)), and when the deuterium nuclei are spin decoupled the sharp pair of fines observed (see Fig. 8.45(b)) have a separation of 5"8 cycles see -1 ( , / ~ ) .
(a)
(b)
| --I0
~ 0
I I0
C y c l e s sec. -I
FIG. 8.45 (a) the *H resonance spectrum of 2,4,6-D3-fluorobenzene at 30 Mc sec-x (b) as in (a) but with strong irradiation of the deuterium nuclei at 4.6053 Mc sec-x. Bak, Shoolery and Willian~99)
I
I
I
-5"8
0
5.8
Cycles
s e c -z
Pro. 8.46 The ~91=re~nance spectrum at 30 Mc sec-' of 2,4,6.Ds-~uombeazme. Bak, Shoolery and w m ~ , ( 9 9 )
The x9F resonance spectrum of moleeule II shows thirteen lines (see Fig. 8.46) which have been interpreted (99) as arising from a 1 : 2 : 1 triplet (due to coupling with the hydrogennuclei, J ~ a -- 5-8cycles sec-l), witheach component of the triplet split into five lines of intensity ratios 1 : 2 : 3 : 2 : 1 (from coupling with the two deuterium nuclei in the positions ortho to the fluorine atom). F r o m the magnitude of the coupling constant JD* or,ho(=1"38 cycles sec-:) it is possible to calculate that j ~ h o must be 9.0 cycles sec -1.
448
HIGH RESOLUTION NMR SPECTROSCOPY
Another example of the use of deuterium substitution to simplify a complex spectrum is in the analysis of pyridine~*°°) by examination of the spectra of 2,6-D2-pyridine and 4-D-pyridine. Tiers et aL have discussed the use of deuterium substitution in determining chemical shifts cgs~. Isotopic substitution can be used in the opposite sense to that described above, namely, the spectrum can be deliberately made more complicated by replacement of a non-magnetic nucleus in a molecule by one of its magnetic isotopes. This can often result in useful information becoming available from the subsequent NMR investigation. For example, the introduction of a carbon-13 nucleus (spin number, I ffi ½) into the ethylene molecule will cause the four magnetically equivalent nuclei of the unsubstituted molecule to become four magnetically non-equivalent nuclei in the substituted moleculeC*°n (an AA'BB' spin system). Thus the interhydrogen coupling constants in ethylene can be obtained. In the carbon-13 containing molecule II the non-equivalence is due
H~/
I
\I~
H,7
II
\H4
to the hydrogen nuclei not coupling to the same extent with the '3C nucleus, that is J,3c-,2 = J~'c-Ht ~ A3c_H~= A~C-H,. Sheppard and Lynden-Bell"°2) have examined the *H resonance spectrum of a sample of ethylene enriched in carbon-13 and from an AA'BB' analysis of the spectrum of molecule II they found the following coupling constants J,~c-H, = 156.4,
A~c_a, = - 2 . 4 ,
J~'~a, -- + 11-7, J ~ ,
J~'~'~ = + 19.1,
-- + 2.5cycles see-*.
Similarly, the hydrogen nuclei in both ethane and acetylene can be rendered non-equivalent by the introduction of a single carbon- 13 nucleus:
H3C- 13CHs
H C = 13CH
AsB3 spin system
AB spin system
Sheppard and Turner ¢*°s~ have explained the extent to which lc-n coupling can bring about magnetic non-equivalence in substituted ethanes and cyclic ethers. The molecules they examined contained only the 1.1 per cent natural abundance of carbon-13 nuclei and thus the work demanded equipment of high sensitivity. An understanding of the technique can be obtained by considering the case of an ethane of general formula R2CHCHR~ (for example CI2CHCHCI2) where R is a non-magnetic group. In a sample containi-g the natural abundance of carbon-13 nuclei, molecules having two carbon-13 atoms R~
/R
*'CHA--t'CKB R/
~R
THE ANALYSIS
OF HIGH
RESOLUTION
SPECTRA
449
will be very rare and can be neglected. Molecules containing no carbon-13 nuclei will possess two magnetically equivalent hydrogen nuclei and give rise to a single N M R absorption signal. The molecules containing a single carbon-13 nucleus possess two non-equivalent hydrogen nuclei and result in a spectrum
(i)
,,"- J4B-":
I
I i
2
(ii}
•
3
I, i i IT z
JBx
j
Fro. 8.47 T h e A B part of A B X theoretical spectra having '/AX = 0 and (i) ./aX/./AB = 0 (ii) Jax/.)'^B = 5. Sheppard and Turner¢1°s)
"'
[ 3,4
'"'
(iii) ,
8
] 6,7,9,[0,11,12
! Ill
3,4
l ~,4
11,12 6,7 9,10
I III
11,1276 9,10
1,2
]
1,2
I 1.2
5,
FIO. 8.48 T h e A A ' part of an A A ' X X '
spin system having M - 0 and or N (iii) L ~ O , N = 3 L , K = 3 L . Sheppard and T u r n ~ tos)
(i) L = O Cn') L ~ O , N - 3 L ,
K)L
expected for the AB part of an ABX spectrum. Figure 8.47 shows two typical theoretical spectra for the AB portion of an ABX spectrum with two different JAe/Jsx ratios, assuming JAx ----0 (]As = JSH and JBx ----Jcn). The coupling constant between the previously equivalent hydrogen nuclei, ]^~, can now be measured. For the illustrative purpose of calculating the theoretical spectra shown in Fig. 8.47 a chemical shift difference between the A and B nuclei
450
H I G H R E S O L U T I O N NMR S P E C T R O S C O P Y
equal to + ½Jsx (that is, + ½J~3cH) is employed. There is no true chemical shift difference between the A and B nuclei since they are chemically equivalent. It should be noted that the centrally situated bands in Fig. 8.47 are not observed in practice because of the presence of the intense t H spectrum of the hydrogen attached to carbon-12 atoms in molecules containing no carbon-13 nuclei. The outer four bands can be treated as a simple AB type spectrum. Substituted ethanes of the general formula RCH2CH2R have also been investigated by NMR. The t H resonance spectrum of the molecule containing ]0 cycles sec -t (a)
{b)-
ILl
I1
(c)
II II¸
(dl
III
FIa. 8.49 The observed and calculated t H resonance spectra at 40 Mc sec -1 of t s C H satellite bands ( A A ' BB' spin system) of the liquids (a) dioxane ( K ~ 20, L = 3.35, N - 8"79, ½Jtsc~ =" 71.1) (b) 1,2-dichloroethane (K,-, 20, L s O . 8 , N = 13.6, ~ J 1 3 c H " 7/.1) (c) 1,2-dibromoethane ( K = 12± 4, L = 5.4, N - 1 6 . 6 , ½J13Cli = 78.5) (d) ethylene oxide ( K = 20, L = 1.3, N : 7"6, ½Jtsc~ : 87-6). Sheppard and Turner (l°s)
no carbon-13 is a single absorption band. For the molecule containing a single carbon-13 nucleus a complex spectrum of the type expected for an A A ' X X ' or an AA'BB' spin system is observed. The four nuclei are rendered magnetically non-equivalent (despite the averaging process associated with rotation about the single C--C bond) and the measured H - H coupling constants JAB and JAB, are functions of the gauche and trans J.H values (see Section 9.5.5).' When the difference between the ~3C-H coupling constants is large compared with the H - H coupling constants, an AA'XX' analysis can be used to interpret the spectra. Figure 8.48 shows three typical spectra for the AA' part of an AA'XX' system. The labelling of the bands refers to the transitions listed in Table 8.48. However, it is usually better to carry out an exact analysis of the AA'BB' type. Figure 8.49 gives the calculated and theoretical AA'BB' spectra for the 13CH satellite bands of dioxane, ethylene oxide, 1,2-dichloroethane and 1,2-dibromoethane.
T H E A N A L Y S I S OF H I G H R E S O L U T I O N S P E C T R A
451
Mortimer <'°4) has analysed the 13CH satellite bands of ethylene oxide, ethylene imine and ethylene sulphide as A A ' X X ' spin systems. The ~sCH satellite bands of both the ~ and fl hydrogen nuclei of furan H(B) H(B9 H(A)~H(A') dissolved in acetone have been analysed to establish cI°5) the values of JAb and JAB, (1"75 and 0"85 cycles sec -I respectively). The analysis of satellite bands arising from magnetic nuclei of lead, mercury and tin has also b ~ n used to unravel the I H resonance spectra of organometallic compounds (see Section 10.3.2). 8.19.3 Calculation of Spectra with Electronic Computer~ I ~2)
The problem of obtaining chemical shifts and spin-spin interaction constants from a complex NMR spectrum, particularly as given by a molecule of low symmetry, is best tackled with the aid of an electronic computer; indeed, in many cases this is the only practicable method. The method is basically one of trial and error---a computer is programmed to calculate a theoretical spectrum from a trial set of spectral parameters: re-adjustment of parameters is made until the experimental spectrum is reproduced. Many computer programrues have been designed for the analysis of NMR spectra but the details of a particular programme depend upon the type of computer available and therefore will not be dealt with here. However, the general method of computation will be outlined c69, io~). The first step in the calculation is to identify the type of spectrum, that is, to divide the magnetic nuclei present into magnetically equivalent groups such as AB,,ApB t,ABC etc.O 5~),and then to assign the spectral lines observed into specific transitions between energy levels (the double resonance techniques described in Section 8.19.5 can assist in making such assignments). After making an assignment the next step is to adjust systematically the values of the spectral parameters so that the observed and calculated transition frequencies and intensities agree exactly. The computational steps involved lie in the calculation of the matrix elements of the Hamiltonian submatrices, and the subsequent diagonalisation of these matrices to obtain the eigenvalues and eigenvectors. The initial spectral assignment can often be made by carrying out a calculation with estimated values of the chemical shifts and coupling constants obtained either by comparison with similar molecules or by the moment method (see Section 8.18.2). The approximate spectral parameters are then used as the initial values in the iterative procedure. Castellano and Waugh (sg) have shown that the results are ambiguous when obtained from an iterative procedure in which the line frequencies alone are the criteria for agreement between observed and calculated spectra. Other criteria must be included in the calculation, such as the invariance of the trace of the Hamiltonian matrix, and intensity sum rules. It has been shown <2. sg. Io7) that the invariance of the traces of the Hamiltonian sub-matrices leads to the simple rule that the energy separation between two 15,
452
H I G H R E S O L U T I O N NMR S P E C T R O S C O P Y
transitions which have the same origin is given by the energy difference of their final energy states; thus the two transitions A and B in the energy level diagram for a four spin system are separated by the same amount as C and D, and E and F ¢69~.
This means that if transitions A and B are assigned in the spectrum then C and D can be assigned by lookingfor a pair of lines such that [~'A-- ~'sl = [~'C-- ~'B[.Two intensity sum rules are useful in helping to assign the observed spectral lines unambiguously. They derive from the principle of spectroscopic stability<~°s~, which states that the sum of the intensities of all the transitions between nearly degenerate states is independent of the strength of the perturbation¢l°sL One consequence of the principle is that if a small perturbation is applied to a system, such as the spin-spin coupling in a nuclear spin system which splits a spectral line into a number of components, then the total intensity is the same as that of the unsplit line. For a system of N spin ½ nuclei the rule can be stated as(moat N~ A~k = (8.274)
~.~
[L + ½ ( N - 1)1! [ - t.. + ~ ( N - 1)]t
where A j~ is the intensity of the transition involving a change from the spin state having lz = m~ to the one with 1- -- ms. Applying equation (8.274) for up to eight nuclei gives the sums for the allowed transitions (A Iz = 5: 1) as No. of spins
~ Ai~
2
2,2
3
3,6,3
4
4, 12, 12, 4
5
5,20,30,20,5
6
6, 30, 60, 60, 30, 6
7
7, 42, 105, 140, 105, 42, 7
8
8, 56, 168, 280, 280, 168, 56, 8
THE
ANALYSIS
OF HIGH
RESOLUTION
SPECTRA
453
The second intensity sum rule states that the sum of the intensities of all the transitions from a particular spin state is equal to the sum of intensities of all the transitions to that state plus 2 m, where m is the value of 1: for the spin state cl o9~ A,p = ~ A~ + 2m. q
(8.275)
r
In applying equations (8.274)and (8.275) to actual spectra it must be remembered that even the correct assignment will not give exact agreement between experiment and theory owing to the experimental uncertainty inthe intensities, particularly when some of the lines overlap. 8.19.4 Double Quantum Transitions
Double quantum transitions were first observed by Anderson cxao): the theory of the transition energies and intensities has been discussed by Kaplan and Meiboom
EmA"mB"me'=,m^ ~'^ 4" ms ~'s + mc ~c + mA msJ^s + m^ mcJ^c + ms mcJsc. (8.276) Each energy state is described by a single spin function JIIA.,ABIIS.v~ Clc.m¢. Double quantum transitions involve changes in the value of A n ~ of - 1 for two of the nuclei in the system so that the possible transitions are
,t~iA.mA BIB. mB Clc. mc
i~
~ I A , mA - 1 ~ I ~ , m B - 1 C I c , mc
AIA. "A Bt.. ,,,~Clc. "c A~,,..,, Bz...~ Czc. "c
A.~A, mA B.ls. ,,~- l G c , ~'c- 1.
The transition energies are obtained by taking half the difference in the energies of the two states,giving ½(~^ + ,'s)+ ½[I - (m^ + ms)] J,,. + ½mc(~rAc + ~sc) ½(~^ + 'c) + ½[I - (m^ + me)] ~^c + ½m~(:^~ + .rsc) ½(vs + ~c) + ½[1 - (mR + me)] Jsc + ½m^(J^B + J^c). When I^ ffi Is = Ic ffi ½then three pairs of lines are observed at the frequencies
½(~^ + ,B) + ~ I(,T^c+ 3sc)l ½0'. + ~'c) + ¼ I(~^s + .Vsc)l ½("B + ~c) + ¼ I(~r^s + 1^c)l
454
HIOH R E S O L U T I O N
NMR SPECTROSCOPY
so that from the double quantum sp~trum one may readily obtain [JAc + Jn c 1, IJAn + Jncl and IJ^n + JAcl. The normal spectrum yields IJ^nl, IJAcl and IJn c I hence it is possible to deduce the relative signs of all three coupling constants. If JAn and JAC are both of the same sign, then
I/An +/Acl
I/ ,nl + I.rAcl
=
H Br
Hc~COOH HA~-~Hm
y!
L
12 I1 I0
76 5
9
32
Br
(o1
[
(~)
t•'•d0
19
18171615
I
13
14
I
I0Cyclessee-'
FIo. 8.50 (a) t H resonance spectrum at 60 Mc see -1 of 1,2-dibromopropionic acid in benzene. 7Hx/2Z~ ,~ 0-4 cycles see -1, (b) as in (a) but with 7 H t / 2 n ~ 2"1 cycles see-1: the lines are broadened by saturation but sharp double quantum transition lines can be seen. McLauehlan and Whiffen (xa3)
and if of opposite sign, IJAB + / a c l
=
I/A~I -
l/^cl
and similarlyfor the relativesigns of JAs and/scDouble quantum transitions are not observed under the conditions usually used in the nuclear resonance experiment, but if the radiofrcquency field is increased so as to just saturate the single quantum transitions,then the double quantum transitionsmay be recorded. The relativesigns of the three coupling constants in 1,2-dibromopropionic acid, C H 2 B r C H B r C O O H , have been deter-
THE A N A L Y S I S OF H I G H R E S O L U T I O N S P E C T R A
455
mined from the latter type of experiment. The normal hydrogen nuclear spectrum at 60 Mc see -x (Fig. 8.50) is that of an ABC spin systemcxxs~. Bf
8r
When the spectrum is recorded with a radiofrequency field of ~,H1/2~ = 2.1 cycles sec-x the single quantum spectrum is saturated and the double quantum transitions 13 to 20 are observed in their place (Fig. 8.50). The bands 13, 14 and 19,20 are assigned readily and their frequency separation measured as Band
Ptequency
Separation
13,14
½(~A + t~s) -4- ¼J(JAc + Jnc)l
7-8 cycles sec-x
The individual coupling constants are found easily from the normal spectrum as
[JAB] ffi 9"9 cycles sec -1 [JAc[ ffi 4"6 cycles sec -I [Jnc] = 10"9 cycles sec -I
so that JA c and JB c must have the same sign, and JA c and JAn must have opposite signs. Double quantum signals can be observed relatively easily for strongly coupled spin systems but when weak coupling is present, as in AX or ABX spin systems, some f o r m of signal enhancement is desirablecx4x~. Cohen and Whiffen~x4x~ have observed an improvement by a factor of eight in the double quantum signal intensity by following up a suggestion made by Yatsiv cxx2~. The experimentalprocedure is simple in that it only requires audiofrequency modulation of Ho in addition to the conditions described above. The method is equivalent in principle to the use of two radiofrequency fields located near the frequencies of those single quantum transitions whose presence is responsible for the double quantum transition under investigation. Cohen and Whiffen showed that the theoretical explanation of the signal observed can be based on the formalism of the double quantum or double resonance theory. They evaluated the factors governing the choice of audiofrequency and its amplitude. 8.19.5 Double Irradiation <163)
The theory and experimental details of double resonance experiments are discussed in Section 6.8: further information may be found in the comprehensive review of nuclear magnetic double resonance written by Baldeschwieler and
456
HIGH
RESOLUTION
NMR
SPECTROSCOPY
Randall( ~~4). In this section examples are given to indicate how double resonance may assist in spectral analysis. Finally, the application of the technique to the determination of the relative signs of coupling constants is discussed. Decoupling in molecules containing two or more isotopic species of magnetic nuclei. The most obvious application of double resonance is to provide decoupling of a group of interacting nuclei so as to reduce the multiplicity of a neighbouring interacting group, and thus to simplify an otherwise complex spectrum. The nuclei are decoupled by irradiating with a radiofrequency (HI), at which the nuclei resonate in the applied field, with a magnitude of H1 equal to or
H I
D
f ~H~
.j,v
a C b
(Q)
Q
~
J \ . (bl
(c)
FIQ. 8.51 (a) The 1H spectrum of pentaboranc-9 at 40 M c see-l; (b) as in (a) but the H B nuclei have been decoupled by irradiatingat 9.6257 M c see-l; (c) as in (b) but the irradiationfrequency was reduced by 500 cycles sec-I. Schaefer, Shoolery and Jones(11s)
greater than the magnitude of the coupling constant concerned. A classic example of this type of application has been presented in Section 8.19.2. Figure 8.51 shows the IH resonance spectrum of pentaborane=9 at 40 Mc sec =I. The bands in the spectrum may be assigned on the assumptions that the four basal hydrogen nuclei are equivalent and interact only with the attached 11B nuclei (spin = i), thus giving the four bands of equal intensity labelled " b " . The apical hydrogen nucleus also interacts only with the attached ~B nucleus and gives rise to the four small bands labelled " a ' . The remaining hump " c " arises from the four bridge hydrogen nuclei which also interact only with the HB nuclei. This assignment was confirmed (11s) by a double resonance ex= periment in which the XlB nuclei were decoupled by irradiating at a frequency of 9.6257 Mc sec-: whilst simultaneously observing the XH resonance at 40 Mc sec -I giving the spectrum shown in Fig. 8.51(b). The I~B nuclei have been decoupled completely leaving two bands b' and c' each at the centre of the corresponding multiplets b and c. Reducing the irradiating frequency by
THE
ANALYSIS
OF
HIGH
RESOLUTION
SPECTRA
457
500 cycles sec -I to the resonance frequency of the apical ~IB nucleus gives the spectrum shown in Fig. 8.51 (c) where the apical :IB nucleus is decoupled completely and the basal 1XB nuclei only partly. In most examples of double irradiation it is usual for the number of lines in the spectrum to be decreased; however, in some cases an increase is found. Figure 8.52(a) shows the 1H spectrum of pyrrole at 40 Mc sec -~, while Fig. 8.52(b) shows the same spectrum run in the presence of a second radio-
H "!
~'H
N
H (a)
H
\__ )
FIG. 8.52 (a) The I H resonance spectrum of pyrrole at 40 M c sec-1; (b) as in (a) but with strong irradiation of the 14N nucleus. Varian Associates(zle)
frequency field (/-/2) at the 14N resonance frequency. The extra bands derive from the resonance of the hydrogen nucleus bonded directly to the nitrogen atom: the triplet structure arises from further coupling with the two neighbouring equivalent hydrogen nuclei. The low field triplet is absent from the normal spectrum because rapid electric quadrupolar relaxation of the 14N nucleus partially decouples it from the hydrogen nuclei and causes the fines to be broadened almost beyond detection. The partially decoupled spectrum of the hydrogen atom bonded to the nitrogen atom is a very broad multiplet which goes unobserved, but strong irradiation at the I"N resonance frequency completes decoupling thus allowing observation of the fine structure. Large values of H~ were used in the above examples; however, Baldeschwieler(117) has shown how extremely accurate values of the resonance frequency of a nucleus may be obtained from double resonance experiments in which the nuclei are only partly decoupled using intermediate values o f / / 2 . In these •n ~ .
15-
458
H I G H K E S O L U T I O N NMR S P E C T R O S C O P Y
experiments separate oscillators were used to provide HI and H~ and the spectrum was recorded by sweeping the main magnetic field. The form of the partly decoupled spectrum is very sensitive to the magnitude of ~2. If ~2 corresponds exactly to the resonance frequency of the second nucleus then a symmetrical spectrum is observed, if not, the spectrum has a predictable asymmetry. For
w_~::8.6t
II
,
-'~:H
0
I!
J'~H
Cycles sec -~
FIG. 8.53 The t H double resonance spectrum of ~4NH~ at 40 M¢ see - t where ,,IN - 0"0 cycles sec - t and ~2N = 8.61 cycles se¢ -1. Baldeschwieler c12°)
~,,j: 0-5
,, !1 "'JNH
,
I IIII
0
J~H
Cycles $ec-I F I o . 8.54 The 1H idouble resonance spectrum of ~4NH+ at 40 Mc sec -1 where AN = 0"5 cycles sec -1 and ~=N = 8.61 cycles sec -1. Baldeschwieler(12°)
THE ANALYSIS
OF HIGH
RESOLUTION
SPECTRA
459
example, Figs. 8.53, 8.54 and 8.55 depict the ~H spectrum of = ' N I ~ at 40 Mc sec -1 with double irradiation for the conditions (a) AN = 0 cycles sec -~, • =N = 8"61 cycles sec -1 (b)AN ffi +0.5 cycles sec -1, ~=N ffi 8"61 cycles sec -1 (c) AN = --0"5 cycles sec -1, ~2N = 8"61 cycles sec-: respectively. AN and *=N are defined by AN = *'oN + ~'2
AN=-O.5 v2~ 8.61
ll}i
Hill
JNH
"JNH
Cycles sec "~
Fxo. 8.55 T h e ~H d o u b l e r e s o n a n c e s p e c t r u m o f Z4NH~ a t 40 M e sec -x w h e r e AN =ffi-- 0"5 ~rc.]e$ $ec-1 a n d 1'2N -- 8"61 cycles sec -1. Baldeschwieler (xae)
Thus Fig. 8.53 is the spectrum with the dccoupling field at the I*N resonance frequency and is symmetrical: Figs. 8.54 and 8.55 show the effect of changing this frequency by -I- 0.5 cycles sec -1 keeping the amplitude of H~ constant. It may be seen from the spectra that ~o N may be obtained to within 0-5 Cycles 8¢C - 1 .
Baldeschwieler(12°) has also shown that the partially decoupled spectrum may be extremely sensitive to small deviations in the symmetry of the molecule. He has calculated the expected partially decoupled spectrum of ISNH~ (Fig. 8.56) in which the ion has (a) tetrahedral symmetry and (b) C3~ symmetry with one hydrogen nucleus chemically shifted 2 cycles see-: upfield from the other three. In both cases, all the JNn coupling constants are equal to 73.7 cycles see-:, in Co) all the J a w coupling constants are equal to the value in methane of 12.4 cycles sec -1. The calculations are for the conditions AN == 0 cycles sec -1 and ~=N ---- 39"1 cycles sec-L When a large perturbing radiofrequency field is applied to remove multiplet splittings originating from a group of magnetic nuclei, in all but the simplest 15=*
460
HIGH
RESOLUTION
NMI~ SPECTROSCOFY
cases a residual effect of the coupling remains (t2~). This residual splitting can m a k e the spectrum as complicated or m o r e complicated than the original pattern, especially when several groups are coupled together. Anderson and Nelson (14s) found that the residual splitting can be greatly reduced or eli. minated by frequency modulating the strong second radiofrequency field. These workers extended the treatment o f Anderson and Freeman (t2~) to derive the conditions necessary to obtain o p t i m u m decoupling in weakly coupled spin systems.
(a)
v2,=39.I 1. AN=0
[ [i
r
,,,
-j
r J
I (b)
v=,~39. i
-d
Jk
I
0
I
Cyclessec-~ Fzo. 8.56 (a) The ~H double resonance spectrum of I~NH+ at 40 Mc sec-1, where AN = 0.0 cycles sec-1 and ~2N = 39.1 cycles sec-t. (b) The ZH double resonance spectrum calculated for distorted ~5~N-H~where the signal for one ZH nucleus is 2 cycles sec-1 upfield from the other three 1H nuclei, JHx' = 12.4 cycles sac-1 and JN, = JN," = 73"7 cycles sec-1. Baldeschwielex(liT)
Decoupling of identical species of magnetic nuclei. The examples of double resonance discussed so far have concerned decoupling of a nuclear species of an isotopic type different from that under observation; the same methods m a y be applied to isotopically identical nuclei. By far the most important application is to the decoupling of hydrogen nuclei but other examples will be found in Chapter 11. It is usual to employ a sideband method (see Section 6.8) since for 1H spectra the two frequencies ~1 and ~2 m a y be separated by as little as 50 cycles sec -~. The foUowing varieties of information m a y be extracted from hydrogen nuclear double resonance spectra (11s).
THE ANALYSIS
OF HIGH
RESOLUTION
461
SPECTRA
(a) Analysis may be simplified by collapsing part of the spectrum. Co) When a molecule has two coupling constants that are almost equal in magnitude, a decoupling experiment may be necessary to resolve the ambiguity. (c) In large molecules, spin decoupling is useful in obtaining the chemical shift between two coupled groups of nuclei when one of the groups is obscured by another part of the spectrum°lgL (d) The relative signs of coupling constants can be determined. Normally, applications (a) and (b) require a decoupling field I-/2 large enough to satisfy the condition
~,~/2~ >> IJl O.Id'
t
(a)
4
0.2J
0.3J
' 0"4J
0"60
O'8J
lJoJ
~'~
2.0J
1
I
l,l ,li ,,I I dl
A 0
(~)
JY~
~
I
,11 il ti11111LII I It
FIG. 8.57 The A part of the IH resonance spectrum of dichloroacetaldehyd¢ (AX spin system) when a second radiofrequency field (YxH2/2.~s JAX/2) is applied near the X resonance. H2 is offset intentionally from the centre of the X doublet by a quantityA orA' expressedin units of J ( - JAr.). (a) The field sweep method for various values of ,4' =. ~x -~ t'2 - 6AX. Co) The freqmmcy sweep method for various values of . i . 0,xHo/2=) ~,,. -
T h e calculated p a t t e r n s a r e 8iven u n d e r e a c h set o f spectra. ~2 ffi 60 M c sec =I JAx -- 2"7 ~ 0"1 cycles sec -~, (~AX"ffi 3"27 p p m . F r e e m a n a n d Vfl'h~en (1111)
when the multiplet then takes on its simplest form. Note that this condition is quite different from the condition for saturation of the nucleus in question, where
~2R]TIT~ ~ 1. Saturation can be shown to be a side effect in the decoupling process (11"). When (c) is to be used for precise chemical shift determinations and for
462
HIGH RESOLUTIONNMR SPECTROSCOPY
nearly all examples of (d), H2 has to be kept small and a compromise must be reached with the condition ~ H2/2~ ~
I~1.
Freeman and Whiffen(~s) have developed a theory for the AX spin system (both nuclei i~-ing XH) to enable spectra to be predicted for conditions slightly 3
-I -
2
~
-2 ~ 5 -3 2.0 1,6 1.2 L
0-8
~
0.4 0 -3
I~o. &58 The t ~ Q o n
I -2
I -i
0
1
2
f ~ . Q ~ c ~ Q ~-d m ~ i f i ~ L for the A resonance of
an AX2 spinsys~m plotteda~!n~ the offsetparameterA for y H 2 / 2 ~ - 0 . 2 J'AX, whereA - - ( ~ 2 - - mx.)/(2nl J',x I) a n d Q - (co x - c o ~ / ( 2 ~ z I'TAX I). Anderson and
Freem(a1n21)
different from those required for best decoupling. They expressed their results in graphical form for both the field sweep and the frequency sweep method (see Section 6.8) of carrying out the double resonance experiment. The latter method offers a simpler set of conditions for spectral interpretation and there are no experimental difficulties using a field/frequency locked spectrometer: the main magnetic field Ho is held constant, ~2 is set near resonance for the nucleus to be irradiated while ~1 is varied uniformly over the range necessary to record the resonance ofthe other nucleus. Figure 8.57illustrates the two types
THE A N A L Y S I S OF H I G H R E S O L U T I O N S P E C T R A
463
of spectra recorded under the two methods of sweeping. For most instances, there are four transitions for the frequency sweep method and three for the field sweep method. Freeman and Whitfen (11s) concluded that there is no advantage to be gained in using the frequency sweep method when spin deeoupling is to be used for the determination of the relative signs of coupling constants or for L~
I
O
-1
-2
-3 2.0 ~.6 1.2 L 0"8
3,4
0.4
o
-2
-!
0
I
2
3
FIG. 8.59 The transltion frequenciesD and inte~ifies L for the A rmonanoe of an AX2 spin system plotted against the offsetpm'ameter ,4 for 7Ha/2~ - ?AX. /1 and .O are defined in the caption to Fig. 8.58. Anderson and Freeman(1al) the accurate determination of chemical shifts c12°). There may be an advantage in the method, however, in the simplification of complex spectra (but see reference 154). Anderson and Freeman c1~I) have extended the calculations made on AX spin systems, to A , X , spin systems, where m, n ~ 3 and the X nuclei are subjected to the strong field H~. The spin decoupling behaviour was again presented in graphical form--this time in such a way that A and X may be identical or different nuclear species. Examples of the AX2 spin system are shown in Figs. 8.58 and 8.59 (these diagrams suffice in fact for aI1A~X2 molecules). The experimental variables are o>1, oJ2 and Ho. In the frequency sweep experiment,
H I G H R B S O L U T I O N NMR S P E C T R O S C O P Y
¢o2 and Ho are held constant or locked together and o~1 is swept through the A resonance. The resulting spectrum may be predicted by drawing a vertical line on Figs. 8.58 and 8.59 atthe requisite value ofA. The field sweep experiments entail holding cox and ~o2 constant while slowly varying Ho, corresponding to varying ~2 and A simultaneously. On the figures a line can be drawn of slope 7A/TX and the intercept on the A axis is called A', corresponding to the deviation from the resonance condition of o~z when o~, = O~A. For 1H-XH decoupling the slope of the line is very close to 45 ° and A' = [2~ c~AX -- (C01 -- C02)1 1 ( 2 ~ IJl)
that is, the discrepancy between the chemical shift and the frequency difference between the two oscillators. Figure 8.58 relates to a weak perturbing field from oJ2; here the frequency sweep method produces spectra which are easy to interpret but the field sweep technique gives complicated results. Double irradiation of AXm type spectra with intermediate values of H2 gives an easily recognisable pattern for A' = 0 (essentially a singie line in the field sweep experiment); for A,X,, spin systems there is a residual splitting which makes recognition of the exact resonance condition considerably more difficurt. Strong//2 fields applied at exact resonance cause muRtiplets to collapse to a single line in AX~ type spectra: A,X,, systems having n > 1 have a residual splitting of the resonance line that persists as/-/2 is increased. The splitting is (2#JAx) 2 (mA -- ½)/7H2 tad sec -1 for 7H2 >> 2.~ IJ]. When the fields are sufficiently strong to satisfy the condition )'H2 > ½ IOJA -- OXl this splitting begins to increase with increase of H2, reaching 2Z~JAx rad sec -1 in the limit. These results illustrate that caution shourd be applied when using the simple pictures of the spin decoupling process--usuaUy the strong field H2 is regarded merely as causing rapid transitions of the X nuclei thus "washing out" the A group muRtiplet ct22). Electric field perturbation or moduRation of a high resolution N M R spectrum appears to be a promising technique o5 x): the theoretical spectra from spectra of spin ½ nuclei, where quadrupole effects are absent, have been calculated for the AB, AB2 and AX3 spin systems. An example wiU now be given of the use of double irradiation in evaluating the chemical shift difference between two coupled groups in a complex spectrum. Turner cx2s) has reported a singie sideband technique (see Section 6.8) which gives decoupled spectra free from unwanted sideband resonances. The modulation frequency ~m is fixed and the spectrum is recorded by sweeping the applied magnetic field; under these conditions only those nuclei are decoupled whose resonances are separated exactly by ~'m. Turner c~2a) demonstrated the method with clerodin, which has the partial structure
c-cI}(
THE A N A L Y S I S OF H I G H R E S O L U T I O N S P E C T R A
465
The ~H resonance spectrum is shown in Fig. 8.60. The lowest field group is a triplet (3 ffi 3.80) assigned to the ~-hydrogen; the next group of bands is a doublet (3 = 4.09) arising from the hydrogen nucleus flanked by two oxygen atoms, and which is coupled to the p-hydrogen. The resonance of the/~-hydrogen nudens could not be separated from the complex spectrum but the signal would be expected to appear at either ~ _~ 8 or at about z = 6 to 7, depending on whether the/~- and ),-hydrogen nuclei are equivalent or not. The position of the p-hydrogen band was found by observing the low field doublet with a H
I3~' cycles sec"
/~ #Yl
(o)
t" = 3"80 4"09
-f'130
136
1.31
137
6" 53
132
138
13,:t
139
135
(b)
140
Cycles sec "z
FIG. 8.60 (a) The XHresonance spectrum at 56.4 Mc sec-z of clcrodin, (b) the low field portion of (a) in the presence of the indicated ~deband froqumcies. "~r1~123)
radiofrequency ~i while applying a strong field at ~I + ~,. The doublet will show optimum decoupling when ~m is equal to the chemical shift difference between this hydrogen nucleus and the /~-hydrogen nucleus. Figure 8.60 Co) shows the low field spectrum for different values of ~=, from these it was found that the chemical shift difference is 137 cycles sec -~ giving T = 6.53 for the ~-hydrogen.
Signs of coupling constants. A valence bond treatment of spin-spin coupling constants predicts that coupling constants may be either positive or negative depending on the molecule considered. The simple nuclear resonance experiment is unable to provide the absolute signs of the constants; this may be seen by observing that changing the sign of the Hamiltonian leaves the theoretical spectrum unchanged"='). It is possible in certain cases, however, to obtain the relative signs of the coupling constants by (i) a full analysis of the spectrum of a strongly coupled spin system, (ii) spin-decoupling of nuclei in a particular spin
466
H I G H R E S O L U T I O N NMR S P E C T R O S C O P Y
state c12s), (iii) examination of double-quantum transitions (see Section 8.19.4). All three methods require that there are at ]east three groups of non-equivalent nuclei(1 s. 126).
(i) Signs of coupling constants from spectral analysis. The details of the analysis of common systems have been discussed fully; the information obtainable on the relative signs of coupling constants in a particular spin system may be found by reference to the relevant section in this chapter. The relative signs affect only those systems in which at least two groups are strongly coupled so that the simplest system of interest is ABX. Here it is possible to obtain the relative signs of J^x and JBx, but not of JAn with respect to JAX or JBx, by comparing t h e distribution of line intensities. In general, the systems A , B ~ and A.BpCq can yield the relative signs of all three coupling constants but the differences between spectra computed with different sign combinations may be very small. (ii) Signs of coupling constants from spin-decoupling experiments. Weakly coupled spin systems are particularly suited to the spin-decoupling method because unequivocal results are obtained c~3°). Between the extremes of weak and strong coupling lie many spectra of intermediate complexity in which some of the observed transitions are recognisable in terms of first order theory. Here, and in the case of strongly coupled systems, double resonance cannot be used in its simple form. The necessary refined experiments are described later in this section. The spin-decoupling method t o b e described now aims at decoupling only those nuclei in a particular spin state, for example, 0c or/~ for nuclei of spin 3. It is essential that there should be no mixing between different product spin functions, that is, the interactions must be first order. The simplest system is therefore APX which contains three weakly coupled nuclei; systems such as ABX introduce further complications. Section 8.19.1 describes how spin systems may be converted to the weakly coupled type. For an APX system, the Hamiltonian may be written ,~ ffi~A[A, "I-1'p[pz -l-~x/xz 4- JAplA, Ipz -I"JAX[A~[Xz "~ YpxIp, lxa and the transition energy for a group A transition is ~A + m x J A x +
mpJ~.
(8.277) (8.278)
Similar expressions can be written for X and P transitions. W h e n IA = Ip = IX = ½ then mA, rap, m x can have the values + 3, that is, o~ or : states, and there are four transitions in each group. If all the coupling constants have the same sign (positive)and J^p > JAx, then the four .4 transitions in order of
decreasing frequency are
THE ANALYSIS OF HIGH RESOLUTION SPECTRA
467
In transition A 1, mx and m~, are both + ½, corresponding to the nuclei P and X both being in the spin state o~. Table 8.63 shows the spin states of P and X for each A transition, also of A and X for P transitions, and A and P for X transitions--assuming that all the coupling constants are positive. The transitions in each group are numbered so that number 1 is at the highest applied field. T~SL~ 8.63 S ~ STAT~Sor N~OHBOLraS~ AN A P X S~'r~M Neighbouring nucleus
Al A2 A3 A4
A X
~
~
Pz P2 P~ P~
Xz X: X3 X~
=~p~
~
~
~
~
In Table 8.63 it has been assumed that JAP > JAx J'~P > Jpx YPx > YAXNow it is possible to irradiate at a frequency midway between transitions AI
and ,42 with a radiofrequency field strong enough to decouple the A nuclei but without disturbing transitions A3 and A4. This has the effect of decoupling . only those nuclei in the A group which belong to the spin states having P nuclei in the spin state ~, so that if the X group transitions are simultaneously observed then transitions X2 and X1 are.seen to collapse to a single line, while transitions Xs and X4 remain unaffected. If all the coupling constants have a negative sign then o~ must be interchanged with/~ everywhere in Table 8.63 and there is no difference in the de,coupled spectra. If, however, J ~ and Jl, X are of opposite sign such that J~F is positive, then in Table 8.63 the0~ and/~ must be interchanged in r o w P of column X, and row X of column P; irradiation of transitions A1 and Az causes transitions X3 and X, to merge. Hence it is possible to ascertain the relative signs of JAP and JPx. Similarly, by irradiating at a frequency midway between transitions P1 and P2 and observing the X transitions one can decide the relative signs of Jxx and JAr; like signs cause X1 and X3 to coalesce while unlike signs cause X2 and X, 4o merge. In any particular APX system the ease with which the signs of coupling constants may be determined depends upon the proximity of transitions At, A2, As, A4 etc., and upon how good is the first order approximation for the interactions. Consider the IH spectrum of 2-furoic acid at 60Me sec-' ; Fig. 8.61 shows that the spectrum is almost first order. The magnitudes of the chemical shifts and coupling constants obtained directly from the spectrumc~°) are cycles sec -1 cycles sec -1 IJAXl = 0"8 cycles sec-1 ~A -- *F ----39"2 cycles see-' ~'x -- ~A ----31"6 cycles see-1. IJ^PI =
3.5
IJ~xl =
1.8
468
HIGH RESOLUTION
NMR SPECTROSCOPY
If JAe and Jl, x are of the same sign then irradiation midway between transitions 5 and 6 with a radiofrequency of strength y H I > = IJAx[ should collapse the transitions 9 and 10. If JAe and Jex are of opposite sign the same experiment would collapse lines 11 and 12. It was found that lines 9 and 10 coalesce (see Fig. 8.62). The relative signs of JAx and JAP may be obtained by irradiating midway between transitions 1 and 2 and observing the X part of the spectrum. For like signs lines 9 and 11 would coalesce and be superimposed upon line 10,
H x~ 0 / , ~ C OOH
I0
8(111
4 .
FIG.8.61 The IH resonancespectrum of 2-furoicacid at 60 Mc sec-2. Freeman and WhilTen 0"27) H
(Q)
9 I0 II
(b)
(c)
I0 II
J2
(a)
FIo. 8.62 The X part of the spectrum shown in Fig. 8.61 but with simuJtaneous in~diafion of (a) transitions 5 and 6 Co) transitions 7 and 8 (c) transitions I and 2
(d) transitions 3 and 4. Freeman and Whiffon(127) hence the spectrum would appear as two lines with tine 10 three times its normal intensity. If JAP and JAx are of unlike signs then line 11 would appear to be three times its normal intensity. Figure 8.62 shows the observed X part of the spectrum under various conditions of irradiation and it is evident that JAX and dAe have the same sign. When the interactions in a molecule are not entirely first order, for instance in an ABX system, then the interpretation of the spin-decoupled spectra is more difficult. It is not possible to separate the transitions into pure A, B or X types; attempts to decouple A and X nuclei by irradiation of A transitions also decouples B and X. trans.Crotonaldehyde provides an example of a limiting
THE ANALYSIS OF HIGH RESOLUTION SPECTRA
469
case a n d illustrates h o w sensitive spin-decoupling can be (.28). The ~H spectrum at 60 M e sec -x is o f the A B P ~ X type. H(X) (A)I-L,\
~C=O
/C=C, cH3 \H(B) (PD H
S
X
t
v
A
X
B
yz
P
F]~. 8.63 The 1H resonance spectrum of liquid trans-crotonaldehydeat 60 Me sec-1. The vertical arrows indicate the location of the strong field H2 used in the double irradiation experiments. Freeman (12s) Pople and Schaefer (sl) have analysed the spectrum and obtained the following coupling constants and chemical shifts:
IJABI = 15..6 cycles see-*
]JAe] =
6"8 cycles sec -1
IJBP] =
1-6cycles sec -1
IJexl =
7"8 cycles sec -1
[JA x I =
0.0 cycles sec- 1.
It was established t h a t JBP a n d JAP have opposite signs. The frequencies measured f r o m the centre o f the X doublet are s = 144.0 cycles sec-* t -- 159.6 cycles sec-* u = 193.7 cycles s e c - ' z~ = 201.4 cycles sec -a w = 209.1 cycles sec- 1 x = 216.9 cycles sec-* y = 450.3 cycles see=' z -- 451.7 cycles sec -1.
All frequencies are precise to + 0"2 cycles sec-*.
470
HIGH
RESOLUTION
NMR
SPECTROSCOPY
Applying the same method as that used for 2-furoic acid, it would seem most convenient to locate the strong de.coupling field/-/2at s or t and to observe the P part of the spectrum: a and t are separated by 15.6 cycles sec -~ and are easily distinguishable. If JAB and JsP are of like signs, applyingH2 at a would collapse lines 1 and 3, whilst the same two lines would collapse for unlike signs with//2 applied at t. However, irradiating at s or t does not produce such a simple pattern at P since besides eliminating the coupling Jm, it also affects the coupling between the B and P groups. The problem may be overcome by applying
.,.
,,
/~, x!<
g
×
~. X
xl ....
\
--X . . . . . . . . . .
/
/~d
,x. . . .
X
X \
X./ L'<
/ I 288
® i 290 Modulo)ion
@ " ××× I 292
frequency,
cycles
294 s e c -I
Fxo. 8.64 The peak height of the line from the collapsed high field 'A' quartet of Fig. 8.63 plotted against the modulation frequency. The point y represents the freqtmncycalculated for opposite signs of J^a and -geFwhile z is calculated for like signs. Below the dotted line, the resonance signal may be resolved into two bands. Fr~man (x2a)
H2 at P and then observing the A part of the spectrum. The high field quartet of the A part of the spectrum coalesces w h e n / / 2 is located at z for like signs of dAB and JBP or at y for unlike signs. Now, y and z are separated by only 1-4 cycles sec -t so that i f / / 2 is too large it may decouple all the P nuclei: the condition must be observed that 7/t2 ~ ~ IJAPI and it is essential t h a t / / 2 is accurately calibrated " s ) . This strong irradiation was set at a value o f ] I JAPI and the spectrum recorded in the following way. A constant modulation frequency was maintained with Hx being provided as a sideband of the decoupling f i e l d / / 2 . The spectrum was traversed by varying the main magnetic field. Decoupling is at its optimum value for the high field
T H E AISALYS1S OF H I G H R E S O L U T I O N S P E C T R A
471
quartet of the A part of the spectrum when the modulation frequency is (t - z) for like signs and (t - y) for unlike signs of J x . and J=~,. As decoupling becomes more efficient the two central lines of the quartet move together and eventually coalesce; a good indication of the extent of decoupling is the height of the coalesced peak or the height of the minimum between them before coalescence. Freeman <x2e) recorded the spectrum for different values of the modulation frequency and measured the peak height of the coalesced quartet for each frequency. Figure 8.64 shows the peak height plotted against the modulation frequency--it is seen that the maximum occurs when H . -- (t - y) and this position is easily discernible from the position H= = (t - z) confirming that JxB and JBr have opposite signs. It is important to note that the sideband and centre band (//~ and H2) are not separated by exactly the modulation frequency~18). If o~. is the modulation frequency in radiaus sec -~, 7 the magnetogyric ratio of the proton, and H= the centreband radiofrequency field in gauss, then oJ, the separation of the sideband and the centrehand is given by 2 -- ~,,2//2. 032 -~- r..Vm
Under the conditions used of a large value o f / / 2 , oJ is just significant (0"2 cycles sec-:) and the modulation frequency must be adjusted accordingly.
The application of a weak second radiofrequency field, H=. By devising some very sophisticated experiments, Freeman and Anderson c~29) have shown that the application of a weak second radiofrequency.field provides information on the ordering of energy levels and the relative signs of coupling constants. Since the second field is merely a perturbation, spin-decoupling does not occur and the phenomenon is best described as double resonance. The frequency sweep method of recording is best used in conjunction with low values of ti2 because one wishes to examine the effects of perturbing each spectral line in turn. Thus when frequency o~1 is swept to record the spectrum, w2 remains on resonance for a particular line or group of lines being perturbed. The main value of the experiment is that it provides information on strongly coupled spin systems and Freeman and Anderson (x=9) have developed an appropriate theory to interpret their results. Information on the arrangement of the energy levels in a spin system can be obtained from the three rules: (i) Any transition which has an energy level in common with a non-degenerate transition excited b y / / 2 will be split into a doublet. (li) If the magnetic field inhomogeneity exceeds the natural line width, the doublets may be either well or poorly resolved; the behaviour is determined by the arrangement of energy levels. Should one of the energy levels be common to both transitions then the other two levels may have either the same magnetic quantum number, giving a sharp doublet, or quantum numbers differing by two units, giving a broad doublet. ('Ri) The magnitude of the splitting in the well resolved doublets depends not only upon H= but on the square root of the intensity of the irradiated line.
472
HIGH RESOLUTION NMR SPECTROSCOPY
The simplest strongly coupled spin system is the type designated AB (Section 8.13.1). The energy level diagram and transition frequencies are shown below: The effect of applying a weak second radiofrequency field (to=) to line ,4= and
Az Az
Bi
Bz
then investigating the remainder of the spectrum by frequency sweeping (tot) is shown in Fig. 8.65. The beat frequency occurring in the spectra is caused by the passage of tot through oJ2 during the scan. When oJ2 is set on line At, A2 is unaffected having no energy level in common with At; B1 is split into a
H __.__H
A~
E}=
Cl". '~3Br "\S /
{Of -,
J
l (b)
(c)
Fla. 8.65 The fre,q u e a ~ swept IH spectrum of 2-bromo-5-chlorothiophene (AB spin system) (a) unperturbed, Co)with a second radiofrequency oJ~ centred on lineAr, (c) as in CO)but with oJ2centred on line A2. Freeman and Anderson ~129)
473
THE ANALYSIS OF HIGH RESOLUTION SPECTRA
well-resolved doublet, and B2 into a broad doublet as predicted by the theory. When to2 is set on line .d~, AI remains unchanged; BI is split into a poorlyresolved doublet and B~ into a sharp doublet. These results confirm the prediction that the matrix elements which determine the relative intensifies of the AB lines also appear in the expression for the splitting. The square of the ratio of the splittings of the well-resolved doublets of Fig. 8.65 Co) and (c) is approximately equal to the ratio o f the intensity of line A~ with respect to line At B
C
A
t01 m
J J:J j _ iA
+
-
)B
+
+
Ic
4"
-
-
+-
+
÷
+-
--
-I-
-
+
+
-I'-
-
FIG. 8.66 The A B C pan of the*H resonance spectrum of 2,3-d/bromopropioni¢ acid, CH2BrCHBrCOOH. The spectra were obtained by sweeping the frequency wl : an indication is provided of the A, B and C quartets and of the spin states of the two neighbour hydrogen nuclei during the transition. The four lower spectra were recorded with double irradiation of lines (24, CI, B4 and A4 with the second radiofrequency o~2. Freeman and Anderson(~29)
474
HIGH RESOLUTION NMR SPECTROSCOPY
(0.43 compared with 0-403. Exactly comparable conclusions obtain for irradiation of lines Bx and B2 by coz. Freeman and Anderson extended this series of experiments to an asymmetric three spin system ABC. They chose to study a rather weakly coupled spin system, namely three of the hydrogen nuclei in 2,3-dibromopropionic acid. Figure 8.66 shows some of the results obtained from the double irradiation. From a single experiment of this type it is possible to deduce the relative signs l(a)
C4 C3
C2 Cl
B4
B3,B2
(b)
BI A4 A3 A2 AI
l
i
(c)
FIe. 8.67 (a) Frequency swept XH spectrum of 2,3-dibromopropionicacid, (b) double resonance spectrum with co2 centred on line Ba, (c) double resonance spectrum with oJ2 centred on line B2. Freeman and Anderson(x29~ of the coupling constants because the observed spectrum corresponds to only one of the four possible arrangements of the spin table shown at the top of Fig. 8.66. This is a refinement of the usual double resonance technique for sign determinations since there is no longer the requirement that the chemical shifts be large compared with the coupling constants. A disadvantage of the spin decoupling method described earlier in this section is that it is often impossible to perturb large spin couplings without also affecting smaller couplings. An excellent example of the sensitivity of the method under discussion is illustrated in Fig. 8.67. Here, it is possible to distinguish the optimum setting of~o2 for line Bz compared with line B~ despite the fact that the two lines are not resolved in the normalspeetrum. The two settings of ~o2in Fig. 8.67 differ by only 0-4 cycles see- x. The weak radiofrequency field technique has been applied to the detection of hidden or weak signals (144). Here it may be assumed that this field perturbs only one of the hidden transitions at a time and any transition having an energy
THE A N A L Y S I S OF H I G H R E S O L U T I O N S P E C T R A
475
level in common with the irradiated transition will be split into doublets ¢Ia9). The behaviour resulting from irradiation of degenerate transitions can be predicted from the theory of Anderson and Freeman (x21). The experimental arrangement requires a field-frequency control loop to be incorporated in the spectrometer. In this type of double resonance experiment two types of sweep are utilised. The first maintains constant the perturbing frequency w2 set on a transition; other transitions are examined by sweeping the main radiofrequency col. Any line which has an energy level in common with the transition perturbed by oJ2 will be split into a sub-multiplet. Additionally, the Overhauser effect may be observed--intensities of some of the lines may be changed because of a population redistribution caused by the transition saturated by the perturbing field H2. In the second method, COl is kept at the peak of a given transition thus monitoring this signal while the perturbing radiofrequency is swept through another part of the spectrum. Whenever the perturbing radiofrequency field sweeps through a transition which has an energy level in common with the transition monitored by oJ~, the intensity of the monitored signal decreases because the line itself has been split and parts of it have moved away from the monitoring frequency. The method promises to be valuable in obtaining 13C coupling constants and chemical shifts. Freeman and Anderson (~4'~) applied the method to a study of 1,l,2,2-tetrabromoethane. The 1sO satellites in the ~H resonance spectrum were examined while the lsC nucleus was subjected to a weak perturbing field. The magnitudes of the xsCX2CH coupling constant and the IsC chemical shift were evaluated. The experiment also provides information on the relative signs of coupling constants: J13c~2cn has the same sign as J13c~ and J , , . Because the second radiofrequency field was derived by audiomodulation of a subharmonic ofoJ x , it is possible to relate the ~sC chemical shift to any XH reference signal with an accuracy of about 1 in 10e. More complicated spectra could be studied by irradiating each xsC site in a molecule in turn since any given molecule contains only one xsC isotope CxSs). Kaiser ¢x.5) has used the internuclear double resonance method (INDOR) for tackling the problem of assigning energy levels: intensity changes can be obtained from the INDOR spectrum without the need for comparison with the conventional spectrum. Transitory irradiation can also help in assignments of energy levelsCX56"x57).' Anderson, Freeman and Reillyc~'~) have indicated the close parallels between double resonance with weak perturbing radiofrequency fields and double quantum transitions excited by a single strong radiofrequency field. The latter experiment may be regarded in some ways as a special case of double resonance. Both methods may be applied to the problem of assigning NMR transitions to an energy level diagram--for strongly coupled spin systems it is an important step in the analysis, for weakly coupled systems the relative signs of coupling constants can be obtained from the spectra. The choice of method depends upon the instrumentation available. Baldeschwieler has presented a density matrix description of double reso]~C~(I $8,1$9).
476
HIGH RESOLUTION NMR SPECTROSCOPY
Spectra of intermediate complexity. The analytical method can be used to determine the relative signs of coupling constants in spin systems having intermediate coupling but it may be necessary to increase the strength of the coupling by recording the spectrum at lower field strengths (see Section 8.19.1) to confirm the result. Another approach is to use spin-decoupling. The influence of the second radiofrequency on the spectrum has to be interpreted by solving the double resonance Hamiltonian applicable to the particular molecule and for a given irradiating power and a given frequency. The method is described both above and in Section 6.8. Baldeschwieler And his co-workers (131" 132. 136. 13v) have extended the calculations to the nuclear magnetic double resonance spectra of general three-spin systems. A third method is to apply a very weak perturbing radiofrequency field in a double resonance experiment (133). The theory of the technique has been given above c129). The method differs from the spin-decoupling experiment (7H2/ 2# ~ J) in that the spectrum approximates to the weak coupling case, providing that the lines irradiated can be identified on a first order basis. Freeman and Bhacca (133) applied the method to analyse an ABCX3 spin system (the XH resonance spectrum at 60 Mc sec -x of 1,2-dibromopropane). The signs and magnitudes of the coupling constants found are in agreement with those determined by Finegold (134) who recorded the 1H spectra at two different spectrometer frequencies and analysed the spectra with the aid of an electronic computer. Triple resonance. In a nuclear resonance spectrum of an APX spin system, the relative signs of JAe and Yrx cannot be determined by double resonance-if JAX is unobservably small or zero (135). The situation arises when long range coupling constants are involved as for example in the 1H resonance spectrum at 60 Mc sec -1 of 3-bromothiophene-2-aldehyde, HA Br IJex[ = 1"4 cycles sec -1 ~_/ He_,// I?_CHxO lJAel = 5"1 cycles sec -1 s I JAxl < 0"2 cycles sec -1. Here the application of a weak radiofrequency H2 does not remove the degeneracy of the transitions (135). The problem can be solved by simultaneous irradiation with three radiofrequency fields(1~°). The interpretation of a triple resonance spectrum is simple only when/-/3 can be regarded as a small perturbation compared with//2, while/-/2 must be sufficiently small to perturb one P transition without appreciably affecting any of the others. These conditions require extremely high resolution, and sweep rates of the order of 3 cycles sec -1 min -1. Sweep rates as low as this are normally attainable only when a fieldfrequency control loop is present. The triple resonance technique was used to establish that JPx and JAP have the same sign (probably positive) in 3-bromothiophene-2-aldehyde (135). The determination of signs of long range coupling constants forms an important step in the understanding of the mechanism of such coupling. Transitory selective irradiation methods can be used when JAX is small in an APX system(ls6"1sT).
THE ANALYSIS OF HIGH RESOLUTION SPECTRA
477
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CHAPTER
9
T H E E F F E C T S OF C H E M I C A L E Q U I L I B R I A AND MOLECULAR CONFORMATIONAL MOTION ON NMR SPECTRA IN CHAPTER2 it was seen how assemblies of magnetic nuclei behave, without reference to the nature of the molecules containing them. Two important effects were noted, namely, that if thermal relaxation times are too long the spectrum is easily saturated and cannot be observed; if the relaxation times are too short, the stationary states become broadened and wide spectral lines are recorded. The spin-lattice relaxation time TI must be longer than 0.1 sec if one is interested in obtaining high resolution spectra. Thus the remarkably long timescale inherent in nuclear resonance spectroscopy allows many phenomena occurring in shorter times to affect the resonance signal. It is obvious then that a full understanding of an N M R spectrum requires not only a knowledge of analysis (as described in Chapter 8) but an awareness of spectral changes brought about by chemical kinetic effects.(sTs-ssl) This chapter deals with the application of N M R spectroscopy to the study of molecular and electronic motions in liquids arising from electron transfer, bond switching (as in hydrogen bonding), molecular association, ion association and ion solvation, internal rotation and conformational motion of organic ring compounds. First, the basic theory will be given; secondly, examples of the above physico-chemical processes will be presented. 9.1 THEO~HC~ CONSIDERATIONS 9.1.1 Introduction
Chemical reactions which proceed slowly to completion can be studied by nuclear magnetic resonance simply by adopting a normal analytical procedure to measure the concentrations ofr eactants or products (see for example reference 1) from absorption band areas. The shortest time required to make such a measurement is of the order of one minute. This restriction excludes the study of transient species, where there are further restrictions due to the lack of sensitivity and to the short lifetimes which cause uncertainty principle broadening. The power of the N M R method is really felt in its application to equilibria in which the forward and backward reactions are occurring very rapidly. When a magnetic nucleus can undergo exchange between two structural sites in a HRS.
16
481
482
HIGH
RESOLUTION
NMR SPECTROSCOPY
moleculeit may have a different chemical shift in each of the two environments. If the process is slow then two separate signals wiU be observed for the resonance of this nucleus. If now the process can be accelerated, say by raising the tempefuture of the system, then eventually the rate of exchange will be sufficiently rapid to give a single resonance line appearing at an intermediate position. The rate required for the averaging of the different electronic environments is determined by the magnitude of the chemical shift difference 0: the time for the exchange will be of the order of 0 -1. Similar reasouing applies to the averaging of spin-spin multiplets. The Arrhenius equation applies as usual 0 = 0o
exp(-zlH/R~
(9.1)
where 0o may be identified with the familiar frequency factor (kT/h) and evaluated as 6.25 x 10tz at room temperature when a rough estimate of the heat of activation A H is required at this temperature. When several temperatures are accessible then 0o can be determined experimentally. In practice, 0 can range from several cycles to several kilocycles. Since 0 is dependent upon the main magnetic field strength, it is possible to obtain a suitable value of 0 for an exchanging system by this means as well as by adjusting the temperature of the system. Development of a theory of spectra observed from intermediate rates of exchange was first accomplished by Gutowsky, McCall and SlichterCZL Extensions of the theory were made by Gutowsky and his co-workersTM 4) and by Meiboom and his associatescs' 6, 7). McC6nnelFS) has used a more direct treatment in which he has shown how the Bloch equations (see Section 2.9) can be generalised directly to include the effects of chemical exchange. McConnell's arguments will now be considered. The reaction was supposed to be the rapid reversible exchange of the magnetic nucleus X between the two molecular environments A and B. It was assumed that the transition state was sufficiently short-lived for magnetisation changes during the transfer of X to be disregarded. These assumptions rule out effects which might result from exchange between the same sites. The first order lifetimes of X in A and B are denoted by TAand by "re, that is, the probability of X at A moving to B is tgx and vice versa: u and v are the components of the X nuclear magnetisation which are in phase and out of phase respectively with the rotating radiofrequency field, and Mz is the component of the X magnetisation in the direction of the main magnetic field. Reactions of any order can be included but then the lifetimes of TA and ee have to be related to pseudo first order rate constants. The magnetisations can be written as the sum of the contributions of the A and B systems as follows u -- uA + us
(9.2)
v = vA + vs
(9.3)
Mz = M~ + M B z
(9.4)
THE EFFECTS OF CHEMICAL EQUILIBRIA
483
The Bloch equations as modified by McConnell are d UA
dt
--
UA
+Aco^v^ =--
UB +
(9.5)
Y'2 A d uB dt
- uB = - T2 B
+AcoBVB
"CB U^
+
(9.6) "~'A
d vA - v^ ve -- A w^ u^ = 4" - - - wl MzA d t T2 A rB d vB - -
dt
-
d co. ue = ~
vB
32 n
dM~ dt
M~ oJ1 v^ -- -TI- ^
dM~ dt
-
wl ve
M~ Tie
- -
v^ + --
T^
(9.7)
~1M~
(9.8)
- -
M~ zl ^
+
- -
M~ re
(9.9)
M~ r~e
+
M~ TA
(9.10)
where M~ and M~ are the equilibrium Z magnetisations of the X nuclei in sites A and B, A co^ -- A cob is the chemical shift difference of X in the two environments, measured in radians sec-t(A oJ^ ffi coA -- oJ and .A a~B = cob - o~), wl ffi 7Hi and -1 -1 ~xA ffi T~A + TAx (9.11) T2~ = T2~ + ~ 1 .
(9.12)
These equations represent the modification of the spin-spin and the spin-lattice relaxation times 2"2A and TI A of X in the A environment. Two similar relations apply when X is associated with B. It is assumed that Tlx, T2A, Tx~ and T2n are independent of 3^ and rB and that only chemical exchange effects bring about relaxation between the A and B systems. Equation (9.5) differs from the corresponding Bloch equation by the inclusion of the two extra terms - UA/rA and us/re to allow for the exchange process. The term - u~Jr^ measures the rate at which u^ decreases due to the chemical transfer of u magnetisation out of the A system. The term uu/rB measures the rate at which uA increases due to chemical transfer of u magnetisation from the B into the A system. Similar terms have been added in the equations (9.6) to (9.10). One of the simplest applications of the equations is to the slow passage case, as normally encountered in high resolution NMR spectroscopy, where dUA dt
dub dt
dv^ dt
dye dt
dM~ dt
dM~z dt
McConnell applied equations (9.5) to (9.10) to the system evaluated first by Gutowsky, McCalland Slichter(2~,to account for the absence of spin multiplets in spectra where they might have been expected. The particular case they worked 16.
484
HIGH
RIiSOLUTION
NMR
SPECTROSCOPY
out was the broadening of a spin doublet. If two nuclei A and B are coupled together then rapid random transitions can smear out the multiplet structure in the resonance spectrum of either A or B. This situation is discussed in Section 6.8 in connection with quadrupolar relaxation and double irradiation experiments. Here, the following simplifications result T2A
T2. -- 7'2
=
'~A ~
~B ~
~'
M~ = M~ = Mo/2.
Providing that saturation is avoided, then M zA and AI~ can be replaced by ½Me. Combination of equations (9.5) and (9.7) gives (T~* - iA oJ~ GA = GB zg 1 -- ½i O~l Mo
where
GA =
and
Gn - uB + i r a .
(9.13)
u^ + i v^
Similarly, equations (9.6) and (9:8) give (z[ 1 - iA cot) Ga = GA r~t _ ½i oj I A/[o"
(9.14)
The total complex.moment G is G = GA + G ~ ] =u+iv
(9.15)
Equations (9.13) and (9.14) can be solved for G when G = -
i c o l g o z{2 + [T~ 1 - ½i(A COA+ A coB)]z } ,2(w]~ _ iA co,) (,~1 -- iA e%) - 1
(9.16)
McConnell's result is similar to that obtained previously by Gutowsky, McCall and Slichter {2~. It is now necessary first to see how the rate of exchange can affect lines separated by a chemical shift and then to find its effect on spin multiplets. 9.1.2 Exchange between Two Different Chemical Environments Slow exchange. At the beginning of this chapter it was pointed out that when no exchange is taking place, two separate signals will be observed. Providing that the chemical shift difference between the sites (A ojA -- A a~a) is sufficiently large, two distinct signals will also be seen when the rate of exchange is slow (rA >> (,40~A - - A ean)-1 ,~ ZB) and the separation will be approximately A o~A --A cos. When the resonant frequency is centred at A, the complex moment GB due to site B can be neglected and therefore only GA is required. In terms of signal strength only the out-of-phase component VAof GA is needed.
THE
EFFECTS
OF CHEMICAL
EQUILIBRIA
485
This may be determi, ed by solving equations (9.5)and (9.7)by putting vB equal to zero, when (ol M~ Z2A (9.17) V ~ V ^ = - - ~2A A w 2A + I Taking into account the fractional populations P^ and Pa at the A and B sites, then
M~ = p^Mo TA
p^=
where
and
M~ = Pe Me
and
PB =
-f^ + T B
TB
(9.18) (9.19)
T A -{- T~
Equation (9.17) then becomes v =
COl pA M o "¢2A 2 + 1 z],~ A co^
(9.2o)
At the exact resonance position A, A oJ^ is zero hence the signal at A will be broadened according to the parameter z2 A, the line width being z~. From equation (9.12) it will be seen that the line broadening in excess of that from the normal spin-spin mechanism has been increased by the chemical exchange process. The resonance at position B has similar properties and the broadened signals at both A and B can be used to estimate ZA and zs from line width measurements prodding the absorption bands do not overlap.
Very fast exchange. The expression obtained by Gutowsky and Saika (a) corresponding to equation (9.16) is G = -
ioJxMo[(~^+ ~B) + TAZB(~ApB + 0~px)] (1 + ~ ^ T ^ ) ( 1 + ~ a r e ) - - 1
where
~^ = T2~ - i(co^ - to)
.,,,
r~
i(co~
o~) J'"
(9.21)
(9.22)
For conditions of rapid exchange zA and za will be very Small hence terms containing Z^ Z* can be neglected and equation (9.21) approximates to the relationship i col Mo (z^ + ZB) G = (9.23) oc^ T A q- 0¢a T B
The imaginary part v is given by v= -
col Me T2 1 + z2(p^ o~^ + PB con - c°)2
(9.24)
Thus resonance absorption occurs when the radiofrequency 60 has the value co = p^ co^ + PB co~.
(9.25)
The position of the single line depends upon the population of the two sites. When exchange is sufficiently rapid, equation (9.12) reduces to ~ ffi T]~, that is, the line width is independent of the exchange process and is determined by the spin-spin relaxation mechanism only. For this condition the line width is given by
z] 1 = T] I = p^ T]~ + P. Tin1.
(9.26)
486
H I G H RESOLUTION NMR SPECTROSCOPY
When the exchange rate does contribute to the line width, equation (9.26) has to be replaced by c9) 2 2 A COA-- A cos) (3A + ZS). T~t = p A T ~ + Ps T -zsI + P~d~s(
(9.27)
Since the line width in the high resolution experiment is limited by inhomogeneity of the main magnetic field, clearly there is a limit to the contribution from exchange which can be measured. Faster rates can still be determined, however, because the natural line width (~T2) -x can be measured by transient methods such as spin echoes and adiabatic fast passage. Intermediate rates o f exchange. The limiting cases just described allow considerable simplification but now the full equation (9.21) must be taken. Gutowsky and Holm c`) expanded this equation and found the imaginary part to be
coxMo[(1 + TT2 x) P + QR] v =
where
p 2 "b R 2
(9.28)
P = 3 { / ' 2 2 - [½ (coA + cos) -- col2 + ¼ (coA -- coB)'-} + T]t
(9.29)
O = 3 [½ (coA "3U coB) -- co -- ~ ( P A -- PB) (coA -- coIl) ]
(9.30)
R = [½ (e~A + coB) -- co] (1 + 23 T2 x) + ½ (PA -- PS) (OJA-- coS). (9.31) We now define 3 so that l~A T B 3 = (9.32) TA + 3B
In practice one would measure the separation between two absorption lines and from it calculate ~'. The positions of the two components and their dependence on • are given by the maxima of equation (9.28). Gutowsky and Holm <4) used the simplifications 1
PA = PIl = -2
(9.33)
3A = 3B = 23
(9.34)
T2ta =
T~
= 0.
(9.35)
On differentiating v with respect to ½(coA + ton) -- ~ the following result is obtained I~'T2 t [½ (COA + COB) -- co]s -I- 232 s ( l q- 3 T ~ t) [½(COA + ('OB) -- (0] 3
+ [(I + rT2 ~) (1 + 2zT2t) 2 - 3s(2 + 33T;.~)]s[½ ((~A + roB) - c o ] = 0 (9.36) where
s = IT2 t + "r T] 2 + ¼ 3(o~A - coB)2].
(9.37)
One solution of equation (9.36) is when (COA+ COs)/2 = co that is, when the radiofrequency has a value midway between the absorption lines at A and B. This solution corresponds to a minimum (two separate lines) when ~ is large,
THE EFFECTS OF C H E M I C A L E Q U I L I B R I A
487
and a maximum, (a single fine), when z is smaU. Applying the simplification (9.35), prevailing when the line widths are small compared with their separation, the observed separation is given by ~obs ---- (CO^ + COS)-- 2CO --'- [I -- 2 ~-2(COA -- COS)2]t½(COX-- COS) (9.38)
and
~o~ -- 0
if
z(co^-cos) > 2
ff
½z(co^ - cos) _-< ~/2.
(9.39)
The critical value for • requires correcting (1°) so that it is compared with V2l(co^ - cos).
Gutowsky and Holm also derived the general non-zero solution of equation (9.36), which includes the effects of overlapping of bands 1 2 (COA+ ~Os)
+
CO
- s(z -1 + 2"2 ~-2) + s~
(con - cos) (T] ~-3 + 4T2 z -2 + 4z-1) ½ • (9.40)
The dependence of ~ob, on z and 7"2 was computed and plotted as shown in Fig. 9.1 from. which values of [t(COA -- COS)]-1 can be interpolated from the experimental measurements of ~o~(coA -cos). The analysis is based on the assumption that the exchange rate does not influence T2. When the temperature dependence of the rate is being studied then precise work requires a correction (`) for the effect of temperature on T2--see Section 2.7 and reference 11. The treatment of Gutowsky and Holm based on equation (9.21) is not completely general in that it does not account for an asymmetric pair of fines. Rogers and Woodbrey (1~ have developed a general treatment and they have devised three computer programmes to (i) generate doublet shape functions for different rates of exchange averaging, (ii) calculate frequency positions and relative intensities of the maxima and the central minimum for different rates of exchange averaging: this programme is applicable to rates corresponding, to before, during and after coalescence of the fines, (ifi) calculate the effect of overlap on the apparent separation of any two Lorentzian fines in the absence of exchange averaging. Rogers and Woodbrey also obtained a useful simplified expression for equal mean lifetimes at sites A and B • (co^ -
cos) =
-+ - -
1 ~/2
Iv + (v 2 -
v)~]*
for ~ ]/2r(co^ - coB) > 1. The symbol ris the intensity ratio of a maximum to
HIGHRESOLUTIONNMRSPECTROSCOPY
488
the central minimum in the pair of overlapping lines. It was assumed that the effect of overlap of the two lines is negligible.
Several chemical species. Gutowsky and Saikac3~have outlined the procedure for dealing with systems involving exchange among three chemically different sites. As before, a single resonance line wig be observed if the exchange is fast; three lines will appear, having intensities governed by the population
T
0.6
~ 0-4 2
(?'21 0
0 "2
0,4
0 -6
[r C~A-~os)]"~ FIO. 9.1 Graph for the interpolation of [T(O)A-- COB)]-1 from the experimental measurements of dob=(COA-- ~%)-1.Gutowskyand Holm(4) densities of the sites, if the exchange is slow. The critical frequency of exchange is of the same order as the chemical shift difference of the components in the absence of exchange. Other general treatments of exchange narrowing for several sites have been reported c13-1s).
9.1.3 The Effect of Exchange upon Spin-Spin Multiplets In the previous section, spin coupling was neglected; now we shall consider absorption lines which consist of spin multiplets. Under conditions of slow exchange the components of spin multiplets are sharply defined. As the rate of exchange is accelerated they are broadened and caused to overlap until finally a single line results at high rates of exchange. The existence of spin multiplets requires that nuclei with fixed orientations of their magnetic moment vectors should be coupled with the nucleus under observation for a time greater
THE EFFECTS
OF C H E M I C A L
EQUILIBRIA
489
than I/J, Any rapid exchange process which continually replaces an interacting nucleus with others of random orientation will lead to confusion of the spin statesand to collapse of the multiplets.The treatment given below applies only if the chemical shift is large compared with the spin-spin interaction. The theory is also inapplicable if the exchange rate is comparable with or greater than the chemical shift.Using the restrictionof a large chemical shift,Solomon and Bloembergen (16) worked out the line broadening for fast exchange. The value obtained is twice that deduced by Gutowsky, McCall and Slichterc~) for slow exchange. Kaplan exv)has put forward a general theory which has the fast and slow rates as limiting cases. Furthermore, the theory can bc extended to include the effect of saturation, fast passage and multiple radiation on an exchanging system and also to include hindered rotation. The procedure used by Kaplan has been employed by Alexander c33"~ to treat cases where exchange takes place between differentuncorrelated molecules.
Doublet. The treatments given in Section 9.1.1 for a symmetrical pair of lines are equally applicable here. The initial quantitative theory of Gutowsky, McCall and Slichter(2), as applied to a doublet, has already been discussed along with the later theory of McConnell (s). More recently, Takeda and Stejskal¢7) have systematised the effect of chemical exchange on a doublet originating from spin coupling. They used as a starting point the total complex moment evaluated by Gutowsky, McCall and Slichterc2) G=
- c o , M e ~ [ 2 + (T2 ~ {I + ~ [ r ~ 1 -
izl o , ) 3 ]
i ( ~ o, + ½~)]} {1 + ~ [ r ~ 1 -
i ( 4 co - ½~)]} - 1
• (9.40
Where A co is the angular frequency of the applied alternating magnetic field measured about the frequency of the centre of the doublet, J is the spin coupling constant (in angular frequency units) and T is the average lifetime of the magnetic nucleus producing the doublet. The following definitions were adopted
Q=7~J [ t = zJ
/
(9.42)
Z = A ¢o/J 1" These were incorporated into equation (9.41) to give for the part of G proportional to the observed signal - - coi M e
j-I
Q{Z 2 + (2t-1 + Q-I) [(2t-1 + Q-l) + ¼Q]} Q2Z4 + Q2Z2[(2/t + Q-1)2 _l_ Q - 2 _ ½] + [(2It + Q-I)+ ¼Q])2.
(9.43) Large values of z and T2 will give two lines of equal intensity separated by J radians see-:. If T2 is large and • is infinitely large, then expression (9.43) predicts two separate lines having a separation less than J. As we have seen before, if T2 ~t$.
16a
490
HIGH
RESOLUTION
NMR
SPECTROSCOPY
remains finite as • approaches zero then a single line will result. A doublet will be resolved providing that [2t-1 + Q-I + (2Q)-1 _ 2(2t-1 + Q-l)s] > o. (9.44) The separation ~ol,, between the maxima of the absorption lines is given by
ao . __ {[4Q(2t-1 + Q_1)3 + (Q2 + 8)(2t -~ + Q_~)2 + (Q2 + 2) J
(2t-~ + Q-~) + 1]t _ 4(2t_1 + Q-1)2 _ (2t-1 + Q_I)Q}t.
(9.45)
This relationship was produced by differentiating (9.43) with respect to Z and by putting the differential equal to zero. Equation (9.45) becomes indeterminate as Q approaches infinity, and then 6ot,/J approaches (1 - 8 t-z)L In the region of moderately rapid exchange, 6oh,vanishes and one has to use the line broadening as a measure of the rate of exchange. The line width at half height ~ can be obtained from the equation J
= {[(t-t + Q-t)z (212t-1 + Q-X] _ 8t-1)2 + (4Q-l[2t-1 + Q-X] + 1)z]÷ + (t-1 + Q:I)(212t-1 + Q-l] _ 8t-1)}½.
(9.46)
As t approaches zero, equation (9.46) becomes indeterminate but -7 = Q
(9.47)
Equation (9.46) is not restricted to a singlet but may be used for a poorly resolved doublet. A similar method has been evolved by Meiboom and "his coworkers(S, ~s) The intensity of the signal can be utilised in the region of rapid exchange assuming that saturation is avoided -
I (2t -~ + Q-~) = Io (2t-1 + Q-I) + ¼Q -
(9.48)
where I is the intensity of the singlet and Io is the intensity of the line expected for z = 0. The absence of saturation can be ensured by checking that I/Io is independent of H~ and the sweep rate. The various expressions given above are cumbersome to apply, hence Takeda and Stejskal(:) employed graphical methods similar to those of previous investigators. Figure 9.2 gives the evaluation of ~ob,/J from equation (9.45) by giving ~obs/J as a function of (¢J)-Z for various values of T2J (see reference 3). Equation (9.46) is represented by Fig. 9.3 which is a plot of ~÷/J as a function of-cJ for various values of T2J (see reference 18). Figure 9.4 is a plot of I/Io against z J for various values of T2J. 7:cannot be evaluated without a knowledge of the spin-spin relaxation time 7'2 and the coupling constant J. Tz can be found from a portion of the spectrum unaffected by chemical exchange, or by making a measurement of T2 under conditions such that = = 0. J can be found from the spectrum for the condition that • approaches infinity, by successive approximation to Figure 9.2. lo can be determined from some part of the spectrum
491
THE EFFECTS OF C H E M I C A L E Q U I L I B R I A
unaffected by exchange, and having an intensity known to be simply related to lo, or by choosing the condition that T ffi 0 for one experiment. An alternative method for the region of slow exchange was developed by Loewenstein and Meiboom
il 0
0-2
0.I
O.B
0-4
( r j ) -~
FIG. 9.2 The evaluation of
~ot~/Y.Takeda
"5F
and Stejskal (~)
T~j 2.5
"
1.0
0.5
0
0.5
I,O
1.5
2"0
Td FIG, 9.3 The evaluation of
~/J.
Takeda and StejskaF 7)
from the parameters T2 and 7. The characteristic feature used to evaluate experimental data was the intensity ratio of a llne maximum to the intensity of the minimum between the doublet components. Figure 9.5 shows a plot of this ratio against TJ for various values of 2/T2J. Tables of exchanged broadened doublets have been compiled tag). Triplet. The earliest example of a triplet collapsed to a single line by an exchange process is that of the resonance of the hydrogen atom in the hydroxyl 16s*
492
H I G H R E S O L U T I O N NMR S P E C T R O S C O P Y
group of ethanol. In his classic work on the NMR spectrum of ethanol, Arnold (2°) added small amounts of sodium hydroxide or hydrochloric acid to promote the proton exchange: very pure ethanol shows a triplet for this resonance. Arnold derived the following general expression for the corn ~lex moment 1 + TT~ 1 + i(co -- co,) 7"
v+iu=-~'MoH:Tz[~1p,[l+i(co-co,)T2]+ '
"~T21
+ /(co - co~)~']
1,0
o "~
0'5-
0
.5
1.0
2.0
I•
rJ
Fie. 9.4 The evaluation of
i/J lifo.
Takeda and Stejskal (7)
o
_
°
o
I
I
T
2
4
I
', 6
',
I 8
,[ ~0
"rJ
FIe. 9.5 Theoretical curves for the ratio of intensities of the line maximum to the minimum of a broadened doublet for various values of ~J and 2/T2J. Loewenstein and M e i b o o m (is)
(9.49)
THE EFFECTS OF CHEMICAL E Q U I L I B R I A
493
where o~ is one of the severalpossible values which the resonance frequency can assume within one group. The probability p, corresponds to oJ, and has the one value of ¼, because there are four possible spin states for the methylene group and it can be assumed that on leaving one molecule a hydroxyl proton has an equal chance of attaching itself to another molecule in any one of the four orientations. Here ~ is the mean lifetime between the exchange events.
C'?
(o)
0"2 0"i
W,
W 2 to
W~
to.;
0-5 0.4
(b)
C.~ 0"2 0-; 0 [
/
~ -.
,
0.5
0"4 0"~
(c}
0-2 0.I 0
FIQ. 9.6 The line shapes o f the hydrogen resonance of the hydroxyl group of ethanol at the exchange times (a) g -- 0.21 sec (b) z -~ 0.052 sec (c) T -ffi 0.013 sec. Arnold (20)
From equation (9.49) line shapes were calculated for three values of the exchange time T assuming that T2 = 2 sec (see Fig. 9.6). From the line shapes, could be determined in the range 1 to 100 sec for various concentrations of the acid or alkali catalysts. In pure ethanol the exchange rate is about 1 sec. The triplet has also been dealt with by Loewenstein and Meiboom (Is) for conditions of slow exchange. They required the analysis in order to interpret the results obtained for the protolysis of the dimethylammonium ion: the hydrogen resonance of the methyl group is a triplet at low rates of exchange. The method they used to determine the line shape will be exemplified in the section
494
HIGH
RESOLUTION
NMR
SPECTROSCOPY
on the quartet which follows. The theoretical curves derived for the triplet, after it has collapsed to a broadened single line, are shown in Fig. 9.7. 7"2, the spin-spin relaxation time in the absence of exchange, was measured from the decay of the "wiggles" after fast passage through a sharp water resonance line (see Section 2.12). Tables of exchange broadened triplets have been compiled¢x9).
3.0
2"0
(M
2/rzJ 1.0
0.1
i 0"2
T 0"3
[ 0.4
i
I 0.6
I
I 0.8
Td
Fro. 9.7 Curvesfor the determinationof the exchangerate for a triplet which has collapsedinto a single line. Loewensteinand Meiboom¢ts~ Quartet. In their study of the fast protolysis of methylammonium in water, Grunwald, Loewenstein and Meiboom<s) utilised the hydrogen resonance spectra of the species CH3, NH~ and H20 in order to measure half-times in the range 0.002 to 0.2 sec. The species give rise to a quartet, a triplet and a singlet respectively. The quartet of the methyl group originates from the spinspin coupling with the hydrogen nuclei of NH3 in CH3NH~: there is no splitring from I ' N because the quadrupole relaxation of this nucleus is rapid compared with its spin-spin interaction with the CHs group. The triplet of the NH3 group resonance is from the I ' N nucleus, the coupling with the CH3 being obscured by quadrupole broadening. Grunwald, Loewenstein and Meiboom~s) employed the theory of exchange broadening, given at the be~nning of this chapter, to calculate line shapes of a quartet for various rates of exchange. It was reasoned that protons exchange one at a time in the NH3 group of CH3NH~, which leads to a total complex moment 2i~olMo ~[1 + s - 9(s 2 + 3 r 2) - - 9s(s z + 7r2)] G= (9.50) [s2 -- (1 -- 9rZ)] [9s 2 - (1 - 9rZ)] where s = 1 + 2z[T[: - i(coo - o~)] (9.51) and r = TJ
THE EFFECTS OF CHEMICAL EQUILIBRIA
2/TF
TJ
O-I} 0"2
~
0
"
1
}
2
0"2
~ 0 . 1 }
I
0"2
~ 0 " 1
0"2
~ 0 " 1
B
} _r 2
i
0.2}
-e
Fro. 9.8 Calculated line shapes for the CH3 quartet in CH3NH~ for different exchange rates. Grunwald, Loewenstem and MeiboomCs)
2/T2J 0
.9
I
~
I
I 2
I 3
I
I 5
t
I
I 8
TJ FIo. 9.9 Ratio of intensities of the maximum to the central minimum of the CH3 quartet in CHsNHa+ as a function of the exchange rate. Grunwald, Loewenstein and Meiboomcs)
495
496
HIGH RESOLUTION NMR SPECTROSCOPY
Once again, T2 is the transverse relaxation time in the absence of exchange and r is the mean time interval between proton exchanges in the N H ; group. The imaginary part of G is the absorption mode and is therefore used to generate the line shapes shown in Fig. 9.8. From this diagram it is seen that the slower exchange rates leave the quartet still resolvable hence it is possible to use as a characteristic parameter the ratio of the intensities of the maximum to the central minimum. The ratio can be obtained from theoretical curves like those in the upper part of Fig. 9.8 and then can be plotted as a function of r J for a
ZlT2J
*
i
i
I
TJ
Flo. 9.10 Haft width ~t of the CHa quartet in CHaNH~ as a function of the e x e l ~ g e rate. Gnmwnld, L o e w ~ t . ~ and MeiboomIsl
number of values of 2/T2J, as shown in Fig. 9.9. Faster rates of exchange cannot be measured by this method because the central minimum disappears. Then the width of the !ine at half height, 5~, can be used to determine z from plots such as those shown in Fig. 9.10 which are plots of 2t~t against ~J for various values of 2/7zJ. Grunwald, Loewenstein and Meiboom ts~ used the decay envelope following rapid passage through the hydrogen resonance line of H 2 0 in order to measure 72. A tabulation of exchange broadened quartets has been published t19).
Quadrupole interaction. In the foregoing sections, spin multiplets were seen to be coalesced by chemical exchange but it is seen (Section 6.7) that this can also occur by double irradiation or by quadrupolar relaxation. An example of the latter is provided by chlorine whose two abundant nuclei 3sC1 and aTC1 each
THE EFFECTS OF C H E M I C A L E Q U I L I B R I A
497
have spin values of ~. These nuclei are deeoupled in almost all molecules because of rapid quadrupolar relaxation induced by fluctuating electric field gradients (see Section 2.5.2). In the instance of a'N, which has a nuclear spin value of 1, one encounters an intermediate ease in which the resonance spectra are not completely decoupled. Pople (24) has developed a theory of the effect of quadrupole relaxation on N M R multiplets. In the absence of this type of relaxation, spin coupling between say a nucleus of spin ½ and one of spin 1 would give spectra comprising 3 lines and 2 lines respectively for the resonance spectra of each of the two nuclei. If the rate of quadrupole relaxation is slow, the individuallines of the two groups of multiplets will broaden. Higher rates of relaxation will cause the multiplets to coalesce into broad lines. For more rapid relaxation, the spectrum of the nucleus of 1 = 1 will broaden further while the spectrumof the nucleus of I = ½ will become a single sharp line. Pople found that the shape function of each line in the triplet is given by 2Tin gn,(~)= 1 +4~z 2 z ~ ( ~ m - ~ )
m--0,
_1
(9.52)
where T, is the lifetime of the nucleus having I = 1 in the state m. The transition probability for d m = +_ 1 is 3
40 ~ q2 Q2 h lr¢ and f o r d m = _+2is 3
e4 q2 Q2 h z~..
20 Here, Q is the quadrupole moment, q is the electric field gradient and Zc is the correlation time for molecular rotation. Because of the magnitudes of the probabilities, the lifetimes of the states m ---- :i: 1 are shorter than the lifetime of m -- 0 by a factor of ~. A consequence of this is that the broadening of the outer lines of the triplet is t as great as that of the central line. The theory has been tested on the 1H and 14N resonance spectra of anhydrous 14NHa reported by Ogg and Ray ¢zs}. The ratio of the heights of the inner to outer components of the triplet of the 1H spectrum was found to be 70 : 47.5. The lines of the 1 : 3 : 3 : 1 quartet observed for the I"N resonance were broadened more than those of the XH spectrum. Pople extended the theory to more rapid rates of quadrupolar relaxation by using the appromixation made by Gutowsky, McCall and Sfichtert2). He was able to show the collapse of a triplet for the resonance spectrum of a nucleus having I -- ½ when coupled to a nucleus having 1 = 1. Line shapes have been computed for I = 1, ~, 2, ~ and 3 ta6a).
9.1.4 Rapid Passage Experiments All the experiments described so far in this chapter have been of the slow passage type. However, McConnell and Thompson (~6) have evolved a resonance technique of the rapid passage type for the measurement of the rates of
498
HIGH
RESOLUTION
NMR
SPECTROSCOPY
fast molecular processes. A given nuclear species having two distinct environmerits A and B can be transferred between them by some dynamic molecular process such as chemical exchange or internal rotation. An adiabatic rapid passage experiment will give a non-equilibrium nuclear spin magnetisation in the A system, say. The B system wiU then take up this non-equilibrium magnetisation provided that the spin-lattice relaxation time T, is greater than the lifetime of transfer. The time interval required for B to acquire the non-equilibrium spin magnetism depends upon the rate of the transfer process and the relaxation times of the nuclei in the A and B systems. These systems may correspond to different chemical environments in a molecule, or to sites that are distinct because of electron coupled spin-spin interaction. McConndl and Thompson applied this new technique to the estimation of the rate of proton exchange between ammonium ions in acidified aqueous solution. Two proton exchange processes are possible (NI-Ia.*H) + + N H ~ ~-~N I ~ + (NHa. *H) ÷ (NH3"
*H)+ + H20 ,-~ ~
+ *H" OH.
The second process can be neglected for solutions of pH less than 2"5. Conventional slow passage nuclear resonance experiments were used to obtain the rate constants for the transfer of proton spins between different ammonium ions. The line shape was calculated as outlined in Section 9.1, treating the case of three equally probable magnetic environments corresponding to the three almost equally "probable oden~tions of the 14N nuclear spin. The general theoretical equation for the line shape is G=
where
-
7HIMo
(C + iD)
d = 363(coo - co) + 2432(coo 363
B = 27 + ~
C=
co)/T2
1232(co°- °~)2 + 32J2
(9.54) (9.55)
9 123 432 7'2 + T-'~'- 123(coo-CO) 2 + r J 2 + T2a 1232 _ ~
T2
D
1232 T 22
-t
(9.53)
432 j z (COo _ CO)2 + ~
= - 9(COo -- co)
(9.56)
(4T2)
243 ~'2 (COo- CO)
"1- 4"~2(CO0 - - co)3 __ Z2(CO0 __ CO),]2.
12'r2 T22 (COo- co) (9.57)
The line shape found from the imaginary part of G was found to correspond to a second order rate constant of 2.3 M-x see -~ or a pseudo first order rate constant of 16.6 see-* (this implies a proton lifetime of 0.06 see). A value of 0.21 see was used for/'2. The lifetime and the rate constant k refer to proton transfers between ammonium ions with different X4N nuclear orientations: the lifetime
499
THE E F F E C T S OF C H E M I C A L E Q U I L I B R I A
and rate constant for transfer between any ammonium ion are (~) z and (]) k respectively. For the fast passage experiments there is still justification for neglecting the effect of proton exchange with water molecules. There are three possible magnetic environments in which a proton can be situated, corresponding to the three possible internal magnetic fields acting at a proton, due to spin--spin interaction with the ~'N nucleus. These three environments will be labelled A, B and C. Let N A be the number of protons in state A with spin up and let N ° be the number of spins in A, B or C under equilibrium conditions. The quantities N .A, N+I t etc. are defined similarly. The various possible "states" for the proton and the appropriate instantaneous number of spins in these states may be represented /~+A . N+B Nc_ _ N_A
N_~
Nc
A
B
C
Two typical differential equations governing the rate of change of the number of spins in the various states are
aN~+
=
_ T;,(N+~
_
NO)
_
-
N °) -
dt
-
dt
k(N+~
_
N~)
_
k(N+~
_
N?)
(9.58)
kiN_
-
N_
-
k
N_" -
NS)
(9.59)
k gives the specific rate constant for proton transfer between the environments A and B. It was assumed that the proton does not relax in the transfer process since the intermediate NHs is very short-lived and is not paramagnetic. Introducing the notation ~,, = N~+ - N ° ~o then
d0 A =
or
o =
_T?X(6^
_
Oo~ _
N+
(9.60)
_ N °
k(OA _ dB) _
(9.61)
k(bA
-- ~c)
(9.62)
dt dd A ~ffi dt
_(T~
+ 2 k ) 6 ^ + T~X 6o + kf6B + (~c)
(9.63)
d~ B
dt
= - ( T ~ ' I + 2k)~S + T~-l~° + k(~^ + ~c)
(9.64)
= _ (T~-~ + 2k)~C + T~-I 8o + k(~A q_ ¢~s).
(9.65)
d/~c
dt Defining and
A =~^+~e A ° ---- 3~ °
+~c
500
H I G H R E S O L U T I O N NMR S P E C T R O S C O P Y
t h e n f r o m e q u a t i o n s (9.63), (9.64) a n d (9.65) dA =
-
T~(A
- d e)
(9.66)
dt d~ B
d6 A -
-
dt
= - ( T ~ "t + 3 k ) ( ~ A _ 6B)
(9.67)
= - ( T ~ -1 + 3 k ) (6a - be).
(9.68)
dt
d6 a
d6 c -
dt
-
dt
M c C o n n e l l a n d T h o m p s o n (26) solved these first o r d e r h o m o g e n e o u s l i n e a r differential e q u a t i o n s a n d f o u n d /1 - d ° = (d - ~1°)o e x p ( - t/T1)
(9.69)
6a - 6 B = (6a - 68)o e x p [ - ( T ~ - t + 3 k ) t]
(9.70)
6B - 6 c = (6B -- 6C)o e x p [ - ( T -1 + 3k) t]
(9.71)
w h e r e the s u b s c r i p t z e r o indicates the values o f these quantities a t t = 0. Solving e q u a t i o n s (9.69) t o (9.71) for 6 ^, 6a a n d 6 c 6 A = 6 o + -/s[26 A - 6 B - 6C]o e x p [ - ( T ~ "1 + 3 k ) t ] + {[6 a + 6 B + 6 c - 360]0 e x p ( - t/Tt) 6s = 6 o + { [ 2 b a - 0
A.-6c]oexp[-(T~
(9.72)
"~ + 3k) t] + { [ 6 A + 6 ~ + 6 c 360]0 e x p ( - t/Tl)
-
(9.73)
~c = ~o + ~[2~c _ ~A _ ~B]o e × p [ - (T? ~ + 3k) t] + ~[~A + ~B + ~C - 36o]0 e x p ( - - t/T1). (9.74) T h e f o l l o w i n g time scale was a d o p t e d . A t t = - e the A r e s o n a n c e signal was e n t e r e d , the centre b e i n g r e a c h e d at t = 0. A t t = 8 the A signal has b e e n p a s s e d a n d a t t = 2 - 8 the B signal is entered, the c e n t r e being r e a c h e d at t = L Similarly, the centre o f t h e C signal is r e a c h e d a t t = 22. I t was a s s u m e d in the c a l c u l a t i o n s t h a t ;t ~. e. F r o m e q u a t i o n s (9.72) to (9.74) the spin m a g n e t i sation c a n b e d e t e r m i n e d at a n y p a r t i c u l a r t i m e : At timer = -e
6~,=~o
6a=~o,
At timet=e
6A = 6 ° c o s 0 ,
6c = 6 °
68 = 6 °,
(9.75)
6c = 6
°
(9.76)
U s i n g the 6 values f r o m e q u a t i o n (9.76) to calculate the s u b s c r i p t z e r o terms in e q u a t i o n s (9.72) to (9.74), a n d p u t t i n g 2 - e ,~ it, at t i m e t = it - e 6^ = 6 ° + { [ 2 6 ° ( c o s 0 -
1)] e x p [ - ( T ~ -1 + 3k)it] + } 6 ° ( c o s 0 exp(-it
Tt) -1
1) (9.77)
6a = 6 ° + {-[6°(I - cos0)] e x p [ - (T~ "1 + 3k) it] + ½6°(cos0 - 1) e x p ( - i t T t ) - 1 (9.78) 6 c = 6 ° + 316°(1 - c o s 0 ) ] e x p [ - (T~-1 + 3k) 2] + -Jr6°(cos0 - I ) e x p ( - i t T t ) - t (9.79)
T H E E F F E C T S OF C H E M I C A L E Q U I L I B R I A
50l
At time t = 2, a resonance.signal proportional to ~e given by equation (9.78) is observed. At time t -- 2 .4. e, ~A and ~B have the values given by equations (9.77) and (9.79) while the B magnetisation has been tilted through angle 0, hence ~s _- ~o cos0 .4. ½cos0{6°(l - cos0) e x p [ - TI-I + 3k) ).)]} 4½/~o cos0(cos0 - 1) exp ( - ~.T~- 1)
(9.80)
/ H 9
Fro. 9.11 Oscillograph tracing of the dispersion mode of a spectrum obtained under rapid passage conditions. McConnell and Thompson(a6)
McConnell and Thompson then calculated Oc at t =- 22 by putting the OA, Oe and ~c values in equations (9.72), (9.80) and (9.74) as initial values in equations (9.72) to (9.74) and then setting t -- ~, in these latter equations. Then ~c _- 6o + ~O(cos 0 _ 1) exp(_~T~-~) [I - e x p ( - 3k ~)] x (~ exp(-2T~'l)[1 - exp(3k 2)] cos0 .4. 1 .4. ~ e x p ( - 2 T ~ -1) x [1 .4. 2 e x p ( - 3 k 4)]}.
(9.81)
Figure 9.11 represents an oscillograph tracing of a rapid passage spectrum demonstrated to be at 0 = 180 ° since the low to high field spectrum is the inverse of that shown in Fig. 9.11. This condition is attained by adjusting the intensity of the radiofrequency field. The sweep rate corresponds to 2 -- 0-41 sec. In the theoretical equations (9-78) and (9-81) the integrated intensity ratios for the three signals depend upon 0 ( = 180°), e x p ( - 2 k) (which is negligible for these equations) and e x p ( - g T i 1) (~ 0.37) giving calculated ratios of 1 : 0.75 : 0.70. The observed intensity ratios of 1.0 : 0.8 : 0.6 are in good agreemeat. Since the findings of McConnell and Thompson were published another chemical system has been shown to exhibit magnetisation transfer, namely, in the 13C resonance of the CO~-HCOg system (z".
~02
H I G H R E S O L U T I O N NMR S P E C T R O S C O P Y
9.1.5 Double Resonance The N M R techniques usually employed for measuring reaction rates are based upon line shape determination, thus limiting lifetimesto less than one second. A new method (3,t)involving double resonance has been demonstrated that is suitable for estimating lifetimes of several seconds. The method is applicable to systems in which a nuclear spin is transferred reversiblybetween two non-equivalent sites A and B. The lifetime and spin relaxation time of site A is obtained in a single experiment by observing the decay to a new equilibrium value of a signal corresponding to A upon the sudden saturation of signal B. The technique can be applied to exchange among several sites. 9.2 ELECTRONEXCHANGEREACTIONS Reactions in this class may be defined as those taking place between similar species in which the net effect is the transfer of electronsmit cannot be inferred that the mechanism is necessarily one of electron transfer, because atom or group transfer could produce the same result. Kinetic studies of reactions of this type have led to a greater understanding of the detailed mechanism of electron transfer processes(2s). In oxidation-reduction reactions involving metal ions, the advent of radioactive tracer methods has provided a simple means of studying slow reactions, for example *Fe2+aq + Fe3~-aq
, *Fea*aq + Fe2+aq.
Reactions of this type are often rapid and isolation techniques cannot be employed. From the preceding section it is obvious that nuclearmagnetic resonance can play a part here because absorption line widths provide a measure of the rates of fast processes (for very fast reactions, electron magnetic resonance can be used). The first example of this particular application was a study of aromatic hydrogen resonance line widths of mixtures of IV, N'-tetramethyl-p-phenylenediamine and its positive ion (Wurster's blue) in water (ella) a N - - O
--
N(CH) al
+
[(CH,),N- - ~
~
N(CHs)]+--" 2 ~-
Bruce, Norberg and Weissman (29~ estimated the bimolecular rate constant to be 2.5 x 104 M-t sec -t at room temperature. Broadening occurred for the ring and methyl hydrogen resonances, while the line width of the solvent hydrogen signals remained constant, thus indicating that the broadening is due to chemical exchange and not to thermal paramagnetic relaxation. The affected line widths were found to be linearly dependent upon the positive ion concentration. The method was estimated to be applicable to exchange reactions for which the rate constant is greater than 3 x 103 M-1 sec -1. :H resonance spectra have been utilised to measure the electron exchange rates in aromatic complexes of Os(II) and OsCIII), and of Fe(II) and Fe(III) (335).
THE EFFECTS OF CHEMICAL EQUILIBRIA
503
The method has also been applied to the inorganic equilibria between Cu (I) and Cu (II) and between V(V) andV (IV) by McConnell andhis co=workers c3°' a 1). This group termed the reactions "paramagnetic pulse reactions" from the following considerations. If, for example, the resonance of a ~3Cu nucleus is being examined and if this is in the cuprous state then after time tc~a) an electron is lost to a cupric ion and the eaCu nucleus is incorporated in the cupric state. The odd electron of the paramagnetic cupric ion will subject this nucleus to an intense magnetic field. After a further interval tc~(n), the lifetime of the Cu (II) state, this field will be removed as the ~3Cu nucleus reverts to the cuprous state. If the magnetic field from, and the lifetime of, the cupric state are great enough, the "paramagnetic pulse" will put the copper nucleus out of phase with the other precessing nuclei. McConnell and Weaver (a°) estimated that the eSCu nucleus loses its phase memory whenever tcutm -~ lO -9 sec. Since tc~a) ~) tc~m), the phase memory time, that is the relaxation time associated with the electron exchange process, is given by 7"2(/)) -- tc.a).
(9.82)
Because this relaxation mechanism and the usual ones due to spin-spin and spin-lattice interactions are present and are independent of one another r ; 1 = [r2(a)] -1 + [T2(b)] -I
(9.83)
where T2 is the overall spin-spin relaxation time and Ta (a) is the appropriate T~ for a.solution containing only cuprous ions. T2 (a) was found to be 2 x 10 -5 see for 1 M cuprous chloride in 12 M hydrochloric acid. The following expression was evaluated T ; 1 = ~/I H ~/3/2 (9.84) where y is the magnetogyric ratio of 63Cu and A H (in gauss) is the peak to peak separation of the experimental Lorentzian derivative curve traced out by a wide line spectrometer. In a like manner,/'2 was determined for mixtures of cupric chloride and 1 M cuprous chloride in 12 M hydrochloric acid. If N is the number of cuprous-cupric electron exchange processes occurring every second in a litre of a solution containing cuprous and cupric ions, then N = k [Cu(II)] [Cu(I)]
(9.85)
tc.,,, = { k [ C u ( n ) ] } "
(9.803
tc.(,,) = {k [Cu(I)])-'..
(9.87)
The following numbers were extracted from the equations: k=5
x 10~M-xsee -~,
t c ~ a ) = 2 x 10 -ssec
and
tc~tm---2 x 10 -Ssec.
For the above considerations to hold, both tc~(n) and the ©lcctron magnetic relaxation time must be greater than or equal to 10 -9 sec. Electron magnetic resonance experiments have confirmed that this condition holds.
504
HIGH RESOLUTION NMR
SPECTROSCOPY
The rapid electron exchange reaction between vanadium (IV) and vanadium (V) has also been investigated in chloride media (3t) and therefore any detailed mechanism must take into account the possibility of the intervention of chloride complexes. The overall mechanism of the reaction proved to be more complicated than that of the Cu(I) - Cu(II) reaction in thatdimers of vanadium Of) had to be postulated 2V(V) ~ [V(V)lz
WCC)h + v(Iv) .
v(Iv) + W(V)h.
The above mechanism was deduced from the rate of electron exchange being first order with respect to vanadium (IV) and second order with respect to vanadium (V). The rate constant was found to be 1.5 x 106 M -2 sec -I at room temperature using as solvent a 6"5 M mixture of hydrochloric and perchloric acids. In the solutions studied, the time intervals between the electron transfer events were in the range 0.81 x 10 -4 to 14.7 x 10 -4 sec for vanadium (V) and in the range 1.0 x 10 -~ to 6.7 x 10 -5 sec for vanadium (IV). McConnell and Berger °2) have developed a theory based on the Bloch equations (see Section 2.9) for reactions involving both paramagnetic and diamagnetic species: this will now be presented. The magnetic nucleus X can be said to spend part of its time in a paramagnetic species P andpart of its time in a diamagnetic species D. If, for simplicity the electronic state of P is taken as a doublet (one unpaired electron), then ~hen X is in the P environment the electron spin may be up or down. The sixpossible routes by which the environment of X can change may bc represented diagrammatically
The specific probabilities (in sec -t) are given as labels to the arrows. W h e n X is contained in D then it is subjected to a magnetic fieldessentiaLlyequal to the intensity of the externally applied magnetic field.W h e n X is in the P+ environment an additional field of strength ~co(27) -I will be present where 8o~ (rad sec-t) is the hyperfine splitting of the electron magnetic resonance of P due to X. A time ~"is introduced such that ~- t is the specificprobability that X leaves the P+ environment, proceeding either to the D state or the P_ state, that is ¢-1 = 3;,' + 1:~,I
(9.88)
THE EFFECTS OF CHEMICAL EQUILIBRIA
505
The method used for the solution of the Bloch equations is similar to that of Gutowsky, McCall and Slichterc2) given previously. For the present problem the Bloch equations have the form
dad
"t- ocvGD ----- -- i t O l M o
(9.89)
dG÷ + a÷ G. = - i to1Mo dt
(9.90)
dG_ + o~_G_ = - i w l M o dt
(9.91)
dt
o,x, = (Y~; 1)v - iA tO
(9.92)
o~÷ = (T~" ~)v -- i ( d to + ½t~ tO)
(9.93)
a - --- (/'2 I)D -- i ( d to - ½8 to).
(9.94)
When X is in the D environment, the resonance signal is centred at tOo, and d tO in equations (9.92) to (9.94) is tOo - to: to is the angular radiofrequency. The subscript D in these equations denotes that/'2 is the line width relaxation time for X in D with no P present. Saturation is assumed to be absent and M: ~ Mo. Solving equations (9.89) to (9.91) gives for the net average nuclear magnefisation of X x~ (1 - x T ; 1) [T2x; 2 - (1 + a+ 27) (1 + ' a - z)] i ~1 +~v.~0 i (27 " TD-- T27D~;1){( 1 + ~DTD) [272~; 2 - (l + ~+ ~)(1 + ~-~)1 I
- 2 ( 1 - TT;I) (27T~l + 1 + O~D27) 27TD -- 272(I Jr" 0~DWD) (2727; 1 "~" I "~- ~D T) + (1 - ~ ; , ) ( 2 7 ~ ; 1 + 1 + ~D27)} (9.95) This is the general solution of the problem which may be simplified by applying the conditions already discussed, namely • v >> 27
(9.96)
(a to ~p/2) 2 ~ 1
(9.97)
~'p ~, ~p
(9.98)
A simple Lorentzian line shape then results 6 =
- i tol M o
v~,
+ ( T ; l)V -- iA to
(9.99)
This expression leads to a transverse relaxation time (T~)ve for a mixture of diamagnetic and paramagnetic environments such that (r;%p
= ~;x + (7;')D.
(9.10o)
From this equation and the relation zv --- (k[P]) -x
(9.101)
506
H I G H R E S O L U T I O N NMR SPECTROSCOPY
one obtains the expression /6
~.
[p]-1 [ ( y ~ l ) D p __ (T~I)D]
(9.102)
which is the same result as that derived by physical arguments for the Cu(I) - Cu(II) example presented above. A more general, less stringent condition than those of equations (9.96-9.98) is (½3 ~ o~)2 >> 1
(9.103)
providing that z is appreciably smaller than 3p, Condition (9.103) can be checked experimentally for most systems since it must be satisfied if the nuclear hyperfine splittings due to X are to be resolved in the electron magnetic resonance spectrum of the DP mixture. The case of 3t) ~ 3p is a difficult one to treat since allowance must be made for the Boltzmann population distribution between the P+ and the P= states so that the line shape formula gives both the broadening effect and also the paramagnetic shift of the nuclear resonance signal. A simple application of the NMR method to the class of reactions under consideration is that of the TI(1)=TI(III) electron exchange system(33) in the form of molten TI2Br4, Rowland and Bomberg found two absorption lines for the nuclear resonance of 2°5TI and ascribed them to the TI(1) and TI(III) valence states at high and low fields respectively. Both states are diamagnetic but the broadening of the lines with temperature enabled the chem/cal lifetimes of the states to be estimated (I0 =5 sec at 500°K), Here the line shape was cal= culated for a chemically exchanging system of two species having equal con= centrations. The validity of the method has been checked by studying the kinetics of the MnO~-MnO~- exchange reaction (in sodium and potassium hydroxide solu= tions) both by tracer (2i) and by NMR (22) measurements. The two sets of results were found to be in agreement. The NMR experiments were based on SSMn resonance which gives no signal in the paramagnetic manganate environment. A further set of results, in agreement with the others, was obtained by making measurements of the relaxation times TI and T2 with the aid of spin echo techniques(23). Johnson 04°) has suggested that a full treatment of the problem investigated by McConnell and Weaver(3°) should take into account both the hydrogen nucleus-=electron dipole-dipole interaction and the isotropic exchange inter= action. The former is not very important for free radicals because the correla= tion time 3# for the interaction is of the order of I0 =11 sec. However, the corre= lation time 3p for the isotropic exchange interaction is equal to the lifetime 3A of the paramagnetic species of the spin-lattice relaxation time ~ of the unpaired electron, depending on which is the shorter, and may be of the order of 10 -4 sec. Johnson applied the exchange broadening theory developed by Kaplan (I 7) and by Alexander(334) to derive an equation for the transverse relaxation times of the nuclei. It was shown that under certain conditions, when more than one NMR signal is observed, the rate constant and the concentration of
THE EFFECTS OF CHEMICAL EQUILIBRIA
507
paramagnetic molecules can be determined from the line widths if the concentration of the diamagnetic species is known. 9.3 ION AND GROUP EXCHANGEREACTIONS At the beginning of the previous section it was pointed out that the detailed mechanism of an electron exchange reaction might involve atom or group transfer. This section, however, deals with those reactions in which valency changes are absent and in which there is little doubt that ions or groups participate in the chemical exchange process. Obviously, a wide variety of equilibria fall into this category; these will be presented in turn, except for hydrogen bonding which will be deferred until the next main section.
9.3.1 Protolysis Reactions One of the main applications of the basic theory given at the beginning of this chapter is to the study of protolysis reactions. Dissociation of protie acids is included in Section 9.3.2. Most protolysis reactions studied by the NMR method have been concerned with amines or amides. However, interesting exceptions are the protolyses of methanol(34) and ethanol (35). The rapid exchange of the hydroxyl proton in alcohols is accessible by the method. The reaction is both acid and base catalysed and is accelerated to a rate corresponding to a lifetime of the alcoholmolecule of the order of 10-? sec by the addition of acid or base in the concentration range 10 -6 to 10 -5 M. Figure 9.12 shows typical spectra used in the rate determinations. The methyl group resonance is not included because it is unaffected by the proton exchange process. For simplicity, only the methylene and water hydrogen resonance signals were utilised in the interpretation. The width of the methylene line is a measure of the total rate of proton exchanges (i) between alcohol hydroxyl groups (ii) between water and alcohol hydroxyl groups. The width of the water line is a measure of the rate of the second process only. Luz, Gill and Meiboom defined the s p e c i e rate of exchange Rx of a given molecular species X as the reciprocal of the mean lifetime, ~rx, of a molecule X between successive exchanges Rx = ~x I -
l -IX]
d[X] (9.104) at
The full width of the water line at half height, zl ~,, was measured as a function of acid or base concentration in the range where separate water and hydroxyl resonances were observed. A 1, is related to the specific rate of exchange of a water proton by the expression Rw = ~w1 = ~A 1, - T~"1
(9.105)
508
H I G H R E S O L U T I O N NMR S P E C T R O S C O P Y
where/'2 is the spin-spin relaxation time in the absence of exchange and can be determined from the decay of "wiggles" after rapid passage through the water line. The total rate of exchange RT was found from the methylene resonance using the procedure for a quartet, outlined in Section 9.1.2.The specific rate of exchange RA between ethanol and water was obtained from the broadening of the water line using the equation RA . .
Rw
rw . .
~A
Ci z
2 [HzO] . [EtOH]
H20
i
(9.106)
OH
,o,
H FiG. 9.12 Hydrog¢n resonance spectra of ethanol containing 4"3 per cent w/w wator in (a) basic solution (b) neutral solution (c) acidic solution. Luz, Gill and M¢iboom caS)
Since RT ---- RA + RB (where RB is the rate of direct exchange between ethanol molecules), RBcan thus be determined. The following reaction scheme was found to be compatible with the kinetic data kt ROH + O H - ~--- R O - + H 2 0 k2 R O H + R O - ~ RO- + R O H k3 ROH + HaO+ ~- ROH~ + H 2 0 k4 R O H + ROH~ ~ ROH~ + ROH k~ ROH + (H)OH ~ RO(H) + HOH
THE EFFECTS OF C H E M I C A L E Q U I L I B R I A
509
Paths 1, 2 and 5 apply to the base range, while paths 3, 4 and 5 apply to the acid range. At 22°C, the mean values of k l , k2, k3, k4 and ks were found to be 2"8 x 106, 1-4 x 106, 2-8 x 106, 1"1 x 10~ and 0.8 iv1-1 sec -I respectively. The rate of proton exchange in neutral anhydrous ethanol (reaction 5) was found to be 2.7 M-1 sec -1 at 42 ± 1°C by a direct method(~37): a graphical procedure was devised which gave rapid evaluation of the average lifetime of a hydroxyl proton between exchange events. In the methanol experimentsc3~ measurements were made on buffered solutions. The results obtained at 24.8°C after extrapolation to zero buffer concentration can be summarised by the equation: rate of proton exchange = 8.8 x 10l° [MeOH,+] + 1.85 x 101° [MeO-]
(M-I see-l). These measurements have been extended to rate determinations for proton transfer between methanol and the components of benzoic acid-sodium benzoate buffers in dilute solution°6' 351.3s2). T w o main terms appear in the rate law; one is a termolecular process involving benzoic acid, benzoate ion and a methanol molecule, the other involves one benzoic acid molecule and two methanol molecules. The number of methanol molecules taking part in each kinetic process was determined by comparing the O H proton exchange rate, as determined from the IH resonance of the C H ~ group, with the exchange rate between methanol and benzoic acid, as obtained from the carboxyl or from the coalesced carboxyl-hydroxyl resonance. Proton exchange in various other alcohol-water systcrns has been investigated(3S7. 3ss) F r o m rapid passage experiments, Mciboom (37)has made a detailed examination of proton transfer in water. Formally, the proton cxchangc reactions can be written H~0
kj + (I'hO) + ~ - ( H 2 0 H ) + +
H20
k2
OI-h + OH" ~ OH- + HOH. The transverse relaxation rate (l/T2) of the hydrogen resonance was found to be pH dependent due to the partially exchange-averaged spin-spin splitting of the resonance by the magnetic nucleus 170 (spin ~) .The increase of relaxation rate observed in natural water (0"037 per cent 170) is greatly increased in 170 enriched water. The conclusions were confirmed by a study of the dependence of the width of the 170 resonance signal on pH. The rate constant kl has a value of (10.6 ± 4) x 109, while k~ is (3.8 -k 1.5) x 109 ~-1 sec-L The spin-spin coupling constant between ZH and 170 was found to be 92 ± 15 cycles sec -1. A more accurate value of 73.5 ± 2-1 cycles sec -1 has been obtained for JJToHfrom the 170 resonance spectrum by reducing the exchange rate °sl. This was accomplished by diluting the water (containing 8 atom per cent ~70) with a large excess of carefully purified acetone (see Section 12.4.3). The quadrtlpole moment of the 170 nucleus prevents observation of multiplet structure in the IH resonance spectrum. Loewenstein and SzSke °91 have re-determined kl and k2 at 25°C and they have obtained values similar to those found by Meiboom. Also they measured
510
HIGH RESOLUTION
NMR SPECTROSCOPY
the respective heats of activation and found them to be 2-6 4- 0.3 kcal mole- 1 and 4.8 + 0.5 kcal mole -1. The rates of inter-water proton exchange have been measured(3sl) for water in various organic solvents. Another system to be investigated in which hydrogen is attached to an oxygen atom is the exchange of hydrogen between water and hydrogen peroxide (4°). The spectrum is never divided into two absorption lines, because the rate of exchange is too rapid at room temperature. However, the line width was seen to be dependent on pH and the hydrogen peroxide concentration. In the pH range 2.5 to 4.5, the reaction involves H30 ÷ and H202 while in the range 4.5 to 6-5 the participating species are HO~, H202 and H20. Protolysis kinetic studies have been carried out on ammonium(2~' ~1-43), methyl, dimethyl and trimethyl ammonium ions (5. ~8.,s. 44. 3 5 3 , 3 s 4 . 3 5 5 ) A kinetic study of the related protolysis reactions of the trimethylphosphonium ions in aqueous solution has also been made ('5). The procedure used is similar to that indicated in the example described above. The addition of diethyl- and triethylamine to acetylacetone has been shown to alter the keto-enol equilibrium position(4~). Proton exchange takes place between the O H and = C H groups of the enol tautomer. Kinetic data has been obtained for amides: the rate of proton exchange for N-methylacetamide and N-methylformamide has been measured in solutions at different pH values (7), the reaction being kA
RCONHCH3 + H30 + ~ (RCONH2CH3)+ + H20. From the N-methyl band (a doublet for pH values 1 to 4) in the NMR spectra of these compounds at 25°C, kA was found to be 200 + 70 M-~ sec -~ and 10 _ 3 M-~ sec -I for N-methylacetamide and N-methylformamide respectively. The former value is in good agreement with the results of other workers (47). Saika (4s) has determined the heats of activation for the acid hydrolysis to be 15 + 3 kcal mole -~ and 13 4- 3 kcal mole -z respectively. In the presence of alkali the reaction becomes kB R C O N H C H s + O H - ~- ( C H 3 C O N C H 3 ) - + H z O
k~) is known only for N-methylacetamide(4:), having a value of 5"2 × 10~ M-t se¢ - 1.
GiUespie and Birchall(49) have investigated the site of protonation for acetamide, N, N-dimethylacetamide, formamide and N, N-dimethylformamide. They used fluorosulphuric acid as the source of protons and they varied the temperature from - 9 8 to 25°C. At the lower temperatures the proton exchange rate decreased sufficiently to permit observation of a new band in the 4-
H resonance spectrum. The band was assigned to the group > C = O H , indicating that protonation had occurred on the oxygen atom. A similar conclusion has been reached by Herbison-Evans and Richards (5°) who measured the line widths in the ~'N resonance spectra of N, N-dimethylformamide in trifluoroacetic acid. Increase of line width (a measure of T~ 1 here) with increase of acid concentration provides evidence for increase in the electric qua-
THE EFFECTS OF CHEMICAL EQUILIBRIA
511
drupole coupling constant. Measurements have been extended to include other amides, thioamides and sulphonamides13s6~. Evidence has been found for diprotonation of thiourea and N-methylthiourea in fluorosulphuric acid. A series of interesting proton exchange reactions between hydrogen fluoride and the weakly basic, substituted benzenes has been studied c5D. In the instances of hexamethylbenzene and durene it was found that the proton may exchange intramolecularly. CH3 CH3 ~I"13, I jH CH3. [ < C H 3 \/'~/ \/~/H I + t\CH ~ --~ i + lJ "CH3 CH3
CH3
H CH3\/[x./CH3 I+1
cs/)<\cm H
H
H H CH3\ ~ / .,C~'13 '/~'/ ~ ~+/~
cH/ [<
\CH3
H
Here the proton exchanges between positions of equal proton affinity while remaining on one and the same carbonium ion. Mesitylene, m-xylene and anisole showed intermolecular proton exchange only. All the systems examined proved to have heats of activation of about 8 kcal mole -a which is thought to be due to the fact that an exchanging grouping, for example ~ C / H + FHF- ~ >'~2--H + HF
.....HF
is aliphatic in type and not likely to form strong hydrogen bonds with the solvent HF. This is in marked contrast to the proton exchanges of low activation enthalpies, such as H20 + H30 +~-'H30 + + H20 and H~O+ + FHF-~-H20 + HF .........HF in which hydrogen bonding is of prime importance. The NMR spectra of 12TI and S*Br have been employed in order to measure the rate of exchange of halide ions with halogen-containing complexes of Hg (II) in water (34¢). Line width measurements permitted the rate constants of the following reactions to be evaluated HgI~- + I- ---* I- + HgI~HgI~(H20)- + I- ---* I- + HgI3(H20)-
HgI3(H20)- + I- --* H20 + I-IgIi-
9.3.2 Solutions of Strong Electrolytes This section includes organic and inorganic compounds, but salts of multi-valent metals are left to the following section on complex formation. Although there are three distinct sites for a water molecule--near an anion, near a cation and isolated among other water molecules--only one resonance line is observed in all cases. The single line represents an average of the shifts arising from the three different environments, since the exchange rate of the protons is much greater than the chemical shift differences. The dependence of the shifts on salt concentration should be linear until water molecules are shared
512
HIGH RESOLUTION NMR S P E C T R O S C O P Y
among anions. The points of departure from linearity can yield information about the packing of ions in solution and also about the maximum distance at which an ion affects the shielding of the hydrogen nucleus. On the average, the water molecules surrounding a positive ion are orientated with the oxygen atom nearest the ion. The inverse will be true for a negative ion but in both situations the hydrogen atoms of the solvent molecules will still be less shielded. This is because electrons would be repelled and are therefore more likely to be loealised on the oxygen atom. A separate factor enters because the addition of an electrolyte to water promotes breaking up of the hydrogen bonded strut-
i
-~o
p
i
i
L
5
I0
15
Concentration,
4
..... 20
mole/lOOOg
25
30
of HzO
FIG. 9.13 Negative chemical shifts d of the hydrogen resonance of some aqueous electrolyte solutions as a function of concentration. Shoolery a n d Alder (s2)
ture of the liquid. Disappearance of the mutually induced dipole increases the electron density around the hydrogen atoms. These effects have been confirmed by Shoolery and Aider (s2) who found that the intense polarisation produced by small ions gave shifts to low magnetic fields but that large ions, having a greater structure-breaking effect, gave a shift of the hydrogen resonance signal of water to high fields. On the one hand, uni-univalent electrolytes tend to fall into the latter class while on the other hand, multivalent ions polarise the water. Figures 9.13 and 9.14 reproduce the data obtained by Shoolery and Alder in terms of the chemical shift ~ defined by a ---- 10 7 X (HHzO --
H,.mo,.)/H.~o
(9.107)
where HH,O and H, mpx, are the fields at which resonance occurs, for a fixed frequency of 30Me see -x, for an external water reference and the electrolyte solution respectively. From these data it is possible to assign to each ion a chemical shift value representing its contribution to the total shift of a given
513
THE EFFECTS OF C H E M I C A L E Q U I L I B R I A
electrolyte. Since the behaviour of concentrated solutions is not characteristic of the interpolated chemical shifts of the ion at one molar concentration, the chemical shift o f the hydrogen resonance of an electrolyte is defined by (9.108)
6 = m(n +6 + + n-6-)
oBe CL2
o Be (NO3)2
20
15
o
HCL
oALCLs o At (NO3)3
B
/ N o O H
lo
tz(SO4h" ~ "
/
/
~C°CLz
0AgN03
~%PO4
I~/
,~'CoBrz~
5
I0
Li Cl
15
20
2~
30
Concentration, mole /10009 of HzO
Fzo. 9.14 Positive chemical shifts 6 of the hydrogen resonance of some aqueous
electrolyte solutions as a function of concentration. Shoolery and Alderca2) where ~÷ and 6- are the relative molar shifts produced by the positive and negative ions, and n + and n- are the number of moles of these ions formed by the dissociation of one mole of the salt. The concentration m is expressed in moles per 1000 g of water. The perchlorate ion was assumed to have 6- = - 0.85 in order to make shifts clue to the polarisation effect alone small and positive.From Table 9.1 it can be seen that the negative ions have a more pronounced effect than the positive ions on the hydrogen resonance. Fabricand and Goldberg (s3) have repeated some of the measurements with the aid of a N M R spectrometer operating at 60 Mc sec -a and using a coaxial sample tube (see Section 7.3.1). Despite the improved precision of their measurements, the difficulty of making the large correction for the bulk diamagnetic susceptibility remains. H ÷ and m~
17
514
HIGH RESOLUTION NMR SPECTROSCOPY
OH- ions are not included in the table as they are special cases due to proton exchange with water molecules. The concentration dependence of the shifts fo~ the multivalent electrolytes indicates the presence of cation-ion interactions at high formal concentrations. The possibility of exchange of protons among the various kinds of environment must also be taken into account. Only a single resonance line was observed for all solutions therefore rate data are available only through line width measurements. The only line broadening to be detected (compared with pure water) occurred for solutions of Al3 +, Be2 +and of concentrated ZnClz. This means that the time for proton exchange is less than 10 -4 see in the other solutions. The high viscosity of the solutions is responsible (see Section 2.7) for the line broadening caused by ZnC12 and BeCI2, while the TABLE9.1 R r ~ A ' r v ~ M o L J , R CHEMICALStuFfs oF arm 1H Rr~SONANC~or WATERCAUSEDBY TH~ ADDITION OF E ~ O L Y ' r E S (52) Ion
Ion
~+ --0"71 -- 0"57 0.09 0"28 -- 0"23 0"45 2'08 4"89 1-24 2"39 5.19
l-
-0.35
K+
Br-
-- 0"21 - - 0"01
Na+ Li+ Ag+
CIFCIO7~ NO~
so~,ro,~-
1"20
0"85 -0"09 1"17 -
3"83
Ba,+
Ca2+ Mg2+
B¢2+
ZII2+ La3+ AI3+
increased line width caused by A13÷ indicates a rate of proton exchange of about 100 sec -1. Fabricand and Goldberg c53) plotted their values of the ionic shifts for a given salt concentration against the ionic crystal radii of anions and cations and observed a linear relationship. This implies that the average number of hydrogen bonds in water near an ion is proportional to the ionic radius and that the number decreases with increasing radius. They also gave evidence showing that the time-averaged packing distance of the ions is closer than if they were arranged on a cubic lattice and that the ionic effects on the hydrogen nuclear shielding do not extend past nearest neighbour water molecules. Hindman(54) has made a study similar to, but more intensive than, that of Shoolery and Alder on electrolyte solutions. He used cyclopentane as an external reference and, using a value of 141.2 cycles sec -1 at 40 Mc sec -1 for the cyclopentane-water chemical shift difference, referred his shift measurements to water. These were obtained for univalent electrolytes at 25.0 _ 0.2°C and were corrected for bulk susceptibility effects according to the equation H.~o = H s "Jl" '~2~ = 1061.~'H"2° - ~ (.~.I -- XH20) }
(9.109)
THE EFFECTS OF CHEMICAL EQUILIBRIA
515
where Hs2o and Hj are the field strengths at which the signals of water and the sample occur; ZH20 and Xa are the volume susceptibilities. Infinite dilution shifts (6~at) were derived by plotting either 6 or ~ob, against m, the concentration in moles per 1000grams water. The data for both could be represented by the empirical equation or 0o~. = ~°..(m) + B(m) 2. (9.110) Some of the values obtained are given in Table 9.2 (in Table 9.1 given earlier, bulk susceptibilities were neglected). The results were also expressed in terms of individual ionic effects as were those of Shoolery and Alder, 6,° , -- (8°M+v÷ + ~x°- v-)
(9.111)
where ~o. and 6°- are the infinite dilution shift values of the individual ions, and v÷ and ~- are the number of ions of each kind into which the salt dissociates. Clearly, this relationship must fail when the solute concentration is very high. It may be seen from Table 9.2 that there is good agreement between ~o values calculated for the univalent ions as found from measurements made on different salts. Hindman has considered the origin of the various contributions to the individual ion shifts in some detail. He summarised the conclusions on univalent cations as follows: (i) The data are not compatible with the concept of a complete hydration shell of tightly bound water molecules. (ii) The effectiveness of the ions in' coordinating .water decreases strongly and regularly with increasing ionic radius. (iii) The "effective" hydration numbers calculated are similar to those estimates of primary hydration obtained from other types of measurement. It was also suggested that, of the halide ions, only the fluoride ion forms a hydrate in the chemical sense. The larger halide ions act primarily to break down the water structure, the effect increasing with increasing anionic radius. IH resonance measurements on salts in 50 per cent dioxane-water mixtures showed that preferential solvation by water occurred for the cations of berylHum, lanthanum and thorium °4s). The magnetic resonance spectrum of the ion itself, rather than that of the solvent is a more meaningful parameter but there are only a few magnetic nuclei suitable for this purpose. Shoolery and Alder (s2) and Connick and Poulson (ss) have examined the 19F resonance of aqueous fluoride ions and Wertz and his co-workerseSe, sT) and Hindman (s4) have obtained the 2SNa and 3sc1 resonance spectra of aqueous Na ÷ and C1-. The 2SNa nucleus has a spin value of t and therefore possesses a quadrupole moment. No chemical shift differences for Na ÷ were observed over a wide range of concentration of various electrolytes; there was, however, a change of line width. Wertz and Jardctzky ¢s7) concluded that the resonance signal they obtained was due entirely to ionic sodium and that the increase of line width with concentration is a reflection of ion-pair or complex formation producing departures from the effective spherical charge distribution of the ion. Similar results were encountered with the chloride ion. 17"
5]6
HIGH RI~SOLUTION NMR SPECTROSCOPY
ooNN
N~
N
~N
Itll +
0 ...
4-
~
~
II ~
II
t~ 0
~"II
~
11
o
+
~
o.
..o
.
o
oo
ooo
o
0
8
~'~
°~ Zell
ooo
o o o o
111
e~
Z
~o
<
.oooo
~
~
666
~" ¢
I~
I l t l
~ ~6
i~
11
~o
1
~o
o
g
o$
~
~
~
~
~
o-
66
~" " 6
66
66
~6
~6
~
z~c
r~ [.., Z .<
o
.>
-g
Z
=
'i'
~E 0,£
r~
,-' +
0 0 0
UzU
0 ~ 0 0
0 0 0 0
II]
I l l l
ozo~
UzU
0 ~ 0
O0
II
It
~
0 ~ 0
tl
~ 0
,
THE EFFECTS OF CHEMICAL EQUILIBRIA
517
A comparison has been made of the 12~I chemical shifts of the alkali metal iodides in aqueous solution and in the solid state c5.). 79Br, 81Br and 2~Na resonances have been used to investigate bromides dissolved in water and in water-methanol mixtures (3.7). Line width measurements on 19Br and SlBr NMR signals from sodium and calcium bromide solutions indicated that ion-ion interactions are negligible compared with ion-solvent interactions. The former interaction proved to be important in the instance of caesium bromide solutions. Bromide chemical shift measurements reflected the same trends. The largest number of experiments on electrolytes have been devoted to the dissodation behaviour of strong acids in water. The reason is that nuclear magnetic resonance spectroscopy ranks with Raman intensity measurements as one of the more important methods of determining the degree of dissociation of strong electrolyte solutions. Nitric acid solutions have been studied extensively by several groups of workers (3' sg-s2). Hood, Rcdlich and Rcillyis°) calculated the dissociation constant of nitric acid from the hydrogen resonance spectral data in the following manner. The chemical shift difference 6 brought about by the presence of the acid is given by 6
-
HH20 ~ H-1
x
l0 s + g.
(9.112)
Where HH2Ois the resonant field of an external water reference sample, H is the resonant field of the electrolyte solution and g is the diamagnetic susceptibility correction obtained from the expression' g = (2:~/3)~
-
ZH,o) x
l0 s
(9.113)
where X is the volume susceptibility of the solution and Xa,o that of water. The observed shift is a linear function ofp the stoichiometric or apparent mole fraction of the hydrogen in H30 ÷ on a total hydrogen basis, according to the relationship
6 ----0"5(~3 -- ~2)P + 1"562 -- 0"5~3
(9.114)
where ~2 is the observed chemical shift of the monohydrate H3NO4 (62 = 4"25) at p = 1 and 63 is the shift of the anhydrous acid HNO3 (63 -- 7.00) at p -- 3. The quantity p is given by 3x p = ~ (9.115) 2-x x being the stoichiometric mole fraction of the acid. In the range p = I to p -- 3 only these two species of nitric acid appear to be present, although there is some evidence of nitrate and hydrogen ions at the lower concentrations and of nitrate and nitronium ions at the higher concentrations. A plot of 6 againstp gave a straight line at low values o f p (0.0 to 0.3) of limiting value 11.82 which was designated 61. For values o f p < I, the degree of dissociation 0c of the acid may be obtained from the relation 6 - - = 61o~ + 6 2 ( 1 - - ~). (9.116) P
518
H I G H R E S O L U T I O N NMR S P E C T R O S C O P Y
The thermodynamic dissociation constant K can be found from the expression
K [~ = a/c(1
-
oO
(9.117)
where a is the activity of nitric acid ce3), c is the concentration in moles litre -1 and ~ is the activity of the undissociated molecule. Figure 9.15 depicts the data of several investigators in the form of a graph of 1ogK~ versus c. A value of 22 was obtained for K by Hood, Redlich and Reillyc6°) in good agreement with 3.0
A
/
2.6
~_
,~//
z.2
I.8
/~/
//fjV
/~/II;
1.4
0
', 4
I , 8
¢,
mole
J 12
I 16
k-~
FIO. 9.15 The determination of the dissociation constant of nitric acid by seve= ral invcsti~tors ORedlich and Bigeleis~ c6')), AGutowskyand $aika<3),~ Masuda and Kanda(sg), o Hood, Redlich and R©illy(6°) the value of 21 obtained previously by Redlich and BigeleisenC64) from Raman line intensities (the data is represented by the dotted line in Fig. 9.15). The remainder of the data in Fig. 9.15 is derived from N M R measurements using the hydrogen resonance line (except in the work of Masuda and Kanda csg) who used nitrogen resonance). Hood, Redlich and Reilly consider that the data of the latter authors is in need of revision. Detailed measurements of hydrogen resonance shifts have also been made on nitric acid solutions more concentrated than 0.5 mole fraction of acid c62). In this concentration region the precise data accrued indicated that there is a small concentration of either trimeric species or cyclic nitric acid dimers. The high precision attained in this study enabled the nuclear magnetic resonance method to be used for the precise analysis
THE EFFECTS OF CHEMICAL EQUILIBRIA
519
( ± 0.02 per cent) of concentrated nitric acid solutions ( > 0.3 mole fraction HNO3). Happe and Whittaker(6~) found that the addition of KNO3 to anhydrous HNO3 resulted in the hydrogen resonance being shifted considerably to lower applied fields. The effect was interpreted as originating from the formation of a solvated nitrate ion (NO~. 2 HNO~)= involving strong hydrogen bonding between the ions and solvent molecules. The effect of the addition of up to 1"5 ~,lal,minium nitrate on the disso~ation constant of nitric acid has also been studied(65). At low salt and low acid concent.rations ( < 0.5M), the data becomes difficult to interpret due to the onset of the hydrolyses AI(H,O)~ + + H ~ O ,-~ H ~ O + + [AI(OH)(H20)s] 2+ |AI(OH)(H20)I] + + H 2 0 ,-~ H s O + + [AI(OH),(H20)4] +.
At room temperature, the results agree with the simple theory combining a common ion effect from the added nitrate ion with competition for water molecules by the aluminium ion. The simplifying assumption was made by Axtmann, Shuler and Murray ~6s) in their treatment that a primary salt effect is absent. At temperatures above 23°C, aluminium nitrate proved to be even more effective at diminishing the dissociation of nitric acid. Measurements on perchloric acid °. 40) led to a thermodynamic dissociation constant of 38 M. It appears that at moderate concentrations, perchloric acid is not much more dissociated than nitric acid but the degree of dissociation is high even in very concentrated solutions. Other acids to be inv~tigated by the NMR method are sulphuric (3' ~6-~s), hydrochlorico. so), trifluoroaceticC~9), (1H and 19F resonance, K = 1.8 M), acetic(a), heptafluorobutyric(7°) QH and ~gF resonance, K = 1.1 M) and iodic (K -- 0.18 M)¢7~) acids. The latter acid is of interest since its thermodynamic dissociation constant is near the upper limit of the region where classical methods can be expected to yield meaningful dissociation constants of univalent electrolytes(72). Aqueous solutions of polystyrene sulphonic acid have been demonstrated to be almost completely dissociated(73). NMR measurements have been used to determine the dissociation constant (22 ± 3 M at 25°C) of the related substance, p-toluenesulphonic acid (74). By using an aprotic solvent like liquid sulphur dioxide<75)it has been possible to correlate the chemical shifts of acidic hydrogen atoms with the p Kvalue of the acid in water (see Table 9.3). The method thus offers a means of obtaining the dissociation constant of an acid in water from a single chemical shift measurement. The anomalous shift produced by HBr was attributed to the formation of a 1 : 1 complex with SO2. In liquid sulphur dioxide, a single resonance line was observed for the hydrogen resonance spectra of aqueous perchloric acid due to the rapid proton exchange between the acid and the water. Two lines are observed when the concentration of perchloric exceeds 50 per cent; from their separation, proton exchange rates were calculated. The rate constants were found to be k,,o = 2.05 sec -1 and kHclo, = 15.0 sec -1 for a 2.13 molal solution of 60 per cent aqueous perchloric acid.
HIGH R E S O L U T I O N
520
NMR
SPECTROSCOPY
The determination o f the dissociation constants of the acids discussed above is made possible because the signal from H 3 0 ÷ (and from O H - ) is shifted about 10 p p m to low fields of the XH resonance of HzO. Musher C7~) has explained this large shift in terms of a model in which all the ionic charge is located on TABLE 9.3 THE 40 Mcs~e-* CSm~CAL SHIRTS (EXTm~AL B~'NZEN~ RrazRm'~cz) or SOME ACIDS IN Llqtnn) SO 2 AND ~ p K VALUES FOR WAT~'R SOLImONS (*s)
Acid
pK
Molality
[CHs)3C'COOH L-'HsC00H [-IC00H L"rI2CI'C00H CHCI2"C00H CCI3COOH CFsC00H CHsS0sH
5"1 4"75 3-68 2.81 1.30 0-52 - 0.26 0.6
1.46 2-15 4-05 0-98 2.16 2-19 4.70 1"71
H2S04
-
HCI04 (70~ aqueous)
-
[-IBr HI
-
3 10 9
Corrected chemical shift of acidic hydrogen atom
1.03
2-12 18.4
10
152-1 cycles see-, 154-2 cycles see-x 156-2 cycles see-x 156.7 cycles see-x 147.3 cycles see-x 143.6 cycles see-* - 128-9 cycles see-* - 126.6 cycles see-x 83.2 cycles see-t + 189 cycles see-x + 375 cycles see-t + 202 cycles see-t -
-
the oxygen atom for both HsO + and O H - . Thus the 1H nuclei and their 1 s electrons " s e e " an electric field of + e/r~Hwhere roll is the O - - H internuclear distance. Since the deshielding effect is given by
A~ = - 7 . 3 8 x lO-19 eZ/r~oH the shift to low fields by about 10 ppm of the O H - and H 3 0 ~ signals can be accounted for.
9.3.3 Complex Formation This section deals with all types of complex formation ranging from simple ion association and ion solvation to strong covalent bond formation. Ion pair formation has already been mentioned and now further examples will be given with more detail. Chemical shift and line width measurements. The sodium ion is useful in the search for weak complex formation because ~3Na is a magnetic nucleus (/--- ~). Comparatively weak interactions are detectable by nuclear magnetic resonance due to the fact that the nucleus has a large magnetic moment and an appreciable quadrupole moment. The charge distribution and hence the electric field of isolated N a ÷ is spherically symmetrical and therefore line broadening will occur if distortion is introduced because the nuclear quadrupole moment will then interact with the gradient of this asymmetric electric field ~365~ (see Section 2.5). Either ion association or covalent bond formation could produce an asymmetric field. Jardetzky and Wertz C77~ have measured line widths from solu-
THE EFFECTS OF CHEMICAL EQUILIBRIA
521
dons of NaF, NaCI, NaBr, NaNa, Na~SOa, N a C I O , , N a a P O , , N a z S O , and N a O H , having found no chemical shift differences for the 2SNa resonances. The results indicated that very strong elecU'ic interactions broaden the line beyond detection and hence signals from sodium ions only appear when their electric field is not greatly distorted. The compounds listed fall into two classes (i) those whose signal amplitude but not line width is concentration dependent (ii) those whose line width increases with concentration while the intensity passes through a maximum. The second category was found to contain the hydroxide, phosphate, hydroxy and keto acids and alcohols: the effect was attributed to weak complexes formed between the sodium ion and the anion. Regardless o f the anion, some diminution in intensity and some increase in line width was observed for solutions of concentration greater than 7 to 8 N. Solutions as concentrated as this probably reflect various stages o f ion pair formation since the presence of four or fewer water ~ olecules per sodium ion can prevent the formation of a single hydration layer. The narrowness o f the 23Na resonance line suggests that the distribution of water molecules around a sodium ion is highly symmetrical and/or that the distortion of the sodium electron shell produced by individual water molecules is rapidly averaged out. x9F chemical shifts of a number o f metal fluoride complexes in aqueous solution have been measured by Connick and Poulson tTs). They defined their shifts according to the expression t5 = ( H / H v , 1) x I 0 e (9.118) -
TAeLE9.4 19F CHF.MICALSmYI~ (40 Mcsec-~) TNAQUEOUSSOLtmONOF M~AL FLUOP.mE • Coro~'x~.s (GASEOUSF2 USEDASEXTERNALREFERENCE)(~s) ]viinlmBlrl
Principal fluoride species ThF a+ UOzF + UO2F~? SnF +, SnF2? FAsF, F siF
-
TiFCa"-I)÷, F-?
ZnF +, FA1F~, A1F2+
Solution I'0M Th(NOa)4, 0"5 MNaF 2.0 MUOu(NOa)2, 2"0 MNaF 1"3 MUOu(NOa)2, 2"6 MNaF 0"8 M SnCl2, 1 MNaF, 0.3 MI-IF dilute NaF or KF 1 M A g N O 3 , 1 MNaF saturated (N'I-I,)2SiFe saturated KzTiF6 saturated ZnFa 2 MA1(NOa)a, 2 MNaF
(cycles sec-1) 336 395 401 527t, 588 547-7 550 557-3 573, 550 579, 552t 588, 588.8t
exchange time (sec)
7 × 10 -5
2 × 1 0 -4 2xl~ 4 5 × 1 0 -3
t denotes the more intense band. where HF2 is the field at which the external reference o f fluorine gas appeared at an irradiating frequency o f 40 M c sec -1. Table 9.4 lists the chemical shifts along with the compositions o f the solutions used and the principal fluoride species expected to be present as computed from a knowledge of formation constants. The most striking feature of Table 9.4 is the small shielding produced HBS. 17 a
522
H I G H R E S O L U T I O N NMR S P E C T R O S C O P Y
by thorium and uranyl ions. In a few cases, more than one resonance line was observed and these were attributed to slow exchange between different fluorinecontaining species. The lower limit for the exchange time is included in the table. The 6 values of 550 and 552 cycles sec -1 measured for titanium and zinc are very likely due to the fluoride ion itself. No simple correlation between the chemical shifts and the electronegativity of the metal ion or the formation constant of the complex could be found. Tiffs result is in contrast to the explanation of the correlation between electronegativity and 19F chemical shifts of fluorine compounds put forward by Saika and SlichterC79~(see Section 4.6). More detailed measurements have been made on the 19F resonances of aluminium fluoride complexes in aqueous solution~55. so). The coalescence of the absorption bands due to A1F2÷ and A1F~ at high added salt concentrations is thought to be due to a slightly greater electrolyte effect on the A1F~ than on the A1F2 +resonance: the increase of the exchange rate of F- was regarded as being less likely. Potassium and sodium ions were shown to be different in their behaviour at low concentration, their effects being opposite in sign. A possible explanation is that sodium ions form a covalent complex with fluoride ions while potassium ions are separated by a layer of water molecules from the fluoride ions (solvent-separated ion pair formation). At high concentrations, the decrease in water content would force the K+F - ion pairs to change over to the direct contact type resulting in positive chemical shifts. Chemical shifts of the ~gF resonance of the fluoride ion in water and in other solvents have been measured and compared with the chemical shifts of the ~33Cs resonance~81~. The 19F shifts were found to be dependent upon both the nature of the added anions and the added cations while the ~33Cs shifts are influenced almost entirely by the added anions. These effects were ascribed to the fortuitous presence of contact ion pairs, shifts being brought about by the increase in paramagnetic circulation occurring via the excited states of the contact ion pairs. The difference between the two sets of results is probably due to the influence of the electron donating power of the fluoride ion. Several papers have been published on the association of thallous and of thallic ions with anions in aqueous solutionCs2-ss). The 2°3T1 and 2°5T1 resonances employed in these studies are rather broad hence a wide-line method was used. The chemical shifts of thallic ions in the presence of chloride and of bromide ions were found to be consistent with the stepwise formation of complex ions ¢s4), in agreement with the findings on the electron exchange reaction between TI(III) and Tl(I) Cs~ and the reaction between Tl(III) and Hg(I) cs~. Different chemical shifts were observed for TI*, T1CI2 and T1CI~-. The Tl(I) state showed much less tendency than TI(III) to form strong complexes as evidenced by the non-linear variation of the chemical shift with anion concentration for solutions of low concentration containing thallous hydroxide, fluoride, acetate, formate, nitrate and perchlorate ¢s3). More concentrated solutions gave a linear dependence between the two functions which was explained as interaction between the ion pairs and the added electrolyte. Covalency in the bonding between the two ions can be detected by employing a paramagnetic anion. Gasser and Richards Css) have used this technique for aqueous solutions of the thallous ion
THE EFFECTS OF CHEMICAL E Q U I L I B R I A
523
by adding ferricyanide ions. Appreciable covalency in the ion pair would cause some unpairing of the electrons in the thallous ion thus reducing thediamagnetic shielding of the nucleus. The recorded chemical shifts from the thallium resonance were about twenty times greater than those produced by simple diamagnetic anions, but this is not sufficiently great to be regarded as evidence for covalent binding. Measurement of the spin-lattice relaxation time T1 of these solutions is in agreement with this conclusion. Several N M R investigations have been made on non-labile complexes, mainly of cobalt (III) Css-91). One of the most interesting applications of nuclear magnetic resonance spectroscopy is to the study of the hydration of cations: Axtmann ¢92) has measured the chemical shifts of the IH resonance of a series of metal nitrates dissolved in water. The shifts for the ions Ca 2÷, Cd 2÷, Ga 3+, Mg 2+, Sr 2÷, Th 4÷, Zn 2÷ and ZrO 2÷ give a good correlation with the p K values for the equilibrium M(H~0)~ + + H 2 0 ~- H 3 0 ÷ + [M(OH)(HzO)s] ~:c-1)+'
By contrast, the ions UO22÷, Pb 2÷ and A13÷ did not fit into the correlation. It was suggested that, on the one hand, the former group of cations show the properry of forming simple ion-dipole complexes and, on the other hand, some covalent character is present in the latter group. Another contributory factor to the anomalous behaviour of the plumbous ion is that its coordination number may only be three or four rather than six c93). Axtmann has pointed out that a measurement of the chemical shift and the pKvalue could be used as adiagnostic test of complete or partial electrostatic binding in the forces of solvation. Jackson, Lemons and Taube ¢9") have used a different approach, namely they have measured the chemical shifts of the 170 resonances of enriched aqueous solutions. The advantage here is the greater magnitude of these shifts compared with those of 1H. In exploring the method, the ion [Co(H20)(NH3)5] 3÷ was chosen since the number of covalently-attached water molecules is known and the rate of exchange of the bound with the solvent water molecules is slow (t½ ~ 28 hours at 25°C) c95). Two separate signals appeared for the cation and the solvent 170 resonance absorption; the former signal was located at the higher magnetic field. The addition of cobaltous perchlorate to the system did not influence the chemical shifts of [Co (H2 (3) (N H 3)s] 3 ÷ but some broadening of the 170 absorption line occurred which is suggestive of the formation of a small proportion of a dinuclear species such as H
[(NHs)sC~3C.o(H20),P÷. H
Measurements were also made of the 170 absorption of solutions of the nitrates or perchlorates of H ~, Li +, Be 2÷, Mg 2+, Ba 2÷, Sn2+, Hg ~+, A1s÷, Ga s4 and Bi2+ in D 2 0 enriched in 170. Hydrolysis of the cations was suppressed with perchloric or nitric acid. This time only a single 170 resonance line was observed. However, addition of the paramagnetic ion Co s+ enabled hydrate water to be distinguished from solvent water in that the solvent absorption 17a*
524
HIGH RESOLUTION
NMR SPECTROSCOPY
line was shifted by the paxamagnetism leaving the cation hydration unaffected. The effect was noticed for Bi2÷, Ga 3+ and AP+; the remaining ions gave only a single water line, considerably shifted from its position in the absence of Co 2+. The conclusion was drawn that the half life of the exchange of water molecules between solvent and hydration shell for the former three ions is greater than 10-4 see, and less than this figure for the other cations. The result for A13÷ is in line with the findings described above in which covalent binding was indicated, and also with isotopic dilution experiments(96~ which showed that the half life for exchange exceeds 0.05 see. The employment of water more highly enriched in tTO makes it possible to determine the coordination number of hydrated ions. Connick and Fiat ¢35o>increased the 170 resonance signal-to-noise ratio at 5.415 Mc see -t by using water enriched to 11.48 per cent tTO. They found the hydration numbers of A13 ÷ and Be 2 ÷ to be 45-9 + 0"2 and 4"2 _+ 0"2 respectively. Jackson, Lemons and Taube c33s~ have found that Cr: ÷ produces a low field shift in the tTO resonance signal of CIO; in addition to the expected low field shift of the tTO signal of H20. Evidence was obtained for preferential replacement by C10; of water molecules occupying axial positions in the first coordination sphere of Cr 2÷. A very effective means of obtaining information on the solvation of ions or the exchange ofligands generally is the measurement of relaxation times. Relaxation times are usually estimated by pulse methods (see Section 2.12.3) and therefore the topic lies on the borderline of the scope of this monograph. An account of some line width measurements made by means of a high resolution NMR spectrometer will be given below followed by a summary of the results obtained by pulse techniques. Results obtained from line width measurements. Pearson and his co-workers (97~ measured line widths, d ~, directly from the oscilloscope trace and they converted these to transverse relaxation times using the relation, 7"2 = (TrA ~)-~. A magnetic nucleus residing within a paramagnetic ion differs in two respects from other nuclei in the solution. Firstly, they are acted upon much more strongly by the local magnetic field set up by the ion so that relaxation is more complete, and secondly, the molecules (solvent or other ligands) surrounding them are less free to exchange with other molecules in solution. Here then, is a means of measuring rates of exchange of lJgands (gs' to2). Taking water as an example, several first order rate constants govern the signal width and intensity. There is the exchange rate constant z~ t, the relaxation rate constant of a hydrogen nucleus in a coordinated water molecule T~ta and the corresponding constant T ~ for a free water molecule. Several other time dependent variables such as the electron spin relaxation time %, the dipolar correlation time of the tombling hydrated ion, and the frequency of the applied radiofrequency field are also important. When 7"2n > zB only a few hydrogen nuclei are influenced by the paramagnetic ion in a time during which the magnetisation is established. Fast rates of exchange and slow rates of relaxation lead to a small effect on each 1H nucleus. Pearson and his co-workers have analysed mathematically
THE EFFECTS
OF C H E M I C A L
EQUILIBRIA
525
the system of a magnetic nucleus having the three environments, a diamagnetic ion labelled A, a paramagnetic ion (possessing a single unpaired electron) with spin up labelled B and spin down labelled C. They adopted McConnell's procedure (s) described at the beginning of Section 9.1. Assuming that the solution is dilute and that there is rapid spin exchange for the electron, then the experimental width of the line 2"~1, unshifted from the pure solvent signal, is given by T~ 1 -- T ~ + Pe/(T~.e + za) (9.119) where Pe is the mole fraction of water molecules coordinated by the paramagnetic ion and it has the value 6 M/55 for a coordination number six and an ion concentration of M moles litre-L The equation shows that the relative values of the exchange lifetime ze and the paramagnetic relaxation time T2 B determine the effect of the paramagnetic ion. If zB >> 7'2 B then the exchange is rate determining, the line width being given by T~ x -- T ~
-1 + "rA1 ffi T2A "{- PB ZB 1"
(9.120)
If Tae >> ze, then relaxation predominates and 2"2 is obtained from T21 ---- T ~ + Pe T 2-1 B.
(9.121)
The relaxation time T2 A of a nucleus in a diamagnetic environment (bulk solvent) is not the same as that for pure water since dipole--dipole interaction occurs through several layers of water molecules surrounding the ion. A concentration dependence is present of the form • 1 T~2A ---- T]~ + Be T],~A
(9.122)
where 2"2o and T2,A are respectively the spin-spin relaxation times for pure water and the relaxation time for the second and higher layers of water molecules. Since T ~t is about ten times greater than T~2,xA,T~2A t can be estimated without serious error. Experimentally, the relative importance of ~ and 2"2 can be assessed in several ways. If the chemical exchange is rate determining then the addition of other reagents will have an effect. Another method is to study the temperature dependence of the experimentally-determined T2. Increasing the temperature will reduce ~e since there will be associated with it a heat of activation according to the expression z~ t o~exp ( - ._4H / R 7")
(9.123)
where/! H is the heat of activation. By contrast, T2 e increases with increase of temperature ix°e) if molecular rotation limits the correlation but if electron spin exchange limits the correlation then 2"2a will be reduced slightly by increase of temperature. A difficulty arises from the fact that zl H is normally rather small. Pearson et al. applied their treatment to the line widths of the hydrogen resonance signal from solutions of a variety of chromium (HI) complexes at ,several acidities. The rate constant k for the proton exchange reaction Cr~NH + OH- --* Cr~N- + H 2 0
526
HIGH RESOLUTION NMR SPECTROSCOPY
was calculated on the assumption that this reaction is responsible for an line broadening in strongly alkaline solution. Table 9.5 fists some of the results obtained from the equation ~gt _- k[OH-]. (9.124) The value of T2A was found by using the results for acid solution where exchange is negligible. Cr(en)s(CIO,)s, Cr(NH3)6(CIO,)3, [Cr(NH3)sH20] (CIO,)s, and K[Cr(NHs)2(SCN),] were chosen for study because independent measurements have been m a d e at low p H values~t°-"),where the rates of exchange are slow enough to be estimated by conventional methods. G o o d agreement was found; for example, Cr(en)] + proved to have a rate constant of TABLE9.5 N - H EXCH~OE RAT~S ~OR Cr(en)s(CIO,0s
WATERAT25°C [OH-I
I"2(see)
pH 2"5 pH 6"1 pH 9"5 pH 9.8 0.0044 0.022 0.044 0.044
0-040 0.044 0-043 0"030 0.024 0"0090 0"0052 0.0066
TB (s~)
1
×
5"7 × 1'2 × 6"5 × 8"6 ×
10-2 I0-s I0-s 10-6 10-6
k (M-~ sec-~)
2 × 3"8 x 3"7 x 3"5 × 2"6 ×
102 106 10~ 10¢ 10~
2"3 x 10s M-t sec -t. There was not good agreement in the values of the heat of activation: unforeseen effects in the N M R work were held to be responsible. Pearson et aL also accumulated data for some paramagnetic ions [Cr(ITI), Fe(HX), Mn(II), Co(H), Ni(1I), Cu(II), Ce(III) and Gd(II1)] dissolved in methanol acidified with hydrochloric acid. By observation of the nuclear magnetic resonance signals for both the CHs and the O H groups, the two rate constants were seen to be almost equal. This is evidence that Ts is being measured and in the instances of Ni(II), Cu(H) and possibly Fe(III) it seemed that the rate of exchange of methanol molecules had been measured. Comparison with results obtained for aqueous solutions (99' to4. tos) showed that the rates of methanol exchange are slower than those of water molecule exchange--as expected. Ethanol solutions gave less clear-cut results but generally the exchange of ethanol molecules is faster than that of methanol. It is worth mentioning in passing that line width measurements on electron magnetic resonance signals given by paramagnetic ions can give useful information about the symmetry of the electric fidd generated by the complexing groups (tee). Connick and his co-workers (z°°' tot) have made measurements of t~O resonance fine widths in order to obtain T2; they also found Tt. They employed solutions of Mn 2 *, Fe 2 *, Co 2 *, Ni 2 * and Fe z * in 0" I M HCIO4 in water enriched with t e e . The rates of exchange (and the corresponding thermodynamic quantifies) of water molecules between the bulk of the solution and the first coordi-
THE EFFECTS OF CHEMICAL EQUILIBRIA
527
nation sphere of these paramagnetic cations were determined(1°1). The rates at 25°C are: 3.1 x 107 sec -1 (Mn2÷), 1.0 x 104 sec -1 equatorial and 2 × l0 s sec -1 axial(Cu2÷),2.7 × 104 sec -x (Ni~÷),l.13 × 10s sec-l(Co2÷)and3.2 x 106 sec-l(Fe2÷). The two exchange rates shown by the cupric ion were interpreted in terms of a distorted octahedral hydration shell. The line broadening of :4N resonances has been used to study the exchange of N H 3 with Ni(NH3)~ + in liquid H 2 0 and NI-I~°'3). In the aqueous exchange, the reaction is first order with respect to the concentration of the complex: itwas found that the enthalpy and entropy of activation are 9.5 + I.I kcal mole -I and - 5 + 4 cal mole -x deg -I respectively. The rate of exchange in anhydrous ammonia is similar at 25°C and so are the energetics (/II-I* = 10 + I kcal mole -x, ,4S* = - 3 4- 4 cal mole -x deg-1).
Results obtained from pulse methods. As discussed in Section 2.5.1, a small concentration of electronically paramagnetic ions dissolved in water causes a marked reduction in the spin-latticerelaxation time T~ of the hydrogen nuclei. Thus TI values can provide information about paramagnetic ions or complexes dissolved in water or any other solvent containing a suitable magnetic nucleus. The most convenient means of measuring TI is by pulse techniques (Section 2.12.3). A n early application of the method was to the study of the rate of reduction of the aqueous Eu(III) ion by zinc(I°7).Most measurements of 2"I and T2 of paramagnetic solutions have been made with a view to providing detailed verificationof the current theories of hydrogen nuclear relaxation or information on the electronic structure of the central metal ion from its effective magnetic mo m e nt.Examples are theions of Fe(TI)(108) ,Fe(III) (109) ,Co(II) (110) ,Nd(III) (110) , Cr(III) (11°), Cu0T) (111), Gd(III) "°2), Ni(II) c112) and Mn(I~ "12). Aqueous solutions of these ions have T~/T2 ratios near unity except for Mn(II) which has 2"1/2:2 ~ 10 for hydrogen resonance at 40 Mc sec-l. King and Davidson (~~3) noticed that the addition of ethylene diamine tetra-acetate or nitrilotriacetate increased T2 to give 71/72 ~ 1. The results obtained for manganous ions in water show that in some cases the temperature dependence of Tx and Te provides information about the hydration of paramagnetic ions. From such data for Mn~÷aq Bemheim and his co-workers (1°2) found the heat of activation for the dipolar relaxation mechanism to be 5.5 kcal mole -1, and that for the chemical exchange to be 8.4 kcal mole -~. The results did not indicate whether chemical exchange involves only the protons or whether water molecules as such participate. However, the low activation heat of 8.4 kcal mole -~ suggests that the mechanism mainly comprises proton transfer. Aqueous solutions of Cu(II), Co(II) or GdffH) proved to have negligible chemical exchange contributions to T1 and 7"2 hence only a dipolar heat of activation can be determined. The values are 2.7, 1"45 and 2.4 kcal mole -1 respectively. The addition of a variety of simple anions to the cupric ion solutions did not produce a significant effect, but the addition of ethylenediamine(en) or 2,2'-dipyridine does have a marked influence on T1 and Te "~'). The data enabled the first order rate constant of 2.4 x 107 M-I sec -~ at 27°C to be found for the reaction C u ( e n ) 2 ( H 2 0 ) i + + en -*.
528
H I G H R E S O L U T I O N NMR S P E C T R O S C O P Y
Measurements of Tt and T2 at various temperatures (~tS) led to a heat of activation of 4.5 kcai mole -~. The effective barriers to tumbling for the ions Cu(H20)+2+ , Cu(en)(H20)~ + and Cu(en)2(H20) 2+ were also obtained having the values 5"1, 4"5 and 3"9 kcal mole -~ respectively. Spin-lattice relaxation time studies c~~e) on aqueous solutions of substitutioninert chromium (III) complexes showed that the ions having greatest tendency to interact with solvent water molecules (by virtue of possessing exchangeable protons, for example, Cr(H20)~ +, or by participating in hydrogen bonding with the solvent, for example, CrF~-) have the smallest spin-lattice relaxation times. The order of increasing TI is Cr(H20)~ +, CrF~-, Cr(NHs)~ +, Cr(C20+)~-, Cr(en)] + and Cr(CN)~-. Results obtained from Cr(H20)6Cls are dependent on the thermal history of the solutionsc117). Gasser and Richards c91) added ethylenediamine to aqueous Co(NH3)~ + and observed ligand exchange according to the overall equilibrium Co(NH3)t+ + 3 en ~ Co(~a)~+ + 6NIt3. The low resolution SgCo resonance spectrum showed two separate absorption bands corresponding to the two Co(IH) entities. The exchange process is sufficiently slow to enable the growth of one absorption band, at the expense of the other, to be measured on the addition of ethylenediamine. The experi= mental data gave a value of 22 4- 2 kcal mole -~ for the overall heat of activation of exchange. The first order kinetics show that the first substitution reaction is rate determining and that the subsequent substitution reactions are rapid. The two 59Co resonance lines recorded for sodium cobaitinitrite in water were attributed, one to a nitrite form of the complex and a hydrolysis product and the other to the nitro isomer. The enthaipy change associated with the isomerism was estimated to be 8 + 1 kcal mole -t.
Addition compounds. Nuclear magnetic resonance has proved effective in the study of complex formation between the strong accepter molecule boron trifluoride and donor molecules containing oxygenc1t s. 121.3~1. a62). The fluorine resonance of a liquid mixture of boron trifluoride and two donor molecules, for example methanol + ethanol, consists of two absorption signals when sufficiently low temperatures are employed. Band area measurements showed that reversible equilibria are established between the two donors and their respective complexes: it is possible to calculate the relative concentration equilibrium constants. Hydrogen resonance spectra were found to give similar data. From the results it was evident that affinity for boron trifluoride decreases in the order water, methanol, ethanol, n-propanol and n-butanol "~s" 1~9) When the temperature was raised to S0°C, the two lines merged to a single line because of rapid exchange of BFs among the donor molecules. The rate of exchange depends upon the boron trifluoride concentrationcIx9) and leads therefore to the mechanism BF3 + CH3OH.BF3 - * B2F¢ + CHsOH B2Fe -+ 2 B F s
slow
rapid
C H 3 O H + BF3 - * CHsOH.BF3
rapid.
THE EFFECTS OF CHEMICAL EQUILIBRIA
529
From a study of the IH resonance of these addition compounds, Di©hF 1~°) has obtained evidence for aggregates greater than 1 : l for alcohols; some may have the structure
The second alcohol molecule is held by a hydrogen bond. Diem also studied the more pronounced changes in chemical shift in the fluorine resonance spec= tra of complexes of boron trifluoride with alcohols, diethyl ether, water and acetic acid. Line width data were used to evaluate the chemical exchange times which in turn allowed the heat of activation to be calculated: 7"3 + 1 kcal mole -1 for the exchange of BF3 between methanol and ethanol. Observations of the XgF spectra of the water-boron trifiuoride system (BF3 concentrations from 0"5 to 11 ~) led to the identification of the species H B F , , HBF3OH, HBF2(OH)2 and HBF(OH)3. One can expect N M R investigations to be extended to other donor molecules such as ketones, and carboxylic acid anhydrides. Nitrogen-containlng donor molecules are also amenable to study ff interference from the nitrogen nuclear quadrupole moment is avoided by application of the double resonance method (see Section 6.8). Happe c122) has made measurements on the association betwcen pyrrole and pyridine from the 40Mcsec -1 XH resonance spectra with simultaneous saturation of the nitrogen nuclei by irradiation at 2.9 Mc sec -~. The large low field shifts of the N H multiplet of pyrrole on the addition of pyridine are good evidence for a N - - H m N linkage. The enthalpy change associated with the equilibrium was found to be - 4-3 kcal mole -~.
Aromatic complexes. These complexes are encountered in nuclear magnetic resonance spectroscopy whenever a polar solute is dissolved in an aromatic solvent. Hydrogen bonding can be a factor cx23) in determining the extent of the solvent-solute interaction. However, there undoubtedly exists a weak specific solvent-solute interaction ct23' ~27). The presence of a molecular complex according to the rapidly established equilibrium solute + solvent ~ molecular complex implies that the chemical shifts of the solvent magnetic nuclei are not only concentration dependent but temperature dependent as well. The latter effect has been studied in detail by Hatton and Schneider c12". It is believed that in complexes between polar solute and non-polar aromatic hydrocarbons, the solute hydrogen nuclei tend to be located above the plane of the ring and ad= jacent to the symmetry axis. This topic is discussed more fully in Sections 3.10 and 10.39 under the general classification of solvent effects. A brief study has been made (12s) of aqueous silver ion complexes of cyclohexene, c/s-2-butene, benzene and toluene. The olefmes appear to form a single non-labile l : 1 complex. Evidently rapid exchange is present in the aromaticsilver solutions since every aH resonance spectrum featured only a single
530
HIGH RESOLUTION NMR SPECTROSCOPY
absorption band due to a particular aromatic hydrogen nucleus. Both 1 : 1 and I : 2 complexes are formed with the former predominating.
9.3.4 Miscellaneous Atom and Group Transfer Reactions The largest group of compounds failing under this heading are the inorganic fluorides which are particularly suited for nuclear magnetic resonance studies. The first work of this kind to be reported is that of Muetterties and Phillips (129) who observed the temperature dependences of the xgF spectra of chlorine trifluoride, bromine trifluodde, bromine pentafluoride and iodine pentafluodde. Impure CIF3 gave a single relatively broad absorption band affected little by temperature variation. The C2, symmetry of the molecule (xa°' 1ai) leads to the expectation that the spectrum should originate from two sets of non-equivalent fluorine atoms and should consist of a doublet and a triplet for an irradiating radiofrequeney of about 40 Me see -1 or higher (a6~). Extreme precautions in the removal of fluoride impurities resulted in this type of spectrum being observed at - 60°C. Raising the temperature gave the characteristic feature of the onset of rapid chemical exchange, namely only a single broad concentration-dependent band was recorded at 60°C. The lifetime of the exchanging fluorine nuclei was estimated to be 2.4 x 10 -5 see at the latter temperature and the heat of activation was estimated to be 4.8 kcal mole -1. The 19F resonance spectrum of bromine trifluoride differed in that above its melting point (8.8°C) only a single sharp line was exhibited. The inability to detect even line broadening was ascribed to a heat of activation lower than that of CIFa. The mechanism put forward for fluorine exchange is
F\ //F,~ / F _ F \ / F \ / F F/CI\F / CI~,F ~-"F/CI,,,F/CI(,,F since association of the halogen halides has been established (132"134). Hamer (I 3s) has cast some doubt on the above results by demonstrating that even greater care in the removal of hydrogen fluoride from chlorine trifluoride raised the temperature at which line broadening occurred. Furthermore, the precise temperature at which multiplet splitting disappeared varied from sample to sample. It may well be that exchange between chlorine trifluoride molecules does not occur in the absence of hydrogen fluoride but even if it does the heat of activation must be considerably greater than 4.8 kcal mole -1. Muetterties and phillips (:29) also collected evidence for fluorine exchange in the N M R spectra of bromine pentafluoride and iodine pentafluoride and were able to evaluate the enthalpy of activation of the latter compound. These workers also extended their investigations to a wide range of inorganic fluorides (~3~). They found that the spectra of all fluorides that are of tetrahedral or octahedral symmetry are relatively insensitive to fluoride impurities by contrast with fluorides of lower symmetry. Sulphur tetrafluoride, of C2, symmetry, shows exchange behaviour and the temperature dependence of the spectrum of this compound is illustrated in Fig. 9.16. Fluorine atoms appear to exchange between the two non-equivalent fluorine environments of the pseudo trigonal
THE EFFECTS
OF CHEMICAL
531
EQUILIBRIA
bipyramid. The activation heat for the process was calculated to be 4.5 + 0.8 kcal mole -1 in good agreement with earlier results "3~). A dimer was suggested once more as a means of effecting exchange F
F
F
F
23°C
SOF:-- , / . ~ . ,
.
-~O"C
H II SO~
SF~
;
tl'SOE
-50~C
-45°C
l-soF:
I I
I_SOFo It
-58°C "w
~
j
-98°C $ = 52 I:}prn d = 78 cycles sec-L
Fxo. 9.16 The temperature dependence of the 19F nuclear magnetic resonance spectrum (30 Mc sec-1) of sulphur tetrafluoride. Muetterties and PhillipsO3e)
Lone pair p electrons occupy one of the coordination positions. A similar set of findings was obtained for selenium tetrafluoride, while teUurium tetrafluoride gave a single sharp line down to - 100°C. The rates of fluorine exchange follow the order SF, < SeF, < TeF, which is the same as that for increasing molecular association as evidenced by melting and boiling points. Some evidence was obtained for the acid-catalysed exchange between F - and SiFt-. Myers (lss) has used the quadrupole broadening of the lzTI resonance line of the iodide ion brought about by the presence of iodine molecules to investigate the kinetics of the tri-iodide equilibrium. In aqueous solution the iodide ion suffers a strong quadrupole pulse as it undergoes chemical reaction with an iodine molecule: there is no longer a spherically symmetrical charge distribution in the tri-iodide ion. Quadrupole couplings in the iodine molecule and
532
H I G H RESOLUTION NMR SPECTROSCOPY
the tri-iodide ion are so strong that the 12~I resonance line is broadened beyond detection. On the basis of the simple equilibrium
I- + I2.q ~ I~ the rate constants for the forward and backward reactions at about 35°C were found to be (4-1 + 0.4) x 101° M-1 sec -1 and (7"6 + 0"8) x l0 T sec -1. The former rate constant is one of the largest reported for a reaction in aqueous solution involving a neutral molecule. Nevertheless, the lifetime of an iodide ion is many microseconds. The heat of activation must be small since the rate approaches the collision frequency. An interesting exchange system is that of allyl magnesium bromide which has a bond switching mechanism within the moleculec~39). It was noticed that the ~H resonance spectrum is much simpler than is normally encountered with allyl compounds due to the merging of the signals from the hydrogen atoms in the 1 and 3 positions thus giving a spectrum of the AX, type. The spectrum can be interpreted reasonably in terms of the rapid equilibrium (tl/, _~ 0.001 sec) B r M g - - C H 2 - - C H ~ C H 2 ~- C H 2 ~ C H - - C H 2 - - M g B r
Other allylic Grignard reagents show this behaviour (1.°). Aluminium trialkyls exist as dimers due to the formation of two A1--C--A1 bridges. Two separate IH resonance signals have been detected(1.~' ~42) from the bridged and outer methyl groups of aluminium trimcthyl at about - 7 5°C: at about - 20°C the mean lifetime of a configuration is about 3 x 10 -3 sec. In mixtures of aluminium trimethyl and aluminium tri-isobutyl an exchange of alkyl groups takes place (~43). The resultant mixed aUcyls are strongly associated as dimers by moans of methyl group bridges. From the concentration dependence of the chemical shifts, Hoffmannc~43) has estimated the T values of the bridge and outer methyl groups to be 9"69 and 10-59 ppm respectively. As the proportion of aluminlum tri-isobutyl is increased, the stage is reached when only one methyl group is available for dimer formation, the structure being i Bu\
/iBu,.
/ i Bu
iBuz
~M¢/
\iBu
/AI,:I
)AI\
Hoffmann has also investigated exchange reactions between AIMe3 and AIMe2H, A1Me2C1, A1Me,OMe, A1Me,OEt, A1Me2t-Bu and A1Me3 • NEt3. The remainder of this section will be taken up with exchange processes sufficiently slow to give distinct spectra of the species taking part. When a mixture is made of boron tri-fluoride, chloride, bromide and iodide rapid equilibrium is established between all possible halide species c144, 3~v). Rapid disproportionation to the BX3 forms prevents isolation of the mixed halides but observation of the 19F resonance allowed the spectrum of the new compound BBrC1F to be recorded( ~ and its chemical shift to be measured (see Section 11.19.1). Intermediate species have also been reported ~45) from the ~~9Sn spectra of mixtures of SnCl,, SnBr, and SnI, (see Section 12.8.2). Koski, Kaufman and Lauterbur(.46) supplemented their infrared(~4~) and mass spectrometric(~48)
THE EFFECTS
OF C H E M I C A L
EQUILIBRIA
533
studies of the exchange of deuterium between diborane and pentaborane with hydrogen nuclear magnetic resonance measurements. In these, the exchange of deuterium between BzD6 and BsH9 was measured by allowing mixtures of them to equilibrate for several hours before examining their N M R spectra. Exchange was found to occur preferentially in the terminal hydrogen positions of pentaborane. The rate of exchange of the apex hydrogen atoms appeared to be within __ lOper cent of that of the base terminal hydrogen atoms: bridge hydrogen atoms in pentaborane did not participate in the exchange. A similar investigation has been carried out on the deuteration ofdecaborane by deuterium oxide (149). The results of previous work else) were confirmed in that rapid exH
t
Reoctlon time
O" O0 hr I I
I I
[ i
0. 15hr
0' 5 0 h r I
I
I • O0 hr
^'teSJ'l ,.,
I
I
1[ ,5,74
1241 o
I,
tL ~01
Fro. 9.17 ThetlB nuclear resonance spectrum of dccaborane in DzO as a function of reaction time. Shapiro, Lustig and Williams(149) change in the bridge positions occurred followed by slower exchange at the terminal sites. Figure 9.17 shows the I*B resonance spectrum at various times; the deuteration of the terminal positions becomes apparent at long reaction times. The 1H resonance spectrum showed similarly that exchange takes place at the bridge positions in less than 0.15 hr. The nuclear magnetic resonance technique has been applied to the study of the well-known hydration reaction of aldehydes ('sl)
H20+R--C<~ ~ R--C~oHHH D20 or H 2 0 was added to the aldehydes, acetaldehyde, propionaldehyde and i-butyraldehyde. Pure acetaldehyde has a ~H resonance spectrum comprising
534
HIGH RESOLUTION NMR SPECTROSCOPY
a quartet of chemical shift - 4"3 ppm (to low fields of an external water reference) and a doublet 4.1 ppm to high fields. The addition of D 2 0 produces extra bands, namely a quartet and a doublet having chemical shifts of 0.7 ppm and 5"0 ppm respectively. These absorption bands are due to the formation of a new species (the partially deuterated hydrate) and they confirm that the cquilibrinm is relatively slow. Intensity measurements enabled the formation constant of the hydrate to be calculated--a value of (2"7 + 0"1) × 10 -2 M-1 was found for acetaldehyde with added H 2 0 or D20 at 25°C. The results are in good agreement with those obtained spectrophotometrically by Bell and Clunie(t 52). 9.4 HYDROGEN BONDING
9.4.1 Introduction
Hydrogen bonding has been mentioned several times already in this chapter because of the fact that it is bound up with a host of solution phenomena. The present section reviews nuclear magnetic resonance studies which have been aimed specifically at investigating hydrogen bonding. The growth of interest in the topic as a whole is very great as will be realised from the publication total of more than 3000 papers. An excellent monograph covering most aspects of the subject has been written by Pimentel and McClellan ¢153). A full knowledge of hydrogen bonding is necessary if we are to understand complicated organic substances (intramolecular interaction) and one of the most important solute-solvent interactions (intermolecular interaction). Hydrogen bonding occurs in aU states of aggregation but our concern with high resolution N M R spectra necessarily limits us to pure liquids and solutions. In passing, it should be mentioned that the low resolution technique can provide much information about hydrogen bonded solids (1s4). The hydrogen bond is an associative interaction of molecules usually containing nitrogen, oxygen or fluorine atoms; however, the base does not necessarily contain a highly electronegative group. The base must be a good electron donor hence aromatic molecules can take part in hydrogen bonding. For this reason, the selection of a suitable solvent is difficult with dioxane and benzene not being inert to proton donating substances. The clusters of molecules resulting from hydrogen bonding are often of well-defined stoichiometry and are located in specific orientations. The justification for including hydrogen bonding in a chapter on dynamic processes is that equilibria between clusters and their component parts are rapidly established and reversible. Many examples involve fairly simple equilibria in the readily accessible temperature range of 0 to 200°C. The hydrogen atom concerned in the bonding usually gives rise to only one resonance signal because movement of a molecule from one hydrogen bonding environment to another takes place in a time short compared with that of making a m e a s u r e m e n t (10 -3 see). Hydrogen bonded systems can be found having lifetimes of the order of 10 -3 sec on cooling. Infrared measurements have played an important part in the determination of equilibrium constants and thermodynamic functions of hydrogen bonded
THE EFFECTS OF C H E M I C A L E Q U I L I B R I A
535
systems. The method is based upon intensity measurements of the band assigned to the monomeric species, it being assumed that polymers do not absorb radiation at the same frequency. This assumption holds providing the polymers are cyclic structures without terminal groups "53). Examples of compounds forming cyclic structures are alcohols, phenols, carboxylic acids, laetams and some amides. Infrared spectroscopy enables different hydrogen bonded complexes to be distinguished. Relatively low concentrations can be used and gas and solid phases can be studied readily. Nuclear magnetic resonance scores in the ready detection of hydrogen bonding because one observes the specific role of the hydrogen atom in the interaction. The most powerful approach to hydrogen bonding problems is to combine both these physical methods. NMR measurements yield the following types of information: (i) chemical shifts of the hydrogen bonded hydrogen nucleus; (ii) hydrogen bond dissociation and exchange times; (iii) changes in the relaxation times T~ and T2; (iv) hydrogen atom positions in hydrogen bonded crystals. Only the first two types fall within the scope of this monograph. Type ('hi) provides an indirect observation of hydrogen bonding between large molecules<~s-~)in which the presence of such an interaction results in aggregates of high molecular weight being formed. The molecular tumbling frequencies are reduced to the critical region where they do not completely average out direct dipole-dipole interactions between neighbouring hydrogen atoms in the molecule. A sharpening of all resonance lines then occurs when the temperature is raised (thus disrupting aggregates) or by replacing the hydrogen bond-forming hydrogen atom by an inactive group, as for example by converting a hydroxyl group to an acetate derivative.
9.4.2 The Nature of theHydrogen Bond Although the existence of intermolecular and intramolecular hydrogen bonds has been recoguised for some time, theoretical interpretations are of recent origin. Further developments of the theory may well have considerable impact on the general theory of the chemical bond. The electrostatic model is a modified point charge array which includes the charge distribution of non-bonded electrons as inferred from bond hybridisation. The calculation of the hydrogen bond dissociation energy of water (156' ls7) gave a reasonable value (6 kcal mole -1) thus encouraging use of this model (~58" ls9>. The electrostatic description does not account for all the phenomena associated with hydrogen bonding, the more important discrepancies being( xs 3) (i) the increase in intensity of the infrared band corresponding to the stretching mode A - - H ; this increase is much too large; (ii) the absence of correlation between hydrogen bond strength and the dipole moment of the base; (iii) the decrease in intensity of the infrared absorption band corresponding to the bending mode of A--H.
536
HIGH
RESOLUTION
NMR SPECTROSCOPY
Attempts to calculate the covalent contribution to the hydrogen bond have not been conclusive (1~°" 1~). A qualitative molecular orbital description has been put forward by Pimentel (162) for hydrogen bonding in the ion HF~. One pair of electrons takes part in bonding in an orbital extending over two bonds (three-centre orbital), each bond then being relatively weak. The other electron pair occupies a non-bonding orbital which places the excess charge on the fluorine atoms. For the general case of any two atoms separated by a hydrogen atom, the location of the charge is dictated by the relative electronegativities of the two atoms. Hydrogen bonding will only be favoured if these atoms are sufficiently electronegative for the non-bonding orbital to accept an electron pair. Pimentel and McClellan (153) have made the intriguing suggestion that this description also applies to atoms of low electronegativity providing that the compounds are electron deficient thus leaving the non-bonding orbital unoccupied as in the case of the boron hydrides. The carbonyl group in benzophenone, for example, has different hydrogen bonding properties when dissolved in hexane as opposed to ethanol. In the latter case a hydrogen atom of the solvent behaves as ff it has a good deal of mobility whereas in the former case it is strongly bound. Thus, the hydrogen atom in the C - - H groups of hydrocarbons is in a valence state involving mainly 1s atomic orbitals. Hydrogen atoms in hydrogen-bonded groups like N - - H or O - - H can have contributions to their valence state from 2s and/or 2p atomic arbitals (tes' le,~). In his molecular orbital treatment, Paolini (~ts) has used this idea of n = 2 orbitals in conjunction with the Lippincott-Schroeder (~ts) potential function description of the hydrogen bond. The strength o f this bond is regarded as being determined by the distance apart of the atoms A and B between which the hydrogen atom establishes a link. The theory was developed for (i) the weak hydrogen bond normally encountered in liquids or solutions and (ii) the strong hydrogen bond present in many crystalline substances and having rather small A B distances. A suitable choice made for evaluation was the O--H. . . . O system. Starting with two orthogonal atomic orbitals on the hydrogen atom, one was selected as being an excited (Rydberg) orbital to help to emphasise the covalent character of the H B bond. This extra orbital on the hydrogen atom is more favourable to bonding, due to its location in space, than is the alternative possibility of an anti-bonding orbital on the A - - H group. The atomic orbitals of A and B were defined as hybrids built from s and p orbitals. The calculations indicated that hydrogen bonds can be formed with very little charge migration. It was also shown that the theory can account for the increase in width and intensity of the infrared absorption band of the O - - H group caused by hydrogen bonding. The large variation of the electronic charge on the hydrogen atom (assuming the O O distance remains unchanged) implies that there is a large probability of the bond being broken within the period of several vibrations. However, there is a series of O - - - O distances permitted by the various parameters so that a hydrogen bonded liquid probably corresponds to a certain statistical distribution of these distances. Since each distance corresponds to a given bond strength, there will be a given shift of the O - - H stretching frequency. The broadening of the bands is due to the overlapping of the various
THE EFFECTS OF CHEMICAL EQUILIBRIA
537
shifted bands. Paolini's theory is also capable of predicting the observed increase of dipole moment brought about by association caused by hydrogen bonding. Calculated values of the bond dissociation energy followed the trend of the experimental values only if the amount of s character of the atomic orbitals of the oxygen atom Ob in O~----H----Oe increases as the bond becomes stronger. Because of the interdependence of the lone pair hybridisation, the increase in s character of one of them brings about a decrease in the other. The unsymmetrical distribution of s character produces Oo--H and H ....Ob bonds of different strengths: the final situation is governed by the groups attached to Oo and O~.
9.4.3 The Origin of Chemical Shifts due to Hydrogen Bonding A comprehensive coverage of the theory of chemical shifts is given in Chapters 3 and 4 but the effect due to the extra complication of hydrogen bonding was omitted: this section attempts to fill the gap in our treatment. The hydrogen resonance chemical shift accompanying hydrogen bond formarion is towards lower fields except when an aromatic compound is the base. This apparent reduction in electron density surrounding the particular hydrogen nucleus is somewhat surprising if one argues from the standpoint of electronegativity of the electron donor. The lack of correlation between the shift and the electronegativity (position in the Periodic Table) may be seen from Table 9.6 which lists the experimental data by Schneider, Bernstein and Pople tle6~. TAaL~ 9.6 NET OmMXCAL S~tS DUE TO THE LIQUID STATE (166) Compound
Liquid shift ppm (to low fields)at t°C
t°C
m.p. °C
CH4 C2H6
0.00
- 98
--184
0-00
-
¢2I'~
0.43
- 60
--172 -170
C2H2 NFI3 H20 I'IF PHa HC1 HI HCN
1-30
- 82
-
1"05
- 77
--77
4.58
6.65 0-78 2"05 2"55 1-65
88
82
0
- 6O -- 90 - 86 - 5 - 13
0 --92 -
133
-
112 50
-
14
They measured the chemical shifts of a series of gaseous hydrides from a methane gas internal reference: shifts of the same hydrides in the liquid phase were measured from an external cyclopentane reference. The liquid measurements required correction for the bulk susceptibility, then the net shift attributable to the liquid state was calculated by subtracting the shift in the gas phase from that in the liquid, both being referenced to methane gas. However, these authors found little correlation between the gaseous chemical shifts and the electro-
538
H I G H R E S O L U T I O N NMR S P E C T R O S C O P Y
negativity of the central atom' For first row elements of the Periodic Table, the trend of the hydrogen resonance signal shift was to lower fields but elements of the second row showed the reverse trend. The net shifts of the liquid hydrides given in Table 9.6 are substantial for all except the saturated hydrocarbons. One can readily appreciate that the N M R method lends itself to the study of association. Liquids exhibiting strong hydrogen bonding properties like H F and H 2 0 have much greater shifts to lower fields than other substances. Another interesting feature of the table is its demonstration of hydrogen bonding of a hydrogen atom in one molecule with the ~ electrons of a neighbouring molecule in the cases of acetylene and ethylene. Earlier studies of a variety of hydrides c1~7" zts) produced no clear correlation of chemical shift with ionicity or hybridisation. The experimental data just presented leads, in the absence of anisotropic or electric field effects, to the conclusion that the electron density probably increases in the neighbourhood of the hydrogen atom responsible for intermolecular association. The electron distribution is such that the magnetic shielding decreases. The remainder of this section PrOvides an outline of the theoretical treatments of the factors influencing shielding of hydrogen bonded magnetic nuclei. Schneider, Bemstein and Pople (z66)considered the difference between the secondary magnetic field due to induced currents both for a gaseous molecule containing the A - - H bond and for a hydrogen bonded system A m H . . . . B to consist of two contributions: (i) the magnetic field experienced by the hydrogen nucleus due to the direct effect of induced currents in B; (iii the polarisation of the electrons in the A - - H bond caused by B, which in turn affects the shielding of the hydrogen nucleus. The theory of effect (i) is very similar to that of the induced current model discussed previously in Section 4.3: the theory gives a general understanding of the main features of hydrogen shielding. Effect (i) depends upon the nature of B and for linear molecules forming linear hydrogen bonds, increased screening is caused by hydrogen bond formation. This means that, except for aromatic systems, the observed chemical shift results from the predominant effect (ii). Treating the hydrogen bond as being electrostatic, effect (ii) manifests itself as a paramagnetic contribution from B thus providing an asymmetric electric field in the vicinity of the A - - H bond. Schneider, Bernstein and Pople (1~6) estimated that hydrogen bond shifts can be interpreted largely in terms of the reduction of the diamagnetic circulation in the A m H bond by the electrostatic field of the donor Y. Hameka ¢~9) has also regarded hydrogen bonding as affecting shielding constants by the opposing intermolecular and polarisation effects. He applied his method of calculating hydrogen resonance shifts (see Section 4.2) to assess the effect of hydrogen bonding on the hydrogen shielding constants of the ammonia molecule, taking the two contributions separately in order to determine their relative importance. (i) The intermolecular chemical shift.
THE EFFECTS
OF CHEMICAL
EQUILIBRIA
539
The N - H axis of one ammonia molecule was assumed to be collinear with the lone pair orbitals of another molecule, the N----H distance being taken as 4.5 ao
H
N
J The intermolecular shielding constant a' of the hydrogen bonded hydrogen nuelens was shown to be given by [0.1985(1 + 2A) - 0-0093(2' + 2A)] × 1.775 x 10-5 2(1+22A +22 ) where A ( = 0.5955) is the overlap integral for tetragonal symmetry and 2 is an electronegativity parameter found to have the value of 0.724 by calculation from the dipole moment of the ammonia molecule. Hencecr' = - 1"02 x 10 -6. (ii) The polarisation chemical shift. The polarisation of the ammonia molecule from liquid association was represented by the change of 2 caused by hydrogen bonding. The dependence of the shielding constant on the variation of ~ was then estimated to be (0-55432 z + 2.96042 + 2.1704) x 1.7748 x 10-s 2 z + 1.19104 + 1 ;t was found to change by 0-046 because of the polarisation shift giving a corresponding change in the shielding constant of - 0.45 x 10 -6. The total change in the shielding constant is thus - 1.47 x 10 -6 as compared with the experimental value ~166~ of 1-05 x 10-6. Hameka's results show that, in the case of ammonia, the intermolecular shift is the main effect whereas those of Schneider, Bernstein and Pople c166~placed the emphasis on the polarisation shift for linearly-associated molecules. Although the two situations may be quite different, Hameka believes that the intermolecular contribution predominates in all liquid association shifts.
9.4.4 Interpretation o:[ N M R Data General Pimentel and McClellan¢1s3~have drawn attention to a set of practical rules cssential to a thorough investigation of hydrogen bonding. These are (i) A really pure solvent should be selected paying due regard to its hydrogen bonding properties. (ii) The concentration should be varied and the sensitivity of the spectrum to this variable determined. Sufficiently low concentrations should be used to
540
HIGH RESOLUTION NMR SPECTROSCOPY
enable control over intermolecular association to be exercised and, if possible to give complete dissociation to monomer. (iii) The chemical shifts should be measured at different controlled temperatures. (iv) Deuterium should be substituted for hydrogen to verify the assignments. The earlier nuclear magnetic measurements were made before all of these precautions became evident. One of the first systematic studies was made by Huggins, Pimentel and Shoolery(~7°) on the concentration dependences of the chemical shifts of various phenols in carbon tetrachloride. The data were interpreted in terms of a monomer plus dimer, the observed chemical shift ~ being the weighted mean <3) of shifts characteristic of the monomer, dM, and the dimer, dD = c, ~M + (1 - 0,) ~x,
(9.125)
where o~ is the fraction of hydroxylic hydrogen atoms present as monomer. The limiting value of ~ and of the slope (d~/dx)o at zero phenol concentration are given by ~o = ~M (9.126) and
(dt/dx)o = 2K(tD -- 6M)
(9.127)
where x is the apparent mole fraction of total phenol and K is the association constant in mole fraction units. If the experimental data is reliable, dM and 6D should be temperature independent. An expression similar to equation (9.125) has been used to represent the data for association between different molecules, for example, chloroform in acetone and in triethylamine(171' 17z) Saunders and Hynec173) have taken the interpretation of data further by considering the general case of association to polymers containing n monomer units. The frequency v of the weighted mean is given by v=
v i M 1 + 2~2KzM~ + 3~'3K3M~ + "'" nv,, K. M"I M~ + 2 K 2 M 2 + 3 K 3 M ~ + ... nK,,M]
the total concentration
c = M~ + 2 K 2 M 2 + 3 K 3 M ~
(9.128) (9.129)
where ~,, is the frequency of the pure n-mer, K, the corresponding equilibrium constant and M~ the monomer concentration. Strictly, solutions should be ideal and the equilibrium constants are thermodynamic constants; however, providing the activity coefficients do not vary in the concentration range studied, the equilibrium constants will be concentration constants. Equation (9.128) is not as intractable as it appears at first sight since after the first few terms in both the numerator and denominator a limit should be approached in a uniform manner. Making the simplifying assumption that all the equilibrium constants but one are insignificant then v=
Kn g ] ) (M~ + nK,,M"t)
(~lnl
+
IZ V n
c = M, + nK,,M].
(9.130) (9.131)
THEEFFECTSOFCHEMICALEQUILIBRIA
541
Tabulation of arbitrary values of M1 enables both • and c to be calculated for any value of n by introducing appropriate constants. Any equation of the above kind containing ~,, wl, ~., c and K. must be of such a form that c and Kj may be replaced by the dimensionless term cK~a-")l ". Thus from a plot of frequency versus concentration for a given K. value, the curve for any K. can be derived by multiplying the concentration scale by a suitable factor. Further simplification is possible by plotting frequency against the logarithm of the concentration, since multiplication of the concentration scale can then be
i 0 0 ~ u 5O
I -I-0
0
1.0
1o9 C
Fit. 9.18 The XHchemicalshifts(40 Mc see-t) of the hydroxylgroup of t-butanol in carbon tetrachloride as measuredfrom an external water refeze.nce.The curves showthe best theoreticalfitsfor A, monomer-dimer,B, monomer-trimer and C, monomer-tetramerapproximations. Saunders and HyneO73~ achieved by simple lateral displacement of the curve. An example of the method devised by Saunders and Hyne is shown in Fig. 9.18. Here, the dots represent experimental data while the full lines are theoretical curves adjusted to give the best fit. The monomer-trimer curve is the best and is surprisingly good over
542
HIGH RESOLUTION NMR SPECTROSCOPY
a wide concentration range. The strong preference for trimers tlT't, x75) suggests a six-membered ring structure with three mutually reinforcing hydrogen bonds. T h e linear portion of the curves at intermediate values of log c was adjusted by varying ~ noting that the slope is proportional to ~ - ~ . Hence it is possible to determine ~ in addition to K~ f r o m the curves. Table 9.7 lists the results obtained by the above authors f r o m their own data on methanol in carbon tetrachlofide, t-butanol in carbon tetrachloride, phenol in carbon tetrachloride and f r o m the data of Becker, Liddel and Shoolery (~Te) for ethanol in carbon tetrachlofide. The values of ~1 for the hydroxyl group in the aliphatic alcohols are close and no doubt characteristic of the intramolecular environment of such a group. Despite the excellence of the data it must be borne in mind that further evidence is required before these constants can be assigned unambiguously to a given theoretical model (173). This is especially true as the method expresses the equilibrium as a mixture of m o n o m e r and " a v e r a g e " polymer. Reid and Connor (~77) provide reasoning in favour of a cyclic trimer for simple aliphatic alcohols: the model allows a "circulating c u r r e n t " loop to be set up outside which lie the hydroxylic hydrogen atoms and thus they suffer a paramagnetic shift. Comparison o f the data in Table 9.7 with the results of Huggins, Pimentel and Shoolery (~7°) is interesting. Saunders and Hyne TAnLE 9.7 CALCULATED EQUILIBRIUM CONSTANTS AND FREQUENCI~ OF SOME HYDROXYLIC
COMVOtn~Sn~ CA~ON. TeTRACnLOPaDEAT21 =t: 1°C MeOH K3 /a Pl P2 P3 )'4 The
EtOH
0.60M -1 0"70 M-1 3"83 M-2 5"19 M- 2 28.4 M-a 44.9 M-a + 169.5 cycles sec- 1 + 174.0 cycles sec-x - 199"3 -- 155.0 --36.5 - 63"8 -18-3 +4*6
t-BuOH
PhOH
0 . 7 2 M -1
0"67 M-I
M-2 50"3 M"a + 164,0 cycles sec-I -- 96"7 --2"7 +29"8
4"78 M-2
5.60
39"6 M-3 +19"5 cycles sec-x -- 204"5 -- 123'8
-- 95"8
numbers in bold type give the best fit with the experimental data.
consider that the successful interpretation of the phenol data in terms of m o n o m e r and dhner only, is due to not extending measurements to below 0-I M: the data included in the table was obtained f r o m solutions down to 0"01 M which was accomplished by using 12 m m spinning sample tubes ¢17s). The method of treating experimental data discussed above suffers f r o m the difficulty that it is insensitive to aggregates greater than tetramers (lv9). Some general methods of treating N M R data have been worked out by Mavel cxa°) who considered association to dimers, association to higher polymers and association of molecule A with molecule B. We will now consider the chemistry of hydrogen bonded systems f r o m the nuclear magnetic resonance aspect. Unfortunately, the number of units forming an aggregate is not clear in m a n y cases because sufficiently dilute solutions were not examined.
THE EFFECTS OF C H E M I C A L E Q U I L I B R I A
543
Hydroxylic compounds. These form the largest class of substances studied so far. Water has been shown to exist in a monomer--direct equilibrium in various mediaCX81.33x~. A similar conclusion was reached for acetic acid at mole fractions less than 0.1 in a range of solvents of widely differing dielectric eonstantsCt82, xs3~: the data from more concentrated solutions are best explained in terms of polymer--dimer equilibria. An unusual effect occurs with this system, contrary to the behaviour of other hydrogen bonded liquids. The resonance signal of the hydroxyl group moves first to low fields but eventually, as solutions become more dilute, it moves to a position higher than that of pure acetic acid. The phenomenon is consistent with the presence of polymers in concentrated acetic acid having weaker individual hydrogen bonds than those of the dimer. In extremely dilute solutions or in solvents of high dielectric constant, dissociation of the dimer occurs. The effect of temperature on the chemical shift due to association indicates that the hydrogen bond strength of acetic acid is comparable with that of liquid hydrofluoric acid tt6e>. The association constants of dimerisation at 25°C were found to be 6.5 x 10 -5 mole in cyclobexane (dielectric constant = 2.05), 2.6 x 10-* mole in carbon tetrachloride (dielectric constant = 2.24) and 6.11 x 10 -2 mole in 1,1-dichloroethane (dielectric constant = 9.1). Experiments designed to determine the chemical shifts of the monomers of formic, acetic and benzoic acids in benzene exs4~ provided evidence for association between solute and solvent. The hydrogen bond shift of the polymeric acids was suggested to originate from increased diamagnetic shielding caused by a reduction in the O--O bond distance. There are indicationscls5) of strong intramolecular hydrogen bonding in the mono anions of dicarboxylic acids which have an anomalously high ratio of their first to their second dissociation constants. From a tH resonance investigation of the hydroxyl hydrogen nucleus in a series of substituted acetic acids, Reeves ¢186) has concluded that self-association takes the form of chainlike hydrogen bonded polymers. He also suggested, from infrared measurements, that in dilute solution monomer molecules are associated with donor centres in the solvent molecules. An examination of the NMR spectrum of ethanol at several temperatures by Arnold and Packard cxs7) pioneered the study of hydrogen bonding by this technique. Investigations on other substances quickly followed but alcohols have remained a fruitful source of information. The interesting results obtained for these compounds by Saunders and Hynec173) have already been described. Measurements on methanol, ethanol, 2-propanol and t-butyl alcohol in carbon tetrachloride at mole fractions as small as 5 x 10 -3 mole fraction provide clear evidence for the existence of dimers in dilute solutions¢tss~. At higher concentrations, the treatment devised by Saunders and Hyne was not found to be satisfactory in that an extensive array of associated species are probably in co-existence. The hydrogen resonance signal of the hydroxyl group of water or methanol was used to give an indication of the affinity of these aeeeptor molecules for the donor molecules acetone and dimethyl sulphoxide¢189~. The order of
544
H I G H R E S O L U T I O N NMR S P E C T R O S C O P Y \
HO/, in.easing bona was to be F O . . . . H O / , ?SO \ / H e - - - - H e . An ingenious utilisation of the hydrogen bonding phenomenon is in the detection of spin-spin multiplets in methanoF :9°). The multiplet splitting has long eluded detection because of chemical exchange effects. However, methanol solutions containing sufficient acetone to increase the lifetime of the protons in the hydroxyl group produce a tH resonance spectrum containing fine structure as may be seen from Fig. 9.19. Such experiments should enable the effect of the acetone concentration on the exchange rate to be determined. The opposite effect, acceleration of proton exchange, occurs when strong acids . . . .
H
,,Ill
Theoretical spectrum of.CH30H for J : 0 . 2 1
F=o. 9.19 The t H spectrum of methanol at 40 Mc sec- t . The fine structure is
revealedby the suppressionof chemicalexchangedue to the presence of acetone (spectrum not shown). Corio, Rutledgeand Zimmermanogo)
or bases are added to aqueous alcohol solutions. The triplet structure of the hydrogen resonance of the hydroxyl group of ethanol collapses to a single sharp fine at quite low concentrations of acid or base. Using a method similar to those outlined at the beginning of this chapter, Arnold~2°~has estimated the mean lifetime • of the proton in one of its environments at various concentrations of acid or base (see Table 9.8). The hydroxyl multiplet of pure ethanol also collapses to a singlet when the temperature is raised to 80°C (~ ~-, 0.03 sec). When small amounts of water are added to ethanol the ~H NMR spectrum of the mixture contains bands corresponding to two hydroxyl groups. These bands coalesce when the water concentration is increased to about 50 mole per cent at room temperature ~91). Raising the temperature accelerates the exchange process to give a single hydroxyl signal for concentrations of water less than 30 mole per cent cxg=~ The heat of activation for proton exchange is about 8 kcal mole -I.
THE EFFECTS OF CHEMICAL
EQUILIBRIA
545
IH resonance measurements o4"thedilution shifts of 2,2,2-trifluoroethanol (~gs) and of 2,2,2-trichloroethanol (x~') indicate that self-association is less than it is for ethanol " ' s ) . The similar magnitudes of the dilution shifts of these halogenated ethanols is best explained in terms of the similar polarities of the CFs and the CCls groups" 9~) rather than as being due to intramolecular hydrogen bonding. T,~x.~ 9.8 THE ~ ~ • OF A PROTON ~ TEUBHYDROXYL GROUP OF ETHANOL¢~0) Concenn'ation 1"9 5 9 1 3"8 12.5
× x x x x ×
10- e N HC1 10 -6 N HC1 10 -6 N H C I 10- 5 N H C I 10- e N N a O H 10- s N N a O H
T 1"2 sec 0"8
0-025 0"012 0.3
0.006
Hydrogen bonding seems to have an appreciable influence on the chemical shift of alkyl hydroperoxides cx97>. Phenols have provided some interesting results, in particular, the intramolecular hydrogen bonding of ortho substituted phenols (19~-199, 336). The hydroxyl group of phenol itself has a chemical shift of 3"93 ppm to low fields of cyclohexane (internal) at infinite dilution in carbon tetrachloride at 25°C "~3. 2ol). Dimer formation is said to occur at high concentrations of phenol (xl°' 202) but Saunders and Hyne (lv3) contend that the monomer-trimer equilibrium is finite--the trimer probably having a six-membered ring structure. An investigation of several substituted phenols dissolved in carbon tetrachloride showed(XTo) the concentration dependence of the chemical shift of the hydroxyl group in ortho-chlorophenol to be less pronounced than for other phenols. This is certainly caused by intramolecular hydrogen bonding of the monomer reducing the tendency to form higher aggregates. The chemical shift at infinite dilution is about 1 ppm to low fields of the corresponding phenol signal. Dilution chemical shifts have been measured "~9" ss6) for the ~H resonance of the OH group in all the ortho-halophenols over a concentration range of 1 to 5 mole per cent in CS2 and over a temperature range of - 53 to + 107°C. In ortho-fluorophenol no evidence was found for intramolecular hydrogen bond interaction between the hydroxyl and the ortho.fluoro groups presumably because the H - F separation is too great even for the c/s configuration. The N M R results led to the evaluation of the equilibrium constants and their associated enthalpies for cis-trans conversion for the remainder of the orthohalophenols. As expected, the c/s form predominates. The rates of exchange between a solvent proton and one in an intramolecular hydrogen bond have been found to depend upon the strength of the internal hydrogen bond (se°). The activation heat for proton transfer between dry methanol and ortho-chlorophenol was found to be 4.58 kcal mole -1. B:RS.
18
546
HIGH RESOLUTION
NMR
SPECTROSCOPY
Sterio hindrance by ortho substituents such as methyl groups also causes the hydrogen bond of polymers to be weaker <17°"zo4). Porte, Gutowsky and Hunsberger(2or) have singled out chelated derivatives for special study. A striking feature of their results is the shiftsto low fields accompanying chelation (see Table 9.9).The chemical shiftsof the hydrogen nucleus of the hydroxyl group were observed to be concentration independent, therefore only the monomeric chelated form exists in solution at room temperature. The chemical shifts of TAB~ 9.9 Tim XHI~soN,,~cm ~ C A L SHnn'soF Sot~mA R O M A ~ C HYDROXY D]m~VATIV]~(2OI)AND~ CHELATESIN CARBON TzrtAcm.oRn3B AT L'qt~T~ Dmtn'xoN AND AT 25°C. TH~ StuFfs TO LOW Fn~ns or C Y c L o m ~ CI~'rE~AL) Compound
Hydroxyl chemical shift - 3.93 ppm - 9-14 ppm 9.55 ppm 10"63 ppm
Phenol Methyl salicylate Salicylaldehyde
-
-
o - H y d r o x y a c c t o p h e n o n e
2-Naphthol Methyl 2-hydroxy-I -naphthoate 2-Hydroxy- 1-naphthaldehyde 2-Hydroxy-l-acetonaphthone Methyl 3-hydroxy-2-naphthoate 3-Hydroxy-2-naphthaldehyde 3-Hydroxy-2-acetonaphthone
- 4.3 ppm - 10-72 ppm - 11.48 ppm 11.85 pprn 8.74 ppm 8.74 ppm - 9.85 ppm
9-Phenanthrol 10-Hydroxy-9-phc~anthrenecarboxyaldehyde Methyl 10-hydroxy-9-phenanthrylketone
-- 4"61 ppm 12.95 ppm 13.25 ppm
-
-
-
-
-
Shifts can be conve)ted to approximate z values using the expression r =. 8"56 + ~c6m2 2-naphthol and 9-phenanthrol given in Table 9.9 had to be computed f r o m those measured at infinite dilution in benzene owing to their low solubility in carbon tetrachloride. The interactions responsible for the large chemical shifts of the intramolecular hydrogen bonded derivatives are probably similar to those producing the large shifts of say concentrated phenol solutions which contain intermolecular hydrogen bonded polymers. The sequence of the chemical shifts is hydroxy ester < hydroxy aldehyde < hydroxy ketone. The amount (A ~) by which the shift of the chelate differs f r o m that of its parent molecule m a y be used as a measure of the strengths of the intramolecular hydrogen bonds. These values were found to be proportional to the bond multiplicity of the C - - C bond in the structure
\c=c /
THE EFFECTS OF C H E M I C A L E Q U I L I B R I A
547
A related property of the bond order is the C-- 0 stretching frequency. A striking linear relationship exists between A ~ and the corresponding wave number difference/1 ~ as may be seen from Fig. 9.20. The better correlation between the mobile bond order and/1 ~ is probably an indication that steric factors need to be taken into account. A similar study of phenols and naphthols has been carried out by Reeves, Allan and Stremme (2°5), and of 2-hydroxybenzophenones by Merrill (198).
E
a; <3 6,0
4..
020
i 60
40 A//,
c m -I
F10. 9.20 The linear relation between A 6 a n d A ~. Line A represents chelated
aldehydes, B chelated ketones and C chelated esters. Pone, G u t o ~ ' and Hunsberger(2OZ) Para-substituted phenols display intermolecular hydrogen bonding and form dimers. At infinite dilution the 1H chemical shifts of the OH group have been found to be almost independent of the nature of the substituent(2°~). The respective shifts are 2.84, 2-90, 2"91 and 2-92 ppm for the substituents H, C1, Br and I using carbon tetrachloride as the solvent (the shifts are to low fields of the cyclohexane internal reference). Meta-substituted phenols can form comparatively stable dimers in non polar solvents when the meta substituents are methoxycarbonyl or nitro groups (2°7). The addition of a polar solvent is required to break up the dimers even when the solute concentration is low. Measurement of ~H chemical shifts of the OH group in the hindered phenols having substituents in the 2 and 2,6 positions has enabled dimerisation constants to be determined(a59). Here it is possible to observe separate OH resonances in solutions containing another hydrogen bonded species. Additionally, 1 : 1 association constants with ethanol were measured. 18"
548
H I G H RI~SOLUTION N M R S P E C T R O S C O P Y
Mercaptans. Hydrogen bonding of the type S--H----S has long been considered unlikely but a nuclear magnetic resonance examination of the hydrogen atom in the SH group of ethyl mercaptan has produced evidence (2os) in support of infrared measurements(2°9) which indicate the existence of such hydrogen bonding. A linear dependence of chemical shift on concentration obtains, there being a total change on dilution of 0"38 ppm. The linear dependence carries the implication that monomers and dimers dorni,ate the system. Amines. Although the hydrogen link between ammonia molecules is well established(~, 6), hydrogen bonding in amines has been the subject of controversy(~1°-21e). The main difficultyhas been that the infrared method is too insensitive to yield reliable data on amine systems. The advent of NMR spectroscopy has enabled amines to be studiedin detail(1~9, 217, 3+8). The maximum magnitude of the hydrogen bond chemical shifts is about I ppm. The choice of a suitable solvent presents some problems because unfortunately, the preferred solvent (carbon tetrachloride) reacts with most amines. Ethyl and diethylamine are tractable in that they react sufficiently slowly with carbon tetrachloride for unperturbed spectra to be recorded(217). The association shifts were found to be 39.8 and 39.6 cycles sec -1 at 40 Mc sec -x irradiation frequency; these are close to the value of 42 cycles sec -t determined for ammonia(tee) . In order to interpret the data for the progressive dilution of mono- and of diethylamine, it was necessary to postulate a monomer-tetramer equilibrium. The resulting association constants were found to be 2-5 × 10 -3 and 2.5 × 10 -+ M-3 respectively. The heat of formation of .the tetramer of diethylamine was estimated to be - 6.8 kcal mole -~ of tetramer, n-Mers greater than tetramers are probably additionally present at concentrations above 8 M of amine. The tH resonance band of the NH~ group of aniline was not shifted by dilution with carbon tetrachloride(2~'. Dilution experiments on pyrrole in cyclohexane have revealed the existence of a monomer
THE EFFECTS OF CHEMICAL EQUILIBRIA
.r~49
tone but only twofold in bemzene (2~2). The self-association chemical shift of chloroform in, and measured from, cyclohexane is 0.17 ppm to low fields while the corresponding shift of silicochloroform is 0.03S ppm to low fields. The ¢~istence of multiple d~ - p ~ bonding between silicon and chlorine may be responsible for these observations. The total dilution shift of chloroform in carbon tetrachloride is 0.022 ppm suggesting that association takes place between the two molecules (223). A s s u m i n g that chloroform is self-associated in cyclohexane to form only dimers, the self-association constant (22s) at 25°C is 0"16 + 0.006 (mole fraction) -1. The character of the Si--O--Si bond has been investigated using chloroform as the donor molecule <224). Compounds cont~inlng the Si--O--Si bond show a progressive decrease in basicity through the series ether to alkoxysilane to siloxane, thus confirming the findings from some related infrared measurementsC225).
A systematic study has been made of halogenated benzenes dissolved in acetone or benzene (22z). For some compounds the hydrogen bond strength was estimated to be comparable with that formed by chloroform in say ace= tone c227) ( A H -- 2"7 -)- 0.1 kcal mole-l). The strength of the hydrogen bond formed by deuterium has been found to be stronger than that formed by hydrogen from a study of the 1 : 1 complex formed between fluoroform (or fluoroform-d) and tetrahydrofuran c~2s). 9.4.5 Keto-Enol Tautomerism
Most stable enols depend upon strong intramolecular hydrogen bonds for their existence, hence, like some of the ortho substituted phenols discussed above, only monomeric forms are to be expected. Furthermore, solvent interaction is negligible as evidenced by the unchanged chemical shifts of the hydrogen nuclei in acetylacetone as found in the pure liquid and when it is dissolved in acetic acid and in triethylamine (192' 229-234!. Chemical exchange is sufficiently slow at room temperature (and the individual signals are collapsed) to observe that acetylacetone is a 4 : 1 mixture of keto and enol forms: CHs CI-~ CHs CHs CH CI-h \c/",,c / __, \ c . / ' ~ c . / Jl H "-H I 0
0
keto
0 0 enol ""'H/
For the detection and estimation of tautomers the N M R method is generally superior to other physical methods, for example, those based upon refractive index, infrared or ultraviolet absorption. The effect of temperature on the nuclear magnetic resonance spectrum enables the enthalpy of keto-enol conversion to be determined. In the case of acetylacetone, Reevesc~3s) obtained a value of 2.7 4- 0.1 kcal mole -~ from measurements made over the range - 16 to 116°C. Also the position of equilibrium is, of course, disturbed by changing the dielectric constant or by introducing a substance which can form strong hydrogen bonds with one of the tautomers (4e" 2ss). The hydroxyl chemical shift of the enol of acetylacetone is shifted to high fields by as much as 7 ppm by the
550
HIGH RESOLUTION
NMR SPECTROSCOPY
addition of triethylamine. Complete conversion to the enol tautomer is caused by strong hydrogen bonding between the amine nitrogen atom and the enol hydroxyl group. A similar strong interaction between diethylamine and acetylacetone results in the formation of a molecular complex ~4~. More complicated molecules may be enolised completely in the pure liquid as was noted for triacetyl methane and for some 2-acylcyclohexane-l,3-diones ¢'3s). All the compounds have the characteristic low field chemical shift of enolic hydrogen nuclei--from 12.6 to 13.8 ppm to low fields of the external water reference. The exceptional properties of the conjugated system of these enolised ~-triketones is underlined by ceroptene which has an enolic hydrogen shift with the extraordinary value of 14.1 ppm: ceroptene has two enolic tautomers. These chelates show a good correlation between the enol chemical shifts and the chelate C = O stretching frequencies (compare the data on aromatic chelates given above). Additional indication of the very strong intramoleeular hydrogen bonding present in these compounds is the low rate of exchange between heavy water and say acetyldimedone. The behaviour is in marked contrast to that of a simple ~-diketone such as acetylacetone. Internal hydrogen bonding is largely responsible for the stability of structure II in the compounds formed from condensation of a ~-diketone with a diamine ~23~) or a monoamine (237). R
R
\
R
\
C=O
\
C=O
/
/
CH~
--",--- C H
\
\
C=N
/ R
\ R'
I
",, /
H
C--N
/
\
R
R' II
C--O
/
~-';" C H
",, H
,,"
\
C=N
/ R
\ R'
Ill
The ketamine form (II) exists to an extent greater than 95 per cent irrespective of variation of substituent R' and of large changes in the nature of the solvent. Hydrogen bonded tautomers involving nitrogen as opposed to oxygen have been detected in substituted formazans ~23s).
Proton exchange effects. The hydrogen exchange processes in fl-diketones comprise exchange with the enol group only and a second type involving the ketc>-enol interconversion. Exchange of the enol hydroxyl proton in acetylacetone has been demonstrated by the chemical equivalence of the methyl groups of the enoF 233). A mixture of acetic acid and acetylacetone at room temperature has a hydrogen resonance spectrum containing separate signals from the hydroxyl groups of these compounds. The signals are broader than those of the isolated compound thus confirming that some proton exchange is occurring c192). The two signals coalesce at 64°C for a solution containing 58 mole per cent acetylacetone: the lifetime of a proton at a given site was estimated to be 1.5 x 10 -3 sec. The process of keto-enol tautomerisation can be regarded as hydrogen atom transfer. From the separation between the absorption signal of the enol
THE EFFECTS OF C H E M I C A L E Q U I L I B R I A
551
hydroxyl group and that of the keto methylene group, the lifetime of each tautomer would have to be less than 5 x 10-" sec to give a collapsed spectrum. Even heating acetylacetone to 150°C does not accelerate the rate sufficiently to produce an averaged spectrum c4e). Acceleration can be brought about, however, by the addition of hydrogen bonding substances. Reeves and Schneider ('6) found that amines greatly increase the keto-enol conversion rate. The rate produced by diethylamlne is greater than that produced by triethylamine presumably because the tertiary amine lacks an exchangeable hydrogen atom. Williams (239) has pointed out that rapid tautomerism is a characteristic feature of boron hydrides and their derivatives. He showed that if exchange of groups is taking place intramolecuiarly in these compounds then, even though it is very rapid, collapse of spin multiplets may not occur. 9.5 HINDEREDINTERNAL ROTATION Various chapters in this monograph have provided numerous examples of the way in which nuclear magnetic resonance spectroscopy has made a si~,-ificant contribution to many branches of physical chemistry: yet another example is the study of hindered internal rotation of molecules. Of course, other physical techniques such as electron diffraction, dipole moment measurements, infrared, Raman and microwave spectroscopies are used to investigate the phenomenon(24o. 241). All are needed ff we are to understand the nature of the forces restricting intramolecular motions. Rotations about several types of chemical bond are classified below. 9.5.1 The N - - N Bond The simplest case of rotationalisomerism about a single bond is that in which not more that two atoms or groups are attached to the atom at each end of the single bond. Examples of this type are to be found in the nitrosoamines
where R1 and R2 are usually alkyl or aryl groups (242). When R1 and R2 are equal, then there are two structurally equivalent rotational isomers caused by some partial double bond character in the N w N linkage, which maintai~ the molecules in the planar configuration shown above. If rotation about this bond is slow, the hydrogen resonance spectrum will show two sets of absorption bands corresponding to the two different environments of the R group. Increasing the temperature will increase the rotational rate thus causing the bands to approach one another until, at some critical rate, they merge into a single set of bands (see Fig. 9.21). At 25°C, the chemical shift difference between the resonances of the CH2 groups of the two isomers of N, N-diethylnitrosoamine is 0"8 ppm and that between the CH3 groups is 0-3 ppm. Thelatter figure tells us that the average rate of re-orientation about the N m N bond must be less than 36 sec -1 at room temperature. Earlier in this chapter it was seen
552
HIGH RESOLUTION
NMR SPECTROSCOPY
that Gutowsky and Holm(4) have developed expressions describing the behaviour of a system containing two non-equivalent nuclear environments between which exchange of nuclei is possible. When the numbers of exchanging nuclei in the two environments are equal and the resonance line widths are small compared with the chemical shift difference, 0, between the two sets of nuclei, the observed chemical shift, Oob., is given by (~o,,, =
(9.132)
[1 - 2/(T (~)2]~ ~
where • is the average time an exchanging nucleus spends in a given environment. Rotation can be described as slow when T ~ ~ 1. Expression (9.132) has been applied to the suitable compound N,N-dimethylnitrosoamine (24e). H CH 2
CH3
]
i
I
:
i J~
8 ....
FIG.9.21 The moderateresolution hydrogen resonance spectrum of N,N-di©thylnitrosoamineunder conditionsof (A) very slow internal rotation and (B) rapid internal rotation Assuming that re-orientation is a typical rate process,- then the rate constant k in the relation k = A exp(-AE/RT) (9.133) can be replaced by ~-1. The frequency factor A is the number of times per second the molecule tries to penetrate the barrier to rotation. Thus A E, the height of the barrier, can be found by determining • at several temperatures: Looney, Phillips and Reilly<242) obtained a value of 23 kcal mole -1 for N,N-dimethylnitrosoamine. The magnitude of this barrier provides the main evidence for the partial double bond character of the N - - N bond. Partial double bonding (p#---p~) is also held to be responsible for the occurrence of cis-transisomerism due to restricted rotation about the B--N bond in C6H5 (C1) B. N(CH3)2 c243) and in (CHs)~ B. N(CHs) CeH5 (2~). When the substituents Rx and R2 are different groups then the spectrum of the very slowly rotating molecule consists of four sets of bands, two associated with R, and two with R2 These arise from the existence of two non-equivalent rotational isomers. The relative intensities depend upon the relative residence times (or lifetimes) of the two isomers. Again, when the interconversion rate
THE EFFECTS OF CHEMICAL EQUILIBRIA
553
exceeds some critical value, the spectrum simplifies to two sets of bands, one for RI and one for R2. Since the relative residence times of the two isomers depend upon the temperature (lifetimes approach equality as the temperature is raised), the time-averaged chemical shifts change with temperature. Between 20 and 150°C, the interconversion rate of benzylmethylnitrosoamine (242) is sufficiently slow to give absorption bands of two distinct isomers. The relative intensities of the bands corresponding to the isomers remained unchanged over the temperature range quoted. The c/s: trana ratio was estimated to be 1 : 3, c/s being defined by the methyl group being c/.vto the oxygen atom. The constancy of the ratio shows that the energies of the two forms must be similar, the heat of activation for the process being less than 1 keal mole -1. 9.5.2 The C - N Bond
One of the earliest applications of N M R to problems of restricted rotation was the confirming (24s) of the postulate (2.6) that there is not free rotation about the C - - N bond of amides. If there were free rotation about this bond in N,N-dimethylformamide H O \Of
I CH, CI-I, the average electronic environments of the two CHs groups would be identical and their hydrogen resonances v)ould be coincident. Even though both methyl groups are bonded to the same nitrogen atom, a chemical shift difference is expected between them in the event of restricted rotation. The different shieldings of the hydrogen atoms of the two CHs groups in the planar molecule are are suit of asymmetric polarisation of the C - - N bond which may be represented H O\C/
H
/N+\ CH, CEL~ The Chemical shift difference of 6 cycles sec -1 (at 30 Mc sex -1) enables the upper limit of 38 sec -1 to be placed on the rotational velocity at room temperature, thus indicating that the barrier to rotation is high. A single hydrogen resonance signal was observed (2*s) for the methyl groups of N-methylformamide and N-methylacetamide indicating that only one of the two possible rotational isomers of these compounds is present (normally the tram formtST°)). The parent molecule, formamide, has a hydrogen resonance spectrum containing a rather broad doublet overlapped by a very broad signal arising from the NI-I2 group. The diffuseness of the spectrum originates from the quadrupole coupling of the X4N nucleus. A method of surmounting the difficulty is to apply the technique of double resonance (~*" (see Section 6.8). A more satisfactory method has been adopted by Sunners, Piette and Scimeider(2*s)
5.~4
HIGH RESOLUTION NMR SPECTROSCOPY
who substituted the ZSN isotope of spin value ½for Z4N. The absence of quadrupole broadening allows a spectrum featuring narrow lines to be recorded as may be seen from Fig. 9.22. AH the expected twenty-four lines are clearly resolved at 25°C. A microwave invvstigation of the gaseous molecule has shown the amide group to be pyramidal, the bond angles CI~HA and CIqHa having the values 120° 17' and 117° 11' respectively(=49). The difference in bond angles
!
I
!J
x
H×
o//
\H.
HA H8
lJ
HA HB
H~,,HB
HA*HS
FIG. 9,22 The 60 Mose¢ -t hydrogen resona,uce spectrum of HCOZSNHz dis-
solved in acetone (10 mole per cent). Sunners, Piette and Schneider(z4s) probably accounts for the observed difference in the coupling constants JNaA and Jsam. In the liquid state the molecules are strongly hydrogen bonded which might affect the planarity of the molecule. The coupling constants and the chemical shifts undergo large variations when the hydrogen bonding interactions are modified by dissolving the compound in water or acetone. The pure liquid does manifest hindered internal rotation, there being a coalescence of absorption bands at temperatures above that of the room. Unfortunately, the situation becomes complex due to the onset of proton exchange between the NH= groups providing an additional source of line broadening and coalescence. The experimental data enabled the heat of activation of proton exchange to be determined as 10 + 3 kcal mole -t. Sunners, Piette and Schneider eliminated t
THE EFFECTS OF CHEMICALEQUILIBRIA
555
the complication by dissolving formamide in acetone. Under these conditions, the formamide molecules are isolated from each other and are hydrogen bonded to the acetone with which they cannot exchange. Over the whole temperature range, the spectrum changes from type A~XY to ABXY (see Section 8.2 for definitions) by virtue of rapid rotation of the NH2 group. This fact suggests that at the highest temperature employed, proton exchange is still relatively unimportant. In order to evaluate the barrier hindering rotation, the lifetimes of the hydrogen atoms at positions A and B were estimated from the broadening of the lines in the X part of the spectrum, and the relationship "r-1 = (T') -1 - T~ I
(9.134)
where (/'2') -I is the half width of the broadened line and T2 -~ is the effective line width in the absence of rotational broadening (obtained from the outer unbroadened lines in the X part of the spectrum). Values of z were also calculated from the expression
42
- *, z = -2:~d
(9.135)
where 3 ~ is the separation in cycles sec -~ between appropriate components of spin multiplets in the AB part of the spectrum whose coalescence is observed. The heat of activation required for rotation was found to be 18 :i: 3 kcalmole -1. Salka (2s°) has obtained a correspor~ding value of 14 -/- 2 kcal mole -I for both N-methylformamide and N-methylacetamide in the liquid state. In their classical studies of rate processes, Gutowsky and Holm (~sl) determined the energy barriers to re-orientation of N,N-dimethylformamide and N,N-dimethylacetamide to be 7 -F 3 and 12 -J- 2 kcal mole -1 respectively. Rogers and Woodbrey(12) have found that the most precise measure of z, the lifetime of a group, is the line shape parameter r, given by the ratio of maximum to central minimum V-mode intensities for a doublet (see Section 9.1.3). They studied the exchange of the methyl group between sites A and B in series of substituted N,N-dimethylamides
O7 /CHa(A) I~/C--N\cHa (B) Table 9.10 lists some of their results derived from 1H resonance spectra obtained at 60 Mc see-l, Small barrier heights were found for dimethylacrylamide and N,N-dimethylcarbamyl chloride. In these cases, the double bond character would be shifted from the C - - N to the C - - R bond. The compounds
C2HbO)c--N(=:
O~/C--N('CF3 CH3/
\CF3
O\r~ N/CFs CFa/'~
\CF3
show no observed chemical shift difference between the NR2 groups at temperatures down to their freezing points. The actual chemical shift differences may be zero or the exchange rates between sites may remain rapid even at low temperatures. The effects of solvents on the barrier heights of some disubstituted amides have been investigated(252). The barrier height appears
181*
556
H I G H RESOI~UTION NMR S P E C T R O S C O P Y
to decrease monotonically with increasing dilution using a non-polar solvent like carbon tetrachloride. The effect is probably due to the decrease in stability of the more polar ground state relative to the less polar excited state with increasing dilution. N o evidence has been found for an appreciable barrier to rotation in orthosubstituted N,N-dimethylanilines c253~. The low barrier has been explained on the basis o f rotation about the C - - N bond being accompanied by inversion about the nitrogen atom. However, an exception may occur in the less-substiTABLE9.10 DATA FOR HINDERED INIIu,,NALROTATION ABOUTTHE C ' - - N BOND IN AMIDES(12)
Heat of activation 0teal mole-1)
Log (frequen~ factor)
Free energy of activation (kcal mole-1)
Coales= ¢~.nce
temperature
(°K) N,N.Dimethylformamide N,N-Dimethylacetamide N,N-Dimethyipropionamide N,N-Dimethyltrifluoroacetamide N,N-Dimethyltrichloroacetamide N,N-Dimethylacrylamide N,N-DimethyIbenzamide N,N-Dimethylcarbamyl
18"3 ~ 0"7 10"6 -4- 0"4
10"8 ± 0"4
7-8 ~ 0-2 7"3 ~ 0.5
9"2 ± 0.7 9"3 ~- 0"6 9"9 -~ 0"3 6"8 ± 0"7 7"7 ± 0"5 7-3 ± 0.5
6-8 4- 0"4
9"1 -4- 0.2 6"0 ± 0.5 7"2 ± 0.4 6"1 + 0.3
21"0 17"4 16"7 17"6 14"9
16"1 15"3 16"5
421"6 360"3 334"4 367"9 287"1 284"9
326"0
tuted ortho-nltroaniline because both the substituents can contribute to the resonance interaction with the benzene ring and there is evidence for hydrogen bonding between the two groups (254, 371). The syn: anti ratios o f some aldoximes / N
OH
/
N
II
II
C 1t
OH
C H
sy.
H
R
ant/
have been calculated from their resonance spectra (255). Of the two signals observed for the i H resonance of the C H group, the one at low fields was assigned to the syn isomer,
9.S.3 The N--O Bond The existence of rotational isomers of alkyl nitrites has been established by Tarte c256) and by Haszeldine and co-workers c~57' 25s) using optical spectroscopic methods. Nuclear magnetic resonance spectroscopy has also been useful in the detection of these isomers c2s9' ~o). N M R studies have shown that temperatures of about - 50°C cause the hydrogen resonance lines of theo~-methyl-
THE EFFECTS OF CHEMICAL EQUILIBRIA
557
erie group to be doubled, probably due to the presence of c i s - t r a n s rotational isomers
R~
R~
O
O
N
N ~
I
of
I
cis
trans
At still lower temperatures, the c/s isomer predominates over the trans to a slight extent. The chemical shift difference between the oc-methylenic hydrogen atoms of the two isomers was found to be about 1 ppm. The signals from these hydrogen atoms are located at lower applied fields for the cis isomer as a result of hydrogen bonding. Phillips, Looney and Spaeth t2e°~ measured the d s : t r a n s ratios over the temperature range - 100 to 20°C. Their values disagreed with those obtained by Tarte c256~ from optical measurements. They calculated an average value of - 130 cal mole -1 for the heat of c i s - t r a n s conversion. The low value led these workers to suggest that the barrier to rotation about the N - - O bond arises not as a result of C - - H . . . . O hydrogen bonding in the cts form t256~ but from partial double bond character in the N - - O bond. The following structure will contribute appreciably to the resonance energy of the nitrites
R~ O÷
II
N
-O/ Steric factors may also be important because the c/s isomer is largely excluded in t-butyl nitrite, and isopropyl nitrite where a planar configuration cannot be accommodated. From the temperature variation of the c/s-trans chemical shifts, the free energy of activation for internal rotation was found to be 10 kcal mole-L This is considerably less than the values associated with nitrosoamines and amides discussed in the previous two sections. The double bond character of thebond forming the rotation axis maybe the least in this group of compounds. The frequency separation method is not particularly suitable for the alkyl nitrites because of the rapidity at which interconversion takes place: furthermore, the temperature range over which the transition occurs is only about 5°C. However, the effects of re-orientation on the line width are much more gradual and take place over a greater temperature range. The latter method has been adopted by Pictte and Anderson t~61~ who developed a procedure for treating the system, ~imilar to that used by Gutowsky and his co-workers ts''~ (see Section 9.1.1). For the slow exchange approximation they obtained the relationship log[~zA, -- (T]X)j] -- log2A(1 - P j ) - A E / ( 2 " a O 3 R T ) where A ~ is the half-height line width o (T~21)yis the effective natural width A is the Arrhenius frequency factor P j is the fraction of nuclei in environment j ,4 E is the potential energy barrier to rotation.
(9.136)
558
HIGH
RESOLUTION
NMR SPECTROSCOPY
A plot of log [z~Av - (T/zt)j] against 1/T thus allows A E to be calculated. The fast exchange expression approximation yields the expression log(hA v 7"2 - 1) ~- log(2A/v/'2) + A E/(2-303RT)
(9.137)
where V is the second moment of each resonance line. T2 = (T2)~ for aH j. This time a plot of log(:~A~,T2 - 1) against 1/T gives A E from the slope of the resulting straight line. Ideally, two checks on the energy can be obtained by following line width changes during the one extreme of slow exchange and then during the other extreme of fast exchange. The experimental conditions required for the alkyl nitrites permitted only the region of fast exchange to be used. In their experiments, Piette and Anderson measured T2 by the method of Wangsness and 1acobsohn ¢262~ (see Section2.10) in which the exponential decay of the damping of the resonance line is taken as a direct measure of T2. The spin-spin relaxation time Tz was assumed to be dependent solely upon the inhomogeneity of the applied magnetic field. Hence the decay of the 'wiggles' of the phenyl resonance of the internal reference compound could be employed. The line width measurements were made on the hydrogen resonance signal of the o,-CH2 or CH group of the alkyl nitrite under examination. A noticeable change in the line width takes place at 15°C and continues linearly to the transition temperature at about - 450C. A Ewas estimated to be 9 +_ 2 kcal mole -1 for methyl, ethyl and n-propyl nitrites while isopropyl nitrite has a lower value of 6 + 2 kcal mole -t. Gray and Reevesc263) have obtained similar results from their measurements on methyl nitrite. Measurements of the dielectric constants of liquid primary, secondary and tertiary alkyl nitrites have been made over an extensive temperature range and from them the relative amounts of the c/s and trans isomers have been estimaed¢264.26s). In contradiction with the above, early N M R results, it has been coneluded that the tramsform predominates in all three instances c372}.Thus the assignments of the N M R absorption bands should be reversed in the papers by Pierre, Ray and Ogg¢2s9~, Piette and Anderson ¢26t~, and Gray and Reeves¢2e3). From such a re-assiLmment one can infer that hydrogen bonding is less important in these nitrites than is the more efficient electron-withdrawing power of the NO2 group when a molecular orbital of the trans type is present.
9.5.4 The S--S Bond The dihedral angle in organic disulphide compounds can produce two different molecular conformations I R' I
IR~ I
THE EFFECTS OF C H E M I C A L E Q U I L I B R I A
559
The interconversion between the enantiomers of 1,2-dithiane has been studied by nuclear magnetic resonance spectroscopy~266). In order to obtain a simple spectrum a deutero derivative was prepared
CD2--CD~
i/
CH.,
CH2
\S__S ~ At temperatures below - 5 0 ° C a 1 M solution of the compound in carbon disulphide gives a spectrum (Fig. 9.23) consisting of a quartet of the AB type H
25°C
) 4-6(~pm)
3°C ~ o 6 5
°C
FIG. 9.23 The XH resonance spectrum of 1,2-dithiane-4,5.D4 at 60 Mcsec -1.
Cbe~cal shifts are measured from benzene used as an external reference. Claeson, Androes and Calvin(16e)
resulting from a Coupling constant of 13.0 cycles sec -1 between the non-equivalent geminal hydrogen nuclei. Coalescence of the multiplets occurs at - 43°C. F r o m the spectrum recorded at - 65°C, the chemical shift ~.. between the axial (a) and the equatorial (e) hydrogen atoms was calculated to be 18.5 cycles sec -I at 60 Mc sec -1. F r o m the relationship k = ~ ~t,,/~/2
(9.138)
the rate constant k was found to be 40 sec -l.
9.5.5 The C--C Bond Many N M R studies have been made on hindered rotation about this bond due to its frequent occurrence in aliphatic groups, particularly as partial double bond character is no longer a requirement. The situation can be complex as a
5~)
HIGH
RESOLUTION
NMR
SPECTROSCOPY
consequence of the increased number of isomers possible. Taking substituted ethanes as our model, there are three rotamers of the meso and dl forms when three different substituents are attached to each of the carbon atoms. The energetically favoured forms are the staggered conformations shown below O
(3
l.°
(=
°\
b'/~c I a I
b'J~c I c TT
b-
I c I b TTT
Internal rotation occurs in ethanes at room temperature hence all three forms are present at this temperature. Since the enthalpies of the three forms are not necessarily equal the three retainers may not be present in equal concentrations. Furthermore, the relative concentrations will be temperature dependent. The factors governing the N M R spectrum resulting from a set of rotamers include (i) the relative energies of the rotamers (ii) the temperature and (iii) the chemical shifts and coupling constants characteristic of each isomer. Thus even if one is not interested in the energetic aspects of internal rotation it is necessary to appreciate its effects in order to understand spectral analysis fully.
Spectral analysis oJ compounds containing C--C bonds. Pople t2e7) has described the principles involved for ethanes containing not more than two different magnetic nuclei. Concurrently, the N M R spectral expectations for ethanes having as many as six magnetic substituents were worked out (26s~. In predicting the type of spectrum which will be observed, account has to be taken of the presence of slow (or zero) rotation giving a summation of up to three different spectra, or whether the molecule is undergoing rapid internal rotation thus giving a time-averaged spectrum. The time scale is, of course, set by the N M R experiment. Five different types of behaviour can be recognised in the nuclear magnetic resonance spectra of substituted ethanes: (a) The three rotational isomers are equivalent. The spectra of the rapidly or the slowly re-orienting molecules are identical and independent of temperature. Some examples are a X~
Y
Y
I /i Y
a Y ..
I
y
I
8
I
B
Y ,
..
I
Y
where the small Roman letters indicate atomic substituents possessing magnetic nuclei and the Greek letters signify non-magnetic substituents.
561
THE E F F E C T S OF C H E M I C A L E Q U I L I B R I A
(b) The three rotamers are equivalent but the spectra of both the rapidly and slowly re-orientating molecules are different: the spectrum of the rapid form is independent of temperature. Examples are: O'
O"
O"
~"
0
0"
O"
7"
0 "
l
0 "',~
"0
0 -
i
-O"
I 0"
0
0
J
i
I
I
•
~
0"
T
The primes indicate nuclei of the same species which arc magnetically nonequivalent. In the example shown on the extreme right only one coupling constant ½(3~,, + Ja,=,,), will appear in the spectrum if rapid intercooversion between rotamers is occurring. If the interconversion is slow then six coupling constants characteme the spectrum; these are J ~ , ~ , J ~ ' , , J,,,o,, J , , , J,,,,,, and J u . A very commonly occurring example of this kind is the ethyl group: the IH resonance spectrum of the group is of the type A=B3 (see Section 8.13.9) because of the rapid internal rotation about the C--C bond at room temperature. If the rate of rotation could be reduced suf~ently then the spectrum would change to one of the type AA'BB'C (see Section 8.17.1). (c) Two rotational isomers are of equal probability (gauche forms) while the third is unique (trans form). The spectrum of the rapidly interconverting molecule is temperature dependent and is of the same type as the tran.v isomer but is different from that of the other two. Some possibilities are ¥
~
o...
o
?"
/.o
~.....
~.I
o,,.
oI
Trons '
Gauche
b
-0
~?"
Spectrut~ o f tyl~e AA' BB'
b
|
[ o!
Gauche
One bond in spectrum
O"
,
O~
'
b
I
"0
0 ~
I
I
!
J
b
o
o
Trans Spectrum of type AA' A~A " BB'or AA'A'~'Xx'
t
Gauche
~0
Gauche
Spectrum "of type AA'BB'CC' or AA'BB'XX'
562
HIGH
RESOLUTION
NMR
SPECTROSCOPY
The gauche forms have indistinguishable spectra. The largest group of molecules exhibiting this behaviour have two identical substituents on each carbon atom. An example of such a compound is CF2BrCBrC12. Nair and Roberts (=~°) have obtained the 19F resonance spectrum of the compound at room temperature and at - 80°C. At the latter temperature the spectrum comprises both A2 and AB types in the ratio 1.4: 1, therefore both trans and gauche forms are populated at the low temperature. The result impfies that the room temperature spectrum (of the A2 type) is characteristic of rapid interconversion Br
Br
I
er
I
CL Gauche
I
CL Gauche
Br
"Irons
(d) The rotational isomers are non-equivalent and may be of unequal probability. The rapidly re-orientating molecule has a spectrum of the same type as that of all the isomers and is temperature dependent. Such a behaviour is shown by structures I, II, and HI given above and by the following systems a 8\\
a //~
y~\
..~.3
8,,.
o II
/y
These three retainers will give NMR spectra of the APX type: the d and ! optical forms always have indistinguishable spectra. It is important to note that while each isomer and the rapidly interconverting molecule give NMR spectra of the same type, the spectral parameters are not identical. This may be seen more readily by taking the simpler example
..3 "x y'
g b..~-
Trans
J
b~.
.$
Y
Y S
Gauche
Here, both retainers give rise to different spectra of the AX type and the rapidly rotating molecules also have a spectrum of this type. If the numbers of mole-
THE EFFECTS
OF C H E M I C A L
EQUILIBRIA
563
cules in the trans and gauche states are represented by nt and no respectively and if J, and 2o refer to the coupling constants in the f i x e d rotamers, then the averaged value of the coupling constant < J > measured from the observed spectrum of the internally rotating molecules is given by
<.r>
{., J, +
=
J'o. C F~Br CF Br CL
66 °C
J,L__j
L
i
I
li
If
tI
-120%
ill
,,
Yrr ]1-
F x"
i "CL Br
I
III
"
II
B~
I!
t "St Br II!
i [I
,,
F×
CL"
(o)
CL
8r"
I "F x Br II
FIG. 9.24 The ZgF resonance spectrum at 56.4 Mc sec -1 of CF~Br CFBrCl in CFCI3. The individual rotamers are AB X spin systems but one approximates to an AA' X spin system. Ramey <se4)
The rotamer fractions nt and n o are temperature dependent: the functions describing the resulting temperature dependence of < J > are given later. An actual example of this type of behaviour is shown in Fig. 9.24. The 19F spectrum of CF2BrCFBrC1 at room temperature is of the A B X type, an average of one
564
HIGH RESOLUTION
NMR SPECTROSCOPY
from each rotamer. The spectral parameters are different in each case. The isomers have the following coupling constants ¢269) { I, } = - 11 "8 cycles sec -1 I Jg = - 14.Scyclessec -x II HI
½(2g + Jr) -- - 14.0 cycles see -I { J, } = - 2 0 " 3 cycles see -1 $g = - 2 1 . 3 c y c l e s s e e -~
Manatt and Ellemanc27s) used a double irradiation technique in a study of the compound CF2BrCBr2F at - I10*C. They found the F - F geminal and vicinal coupling constants to be opposite in sign. The spectrum indicated that the most populated conformation is the one having two of the bromine atoms trans to one another. (e) The three rotamers are equivalent. The slowly and the rapidly interconverting molecules give rise to spectra containing the same chemical shifts but more spin-spin coupling constants characterise the spectrum of the slowly re-orientating molecule. The rapidly interconverting form has a spectrum which is temperature independent. There is only one possibility, namely o
I
b
The two halves of the molecule have C3° symmetry and all the atomic substituents have magnetic nuclei. From the foregoing it is evident that the symmetry of rotamers and their time-average is all-important in determining the NMR spectrum. Gutowsky(27~) has stressed the central role of asymmetry in the occurrence of magnetic nonequivalence. He has given an algebraic analysis of the (d) behaviour listed above. Taking the three dl pairs b
o
y
b"
I I
8
7
"a"
a'"
(I)
a'
I I
~
7
a
a ~
(2)
8
I I
"b
(3)
the rotationally averaged chemical shifts < ~> are given by
<,.> and
=
Zx.,: |
<~,,> = ~* x. ~' J iI
(9.139)
THE
EFFECTS
OF
CHEMICAL EQUILIBRIA
565
where x. is the mole fraction of rotamer n and ~. is the chemical shift of the ith nucleus in rotamer n. The net chemical shift between a and a' is given by
(,/1~,~>=
#,~,., -- ~,,,,> = ~.,x.O'~
(9.140)
-- ~':')
n
which can be written as
=
x.(a
o
+ a ,,.)
(9.Pa)
n
where A ~o = (~ _ ~,) for the parent compound in which magnetic nucleus b is of the same kind as the a magnetic nuclei, that is, all three rotamers and their populations are identical. A ~. is defined as the change in ,4 ~o caused by substitution of a for b. Gutowsky obtained the expression
(9.142)
< z l , > f Y,(x.--'})d~,° + ~,(x.--"})A,. + ~½zt,.. II
II
II
The first term depends upon the chemical shift differences caused by the initial asymmetry of the parent compound (when b = a) and the population differences introduced by having b @ a. The second term depends upon the population differences and the shift changes ,4 ~, caused by b ~ a. The last term is the "asymmetry effect "--the net shift which would result if the rotamer populations were identical. By choosing spin systems which can be analysed relatively easily it is possible to evaluate coupling constants and obtain qualitative information on rotamer populations. Many substituted ethanes (25s, 27o-274, 277-282, 373) have been examined, providing examples of the Various classes of behaviour defined above. A particular example will now be described. Anet (2s2) has examined the 1H resonance spectra of the meso and d! isomers of 2,3-dibromobutane and has analysed the spectra as AA,V ~3~,s v , spin systems having Jxxo = O. The meso isomer exists as a trans and two gauche forms CH3
CH5
H.. I
CH3
I
CH .. I ....B,
I
CH3
I
Br
H
"Irons I
Gouche IT
The d/isomer is a mixture of distinct d and I molecules each of which can have three rotamers CH3 Br,,
I //CH 3
I
H Trr
CH~ H,,
I .p...Br
I
CH3 ~Z
Cl-~ CH3 .
I
I
Br ~Z
..-H
566
HIGH RESOLUTION NMR SPECTROSCOPY
From intensity measurements of the multiplets featuring the comparatively large values of JAA' (7.85 cycles see -a) in the spectrum of the meso isomer, it can be shown that rotamer I predominates (66 -t- 10 per cent) but that appreciable amounts of II are also present. Changing the solvent changes the dielectric constant thus affecting the conformer populations (z72, 2s3, 29e~ since I and H have different dipole moments. The low value of JAA" ( = 3"15 cycles see -a) found for the dl isomer indicates that V must be present in small amounts (14 + 10 per cent) while the remainder is mainly IH. A survey has been made c2e') of A_&'BB' type spectra given by hydrogen nuclei in 1,2-disubstituted ethanes (denoted XCH2CH2Y). A large number of such compounds give IH resonance spectra which can be classed as "deceptively simple", that is, seven or three line spectra are observed instead of the possible ten lines for half the spectrum. Ten lines are observed when ~A -- VS (ffi ~0~) and JAA' -- JBS" (ffi M) are small. Five line and three line spectra occur as a consequence of VA -- ~'Band JA A' -- JB 3, being large compared with JAB "F JAS' (---- N). The rotamers are
x H(B).., I H( A )\.,.' ~ / / I y
H(A} H(B)--.. ...H(e~}
..H(B') H(A')
H(A') H(B).. [ ..H(e') X~" " / / I Y
H(A,) ,"/ ~ , .i, ~ \ X
I Y
Trons
H(A)
GaUche
The spectral expectations for these conformers can be obtained under behaviour (c) given above. Assuming the trans (Jr) and gauche (Jg) coupling constants are invariant (but see reference 344) for a given compound then JAB = n, Jg + ½ng(J, + Jg) and It may be shown that and hence
(9.143)
Jan" = JA'B = n, Jt + noJo
N = J + J' = ½[J, + 3J, + n,(J, - Ja)]
(9.144)
L = J-
(9.145)
J' = (½ - ~n,)(Jr - Jo)
2JA~ + Jaw = Jt + 2Jg = I N + ½L
(9.146)
Abraham and Pachler(2s4~ arrived at several important conclusions. First of all, ] N + ½L should remain constant for a given compound and should be independent of the rotamer populations. From the spectrum analysis, only the magnitude of N and L may be found but by assuming N to be positive, the sign of L may be determined under conditions giving different conformer populations (e.g. by varying the temperature or the solvent). Hence Jf + 2 J o can be determined uniquely. Also the sign of L tells us whether the trans or the gauche form is the more stable, because Jt > Jg, so that if for example L > 0 then nf is less than 3, that is, the gauche isomer is the more stable.
THE EFFECTS OF CHEMICAL EQUILIBRIA
567
I t m a y b e seen t h e n t h a t ff the r o t a m e r s are o f e q u a l energy, L = 0 a n d the h a l f s p e c t r u m c o n t a i n s five or three lines. N o t e that the value o f [ N + ½ L is n o t exactly e q u a l to ~(J, + 2Jo), the value for L = 0, TABLE 9.11 H-H COU-PX~GCONSTANTSAND CHEMICALSl'uyl~ IN 1 , 2 - D ~ ETIIAN~¢2a't) Compound
Yo6
M
N
L
~N ± ½L
J, + 2J.
1-Chloro-2bromoethane liquid 1:lin acetonitrile
12-60
0-95
15"00
3"07
24"0 (21"0)
20-21
1"02
14-00
1"58
21"8 (20"2)
2-Odoroethanol Liquid I : 2 i n CCI,t 1 : 1 in acetone
11.24 13.20 10.83
@81 0-76 0-80
11"18 11"82 11"43
1"90 2-15 1"22
17-7 (15"8) 17-3 (15"2) 17-8 (16"5)
17.6
22.8 25"0 25"65 21.8
1"94 1"71 1"77
11"72 11"30 11"64 12-09
1"20 1"48 1"05 0
18"2 (174)) 17"7 (16"2) 18"0 (16"9) 18'1 (18"I)
18.0
11"70 12"20
9"61 9"73 9"66 10"02
2.24 2"80 2"82 2"08
15"5 (13"3) 16"0 (13-2) 15-9 (13"1) 16-1 (14.0)
15-8
11-52
1-53 1.87 1"24 1"52
34"8 35"4
0 0
11"06 11"52
2"10 1-76
17"6 (15"5) 18"1 (16"4)
17"9
36-3 35-8 35.9
0-83 0-89 0"85
9"68 9-80 9"73
2"83 2"79 2"88
15"9 (13-2) 16"1 (13"1) 16-0 (13"2)
I
10.72 15-36
1"10 1.18
13"17 12"90
1"35 1-91
20-4 (19"1) 20-3 (28"4)
}
14.40 13-87 15.39 15-4
1-15 1"08 1"26 1"73
13"07 12"96 13"12 12-78
1"54 1"38 1"88 1"93
20"4 (18"8) 20-1 (18"8) 20"6 (18-7) 20-1 (18"2)
20"4
29"3
2-82
15-89
3-94
25"8 (21-9)
21"9
20.6
2-Bromoethanol liquid 1 : 1 in CHCI3 I : 1 in CS2 1 : 1 in acetone
-
J
2-Methoxyethanol liquid I : 2 i n CC14 I : I in D 2 0
1 : 1 in acetone
8"98
2-Chloroethyl ~tate liquid : 1 in CC14
2-Methoxyethyl ~cetate liquid : 1 in CC14 : 1 in acetone
~aevulic acid : 1 in CHOa : 1 in D20
16"0
Sthyl laevulate iquid : 1 in CHCIa : 1 in MeOH ! : 3 in CFaCOzH
t-Cyano-2,2~methyl 'mtyraldehyde iquid
568
H I G H R E S O L U T I O N NMR S P E C T R O S C O P Y
but it is a function of the rotamer proportions. Hence the range of N values observed may be due in part to the substituent effects but it also reflects the rotamer populations. Table 9.11 lists the data obtained by Abraham and Pacifier c2a4). The table shows that the gauche isomer of a 1,2-disubstituted ethane is often more stable than the trana isomer, a result which is opposite to that found in the haloethanes. However, low resolution N M R observations on solid halogenated ethanes c'-93) have shown that CFCla. CFC12 exists mainly in the gauche form while CFzC1. CFzC1 and CF2Br. CF2Br are probably in the trans form. Some of the solvent effects are surprising, for instance, the rotamer fractions of 2-methoxyethanol are virtually identical in CC14 and in D20. From Table 9.11 it may be seen that the value ] N + ½L is not quite constant for a given compound and it increases as the fraction of the trans isomer increases. Hence J, + 2Jr is not the same for the two isomers of each compound. Abraham and Pachler have explained the effect as being due to the dihedral angies of the gauche isomer differing c2ss' zs~) from 60 ° and 180°. The fact that the M values change with the solvent leads to the conclusion that geminal coupling constants are more sensitive to environmental changes than are vicinal couplings. In the above discussion it has been assumed that the two gauche isomers are indistinguishable, but Harris and Sheppard czsl) found evidence contrary to this in their study of the 19F spectra of 1,2-disubstituted perfluoro-ethanes. They utilised the ingenious method of analysing the tsC satellite signals of the main tgF resonance (z~2) (see Section 8.19.2).
Solvent effectS. The discussion of solvent effects presented so far has not included a quantitative treatment. Whipple ¢z87) has reported the variations of cgs-l,3 ~H coupling constants (J) in 2,3-dihalopropenes. The two h"kely conformers were deduced to be z
z¢cH=l
I
I Y rr
J is given by J -- nx Jx + nxx J , where n~ and nn are the mole fractions of the isomers. Also, it was assumed that J~ = ½Y' and Jn ffi IJ', J' being the 1,3 coupling constant in the appropriate 2-halopropene. The equilibrium constant K was calculated from nx and nn thus
K
ffi nl/nlt =
2(ZdZnL)exp(--AE/RT)
(9.147)
THE EFFECTS OF CHEMICAL E Q U I L I B R I A
569
where ZI and Zn are the partition functions of the two isomers. The energy difference/I E betwcen the two isomers in a solvent is given by approximately H E ffi A E o -
where
(8=Nd/3M)[(e
-
1)/(2e + 1)] (p~ - / ~ )
(9.148)
d - density of solute
M = molecular weight of the solute • ffi dielectric constant of the medium Px and Pn -- dipole moments of the conformers. Thus there should be a linear relatiouship between log K and (~ - I)/(26 + 1); this was confirmed by measurements made on the isomer ratios of 2,3-dichloropropene (2e~). Snyder(s'9) has pointed out that . / ~ can be affected by solvent without conformational changes being involvedisis).
Temperature studies. Microwave spectroscopy has provided a good deal of information on the barriers to molecular internal rotation. Unfortunately, the restrictions inherent in the method allow measurement of barriers only of methyl groups rotating against various other groups. Barriers tend to be small and roughly constant at the value 3 kcal mole -I for molecules of this type and it seems that steric repulsions make little contribution to the barrier. Heights of barriers of rapidly rotating isomers have also been measured by ultrasonic relaxation (29e). Enthalpy differences between gauche and trans isomers can be small: the value for n-propyl chloride has been estimated tO be 50 -I- 150 cal mole -1 by spectroscopy(294) and 300 -I- 200cal mole-x by eieetron diffraction(295). Substitution at both carbon atoms in ethane can raise the barrier substantially, for example the barrier (2.°) for CCIsCC13 is 10-8 kcal mole -x. Barriers of thi.q height can be measured in molecules containing hydrogen or fluorine using nuclear magnetic resonance spectroscopy. By measuring the relative areas of the lines due to the three rotamers, the relative energies of the three potential minimst can be obtained. Comparison of the observed spectra at transition temperatures with calculated spectra, assnming likely barriers to internal rotation, can provide much information about the heights of potential maxima. Potential maxima have been measured(2") for CFC1BrCFC1Br where they are all about 10-0 + 0.5 kcal mole -1 above the potential minima, consequently the low temperature 19F spectrum is a superposition of the spectra of three different rotamers of each of the meso and the d/isomers. The estimation of the relative energies of all three forms of ethanes having an asymmetric carbon atom is one of the triumphs of NMR spectroscopy. A method of evaluating the energies in the situation prevailing when separate rotamer spectra can be observed has been reported ¢2s8). In most substituted ethanes, rotational averaging occurs which simplifies the spectrum but reduces the information content. Gutowsky, Belford and McMahon (~sg) have considered how valuable data can be acquired from rotationally-averaged spectra. Their procedure is given below.
570
HIGH
RESOLUTION
HMR SPECTROSCOPY
The experimentally observed average resonance frequency (chemical shift) of the tth nucleus is given by !
(,,> -- xx "l + x2 ~'~ + x3 ,~
(9.149)
and the averaged coupling constant between the tth and jth nuclei is given by
(a',~>
__ x ~ J ; ~ + x~S~~ + x~ j u
(9.150)
where xt, x~ and x3 are the mole fractions of the three retainers. The mole fractions may be represented in terms of the relative energies of the retainer (see Fig. 9.25) ;these are definedby putting A El = Ex - E3 and A Ez = E2 - E3. The ratios of the mole fractions can then be expressed
xt: xz: x3 = Q't e x p ( - A E t / R T ) : Q ; exp(-AE2/RT): Q'3
(9.151)
E 3
F 0
i
', ~
2~
FIG. 9.25 T h e energy o f a s u b s t i t u t e d e t h a n e as a f u n c t i o n o f t h e dihedral angle between C--a a n d C--b b o n d s in a g r o u p a--C--C-b
where Q" is the partition function for a particular retainer n, excluding the internal rotation coordinate ~. To a good approximation Q~ ffi Q~ = Q~, hence 0 ' , ) -- Q~' [v~ e x p ( - A E , / R T ) + ~ e x p ( - A E a / R T ) + ~,~1 (9.152) where Q~, is the internal rotation partition function for the three retainers at thermal equilibrium
Q~ = e x p ( - d E t / R T ) + e x p ( - d E 2 / R T ) + 1
(9.153)
Also, assuming coupling constants are temperature independent ¢3°~), ij
(Ju) = Q~t[J~4 exp(-AE~/RT) + Jz e x p ( - A E 2 / R T ) + J~].
(9.154)
Thus if the temperature dependence of (r~) and ( J u ) is dependent only on the equilibrium proportions of the rotamers then the relative energies of the rotamers and the individual chemical shifts and coupling constants of each rotatio-
THE EFFECTS OF CHEMICAL EQUILIBRIA
571
nal isomer can be calculated. Coupling constant measurements are preferable because they'are unaffected by molecular association; there may be temperature effects, however (344). The resulting equations are difficult to solve and the magnitude of the effect of temperature on (,/tj~ is only about I cycle sec -I in ~H resonance spectra over a range of 200°C (while (~i) changes by about 2per cent over a similar range). Despite the difficulties it is possible in favourable cases to extract all the unknowns (374). The situation will improve as instrumental techniques advance. The dependence of JHs upon the dihedral, angle has been calculated to be (see Section 5.3.1) d , s -- 9 cos2~ - 0.3 (cycles sec -x) (9.155) but Fessenden and Waugl I(29°) doubt if a similar function can be applied to coupling between hydrogen and fluorine nuclei. Fitting of equations (9.152) and (9.154) to the observed temperature dependence of (vt) and ( d ~ ) is best done with the aid of a digital computer. This assistance is necessary even when the two gauche forms are identical so that the unknown parameters are reduced from five to three, that is,
•4 E ( = E 2 - E 3 = E , - E o )
plus dt
and
do
or vt and ~o. Thus equation (9.154) takes the form
(s)
=
2J, + J, e x p ( - A E / R T ) 2 + exp ( - .4 E]RT)
(9.150
in which it is implicit that Ez > Ea. If a A E value is assumed or obtained from another set of experiments, such as infrared spectroscopy, then the coefficients o l d a and Jt in equation (9.156) can be calculated for each temperature,/'1...Tt at which there is a value for (d). Hence there are a set of k equations of the form (.It> = a,d, + b,d, (9.157) which is linear in the unknowns ,/g and dz, The reduction of the equation to a linear form permits the application of least squares methods to the evaluation of best trial values for d o and $ . With these and/I E it is possible to calculate (d~ as a function of temperature from equation (9.156) and thence obtain the sum of the squares of the deviations of the experimental points from the theoretical curve. The sum is minimised by repeating the process with systematic variation of A E. The above procedure has been applied by Gutowsky, Belford and McMahon to studies on CHC12CHC12, CHCI2CHF2, CF2CICFCI2, CHCI2CI-IFCI, CF2BrCFBrC1 (the dvalues found for this compound are not in agreement with those reported by Newmark and Sederholm (269) who froze out the rotamers) and CFC12CHClz. They found that the procedure could not be utilised in treating data for CH(C6Hs)2CH2(COC6Ha), CHBr2CH2Br, CH2(CeHs)CH~C1 and CH2(CeHs)CH2Br because the low value of A E (35 to 90 cal mole -x)
572
HIGH RESOLUTION NMR SPECTROSCOPY
causes ( J ~ ) to be virtually temperature independent. The following approximate expression was used for such systems: <~D = ½ I~, - Jol (AE/RT) + I
(9.158)
where A E -- Et - Eo and 1 is a constant. The results from chemical shift data appear to be the least reliable because of effects from molecular association. The agreement between A E's found from N M R measurements with those from vibrational spectra is good, hence the approximations are probably justifiable. The best example is CHC12CHC12, where the value 1 1 0 0 _ 35 cal mole -1 from (~H) agrees very well with 1050 + 30 cal mole -t from < a s s ) and 1080 + 40 cal mole -1 from vibrational spectra even though (~K) is affected by hydrogen bonding. Fessenden and Waugh ¢29°) have pointed out that it is somewhat disturbing that Jt and Jo for CHClzCF2CI and for CFClzCHC12 ezs°) axe markedly different, namely 10 and 5 cycles sec -x, and 18-1 and 1-0 cycles sec -t respectively. Comparing CF2BrCF2Br c2sx) with CF2BrCFBrz ¢278) there is a considerable increase in both Jo and Jt brought about by substituting fluorine for bromine-Jt is increased more in magnitude than is Jo. From the values of A E obtained, Gutowsky, Belford and McMahon decided that two factors govern the relative stability of the rotational forms. For the related compounds, CHClzCHClz, CHC12CHF2 and CFC12CHC12, the rotamer of lowest energy is the one in which the two hydrogen nuclei, and in the latter the fluorine and the hydrogen nuclei, are gauche. These have the largest dipole moments hence the steric effect is small. Steric as well as dipole interactions tend to stabilise the gauche form of CF2CICFC12. The compounds having low values of 3 E discussed above have their most stable form determined by steric factors. Interesting chemical shift regularities were found. For CHClzCHClz, the form with the 1H signal at lowest fields is that with both hydrogen nuclei adjacent. Similarly, for CF2C1CFC12 the resonances of both sets of fluorine nuclei are farthest downfield in the trans rotamer, in which all three fluorine atoms are adjacent. In both compounds, the rotamer with the downfield shift(s) is apparently that in which the vicinal substituents would produce the largest net electrostatic interaction ¢29:). Such correlations might assist in establishing unknown molecular configurations. Caution is required because Thompson, Newmark and Sederholm c2ss) obtained the xgF chemical shifts of the rotamers of CFCIBrCFC1Br and found that they occur in the following order: farther to low fields when a F atom is gauche to a F and a C1 atom, a F atom is gauche to a Br and C1 atom, and a F atom is gauche to a F and a Br atom. Thus the high field shifting power of the individual atoms is in the puzzling order CI > F > Br.
9.5.6 The C ~ P bond The conformational studies made so far have been confined to compounds of the t y p e (292) RI\ ~:0 Rz%C--P&Y Ra / \X
THE EFFECTS OF C H E M I C A L E Q U I L I B R I A
573
where R is a hydrogen atom, a phenyl group or an alkyl group, while Y is OH, (21, OCHs or OCH2CH3 and X is either CI or a phenyl group. Clcarly, the principles governing the isomerism of the C--C bond are also applicable here. The evidence from the XH resonance spectra is that rotation about the C--P bond is rapid at room temperature. 9.5. 7 The Nature of the Barrier to Internal Rotation
The existence of potential barriers to internal rotation has been known since the 1930's but as yet there is no complete understanding of their properties. The magnitude of the barrier is relatively constant for compounds of the type CHHs. CH,X,_ s where X = H, F, C1 or Br. Because of this it has been suggested c2.°) that a single interaction is mainly responsible. However, there remains a paucity of experimental data of adequate accuracy for a sufficiently large number of molecules. Regarding the C--C bond in ethane as being cylindrically symmetrical precludes an angular dependence of the energy barrier to rotation. Invoking higher energy orbitals such as carbon 4.1 levels is not particularly helpful(2"), especially as greatest overlap of bonding orbitals would favour the eclipsed configuration: it is known that the staggered configuration is the stable one. A difficulty in the problem is that mixing d or/character into the carbon orbitals may be important but the crudity of the available quantum mechanical methods obscures the issue. Another contribution which may be si~ificant is some double bond character in the C - - C bond (usually described as hyperconjugation) but again the eclipsed configuration appears to be favoured. Empirical data such as bond dipole moments and steric repulsive forces have been used to treat the problem. These attempts to account for rotational barriers by the interaction of dipoles, quadrupoles, octupoles etc., at the opposite ends of the molecule do not provide the whole answer(24°). However, electrostatic effects of this kind will contribute when polar groups are involved. Wilson (2"°) has concluded that the most likely picture of the ethane barrier arises from the C - - H bond orbitals on the carbon atoms being more concentrated than sp hybrids. A mode of attack which may prove useful is the development of theories to enable nuclear coupling constants J , , , to be calculated (and compared with experimental data) for a number of values of the HCC'/CC'H' dihedral angle (see Section 5.3.1) (299-3°1). A method has been described(so2) for direct calculation of the barrier to internal rotation in ethane and ethane-like molecules. The electron distribution is taken to be delocalised with cylindrical symmetry about the central bond. This model predicts energy differences between staggered and eclipsed forms, composed solely of nuclear-nuclear repulsion energy, that are consistently about 5/3 of the experimental barrier heights. A self-consistent-fieldapproximation has been used (34s) to calculate the barrier to internal rotation in ethane 0"3 kcal mole-Z).
574
H I G H R E S O L U T I O N NMR S P E C T R O S C O P Y
9.6 INVERSIONOF NITROGENCOMPOUNDS Kincaid and Henriques ¢~°s) have estimated the heat of activation for the inversion of N-methylethylenimine (1-methylaziridine) CH~ CI-I, to be 25 kcal mole-L They noted that resolution of the isomers would be practically impossible ff the rate constant is less than 10.5 sec-' : this value is about that obtained when 25 kcal mole -~ is substituted into the Arrhenius equation k -- 10-Is e x p ( - / I
E/RT).
(9.159)
However, attempts to resolve this and other ethylenimines into their isomeric forms have been unsuccessfuFs°4-s°6). Steric effects from branched groups such as t-butyl attached to the nitrogen atom were found to lower the energy barrier. Cyclic compounds with the nitrogen atom incorporated in unstrained five- or six-membered rings should undergo inversion with practically the same ease as the analogous open chain compounds. Kincald and Henriques calculated the upper limit of the heat of activation for inversion of trimethylamine to be 15 kcal mole -1. Thus, rates of inversion are suitable for investigation by nuclear magnetic resonance. Bottini and Roberts (seT) found that at room temperature there are two distinct signals in the spectrum of N-ethylethylenimine which can be assigned to the ring CH2 groups being either c/s or trans to the ethyl group. On heating to 120°C the absorption bands coalesce to a single line. By contrast, due to inversion being more rapid than 27 times a second, these workers found (s°s) that the compound CI~|
CHz shows only one ~H resonance line for the ring methylene group at room temperature: two components appear below - 77°C. A number of other ethylenlmine derivatives have been examined at various temperatures and the rate constants found by taking them as being equal to 4~/V2. The results obtained so far show that conjugation with nitrogen is very important in increasing the nitrogen inversion rates. Bottini and Roberts (3°s) found that inversion rates are increased by bulky groups whether they are attached to the nitrogen or carbon atoms of the imine ring. Substitution of alkyl groups for one hydrogen or two cis hydrogen atoms seemed to give preferred configurations with the nitrogen substituent traws to the ring substituent(s). The nitrogen inversion rates of N-substituted trimethylenimines and larger ring imines were too great to be measurable by the NMR method at temperatures above - 77°C.
THE EFFECTS O1= CHEMICAL EQUILIBRIA
575
9.7 CONFOme~TIONOF SATURATEDRING CObl]POLINDS
9.7.1 General Considerations NMR investigations of the conformation of ring compounds have been centred mainly on cyclohexane derivatives. At room temperature, cyclohexane itself has a single ~H resonance absorption band originating from the rapid chair-to-chair interconversion
H~,
Ha
e / a H~ M
•
H
-
H= _ ~ He Ho
He
' -- ' He
H~, H=
Ht
He H=
The rapid equilibrium averages the chemical shift difference between the axial and equatorial hydrogen atoms since the process of interconversion (brought about by partial rotation of each carbon-caxbon bond) puts an equatorial H atom into the axial position and vice versa. Proof of the postulate came from the following experiment: solutions of cyclohexane in carbon disulphide when cooled show a progressive broadening of the absorption band until at - 70°C two distinct bands appeareS°9): the signal at lower fields being assumed to be that of the equatorial hydrogen nucleica1°). The chemical shift difference between ax/al and equatorial hydrogen atoms probably has its origin in a long range shielding effect connected with the diamagnetic anisotropy of the carbon carbon bonds cszx). At lower temperatures still, the band separation remains constant at about 28 cycles sec -~ for an irradiating frequency of 60 Me sec -~. At - 66-5°C, the rate constant for chair-to-chair interconversion was estimated to be 60 sec -x. The free energy of activation was found to have a revised value(3 x2, sle, s~7,sTs, s76) of 10.1 kcal mole -x which can be assumed also to be the value of the enthalpy of activation(3°~. Harris and Sheppard (s x7) estimated the entropyofactivation to be - 7-9 _+ 1.0 ca1 mole -x deg -~. Similar measurements have been made on perfluorocyclohexane dissolved in trichiorofluoromethane and trifluoromethylbenzene(~z). The exchange rate constant was calculated to be 3.5 x 10' sec -x from the width of the broadened absorption band at 25.5°C. Temperature variation led to values of 7.5 + 0.3 kcal mole -z and - 10-7 ca1 mole -~ deg-' at 25°C for the heat and entropy of activation respectively. It is of interest that the estimated vaiue of the rate constant of perfluorocyclohexane is almost identical with that measured for cyclohexane at - 66-5°C. Tiers (s~2) explains the high rate of conformational interconversion of the former compound as being due to the existence of intrinsic torsional strain resulting from steric or electrostatic 1,3 diaxial fluorine-fluorine repulsions. This strain is diminished in the transition state.
576
H I G H R E S O L U T I O N NMR S P E C T R O S C O P Y
In work of thig kind, it has not been necessary to invoke the less energetically favourable boat form of cyclohexaneC313): appreciable amounts of the boat form exist only in molecules with unusual 1,4 interactions¢3~'). .Almost all NMR investigations of cyclohexane derivatives have been designed to measure the average chemical shift at room temperature and the individual axial and equatorial chemical shifts have been determined from related molecules having fixed conformers. Difficulties arise using this procedure because of the profusion of spin coupling, aggravated by small chemical shift differences amounting to only 0.1 to 1.0 ppm. The half-width of the ~H resonance signal has also been used to indicate conformationc3~5>. Here the axial-axial conformer has a much broader line than that of the equatorial-equatorial form. In fixed conformations the spin-spin coupling constant between neighbouring hydrogen nuclei in the axial-axial position is about 2 to 3 times larger than that
(o)
(c)
(t))
(~
Fla. 9.26 Tb¢ t H resonance spectra at 40 Mc sec-1 of the CHX group in (a) bromocyclohemmc at - 88°C Co) bromocyclohexane at - 97°C (c) chlorocyclohexane at - 80°C (d) ehlorocyclohexan¢ at - 92.5°C. Reeves and StrommeO13)
of neighbouring hydrogen nuclei in other orientations ¢31°). This result is similar to that found for the coupling constants of c/s and trans ethylenes and gauche and trans ethanes ¢21~). Correlations between chemical shift and substituent position have been found c31°). For example, the hydrogen atoms of methyl groups in acetoxy substituents usually give rise to signals at lower fields in the axial orientation than when in the equatorial orientation. Cyclobexyl halides h a w been studied extensively using infrared spectroscopy but the results obtained are the subject of controversy~3~s"3~,). Reeves and Stremmeesz3) have carried out population studies on two isomers in equilibrium by making relative intensity measurements on the two absorption bands produced from the o~ hydrogen nucleus by cooling the system. Figure 9.26 shows the two signals obtained from the hydrogen nuclei adjacent to the halogen atoms in bromo- and chiorocyclohexane at low temperature and dissolved in carbon disulphide. The temperature at which coalescence of these bands occurs was used to estimate the approximate energy barrier to interconversion for these and other cyclohexanederivatives(32°'377) (see Table 9.12). It can be seen from Table 9.12 that the conformer with the higher dipole moment is favoured by the solvent with the highest dielectric constant. Thus the population of the
THE EFFECTS OF CHEMICAL EQUILIBRIA
577
ee form of 1,2-transdibromocyclohexane increases with increase of the acetone concentration. 1,2-transchloroiodocyclohexane has been shown to exist 68 5 : 3 per cent mole in the diaxial halogen form (321). Interconversion of substituted cyclohexanes will occur if both conformations have the same number of substituents opposing each other in axial positions, since otherwise there is an appreciable energy difference between the two forms. Six identical substituents in the 1, 2, 3, 4, 5, 6 positions place each ring hydrogen atom in the same inductive environment but long range shielding effects can TABLE9.12
Compound
Chlorocyclohexane Bromocyclohexane 1,2-trans-Dichlorocyclohexane
Concentration mole ~e 54.5 55"4 40-0 29.0 28.0 53.0 69-0 21"0
1.2-trans-Dibromocyclohexanc
34.0 59"0 33"0
Solvent
Equilibrium
C~13~i3tration of one
form CS 2
c~,~c~
CS2 CS2
77% C~ 82% Br.
35% o , o ,
cyc]ohexane}
33~0 CI,C~
70-9% Br,Br, CS2 CS2 acetone acetone [ acetone)
Br.Br. ~ Br,Bro
71.6~ Br~Br, 70-2~0 Br,Bra 49-9~e Br,Br. 54"2~ Br~ro 61"3~e Br,Br, 64"0~e Brolka
Energy barrier lwad mole-1 10"9 10"9 12-0 12"0 12"0 124) 12"0
produce differences in the magnetic resonance spectra of the different isomers (32~). Brownstein (3~2) observed that the spectra are similar if the six substituents are chlorine atoms or hydroxyl groups. He was able to correlate the positions of the absorption bands of the various isomers with the proximity of substituents, the m o r e stericatly hindered hydrogen nuclei being the least
shielded. The magnitude of the spatial effects in these compounds can be as much as 0.7 ppm. Even greater spatial effects are to be expected for the XgF resonance of 1, 2, 3, 4, 5, 6 hexafiuorinated cyclohexanes because in this instance the magnetic nucleus is at the centre of overlapping lone pairp orbitals. These compounds are not yet available for study but Feeney and Sutcliffe¢32s.324) have studied the spatial effects in other fluorocyclohexanes, namely, undecafluorocyclohexanes and decafluorocyclohexanes. They concluded that the main influence on the chemical shielding of a fluorine nucleus comes from the influence of a hydrogen atom through bonds. A spatial effect is operative when the immediate neighbours of a fluorine nucleus are all fluorine atoms. From the temperature dependence of the resonance spectra, the conclusion was drawn that one isomer of chloroundecafluoro- and perfluoromethylcyclohexane predominates at room temperature. The analysis of the spectra of fluorinated cyclohexanes is simplified by the fact that spin--spin interaction is largest for the geminal n~tgnetic nuclei. Hence the identification of an AB quar•r:r~. 19
578
HIGH
RESOLUTION
NMR SPECTROSCOPY
tet(312. 32,~ (see Fig. 11.11) in a 19F spectrum is proof of the location of an axial and an equatorial fluorine atom attached to the same ring carbon atom especially when the coupling constant Ja, has the value -,, 280 cycles sec -1 and the chemical shift difference is ~ 20 ppm. By working out the spectral expectations for each conformer and for that of the average of the two, additional information on the conformational state of an isomer was obtained(323, 33s). It was deduced that for some of the isomers of dihydrodecafluorocyclohexane the c/s- 1 H : 3 H compound is probably fixed with the hydrogen atoms in the ax/al positions; cis- 1 H : 4 H and cis- 1H : 2 H rapidly interconvert at room temperature; trans- 1H/2H exists as a fixed Helle conformer over the temperature range - 3 0 to + 80°C; the spectrum of trans1 H[3 H indicates that the molecule is rapidly interconverting. Homer and Thomas c333) have also obtained ~gF resonance spectra of some mono- and disubstituted perfluorocyclohexanes and they extended their measurements to tri- and tetra-substituted compounds. From the correlations obtained they concluded that the cis- 1 H : 3 H isomer is inverting rapidly at room temperature. Three types of behaviour were recognised: (a) molecules which are conformationally rigid (or possibly which are interconverting rapidly but which have a very low population of one conformer); Co) molecules inverting rapidly between equivalent conformations; (c) molecules inv.erting rapidly between non-equivalent conformations. Behaviour (a) is shown by compounds having a 1,2 or a 1,4 diaxial arrangement of hydrogen atoms. Also included in this category are derivatives which have a CFs group in the equatorial position. Further discussion of fluorinated cyclohexanes is given in Section 11.13.3. The IH resonance spectra of dimethylcyelohexanes and related compounds can2) have been examined between - 130 and + 130°C. It was found that only 1,1-, 1,2-cis-, 1,3-trans- and 1,4-cis-dimethylcyclohexane and cis-decalin produce spectra which vary with temperature. For each of these, it is evident that the molecules undergo rapid ring inversion at room temperature but that below - 100°C the spectra are those of the conformationaUy fixed species. Observation of a relatively narrow region of ring hydrogen resonances did not necessarily imply that rapid interconversion was occurring. Deuterium substitution has' been employed to confirm the conformational analysis of trans-5-acetoxy-2-phenyl-l,3-dioxanCa25):
H
0/~
H
"-Ph //
HI AcO H
d
579
THE EFFECTS OF C H E M I C A L E Q U I L I B R I A
From the IH N M R spectrum it was decided that the isomer is fixed at room temperature in chloroform solution. The cis isomer has a simpler spectrum from which it was concluded that rapid interconversion is present: however, lower temperature measurements would provide additional evidence ff line broadening occurs (326). Anet (32~) and Premuzic and Reeves(32~) have also used deuteration to simplify the ~H spectra of cyclohexane derivatives. Another heteroalicyclic six-membered ring compound to be investigated by N M R spectroscopy is NN'-dimethylpiperazine (32s)
H H N
~
N~IVeMe---N~ , ~ , j -- '~:
H
H H~
At room temperature, the IH resonance spectrum of the compound dissolved in CH~CI, comprises a fairly sharp low field band from the ring methylene hydrogen nuclei and a sharp higher field band from the six methyl hydrogen nuclei. At 40°C, the low field band resolves into an AA'BB' type spectrum (see Section 8.16.2). From the coalescence temperature and the exchange broadened single peak near the fast exchange limit (equation 9.137), the energy barrier for interconversion between the two identical chair conformations was computed to be 13.3 4" 0"3 kcal mole -~. Reeves and Stremme °2s) noted that increasing bulk of the substituents may be responsible for the significant increase in energy barrier (per mole) in the series: cyclohexane ~ 9"7 kcal, chloroand bromocyclohexane ~ 10.9kcal, trans-l,2-dichloro-and trans-l,2-dibromocyclohexane ~ 11.8 kcal, and NN'-dimethylpiperazine 13.3 kcal. The AA'BB' analysis yielded the following constants (by assuming that J,1o2 = 7.43 cycles sec-1) Jalei -- 13"2cycles sec -1, J,~,2 = 2.1 cycles sec -~, J, lo~ = 2.4 cyclessec=1 and the chemical shift difference between the axial and equatorial hydrogen nuclei is 0-27 ppm. The inversion frequency about the nitrogen atom in a sixmembered ring is similar to that for ammonia therefore the methyl resonance remains sharp at all the temperatures used. N M R measurements have been used to investigate the ring inversion of cyclo-octane (339). -
9.7.2 Fused Saturated Ring Compounds Musher and Richards (329) have found that the high resolution IH N M R spectra of c/s- and trans-decalin are completely different as may be seen from Fig. 9.27. The c/s- compound always gives a narrow absorption line, while the trans form gives a broad line. The trans form of decalin has a relatively'rigid carbon skeleton enabling the hydrogen atoms to be classified as axial or equatorial. Spin coupling between the hydrogen nuclei leads to the broad unresolved signal observed. Rapid simultaneous interconversion of both chairs of 19"
580
H I G H R E S O L U T I O N NMR S P E C T R O S C O P Y
c/s-decalin gives sufficient averaging to produce a sharp absorption band. From the experimental data, it may be concluded that ring inversion must occur at frequencies greater than 30 cycles sec -1. Homer and Thomas (33°) have examined the zgF spectrum of c/s- and trans. perfluorodecalin at 34°C (see Fig. 11.12). As discussed above, the greater chemical shifts involved permit a more complete spectral analysis to be made from the ready recognition of AB quartets. The spectrum of the c/s isomer indicates rapid interconversion between conformations resulting in loss of axial--equatorialidentity of the nuclei, hence ring position becomes the sole factor affecting shielding. Cis
Trons
L _J !
I I I
k
k t
I
i~,--5 0 --D,,~ Cycles sec -t
'
I
'
r
I I I
~
I 1
I I
I
I
i
i Z
5 0 -.-~ cycles Sec-t
FIG. 9.27 The ZH resonance spectra at 29.92 Mc sec-t of carbon disulphide solufionsof A: c/s-decalin, B: trans-decaFm, C: c/s-decal-c/s-l-ol,D: transdecal-trans-l-ol, E: cis-decal-cts-2-ol, F: trans-decal-c/s-2-ol, G: c/s-decaltrans-2-ol. Musher and Richards(329)
C/s- and trans-decalin have been studied at low temperatures and the resulting spectra have been compared with those of c/s- and trans-hydrindan obtained under similar conditions(33z). The spectrum of cLv-decalin at temperatures as low as - 12 I°C shows no indication of the broadening which occurs in the case of cLv-hydrindan at higher temperatures. Moniz and Dixon (316) assumed that the energy barrier of interconversion is lower in c/s-d¢calin than that in c/.~-hydrindan and that the non-bonding interactions present in cis-decalin raise the ground state energy to a greater extent than do the strains present in the six-membered ring of c/s-hydrindan. The nature of the ring fusion of transd¢calin and tran$-hydrindan does not allow chair-to-chair interconversion of these molecules, hence their ZH resonance spectra are broad and ill-defined. The
THE EFFECTS OF CHEMICAL EQUILIBRIA
581
NMR spectrum of c/s-bicyclo [3.3.0] octane (two fused saturated five-membered rings) was unusual in having a low field signal 2.4 ppm from the tetramethylsilane internal reference. The absorption band was assigned to the bridgehead hydrogen atoms. The resolved structure in the spectrum provides evidence for a rigid carbon skeleton. An investigation ~326> of the four isomers of 10-methyldecalol-2 showed that the ring configuration greatly affects the resonance signal location of the angular methyl hydrogen atoms and that there is also a smaller effect due to the hydroxyl group being either c/s or t r a n s to the methyl group.
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582
HIGH RESOLUTION NMR SPECTROSCOPY
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HIGH RESOLUTIOIq NMR SPECTROSCOPY
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THE EFFECTS OF CHEMICAL EQUILIBRIA
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588
HIGH RESOLUTION NMR SPECTROSCOPY
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APPENDIX
589
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T A B L E OF P R E D I C T E D V A L U E S OF S H I E L D I N G C O N T R I B U T I O N S FROM RING CURRENTS IN AROMATIC HYDROCARBONS Calculation of Nuclear Magnetic Resonance Spectra of Aromatic Hydrocarbons, by C. E. JCmNSONand F. A. BOVL~Y,J. Chem. Phys., 29, 1012-1014 (1958). HOW
TO USE THIS
TABLE
The z-axis is the axis normal to the plane of the aromatic ring at its centre, and is measured in ring radii (1.39A). The values range from z equals 0.00 to z equals 4.00 in increments of 0.1. For each value of z, the p axis (in the plane of the ring) ranges from p equals 0.00 to p equals 4.00 in increments of 0.1, also measured in ring radii. These values start in the upper left-hand corner and increase from left to right. Negative values correspond to a reduction 3n the apparent field; positive to an increase in the apparent field. Example: For a p value of 1.1 and a z value of zero the shielding contribution is -2.74135 ppm.
595
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~6~6
o6~6
~
I¸ I ~DC~
~
.
0 0 0 0
l . ~ o o o
C~ C 3 0
C~
C
APPENDIX
C O M P I L A T I O N OF V O L U M E DIAMAGNETIC SUSCEPTIBILITIES OF O R G A N I C C O M P O U N D S (The data in this appendix is published by kind permission of Dr. G. W. SIm~, Laboratories of Cnme~ Motors Corporation, Michigmn, who compiled the mmceptibilities from an extensive review of the literature.) In Secdon 7.2.1dealing with external rcfe~e,ncing procedures the need for aocura~ ~ susceptibility data is ¢maphasised. Appendix C provides a comp~.hmmive list of valuta for more than 750 organic compounds. If a substan~ with a magnetisation I (magnetic monmat per unit volunm) it placed in a magnetic field of strength H, the magnetic induction, B, is given in cSs Gat~_~n units by B-- H+
4~I
which can be rewritten
B = H + 4 ~ z , H = H(I + 4nX,)
where Z, is the volume susceptibility, a di~onless quantity. Also included in this appendix axe values of X and XM. The specific, or mass susceptibility, X, is defined by ;~
=
z,/e
where Q is the density of the sample in grams cm -3.
The molecular susceptibility ZM is ZM = M X ffi M X,/#
where M is the molecular weight of the substance in grams.
605
606
APPBNDIX C Compound
Acenaphthene Acetal Acetaldehyde Acetamide Acetic acid Acetic anhydride Acetone Acetonitrile Acetonylacetone Acetophenone Acetoxime Acetylacetone Acetyl chloride Acridine Allyl acetate AHyl alcohol n-Amyl acetate iso-Amyl acetate n-Amyl alcohol ~,-Amyl alcohol iso-Amyl alcohol sec-Amyl alcohol tert-Amyl alcohol n-Amylamine isc~Amylamine n-Amylbenzene iso-Amyl bromide iso-Amyl-n-butyrate iso-Amyl chloride iso-Amyl cyanide iso-Amylene iso-Amyl ether iso-Amyl formate Amyl iodide n-Amyl methyl ketone iso-Amyl propionate n-Amyl valerate Anethole Aniline Anisidine Anisole Anthracene Anthraquinone Arabinose Asarone Asparagine Aspartic acid Azobenzene m-Azotoluene Benzaldehyde Benzaldoxime
- X M x 10e 109"3 81'39 22"70 31"5 31"54 (52"8) 33"78 28"0 62"51 72"05 44'42 M'88 81"5 (123"3) (56-7) 36"70 89.06 89"40 (67"5) (71.o) 68"96 69"1 (70.9) 69"4 71"6 112"55 (88"7) 113"52 (79"0) 73-4 (53"7) (129) 78"38 (118"7) 80"50 101-73 124-55
(96"0) 62-95 (80"5) 72"79 (130) (119"6) 85"70 131"4 69"5 64-2 ± 0-4 (106.8) (127"8) 60"78 (69"8)
- X x 106
- X , x 106
0.709) 0.688 (32 °) 0"5153 (0-533)
(0.726) (99°) (0-568) (32°) (O'4O3) (18°) (0.618) (2O°) (0.551) (32°) (0-562) (15°) (0-46O) (20°) (0.534) (20°) (0.531) (20°) (0.615) (20 °) (0.480) (2o ° )
0.525(32 °) 0.517 0"5814 (0.682) 0.5476 0.5998 0.6076 0.5481 (1"038) 0.688 0-566 0.632 0.6845 0.687 0-766 0.8060 0"7823 (25°) 0"785 0.804 (0"796) (0"821) (0.759) 0"587 0-7174(25°) 0"741 (0-755) 0.766 0"813 0"6748 0"5996 (18°)
(0-535) (20°)
(0.548) (20°) (0"757) (20°) (0"525) (200) 0.540 (20°) 0.5979(20.7 °) 0"599 (20°) (0"624) (200) (O.655) (250) (0.64) (150) (0.635) (200) (0.654) (15 °) (0.606) (20°) (0.616) (20°) (0.652) (20°) (O.706) (20°) (0.616) (25 °) (O.662) (20°) (0-6o9) (20°) (0"480) (20°) (0"635) (15°) (0.591) (25°) (0.910) (20°)
0"7053
(O-58O) (150)
0"7054 (25°) 0"7239 0.648 (0.676) 0.654 (0-673) 0.731 0"575 0"571 (0"631) (O-526) (0"482) 0-586 0"608 (0.573) 0-576
0"609) (25°) (0"638) (00) 0"644) (15 °) (0"691) (20°) (0"72) (2O°) (0.672) (15 °) (0.914) (27°) (0.825) (20°) (o.9o5) (20°) (0"735) (18") (o-812) (15°) (0.800) (12°) 0.611 (70.50) 0.643 (58 °) (0.602) (15°) (0"639) (200)
APPENDIX C
607
(continued) Compound
Benzamide Benzene Benzidine Benzil Benzoic acid Benzoic anhydride Benzonitrile Benzophenone Benzoyl acetone Benzoyl chloride Benzyl acetate Benzyl alcohol Benzylamine Benzyl chloride Benzyl formate
BenzyHdene chloride Benzyl methyl ketone
Biben~1
m,m'-Bitolyl Borneol Bromobenzene Bromo chloromethane Br0modichloromethane Bromoform ~-Bromonaphthalene m-Bromotoluene Bromotrichloromethane 1,4-Butanediol n-Butyl acetate is~Butyl acetate n-Butyl alcohol iso-Butyl alcohol sec-Butyl alcohol ten-Butyl alcohol n-Butylamine iso-Butylamine n-Butylbenzene iso-Butylbenzene tert-Butylbenzene n-Butyl benzoate Butyl bromide iso-ButyI bromide l-n-Butyl chloride 2-n-Butyl chloride n-Butyl cyanide ten.Butyl cyclohexane
--ZM x I0 e
(72.3) 54"84 110"9 (118"6) 70"28 (124.9) 65.19 109.60
(95"0) (75.8) 93.18 71 "83 75"26 81.98 81.43 (97.9) 83"44 (126-8) (127"4) 126"0 78 "92
--Z x I0 e
--Z, x I0 e
0"597
(o-8ol) (4 ° )
0-702 (32°) 0.603
(0.754) (20°)
0.564
(0.575) 0-552 (0.632) 0.6013 0"586 0.539 (200)
(0.62O) (O.664) (0-702)
(o-647) (0.598) 0.608
0-611
0.616 (100°)
(0.728) (150) (0"662) (15") (0"638) (15 ° ) (o-66) (50°) (0"639) (600) (0.657) (15 °) (0-655) (16") (0.697) (15 ° ) (O-69O) (19 °) (0"713) (18 °) (0-646) (20o) (0.763) (14°) (O-624) (20 °) 0.671 (54.5 °)
0.6219 0.696 0.6993(27.4 °) (0.817)
(0.699) (16") (0.826) (20°)
0.50.30(20°)
(0"753) (200)
55"0 ± 0"6
(0.425)
(0-846) (19o)
66.3 ± 0.3 82.60 115"90 (93 "4)
(0.405)
(0.812) (i5 °) o.948 (200) o.840 (2oo) (0-770) (20°)
73"1 ~ 0"7 61"5 77"47 78"52 56.536(20 °) 57-704(20 °) 57.683 (20°) 57"42 58"9 59"8 100-79 101-81 102.5 116.69 77.14 79"88 67"10 67.40
(0~369) (0"682) 0"6669(25 °) 0"6760 (0"7627)
0-327 0"560 0"546
(0"785) (0-7782) 0 7747 (25°)
(O.805) (0"818) (0.751) (0.759) (O-764)
(0.758) (00) (0.696) (2o°)
(0"583) (25°) (0.584) (250) (0.6176)(20°) (0"624) (200) (0"629) (20") (0.61 I) (200) (O.596) (2O') (O.599) (20°) (O'646)(2OO) (0.64S) (20°) (0-662) (20°) (O-656) (7.5°) (0.73O) (2O°)
(62.8)
0"6548 0"563 (200) 0.583 (20 °) 0.725 0.728 0.7558(27.4 °)
(O.6O6)(20")
115.09
0.8205
0.667o(2o~
(0.737) (200) 0-642 (20°) 0.635 (20° )
608
APPENDIX
C
(continued) Compound
3-Butyl ethyl ketone 2-Butyl formate so-Butyl formate ~utyl iodide so-Butyl methyl ketone err-Butyl methyl ketone ~-tert-Butylphenol Butyl sulphide Sutyl thiocyanate ~-BuV~aldehyde so-Butyraldehyde ~o-Butymldoxime a-Butyric acid ~o-Butyri¢ acid Butyronitrile ~codyl Camphor Camphor/c acid Camphoric anhydride n-Caproic acid a-Caprylic acid Carbanilide Carbon disulphide Carbon tetrabromide Carbon tetrachioride Carbon tetraiodide Carvacrol Carvone Cetyl alcohol Chloral Chloroacetic acid Chloroacetone Chloroacetylchloride Chlorobenzene Chlorodibromomethane Chloroethylene Chloroform 7-Chloroiodobenzene ~"nloronaphthalene m-Chloronitrobenzene o-Chlorophenol p-Ghlorophenol Cholesterol
-X,~ x 106
- X x 10e
- Z , x lOs
0"7072 0"6446 0.6540 0.5086(18 °)
(0.579) (200) (0.571) (25°) (0"574) (250) (0"822) (2O0)
0.6995
(0.561) (200)
69'86 108-0 (113-7) (79-38) 46"08 46.38 56"12 55"10 56.06 49"4 (99.9) (103) 129"0
0"6979 (0.719) 0.7774(27.4 °) 0.891 (27.40) 0.6394 0.6436 0.6443 0"625 0"6363 (250) (0.715) 0"476 0'68
(0.644)
(0.558) (160) (0.653) (1140) (0"652) (16°) (0.659) (25°) (0-522) (20 ° ) (0"511) (2O° ) (0.576) (2O°) 0"598 (200) (0.601) (250) (0.569) (15° ) (0"689) (15 ° ) (0"67) (25°) (0"791) (20°)
(113) 78"55 101.60 134"05 (41) 93"73
0"620 0"6762 0.7053 (0.6316) 0"54 0-2826(200)
(0"740) (20°) (0.624) (250) (0.642) (20°) (0"783) (200) (0"699) (22°) (0.966)
66"60 (136) (109.1) (92.2) (183"5)
0"433 0"261 0"726 0"614 0.757 (17.50) 0.459 (O-5O9) 0"550 (0"475) (0"6216)
0-691 (20°) (1"13) (20°) (0"709) (20°) (0.59O) (20°) (0.619) (500) (o.694) (2o°) (0.8o4) (20°) (0"633) (200) (0"710) (00) (0"688) (20°)
(0-361) 0"574 0"497 (0"417)
(0"883) (150) (0.528) (liq., 15°) 0-740 (200) (0-786) (57°)
8O'73 65"83 66"79 (93"6) 70"05
(67.7) 48.1 (50"9) 53"7 69"97
75-1 4- 0"4 35-9 59"30 99"42 107.60
0-661
0.789 (200)
(74.8) 77-4 77-6 (284"2)
0"475 (O.6O2) (0.604) 0"735
0"638 (480) (0.747) (180) (0"789) (200)
(0-7s4)(200)
APPENDIX C
609
(con.~u~ed) Compound (~Mnami¢ acid Onnam/c aldehyde Cinnamyl alcohol Citral Conmarin o-Cresol m-Cresol >Cresol a-Cresylmethyl ether m-Cre~lmethyl ether ~ e s y l m e t h y l ether Cunm~
Cyame~de Cyanogen OTanuric acid Cyclobutanecarboxylic acid 1,3-CycJohexadie~e 1,4-C3rdohexadiene Cyclohexane Cyclohexane¢arboxyllc acid
Cyclohexanol Cyclohexanone Cyc,lohexene Cyclooctane Cyclooetene Cyclopentane C~clopcntanecarboxylic acid Cyclopontanone Cyclopropane Cyclopropanecarboxylic acid
- X u x 10e
- X x 10e
--Z, x 10e
0.529 0.566 0.650 0.650 (O.565) 0"655 0-661 (26°) 0-667 (26D 0-671 (40°) O-638 (4o0) 0"648 (4oo) 0-74 {~435 0"415 (0-391) 0.5816(30°)
(0-660) (40) (0-629) (150) (O'679) (200) (O'577) (20°) (0.528) (200) O.7O6 (20°) (0-690) (200) (O.69O) (200) (0-661) (15°) (0-623) (150) (0.629) (19°) (0.642) (0.490) (15°)
48"6 48.7 68.13 83"24
(0"607) (0-608)
(0-510) (200) (~5153 (2O°) (0-627) (2O°) (0.668) (30°)
73"40 62.04 57"5 91"4 84.6 59"18 73.48
0"732 0.6323 (0"700) (0"8153 (0.769) 0.8439 0.6446(30°)
0.694 (20°) (0.599) (20o) (0.567) (200) (O-684) (20°) (O.654) (20°) 0.629O(2O°) (0.677) (30°)
51.63 39"9 45"33
0-6141(30°) (0.948) 0.5271 (30°)
(0-582) (300) (0.683) ( - 79°) (o.569) 0o0) (0-656) (20°) 0-6814(20°)
(107-7) 119"74
0"769 0"7718 0"774 0"779 (0"8416)
80"88 (133"1) 57 (101"4) 81.1 4- 0"4
0'846 (0.50) 0.430 (0-334)
(0-649) (21°) (o-54) (24°)
78.36 (74-8)
(87.2) (98"9) 82"5 70"80 71"4 72"1 81"94 77"91 79"13 (88"9) (56"1) (21"6) 50.5(1o °) 58"16
~"ymene
(lo3.2)
Decalin ds-Decalin trans-Decalin n-Deca~ Deuteroindene Di-iso-amylamine Diazoacetic ester ~Dibromobem~ne Dibrom"ochchlorometl~n¢ 1X.-Dibromoethylene
106"70 (107-0)
1,2-D~"oromo-2fluoroethane 1,2-Dibromotetra¢hloroethane Di-n-butylamine HaS. 20
0-8100(27.5°) 0.6499(30 °)
(0.359) Crlq. 17°)
(0.842) (0O) (o.613) (30 ° )
(0.686) (350) (0-670) (35°) (0"6143(20°) (0"692) (13°) 0-786 (100°) (o-sos) (25o)
(71-7) (78.0)
0-386 0"379
(o.877) (17.5 °) (o.8553 (17 °)
(126.o)
0"387
(1.o49)
103"7
(0-8O2)
(0.767) (2O°)
610
APPENDIX C
(continued) Compound Di-iso-btltylamine Di-sec-butylamine Di-iso-butyl ketone D~utyl pbthalate Dichloroacetic acid o-Dichlorobenzene m-Dichlorobenzene r-Dichlorobenzene Dichlorodittuoro-
- X M x 10e
- Z x 10e
- X , x 106
105"7 105"9 104"30 175"1 58"2 84"26 83"19 82"93 52"2
(0"817) (0"819) 0.7335 (0"629) (0"451) 0.5734 0"5661 0.5644 0.432
(0-609) (200) (0-641) (00) (0.591) (200) (0-657) (210) (0.705) (200) (0.748) (200) (0.729) (200) 0"823) (20.~ (0.642) (-- 30°)
1,1-Dichloroethylene
49"2
cis-l,2-Dichloro-
51"0
0-508 0-526
(0.635) (15") (0.679) (15")
48"9
0"504
(o.638) (150)
44"55 129"31 231"98
(0.530) 0"7776 0"7930
(0.742) (140) 0.6889(20*) 0-7001 (200)
70"71
0"7059
(0"576) (20a)
146"2
(0.676)
(0-663) (160)
118.8
(0.593)
(0.602) (14°)
56.8 (124.5)
(0-777) 0"802
(O.552) (180) (0.699) (00)
115"2
(0.612)
(0"614) (20°)
0"6751 0"5782 0"5595 (0-574) 0.685 0.6035 0"563 0"753
methane
ethylene trans-1,2-Dichloroethylene Dicyandiamide Dicyclohexyl 1,1-Dicyclohexylnonane Diethyl acetaldehyde Diethylallylacetophenon~ Diethyl allylmalonate Diethylamine Diethylcyclohexylamine Diethyl ethylmalonate Diethyl ketone Diethyl malonate Diethyl oxalate Diethyl phthalate Diethyl sebacate Diethyl succinate Diethyl sulphate Diethyl sulphide Diethyl tartrate l-Difluoro-2-dibromoethane l,l-Difluoro-2,2dichloroethyl amyl ether l,l-Difluoro-2,2dichloroethyl butyl ether l,l-Difluoro-2,2dichloroethyl ethyl ether
58"14
(85.5)
0.550 0"382
(0"551) (19°) (0"611) (20°) (0"603) (150) (0.645) (250) (0.661) (20°) (O.628) (20°) (O.667) (15°) (0.630) (200) (O.662) (200) (0.883) (20°)
129"84
(0.587)
(0-694),(200)
119"48
(0"577)
(0"703) (200)
96"13
(0"537)
(0"723) (20°)
(92.6) 81"71 127"5 (177"0) 105"07 (86"8) (67"9)
(113.4)
611
APPENDIX-C
(continued) Compound l,l-Difluoro-2,2dichloroethyl methyl ether l,l-Difluoro-2,2dichloroethyl propyl ether Dihydronaphthalene ~-Dimethoxybenzene m-Dimethoxybenzene ~-Dimethoxybenzene Dimethoxymethane Dimethylacetophenone Dimethylallylacetophenone 2,2-Dimethylbutane ?.,3-Dimethylbutane 2,3-Dimethyl-2butene 1,2 and 1,3-Dimethylcyclopentanes 2,5-Dimethylfuran Z,3-DimethyLhexane 2,5-DimethyLhexane 3,4-Dimethylhexane Dimethylketo tetrahydrofurfurane Dimethyl malonate ?.,4-Dimethylnonane L4-Dimethylnonane 1,5-Dimethylnonane Z,6-Dimethyloctane Dimethyl oxalatc 1,2-Dimethylpentane 1,3-Dimethylpcntane 1,4-Dirn~hylpentane Z,2-~ylpropane Z,4-Dimethylpyrrole Z,5-Dimethylpyrrole Dimethyl succinate D/methyl sulphate Dimethyl sulphide ~',N-Dimethyl urea ~r,N~.Dimethyl urea 7-Dinitrobenzene ~-Dinitrobenzene 7-Dinitrobenzene 1,4-Dinitrophenol 1,4*Dioxane Diphenyl Diphmlylumlne 20*
--ZMx 10e
--X x 10e
--X,x 10e
80"68
(0"489)
(0.696) (200)
107"19 (85"1) 87"39 87"21 86"65 (47"3) 96"8
(0"555) 0.6.54 0"6329 0"6316 0"6275 0"621 (0"653)
(0.701) (2o°) (0.652) (12°) (o.686) (25 °) (0.682) (00) (0-661) (55 °) (0.532) (o.645) (16° )
122"4
(0.650)
(0.635) (160)
76-24 76"22 65-9
0'8848 0.8845 (0.783)
o.5744(20*) o.5853(20*) (0.557)
81"31
0.8281
0-6224(200)
66"37 98"77 98"15 99'06 (68"5)
0.687 (2o') 0.8648 0.8593 0-8673 0.600
(0.620) 080)
69-69 154.68 134.70 134"52 122.54 (55"7) 86.97 87"51 87.48 63"1 69"64 71"92 81"50 (62.2)
0"5277 (O'862) (O'862) (0.861) (0"861) 0"472 0.8680 0.8733 0-8732 (0.875) 0.732 (200) 0"756 (20 °) 0.5581 0.493 0"723 (0.625) (0"639) 0.3921 0.4197 0-4064 0"397 0"592 (320) 0"676 0.648
(O.6O9) (20°) (0"630 (20°) 0-647) (2O°) (O'647) (2O') (0"627) (20°) 0"542) (540) 0.5849(200) o.6o7o(2o*) 0.5876(20*) (0.536) (00) (0..679) (14°) (0.707) (2O') 0-625) (18") (0-657) (20 ° ) (0.612) (210) (0-7S4) (0.730) (0"614) (17") (0"659) (00) (O.660) (30 ° ) (O.668) (24*) (O-6O6)(32'9 0.664 (73") 0"686 (55"5°)
(44.9) 55"1 56"3 65-98 70-53 68-30 (73"1) 52-16 (104"2) (109"7)
0.6164(200)
0.5969(200) 0-6240(20 °)
612
APPENDIX C
(continued) Compound Diphenylchloroarsine l,l-Diphenylethylene 1,6-Diphenylhexane Diphenylmethane 1,1-Diphenylnonane N,N-Diphenyl urea N,N'-Diphenyl urea Di-n-propyl ketone Di-iso-propyl ketone Dipropyl oxalate Dodecyl alcohol Duldtol Elaidic acid Erythritol Ethane 1-Ethoxynaphthalene 2-Ethoxynaphthalene Ethyl acetate Ethyl acetoacetate Ethylacetophenone Ethyl alcohol Ethylallylacetophenone Ethylaniline 9-Ethyl anthracene Ethylbenzene Ethyl benzoate Ethyl benzoylacetate Ethyl bcnzylmalonate Ethyl bromide Ethyl bromoacetate Ethyl butylmalonate Ethy n-butyrate Ethyl iso-butyrate Ethyl chloroacetate Ethyl cinnamate Ethyl isocyanate Ethyl cyanoacetate Ethylcyclohexane Ethyldiallylacetophenone Ethyl dichloroacetate Ethyl diethylacetoacetate Ethyl diethylmalonate Ethylene Ethylene Ethylene bromide
--ZM x 106
--X x 10e
--Z, × l0 s
(145"5) (I18"0) 171.81 (115.7) 206.32 126.3 127.5 80'45 81"14 105"27 147"70 112-40 204.8 73"80 27"37 119'9 119-2 54"10 71"67 95"5 33.60 122"5
0"550 0-655 0-7208 0-688 0.7357 (0"595) (0.6OO) 0"7050 0.7110 0"6046 0"7849(20.7*) 0"617 (0-725) 0.604 (0"910) 0"696) (0"692) 0"614 0"5508 0"644) 0"728 (0.651)
(o.871) (40 °) (0"68O) (14°) 0.6877(200) 0"684 (35.5 °) 0"6935 (2O*) (0"759) (0.743) (20°) (0"576) (20°) (0"573) (20°) (0-628) (00) (0.652) (24°) (0.905) (15 °) (0.619) (79°) (0.876) (20 °)
89.30 153-0 77-20 93"32 (115"3) (154"5)
(0"737) (0.741) 0.7272 0.6211 0-600 0.6172
0.709) (0"771) (990) 0"6341 (20°) 0"648) (25°) (0.673) (20°) (o.663) (2o °)
54"70 (82'8) (74-9) (77"7) 78"32 (72"3) (107"5) (45"6) (67"3) 91"09 147"4
0"502 0"496 0"6442 0"6693 0"6743 0"590 0"610 0"642 0"595 0"8118 (0.646)
0.915 (20°) (0-747) (20°) 0"629) (20°) (O'585) (25°) (0"583) (25°) (0.684) (20°) (0.640) (20°) (0"582) (160) (0"632) (20 ° ) 0"6324(200) (O'636) (160)
85"2 (117"9)
(0-543) 0"6328
(0"690 (2O0) (0"615) (20°)
(140.4)
O.6492
(0.641) (20°)
12"0 15.30 18"80
(0"428) 0-546 (32°) 0"419
(0.511) ( - 1000) (0.738) (20°) (0"734) (25 °) 0.554 (20°) (0"565) (20°) (0"639) (16°) 0-575 (20°) (0"634) (16°)
(0.242) ( - 102 °) (0.3o9) ( - 102 o) 0.915 (20°)
APPENDIX
C
61:
(continued) Compound Ethylene chloride Ethylenediamine Ethylene iodide Ethylene oxide Ethyl ether Ethyl ethylacetoacetate Ethyl ethylbutylmalonate Ethyl ethylpropylmalonate Ethyl formate Ethylidene chloride Ethyl iodide Ethyl iodoacetate Ethyl lactate Ethyl methylacetoacetate Ethyl methyl ketoxime Ethyl methylphenylmalonate Ethyl oxamate Ethyl phenylacetate Ethyl phenylmalonate Ethyl phenylpropionate Ethyl phosphate Ethyl propionate Ethyl propylacetoacetate Ethyl-n-propyl ketone Ethyl sulphite Ethyl thiocyanate Ethyl isothiocyunate Ethyl thiolacetate Ethyl tribromoacetate Ethyl trichloroacetate N-Ethyl urea Ethyl is~valerate Eucalyptol Eugenol and isoeugenol Fluorene Fluorenone
Fluorobenzene
--ZM x 10e
- X x 106
- - Z , x 106
0.602 (32°) 0"771 0 2 °) 0.371 0 2 °) 0"743 0"5937
(0"757) (200) (O'686) (20°) (0'791) (100) (0"618) (7°) 0.534 (20°) (O'582) (2O°)
(163"3)
0.6683
(0"650) (200)
(152.4)
0"6619
(O.648) (200)
(72.6) (81"9)
0.580 O.58O 0.4470(17.50) 0.456 0.615 0.5684
0.531 (20°) (0"681) (20 °) (O.864) (20°) (0.829) (13°) (O'633) (250) (0"569) (200)
57-32
0"6580
(o.53o) (20 °)
(153"2)
0"6121
(o.658) (2o0)
62"0 104-27 (142.2)
(0"529) (0"635) 0-6017
(0-427) (19 °) (O.656) (20°) (0.659) (20 °)
(104.2)
0"598
(o.636) (13 °)
(66.5) (105.7)
0"539 0"6514 0"6135
(o.576) (25°) (0.584) (15o) (0"593) (200)
69.03
0"6891
(0"560) (22 °)
0"546 0.6392(27.4 °) 0.6772(27.4 °) 0.6019(27.4 °) 0.368
(0.604) (0"637) (0.680) (O'586) (0"821)
(99.6)
0"520
(0.719)(20 °)
55"5 (91-1) (116-3) (102.1)
(0"630) 0"754 0"622
(0.764) 0.607) (O.699) (O.663)
(lo8-9) (98.9) (58-4)
0"655 0"549 0"608
(O.788) (O0) O.623 (100 °) (O'623) (20°)
59"62 46"26 104.7 30.7 55"10 (84.4)
43"00 (57 "4) (69"7) (97"6)
(98"2)
(75"4) (55"7) (59"0) (62"7) (119-5)
(O.697)
0.700
(o0)
(25 ° ) (15 °) (25 °)
(200)
(18 o) (20 ° ) (20 ° ) (200)
614
APPENDIX C
(contInued) Compound
.."luorodichloro-
--XM × 10e
--X x 10s
--%, x I0 6
48-8
0"474
(0-676) (0°)
72-5
0.485
(0.742) (25')
58"7
0"427
(o.638) (17°)
(18"6) (21.9) 19"90 42"9
0"62 0"486 0"432 (0"549)
(0.51) (-200) 0"551) (200) 0.527 (20°) (O.452) (2O0)
48"0
(0"614)
(0.505) (2o°)
(0-423) 0"633(200) (O.49o) (0"529) 0"658 (0"563) 0"619 (0.537) 0"624 0"638 0"7096 0.8507 0.789 0.680 (0"808) 0-7625 0"8084 0-6900 (0"801) (0"826) 0"294 0"518 0"476 0"8286 0"671 0"8458
(o.692) (2o0) (0.598) (15°) (0.568) (2O°) (0.896) (4° ) (0"610) (20°) (O'869) (250) 0"779 (2O0) (O-846) (5Oo) 0"698 (20°) (o.72o) (21 °) (0"603) (20°) 0.5817(20 °) (O.647) (20°) 0"626 (200) (0.628) (2O0) 0.6528 (20°) 0.6559(200) (0.630) (250) (0"584) (25°) (0"615) (250) (1.124) (20") (1.o59) (24°) (0-995) (200) o.6421 (2o0) (0"462) (20°) 0"6596(200)
79.20 124.23 (20°) 91"42
0.8654(27.4 °) 0"774 (0.767) 0"7131
(0"565) 0"637 (200) (0"658) (20°) (0"583)
102.58
0"7162
(0"634) (20°)
(78.5) 66-47 66"8
0.664 0.6040 (0-547)
(0"639) (160) (0-797) (20") (0.618) (13~')
methane
Fluorotrichloroethylene Fhorotrichloromethane Formaldehyde Formamide Formic acid Fulvene ('Benzene ~M measured to be 49) Fulvene ( x 54"8 49 / Fumaric acid Furan Furfural Gall~ acid Gcraniol formate D-Glucose Glycerol Glycine Glycol Guaiacol Heptaldehyde n-Heptane 4-Heptanol n-Heptanoic acid n-Heptylamine n-Heptyl benzene Heptyl cyclohexane n-Heptylic acid I -Heptyne 2-Heptyne Hexabromoethane Hexachlorobenzene Hexachloroethane n-Hexadecane 1,5-Hexadiene 2,2,4,7,9,9-Hexamethy[decane 1-Hexane n-Hexyl alcohol n-Hexyl benzene n-Hexyl methyl ketone n-Hexyl methyl ketoxime Hydrindene Hydroquinone 7-Hydroxybenzaldehyde
49"11 43 "09 47"1 90.0 (119"9) 101.5 57.06 40"3 38.80 (79.2) 81"02 85.24 91.5 88.60 93.1 134.41 147-40 89-74 77.0 79-5 (148.o) (147"5) (112"7) 187"63
(55"I) 191.52
(74.6)
APPENDIX C
Compound 4-Hydr0xy-2-buta-
--XM x I0e
(52)
--Z x 10s
0.55
615
-;U, x I0e
(0.573) (14°)
none
Indene (natural) Indene Oynthetic) Iodobenzene Iodoform (in soln.) o-Iodotolueam m-Iodotolmme ~-Iodotoluene Maleic acid Maleic anhydride Malonic acid Mannose Mesitylene o-Methoxybenzalde. hyde p-Methoxybenzalde. hyde o-Methoxybenzyl alcohol l-Methoxynaphthalene Methyl acetate Methyl acetoacetate Methyl alcohol l~lethylamine N-Methylaniline 9-Methylanthracene Methyl benzoate Methyl-o-benzoylbenzoate Methyl bromide 2-Methylbutane 2-Methyl-2-butene Methyl butyl ketone
84-79 80"89 92"00 117"1 (112"2) (112-3) 101-31 49.71 (35"8)
(0.730) (0.696) 0.451 0.2974(20 °) 0"5145(300) 0.5152(30 °) (O.465) (0.42S) 0"365 0-4453 0"571 0.7682(20 °) (0"558)
(0.723) (250) (0.690) (25° ) 0.826 (20 ° ) (1-192) (170) (0.874) (2O0) (0.875) (200) (O.78O) (4O° ) (0.681) (20 °) (0"341) (200) (0.726) (15°)
78"0
(0.572)
(o.642) (2o°)
87"9
0-637
(0.664) (25°)
107-0
(0.676)
(0"741) (14 ° )
42.60 59.60 21.40 (27-0) 82"74 146-5 81"59 139"4
0"575 0"5132 0.668 0"870 (0"773) (0"762) 0"5993 (0"580)
0-537 (20°) (0.553) (2OD 0"530 (20°) (0"608) ( - 110) (0. 762) (200) (0"812) (99°) (0"651) (25 °) (0-690) (19° )
(57"3) 64.40 54"14 (69-1) (69"3)
0.603 0"8925 (0.772) 0.690 0-692
(1.044) (00) 0"5531(20°) (0.516) (13°) (0.563) (1~') (0.554) ('2o°)
(70-4)
0"703
(0-.~2) (16°)
Methyl butymto Mo~yl chlor(moot~t~ Mothylcyclo~e 2-Methyk-yclohexa-
(66.4) 58"I 78-91 (74.0)
0"6498 (0"535) 0.8O38 O-66O
(0.588) (16°) (0.661) (20 °) 0.6181(200) (0.610) ( 1 ~
none 3-Methylcyclohexa.
(74.8)
0.667
(o-61o) (2o0)
(63"5)
0.566
(o.516)(24 °)
70.17 65-10 (46-6)
0.8338 0"375 0.549
0.6245C2O0) 0.935 (2o°) (0.733) (2o°)
Methyl isc~butyl
(46-3) 102.90 92,32 76.0
(0.879) (0.665) (7.0o) (0.632) (20 °)
k~one Methyl ~a'~-butyl
ketone
none
4-Methylcyclohexanone
Methylcyclopentane Methylene bromide Methylene chloride
616
APPENDIX
C
(continued) Compound Methylene iodide Methylene succinic acid Methylethylallylacetophenone Methyl ethyl ketone Methyl formate Methylfumar~ acid 3-Methylheptane 5.Methyl-1,2-hexadiene 2-Methylhexane Methyl hexyl ketone Methyl iodide Methylmalei¢ acid Methyl-o-methoxybenzoate Methyl,o~-methoxyiso-butyrate 1-Methylnaphthalene 2-Methylnaphthalene 4-Methylnonane 4-Methyloctane 2-Methylpentane 3-Methyli~mtane 4-Methyl-2-pentanol Methyl phenylacetate Methyl propionate Methyl-n-propyl ketone Methyl-iso-propyl ketone 1-Methylpyrrole 2-Methylpyrrole Methyl salicylate ~-Methyl styrene 2-Methylthiophene Methyl trichloroacetate N-Methyl urea Morpholine Myristic acid Naphthalene 1-Naphthol 2-Naphthol c~-Naphthonitrile ~-Naphthonitrile ~-Naphthoquinone 1-Naphthylamine 1-Naphthylamine
--XM x 10e 93"10 57"57
- Z x 10e
I
- X , x I0 e
0"348 (O.443)
1.156 (20°) (0.723)
133"3
(0"659)
(O'643) 0 6 °)
45"58 (32"0) 56"98 97"99 73-6
0.6322 0"5327 (0-438) 0"8580 (0.765)
(0.509) (20°) (0.519) (20°) (0.642) 0.6056(2o °) (o.553) (19°)
86"24 (93"3) (57.2) 57"84 95"6
0-8607 0-728 0.403 (0.446)
0.5841 (2o°) (0.596) (20°) (0.918) (20°) (0.721) (o.665) (19°)
(81.9)
0"620
102"15 108"83 121"39 109"63 75"26 75-52
(0-718) (0.765) (o-853) (0-855) 0"8734
(0"575)
(0.736) (14°) (0-743) (20°) (O-625) (2O°) (0.618) (2O0) 0.5705(20D O.5823 (2Oo) (0.641) (20°)
92"73
0-8764 0-788 (0"618)
(55"0) 57"41
0"6240 0-6664
(0"571) (20°)
58"45
0-6790
(O.545) (2O°)
58"56 60.10 86.30 (80.1) 66"35 84.2
0"722 (20°) 0-741 (20°) 0-567 0"678 0-676 (200) (0.475)
(0.664) (100) (O.7OO) 0-668 (2O°) (0"620) (200) (0.689) (200) (O-707) (19°)
44.6 55"0 176"0 (91"9) 95"22 98"25 103"3 101"0 73"5 91"85 I01"8
(0.602) (0.631) (0.771) 0.717 (0.660) (o.682) (0"674) (0"659) (0.465) (0-641) (0"711)
0"725) (0"631) O"661) (60°) (0"821) (2O°) 0"834) (4°) (0"819) (4°) (0"753) (5°) (0"721) (600) (0"661) (0"757) (54°) (0"798) (250)
80"4
(0.645) (160)
(0.541) (15°)
APPENDIX
617
C
(continued) Compound 2-Naphthylamine Nicotine o-Nitroaniline m-Nitroanfline ~-Nitroaniline ~-Nitrobenzaldehyde Nitrobenzene o-Nitrobenzoic acid ~-Nitrobcmzoic acid ~-Nitrobenzoic acid o-Nitrobromobenzene m-Nitrobromobenzene ~-Nitrobromobenzene Nitroethane Nitromethane 1- N i t r ~ p h t h a l e n e o-Ni~ophenol m-Nitrophenol ~-Nitrophenol 2.Nitropropane N-Nitrosodiethylamin¢ p-Nitrosodiethylaniline Nitrosopiperidine o-Nitrotohame m-Nitrotohma¢ p-NRrotoluene
-X~x
l0 s
- Z x 10e
- - ~ , × 106
98"00 113"328 66"47 70"09 66"43 66"57 61"80 76"11 80"22 78"81 87"3 89.5
(0"684) (0"68~) (0.481) (0.507) (o-481)
89"6
(0-444)
(0"859) (22 ° )
(35"4) 21"1 98"47 68 "97 65"91 (67"1) 45.73 59"3
0"472 0.3457 (O-569) (0"496) (0"474) 0.482 0"5135 (0"580)
(0.497) (20°) (0-391) (25 °) (0.696) (62°) (0-873) (20°) (0.750) (o.740) (0-509) (20°) (0.546) (200)
92"6
(0"520)
(0-644) (15 °)
(63-4) 72"28 72"71 72"06
0"555 0"5272 0.5304 0.5257
(0-590) (20°) (0.613) (20°) (0.614) (200) (0.670) (20°)
0"4407 0-5O2 0"4556 0.4802 0.4718 (0"432) (O.443)
(0.720) (98°) (o.7o5) (2o °) (0.694) (150) (0.725) (200) (0.691) (14°) (0.507) (0°) 0-604 (20°) (0.718) (4°) (0.717) (4°) (0.731) (32°)
(0.700) (800) (0-755) (200)
(in soln.) n-Nonane n-Octane Octyl alcohol OctT1 chloride Octylcyclohexane Octylene Oleic acid Oxalic acid Oxamide Palmitic acid Paraldehyde Pentachloroethane 2,3-Pentadiene n-Pentane Perhydroanthracene Phenanthrene Phe~mthrenequinone p-Phenetidine Phenetole Phenol HP,8. 20a
108"13 96"63 102"65 (114"9) 158.09 (89"5) 208-5 60-05 (39-0) 198-6 (86.2) (99"1) 49"1 63"05 146"01 (127"9) 104"5 (96"8)
(84.5) 60"21
0.8431 0.8460 0.77C~(200) 0.773 0-8051 0-798 (0"738) (0.4763) 0"443 (0-775) 0"652 0.490 (0.721) 0"8739 0"7592 0"718 0"502 0.7O6 (25°) 0"692
(0.640)
o.6o57(200) 0.5949(20°)
(0.640) (20°) (O.670) (200) 0-6578(20°) (0"570) (17°) (0'661) (18°) (0"787) (0"738) (0-661) (62°)
(0.648) (20o) (0"819) (25 °) (0"501) (200) 0.5472(20 °) 0.7178(20 °) (0"763) (100°) (0.698) (0,749) (15 °) (0.689) (200) (0.675) (45 °)
618
APPENDIX C
(contOm~d) Compound ?henylacetaldehyde ?henyl acetate ?henylacetic acid ~enylacetylene I-Phenyl-l-butene ?henyl iso-cyanate n-Phenylenediamine ?henyl ether ?henylhydrazine ?henyl mercaptan t-Phenyl-2Methylbutane ?henyl propionate ?henylsulphone ?henyl thiocyanate Phenyl isothic~ cyanate [-Phenyl-4,6,6trimethyl heptane ~r-phcnyl urea ?hthalic acid rerephthalic acid ?hthalic anhydride ?icric acid ?iperidine Propane ?ropene ?ropionaldehyde Propionic acid ?ropionitrile ?ropiophenone ~-Propyl acetate so-Propyl acetate ~-Propyl alcohol so-Propyl alcohol 3-Propylbenzene l-Propyl benzoate ~-Propyl bromide so-Propyl bromide ?ropyl butyrate a-Propyl chloride so-Propylcyclohexane Propylenediamine Propylene oxide Propyl formate ?ropyl iodide ?ropyl propionate (ext#ap.) ?ropyl sulphide Pseudocumene
-ZM x 10• 72"01 82"04 82"72 72"01 93"49
(72.7) 70"53 (108"1) 67"82
- Z x 10e 0"5994 (0.603) (0"608) (0-705) (0.7077) 0.610 0.6529 0"635
(O'627)
- Z , x 10e (0-614) (20.) (O'647) (25") (0"657) (80.) (0.655) (20") (0.6239) (20°) (0"699) (20°) (O'723) (58") (0"681) (20.) (0"688) (23°) (0.693) (20°) (O.66O) (20")
(70"8)
0-6425(27.4°)
113"53
(0.766)
93"79 (129"0) (81.5) (86"0)
(0.625) 0-591
0.6027(27.4") 0.6365(27.4")
(0.654) (25") (0.740) (20.) (0"677) (24 ° ) (0"719) (240)
173.90
(O'796)
(o.682) (20")
82.1 83.61 8~;.51 67.31 .84.38 64.2 40.5 31"5 34-32 43"50 38"5 83"73 65-91 67.04 45.176(20°) 45.794 (200) 89"24 105.00 (65"6) (65-I) (89.4) 56"10 102.65
(0"603) 0"5035 0"5029 (0-454) (0-368)
(O-785) (O'802) (20.)
(0-7621) (0.742) (0.64O) 0"533 0"529 0.6867 0"715 0.8131
(0.477) (20.) 0"582 (20°) (0.547) (21 ° ) (0"631) (20° ) (0"569) (25° ) (0-566) (25° ) (0.6O47) (20.) (0"5985) (20.) (0.640) ( ~ ) ( 0 - ~ ' ) (25=) (0"721) (20.) (0"693) (2O=) (0"6O4) (15 o) 0.633 (20°) 0.6528(20.)
(58.1)
0"784 (0'732) 0"6248 0-4958(30") 0.6711
(0.688) (15°) (0.629) (0.) (0.563) (20.) (0-864) (20D (0.593) (20°)
0-7787(27.4")
(o-634) (17'3 (0.740) (20")
42'5
(55"0) (84.3) (77.95) (92"1) (101.6)
(0.754) (o.919) (0.749) 0"5910 0-586 (O.699) 0.6240 0.6453(250) 0.6564
(0.7518)
0.845 (20.)
(0-759) (0"694) (4") (0"649) (O'65O) (20°) (0"538) ( - 45°)
(0.456) (-47°)
APPENDIX
C
619
(continued) Compound Pyrau~ne Prrcne 1~ridine Pyroeateehol Pyrrole Pyrrolidine Quinotine Quinoo¢ [tesorcinol Rhamnose ~frol and iso-Safrol Salicylaldehyde Salicylicacid
Satigenin Salol Stearic acid Stilbene Styrene Succinic acid Succim'c anhydride Succinimide Sulphamide Terpineol 1,1,2,2-Tetrabromoethane 1,1,2,2.Tetrachloroethane Tetrachloroethylene Tetrahydroquinoline Tetraiodoethylene Tetranitromethane Thiazole Thiophene Tolane Toluene o-Toluidin¢ m-Toluidine ~-Toluidine =.Tolunitrile Tri-iso-amylamine 1,2,3-Tribromopropane Tri.iso-butylamine Trichloroacetic acid (in soin.) Trichloroethylene "IYichloronltromethane Triethylamine Triethyl citrate Triethyl phosphate 20=*
[
--ZM X 10e
--X × 10e
--Z~ x |06
37"6 147"9 49"21 68"76 47"6 54"8 86"0 38"4 67-26 99"20 (97"5) 64"4 72.23 76"9 (123"2) 220-8 (120"0) (68.2) (57"9) (47"5) (47"3) 44-4 111-9 (123.4)
(O.469) 0-731
(89.8)
0"535
(0.856) (200) (o.802) (0.715) (o.922) (o.360) (0.714)
017-9)
0"492 0"668 0"309 0"2195 0"595 (2O0) 0.682 (200) 0.667 o.7176 o.71o (240) 0"697 (250) 0.673 (250) (0.656) 0.845 0"420
(156"8) 73"0
0.846 0"44)
(o.646) (25 °) (0-723) (46°)
65.8 (75"3)
0.501 0"458
(o-734) (200) (0.756) (200)
81.4 (161-9) (125.3)
(o.804)
(0.586) (200 (0-666) (200) (O'735) (20D
81"6 (89-0) (164-3) 43"02 50"55 57"38 (118"9) 66.11 76"O 74"6 72"1 76"87 (156"6)
(0"622) 0.6248 (O.709) (0.771)
(0.666) (0-355) 0"6112 0-605 0.601 (0.527) 0"523 0-620 0"575
(0.776) 0"666 0"655 0"4902 0"475 0"477 (0"462)
(o.725) 0.357
0"586 0-688
(o.484) (61°) (0.933) (o0) (0-611) (20°) (0-857) (15°) (0.688) (20°) (0"567) (230) (0.729) (20°) (0.468) (200) (0.785) (15°)
(O.89O)(200) (0.66) (200) (0.615)(200) (0.755)(200) (0. 720) (25°) 0.678 (45°)
(0.657) (69°) (O.646) (1250) (O.594) (20D (O.767) (15°) (O-524) (O-674) ( 1 ~ (0.832) (0"678) (Room temp.) 0"O58) (200)
(15°) (4°) (200) (2o°) (17°)
(0.726) (2O0) (0.644) 0O00) 0.6179(20D (0. 709) (2O° ) (O-689) (200) (0"7O4) (2O0) (0-666) (180) (0.647) (250) (1"023) (230)
620
APPENDIX
C
(contbmed) Compound
- Z M x lOs
Triethylphosphine Triethyl phosphite Trimethylacetophenone 2,2,3-Trimethylbutane 2,2,3-Trimethylpentane 2,2,4-Trimethylpentane 1,3,5-TrinRrobenzene Triphenylbismuthine rriphenylcarbinol rriphenylmethane rriphenylphosphine rriphenyl phosphite Undecane Urea Urethan iso-Valeraldehyde a-Valerio acid !so-Valeric acid ~Vater Water. (value usually used as standard) ~-Xylene ~-Xylene ~-Xylene gylose
Formula ;OC12 .'NC1 .'C12S .'H2N2 .'H,N2S ;2H,OS ;2H,O2S 32H6SIC12 .'3H5OC1 -'3HsN6 ;aliaS ",tHsO2 ;,tHTOCI 74HsN2S
- Z x lOs
(9o.o) (lO4.8)
- Z , x 10s
0"762 0"631 (0"667)
(0"610) (15°) (0.611) (20°)
88"36
0.8818
0"6086(20 °)
99"86
0"8743
0"6261 (20°)
98.34
0"8610
0"5958 (20°)
108'2
74"55 (196.8) (175"7) (165"6)
(166.8) (183"7) 131.84 33.4
(0'591) (20°)
0.447 0"675 0"678 0"636 0-592 (0.8435)
(0-708) (2O° ) (0.802) (20 o)
0"64 0.668 0,6548 0"663 0.7218(20 °) 0.720 (20°)
66.85 (67.7)
(13.0o) (12"97) 77"78 76"56 76"78 84-80
Phosgene Cyanogen chloride Thiophosgene Cyanamide Thiourea Thioacetic acid Mercaptoacetic acid Dimethyl dichlorosilane Propionyl chloride Melamine Propyl mercaptan Vinylacetate Butyryl chloride N-Allyl thiourea
(0.350)
(O.556)
(57) (57.5)
Compound
(o.648) (16°)
o.644o(2o °)
0-7327 0"7212 0.7232 0"565
- Z M x l0 s
0"686 (100 °) (0.759) (0-701) (180) (0.6247)(20 °) (0"742) (200) (0"63) (21 °) (0.536) (17° ) (0.617) (20°) (0.621) (15 °) (O.72O5)(20o) (0.719) (20°)
0.6235 (20°) 0.6226 (20o) (0.862) (200)
- Z x l0 s
- X , x l0 s
48 32"4 50"6 24"8 42"4 38"4 50"0 82"45
0"485 0-527 0.440 0-590 0.557 0.505 0.543 0"6392
(0.675) (0.642) (0.664) (0.639) (0.782) (0.542) (0-720)
51 61"8 58"5 46.4 62"1 69'0
0"55 0.490 0"768 0"539 0"582 0.595
(0"59) (0"771) (250 °) (0"642) (25 °)
(19 °) (4°) (15°) (20°) (10 °) (20 °)
(0.5o2) (0"598) (20°) (0"725) (20°)
APPENDIX C
621
(contained) Formula
~',I-IgONa ~'sH,tOaN4 ~'5HsO2 ;sH,o0 ;sH, , O N ;sHz,0Na
~.$H12ON2
Compound
-ZM x 10e
- g x 10e
- X , x 10e
Acetone semifarbagone Ethyl disulphide Uric acid Methyl metha¢rylate Cyclopentanol Methyl n-propyl ketoxime Ethyl methyl ketone semi¢arbazone
66"29
0"5758
(0-716) (200)
83"6 66"2 57"3
0.684 0"394 0-572
(0.679) (20o) (o.746) (0"535) (20°)
64.0 68"82
0-743 0"6796
(0.705) (20°) (0.618) (20°)
77.93
0"6034
(0.710) (20°)
N, N,N', N'.Tetra-
75"7
0-652
(0"634) (15°)
84"4
0-417
(0.708) (22°)
84-89 84-06 60"3 59.8 59"8 57"2
0"494 0.489 0"648 0"642 0"642 0"696
(0.780) (20.4°) (0"880) (0.616) (15o) (0-617) (15o) (0"614) (15 °) (0-539) (0°)
118"4
0.568
(0.795)
140-6
0.585
(0.755) (200)
79"97
0-6942
(0.623) (20°)
89"47
0.6249
(0.669) (20°)
90.68
0"6333
(0.730) C~ °)
86"21 79-4 83"56
0"8610 0'777 0"534
(0.601) (2o°) (O-564) (20 ° ) (O.824)
99"4
0"594
(0.843) (20 ° )
81"98
0.648
(0.701) (20°)
~HTCI
n ~ h y l urea 1-Chloro-2,4-dinitrobenzene m-Bromoaniline p-Bromoaniline 2-Methyl pyridine 3-Methyl pyridine ,g-Methyl pyridine 2,3-Dimethybl,3butadiene Tetramethyl thiuram monosulphide Telramethyl thiuram disulphide Methyl-n-butyl ketoxime Methyl-n-propyl ketone semicarbazone Diethyl ketone semicarbazone 3-Ethyl pentane iso-Propyl ether o-Chlorobenzoic acid 2-Mercaptobenzothiazole o-Chlorotoluene
"7H7C1
m-Chlorotoluene
80"07
0.633
(0-679) (20°)
'~7HvCI
p-Chlorotoluene
80"07
0.633
(0"677) (2OO)
~HaNaS ~HgON
N-phenylthiourea o-Anisidine
87"4 80-44
0"574 0.654
(0-75) (0.714) (20°)
'~THgON
m-Anisidine
79"95
0"650
(0.712) (20°)
~sHsO4N2CI ~H6NBr .'6HeNBr ~'eH7N ".eHTN ~eHTN ;eH~o ~eH,zNzSa .'eH,2N2S4 .'eHlaON .'6H:aON3 ".eH,aONa .'6H,4 ;eH140 ~HsO~CI ~'vHsNS2
622
APPENDIX
C
(cona'~ed) Formula
Compound
- Z M x l0 s
- X x l0 s
- Z , x lOs
~HgON
p-Anisidine
80.56
0"655
(O.7O2)(55°)
~HgN
2,4-Dimethyl pyridine 2,6-Dimethyl pyridine Methyl amyl ketoxime Methyl-n-butyl ketone semi¢arbazone 1-Heptanol Phenylacetyl chloride o-Toluic acid 2,4,6-Trimethyl pyridine Hexylacetate Methyl amyl ketone ~micarbazone 3-Methyl-3-ethyl pentane 4-Methyl heptane 3-Ethyl hexane i~3-Quinoline Acetone-2,4-dinitro. phenylhydra-
71"50
0.667
(0.633)(0O)
71"72
0.669
(0.630)(0O)
91.24
0-7062
(0.630)(20 °)
100.40
0"6386
(0-643)(20°)
91"7 88"5
0"790 0.572
(0.649)(20°) (0.668)(20°)
80"83
0"594
83.22
0"687
(0.626) (I 12°) (0.630) (20°)
100"9 112"25
0"700 0-6555
(0.623) (OO) (0.687) (200)
99"9
0"875
(0.623)
97"30 97"76 83"9 110-62
0"8517 0"8558 0"650 0"4644
(0.614) (0.614)(20 °) (0.714)(20 °) (0.653)(20 °)
123 "60
0-6662
0-716)(20 ° )
116"9 93"3
0"810 0"706
(0"667)(O°) (0.685)
124"11
0"4921
(0"631) (20°)
101"2 101"31 116"3 135"16
0"754 0"7549 0"653 0"5076
(0.632) (81D (0.651) (20o) (0.663) (17.5°) (0.670) (20°)
135"70
0"5097
(0-623) (20°)
111"6 116"5
0-733 0.477
(0.659)(16D (0.703)(209
:THgN ~HlsON ~HtsONa ;IHxsO ~HTOCI
:sH802 :sHxxN ~aH,s02 ~sHt7ON3
~8H,s ~H,s ~sH,s ~HTN ~H, oO4N4
zone
~gHxgONa ~H2oO ~toH,2 ~1oH1=OtN4
Methyl-n-hexyl ketone semicarbazone Di-isobutyl ¢arbinol 1,2,3,4-Tetrahydronaphthalene Ethyl methyl ketone-2,4-dinitrophenylhydrazone
hoHl,
:,oH,, :,,Ht402 :,,H1404N4 ;**H,404/q~
Durene see-Butyl benzene Benzyl butyrate Diethyl ketone2,4-dinitrophenylhydrazone Methyl-n-propyl ketone-2,4-dinitrophenylhydrazone
:,2Hs ~,2HsO4N2
Acenaphthylene 2,4'-Dinitro-1,1'biphenyl
APPENDIX C
623
(continued) Formula
C,21-I,C12 CtaI-I,Br= C,2H,02N C,2H, oO. Cz2H,oS CI =Hle04N4
C,2H2~0n C, aH, s0,N4 C,4H,,02 CI,H2oO,N,,
C, sHz~O CIsHIsSiCI
Compound
--~m × I 0e
--7. x 10~
--7, x 10e
4,4'-Dichloro-l,l'biphenyl 4,4'-Dibromo-l,l'biphenyl 2-Nitro-l,1'biphenyl Qnlnhydrone Phenyl sulphide Methyl.n-butyl ketone-2,4-dinitrophenylhydrazone Sucrose
133"1
0"597
(O'859) (2O0)
151"3
0.485
(O-92O) (2O°)
109
0"547
(0.788) (2OD
105 119.2 147"65
0"481 0.640 0.5268
(0.674) (2O0) (0.716) (2O°) (O.643) (2OD
189"1
0-552
(o-877) (15°)
Methyl amyl ketone -2,4-dinitrophenylhydrazone Benzyl benzoate Methyl n-beryl ketone-2,4-dJnitrophenyl_hydra. zone Chak,one Triphe~yl chlorosi]ane
156"84
0.5329
(0.651) (20° )
132"2 171"73
0-622 0"5570
(0.693) (18°) (O.655) (2O0)
125.7 186.7
0-604 0.6333
(o-646) (62°) (0.814)(0 °)
APPENDIX
LINE
FREQUENCIES
INTENSITIES
D
AND
RELATIVE
O F AB2 S P E C T R A
(P. L. Como, Chem. Rev., 60, 363 (1960))
Transition amber
J/~o ~
Intensity
0"05
1.89640 2"09615 2"10361 1"90386
1.02628 0"97622 0"97506 1"02494
0.00000 0.00000 0"00000 0.00000
1-10361 0"90386 0-99254 1-00000
0-04872 - 0-05122 -0"00250 0.00000
0.000oo
2.00000
0.00000
2"00250
1-78632 2"18430 2"21366 1"81568
1.05523 0.95474 0.95049 1.04951
1.67123 2.26433 2.32866 1-73556
1"08701 0"93537 0"92664 1"07336
1"21368 0"81570 0.97064 1"00000
0.09477 - 0.10474 0-00997 0"00000
1.32877 0.73567 0.93568 1.00000
0"13799 0"16036 - 0.02237 0"00000
0-00002
2.00997
0.00011
2"02237
1"55300 2"33643 2"44667 1"66324
1.12170 0-91789 0.90381 1.09619
1.43377 2.40100 2.56548 1.59826
1"15936 0"90212 0"88224 1-11776
0.17830 0-21789 0.03959 0.00000
1-56623 0.59900 0.83552 1.00000
0"21.f~4 - 0"27712 0"06149 0.00000
2.03959
0.00074
2-06149
.//1'o6
Intensity
Frequency
0.00
2"00000 2.00000 2-00000 2-00000
1"00000 1-00000 1-00000 1.00000
4 1 2 3
1"00000 1"00000 1"00000 1",00000
9
8 6 5 7
8 6 5 7
8 6 5 7
0"10
0-20
4 1 2 3
1"44700 0-66357 0"88976 1-00000
9
0.00034
0.15
-
-
-
625
0.25
Frequenfy
-
-
626
APPENDIX D
(contlnued) Transition number 8 6 5 7
J/~o 8 0.30
4 1 2 3
8 6 5 7
0"40
Intensity
Frequency
1-31579 2-45859 2.68284 1"54004
1"20000 0"88788 0"86212 1"13788
1"68421 0"54141 0-77575 1.00000
0"25000 -- 0"33788 - 0"08788 0.00000
0.00137
2.08788
1.09175 2.55529 2.90496 1.44142
1.28990 0.86332 0.82657 1.17343
8 6 5 7
0"50
0-60
Intensity
0"35
1.20117 2.50981 2.79660 1.48796
1-24355 0.87500 0.84355 1.15645
1"79883
0.28145 - 0.40000 -0"11855 0"00000
0"49020 0"71321 1.00000
Frequency
O.00223
2"11855
0"98894 2-59566 3.00656 1"39984
1-33890 0"85272 0"81118 1"18882
0"31010 - 0"46332 -0"15322 0.00000
2-01107 0-40434 0"58910 1.00000
0"33610 - 0-52772 -0"19163 0.00000
0.00329
2-15322
0"00450
2"19163
0"89366 2"63151 3"10054 1"36270
1-39039 0-84307 0"79732 1-20268
0"80641 2"66335 3"18647 1"32953
1-44415 0"83426 0.78489 1"21511
2"10634 0"36850 0"53097 1"00000
0-35961 - 0"59037 - 0"23346 0.00000
2"19359 0-33665 0"47688 1.00000
0"38085 - 0"65926 - 0"27842 0.00000
0"00580
2"23346
0"00712
2-27842
0-72727 2.69169 3"26433 1"29991
1.50000 0-82621 0"77379 1"22621
0"65603 2"71695 3"33436 1"27345
1"55772 0"81882 0"76390 1"23610
2"27273 0.30831 0"42737 1.00000
0-40000 - 0-72621 - 0"32621 0.00000
2"34397 0-28306 0.38258 1.00000
0"41728 -- 0"79382 -- 0"37655 0.00000
0.00840
2-32621
O'00960
2"37655
1.90825 0.44471 0"65033 1"00000
8 6 5 7
J/Vo 8
0"45
0"55
0"65
APPENDIX D
627
(cont/nued) Transition number 8 6 5 7
8 6 5 7
J/"o 0"70
0.8O
Intensity
8
6 5 7
0"90
1"00
J/,od
Intemity
Prequency
0.~
0.53542 2.75968 3 -45 294 1.22868
1.67805 0.80578 0.74728 1.25272 0.44695 - 0.93078 - 0.48383 0.00000
0.59227 2.73950 3"39704 1"24982
1.61714 0.81203 0.75510 1-2449O
2"40773 0"26050 0'34245
0-43286 -- 0"86203 -- 0"42917
1 "00000
0"00000
2.46458 0.24032 0-30674 1.00000
0"01069
2"42917
0.01164
2.48383
0"48487 2-77778 3"50269 1"20978
1"74031 0"80000 0-74031 1"25969
0.43998 2.79405 3.54691 1.19284
1.80376 0.79465 0.73411 1.26589 0.47124 - 1.06965 - 0.59841 0.00800
2'51513
8 6 5 7
Frequency
0.85
0"45969
0"~?~
- 1"00000
0"27509 1"00000
-- 0"54031 0.00000
2-56002 0.20595 0-24714 1-00000
0"01245
2-54031
0.01311
2.59841
0"40014 2"80871 3"58623 1"17766
1"80827 0"78969 0"72858 1-27142
0.36477 2.82196 3.62120 1.16401
1.93372 0.78508 0.72364 1.27636
2-59986 0'19129 0.22249 1"00000
0"48173 -- 1"13969 - 0"65796 0.00000
2.63523 0.17804 0.20076 1-00000
0.49128 - 1.21008 - 0.71879 0.00000
0-01363
2"65796
0.01403
2-71879
0-33333 2"83395 3"65235 1"15174
2"00000 0"78078 0"71922 1"28078
0.88579 2.94338 3-90354 1.04595
3.41421 0.73205 0.68216 1.31784
2.66667 0"16605 0"18160 1-00000
0".50000 -- 1.28078 -- 0"78078 0.00000
2"91421 0.05662 0.03984 1.00000
0.58579 - 2-73205 - 2.14626 0.00000
0"01432
2"78078
0-01067
4.14626
0.95
2.00
628
APPENDIX D
(continued) Transition number 3"00
Intensity
Frequency
0"03699 2.97225 3"95651 1"02125
4.88600 0"71221 0.67379 1.32621
2.96302 0-02775 0.01574 1.00000
Intensity 4-00
Frequency
0.02034 2.98365 3-97543 1.01212
6.37228 0-70156 0.67072 1.32928
0.61400 -- 4-21221 -- 3"59822 0.00000
2'97966 0.01635 0.00822 1.00000
0-62772 - 5-70156 - 5.07384 O.O0000
0.00650
5-59822
0.00423
7.07384
0"01281 2"98925 3"98425 1.00781
7"86421 0"69493 0"66927 P 33073
0.00309 2.99717 3-99605 1"00197
15"34847 0"68115 0.66732 1-33268
2"98719 0"01075 0"00501
0"63579 - 7" 19493 - 6"55914
1.00000
0-00000
2"99691 0-00283 0.00112 1.00000
0"65153 - 14"68115 - 14.02961 0-00(300
9
0"00294
8"55914
0.00086 -
8
0"00000 3"00000 4"00000 1"00000
0"66667 0"66667 1"33333
5"00
6 5 7
3-00000
co
0-00000
0.66667 - oo
0.00000 1.00000
0-00000
0.00000
-
10.00
16.02961
APPENDIX E
LINE FREQUENCIES AND RELATIVE INTENSITIES OF AB3 S P E C T R A (P. L. CORIO, Chem. Rev., 60, 363 (1960))
Transition number
J/¢o ¢5
Intensity
Frequency
10 14
0.00
3"00000 3"00000 2"00000 2"00000 4"00000 3.00000 3"00000 4"00000
1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1-00000 1.00000
1-00000 1-00000 2.00000 2.00000 1-00000 1.00000
0.00000 0.00O00 0.00000 0.00000 0.00000 0.00O00
1.16137 0.86101 2-09988 1.90012 0.94000 1-03765
0.00000 0.00000
2.00000 2.00000 •
0.00000 0.00001
2"02928 1"97946
2"65465 3"25724 1-80099 2"19901 4-39218 2-74271 3-34525 3"60767
1.05826 0.95678 1.05249 0.95249 0.95313 1-04687 O.94836 1.05164
2.45004 3.35710 1.70332 2.29668 4.57449 2.64269 3.54948 3-42481
1"09441 0-93949 1-08059 0"93059 0"93252 1.06748 0"92239 1"07761
1.34535 0.74276 2-19901 1-80099 0.86506
0.14174 0.15678 O-04751 0.05249 - 0.06668 0.03184
1.54996 0-64290 2.29668 1.70332 0.78262 1.02522
0"20559 -- 0"23949 0"06941 0.08059 --0"11151 0"03358
2.06668 1.96816
0.00021 0-00049
2"11151 1-96643
8
13 12 7 11 9
15 16 10 14 8 13 12 7 11 9
0.10
1.04698
15 16
0"00005 0"00009
-
629
J/vo ~5
Inten~ty
0.05
2.83863 3.13899 1-90012 2.09988 4.19900 2-86104 3.16141 3.8O099
0.15
Frequency 1-02697 0-97678 1-02562 0"97562 0-97571 1-02429 0"97448 1"02552 0-07303 0"07678 0-02438 -- 0.02562 - - 0"02928 0-02054 --
630
APPENDIX E
(contOured) TransitionI J/~od number 10 14 8 13 12 7 11 9
0"20
15 16 10 14 8 13 12 7 11 9
0"30
15 16 I0 14 8 13 12 7 11 9
0"40
Intensity
Frequency
2.22942 3-44 115 1"60777 2"39223 4"74226 2"55833 3"76904 3"25567
1.13589 0"92450 1"10990 0"90990 0"91402 1"08598 0"89737 1"10263
1"77058 0"55885 2.39223 1"60777 0"69889 0'97375
J/vo~ ]
Intensity
Fr~cluency
2-0000O 3-51186 1"51493 2.485O7 4"89345 2.48717 3 09641 3"10198
1.18301 0"91144 1-14039 0.89039 0.89758 1.10242 0-87400 1.12600
0-26411 -- 0"32450 0"09010 - 0-10990 --0"16302 0"02559
2.00000 0.48814 2.48507 1.51493 0"61841 0.89802
0"31699 - 0"41144 0.10961 -0-14039 - 0-22045 0.00797
0-00052 0"00155
2-16302 1"97441
0.00098 0.00359
2.22045 1.99203
1"77053 2'57143 1-42530 2"57470 5"02745 2"42703 4.22266 2"96425
1"23589 0.90000 1"17202 0"87202 0"88310 1"11690 O'85279 1"14721
1"54997 3"62177 1"33930 2"66070 5"14475 2"37606 4"43912 2"84216
1"29441 0.88993 1"20474 0"~5474 0.87040 1.12960 0.83408 1"16592
2.22942 0"42857 3"57470 1"42580 0"54398 0"80634
0"36411 - 0"50000 0"12798 - 0" 17202 - 0"28310 - 0'01899
2"45004 0"37823 2.66070 1"33930 0-47702 0"70780
0"40559 - 0.58993 0.14526 - 0.20474 - 0"35026 - 0.05474
0"00155 0"00676
2"28310 2"01899
0"00218 0"01092
2-35026 2"05474
1"34535 3"66448 1.25722 2'74278 5"24658 2"33271 4"63898 2"73493
1"35826 0"88102 1"23852 0"83852 0"85929 1-14071 0"81795 I" 18205
1"16137 3-70088 1"17927 2.82073 5"33451 2"29570 4"81807 2-64151
1-42697 0"87310 1"27329 0.82329 0.84958 1"15042 0"80429 1"19571
2"65465 0"33552
0.44174 - 0'68102
2-83863 0"29912
0'47303 -- 0"77310
0"25
0"35
0"45
APPENDIX E
631
(cont~a~ed) Transition [ number I 3 5 4 2
J/,od I 0.40
Intensity
Frequency
2"74278 1-25722 0-41790 0"61041
0"16148 - 0.23852 - 0"42134 - 0"09857
I J/~'o8 0.45
Intensity
Freque~y
2.82073 1.17927 0-36637 0-51986
0.17671 - 0.27329 -- 0-49579 - 0.14965
0.00342 0"02056
2.49579 2"14965 1"57678 0"85967 1"34564 O'79564 0'83363 1"16637 0"78348 1"21652
0"00281 0"01568 •
2"42134 2.09857
1-00000 3.73205 1"10557 2.89443 5"41025 2.26399 4"97487 2.56066
1.50000 0.86603 1.30902 0-80902 1"15892 0.79289 1.20711
0-86101 3-75888 1"03616 2"96384 5.47545 2"23669 5"10993 2-49106
1 6 3 5 4 -2
3-00000 0.26795 2.89443 1"10557 0.32180 0-43934
0.50000 0-86603 0"19098 0"30902 -0.57313 -- 0"20711
3"13899 0"24112 2"96384 1"03616 0-28343 O-36995
0"52322 -- 0-95967 0-20436 -- 0"34564 - 0"65297 -- 0"27009
15 16
0.00396 0-02513
2"57313 2-20711
0.1)1144"3 0"02906
2.65297 2"27009
0"74276 3"78208 0"97101 3"02899 5"53162 2"21309 5"22503 2"43135
1"65678 0"85394 1"38310 0"78310 0"82709 1"17291 0'77575 1"22425
0-64290 3"80225 0"91002 3"08999 5.58012 2"19261 5"32257 2.38022
1"73949 0.84875 1.42134 0.77134 0"82131 1.17869 0-76943 1-23057
3-25724 0"21792 3"02899 0"97101 0.25046 0"31141
0"54322 -- 1"05394 0-21690 0.38310 -- 0.73496 0"33780
3-35710 0"19775 3.08998 0"91002 0"22214 0"26268
0.56051 - 1-14875 0.228~ -- 0"42134 - 0"81881 -- 0.40955
0.00482 0.03221
2.73496 2"33780
0"00513 0"03453
2-81881 2.40955
0-55885 3-81987 0.85308 3-14692 5"62210
1"82450 0.84403 1-46033 0.76033 0"81620
0"48814 3-83533
1"91144 0"83972 1.50000 0-75000 0.81166
15 16 10 14 8 13 12 7 11 9
10 14 8 13 12 7 11 9
0.50
0.60
15 16 10 14 8 13 12
0.70
0"55
0-84108
-
-
0"65
-
- -
0"75
0.80000 3.20000 5"65857
632
APPENDIX B
(e.ont/nued) Transition number
Intensity
Frequency
2.17476 5.40507 2-33645
1.18380 0"76427 1"23573
6 3 5 4 2
3.44115 0"18013 3"14692 0"85308 0"19777 0.22240
15 16
7 11 9
0.70
1
10 14
0"80
8
13 12 7 11 9
15 16 10 14 8
13 12 7 11 9
0.90
Intensity
Frequency
2.15914 5-47489 2.29893
1-18834 0-76005 1"23995
0.57550 - 1.24403 0.23967 - 0.46033 - 0.90426 - 0.48473
3.51186 0.16467 3-20000 0.80000 0.17676 0.18921
0"58856 - 1"33972 0.25000 -0.50000 -0.99111 - 0"56283
0.00537 0.03608
2.90426 2.48473
0-00553 0-03697
2"99111 2"56283
0-42857 3"84895 0-75061 3"24939 5"69035 2"14541 5"53414 2"26671
2.00000 0"83578 1.54031 0"74031 0"80762 1"19238 0.75660 1.24340
0.37823 3.86100 0.70470 3"29530 5"71816 2.13330 5.58460 2"23896
2"08993 0"83216 1.58122 0"73122 0"80100 1.19600 0.75378 1"24622
3.57143 0"15105 3"24939 0.75061 0"15859 0"16186
0.60000 - 1"43578 0"25969 -- 0-54031 - 1"07918 - 0"64340
3.62177 0.13900 3.29530 0"70470 0.14284 0"13927
0.61007 - 1"53216 0.26878 - 0.58122 - 1-16831 - 0.72609
0.00564 0.03729
3.07918 2"64340
0"00570 0"03717
3-16831 2.72609
0-33552 3.87171 0.66207 3"33793 5-74259 2.12257 5.62777 2.21497
2-18102 0-82882 1"62268 0.72268 0.80074 1.19926 0.75146 1.24854
0"29912 3.88127 0"6225 0 3"37750 5.76413 2"11303 5"66488 2"19416
2"27310 0.82574 1"66466 0"71466 0-79781 1-20219 0"74956 1-25044
3.66448 0"12829 3"33793 0-12912 0"12055
0.61898 - 1"62882 0.27732 -- 0"62268 - 1.25838 - 0"81059
3.70088 0"11873 3"37750 0.62250 0.11714 0-10496
0.62690 - 1.72574 0"28534 - 0.66466 - 1-34928 - 0"89665
0-00572 0'03671
3"25838 2"81059
0.00570 0.03600
3.34928 2"89665
0.66207
15 16
J/,o 0-75
0.85
0"95
APPENDIX E
633
(e.ont/nued) Transition [ number [
J/"o
Intensity
Frequency
Frequency
0.05855 3-96396 0.21115 3.78885 5.93683 2"03259 5"92525 2"04352
4.30278 0.79129 2.61803 0.61803 0"77026 1-22974 0"74122 1"25878
0.26795 3.88982 0"58579 3"41421 5-78319 2"10452 5.69694 2.17604
2-36603 0.82288 1"70711 0"70711 0"79516 1"20484 0-74799 1"25201
1 6 3 5 4 2
3"73205 0.11018 3"41421 0"58579 0.10663 0"09191
0-63397 -- 1"82288 0"29289 - 0"70711 - 1.44091 - 0-98406
3"94145 0"03604 3"78885 0-21115 0-02713 0.01503
0"69722 - 3"79129 0"38197 -- 1-61803 -- 3"35284 -- 2"86433
15 16
0-00566 O'O3511
3.44091 2.984O6
0"00345 0"01620
5-35284 4"86433
0-02434 3"98248 0-10263 3-89737 5"97075 2"01548 5"96729 2"01884
6-28388 0-77872 3-58114 0"58114 0-76266 1-23734 0-74250 1-25750
0-01320 3"98970 0-05972 3-94029 5"98324 2"00898 5-98178 2-01042
8"27492 0"77200 4"56155 0"56155 • 0-75913 1"24087 0-74379 1-25621
3"97566 0-01752 3"89737 0"10263 0.01173 0"00550
0"71612 - 5"77872 0"41886 --2"58114 -- 5"32010 -- 4"82526
3 "98680 0"01030 3"94029 0"05972 0-00646 0"O0278
O'725O8 - 7"77200 0"43845 - 3"56155 - 7"30313 - 6-8O6O5
0"00204 0"00837
7"32010 6"82526
0"00132 0-00502
9"30313 8"80605
0"00826 3"99323 0"03884 3"96116 5"98916 2"005 85 5"98841 2"00660
10"26970 0-76783 5"54951 0"54951 0-75711 1"24289 0"74476 1"25524
0"00197 3-99822 0-00993 3-99007 5-99724 2"00151 5"99714 2-00161
20"25961 0"75914 10"52494 0-52494 0"75335 1"24665 0-74712 1"25288
3"99174 0"00677 3"96116
0"73030 -- 9"76783 0"45049
10 14 8 13 12 7 I1 9
10 14 8 13 12 7 11 9
1"00
3"00
15 16 10 14 8 13 12 7 11 9
5-00
2"00
Inten~ty
4.00
10-00
3-99803 0.00178 3"99007
0"74039 - 19"75914 0"47506
634
APPENDIX E
( ~ ) Transition J/'o8 number 5 4 2
5.00
15 16 10 14 8 13 12 7 11 9
15 16
oo
Intensity
Frequency
0"03884 0"00407 0"00166
-- 4"54951 -- 9"29276 -- 8"79463
0"00091 0"00333
11-29276 10"79463
0.00000 4.00000 0"00000 4"0OO0O 6"00000 2"0000O 6.00000 2.00000
~o 0.75000
4.00000 0.00000 4.0O0OO 0.000(30 0.00000 0.00000
0-75000 - oo O-5000O - ao - oo -
0.00000 0.00000
0"5O0OO 0"75000 1.25000 0.75000 1.25000
oo
J I 10.00
Intensity
Frequency
0.00993 0.00098 0"00036
-- 9.52494 - 19"27163 -- 18.77210
0.00027 0.00089
21.27163 20.77210
APPENDIX
F
LINE FREQUENCIES AND RELATIVE INTENSITIES OF ABI SPECTRA
636
APPENDIX
Transition]
numlaa"
I .r/~o~I Intensity
Frequency
F
Tramition[ I number ~Vo~ I Intensity
0.000
Frequency
0.050 0-00000 0.00O0O 0.00000 0.00000 0.OOO0O 0-00000 0.00000 0.00000 0.00000
1 2 3 4 5 7 8 6
0-82120 0-89172 0-97791 1"08555 1.22365 ~71157 2.97762 3"31082 ~00000
0.10232 0-05597 0.00748 0.04347 - 0.09731 0.05122 O-0O25O 0.04872 0.0O00O
9 6
1.00000 1.00000 1.00000 1.00000 1-00000 3.00000 3.00000 3.00000 2.00000
10 ll 12 13 14 15 16 17 18 19 20 21
4.00000 6.00000 4.00000 6-00000 6.00000 4-OOOOO 6.00000 4.00000 6.00000 6.00000 6-00000 6.00000
-
1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1-00000 1.00000 1.00030
10 11 12 13 14 15 16 17 18 19 20 21
4"17880 6"28708 3" 82119 6"30918 5.71290 4"22364 5.69080 3-77635 6"28843 6"31081 5"71156 5"68918
- 0"97732 0.97632 -1.02368 0-97519 1.02481 - 0"97386 1"02614 - 1.02769 - O-97622 - 0-97506 1.02494 - 1.02628
22 23 24 25
0.00000 0-00000 0.00000 0.00000
-
2.00000 2.00000 2.00000 2.00000
22 23 24 25
0.00001 0.00001 0.00001" 0.00001
-2.05597 -2.00748 1.95653 -2.00250
9 6
0-90547 0-94757 0"99440 1"04679 1"10578 2"85286 2.99438 3"15276 2.00000
0-05060 0"02653 0"00187 - 0.02340 --0.04935 0.02531 0.00062 - 0.02468 0.00000
9 6
0"74621 0-83439 0"95139 1-11390 1-35428 2"57629 2"94999 3-47374 2"00000
0.15503 0.08810 0.01676 0.06002 - 0.14371 0-07770 0-00562 0.07209 0.00000
10 11 12 13 14 15 16 17 18 19 2O 21
4.09453 6" 14697 3.90546 6.15257 5'.85303 4.10578 5-84743 3.89422 6.14714 6.15276 5"85286 5.84724
-
0.98810 0-98782 1.01218 0-98752 1.01248 0"98720 1-01280 1.01315 0.98781 0.98751 1.01249 1.01282
10 11 12 13 14 15 16 17 18 19 20 21
4"25379 6"41940 3"74618 6"46804 5-58050 4"35421 5-53182 3-64572 6.42371 6.47372 5-57627 5"52626
22 23 24 25
0.00000 0.00000 0-00000 0.00000
-
2"02653 2.00187 1.97660 2.00062
22 23 24 25
0.00003 0"00007 0"00007 0"00002
1
2 3 4 5 7 8
9
0.025 1
2 3 4 5 7 8
-
0"075 1
2 3 4 5 7 8
-
-
-
-
0"96753 0-96554 1.03446 0-96312 1.03688 0.96011 1.03989 1.04379 0.96520 0.96271 1"03729 1-04041
2.08810 - 2-01676 1.93998 - 2.00562 -
APPENDIX
F
637
(continued) Transition~-~ number I -''-
Intensity
Frequency
1 2 3 4 5 7 8 9 6
0.67950 0.77716 O'91620 1.12970 1.49795 2-44711 2"91192 3"64104 2-00000
0.20863 0.12273 0-02963 - 0.07286 - 0"18839 0-10474 0.00997 - 0-09466 0-00000
10 11 12 13 14 15 16 17 18
21
4"32050 6-54334 3.67942 6"62722 5-45639 4-49771 5-37234 3-50205 6"55289 6"64097 5"44704 5"35896
-
22 23 24 25
0-00008 0-00020 0"00024 0-00007
-
1 2 3 4 5 7 8 9 6
I
Intensity
0-11111
Frequen~
O-2OO 1 2 3 4 5 7 8 9 6
O'47816
0-43007
0-56913 0-72349 1.04679 2.18991 1.99071 2.66929 4-34100 2.00000
0-28215 0-11443 -- 0.08454 - 0-34689 0-21789 0-03959 - 0-17830 0-00000
0-95863 0-95547 1.04453 0-95144 1"04856 0-94607 1"05393 1-06161 0"95474 0"95049 1"04951 1"05523
10 11 12 13 14 15 16 17 18 19 2O 21
4.52184 6"95271 3-47758 7"22981 5"04459 5"18513 4"76329 2.81009 7.00929 7"33999 4"98971 4.65900
- 0-93007 - 0-92201 - 1.07994 -- 0-91027 -- 1-08973 --0-89077 -- 1.10923 -- 1"15311 -- 0"91789 -- 0.90381 -- 1"09619 - 1-12170
2"12273 2.02963 1.92714 2.00997
22 23 24 25
0-00058 0"00212 0-00478 0"00101
.-
0-56732 0.66761 0-82661 1.I 1786 1"82314 2.20701 2.80703 3-98631 2.OOOOO
0"31810 0-19863 0-08568 - 0-08683 - 0-27198 0.16037 0"02237 - 0-13799 0.0O00O
1 2 3 4 5 7 8 9 6
0.4O682 0-48391 0.61937 0-92811 2"57809 1-79701 2-50654 4-69868 2"00000
0-544O4 0"37198 0.17443 -- 0"06580 --0.41230 0.27712 0.06149 -- 0"21564 0-00000
10 11 12 13 14 15 16 17 18 19 20 21
4.43268 6"76507 3"56705 6"93873 5"23382 4"82171 5-05901 3"17686 6-79299 6.98596 5"20667 5-01369
- - 0-94310
-- 0-93743 -- 1.06257 - 0-92962 - 1"07038 - 0"91787 - 1-08213 -- 1-10302 - 0"93565 - 0.92664 -- 1"07336 -- 1"08701
10 11 12 13 14 15 16 17 18 19 2O 21
4"59318 7-10926 3-40585 7"49086 4-88578 5.56674 4-49381 2"42191 7.20299 7-69645 4"79478 4"30132
-- 0-91904 -- 0-90889 -- 1-09111 -- 0-89357 -- 1-10643 -- 0-86620 -- 1-13380 - 1"21270 -- 0-90212 -- 0-88224 -- 1-11776 -- 1.15936
22 23 24 25
0.00027 0-00084 0-00143 0-00034
-----
22 23 24 25
0"00097 0-00399 0-01134 0.00223
-- 2"37198 -- 2.17443 - 1.93420 - 2"06149
g
2"28215 2"11443 1"91546 2"03959
0"250
0.150
2"19863 2-06568 1"91317 2"02237
638
APPENDIX V
(continued) Intensity
Frequency
1 2 3 4 5 7 8 9 6
0"26393 0-30171 0"36581 3'61803 1"33413 1.95100 5-72474 2.00000
0-89443 0-66902 0-40705 0"07967 - 0-55279 0"46332 0"15322 --0-31010 0.00000
0-86212 1"13788 - 1.20000
10 11 12 13 14 15 16 17 18 19 2O 21
4.73607 7"43436 3.26180 8.07068 4"55305 6"57446 3"87529 1.38197 7"66587 8"71487 4"32425 3"27526
-- 0-89443 -- 0"88017 -- 1-11983 -- 0"85786 - 1"14214 --0.81476 - 1"18524 -- 1-44721 - 0"86332 --0"82657 -- 1"17343 -- 1"28990
2"46698 -- 2.24417 1"96878 2"08788
22 23 24 25
0"00214 0.01046 0.04357 0.00988
-- 2"66902 -- 2-~Y/05 -- 2.07967 - 2"15322
0- 30242 0"35168 0.43781 0-63805 3"31270 1.47059 2.13963 5"39648 2.OOO0O
0"77649 0"56625 0"32216 0"01787 -- 0-51453 0.40000 0-11855 0-28145 0.00000
1 2 3 4 5 7 8 9 6
0-23203 0.26028 0.30606 0"39740 3-87164 1"21301 1"76730 6"03319 2"00000
1"01324 0-77470 0"49768 0-15217 -0-58406 0-52772 0" 19163 - 0"33610 0.00000
10 11 12 13 14 15 16 17 18 19 20 21
4.69758 7.34590 3.30063 7.90988 4.64388 6.28003 4.04925 1.68730 7.52941 8-38978 4.46389 3-60352
--0"90149 0-88827 1"11173 • - 0-86764 -- 1"13236 0-82807 -- 1.17193 -- 1-36047 0.87500 - 0.84355 --1"15645 -- 1"24355
10 11 12 13 14 15 16 17 18 19 2O 21
4"76797 7"50769 3.22960 8"20405 4"47773 6-81881 3.73096 1-12836 7"78699 9-01968 4"19950 2"96681
-- 0"88824 -- (>87322 -- 1"12678 --0.84977 -- 1-15023 --0"80472 -- 1" 19528 -- 1"54094 -- 0"85272 -0-81118 -- 1"18882 -- 1"33890
22 23 24 25
0.00179 0.00843 0.03244 0"00670
-- 2"56625 - 2"32216 -- 2"01787 --2"11855
22 23 24 25
0.00243 0"01215 0-05283 0"01351
- 2-77470 -- 2-49768 --2"15217 -- 2"19163
numTranSi~e~n~
Intensity
Frequency
0-30O
0"400
1 2 3 4 5 7 8 9 6
0-34927 0-41185 0-52272 0-78405 2"96072 1.62425 2.32725 5-05263 2.00000
10 11 12 13 14 15 16 17 18 19 20 21
4.65073 7.23888 3.34788 7"71755 4.75355 5.93969 4"25523 2.03928 7.37575 8.04851 4"62012 3.94737
22 23 24 25
0-00139 0-00619 0-02103 0-00412
0.65918
0-46698 0-24417 - 0-03122 0.46803 0-33788 0"08788 0-25000 0.00000 -
-
0.90962 0.89775 - 1.10225 0-87944 1"12056 0"84517 -- 1-15483 -- 1"28197 -
- -
- -
- -
- -
- -
0"88788
- -
- -
-
- -
-
0-50668
0.450
0.350
-
- -
- -
- -
- -
APPENDIX F
639
(¢ont/nued) Transition I
Intensity
Frequency
0.20536 0.22587 0.25706 @31080 4"07648 1"10549 1"59290 6"31902 2-0000O
1.13278 0.88278 0.59307 0.23346 -- 0.60961 0..59307 0.23346 -- 0.35961 0.00000
10 11 12 13 14 15 16 17 18 19 20 21
4"79464 7"56877 3"20270 8"31437 4,41512 7-01702 3"61272 0.92352 7"89451 9-30161
22 23 24 25
Transition I
Fl'~lmmCy
1 2 3 4
(>16378 0.17316 0.18447 0.19310 4.36852 0.02493 1.28210 6-81818 2.00000
1"37361 1"10468 0.79501 0.41602 - 0.64792 0.72621 0.32621 -0.40000 0-00000
2.68098
-- 0.88278 -- 0.86728 -- 1"13278 -- 0.84307 -- 1-15693 -- 0.79732 1"20268 -- 1"64039 -- 0.84307 -- 0.79732 -- 1.20268 - 1.13904
10 11 12 13 14 15 16 17 18 19 20 21
4.83622 7"66306 3-16084 8"48153 4-31908 7-30336 3"43838 0"63148 8"07507 9"79297 3"89972 2"18182
- 0.87361 -- 0.85746 - 1"14254 -- 0.83287 1"16713 -- 0.78814 -- 1"21186 1"85208 -- ~82621 - 0"77379 -- 1-22621 - 1-50000
0.00265 0-01345 0-05947 @01740
- 2.88278 - 2.59307 -- 2.23M6 -2.23346
22 23 24 25
0.00294 0"01492 0.06517 0.02521
-
1 2 3 4 5 7 8 9 6
0.18288 0..19718 0-21707 0.24401 4"23949 1-00994 1"43065 6.58077 2.00000
1.25294 0.69240 0.32187 - 0.63058 0.65926 0.27842 - 0.38085 0.00000
0"14744 0"15294 0-15781 0.15439 4"47079 0"84917 1"14775 7'03190 2"0O00O
1"49473 1"21792 0.90036 (>51479 - 0.66238 0.79382 0"37654 -- 0.41728 0.O0OOO
10 11 12 13 14 15 16 17 18 19 20 21
4"81712 7"61994 3-18005 .8"40569 4"36288 7.17604 3"51651 0.76051 7.99006 9"55941 3.98858 2"41923
-- 0.87794 --0.86201 -- 1"13799 -- 0.83751 -- 1"16249 --0.79196 -- 1"20804 -- 1.74442 -- 0. 83426 -- 0.78489 -- 1"21511 -- 1"44415
10 11 12 13 14 15 16 17 18 19 20 21
4"85256 7"69962 3"14443 8"54482 4"28216 7"40564 3"37482 0"52921 8"15083 1"00031 3"82035 1-96810
-- 0.86973 -- 0"85347 -- 1.14653 -- 0"82898 -- 1"17102 -- 0.78545 -- 1.21455 -- 1.96262 - 0.81882 -- 0.76390 -- 1"23610 -- 1-55772
22 23 24 25
0.00282 0.01436 0"06345 O-02136
-- 2"99288 --2-69240 -- 2"32187 -- 2"27842
22 23 24 25
0.00302 0.01521 0.06515 0"02881
-- 3"21792 -- 2-90036 -- 2"51479 -- 2"37655
number
O.6OO
O.50O 3 4 5 7 8
9 6
-
Ia/,oaI
Intensity
number l J/ o~
408808
- -
- -
-
3"10468 2"79501 2-41602 2"32621
0-650
0"550
0.99288
640
APPENDIX F
(continued)
i I'o ]
Frtmition I
In~mi~
F_nqua~
Inte~ty
Frequency
0"11055 0.10874 0"10288 0"08445 4.67061 0.66667 0"82528 7-54539 2.00000
1.86015 1"56431 1"22885 0.83092 -- 0"69377 1.00000 0-54031 -0"45969 0"00000
0"800
0"700 I 2
0"13335 0.13581
1 6
0.13593 0.12484 4.55226 0"78151 1.02736 7"22319 2"O0000
1-00801 0-61730 - 0.67455 0.86203 0"42917 - 0"43286 0"000OO
10 11 12 13 14 15 16 17 18 19 20 21
4-86662 7.73081 3"13032 8.59793 4.25087 7"48836 3.32290 0"44774 8.21849 1.01911 3-74944 1-77681
-- 0"86622 -- 0-84994 -- 1.15006 -- 0"82570 -- 1"17430 -- 0"78359 -- 1"21641 - 2.07545 - 0.81203 - 0.75510 - 1.24490 - 1.61714
10 11 12 13 14 15 16 17 18 19 20 21
4-88945 7.78071 3-10751 8.68087 4-20135 7.61132 3.24494 0"32939 8.33333 1.05081 3.62933 1-45461
--0"86015 -- 0"84401 -1"15599 -0"82054 -- 1.17946 -- 0"78153 -- 1"21847 -- 2"30623 -0"80000 -- 0"74031 -- 1"25969 -- 1"74031
23 24 25
0"00305 0.01526 0.06390 0"03207
-
3-33238 3.00801 2.61730 2-42917
22 23 24 25
0"00304 0"01490 0.05928 0.03734
-- 3.56431 -3"22885 -2.83092 -- 2"54031
1 2 3 4 5 7 8 9 6
0"12118 0"12123 0-11787 0"10210 4-61766 0"72097 0"92021 7.39374 2.00000
1.73804 1.44789 1"11761 0"72286 - 0.68490 0.93078 0"48383 - 0.44695 0.00000
I 2 3 4 5 7 8 9 6
0"10124 0"09799 0"09035 O.O706O 4.71387 0"61785 0"74142 7.68005 2.00000
1"98250 1"68151 1"34151 0.94105 -- 0"70145 1.06965 0.59841 -- 0"47124 0"0000O
10 11 12 13 14 15 16 17 18 19 2O 21
4.87882 7.75759 3"11812 8-64278 4.22420 7"55582 3.28024 0"38234 8.27903 1.03588 3.68605 1.60626
- 0.86304 -- 0"84681 -- 1"15319 - 0-82291 - 1.17709 - 0"78234 - 1.21766 - 2.19010 - 0.80578 - 0.74276 - 1.25272 - 1-67805
10 11 12 13 14 15 16 17 18 19 20 21
4.89876 7.80078 3.09823 8.71344 4.18165 7.65740 3.21553 0"28613 8.38215 1.06407 3-57853 1.31995
-- 0"85750 -- O-84107 -1.15849 -0"81849 -1-18151 -- 0-78105 1"21895 -- 2"42355 -- 0"79465 -- 0.73411 -1.26589 -- 1"80376
22 23 24 25
0-00306 0"01515 0"06184 0-03492
- 3"44789 -3-11761 - 2.72286 -- 2.48383
22 23 24 25
0.00301 0-01456 0.05647 0"03932
--3.68151 -- 3"34151 -2.94105 -2-59841
5 7 8 9
1.61622 1.33238
0"750
0 " 8 5 0
APPENDIX
F
641
(continued) rransitio~ J/~o6
Intensity
Frequenoy
0.09303 0"08867
9 6
0"05962 4.74954 0"57386 0"66746 7.79957 2.00000
2.10508 1-79940 1"45538 1"05291 -- 0.70814 1-13969 0"65796 -- 0"48173 0.00000
10 11 12 13 14 15 16 17 18 19 20 21
4"90697 7"81830 3.09007 8.74145 4"16458 7"69599 3.19085 0"25046 8"42614 1(>75867 3.53296 1"20043
ITransition I
Intensity
Frequemcy
1 2 3 4 5 7 8 9 6
0"07931 0"07348 0"06329 0"04368 4"80409 0"49816 0"54479 8.00000 2.00(300
2.35078 2"03692 1.68614 1"28078 -- 0"71922 1-28078 0"78078 - 0"50000 0"00000
-- 0"85508 -- 0"83925 -- 1"16075 -- 0"81673 - 1"18327 -- 0"78080 -- 1"21920 -- 2"54186 -- 0"78969 --0"72858 - 1"27142 -- 1-86827
10 11 12 13 14 15 16 17 18 19 20 21
4.92069 7"84721 3"07647 8"78676 4"13671 7"75632 3"15223 0"19591 8"5O184 1.09571 3.45521 1"00000
--
0"85078 0.83536 1.16464 0"81386 1.18614 0"78078 1.21922 2.78078 O-78O78 0.71922 1.28078 2.00000
22 23
0"04090
---
0.00284 0"01324 0"04777 0"04295
-
4.03692 3.68614 3.28078 2.78078
0"08576 0"08057 0"07089 0"05081 4"77920 0"53413 0.60227 7-90569 2.O0O0O
2-22784 1-91790 1"57031 1"16623 - 0"71402 1.21008 - 0.71879 --'0"49128 0"00OOO
1 2 3 4 5 7 8 9 6
0"02513 0"01962 0"01327 0"00629 4"95967 0.16987 0.11951 8"74264 2.0OOOO
4.82843 4.47418 4.09524 3.68556 - 0-76393 2.73205 2.14626 - 0"58579 0.O0O0O
13 14 15 16 17 18 19 20 21
4"91424 7-83366 3-08286 8-76568 4"14972 7"72858 3-16999 0"22080 8.46587 1"08636 3.492O4 1.09431
- 0.85284 -- 0.83721 -- 1"16279 - 0"81520 - 1-18480 - 0-78072 - 1"21928 -- 2"66098 -- 0"78508 -- 0"72364 -- 1'27636 -- 1.93372
10 11 12 13 14 15 16 17 18 19 20 21
4.97487 7.95525 3"02361 8"94350 4.03724 7"94320 3"03403 0-04033 8"83013 1"17106 3"13785 2"57359
- 0.82843 -0.81732 -- 1-18268 - - 0"80374 -- 1.19626 -- 0.78568 -- 1-21342 -- 5"23607 - - 0"73205 -- 0"68216 -- 1.31784 -- 3"41421
22 23 24 25
0.00290 0.01371 0.05062 0.04210
----
22 23 24 25
0"00152 0"00599 0"01648 0"03202
-- 6"47418 -- 6-09524 -- 5-68556 -- 4" 1 4 6 2 6
number
number [ .11,oa 1.000
0"9OO
1 2 3 4 5 7
0-07981
8
0.00296 0"01416 0"05354
24 25
3.79940 3"45538 3.05291 2"65796
•
24
25
11-950
1 2 3
4 5 7
8 9 6 10 11
12
~-RS.
21
2.000
3"91790 3"57031 3"16623 2.71879
642
APPENDIX F
(continued) rrensifion number
Intensity
Frequency
0"01211 0-00876 0-00539 0.00226 4"98336 0-08324 0"04722 8"88904 2.000O0
7-31971 6"95075 6"56377 6"15584 - 0.77689 4"21221 3"59822 -- 0-61400 0"0OO0O
Transition number
~/,o6
Intensity
Frequency
2 3 4 5 7 8 9 6
0.00465 0-00314 0-00178 0.00068 4"99437 0.03224 0.01502 8-96156 2.00000
12"31220 11"93107 11"53835 11"13291 -- 0-78657 7"19493 6"55914 - 0"63579 0.00000
5.000
3"000 1
10 11 12 13 14 15 16 17 18 19 20 21
4"98790 7"97913 3"01124 8"97462 4"01683 7"97552 3"01438 0-01664 8"91676 11"86954 3.06374 0-11096
--0.81971 -0"81134 -- 1'18866 - 0"80169 -- 1"19831 -- 0"79038 -- 1"20962 -- 7"72311 --0"71221 -- 0"67379 -- 1"32621 --4.88600
10 11 12 13 14 15 16 17 18 19 20 21
4"99535 7"99221 3-00427 8.99081 4"00611 7"99143 3"00495 0"00563 8"96776 11"95274 3"02343 0"03844
-0.81220 - 0"80667 1-19333 0.80061 1-19939 - 0"79395 1-20605 -- 12"71343 0-69493 -- 0-66927 -- 1"33073 - 7.86421
22 23 24 25
0-00087 0-00316 0.00784 0-01951
-- 8"95075 - 8-56377 --8.15584 - 5"59822
22 23 24 25
04)0038 0-00129 0.00294 0"00882
-13-93107 - 13-53835 - 13"13291 --8"55914
0-00709 0-00492 0-00288 0-00114 4"99010 0-04905 0.02466 8-93898 2.00000
9"81507 9"43855 9.04790 8"64142 - 0-78301 5"70156 5.07384 - 0-62772 0.00000
I 2 3 4 5 7 8 9 6
0-00122 0-00078 0-0(3042 0.00015 4"99866 0-00848 0"00336 8.99073 2"00000
24.80625 24.41577 24"01919 23-61623 - 0.79344 14"68115 14.02962 0"65153 0.00~0
10 11 12 13 14 15 16 17 18 19 20 21
4"99291 7.98799 3.00654 8.98567 4.00953 7"98646 3.00787 0-00901 8"95095 11 "92629 3"03636 0.06102
-- 0"81507 0-80840 - 1.19160 -- 0- 80095 -- 1.19905 - 0-79256 -- 1"20744 - 10.21700 - 0"70156 - 0-67072 - 1"32928 -- 6"37228
10 11 12 • 13 14 15 16 17 18 19 20 21
4"99878 7"99800 3"00111 8"99770 4.00153 7"99790 3.00120 0-00134 8"99152 11"98816 3-00591 0-00927
- 0.80625 - 0. 80327 - 1"19673 0-80015 1-19985 0-79688 -- 1"20312 - 25.20656 -0.68115 - 0-66732 1"33268 1"53485
22 23 24 25
0-00055 0.00193 0"00453 0-01269
- 11"43855 - 11.04790 - 10-64142 -- 7"07384
22 23 24 25
0-00011 0-00035 0-00075 0"00257
- 26-41577 26-01919 - 25.61623 -- 16.02961
- -
-
- -
-
- -
lO.O00
4.000
- -
-
-
- -
-
-
-
-
APPENDIX F
643
(continued) TransitiOnnumber
3/v°61 Intensity 0-00000 ~00000 0-00000 0-00000 5.00OO0 0-00000 (~00000 9.00000
Frequency
oo
o~
- 4.00O00
-
2.00000
21"
10 11 12 13 14 15 16 17 18 19 20 21
5.00000 8.00000 3.00000 9-00000 4-00000 8.00000 3.00000 0-130000 9.00000 12.00000 3.0000O 0.00000
22 23 24 25
0.00000 0-00000 0-00000
1.00000
0-00000 -
2.00000 2.00000 2-00000 2.00000 2.00000 0-00000 - 4.00000
- 1-00000 - 1-00000 - 1.00000
oo -
APPENDIX G
LINE FREQUENCIES AND RELATIVE I N T E N S I T I E S O F HALFA~B= S P E C T R A (P. L. C o m o ,
Transition I number I
J/,o
Intensity
Chem. Rev., 60, 363 (1960)).
Frequency
Intensity I Frequency J,
1 2 3 4
0"00
5 6 7
8 9 1
0"10
2 3 4 5 6 7 8
9 1 2 3 4 5 6 7
0.20
8
9 1 2 3
O-30
0"05
2.21487 2"19901 1-98654 4-00000 1"81445 1-98413 1-80099
0.45013 O.45249 O-50224 0.50000 0.54988 0"50274 0"55249
0-00000 0.00000
1-45512 1"55487
0"51188 0"60990
2-73999 2"57470 1"90650 4"0OO0O 1"51843 1"83459 1"42530
O-35400 O'37202 0"51543 0-5OOOO 0"64741 O'52860 0"67202
0.00007 0.00003
1"42094 1"61896
0-00037 0"00012
1"39803 1"69144
3"04200 2-74278 1"85526 4.00O00 1-40074 1-70049 1-25722
0.30975 0-33852 0-52322 O.5OO00 0"69445 0-55381 0"73852
3"35787 2"89443 1"80516 4"00000 1"29754 1"53606 1"10557
0.26928 0"30902 0"52998 0"50000 0.74024 0"58805 0"80902
0"00122 0"00028
1"38678
0"00288 0"00050
1"38732 1"85828
3"67373 3"02899 1"75801
0"23322 0"28310
3.97544 3"14692 1"71648
0"20182 0"26033 0"53694
2.0O0OO
0.5OO00
2-00000 2.00000 4-00000 2.00O00 2"00000 2"00000
0-50000 0.50000 0.50000 0.500O0 0.50000 0-50000
o.ooboo
1.5oooo
0.00000
1.50000
2.46251 2.39223 1.95243 4.00000 1.65527
0-40113 O.40990 0.50792 0.50000 0-59915
1"92969 1"60777
0"15
0"25
1"77148
0.53474 645
0"35
646
APPENDIX G
(continued) tradition I number I y/v°~
Frequency
J/~o
Intensity I Frequency
0"50000 0"78486 0"63145 0"88310
0-35
7
4.00000 1"20752 1"35451 0"97101
4-00000 1-12774 1"17049 0"85308
0-50000 0"82843 0"68371 0"96033
8 9
0.00547 0.00075
1"39941 1"95105
0.00885 0"00099
1"42247 2"04909
7
4"25184 3'24939 1.68151 4.00000 1.05648 0.99634 0"75061
0"17500 0-24031 0"53644 0"50000 0"87113 0.74418 1.04031
4"49640 3"33793 1"65329 4"00000 0"99244 0"84015 0"66207
0"15242 0"22268 0"53340 0"50000 0"91313 0.81197 1"12268
8 9
0"01262 0-00121
1"45563 2.15176
0"01634 0-00138
1.49778 2.25850
4"70725 3"41421 1-63173 4.00000 0.93449 0"70542 0"58579
0"13356 0.20711 0"52814 0.50000 0.95459 0-88607 1.20711
4"88597 3-47988 1"61631 4"00000 0"88162 0"59229 0"52012
0-11786 0"19330 0"52111 0.50000 0"99567 0.96550 1"29330
0"01956 0.00154
1.54777 2"36881
0"02217 0-00162
1"60447 2"48227
7
5"03592 3"53644 1.60632 4.00000 0.83322 0"49884 0-46356
0.10480 0"18102 0.51270 0.50000 1'08648 1.04935 1"38102
5"16124 3"58525 1-60092 4-00000 0"78863 0.497~2 0"41475
0"09390 0.17006 0"50330 0.50000 1"07713 1"13682 1.47006
8 9
0.02402 0.00168
1.66685 2.59853
0"02520 0"00169
1"73402 2"71725
5-26595 3-62747 1.59935 4.00000 0.74740 0"35985 0-37253
0.08474 0.16023 0-49322 0.50000 1.11773 1"22725 1.56023
5"35366 3"66410 1.60086 4.00000 0-70915 0"30881 0-33590
0.07701 0'15139 0.48270 0-50000 1"15833 1"32007 1"65139
0.02579 0"00167
1.80521 2.83819
0.02588
1"87979 2"96110
4
0"30
5 6
1 2 3
0.40
4
5 6
1 2 3 4 5 6 7
1 2
0.50
0"60
5 6
I 2 3 4 5 6 7 8
9
0.70
Intensity
0"45
0"55
0"65
0"75
0.00163
APPENDIX
G
Frequency
,, J/~o ~
Intensity
0.85
5"48990
647
(continued) Transition J/~o number 1
0.80
2 3 4 5 6 7
1
Intensity [ 5.42746 3.69600 1.60483
0.30400
0-07044 0.14340 0.47195 0"50000 1"19900 1-41484 1"74340
0"02562 0"00158
1-95724 3"08579
5"54302 3.74831 1.61804 0.60929 0.20386 0.25169
0-05996 0"12956 0"45033 0-50000 1.28072 1.60880 1.92956
0.02436 0.00143
2.11909 3.33985
4.00000 0"67356 0-26696
0"90
2 3 4 5 6 7
4.00000
1"00
3.00
0.95
Frequency
3"72387 1-61071 4.00000 0-64034 0.23246 0.27613
0"06481 0"13615 0"46113 0"50000 1-23979 1"51118 1"83615
0.02508 0-00151
2"03712 3"21209
5.58894 3.76984 1.62637 4.00000 0.58020 0.17954 0.23016
0"05574 0-12355 0"43965 0.50000 1-32184 1"70745 2"02355
0-02359 0.00135
2"20283 3-46893 0-02232 0.06155 0-27475 0.50000 2.23552 3"84835 4"06155
5-62808
0.05205
3"78885 1"63556 4.0O000 0-55292
O"11803 0"42914 0"5OOOO 1"36316
0-15950 0"21115
1"80693 2.11803
5.90634 . 3.94029 1.81161 4.00000 0.23917 0.03365 0.05972
0"02267 0-00127
2-28812 3"59923
0-00894 0"00029
4"14542 6"35862
5"95830 3"97279 1"89741
0"01434 0"04138 0"19571 0"50000 3"16867 5"88705 6"04138
5-97655 3-98456 1"93738 4.00000 0"07557 0.00797 0"01544
0"01061 0"03113 0"15066 0-50000 4-13014 7"91160 8"03113
0-002.50 0"00003
8"07287 12.19240
5.99625 3.99750 1.98897 4.00000 0.01311 0.00125 0-OO250
0"00418 0"01249 0"06212 0"50000 10-05380 19-96286 20-01249
4"00000
0-12550 0"01441 0.02721 0"00431 0.00008
2-00
4.00
6"09710 9"25143 ,.,,,
5"00
5.98500 3.99007 1.95828 4-00000 0-05002 O.0O506 0.00993
0"00843 0-02494 0.12207 0.50000 5"10556 9"92780 10"02494
10.00
648
APPENDIX G
(continued) transition
Intensity
number 8
9
5.00
I
Fr~ucacy I J]%
0"00163 0.00001
10"05831 15"15543
6.00000 4.O00O0 2.00000 4-00000 0.00000 0-00000 0"00000
0.00000 0.00000 0.00000 0.50000 co oo co
0.00000 0.00000
~o co
10.00
Intensity 0-00041 0.00(300
]
Frequency 20.02916 30.07878
APPENDIX
H
LINE FREQUENCIES AND RELATIVE INTENSITIES O F A3B2 S P E C T R A Transition number f 1 2 3 4 5 6 A~ 7 i 8 9 10 11 12 = 13 i
1'05383 1"05122 0-95365 1"00250 0"95128 I'00000 1"05103 1"00301 0-94894 1"00244 1"04862
0.00000 0.00000 0.00000 0-00000 0.00000 0-00000 0.00000 0.00000 0.00000 0.00000 O.00(X~ 0.00000
2.30919 4-19229 1.71292 3.80771 4.20721 3"79279 2"33634 2-07388 1.73116 1"87457 1.88577 2"07618
0"07117 0"02378 -- 0"07865 0.02494 0"02494 0"02628 0"07478 0"02035 -- 0"07484
0"00000
-0"90788 2"03370 1-98376 - 1"10713 3"00753 1"93067 2"07930 - - 1"00997 - - 1"00250
2"06044 1.47134 3"76534
1"18515 1-16037 0"88053
1"00000 1-00000 1"00000 1"00000 1-00000
2-00000 4-00000 2.00000 4.00000 4.00000 4-00000 2.00000 2.00000 2-00000 2.00000 2.00000 2.00000
7 8 9 10
111 12
0-00000
'I 2 3
1"00000
-
0"00000 0"00000 0"00000 0"00000 0"00000 0"00000 0"00000 0"00000
.5 6 7 8
.9 0.10
21 a
1.00000 1-00000 1"00000 1.00000 1"00000 1"00000 1.00000
2-69081 1-80771 3-28708 1.98508 2-20721 12.00000 3-61692 2-97284 3-33633 3"97054 2"73116 2"98175 4"41251
3-00000 2.00000 3.00000 2.00000 2-00000 12-00000 4.00000 3"00000 3-00000 4-00000 3.00000 3.00000 4.00000
6
HRS.
0.05
0-(30
5
1 2
Frequency
Frequ~ey
1
A,
Intensity
Intensity
2 3
B
J/~o 6
J#'o ff
2.37259 1"63141 3.54341
-
-
1"00000
0.00001
2.00000 2.00000
0"00001 0-00001
1.00000
0.00000
3-00000 2-00000 2.00000
0-00000 0"00001 0"00001
1"00000
0.00001
-- 1-OOOO0 1"11554 1.10474 0"91410 649
0"15
1.60202 0.95147
- -
-
-
0"03027 0"02824 0"02071
650
APPENDIX H
(continued) Transition' number
J/~o~
Intensity
4 5 6 7
0-10
1"94128 2"42736 12.00000 3-27099 2.87275 3.75441 3"88909 2"51488 2"94153 4"83939
1.00997 0.90523 1.00000 1"10331 1.01419 0"89675 1"00910 1"09415 1-00635 0.90666
2.62741 4"36860 1"45659 3"63136 4"42732 3"57264 2"75454 2"10145 1"51492 1"70399 1"75615
A
8 9 I0 11 12 13 1 2 3
B! 45 ~ 6 7 8
9 10 " 11, • 12
2.08504
fl 2 3 .~ 4
0"00015 0"00012 0.00014 0"00003 0.00000 0"00028 0.00007 0"00008 0.00005
6 7 8
9
1
2 3 4 5 6 7
A'
8
•
9 10 11 12 13
0"20
1.76808 1"32714 3"95330 1"77953 2"89400 12"00000 2-69509 2"38893 4"83922 3-64181 2-19304 2-85171 5.65470
s/v,, ,s
~tenfi~
Frequency
0.15
1"87135 2"65754 12"00000 2"96460 2-67892 4"25941 3"77276 2.33849 2"89540 5"25994
1.02237 0.86201 1.00000 1.15583 1.03669 0.84548 1"01810 1.13642 1.01089 0.86651
0"13446 0-04526 - 0"16410 0.04951 0"04951 -- 0"05523 0.14803 0"03312 0"14891 - 0"07185 -- 0-06110 0-03059
2"93957 4"52866 1"23466 3-47111 4"65731 3"34246 3"26018 2.09490 1"33864 1"50530 1"62624 2.00081
0"18985 0.06463 0-25553 O-07336 0"07336 0.08701 0"21763 0.04139 0-22187 0"12517 0.09634 0.02642
0"83304 2.08438 1.98485 - 1"22710 3"03050 1"87525 2"16435 1.03960 - 1.00997
0"00093 0.00046 0.00062 0"00010 0'00000 0"00170 0.00025 0"00034 0.00023
-- 0"77767 2"15113 2"00257 - 1"35793 3"06992 1"83716 2"25086 1.08802 - 1.02237
1"50515 1.19800 4"11024 1"67103 3"13245 12"00000 2"45798 2"03210
1"34650
Frequency
-
1.26235 1"21780 0.85208 1"03959 0"82170 1.000130 1.20794 1-O7329 0.79765 1.02684 1.17551 1"01431 0.83149
0"25
5"45443 3"51396 2"07115 2-81489 6-00914
-
-
1.27718 0"82793 1-06149 0"78436 1.00000 1"25049
1-12573 0"75560 1"03273 1"21163 1-01593 0"80153
APPENDIX
H
651
(continued) Transition I
1 2 3 4 5 6 7
B'
0.20
8
9 10 11 [ 12 rl 2 3 .~ 4 E ~
5 6
7 8
9
A
1 2 3 4 5 6 7 8 9
10 11 12 13 1
2 3 4 5 6 B~
7 8
9 10 11 12 21a*
0-30
Intensity
Fr~lu~acy
3"23193 4-67286 1"04671 3-32648 4"89333 3"10600 3"84190 2"06969 1"19341 1.29841 1.50233 1.81624
0-23765 0"08211 -- 0-35208 - - 0"09619 0"09619 --0"12170 0"28112 0.04859 -- 0-29372 - - 0"18985 --0"13251 0.00549
0"00329 0"00110 0"00159 0"00019 0.00000 0"00597 0"00057 0-00082 0-00067
- 0"74357 2-23263 2"03565 - - 1-49787 3-12702 1-81900 2"33486 - 1"15385 - 1"03959
1"27627 1"08283 4-24027 1-55150 3.36842 12.00000 2.24844 1"65864 6-05064 3.40196 1-96723 2"78667 6-31618
1"43681 1"33788 0.80737 1"08788 0"75000 1.00000 1-31026 1"19428 0"72076 1"03427 1"24498 1"01562 0"77641
3-72374 4"91717 0-75974 3-08008 5-36568 2-63158 5"06234 2"01299 0"96842 0"92371 1-28190 1"26810
0"31319 0.11212 -- 0-55737 - - 0.1[3788 0.13788 -0.20000 0"38227 0.07067 -- 0"43467 - - 0"34911 - 0-20532 --0"09126
025
Intensity
Frequency
3.49485 4.80200 0"88977 3"19652 5.13097 2"86755 4"46083 2-03914 1"07188 1-38730 1"55627
0"27850 0"09780 - 0"45298 -0"11776 0"11776 -0"15936 0-33638 0-05772 - 0-36458 - 0.26494 - 0.16890 - 0"03374
0-00803 0-00192 0.00300 0"00030 0.00000 0.01443 0-00102 0.00144 0-00149
- 0"73162 2"32740 2"08245 - 1"64545 13"20264 1-82212 2.41329 - 1"23542 - 1"06149
1"08169 0"98039 4-34769 1"42642 3"59766 12"00000 2"06194 1-31588 6"58156 3"31352 1-87783 2.76641 6"57539
1"53240 1.40000 0"78976 1"11855 0"71855 1.00000 1"36075 1"27769 0.69329 1"03091 1"27580 1-01355 0.75540
3"91831 5"01961 0"65231 2-97593 5"59319 2"40235 5"59901 1"99743 0"87963 0"77185 1"18499 0"99756
0.34260 0.12500 - 0. 6 6 4 7 6 -0.15645 0-15645 - 0.24355 0.41896 0-08789 - 0-50419 - 0.44096 - 0-24194 - 0.16545
1"10080
0"35
APPENDIX H
652 (continued)
Transition number
J/Uo
Intensity
Frequency
0.30
--
9
0.00275
-
-- 0"74134 2"43387 2"14115 - 1"79940 3"29657 1-84628 2"48456 1"33084 1"08788
0.35
8
0"01483 0"00276 0"00464 0"00038 0"00001 0-02652 0"00157 0"00207
1.63246 1-46332
0"45
'1 2 3 6
~5
~
6 7
I
1 2 3 4 5 6 A~ 7 8 9 10 11 12
0"40
0"91886 0"88942 4"43649 1"30067 3"81650 12"00000 1"89490 1"03032 7"02442 3"25156
0"77460
1-15322 0.68990 !-00000 1.41118 1.37367
0.67235 1"02335 1.30430 1.01004 0.73790
1"80014 2-75314
•
6"79072 i
tt
3 4 5 6 7 9 10 11 I 12
~
~ 5 6 7 8 9
0.50
- 0"77084 2"55055 2"20992 - 1"95871 3"40739 1"88964 2"54866 1"43831 1"11855
0.00659
0"03307 0"00418 0.00879 0"00052 0"00006 0"05905 0.00385 0"00302 0"00901
0"67218 0"73699 4"57143
1"84307 1"59307 0-75000
0"59450
-
0"55
-
-
0"78380 0.80868 4-51012 1.17820 4"02213 12.00000 1.74410 0-8O666 7-37932 3"21495 1.73203 2.74568 6"96805
- 0"81730 2"67609 2.28709 - 2"12253 3-53278 1"94941 2"60672 1-55613 - 1.15322
0.02866 0"00394 0.00769 0.00048 0-00003 0.05108 0.00298 0.00291
1 2 3
0'02232 0"00342 0'00627 0"00045 0.00002 0'03976 0-00222 0"00254 0"00447
4'21620 5"19133 0"48989 2"79967 6"01313 1"97787 6"40536 2"00663 0"73541 0"54154 1"01323
1"99530 0"80267 0"64529 1"09579 0"77108
8
F~u~
0"36754 0"13668 -0"77460 -0"17343 0"17343 -0"28990 0"44761 0-10876 -0"57334 -0-53916 -0"27907 -0.25371
4"08114 5"11058 0"56351 2"88284 5"80991 2'18350 6"04688
1
I 2
Intensity
0"57995 0"67329 4"62277
1"73623 1"52772 0"76146 1-19163 0-66390 1"00000 1"46185 1"47960
0"65669 1"01267 1"33064 1"00541 0"72330 0"38877 0"14728 0"88646 0" 18882 0"18882 0"33890 0"46977 0"13215 --0"64226 0"64250 -0"31703 - 0"35314 -
--
--
-
0"87768 2-80930 2"37122 - 2"29017 3.67016 2"02250 2"66034 1-68283 1"19163
-
---
1"95246 1"65926 0.73995
APPENDIX H
653
(continued) Transition number
A
,#
i
4 5 6 7 8 9 10 11 12 13
Intensity 0.50
1
2 3 4 5 6 7
B
I 8
9 t0 [1 .. L2 1
2 3 ~
5 7
I 8 9
l 1 2 3 4 5 6 A 7 8
9 I0 :II I 12 13 1
0.60
Frequency
3/~o 6
Intensity
Frequency
0.95377 4"38718 12.00000 1-48239 0"50982 7.87721 3.20204 1-61820 2"74356 7"23363
1.27842 0.61915 1.00000 ! .56495 1.71207 0-63631 0-98627 1.37742 0.99406 0.70076
4.42005 5"32671 0"37724 2.65905 6-37294 1.61282 6.90614
-- 0.75000 - 0"35611 -- 0-46107
2"06166 0"62358 0"38863 • 0"86541 0"36363
0.42255 0"16574 1-11495 0.21511 0.21511 0.44415 0.50043 0-18216 0"77988 0.86083 0-39652 - 0"57533
0"03536 0-00421 0"00960 0"00051 0"00007 0"06340 0"00478 0-00301 0"01160
0"94918 2-94918 2'46107 - 2.46107 3"81718 2-10611 2.71107 1"81718 - 1-23346
0.03590 0.00410 0-01010 0.00050 0.00008 0.06474 0-00580 0.00288 0"01424
1"02948 3.09486 2-55566 - 2.63477 3.97185 2"19791 2-76021 1"95812 - 1-27842
0.50354 0"61662 4.66600 0"85473 4.54546 12"00000 1"36828 0"41427 8"04950 3-21663 1"57026 2.74690 7-33265
2"06394 1-72621 0'73107 1"32621 0.60000 1.00000 1"61775 1"83505 0"62972 0"97215 1"39813 0"98781 0"69201
0'43997 0"56611 4"70264 0.76517 4"68793 12-00000 1-26374 0.34184 8"18647 3.24000 1-52721 2-75213 7.41487
2"17717 1"79382 0"72319 1"37655 0"58272 1.00000 1"67155 1"96085
0.98140 0.68453
4"49646 5"38338
0.43606 0-17379
4.56003 5-43389
0.44783 0-18118
1"06193 4"21268 12"00000 1"60713 0"63702 7"65848 3-20005 1-67178 2"74285 7"11376
1"23346 0.64039 1.00000 1.51304 1.59307 0.64503 1.00000 1.35496 1.00000 0.71107
4"32782 5"26301 0"42857 2.72539 6"20108 1"78732 6.68659 2.02969 0.67612 0"45715 0"93656 0"46188
0.40693 0"15693 -
0.55
1.00000
- 0.20268 0"20268 - 0"39O39 0"48696 0"15693 -
0"71107
-
0-65
-
0-62468 0.95812 1.41720
654-
APPENDIX H
(continued) 1
~amitionnumberI J/*o I 3
i4
0.6O
5 6 7 8
B,
i 9
i L° 12 1
2 3 4 6
7 8
L 9 1 2 3 4 5 6 A 7 8 9 10 11 12 I 13
I
Frequen~
J/~o
Intensity
0-33400 2"59981 6"52865 1-45455 7"07832 2"10005 0-57674 0"33288 0-79920 0.29082
1-23107 -0"22621 0-22621 -0"50000 0"51114 0"20717 -0"84875 -097433 -0"43843 -0"69419
0.65
0.29736 2.54690 6.66873 1.31207 7.21461 2.14250 0.53484 0.28730 0.73766 0.23635
- 1-34819 - 0.23610 0-23610 - 0-55772 0.51978 0.23150 - 0.91773 - 1-08999 - 0.48192 -- 0.81639
0"03520 0-00387 0"01036 0.00047 0.00009 0-06394 0"00686 0-00270 0"01681
-- 1"11675 3"24564 2"65415 --2.81089 4-13263 2.29607 ~80871 -2.10478 - - 1.32621
0.03373 0.00359 0.01040 0.00046 0-00010 0.06177 0.00799 0.00245 0"01921
- 1.20957 3.40089 2.75591 - 2"98911 4"29832 2"39919 2"85725 -- 2"25644 -- 1"37655
0"38680 0"52101 4-73387 0.68491
2.29186 1.86203 0.71616 1-42917 0.56714 1.00000 1.72640 2"08859 0"62076 0"94448 1.43476 0.97494 0"67808
0-34208 0.48064 4"76065 0"61347 4"92916 12.00000 1"07993 0-24290 8-38625 3.30171 1-45322 2.76609 7"54147
2"40776 1"93078 0.70985 1"48383 0.55305 1"000(30 1"78234 2"21766 0"61767 0-93141 1"45091 0.96853 0"67248
- 0"98686 - 1"20737 --0"52704 - 0"94097
4"65793 5"51936 0"23935 2"45737 6.90588 1"07084 7"41208 2.23240 0"46315 0"21879 0-62769 0"16336
0"46724 0"19422 -- 1-58485 -- 0.25272 0.25272 -- 0.67805 0.53276 0"27714 -- 1"05617 -- 1.32616 -- 0"57379 -- 1"06723
1.30685 3.56011
0"02974 0.00294
-- 1-40776 3.72286
4'81546 0.28623 8.29662
3"26911 1"48837 2"75867 7"48362
9 O A L2
2
0"03183 0"00327
6
7
8
-
12"00000 1"16788
4"61320 5.479O0 0"26613 2"49963 6.79409 1"18454 7.32371 2"18709 0"49716 0"24982 0"68057 0"19509
1
2 3 4 5 B I
0.70
Intensity
0-45814 0-18797 - 1"46616 - 0"24490 0"24490 - 0"61714 0"52686 0.25488
--
O.75
I
Frequency
APPENDIX
H
655
(continued) Transition number
Frequency
J/~o ~
Intensity
2"86039 3"16918 4"46801 2"50621 2"90628 - 2"41249 -- 1"42917
0.75
0.01010 0.00039 0.00010 0"05548 0-01023 0.00197 0"02328
2"96717 - 3.35086 4"64101 2"61631 2-95611 - 2.57243 1.48383
0"85
9 I0 11 12 13
4"78375 O'55O19 5"03026 12"00000 0-99926 0.20864 8.46000 3"33609 1-42133 2"77405 7.59049
2"52470 2"00000 0"70416 1-54031 0"54031 1.00000 1-83936 2"34759 0-61521 0"91903 1"46576 0"96223 0"66758
0"27200 0"41190 4-80379 0"49428 5"12004 12"00000 0"92530 0-18117 8"52134 3-37101 1"39232 2"78228 7-63231
2.64250 2"06965 0.69901 1"59841 0-52876 1-00000 1"89745 2"47806 0"61322 0-90738 1.47940 0"95608 0-66328
1 2 3 4 5 6 7 8 9 10 11 12
4"69578 5"55556 0"21625 2-41956 7-00537 0-96974 7-48451 2"27734 O'43238 0"19290 0.57883 0"13858
0-47531 0"20000 -- 1"70416 -- 0"25969 0"25969 - 0"74031 0-53774 0-29820 - 1-12566 - - 1-44609 0-62213 - 1-19464
4"72800 5"58810 0-19621 2"38569 7-09382 0-87997 " 7"54450 2"32114 0"40444 0"17115 0"53379 0"11894
0-48250 0"20535 - - 1"82401 0"26589 0"26589 - 0-80376 0.54200 0"31805 -- 1.19534 -- 1.56694 - 0"67202 -- 1.32284
0-02762 0.OO263 O-OO982 0.00036 0.00010 0"05203 0"01132 0"00173 0"02489
- 1.51165 3-88875 3"07590 3"53398 4"81677 2"72888 3 -00692 - 2-73580 -- 1"54031
0"02556 0"00233 0~349 0-00034 0"00009 0"04863 0.01236 0-00152 0.02622
- 1"61801 4-05745 3-18630 - 3"71836 4"99486 2"84344 3.05880 - 2-90224 - 1"59841
0.22061 0-35609 4"38657 0.40151 5.27046 12.00000
2.88025 2-21008 0-69005 1-71879 0.50872 1.00000
3"/I'o
6
0.70
3 4 5 6 8 9 I 2 3 4 5 6 A ,I 7
0.80
i 8
B
1 2 3 ~
5 6 7 8
, 9
A.
f
1
~
2 3 4
i 5 l 6
0"9O
Intensity 0-01030 0-00043 0-00010 0.05882 0"00911 0"O0221 0"02138
--
0.3O422 O.AAAAA
O-24439 0"38258 4"82126 O-44497 5-19972 12"00000
- -
2-76106 2.13969 0.69433 1"65796 0.50827 1.00000
0.95
F~u~
APPENDIX H
656
(continued) Transition I number I 7 8 9 10 11 12 13
A
J/~o~
Intensity
Frequency
']'/~o ~
Intensity
Frequency
0.90
0"85753 0"15884 8.57286 3.40561 1.36587 2.79061 7.66821
1"95657 2"60881 0"61159 0.89648 1"49193 0"95010 0.65946
0-95
0.79545 0"14046 8"61654 3.43928 1'34172 2"79889 7.69923
2-01667 2"73969 0.61024 0.88631 1"50344 0-94431 0.65606
4-75561 5"61743 0.17874 2.35531 7-17245 0.80028 7.59471 2.36326 0.37900 0.15274 0.49238 0-10315
0.48894 0.21031 1.94433 0.27142 0.27142 0"86827 0-54568 0.33670 1-26523 1.68856 0-72339 1.45154
4-77940 5"64391 0-16343 2.32803 7.24240 0-72954 7.63713 2.40337 0.35577 0-13706 0.45439 0.09029
-
0.49475 0.21492 2"06505 0.27636 0"27636 0"93372 0.54889 0"35418 1.33531 1.81080 0"77618 1"58055
0.02360 0.OO2O6 0.OO913 0.00031 0.00009 0"04536 0-01334 0.OO133 0.02727
- 1"72643
0.02178 0.00181 0.00876 0.00028 0.00008 0.04228 0.01425 0.00116 0.02806
-
0.20000 0-33211 4.85005 0.36319 5.33334 12.00000 0.73859 0.12515 8.65386 3-47168 1.31962 2.80704 7-72619
3"OOOOO 2-28078 0"68614 1"78078 O'5O0OO I'OOOOO 2.07772 2"87056 0.60911
4"80000 5"66789 0"14995 2"30347 7"30470
1
2 3 4 5 I
6 7
B
8 9
10 11 [ 12 1
2 3 ,
5
~
6 7 8 9 1
2 3 4 5 ! I
6
A, I 7 8
9 10 11 12 13 1
2 B
3
4 5
I'OO
-
-
-
4.22868 3"29812 3"90388 5"17493 2"95961 3"11180 - 3.07141 - 1"65796 -
-
-
--
1.83661 4.40218 3.41117 - 4.90042 5.31673 3.07711 3.16594 - 3.24304 - 1-71879
0.87687 1.51402 0.93872 0"65302
0.05051 0-11325 4-95677 0"07967 5"82843 12.0OOO0 0-21674 0-02886 8.91359 3-82280 1.10843 2.91365 7.92684
5"44949 3.73205 0"64575 3-14626 0.41421 1.0OOOO 3.42349 5-44949 0.60202 0.78246 1"61552 0.36329 0.62531
0.50000 0.21922 - 2.18614 -0-28078 0.28078
4"94949 5.88675 0.04323 2.09190 7.80708
0-55051 0-26795 -4.64575 -0"31784 0-31784
2.00
APPENDIX H
657
(continued) Tramition number '
6
I
J/~o 1.00
7 8 B
~
9
10 11 12 f
1 2 3 5 6 7
I
8
, 9 1
2 3 4 •
5
6 7
A
8
9 10 11 12 13 1 2 3 4 5 6 B I 7
189 tO tl ~ [2
I
1 2 4 5
3.00
Intensity
Frequency
0"66667 7-67326 2-44131 0"33453 0"12361 0"41955 0"07969
1"00000 0-55173 0"37046 - 1"40560 1"93356 - 0"83O3O - 1.70972
0-02010 0"00159 0-00838 0'00026 0"00008 0'03942 0"01508 0.00101 0-02863
- 1.94827 4-57:772 3-52530 - 4-27788 5-54OO1 3"19570 3-22118 - 3"41688 - - 1-78078
0-0:'~":'~ 0-05550 4"98000 0-08148 - 5"92603 "12-00000 0"09497 0"01272 8"96163 3"91785 1"05170 2"95387 7"96693
7"93273 5"21221 0-63104 4"59822 0"38600 1"000(30 4"86824 7"98019 0-60087 0-75671 1.64241 O-82687 0.61651
4"97778 5-94451 0-02000 2"04249 7-91303 0-07397 7"96503 2-92485 0"06437 O'O1306 0"04329 0"00704 0"00239 0"00003 0"00176 0"00001 0.00001
J#'o
Intensity
Frequency
0"17157 7"92048 2"82828 0-12613 0"02989 0-10837 0"01661
- 2"41421 0"57651 0"55051 - 2.84275 - 4"42822 - 2-09052 - 4"27096
0"00550 0.0O014 0-00345 0.00005 0.00002 0-01237 0"01773 0-00008 0"02135
- 4"32247 8.32247 5.91671 - 8"13425 9.36148 5"65545 4"48850 - 7"14395 -- 3"14626
0"01240 0"03270 4"98855 0-01644 5"95932 12-00000 0.05225 0-00715 8"97836 3"95328 1"02984 2"97172 7"98116
10-42443 6"70156 0"62348 .6-07384 0"37228 1"00000 6"34128 10.49496 O'60049 0"74681 1"65270 0"80613 O'61228
0"56727 0"28779 -- 7"13104 - 0"32621 0"32621 - 3-88600 0"5843O 0"61473 -4"31234 - - 6"93522 - 3"49680 -- 6"79502
4"98760 5"96730 0-01145 2.02424 7-95087 0"04068 7-98044 2"95883 0"03871 0"00738 0"02242 O'00382
0"57557 0"29844 -- 9-62348 - 0.32928 0-32928 - 5"37228 0"58819 0"64610 - 5"79493 - 9"44082 - 4-94837 - 9"30628
--6-78116 12"23370 8"37860 -- 12"07443 13"29181
0.00132 0.00001 0.00103 0.000(30 0.00000
- 9"26067 16"19014 10"85923 -- 16"04188 17"25465
-
2.00
4.00
658
APPENDIX
H
(continued) Transition number 6
J/Vo ,~
Intensity
Frequency
3.00
0"00578 0.01268 0"00001 0"01301
8"15260 5"89592 --11"04852 -4"59822
5.00
0"00789 0"02150 4-99260 0"01001 5"97437 12"00000 0"03284 0"00458 8"98615 3"96996 1"01933 2.98097 7"98788
12"91948 8"19493 0"61887 7"55914 0"36421 1"00000 7.82550 13.00345 0.60031 0.74206 1"65763 0.79284 0.60977
4"99211 5"97851 0"00741 2"01562 7'96850 0"02563 7"98753 2"97422 0.02576 0.00471 0.01352 0.00241
7 8
1 2 3 4 5 6 A 7 8 9 10 11 ! 12 i. 13 1 2 3 4 5 6
B]
7
18
9 10 11 12
f
1 2 3 4 6 7 8
t
9
1
2
i 34 A
5 6 7 8
Intensity
Frequency
4.00
0"00340 0.00888 0.00000 0.00846
10.65358 7.34788 - 15.00146 - 6.07384
I0"00
0"00195 0"00566 4-99811 0"00224 5-99382 12"00000 0'00784 0.00115 8.99656 3.99243 1"00491 2"99477 7.99698
25-4O967 15.68115 0.60952 15.02961 0.34847 1-00000 15.29514 25.51929 0.60008 0.73555 1.66437 0.76436 0.60484
0"58052 0"30507 12-11887 - 0.33"07.3 0-33073 6.86421 0.59053 0.66448 - 7-28374 - 11.94489 - - 6.41896 11-81261
4"99805 5-99435 0.00189 2-00394 7.99211 0-00618 7-99690 2.99390 0.00693 0.00113 0.00293 0.00061
0.59033 0"31885 -- 24.60952 - - 0"33268 0"33268 -- 14.34847 0.59524 0-69995 -- 14"75966 - 24.45469 - - 13.85965 -- 24-32398
0"00083 0"00001 0"00068 0.00000 0"00000 0"00221 0"00643 0-00000 0"00588
--11.74843 20-16446 13"34751 -20"2148 21.21357 13"15498 8"81865 18"97363 - 7.55914
0.00020 0-00000 0.00019 0.00000 0.00000 0.00054 0.00202 0.00000 0.00172
-- 24.22411 40-11449 25"82388 -- 39.97870 41"18362 25.65960 16"25956 -- 38.91917 -- 15.02961
0-00(300 0-00000 5-00000 0.00000 6.00000 12.00000 0.00000 0.00000
co oo
0.60000 oo
0.33333 1-00000 oo co
APPENDIX
659
H
(continued) Pransition number
91 10 11 12 13
A:
[
1 2 3 4 5
i 6 B] 7 8 9 lO I1 I 12 {
1 2 3
.5
t
5 7 8 9
J/~o 6
Intensity
Frequency
9-00000 4.00000 1"00000 3.00000 8.00000
0.60000 0.73333 1.66667 0.73333 0.60000
5.00000 6.00000 0.00000 2.00000 8.00000 0.00000 8.00000 3-00000 0-00000 0.00000 0.00000 0.00000
0.60000 0"33333
0.00000 0.00000 0.00000 0-00000 0.00000 0.000~0 0-00000 0.00000 0-00000
- - 0 0
--0-33333
0.33333 0.60000 0"73333 --
OO
--
00
--
OO
--
OO
--
CO CO
--
CO
--
OO OO CO CO
--
OO
--
OO
Intensity
Frequency
APPENDIX ]
MISCELLANEOUS SPIN SYSTEMS-NOTES AND LITERATURE REFERENCES ABe XH resonance spectra of isopropyl derivatives: J. RA~wr, Ann. Physik, 10, 1 (1962). A~B~ The ~H resonance spectrum of propane; the "composite particle" method of analysis was used: D. R. WmTMAN,L. ONSAGER,M. SAUNDERSand H. E. Duns, J. Chem. Phys., 32, 67 (1960). A B T T h e 1H resonance spectra of substituted thiophenes: R. A. H O ~ A N and S. GRONO" WlTZ, Arkivfor Kemi, 16, 50 (1960). Here T is a magnetic nucleus of spin value of I ; it is pointed out that the spin system may be treated as ABX2 by ignoring all spin functions which are antisymmetric in the two X nuclei. The only difference between the observed spectra of AB X2 and ABT is one of relative intensifies of bands. X3AA'X~s(i) (Jxx' -- 0) IH spectra of 2,3-disubstituted butanes: F. A. L. ANEr, J. Amer. Chem. Sot., 84, 747 (1962). (ii) AISodone for ethane enriched with two lsC nuclei: R. M. LYNDEN-B~LL,MoI. Phys., 6, 601 (1963). XnAA'X'(Jxx' ffi 0) General expressions have been derived for line positions and intensities: R. K. ~ , Can. J. Chem., 42, 2275 (1964). ABX3 The 1H spectrum of 1-chlorobutadiene-l,2 : S. L. MANATTand D. D. ELL~.~, J. Amer. Chem. Soc., 84, 1579 (1962)..Saturated aliphatic compounds: F. A. L. ANL~T,Can. J. Chem., 39, 2262 (1961). 2-substituted propenes: E. B. WHnn'LE, J. H. GOLDSTE[Nand L. M~'D~.L, J. Amer. Chem. Soc., 82, 3010 (1960). Cis and trans crotonic acids and methyl esters and also N-methylformamide: V. J. KOWAL~WSKIand D. G. DE KOWAL~VSK~,J. Chem. Phys., 33, 1794 (1960). ABCs Perturbation theory has been used to extend the ABXs analysis to ABCs spin systems and the results were applied to the XH resonance spectrum of tr~s-propenylbcnzene (trans-ff-methylstyrcoe): R. W. FL~SV.NDn~and J. S. WAUGH,J. Chem. Phys., 30, 944 (1959). 9F resonance spectra halogenated propenes: J. D. SWALENand C. A. ~ Y , J. Chem. Phys., 34, 2122 (1961). Here the approximate values of the chemical shifts and coupling constants obtained from an ABXs analysis were used to set up trial matrices which were then modified by an iterative process to find the convergent solutions corresponding to the measured eigcnvalues of the system. Theoretically, the relative signs of all three coupling constants can be derived from such an analysis. A B C X Complete analysis made of the XH resonance spectra of the isomers of CHzFCI-IBrCOOCH3 : J. B. STOTH~S, J. D. TAL~&~Nand R. R. Fl~S~t, Can. J. Chem., 42, 1530 (1964). A B P X 1 9 F and IH resonance spectra of halogenated ethanes: J. L~E and L. H. StrrCLIFFE, Trans. Faraday Soc., 54, 308 (1958).*9F and XH spectra of 1-fluoro-2,4-dinitrobenzene :B. D. N. R.AO and P. VENKXT£SWARLU,Prac. Indian Acad. Sci., 52, 109 (1960); J. Sci. and Industrial Res., 20B, 501 (1961). ; B. D. N. RAO, Mol. Phys., 7, 307 (1964). A B X ] r 1H resonance spectrum of ~-picoline: B . D . N . RAo and P. VENKAIT.SWARLU, Proc. Indian Acad. Sci., 54, 305 (1961). R. J. AngAH~ and H. J. BEI~Slam~, Can. J. Chem., 39, 216 (1961). RIGOS has shown that the 4 x 4 secular determinant in the A B X Y analysis can be solved exa~-tly under one of three limiting conditions; the analysis was applied to the tH spectrum at 60 Mc sec-1 of the pyridine ring in nicotine. N. V. RIGGS, Austrah'an ?. Chem., 16, 521 (1963). 661
662
~PPENDIX I
AB2X The tH resonanc~ spectrum of meta-dinitrobenzene: R. J. A a ~ , E. O. BisHoP and R. E. RICHARDS,MoL Phys., 3, 485 (1960). ABaC Explicit expressions for the transition enersies and relative intensities cannot be written down for this system: a computer can be used to obtain convergence on to the correct eigenvalues of the 3 x 3 and 4 x 4 subnmtric~ encountered in the analysis. A B 2 X trial values of the spectral parameters are used initiallyto set up an approximate matrix which is then sue,ce~ively modified until its roots correspond with the observed eigenvalues. The relative signs of all the coupling constants can be obtained. AA~KL and A B K L tH resonance spectra of 2-pyridines: V. J. KowAt~wsru and D. G. D~ KOWALeWSKX,J. Chem. Phys., 37, 2603 (1962). AB2X2 The 19F and alp resonance spectra of triphosphonitrilic~-l,l-difluoride-3,3,5,5-tetrachloride (PaNaCLtF2): M. L. I-IEF~AN and R. F. M. WHrtm, J. Chem. See., 1382 (1961). AB, tX19F spectrum ofpentafluorosulphur hypofluorite (SFs.OF): R. K. HARMSand K. L PACKER, J'. Chem. So¢. 3077 (1962). A B e X tH spectra of isopropyl compounds: J. RAI,n~r, Ann. Phy~ik, 10, 1 (1962). A A ' P P ' X The analysis has been devised for spin systems containing different isotopes: S. MATSUOg.A and S. HATroPs, Sol. Reports Kanazawa Univ., 6, 33 (1958). The tH resonance spectra oftaC satellites in ethylene oxide, ethylene imine and ethylene sulphide: F. S. MORTXM ~ J. Mol. Spect., 5, 199 (1960). AA'XsXs' The tH resonance spectra of 2,3-disubstituted n-butanes: A. A. BoTm,mR-BY and C. N ~ Cot~, J. Amer. Chem. Soc~, 84, 743 (1962). A A ' A " A " ' X X ' tH and 19F resonance spectra of p-dilluorobenzene: S. MATSUOKAand S. I - ~ r o m , Sci. Reports Kanazawa Umv., (1959). Lynden-Bell has tabulated the matrix elements of the hue,lear spin Hamiltonian for this system: it occurs inpara.difluorobenzeneor ethylene having two xaC atoms. The relative signs of the two .fAxconstants can be determined. Also the relative signs of JAA and Jxx can be determined from the X but not the A part of the spe~rum. R. M. LYNDEN-BeJ.~ MoL Phys., 6, 601 (1963). A A' A"Xa Xa' Xa" The t H and t 9F resonancesof the 1,3,5-trisu bstituted benzenes (C Ha)aCell a and (CFa)aCeHs: J. V. Acmvos, Mol. Phys., 5, 1 (1962). The high symmetry of the molecules enabled the group theoretical method of analysis to be used (see E. B. WItaON, J. Chem. Phys., 27, 60 (1957). A2,42'X with strong cross-coupling. The tH spectrum at 60 Mc sec-x of the ring of cyclopropylaminecanbe simplified from being a n A A ' B B~X to an A2 As' Xspin system by choosing the correft concentration in benzene. H. M. HUTrON and T. S c H A ~ Can. Y. Chem., 41, 2774 (1963). AA' P P ' P " P'".1"2 XH resonance spectrum of bicyclobeptadiene:
H(A) H ( p ~ I ) ~ H(P) H(PII)/H~( AI} "~'H(Pt')
F. S. MORTIMER,3".Iv[el. Spect., 3, 528 0959). As in thepreceding spin system, this symmetrical molecule lends itself to the application of the group theoretical method. AA'BB' X tH and tgF spectra at 40 and 56.4 Mc sec-1 of pars-substituted monofluorobenzenes: G. ARULDHASand P. VENKATr~WAKLU,MoL Phys., 7, 65, 77 (1964).
APPENDIX
I
663
A 3 B C X The IH and 19F resonance spectra of c/s and trmts-l-fluoropropene: R. A. BEAUDFr and 3. D. B ~ C m W I E ~ . R , J. Mol. Spect., 9, 30 (1962). A B B ' C C ~The "direct" method of analysis described by D. WHITMAN,.L Mol. Spect., 10, 250 (1963). A BB I C C ' X and A BB'C C'D The 1H spectrum of monofluorobenzene: S. FUnWXRAand H. Sm~azu, J. Chem. Phys., 32, 1636 (1960). 1H resonance spectra of 1, 2, 3, 4, 5, 6-hexachlorocyclohexanes: R. K. HARMS and N. S~PPA~, Mol. Phys., 7, 595 (1964). A B CD ,I": The 1H resonance spectrum of N-benzylthieno [3,2-6] pyrrole: H. S. Gtrrowsgy and A. L. P o n ~ , J. Chem. Phys., 35, 839 (1961). AsBCD The IH resonance spectra of halogenated aliphatic hydrocarbons: H. Fn~GOLD, Proc. Chem. Soc., 213 (1962). A~,Kn... X,. Double irradiation expectations worked out for general case and applied to 40 Mc sec-1 spectra of CH3CHF2 (A3KX2), CH~CH2F (AsK2X) and CF2CHF (AKPX): M. B A R ~ . n and 3. D. BXLDI~nm.gR, J. MoL Spect., 12, 23 (1964).
NAME
INDEXmVOLUME
Abragam, A. 17, 30, 116 Abraham, R.J. 147, 148, 149, 259, 329, 363, 415, 565, 566, 568, 572, 661,662 A~'ivos, J.V. 662 Abroad, M. 575 Aihara, E. 108 Ainsworth, J. 568 Akahori, S. 176 Alberty, R.A. 565 Alder, B.J. 511 Aleksandrov, I.V. 9 Alexakos, L.G. 237 Alexander, S. 106, 108, 176, 451,465, 489, 510 Allan, E.A. 545, 547 Allen, L.C. 230 Allerhand, A. 481 Allred, A.L. 524, 548, 549 Anbar, M. 510 Anderson, D. H. 109, I10, 111, 127, .154, 160, 164, 165, 172 Anderson, E.W. 575 Anderson, H.L. 237 Anderson, J.K. 556 Anderson, J.M. 476 Anderson, M.M. 524 Anderson, P.W. 488 Anderson, W.A. 23, 116, 117, 121, 177, 191,202, 204, 206, 212, 231, 232, 237, 240, 241,243, 309, 329, 335, 344, 347, 350, 351, 388, 391,427, 431, 453, 460, 463, 466, 470, 471, 474, 475, 557, 558 Andrew, E.R. 4, 21, 31, 39, 46, 217, 256 Androes, G.M. 559 Anet, F. A. L. 174,175, 247, 565, 575, 579, 661 Ankel, Th. 174, 387 Arata, Y. 176 Armstrong, A.M. 522 Arnold, J.T. 23, 206, 208, 236, 344, 492, 542, 543 Aruldhas, G. 662 Axtmann, R.C. 519, 523 Ayant, Y. 26
Baba, H. 150 Bacon, J. 347 xxi
1
Baggett, N. 578 Baflar, J.C. 523 Bak, B. 445, 446, 447, 448 Baker, E.B. 203, 223, 229, 244 Baker, M.R. 127 Baideschwieler, J.D. 190, 241, 247, 267, 371, 398, 456, 457, 459, 461,463, 476, 662. 663 Baldwin, H.W. 524 Bannerjee, M.K. 341 Banwell, C.N. 173, 174, 180, 182, 185, 190, 321,358, 374, 387, 423, 435, 436, 437, 438, 439, 440, 441,442, 566 Barfield, M. 103, 171, 172, 663 Barfield, P.A. 552 Barton, G.W. 230 Basolo, F. 526 Batdoff, R.L. 546 Beaudet, R.A. 662 Becker, E: D. 147, 262, 264, 542, 545 Belford, G.G. 168, 169, 190, 569, 571,572 Bell, R.P. 534 Bender, P. 565 B~,n~, G.J. 63 Benedek, G.B. 28 Bensey, F.N. 530 Benson, R.E. 115, 116 Bentley, P.G. 215 Berber, S.B. 504, 510 Berlin, A.J. 136, 575 Bernheim, R.A. 524, 525, 527, 528 Bernstein, H.J. 25, 90, 92, 124, 125, 127, 129, 132, 141,144, 163, 171,172, 173, 176, 216, 261,283, 326, 327, 328, 329, 335, 363, 404, 405, 406, 407, 448, 467, 486, 537, 538, 539, 548, 565, 572, 575, 576, 661 Bersohn, R. 77, 78, 122, 123 Bha~a, N.S. 9, 466, 476 Bhar, B.N. 549 Bible, R.H. 9 Bigeleisen, J. 518 Birc~all, T. 510, 511 Bishop, E.O. 661 Bitter, F. 203, 240 Bleaney, B. 4 Bloch, F. 2, 16, 20, 25, 29, 30, 31,34, 35, 37, 38, 39, 40, 41, 45, 46, 52, 53, 206, 207, 208, 209, 240, 244, 247
xxii
NAME INDEX--VOLUME I
Bloembergen, N. 20, 23, 24, 25, 27, 30, 32, 33, 38, 41, 46, 48, 76, 185, 210, 217, 218, 224, 233, 489, 526 Bloom, A.L. 56, 202, 240, 242, 244-7 Boden, N. 156, 185, 191, 571 Bonner, L.G. 548 Bothner-By, A.A. 90, 94, 175, 176, 260,662 BSttcher, C. J.F. 95, 98 Bottini, A.T. 574 Bovey, F.A. 23, 24, 82, 140, 141, 148, 257, 259, 268, 575 Bradbury, A. 256 Bradford, R. 223 Bradley, R.B. 147 Braillon, B. 177 Brey, W.S. 571 Britt, A.D. 506 Broer, L. F.J. 2 Broersma, S. 26, 524 Bromberg, J.P. 506 Brooks, H. 74, 161 Brossel, J. 240 Brown, C.J. 173 Brown, H . W . 558 Brown, R.D. 150 Brown, R.M. 63, 237 Brown, T.H. 524, 525, 527, 528 Brownsteln, S. 212, 221,263, 519, 556, 576, 577 Bruce, C.R. 502 Bruegel, W. 174, 387 Buckingham, A.D. 87, 88, 89, 90, 94, 95, 134, 138, 151, 157, 162, 191,444, 464 Burbank, R.D. 530 Burd, L.W. 203, 223, 229 Burgess, ~l. H. 237 Burke, J.J. 532 Bushwell, A.M. 548 Byers, H . F . 548 Bystrov, V.F. 548 C,ady, G.H. 347 Cairns, T.L. 574 Calleja, F. J.B. 549 Calvin, M. 559 C,aznploeLl,A.N. 549 Camponovo, G. 236 Cantaguzene, J. 545 Carpenter, D.R. 549 Can-, H.Y. 55, 56, 160 Carrington, A. 4, 522 Carter, R.E. 149 Carver, T.R. 240 Castellano, S. 378, 379, 380, 381,386, 387, 435, 451
Cavanaugh, J.R. 84 Chandrasekhar, S. 22 Chesnut, D.B. 115 Chiarotti G. 23 Choppin, G.R. 519 Chou, L.H. 574 Claeson, G. 559 Clay, C. 223 Clough, S. 256, 425 Clunie, J.C. 534 Cohen, A.D. 182, 190, 281,287, 388, 391, 448, 455, 476, 541, 566 Condon, E.U. 33 Conger, R.L. 526 Connick, R.E. 515, 521,522, 524, 526 Connor, T.M. 481,542, 545 Conroy, H. 170 Coolidge, A.S. 124, 161 Corio, P.L. 9, 285, 294, 304, 307, 321,329, 334, 341,346, 347, 365, 543 ComwelL C.D. 237 Costain, C.C. 554 Cotton, F.A. 531 Coulson, C.A. 106, 122, 123, 146, 163, 536 Cox, P.F. 25, 30, 174, 182, 411, 527, 528 Coyle, T.D. 532 Craig, N.C. 398 Craig, R.A. 528 Crapo, L.M. 185 Crawford, B. 565 Creswell, C.J. 548, 549 Crook, J.R. 548 Custer, R.L. 217 Cutler, D. 56
Dailey, B.P. 98, 135, 141, 142, 143, 144, 145, 154, 408, 413 Das, T.P. 16, 17, 32, 55, 56, 77, 122, 123, 126, 129, 152, 153, 154, 155, 156, 341 Davidson, D.W. 558 Davidson, N. 527 Davis, D.R. 174, 175 Davis, G.T. 157 Davis, J.C. 542 Dearden, L C. 548, 556 De Boer, E. 116 Debye, P. 22 Deck, J.C. 151 De Kowaiewski, D.G. 277, 661, 662 Dharmatti, S.S. 523 Dickinson, W.C. 59, 65, 68, 90, 92, 120 Diehl, P. 89, 95, 368, 528 Dietrich, M.W. 502 Dinius, R.H. 519 Dime, P. A.M. 79, 109
. ° °
NAME INDEX~VOLUME 1 Dischler, V.B. 407 Dixon, J.A. 575, 580 Dobbs, F.W. 341 Dobinson, B. 578 Dodgen, H.W. 527 Dodson, R.W. 522 Douglass, D.C. 262, 515 Dowling, J. M, 554 Downing, J.R. 548 Dravnicks, F. 522 Drinkard, W.C. 510, 542, 543, 580 Drury, J.S. 528 Drysdale, J.J. 565 Dubb, H.E. 135, 345, 661 Dudek, G.O. 550 Duncan, A. B.F. 183 Fades, R.G. 256 Eaton, D.R. 115, 116 Eberson, L 542 Ehrenson, S. 155,263 Einstein, A. 17 Eisner, M. 24, 25 Eliel, E.L. 576 Elleman, D.D. 176, 185, 190, 244, 277, 564, 572, 661 Elvidge, J.A. 461 Emerson, M.T. 95, 96, 97, 98, 99, 510, 548 Emsley, J.W. 139, 156, 185, 191, 571 Englert, G. 407 Ernst, 1',. 230 Ettinger, R. 501 Evans. B.A. 202 Evans, D.F. 26, 98, 184, 186, 191, 257, 260, 262, 263, 267, 271, 466 Eyring, H. 72, 164, 573 Fabricand, B.P. 513, 514 Farrar, T.C. 25, 172 Feeoey, J. 156,185, 191,273, 358, 543, 548, 571, 577, 578 Fermi, E. 104 Feshbach, H. 238 Fe~senden, R.W. 82, 148, 176, 374, 378, 571, 661 Fiat, D.N. 524 Fi~ovich, G. 257, 266 Finegold, H. 476, 565, 569, 663 Fischer-Hjalmars, I. 536 Fixman, M. 77, 124, 125, 127, 128, 129, 130 Fluok, E. 9 Flynn, G.W. 190, 398 Forbes, W.F. 548, 556 Fors~, S. 502, 542, 548, 550 Forsling, W. 549
Foster, A.B.
XXlll 578
Foster, M.R. 260, 268 Fraenkel, G. 149,215 Franconi, C. 215 Frank, A. 71 Frank, P.J. 154, 572 Fraser, R.R. 176, 180, 661 Fratiello. A. 262, 515 Freeman, R 89, 95, 102, 172, 204, 224, 229, 247, 460, 462, 463,464, 466, 468, 469, 471, 474, 475, 476, 522, 523 Frei, K. 261 Freidman, L. 532 Frost, A.A. 164 Fujiwara, S. 34, 176, 373, 431,545, 663 Fuson, N. 548 Gabillard, R. 43, 217 Garnett, J.L. 445 Gasser, R. P.H. 522, 523, 528 Gassie¢, J. 545 Gates, P.N. 532 George, J.W. 531 Gerlach, W. 2 Ghose, T. 77, 126, 129 Ghosh, S.K. 77 Gill, D. 56, 486, 507 Gillespie, R.I. 347,510, 511,519 Gioumonsis, G. 453 Giulotto, L. 23, 535 Glasel, I.A. 267 Glick, R.E. 90, 94, 120, 155, 260, 263 Golay, M. J.E. 202 Goldberg, S. 513, 514 Goldenson, J. 157.481 Goldstein, J.H. 177, 329, 335, 387, 451,661 Goodman, L. 156, 157 Gordon, S. 98,141, 142, 143, 144, 145 Gordy, W. 126, 127, 154, 548 Gorin, E. 573 Gorter, C.J. 2 Govil, G. 428 Graham, D.M. 565 Graham, J.D. 197 Grab.n, R. 536 Gram, D.M. 103,170, 171, 172, 347, 404, 405 Grant, R.F. 558 Gray, K.W. 256 Gray, P. 558 Griffith, J.S. 157, 523 Grivet, P. 26, 56, 217 Gronowitz, S. 176, 177, 178, 179, 319, 358, 377, 661 Grunwald, E. 482, 490,491,494, 507, 509, 510 Guillano, C.R. 503, 504
xxiv
NAME INDEX--VOLUME 1
Gtinthard, H.H. 203,216, 217, 431 Gutowsky, H.S. 24, 25, 28, 61, 63, 99, 107, 108, 128, 151, 154, 155, 156, 160, 168, 169, 170, 171, 172, 173, 174, 180, 185, 186, 187, 188, 189, 190, 192, 193, 194, 195, 196, 207, 217, 218, 235, 237, 238, 239, 240, 290, 307, 347, 357, 404, 405, 466, 481,482, 485, 486, 487, 488, 517, 522, 524, 525, 527, 528, 538, 545, 546, 547, 552, 555, 564, 568, 569, 571, 572, 573, 663 Guy, J, 77, 78, 122, 134, 146
Hahn, E.L. 29, 40, 52, 53, 55, 56 Halbach, K. 237 Hall, G.G. 83, 144, 145, 146, 147 Hall, L.D. 575 Halpern, J. 502, 522 Hameka, H.F. 75, 85, 121, 122, 125, 126, 127, 146, 538, 539 Hamer, A.N. 215, 530 Hansen, W.W. 2, 25, 46, 208 Happe, J.A. 517, 518, 519, 529, 548 Harbottle, G. 522 Hardisson, A. 83, 144, 145, 146, 147 Harris, R.K. 337, 565, 568, 572, 575, 661, 662, 663 Hartman, J.S. 579 I-Iaszeldine, R.N. 556 Hatton, J. V, 258, 529 Hattori, S. 185, 423, 662 Hawthorne, M.F. 533 Heffernan, M.L. 150, 662 Heidberg, J. 556 Henderson, W.A. 445 Henriques, F.C. 574 Herbison-Evans, D. 510 Hertz, H.G. 222 Herzberg, G. 152 Herzog, B. 29 Hickmott, T.W. 527 Higham, P. 268 Hill, N.E. 24,25 Hindman, J.C. 514 Hiroike, E. 165 Hirschfelder, J.O. 77, 123 Hirst, R.C. 404,405 Hobbs, M.E. 268, 510 Hobgood, R.T. 387 Hoffman, C.J. 128, 151, 155, 156, 238, 538 Hoffman, E.G. 532 Hoffman, R. A. 176, 177, 178, 179, 228,319, 358, 377, 502, 661 Hollis, D.P. 558 Holm, C.H. 27, 115, 185, 187, 357, 482, 486, 487, 552, 555
Holm, R.H. 550 Holmes, J.R. 179, 180, 510, 543, 580 Holt, E.K. 518 Homer, J. 578, 580 Hood, F.P. 576 Hood, G.C. 517, 518, 519 Hornig, J.F. 77 Howard, B.B. 95, 96, 97, 98, 99, 548 Howarth, O. 237 Hubbard, P. S, 24 Htiek©l, E. 83 Huggins, C. M. 540, 542, 545, 546, 548, 549 Hull, R.L. 528, 548 Hume, D.N. 46, 47 Hunsberger, I.M. 545, 546 Hunt, J.P. 527 Hutton, H.M. 196, 405, 662 Hylleraas, E. 67, 68 Hyne, J.B. 540, 541,542, 545 Ichishima, I. 569 Ingrain, D. J.E. 4 Irsa, A.P. 532 Isldguro, E. 77, 124, 162 Ito, K. 72, 122 Itoh, J. 240, 241,517 Jackman, L.M. 83, 144, 146, 147; 176, 230, 267, 329, 461, 575 Jackson, A.H. 259 Jackson, J.A. 523, 524 Jacobsohn, B.A. 41, 42, 224, 226, 558 James, H.M. 124, 161 Jander, J. 556 Jamtsonis, G.A. 556 Jardetzky, C.D. 259 Jardetzky, O. 230, 259, 515, 520 Jarrett, H.S. 549 Jaynes, E.T. 56 Jenks, G.J. 256 Jensen, E. 56 Jensen, F.R. 136, 575 Johnson, C.E. 82, 140, 141,148 Johnson, C.S. 28, 481,506 Johnson, L.F. 177, 191,230, 329, 335 Jonathan, N. 141,142, 143, 144, 145 Jones, A.C. 519 Jones, J. 481 Jones, R. 456 Jones, R. A.Y. 266 Josey, A.D. 115, 116 Josien, M.L. 548 Juan, C. 173, 174, 192, 193, 194, 195, 196 Jumper, C.F. 507, 509, 549
NAME I N D E X - - V O L U M E ] Kaiser, R. 239, 240, 241,242, 475 Kamio, S. 545 Kanda, T. 517 Kaneker, C.R. 428, 523 Kaplan, J.I. 34, 172, 207, 453, 489 Kaplan, M.C. 510 Karabatsos, G.J. 197, 510 Karle, J. 568 Karplus, M. 106, 109, 110, 111, 152, 153, 154, 155, 156, 157, 160, 164, 165, 166, 167, 169, 170, 171, 172, 174, 176, 178, 179, 180, 181, 186, 189, 573 Karplus, R. 237, 239 Kartzmark, E . M . 549 Katayama, M. 545 Katritzky, A . R . 266 Kaufmann, J.J. 532 Kellogg, J. M.B. 120 Kenner, G . W . 259 Khetrapal, C.L. 428 Kimball, G. 72 Kimura, M. 185 Kincaid J.F. 574 King, J. 527 Kirby-Smith, J.S. 548 Kissman, H . M . 574 Kivelson, D. 179, 180, 329, 510, 542, 543, 580 Kivelson, W . G . 329 Klanberg, F. 527 Klein, M.P. 230 Klinck, R.E. 529 Knight, W . D . 59, 60 Koide, S. 77, 124 Komaki, T. 569 Korinek, G.J. 540, 548 Kornegay, R . L . 575 Korringa, J. 247 Koski, W.S. 532 Kotin, L 519 Kowalcwski, V.J. 228, 277, 661,662 Kowalsky, A. 265 Kozyrev, B.M. 4 Krakower, E. 545 Kromhout, R . A . 510 Kruekeberg, F. 174, 387 Kubo, R. 24, 30, 488 Kuchitsu, K. 569 Kuhlmann, K. 476 Kullnig, R . K . 169, 575, 576 Kuratani, K. 569 Kurita, Y. 122 Kurland, R.J. 174 LadeU, J. 134 Lake, K . J . 549
XXV
Lamb, J. 566, 569 Lamb, W.E. 66, 68, 71, 80 Lambert, J.D. 548 Landau, L.D. 66, 67, 79, 80 Lappert, M . F . 552 Lasarev, B.G. 3, 35 Laukien, G. 527 Lauterbur, P.C. 157, 174, 337, 532 Lawrenson, I.J. 28 Leane, J.B. 217 Leztwith, A. 358 Lee, J. 185, 552, 560, 565, 661 L~_,~cc~_,J. 215 LeF/~vre, C.G. 576 LeF6vre, R. J.W. 139, 576 Lemieux, R.U. 169, 574, 576 Lemons, J.F. 523, 524 Lennard-J0nes, J. 535 Lewis, A. 175 Lewis, 1. C. 155 Lezina, V.P. 548 Lhermitte, Y. 545 Li, N.C. 262 Liddel, U. 542 Lifschitz, E . M . 66, 67, 79, 80 Linder, B. 95, 96, 97, 98, 99 Lindstr6m, G. 549 Linnet, W.. 123 Lippincott, E.R. 536 Lipscomb, W. hi. 573 Loewenstein, A. 56, 481,482, 490, 491,493, 494, 509, 510 Lombardi, E. 533 London, F. 83, 84, 85, 95, 96 Longuet-Higgins, H.C. 4, 163 Looney, C.E. 551,552, 553, 556, 557 Lovering, E.G. 162 Lustig, M. 533 Luszczynski, K. 56 Lutz, R.P. 174, 175 Luz, Z. 486, 507, 510 Lynden-Bell, IL M. 168, 170, 174, 192, 197, 284, 448, 661,662 Mackellar, F. 259 Mackor, E.L. 511 Maclean, C. 511 Mahendroo, P.P. 28 Mailer, J.P. 466 Maier, W. 407 Malinowski, E. 193, 194 Manatt, S.L. 176, 185, 190, 244, 277, 564, 572, 661 Mandell, L. 177, 661 Margenau, H. 69, 77
XXVi
NAME INDEX--VOLUME I
Marshall, T.W. 85, 97, 99, 100, 101 Martin, J. 408, 413 Martin, M. 545 Martin, R.J. 529 Marugg, B. 236 Masuda, Y. 517 Matsen, F.A. 120 Matsuoka, S. 185, 423, 662 Matsushima, M. 398 Mattinson, B. L H. 556 Mavel, G. 9, 543 Mayo, R.E. 387 McCalL D.W. 61, 63, 240, 290, 307, 482 McCann, A.P. 202 McClellan, A.L. 534, 535, 536, 539 McClure, G.R. 177 McClure, R.E. 217 McConneIl, H.M. 27, 63, 75, 76, 106, 107, 108, I15, 116, 117, 129, 137, 160, 163, 164, 173, 181,182, 183, 185, 186, 262, 298, 337, 392, 398, 427, 431,482, 497, 503, 504, 510 McGarvey, B.R. 77, 126, 351, 522, 526 McLachlan, A.D. 106, 116, 149 McLauchlan, K.A. 162, 172, 281, 455, 476 McLean, A.D. 63, 185, 298, 337, 392, 398 McMahon, P.E. 168, 169, 190, 569, 571, 572, 573 McWeeney, R. 83, 144, 146 Mecke, R. 541 Meiboom, S. 34, 56, 453, 482, 486, 490, 491,493, 494, 507, 509, 510 Meinwald, J. 175 Meinzer, R.A. 481 Meisenheimer, J. 574 Meisenheimer, R.C. 262 Merril, J.R. 545, 547 Meyer, L.H. 185, 217, 538 Miller, J.J. 533 Miller, R. 576 Milner, R.S. 115 Mitchell, R.W. 24, 25 Miyazawa, K. 569 Mizushima, S. 551,569 Molt, R.Y. 169 Moniz, W.B. 575, 580 Mooney, E.F. 532 Morgan, L.O. 25, 30, 526, 527, 528, 548 Morin, M.G. 268, 519 Morino, Y. 569 Moritz, A.G. 136 Morse, P.M. 238 Mortimer, F.S. 176,388,391,392,451,662 Mott, N.F. 71 Moxon, L.A. 233 Muchowski, J.M. 547
Muctterties, E.L. 520, 530 Muller, N. 157, 177, 191,192, 194, 195, 337, 481, 532, 578 Murphy, G.M. 69 Murphy, J. 25, 30, 77, 527, 528, 548 Murray, B.B. 519 Murray, F.E. 535 Murray, G.R. 102, 523 Murrell, J.N. 266 Murthy, A. S.N. 545 Musher, J.I. 136, 172, 476, 520, 579, 581 Myers, O.E. 506, 531 Naar-Colin, C. 175, 176, 662 Nagasawa, M. 519 Hair, P.M. 562, 565 Narasirrd~m, P.T. 133, 134, 135, 177, 351, 371,372, 392, 442 Nelson, F.A. 460 Newell, G.F. 121 Newmark, R.A. 569, 571,572 Nilsson, M. 550 Nist, B.J. 283 Nolle, A.W. 28, 527, 528, 548 Norberg, R.E. 502 Nordlander, J.E. 532 Nordsieck, A. 121 Noyc¢, D.S. 136, 575
Ogg, R.A. 27, 240, 497, 517, 528, 553, 556, 558 Onsager, L. 89, 135, 345, 661 O'Reilly, D.E. 77, 111, 157, 511 Orgel, L.E. 157, 523 Ormand, F.T. 120 Overhauser, A.W. 116, 247 Pachler, K. G.R. 172, 566, 568 Packard, M.E. 2, 25, 46, 202, 203, 208, 236, 542 Packer, K.J. 337, 520, 662 Pajak, Z. 548 Pake, G.E. 4, 19 PaJko, A.A. 528 Palmer, J.W. 524, 526 Pan/sh, M.B. 530 Paolini, L. 536 Parks, J.R. 157 Parr, R.G. 71, 121,573 Paterson, W.G. 509, 547 Patterson, A. 501 Paulett, G. 268, 519 Pauli, W. 1
NAME INDEX--VOLUME 1 Pauling, L. 81, 106, 109, 110, 128, 167, 180, 553 Pearson, M.J. 558 Pearson, R.G. 524, 526 Petrakis, L. 101, 102, 190, 191, 257, 271 Phillips, W.D. 115, 116, 530, 551, 552, 553, 556, 557, 565 Pierens, R.K. 576 Piette, L.H. 240, 548, 553, 555, 556, 557, 558 Pimental, G.C. 534, 535, 536, 539, 540, 542, 545. 546, 548 Pinkerton, J. 127 Pitcher, E. 151, 191 Pitzer, K.S. 542 Pitzer, R.M. 573 Plovan, S. 550 Pople, J.A. 25, 79, 80, 81, 83, 84, 85, 90, 92, 97,99, 113,114,124,125,127,129,131,132, 141,144, 146, 148, 157, 162, 176, 184, 185, 216, 240, 283, 326, 327, 328, 329, 335, 368. 370, 404, 405, 406, 407, 448, 464, 467, 469, 486, 497, 535, 537, 538, 539, 548, 560, 565 Pone, A.L. 545, 546, 663 Portis, A.M. 226 Post, B. 134 Poulson, R.E. 515, 521,522. 526 Pound, R.V. 2, 20, 23, 24, 25, 26, 27, 32, 33, 38, 46, 76, 210, 217, 218, 221,224, 229, 233 Powell, R.L. 548 Powles, J.G. 4, 50, 56, 63, 130 Pratt, L. 115 Premuzic, E. 576, 577, 579 Present, R.D. 161 Primas, H. 203, 216, 217, 230, 237, 321, 431,435, 436, 437, 438, 439, 440, 441,442 Pritchard, D.E. 177, 191,192, 194, 195, 532 Proctor, W.G. 25, 59, 61 Prohaska, C.A. 572 Prosser, F. 156, 157 Purcell, E.M. 2, 17, 20, 23, 24, 25, 27, 28, 32, 33, 38, 46, 55, 56, 63, 76, 160, 210, 217, 218, 221,224, 233 Purlee, E.L. 510 Quaff, J . w . 347 Quirm, W.E. 63 Rabi, I.I. 16, 50, 120 Ramey, K.C. 563, 571 Ramsey, N.F. 2, 11, 18, 50, 63, 68, 70, 74, 75. 99, 101, 102, 103, 104, 105, 109, 120, 121,122, 124, 125, 127, 151,160, 161, 162, 165
xxvii
Randall, E.W. 241,247, 456, 461 Ranft, $. 661,662 Ransil. B.J. 154 Rao, B. D.N. 407, 408, 411,476, 542, 543, 661 Rao, C. N. IL 545 Ray. J.D. 27, 240, 497, 517, 553, 556, 558 Reddy, G.S. 177, 451 Redlich, O. 517, 518, 519 Reeves, L.W. 258, 481, 510, 542, 543, 545, 547, 548, 549, 550, 551,558, 576, 577, 579 Reid, C. 141, 541, 545 Reilly, C.A. 44, 63, 172, 185, 262, 276, 282, 298, 329, 337, 361, 369, 370, 375, 376, 377, 378, 427, 451, 452, 466, 475, 517, 518, 519, 661 Reilly, E.L. 551, 552, 553 Reynolds, W.F. 196 Richards, J.H. 149 Richards, R.E. 4, 102, 117, 202, 217, 237, 258, 268, 321,357, 408, 417, 418, 419, 420, 421,422, 510, 517, 522, 523, 528, 534, 579, 661 Riggs, N.V. 661 Roberts, J.D. 172, 174, 175, 176, 177, 212, 283. 532, 562, 565, 574 Rocard, J.M. 63 Rodebush, W.H. 548 Rogers, E.H. 227 Rogers, M.T. 133, 134, 135, 177, 351,371, 372, 392, 442, 530, 555 Rollin, B.V. 46 Roper, R. 576 Rose, M.E. 202 Rosen, N. 122, 123, 146 Roux, D.P. 63 Rowland, T.J. 506 Roy, S.K. 56 Royden, V. 240 Ruiner, G. 109 Rutenberg, A.C. 523, 528 Rutledge, R.L. 329, 543 Ryschkewitsch, G.E. 552 Sack, R.A. 488 Sadler, M.S. 549 Saha, A.K. 16, 17, 32, 55, 341 Sa&ka, A. 151, 152, 153, 157, 183, 185, 187, 357, 482, 485, 488, 510, 517, 522, 538, 555, 565 Sallkhov, S.G. 4 Salpeter, E.E. 49, 226 Sandorfy, C. 135 Santry, D.P. 162 Sasson, M. 56
xxviii
NAME INDEX--VOLUME I
Sato, S. 240, 241 Saunders, M. 135,345, 540, 541,542, 545, 661 Schacher, G.E. 511 Schaefer, T. 94, 147, 149, 150, 151, 174, 196, 217, 321, 357, 364, 368, 370, 405, 408, 416, 417, 418, 419, 420, 421,422, 423,444, 469, 549, 662 Schaeffer, R. 456 Schimamouchi, T. 569 Schliiter, J. 527 Schmitz, H. 530 Schneider, W.G. 25, 90, 92, 94, 124, 125, 127, 129, 132, 135, 137, 138, 141,144, 147, 149, 150, 151, 157, 176, 216, 258, 260, 268, 283, 326, 327, 328,335,404, 405, 406, 407, 416, 423, 444, 448, 467, 486, 510, 529, 535, 537, 538, 539, 540, 543, 548, 549, 550, 551, 553, 555, 565, 575, 576, Schroeder, R. 536 Schubnikov, L.V. 3, 35 Schug, J.C. 151, 529, 573 Schug, K. 511, 548 Schumacher, H.J. 530 Schwinger, J. 16, 50 Scmggs, R.L. 262 Searles, S. 550 Scderholm, C.H. 101, 102, 136, 185, 190, 191,257, 271,509, 571,572, 575 Sclwood, P.W. 526, 527 Scnda, K. 185 Shapiro, I. 533 Sheppard, J.C. 506 Sheppard, N. 136, 163, 168, 170, 171,172, 173, 174, 180, 182, 185, 190, 192, 197, 266, 287, 358, 374, 387, 388, 391,423, 448, 565, 566, 568, 572, 575, 576, 663 Shimizu, H. 34, 176, 431, 663 Shimomura, K. 28 Shoolery, J.N. 135, 147, 151,177, 191,212, 240, 242, 244-7, 271, 329, 335, 445, 446, 447, 448, 456, 511,540, 542, 545, 546, 548, 549, 565 Shortley, G.H. 33 Shreeve, J.M. 347 Shuler, W.E. 519 Siddall, T.H. 572 Silvir, B. 510 Sinha, S.K. 77 Skavlem, S. 67, 68 Sliehter, C.P. 9, 63, 151, 152, 153, 157, 240, 290 307, 482, 522 Slomp, G. 202, 228, 259, 351 Smaller, B. 237 Smith, D.C. 532 Smith, D.F. 530 Smith, F. 202
Smith, 1. C. 149 Smith, J. A.S. 202 Sneddon, I.N. 71 Snyder, E. L 176, 177, 569 Snyder, L.C. 71, 121 Sogo, P.B. 533 Sollich, W.A. 445 Solomon, I. 30, 50, 185, 489, 527 Somers, B.G. 547 Soutif, M. 217 Spaeth, C.P. 556, 557 Spalthoff, W. 222 Spedding, H. 5O9 Speirs, J.L. 530 Spencer, R.H. 217 Spiesecke, H. 135, 137, 138, 147, 260, 268 Spurr, R . A . 548 Stafford, S.L. 371 Stanford, S.C. 548 Stehling, F.C. 232 Stejskal, E.O. 25, 482, 490 Stephen, M.J. 77, 78, 90, 91, 92, 93, 94, 95, 111, 112, 123, 124, 127, 129, 162 Stem, O. 2 Sternheimer, R. 152 Stevens, K. W.H. 4 Stewart, W.E. 177, 329. 335 StiUman, A.E. 519 Stone, F. G.A. 151, 191, 532 Stothers, J.B. 529, 661 Stover, E.D. 526 Strange, J.H. 50, 63 Streitwieser, A. 83 Strick, E. 223 Stremme, K.O. 547, 576, 579 Strong, T. 548 Sudhanshu, S. 387 Sugiura~ Y. 123 Sunners, B. 548, 553, 555 Suryan, G. 49 Sut¢liffe, L.H. 156, 185, 191,273, 358, 543, 548, 560, 565, 571,577, 578, 661 Suzuki. S. 150 Svatos, G.F. 337 Swalen, J.D. 172, 282, 329, 361, 375, 376, 377, 378, 427, 451,452, 453, 661 Swift, T.J. 526 Symons, M. C.R. 522 Sz6ke, A. 509 Taft, R.W. 155, 157, 545 Takeda, M. 259, 482, 490, 568 Talman, J.D. 661 Tarbell, D.S. 574 Tarte, P. 556, 557
NAME I N D E X - - V O L U M E 1 Taube, H. 523, 524 Taylor, R.C. 177, 335 Thomas, L . F . 578. 580 Thompson, D . D . 497, 510 Thompson, D.S. 569, 572 Thompson, H.B. 530 Thwaites, J.D. 202 Tiers, G. V.D. 31, 99, 162, 171, 172, 184, 257, 259, 264, 265, 266, 268, 286, 445, 550, 575, 578 Tillieu, J. 77, 78, 122, 130, 134, 135, 136, 146 Tipman, N. 17,. 247 Tolansky, S. 1 Tomita, K. 24, 30, 247 Torrey, FI. C. 2, 40, 52, 53, 217, 221 Tosch, W.C. 578 Townes, C.H. 154 Tsubomuro, H. 536 Turkevich, J. 164 Turner, D . W . 244, 267, 464 Turner, J.J. 168, 169, 182, 190, 277, 287, 448, 565, 566. 576 Tuttle, T.R. 116, 217 Tzalmona, A. 56 Utterbaek, E.J.
548
Vane, F . M . 197 Van Geet, A . L . 46, 47 Van V|eck, J.H. 70, 71,165, 172, 431,432 Vaughan, W.R. 177, 335 Venkateswarlu, P. 407, 408, 411,543, 661, 662 Virmani, Y.P. 428 Wahl, A . C . 502, 506 Walker, S.M. 156 Wallis, R . F . 123 Walter, J. 72, 573 Wang, S. 122, 123, 146 Wangsness, R . K . 36, 224, 226, 558 Waring, C.E. 217 Waugh, J.S. 28, 82, 137, 148, 176, 341,374, 378, 379, 380, 381,386, 387, 435, 451,531, 565, 571,661 Weaver, H.E. 147, 151, 503
I~RS.
22
XXiX
Wegmann, L. 236 Well, J.A. 556 Weinbaum, S. 123 Weinberg, I. 543 Weissman, S.I. 116 Wertz, J.E. 32, 515, 520 West, R. 549 Whatley, L. 549 Wheeler. D . H . 522 Whiffen, D . H . 4, 172, 204, 244, 247, 455, 460, 462, 463, 468, 476 Whipple, E.B. 177, 329, 335, 568, 569, 661 White, J.W. 117 White. R. F . M . 519, 662 Whitesides. G . M . 532 Whitman, D . R . 135, 345, 392, 407, 427, 451,452, 661,662 Whittaker. A . G . 517. 518, 519 Wiberg, K.B. 283 Wick, G.C. 71, 73, 74 Wiley R . H . 176 Williams, D . H . 9 Williams, G.A. 107, 108, 185, 186, 187, 188, 189, 207, 237, 238, 239, 357, 445, 446. 447, 448, 466 Williams, R.B. 203, 211, 224, 225, 226, 232, 233, 234 Williams, R.E. 533, 551 Wilson, E.B. 106, 109, 110; 167, 298, 551, 569, 573, 662 Wimett, T . F . 62, 99, 160 Wishnia, A. 527 Woessner, D.E. 24, 25, 28, 524, 525, 527 Woodbrey, J.C. 487, 555 Yamagata, Y. 517 Yamaguchi, I. 556 Yamamoto, T. 373 Yasaitis, E. 237 Yatsiv, S. 453, 455 Yen, W.M. 506 Yorke, B.A. 517 Yu, F . C . 25, 59, 61
Zacharias, J . R . 120 Zahn, C. T+ 127 Zavoisky, E. 4 Zimmerman, J.R. 260, 268, 329, 547
SUBJECT INDEX The heaviness of the type indicates the importance of an entry: page numbers in bold type can signify that a topic runs on for several pages. Acetaldehyde 273 Acetaldehydes (dichloro-), IH double resonance spectrum 461 hydration of 533 Acetic acid, ~3C spectrum 991. 993 Aoetone, as a solvent 259 Acetonitrile,effect of solwnts on 1H chemical shift 846 Acetoxycholestonanes, IH spectra 706 Acetytenes :3Cc spectral parameters 1001, 1030 effect of solvents on ~H chemical shifts
Alkene.~, 13C-13C coupling constants 1030 cis and trans XH chemical shifts 729 correlation of ZH chemical shifts with group dipole moments 718 Hammett # constants 717 substituent electronegativity 717 correlation of H - H coupling constants with substituent elec~'onegat iviW 714
effect of methyl group sutntitution on XH chemical shifts 733 ~H chemical shifts 727, 731, 732, 736, 749, 848 737, 743, 744, 745 1H chemieal shifts 745 H-H coupling constants 710, 726, 727, H-H coupling constants 746, 748 731,732, 735, 739 Addition compounds of BF3 528 1H spectra 710 ~TAl chemical shifts Alkenes (metal) of alkyls 1094 IH chemical shifts 742 of salts 1094 H-H coupling constants 743 27A1 resonance 1093 Alkyls (metal), coupling constants 689 line widths 1094 Allyl magnesium bromide, IH spectrum 689 Alcohols Amides hydrogen bonding in $42 calculation of internal chemical shift 133 x70 chemical shifts 1045 h~ydrogen bonding of 548 Aldehydes rotational isomerism 553 aromatic proton exchange 510 :aC spectral parameters 1001, 1005 Amines, hydrogen bonding 548 ~H chemical shifts 765 long range H-H coupling constants Amino acids, IH chemical shifts 812 Ammonia 767 calculation of .H-H coupling constant H-x3C coupling constants 1020 183 ~70 chemical sldfts 1045 calculation of hydrogen bond shift 538 Aldopyranoses (acetylated), 1H spectra 704 1aN spectrum 1031, 1093 Aliphatic Ammonium ion acids, 170 chemical shifts 1045 ~H double resonance spectrum 458, 459, fluorocarbons, calculations of coupling 460 constants 183 14N INDOR spectrum 1033 Alkaloids, XH spectral parameters 806 proton transfer 498, 510 Alkanes derivatives, t a n spectra 1038 ~H chemical shifts 666 Analysis H - H geminal coupling constants 677 AB spin system 310 H-H vieinal coupling constants 678 direct method of analysis 438 Alkenes double irradiation (weak field) 472 anomalous IH chemical shifts 719 effect of elecmc fields on 464 x3C chemical shifts 999 xxxi 22*
XXJdi
SUBJECT INDEX
Analysis, energy levels 317 theoretical spectra 320 transition energies 318 AX spin system, double resonance behaviour 247, 462 ABe spin system 320 basic product functions 323 diagonal matrix elements 323 electric field effects 464 energy levels 324 relative intensities 325 theoretical spectra 327, 625 AX= spin system, double resonance behaviour 462, 463 AB3 spin system 329 diagonal matrix elements 331 relative intensities 332 theoretical spectra 333, 629 transition energies 332 AX3 spin system double resonance behaviour 247 electric field effects on 464 AB. spin system 337 eigenfunctions 337 eigenvalues 337 relative intensities 339, 635 spin product functions 336 theoretical spectra 340, 635 transition frequencies 339, 635 ABe spin system 661 AB. spin system 341 general features of spectra 341 relative intensities 343 sub-spectra 342 transition frequencies 343 A2B2 spin system 347 relative intensities 349 theoretical spectra 350, 645 transition frequencies 349 A2X2 spin system 347 A2B6 spin system 661 A3B2 spin system 351 combined transitions 354 relative intensities 352, 353 sub-states 351 theoretical spectra 355, 649 transition frequencies 352, 353 ApB. spin system 344 general features of spectra 344 numbers of transitions 346 A,.X2 spin system, double resonance behaviour
463
AnX,, spin system, double resonance behaviour
463
ABC spin system 372 ABK approximation method
375
Analysis, complete solution 378 diagonal matrix elements 373, 382 double irradiation (weak field) 473 eigenvalues 381 intensity sum rule 380 spectrum (tH) of styrene 381 spectrum (XH) of vinyl bromide 711 subtraction rules 383 Sudhanshu's method 387 trace invariance of sub-matrix 379 ABK spin system 376 energy levels 376 styrene oxide, IH spectrum 377 wavefunctions 376 ABT spin system 661 ABX spin system 357 deceptively simple spectra 363, 364 diagonal matrix elements 357 energy levels 359 relative intensities 360 styrene oxide, XH spectrum 362 subtraction rules 361 theoretical spectra 362, 449 transition energies 360 APX spin system signs of coupling constants 466 triple resonance spectrum 476 AB2C spin system 391,661 off,diagonal elements 392 AB2X spin system 661 ABXz spin system 388 diagonal matrix elements 389 relative intensities 390 spectrum of 2,3-dichioropropene-1 (IH 388 transition energies 390 ABPX spin system 661 ABXY spin system 661 AA'BB' spin system 348, 399 explicit energy expressions 401 1H spectra of benzenes Q~-substituted) 408, 414 ethanes (1,2-substituted) 415, 566 1H spectrum ofbenzofuraTan 407,408, 4O9 benzene (p-bromochloro-) 414 ethane (1,2.bromochloro-) 415 naphthalene 405, 406 matrix elements 400 relative intensities 402, 410 relative signs of coupling constants 404 theoretical spectra 404, 413 transition energies 402, 410 AA'XX' spin system 392 tgF spectrum of benzene (1,2-difluoro3,4,5,6-tetrafluoro) 399
SUBJECT I N D E X Analysis, ~gF spectrum of ethylene (1,1-difluoro) 397 IH spectrum of ethylene (l,l-difluoro-) 397 matrix elements 393, 394, 395 theoretical spectra 396, 404, 413, 449 transition energies 396 AA'KL spin system 662 ABKL spin system 662 ABCX spin system 423, 662 matrix elements 424 ABCD spin system 425 A B K ¥ approximation 427 XH spectrum of glycidaldebyde 428 matrix elements 428 A2A~X spin system 662 ABB'CC' spin system 662 AA'BB'X spin system 416, 662 ~gF spectrum ofpara-fluoroaniline 419 XH spectrum ofpara.fluoroaniline 419 matrix elements 417 relative intensities 419, 420, 421 symmetrised spin functions 417 transition energies 419, 420, 421 wavefunctions 418 AB2X2 spin system 662 AA'PP'X spin system 662 ABC3 spin system 392, 661 ABXs spin system 661 AB4X spin system 662 AA'A"A"'XX' spin system 662 " AsB2C spin system 392 A3B2X spin system 370 XH spectrum of phosphorus triethyl 371 AsBCD spin system 663 AsBCX spin system 662 ABB'CC'D spin system 663 ABB'CC'X spin system 663, 1092 ABC-'DX2 spin sys~m 663 ABCXs spin system, double irradiation of 476
ABPsX spin system, double irradiation of 469 A.BXp spin system 365 general features of spectrum 368 numbers of transitions 369, 370 relative intensities 367 sub-spectra 366 transition frequencies 367 A , . K . . . . X . spin system 663 A.BPcXp spin system 370 AA'X 3 X'3 spin system 661, 662 AA'X.X~ spin system 661 ! II AA I A I I x sXsX 3 spin system 662 A A ' P P ' P " P " 'X2 spin system 662 •
a-es.
224
xxxiii
Analysis, of spectra 280 aids 442 complex particle method 345 direct method 435 double irradiation 455 double quantum transitions
4S3
effect of rotational isomerism 560 electronic computation 4$1 isotopic substitution 445 moment method 431 perturbation theory 428 rules 309 quantitative 211, 234 Anilines (p-fluoro) coupling constants 423 19F spectrum 421 IH spectrum 422 Anisotropy effect of neighbour 131 of shielding constants 94, 113 of C-C and C-H bonds 135, 676, 696 of C-X bonds 136 Annulenes, 1H chemical shifts 745 Anthracene, 1H spectrum 773 Antimony pentafluoride ~9F spectral parameters 928 19F spectrum 929 121Sb resonance of NaSbF6 1097 Aqueous electrolytes 511, 514 Area of bands 234 Aromatic compounds, see Benzenes as solvents 2.58 calculation of chemical shifts 140, 145, 149, 595, 770 calculation of coupling constants 150 charge densities in 149 effect of solvents on IH chemical shifts 851 H - H coupling constants 770 molecular complexes 529 polynuclear, IH chemical shifts 770 ions, 1H chemical shifts 774 Arsenic (VSAs)resonance 1097 Association, see Hydrogen bonding Atomic orbitals, calculation of shielding constant 121 Average energy approximations 71, 106 Azulene calculation of internal chemical shift 146 1H spectrum 778 carbonium ion, XH spectrum 778 11B chemical shifts 971, 974 of boron halides and derivatives 972, 974
xxxiv
SUBJECT INDEX
riB chemical shifts, of boron hydrides and derivatives 972,974 Barrier to internal rotation, nature of 573 Basic product functions 297 symmetry functions 298 consort of 299 of D , point group 304 Benzaldehydes, H-H coupling constants 277 1,2-Benzanthracane (9,10-dimcthyl-), carbonium ion 776 XH spectrum 776 Benzene as a solvent 258 chloro- and deuterochloro-, tH spectra 751 isopropyl-, XH spectrum 6 meta-dinitro-, 1H spectrura in acetone 444 in benzene 444 monofluoro-, tgF spectrum 446 monofluoro-2,3,5,6-D4-, XH spectrum 446 monofluoro-2,4,6-D 3, tgF and ~H spectra 447 ortho-dimethoxy-, 13C spectrum 1008 para-chlorobromo-, XH spectrum 414 para-methyinitro-, effect of solvents o n tH spectrum 853 2,3,5,6-tetrachloro-, tH spectrum with laC satellites 1023 Benzenes xaC-xaC coupling constants 1030 xSC~H satellite spectra 1022 H-H coupling constants 403, 770 ortho-disubstituted, analysis of spectra 403, 405 para-disubstituted; effect of solvents on 1H chemical shifts 852 1H spectra 408 proton exchangewith HF 511 alkoxy, XH chemical shifts 764 disubstituted, 1H chemical shifts correlation with Hammett o constants 758 empirical calculations 754, 1140 fluorinated xgF chemical shifts correlation with Hammett o"constants 897 F - F coupling constants 901 H-F coupling constants 901 halo-, XH chemical shifts 760 hydroxy-, XH chemical shifts 763
Benzenes, monosubstimted 13C chemical shift correlation with Hammett o constants 753 x3C spectral parameters 1002 XH chemical shifts 750 correlation with Hammett ~ constants 752 of meta nuclei 752 of ortho nuclei 752 ofpara nuclei 752 trisubstituted, tH chemical shifts 767 BenzofurnTn,% tH spectrum 407, 408, 409 Binary fluorides, correlation of XgF chemical shifts and coupling constants 880 with electronegativity 882 Biot--Savat law 67, 79 Biphenyls 767 Bloch equations 34, 38 including chemical exchange 482, 505 susceptibilities 39 Boltzmann distribution of nuclei 17 Borane hexa-, l i b spectrum 982, 984 tetra-, t:B spectrum 981 undeca-, XtB spectrum 984 deca-, XlB spectra 982, 984 pentao, 11B spectra 981,985 Borohydrides, xxB spectral parameters 987 Boron alkyls, 11B spectral parameters 975 halides 528, 532, 971, 974, 976, 987 tXB spectral parameters 971, 974 complexes 952, 973 19F spectra of mixed haLides 945 tgF spectral parameters 944 hydrides, ItB spectral parameters 972, 974, 977 deuteration 533 tetr~uoride ion, 19F spe~ral parameters 946 79Br and SlBr resonance of Br~q 517 sxBr resonance of HBr 1096 Bridge circuits 217 Bromine pentafluoride 943 trifluoride 942 Bulk suceptibility correction 260 Butadiene--isoprene copolymer, XH spectrum 832 Butanes (2,3-disubstituted) 1H chemical shifts 690 H-H coupling constants 690 t-Butanol, hydrogen bonding 541 Butene-1
SUBJECT INDEX Butene-1, :H chemical shifts 723 H - H coupling constants 723 i-Butenes (1-substituted) IH chemical shifts 736 H - H coupling constants 737
13C chemical shifts comparison with 29Si chemical shifts 1051 of acetylenes 1001, 1030 of aromatic compounds 1002, 1005, 1030 of carbonyl groups 1009 of ethanes 997, 1030 correlation with substituent electronegativity 996 of metal carbonyls I010 of methanes 992, 995 correlation with substituent electronegativity 996 of monosubstituted benzenes 1002 of phenoh 1006 of simple organic compounds 990, 1030 13C resonance, experimental procedures 988 13C satellites in 1H spectra 448, 450, 475 in 19F spectra of fluoroalkenes 914, 962 Caesium (lS~Cs) chemical shifts of halides 1093 Calibration of spectra sideband method 237, 274 wiggle beat method 276 Carbohydrates (acetylated) contigurational effects on IH spectra 701 Carbonium ions, IH chemical shifts 774 Carbonyl groups, IsC chemical shifts 1009 CAT 230 C-C bond shifts 666, 698 IsC-:3C coupling constants 1029 CDCI3 as a solvent 257 Ceils, see Sample containers lSC-F isotope shifts in fluoro-organic compounds 962 CFC13 as a solvent 257, 266 Character table for D4 point group 301 Charge densities in aromatic molecules 149 Chemical equilibria, effect on spectra 481 exchange 484 Chemical shifts (see Shielding constants), 4, 59, 65 calculation for alkyl compounds 136 amides 133 22 a*
XXXV
Chemical.... aromatics 140, 149, 595, 770 azulene 146 cyclohexane 136 cyclohexa~ (perfluoro-) 139 fluorine compounds 151 fluorob~n=nes 154 heterocyr~ aromatics 145, 789 • paramagnetic contribution for SgCo resonance 1078 paramagnetic contributions for 19~Ig, 2°VPb and 2°sT1 resonances 1098 c/s and trans in olefines 729 contact 826 contributions from ring currents in aro= matics 141, 595, 770 conversion factors for external refe~e~.cin8 263 correction for bulk diamagnetic susceptibility 66, 260 effect of parmnagnetic materials 115, 826 empirical calculations 838 empirical calculations for disubstitoted henzenes 754 equivalence 283 gas-to-solution 97, 841 IH, charts of 1131 IH for organic compounds (Tiers'compilation) 1115 isotope effects 875, 916, 962, 1022, 1092 origin of 59 ortho effect in fluorohenzenes 155 Chemical shift/electron density ratios for hydrocarbon aromatic ions 781 Chlorine 3sCl and 37CI, chemical shifts of inorganic chlorides 1095 3sCl resonance of Cl~ 515 nuclear quadrupole splittings in inorganic chlorides 1096 resonance 1095 trifluoride,19F spectrum 941, 942 Chromium ~ comple~,cs, ligand exchange rates 525 sgCo chemical shifts calculation of paramagnetic contribution 1078 correlation with electronic absorption 1080
of cobalt (HI) complexes 528, 1080 solvent effects 1081 temperature dependence 1081 Cobalt complexes, rate processes 1082 Collapse of doublet by chemical exchange 487, 489
XXXVi
SUBJECT INDEX
Coupling, in fluoroalkanes 1017 Collapse, of triplet by chemical exchange 491 in fluoroalkenes 915, 962 of quartet by chemical exchange 494 in fluoromethanes 883, 962, 1017 2,4,6-Collidine, t3C spectrum 1008 in perfluorocyclobutane 959, 962 Combination transitions 325 . relative signs 1011 Commuting opsrators 291 F-C1 in FCIO3 944 Complex particle method of analysis 345 F-F Computers effect of temperature 878 in the analysis of spectra 451 electron-orbital and orbital-orbital inof average transients (CAT) 230 teraction 184 Configuration of cyclohexanes 703 in aliphatic fluorocarbons 183 Configurational effects in t H spectra of in antimony pentafluoride 928 carbohydrates 701 in aromatic fluorocarbons 183 Conformation in c/~- and trans-l,2-diflunro-l,2-diof fluorinated ethanes 889 chloroethylene 916 of saturated ring compounds 575, 579, m c/s- and trans- N2Fz 948 920 m fluorinated aromatic compounds 901 Conformational motion, effect on spectra m fluorinated cyclobutanes 918 481 m fluoroacyl metal compounds 896 Conjugated polyenes, tH chemical shifts m fluoroalkanes 875, 958 744 in fluoroalkenes 906 Contact shifts 115, 826 m fluoroalkyl metal compounds 895 Copper (eSCu) resonance 1096 in fluorocarbon sulphides 894, 926 Copper(I)-copper(ID exchange reaction in fluoroethanes 886 503, 1096 in fluoropropanes 886 Coproporphyrin-l, IH spectrum 792 in hypofluorites 949 Correlation in interhalogen compounds 940 function 439 tn miscellaneous organic fluorine comapplication to a system of chemically pounds 962 equivalent nuclei 441 in perfluorocyclohexane 921 evaluation of 441 in perfiuoropiperidine 926 time (Te), 21, 31 in perfluorovinyl metal compounds Correspondence principle 291 906 Coupling constants (see Spin-spin coupling) in phosphorus halide derivatives 961 61, 113 in sulphur hexafluoride and its derivaabsolute signs 162, 681 tives 930 angular variation for geminal nuclei 171, in sulphur tetrafluoride and its deriva711 tives 938 angular variation in furam and pyrroles involving perfluoromethyl groups 959 786 long range 879 between H nuclei separated by four a relative signs 888, 914 bonds 174 through-space 190 calculation for HD 160 F-Hg calculation of H - H for methane 163 in mercury fluoroalkyls 897 calculation of v/e/ha/and geminal H - H in in mercury fluoroalkenes 907 ethanes and ethylenes 166 F-M in MFx compounds 880 calculations (theoretical) 103, 105, 106, F-X4N in NTs and c/s- and trana- N2Fa 109, 111, 160 948 dependence on atomic number 63 F-93Nb in NbF~6 1101 B-P in (CH3)~PHBH3 988 F-ZgSi in substituted silanes 1050 xsC--XsC 1029 F--Sn in tin fluoroalkenes 907 F-XXB, in boron halides and derivatives F-P 1061 945 in perfluoroalkyl derivatives 896, 9$9 F_Is c in perfluorotriphosphonitrile 951 additivity effects 1017 in phosphorus halides and derivatives correlation with tgF (t3C-t2C) isotopic 949, 959, 960, 961 chemical shifts 1022
SUBJECT INDEX Coupling..., in phosphoryl halides 1077 H A l in fithium aluminium hydride 1094 H-B in borohydrides 987 in boron halides and derivatives 972, 974
in boron hydrides and derivatives 972, 974, 977, 980, 982, 983 in (CHs)2PHBH3 988 in diborane diammonlate 986 in miscellaneous boron compounds 972, 974 H J 3 C 1011, 1032 additivity rules 193, 1017, 1019 correlation with bond length 1014 correlation with electronegativity 1014 correlation with XH chemical shifts 1020
dependence on hydridisation 1011 dependence on s character 192, 1012 in acetic acid 993 in acetylenes 1001 in aldehydes 1020, 1032 in ~kyl and silyl selenides 963 in alkenes 1028 in aromatic compounds 901, 1003, 1005, 1023 m chloroethylenes 722 m ethanol 993 m ethylene 448 m fluoromethanes 883 m formyl compounds 1014 m heterocyclics 1021, 1032 m hydrocarbons 1012 m methanes 1013, 1014 m monosubstituted benzenes 1003, 1032 in phenols 1006, 1023 in plumbanes 824 in propyne 994 in pyridine 994 in stannanes "824 in unsaturated compounds 196 long range 683, 1024, 1 0 2 7 relation with ~ u and ~ in ethylenes
1020 signs 197, 1011 H-D 1092 H-F in fluorinated aromatic compounds 901 in fluoroacetylene 916 in fluoroalkenes 909, 910 in fluoroethanes 886 in fluoromethanes 883
xxxvii
Coupling..., influoropropunes 8S7 in phosphorus (V) fluoride derivatives 960, 961 in silicon compounds 1050 long range 879, 902 orbital contribution 185 relative signs 888 H-H absolute signs 681, 713 correlation of geminal in alkenes with HCH angle 711 correlation of v/c/ha/ in ethanes with substiment electronegativity 680 correlation with Jc~ 1020 correlation with substituent electronegativity in vinyl derivatives 714 in acetylenes 177, 746, 748 in alkanes, geminal 677 in aikanes, vicinal 678 in alkenes, geminal 711, 714, 722, 728 in alkenes, vieinal 712, 714, 722, 728 in alkenes (metal) 743 in ammonia 183 in aromatic hydrocarbons 180, 682, 770, 1023 m butanes (2,3-disubstimted) 6 9 0 m butene-I 723 m i-hutches (1-substituted) 735 m cyclohexanes 679 m (CH3)ePI-IBHs 988 m ¢yclophane 173 m cyclopropanes 694 ]n esters (~t~.tmsaturated) 739 m ethanes 567, 681 m furans 782 m hexene-I 723 m indene 769 m olefines 710 m picolines 797 m propene-1 723 m propenes (2,3-disubstituted) 732 m propenes (2-substituted) 731, 735 m pyridines 794 m pyrroles 787 m quinolines 800 m silanes, geminal 823 m silicon compounds 1050 m thiazoles 805 m thiophenes 804 long range in acenaphthenes 685 long range in aromatic aldehydes 767 long range in indene 769 long range in olefmes 176, 685, 739, 743 long range in saturated compounds 683
xxxviii
SUBJECT INDEX
Coupling.... signs ofgeminal 172, 682, 695 Crossed coil detection 45 signs of vieinal 172, 682, 695 probe 208 H-Hg trans-Crotonaldehyde in 3-chloromercorffuran 1098 :H spectrum 469 in mercury alkyis 690, 1097 spectral parameters 469 in mercury vinyl 743 Crystal field calculations for SiCo shielding in organomercury compounds 1098 H_t'tN 1078 133~ r~6onanc~ in isouitriles 1040 of aqueous solutions 522 long range 1040 of metal in liquid ammonia 1093 temperature effect 819 of solid hafides 1093 H-xTO in water 509, 1048 H-SXp Cmnene, IH spectrum 830 for directly bonded nuclei 1061 Cyclic compounds, conformation 575 in (CHs)sPI'IBH3 988 Cycloalkanes in dialkyl phosphonates 1072 XH chemical shifts 692 in diphosphine 1074 Cycloalkanones, XH chemical shifts 692 m orgenophosphorus esters 1065 Cyclobutane(l,l-dimethyl-2,2,3,3-tetram pbosphites 1070 fluoro-), XgF spectrum 917 m phmphoryl halides 1077 Cyclobutane(1 -phenyl-2,2,3,3-tetrafluoro-), m phosphorus acids 1069 19F spectrum 917 m phosphorus alkyls 1064 Cyclobutanes, tgF spectral parameters 918 in PjNsCI4(SC2Hs)2 1075 Cyclobexane m symmetrical and unsymmetrical as a reference compound 264 trialkylphosphates 1063 axial-equatorial chemical shift 136 relativesigns 1061 Cyclohexane (bromo-), tH spectrum 576 H-2°TPb in plumbanes 824 Cyclohexane (chloro-), IH spectrum 576 H-19spt in platinum complexes I097 Cyclohexane (10~,3~.dimethoxy-2~acetoxy-), H~gSi in substitutedsilane 1050 IH spectrum 702 H-Sn, dependence on s character Cyclohexane (1~,3/~-dimethoxy-2~-acetoxy-). in stannanes 824, 1086 XH spectrum 702 in tetramethyl tin I013, 1088 Cyclobexane (methyl-), tH spectrum 706 in tin alkyl halides 1088 Cyclohexane (2,2,6,6-tetradeuteromethyl-) in vinyl tin 743 1H spectrum 706 H-TI Cyclohexanes in thallium alkyls 1090 configurations 703 in thallium phenyLs 1091 disubstituted, ~H spectra 708 in thallium vinyl 743 fluorinated 575, 577, 920 relative signs, 1091 XH chemical shifts 696 H-X (group IV element), correlation with XH specU'al line widths 708 atomic number of X 825 H - H coupling constants 700 long range 174 interconversion 575 axp_sIp 1061 monosubstituted, XH spectra 576, 706 in diphosphine 1074 1,2,3,4,5,6-hexachloro-, XH spectra 705 in non-cyclic phosphonitrilic halides perfluoro, interconversion ~$75 1078 Cyclohexyl in perfiuorotriphosphonitrile 951 acetates, XH spectra 703 in phosphates 1071 alcohols, IH specU'a 703 in phosphites 1070 Cyclopentadienyl anion, XH chemical in phosphorus acids 1069 shifts 779 in PsNsCIa(SC~Hs)2 1075 CyclopentadienyLs, IH chem/cal shifts 743, signs from double irradiation 466 780 signs from double quantum transitions Cyclophane, H - H coupling constants 173 454 2°3Tl-2°ST1 in thallium ethoxide 1092 Cyclopropane (l,l-dichloro-2-methoxy-),XH spectrum 693 CoupLing, virtual 814
SUBJECT INDEX Cyclopropane (l-methyl-2,2-difluoro-), XgF spectrum 917 CycloproFanes XgF chemical shifts 916 F - F coupling constants 916 XH chemical shifts 690 H - H coupling constants 694
6, the chemical shift (see Chemical shift) 5 for external refe~er.ce compounds 263 D,t point group, elements and character table 301 De~borane, 11B spectrum 982, 984 DecaUns, 1H spectra 580, 709 D e ~ o l s (10-methyl-), IH chemical shifts 709 Deceptively simple spectra 363 Derivation super operator 436 Deuteration in the determination of polymer ta~cities 836 Deuterium resonance 1092 coupling constants 1092 isotope effects 1092 Deuterochloroform as a solvent 257 Diamagnetic shielding 65 Diazo compounds, 1H chemical shifts 1139 Diborane, xlB spectrum 978 diammoniate, 11B spectral parameters 986 IH spectrum 979 Diboranes, 11B spectra 979 Dienes (conjugated), H - H coupling constants 741 Diethyl mercury, XH spectrum 687 Dioxane, laCH satellite bands 450 Dime delta function 79 Direct method for the analysis of spectra 435 Dished magnetic fields 204 1,2-Dithiane, 1H resonance 559 Double irradiation 240, 446, 447 in the determination of polymer structures 835 of the ammonium ion 458, 459, 460 of trans-crotonaldchyde 469 of dichloroacetaldehyde 461 of 2-furoic acid 468 of heteronuclear systems 455, 1093, 1640 of homonuclear systems 460 of 1,1,2-trifluoro-2-bromoethylene 915 relative signs of coupling constants 461, 466 theory of 244 to measure reaction rates 502
xxxix
Double irradiation, transitory 475, 476 with a weak second radiofrequency field 471 Double quantum, transitions 453 in 1,2-dibromopropionic acid 454 relative signs of coupling constants from 454 Doublet collapse by chemical exchange 487, 489 DSS, for aqueous solution ref~e,acing 265
Electric field effects intramolecuiar on 19F shielding 156, 157 on AB, AB a and AX3 spin systems 464 reaction field effects 88 Electrolyte solutions 511 relative molar chemical shifts 514 Electron densities from chemical shift measurements 781 spin resonance (F.SR) 4 Enthalpy difference between rotamers 570 Epoxides (monosubstituted) analysis of ZH spectra 377 2H chemical shifts 695
Equivalence chemical shift 283 magnetic 283, 308 symmetrical 284 Equivalent nuclei 284, 307 effect of conformational motion 285 Esters 170 chemical shifts 1045 ~fl-tmsaturated, 1H chemical shifts 737 Ethane (1,2-chlorobromo-), IH resonance 415 Ethane (1,2-dibromo-), IaCH satellites 450 Ethane (1,2-dichloro-), x3CH satellites 450 Ethane (1,1,2-trifluoro- l ,2-cfibromo-1chioro-), 19F resonance 563 Ethanes xsC chemical shifts 997 correlation with substiment electronegativity 996 calculation of geminal and vieinal H - H coupling constants 166 enthalpy differences between rotamefs 570
vicinal H - H coupling constants 567, 680 Ethanol 257, 273 13C spectrum 993 chemical exchange 493, 544, 508 IH resonance of aqueous solution 508
xl
SUBJECT
Ethyl derivatives ‘H chemical shifts 672, 673, 686 correlation of 13C chemical shifts with substituent electronegativity 996 correlation of ‘H chemical shifts with substituent electronegativity 670 metal derivatives 13C chemical shifts 999 ‘H chemical shifts 675 H-X coupling constants 688 Ethylene, H-13C coupling constants 448 Ethylene (l,l-dilluoro-) coupling constants 398 ‘H and lgF resonances 397 Ethylene (1,1,2-tritluoro-2-bromo-) lgF decoupled spectrum, 915 lgF spectrum 915 Ethylene [cis- and travel-fluoro-2-(perfluoroisopropyl)-] lgF chemical shifts 913 F-F coupling constants 913 ‘H chemical shifts 913 H-F coupling constants 913 Ethylene oxide, 13CH satellite bands 450 Ethylenes, calculation of seminal and vicinal H-H coupling constants 166 Ethylenes (chloro-) ‘H chemical shifts 722 H-13C coupling constants 722 H-H coupling constants 722 Exchange rate effect on a doublet 487 effect on a quartet 494 effect on a triplet 493 reactions electron 502 ion and group 507
lgF chemical shifts, @+ values 266 calculation of 151,266 correlations with substituent electronegativity 882 effect of solvents on 872 of alkyl and silyl selenides 963 of binary fluorides 880 of boron halides and derivatives 944,952 of fluorinated‘ heterocyclics of S, Se and P 926 of fluorine-containing polymers 954 of fluoroacyl metal compounds 895 of fluoroalkanes 885,957 of tluoroalkyl metal compounds 895, 959
INDEX
lgF chemical shifts, of fluorobenzenes 897 correlation of meta and para shifts with Hammett u constants 897 of fluorocarbon derivatives 879, 890 of fluorocyclobutanes 918 of fluoroethanea 886,958 of fluorohalohydrocarbons 883 of fluoromethanes 883 of fluoronaphthalenes 905 of hypofluorites 949 of inorganic fluorides 530, 880 of interhalogen compounds 940 of miscellaneous fluorine-sulphur containing compounds 939 of metal fluoride complexes 521,952,953 OfnitrogerMuorine containing compounds 946, of phosphorus halides and related compounds 949,959,960,%1 of sulphur hexafluoride and derivatives 930 of sulphur tetratluoride and derivatives 937 lgF (13C-12C) isotopic chemical shit correlation with JcF 1022 Fatty acids, ‘H chemical shifts 812 Fermi contact term 163 Field/frequency control 203, 229 Fig factor 45 First or&r spectra 280 Fluorinated cyclohexanes 920 Fluoroacetylene, lgFspectral parameters 916 Fluoroalkanes lgF chemical shifts 883, 885, 893, 957 lgF coupling constants 883, 885 Fluoroalkenes lgF chemical shifts 906,957 19F coupling constants 907, 909 H-H coupling constants 910 Fluorobenzne (ortLdichloro-) 19F coupling constants 398 lgF spectrum 399 Fluorobenmnes calculation of 19F chemical shifts 154 ortho effect 155 Fluorocarbon sulphides lgF chemical shifts 894 F-F coupling constants 894 Fluorocarbons calculation of coupling constants 183 containing nitrogen, lgF chemical shifts 889 Fluorocyclobutanes 918 Fluoroethanes conformational studies 889 F-13C coupling constants 1017 lgF spectral parameters 886, 958, 1017
SUBJECT INDEX Fluoromethanea F-‘% coupling constants 1017 19F spectral parameters 883, 1017 Fluoronaphthalenes, r9F chemical shifts 905 Fluoropropanes, 19F spectral parameters 887, 892 Fluorosilanes (methyl and ethyl), 19F chemicalshifts 927 Fhtx stabiliser 203 Formamide ‘Hspectrw 241,554 ‘Witradiation 241 Furan and substituted furans, ‘H spectral parametm 782 2-Furfurol, ‘H spectrum 364 ZFuroic acid l-H doubk resonance spectrum 468 ‘H spectral parameter% 467 ‘Hspecmrm 468 g, Land& or spectroscopic splitting factor 14 Gases, medium effects on chemical shifts 857 Gauge invariance 67 Geti coupling constants, see Coupling constants Gemmnes, spectral parameters 823 Glutaric acid (B-methyl-), ‘H spectrum 815 Glycidaldehyde, ‘H specuum 428 Glycidonitrile, ‘H spectrum 282 ‘Hchanicalshifts correlation with Ja 1020 empirical estimation 838 medium&cts 841 of acetylenes 745 solvent effects 749 of alkenes 711, 727,735, 741 anomalous 719 correlation with group dipole moments 718 correlation with Hammett u constants 717 effects of methyl group substitution 733 induced ring currents in annulenes 745 of alkyls 666,670,676, 685 of aromatic compounds 749, 770, 774 aromatic ions 774 correlations with Hammett u constants 752 effect of solvents 749 polyJluclear 770 ring currents 770 of cyclohexanes 696
xli
‘H chemical shifts, of cyclopropanes 690 of ethyl derivatives 670 carrelatioa with substituent electronegativity 670 of heterocychc compounds 782, 787, 794,798 alkaloids 806 amino acids and peptides 812 effect of solvents 798 fatty acids 812 furans 782 pyridines 794 pyrroles 787 quinolines 798 steroids 808 thiaxoks 805 thiophenes 802 of hydrogen attached to atoms other than carbon 816 of metal alkenes 742 of metal alkyls 689 of methyl derivatives 666 of paramagnetic species 826 of polymers 829 of propyl derivatives 676 of water effect of complex formation 520 effect of electrolytes 511 effect of hydrogen bonding 537 for ions at infinite dilution 515 relative molar 514 Hl (radiofrequency field) effect on spectra 230 meawrement 230 Halide ion exchange 511 Halogen fluorides 530, 940 Hamiltonian 290 matrix elements 304 HD, calculation of coupling constant 160 HCtWCpliCS
calculation of rH chemical shifts 145 effect of solvents on ‘H chemical shifts 855 fluorinated derivatives of S, Se and P 926 H-13C coupling constants 1021 ‘H spectra 782 Hexaborane, llB spectrum 982 Hexen* lH chemical shifts 723 H-H coupling constants 723 199Hg and ‘OIHg resonant 1097 chemical shifts of alkyls 1097 coupling constants in alkyls 1097 in 3-chloromercurifuran 1098 in organo-mercury compounds 1088
xiii
SUBJECT
I-U&red internal rotation 551 barrier heights 555,570 nature of barrier to 573 of C-C bond 559 of C-N bond 553 of C-P bond 572 of N-N bond 551 of N-O bond 556 of Ss bond 558 Homogeneity, see Magnets Hydration of ions 1048 Hydrocarbons, calculation of chemical shifts 135 Hydrogen bonding 534,816 chemical shifts due to 537 inamines 548,818 in halogenated compounds 548 in hydroxylic compounds 543 in mercaptaos 548,820 nature of 535 procedures for studying 539 peroxide, proton exchange 510 Hydroxyl groups, ‘H chemical shifts 816 proton exchange 507 Hyperfine interaction constants 115, 177 Hypoffuorites. r9F spectral parameters 948 Z, spin value 1,589 I2 operator 294 Is71 resonance of I’ aq 517,X096 of 1; 531, IO96 Indene ‘H spectrum 769 signs of H-H coupling constants 769 Indoles, ‘H spectra 793,794 INDOR 475,989,1033 Induced current model 79 Inositols. ‘H spectra 706 Integration of spectra 236 Intensities (relative) of multiplet components 7 Intensity measurements of bands 234 sum rule 380,452 transiton 306 Interatomic currents in benzene 82 in ethylene 82 in molecules 81 Interhalogen compounds, 19F spectral parameters 940 Internal rotation, see Hindered internal rotation und Conformation
INDEX
Inversion of nitrogen compounds 574 Iodine pentatktoride, chemical excw 943 Ions aromatic 774 inorganic, infinite dilution shifts 515 Iron carbonyl (pertluoroalkyl-), rgF spectral parameters 926 Irreducible representation 299 Isonitriles, H-i4N coupling constants 1040 Isopropyl benzene, ‘H spectrum 6 Isotopic effect on 19F chemical shifts 875, 916, 962 correlation with Jox 1022 on ‘H chemical shifts 1092 Isotopic substitution as an aid to analysis 445
3-Ketoallopregnrmes, iH spectra 809 Keto-enol tautomerism 549 Ketones, 170 chemical shifts 1045
Lamb diamagnetic shielding 66 Land& (spectroscopic) splitting factor 14 Langevin equation 35 Larmor .equation 13,67 precession 11 Lead (207Pb) resonance 1100 chemical shifts comparison with 19gHg andZosTl 1098 in inorganic compounds 1101 in organic compounds 1101 Lead tetraethyl, ‘H spectrum 687 Leakage 210 optimum value 211 Lime broadening from chemical exchange 485 from sweep rate 224 saturation 34, 40 spin-lattice 31, 32 shape 42 distortion 33 effect of quadrupolar relaxation 496 Lorentzian 505 of collapsing doublet 407. 489 of collapsing quartet 494 of collapsing triplet 493 widths andT2 40 effect of saturation 33 for liquids 30 Gaussian 30
xliii
SUBJECTINDEX Line..., in electron exchange reactions 506 in proton transfer reactions 510 Lorentzian 30,40 mean square 31 measurement of T2 from 49 of ‘*N resonances of ammonia and related compounds 1035 of 14N resonances of nitrites, nitrates and nitro compounds 1036 solvation of ions using 524 variation from quadrupolar relaxation 1102 ‘Li resonance of the metal in liquid ammonia 1093 Long range coupling constants, see Coupling constants Longitudinal relaxation time, Tl (see TJ 19,37 Lorentxian curves absorption (x”) 39 dispersion(x~ 39 Lorentx-Lorentx equation 98 Low resolution NMR 4 Lowering operators 292 Lunacrine, *H spectra 807 Lunine, ‘H spectra 807 2,6-Lutidine, ‘H spectrum 328 3,5-Lutidme, ‘“C spectrum 1008
Magnetic complex susceptibility 36 equivalence 59,283 effect of symmetry on 564 field effect on spectrum 227,442 recurrent sweep 223 shape 204 slow sweep 223 sweep 223 moment of electron 2, 14 of nucleus, see Nuclear moments 589 quantum number, m 14 radiofrequency susceptibility 34.40 screening of nuclei 65 shielding in atoms 66 shielding in molecules induced current model 79 perturbation theory 68 variation theory 77 static susceptibility 35, 65 Magnetisation vector, M 29, 37 Magnetogyric ratio, y 11, 14 Magnets 201 cycling 204
Magnets, field homogeneity 272 shimcoils 202 stabilisatkm 202 Medium efkts on 19F chemical shifts 872 on ‘H chemical shifts 841,857 Mercaptan (methyl), ‘H spectrum 334 Mercaptans, hydrogen bonding 548 Mercurifuran (3chloro-), H-199Hg coupling constants 1098 Mercury alkyls, H-199Hg coupling constants 690, 1097 chemicalshifts 1098 diethyl, ‘H spectrum 687 dimethyl, 19?Hg spectrum 1100 Metal carbonyls 823 ‘JC chemical shifts 1010 “0 chemical sbifts 1047 ethyl derivatives, lH chemical shifts 675 fluoride complexes 9S& 953 Methane calculation of H-H coupling constant 163 effect of solvents on ‘H chemical shift 842 Methanol, ‘H spectrum 544 c&Methyl crotonate, ‘H spectrum 737 Methyl derivatives lsc chemical shifts 992, 995 correlation with substitucnt electronegativity 996 13GH coupling constants 1013,1014 ‘H chemical shifts 666,668 correlation with substitucnt electronegativity 666 of group IVR elements 13C chemical shifts 999 ‘H chemical shifts 685 Methyl halides, ‘H chemical sbifts 667 Molecularbeams 2 Momenta, of spectra 431 Multiplet collapse by chemical exchange 488
‘*N chemical shifts
1033
of nitrates, nitrites and nitro compounds 1036,1037 “N resonant 1031 line widths of ammonia and related compounds 1035 line widths of nitrites, nitrates and nitro compounds 1036 of complex ammines 527
xtiv
SUBJECT INDEX
~4N resonance, of liquid ammonia containing alkali metals 1093 quadrupolar effects 1037 23Na resonance 1092 of N a ~ 515,5.17,520 Naphthalene calculationof chemical shifts 145 IH spectrum 405, 406, 773 H - H coupling constants 407 l,ro~ 19F spectrum 1101 93Nb spectrum 1101 Nickel complexes contact shifts 826 Ni(I1) N,N'-di(6-quinolyl)aminotroponeiminate, XH spectrum 828 Ni(H) ethylene diamir~e, IH spectrum 827 Nitric acid, XH resonance 517 Nitrites (alkyl), rotational isomerism 556 Nitrogen compounds, inversion of 574 Nitromethane as a solvent 259 Nitrosamines, rotational isomerism 5 5 i Niobium (93Nb) F-93Nb coupling constant in NbF~ 1101 spectrum of HbF~s 1101 NMR spectrometers 200 basic requirements 200 commercial 248 cycling of electromagnets 204 magnets for 201 Notation for spin systems 283 Nuclear induction 44 magnetic resonance absorption 45 magnetisation in phase 36 out of phase 36 moments 1, 3, 10, 1102 of IH nucleus 14 paramagnetic susceptibility 3, 35 properties 589, 1109 quadrupole splittings in asCl and ~TC1 resonances 1096 spin, I 1, 589 Nutation 52
lvO chemical shifts 1042, 1044, 1046 correlation with electronic transition energies 1043 of transition metal compounds 1044 effect of paramagnetic ions 1047 of metal carbonyls 1047 lye resonance 1041 aqueous electrolytes 523, 526 170-H coupling constant in water 509
Olefines, see Alkenes and Vinyl derivatives Operators, rahing and lowering 292 Overhauser effect 116, 247
3xp chemical shifts 1053 characteristic 1066 compilation 1143 correlation with bond properties 1056 in perfluoroalkyl phosphorus derivatives 959 in quadruply connected phosphorus compoonds 1059, 1060, 1144 in triply connected phosphorus compounds 1058, 1059, 1143 of cyclic phosphorus compounds 1075 of phosphates (condensed) 1073, 1145 of phosphines 1143 of phusphites 1068, 1143, 1147 of phosphonates 1072, 1149 of phosphonitrilic compounds1075, 1077, 1152, 1154 of phosphorus acids 1069, 1074 of phosphorus-nitrogen containing compounds 1148 of phosphoryi halides 1076 31p coupling constants 1061 relative signs 1061 sip resonance 1051 Paddles 209 Paramagnetic current density 80 susceptibility 35 2°TPb chemical shifts comparison with those of 1~gHg and 2°STl 1098 in inorganiccompounds 1101 in organic compounds 1101 2°VPb resonance 1100 Peaked magnetic fields 204 Pentaborane-9 11B spectrum 981 decoupling of 11B nuclei 456 IH spectrum 456 Pentaborane-ll, ~IB spectrum 981 Pantahorane halides, ~IB spectra 978, 985 Peptides, 1H chemical shifts 812 Perchloryl fluoride and related compounds, ~gff spectral parameters 943 Perfluoroacyl metal compounds chemical shifts 895 coupling constants 896 Perfluoroalkyl metal compounds chemical shifts 895 coupling constants 896
SUBJECT
Perfbtoroalkyl, sulphur hexafiuoride derivatives, tgF spectral pammeters 932 sulphur tetrafluoride derivatives 937 Perfluorocyclohexane axiaJ-eqwtorid chemical shift 139 lgF spectral parameters 920 PerSuorocyclohexanes dihydro-, lgF chemical shifts 922 monosubstituted, lgF chemical shifts 921 lgF chemical shifts 925 PerSuorodecalin (cis- and trans-) conformation 922 lgFspectra 924 Perfluoroisopropyl bromide, lgF spectrum 872 Perfhtoromethylcyclohexane, *‘F spectrum 876,921 Perfhtoropiperhline conformation 925 lgF spectrum 925 Perfluorovinyl metal compounds lgF chemical shifts 906 F-F coupling constants 907 Permanganate-manganate exchange reaction 506 Permutation operator 106 Perturbation theory 68, 428 Phasememory 29 relaxation time, see T2 Phenol (2,4,6tribromo-), ‘H spectrum with 13C satellites 1023 Phenols 1% spectral pammetem 1006 chemical shift/vibration frequency relation 547 ‘H chemical shifts of hydroxyl group 818 hydrogen bonding 545 Phosphate isohypo- ion, 3lP spectrum 1068 tetrapoly- ammonium sal& 31P spectrum 1071 Phosphates 31P chemical shifts 1144 condensed 31P chemical shifts 1145,1073 JlP spectra 1073 triallcyl, H-31P coupling constants 1063 Phosphine (di-), 31P coupling constants 1074 Phosphines, 31P chemical shifta 1143 Phosphite (trimethyl-), 31P spectrum 1054 Phosphites (organic), slP spectral parameters IM8, 1143,1147 Phosphonates, 31P spectral parameters 1072, 1149
INDEX
XIV
tri-Phosphonitrile (hexatluoro-), lgF spectrum 951 Phosphonitrilic (non-cyclic) halides 31P chemical shifts 1152 31P spectra 1077 Phosphorus triethyl-, ‘H spectrum 371 acids, 31P spectral parameters 1069 alkyls, H-31P coupling constants 1064 compounds, compilation of chemical shifts 1143 halides and derivatives, lgF spectral para meters 949,959,960,1055 organic esters, H-31P coupling constants 1064 Phosphoryl halides 31P chemical shifts 1076 jlP spectra 1076 Platinum (lgsPt), H-lQ5Pt coupling constants in platinum complexes 1097 Plumbanes, spectral parameters 823 P3N3Clr(SC2H5)2, 31P spectrum 1075 Polymers ontainiig fiuorine 954 copolymers of butadiene and isoprene 831 segmental motion 829 structure 831, 835 tacticity 833, 836 Poly+l)-methyl-l-alkenes 833 Polymethyhnethycrylate, ‘H spectrum 834 Polypropylene, ‘H spectra of various tatticities 837 Polystyrene 838 ‘Hspectrum 830 Polyvinyl chloride, ‘H spectrum 836 Population of spin states 16 Porphyrins calculation of chemical shifts 147 ‘H chemical shifts 792 Probe (sample holder) 207 Product spin functions 293,297 Propane, ‘H double resonance 476 PropargYl bromide+ ‘H spectrum 747 halides, ‘H chemical shifts 747 Propene-1 ‘H chemical shifts 723 H-H coupling constants 723 lHspectrum 724 eis- and trans-l-fluorolQF chemical shifts 912 ‘H chemical shifts 912 H-F coupling constants 912 2,Edichloro-, ‘H spectrum 388
xlvi
SUBJECT INDEX
Propene-2 (cis- and Mns-2-fluoro-3-chloro-) 1‘F chemical shifts 911 H-F coupling constants 911 ‘H chemical shifts 911 PropeneS
2-substituted ‘H chemical shifts 730 H-H coupling constants 731 2,3disubstituted ‘H chemical shifts 723 H-H coupliag constants 723 Propionic acid l&dibromo-, ‘H spectrum 454 f3-dibromo; doubk resonance 473,474 Propyl (i and n) derivatives, ‘H chemical shifts 672,676 Proppe,
13C spectrum
994
ProtoIysis reactions 507 Proton exchange in alcohols 507 inanlines 1040 Pterocarpin, ‘H resonance spectrum 814 Puke methods 40.50 electrolyte solutions 527 Pyridine as a solvent 259 ‘3C spectrum 994 ‘H spectrum 794 3-bromo-, ‘H spectrum .706 1,2,5,6-tetrahydro‘H decoupled spectrum 798 ‘H spectrum 798 Pylidines 13C spectra 1008 ‘H spectral parameters 794 Pyridinium ion, ‘H spectrum 1040 with l*N decoupled 1040 Pyrrole effect of solvents on ‘H spectrum 856 ‘H spectrum 457,787 ‘H spectrum with ‘*N decoupled 457 Ndeutero-, IH spectrum 787 Pyrroks, ‘H spectral parametez3 787
QUadNpol~ effects in ‘*N spectra 1037 line broadening 970, II02
QMpole
moments 27,589 relaxaton 26,426 effect on ‘H signals of NH groups 819 Quantitative analysis optimum spectr0mete.r conditions 211, 234 of phosphorus compounds 1076
Quantum mechanical formalism 287 Quartet colkpse by chemical exchange 494 Quinoline IH spectral parameters 798 5,7-dichloro-, ‘H resonance spectrum 801 5,7dimethyl-, ‘H resonance spectrum 801 I-methyl-, ‘H resonance spectrum 801 Radiofrequency effect on spectra 227 oscillator 227 power effect on spectra 230 measurement 230 receiver 232 Raising operators 292 Rapid passage experiments 40,497 s7Rb chemical shifts, of solid halides IO93 *‘Rb resonance of the metal in liquid ammonia 1093 Reaction fields 88,95,138,841 in purudinitrobenzene 90 Reference compounds 61,260 scale interconversion 267 Referencing external 260 for ‘“C resonance 991 for lgF resonance 266 for IH rseonance 264 intemal 263 Relaxation quadrupole 26,496 spin-lattice (see uiso TI) 18, 76 spin-spin (see a/so Tz) 28 Representations irreducible 299 reducible 299 symmetry group 298 Resonance condition classical description 10 quantum mechanical description 15 equation 13, 59 Ring currents 141,595, 770 Ringing 40,224 Rotating fields 13 co-ordinates 50,245 Rotational isomerism 551 about GC bond 559 solvent effect on .~ 568 temperature effect on 569
SUBJECT
Rubidium (s7Rb) chemical shifts of solid halides 1093 resonance of metal in liquid ammonia 1093
Sample containers 268 coaxial 268 for small volumes 270 for temperature studies 271 having large filling factors 269 sealing 271 spherical 268 preparation 256 spinning 206 Satellite spectra, 13CH of substituted ben7xnes 1022 Saturation 33,231,461 degree of 38 factor 34 12%b resonance of NaSbFe 1097 screening, see Shielding Secuhu dekrminant 296, 323 equation for NMR 288, 289 factorising 290 of AR spin system 312 Sekction rules 310 Selenium (“Se) chemical shifts of selenous and selenic acids 1097 2gSi chemical shifts 1049 comparisonwith 13Cchemical shifts 1051 2gSi resouance 1048 Sensitivity enhancement 230 SFaOCsH,, lgF spectrum 931 Shape factor 65 fuxtion 30,33 Shielding, diamagnetic contribution
70,75, 78 inalkenes 7% in benzenes 752 in linear molecules 73 in molecules containing heavy atoms 74 paramagnetic contribution 70, 75, 78, 894,895 Shielding constant or ccefkient, u, (see f&o Chemical shift) 61, 66 anisotropy of 94, 113,135, 136 calculation 120 by induced current model 130,595, 770 for halogen hydrides 124 for group VI hydrides 127 for H2 molecule 120
INDEX
xlvii
Shielding. . ., for hydrocarbons 129 effect of, electric field 85 isotopic substitution 99 medium 841, 857, 872 molecular iuteractions 90 Van der Waals forces 95 effects, S, and S,,, in benzenes 754, 1140
of lgF nuclei 874 influence of Iow-lying excited states 894,895 temperature dependence 101 shimcoils 202 Sideband, see Calibration of spectra und Double irradiation calibration of spectra 236 interpolation 275 single 244 superposition 275 Signal shape, see Line shape to noise ratio 233 Signs of coupling constants, see Coupling constants Silanes trisubstituted, ‘H chemical shifts 822 fluoro-, lgF chemical shifts 927 Silver ion complexes 529 Simplification of spectra 277 single coil detection 45 probe 217 “‘Sn chemical shifts 1083 of tin alkyk 1083 of tin halides 1083, 1084 lagSn coupling constants 1086 r ’ ‘Sn resonance 1082 Solvent effects on 13C chemical shifts 992 “‘Co chemical shifts 1081 “F chemical shifts 872 ‘H chemical shifts of acetonitrile 846 aatyknes 749, 848 aromatic compounds 258, 749,851 benrene 258 methane 842 unsaturated heterocyclic compounds 798, 855 rotational isomerism 568 ‘rgSn chemical shifts 1084 spectra 442 Spectral density functions 21 Spectrometers, see NMR spectrometers Spectroscopic
(Landk) splitting factor, g 14
xlviii
SUBJEC TINDEX
spin angular momentum
operators 15,290,291 commutation rules 291 correlation matrix 181 correlation function in benzene and naphthalene 182 decoupling (see uZso Double irradiation) 240 theory 244 densities in tetrahedral Co(U) complexes 828 eigenfunctions 15,293 eigenvalues 15 exchange 29 Spin-lattice relaxation me&anism 18, 76 time (see also Tr) 19 number (or value), Z 1, 589 Spin product functions 293,297 temperature 17 negative 17 system notation 283 Spinning of sample 206 Spin-spin coupling effect of time averaging 65 mechanism 63 molecular orbital calculations 106 valence bond calculations 109 variation method 111 coupling constant, J (see aLso Coupling constants) 7, 61 calculation for HD 160 evaluation 109,111 theoretical calculation of 103, 105, 160 reiaxation mechanisms 28 time (see also T2) 29 Spontaneous emission 18 Starmanes, spectral parameters 823.1086 Stamk bromide and iodide, ilgSn spectra 1084 Steroids, ‘II chemical shifts 808 Strong electrolyte solutions 511 Stytene 'H chemical shifts 719 ‘H spectrum 381 oxide, ‘H spectrum 362,377 Sub-spectra, superposition of 345 Succinic (methyl-) acid, ‘H resonance spectrum 815 Sulphur dioxide as a solvent 259, 519 hexafluoride
Yulphur, disubstituted, lgF spectral parameters 935 monosubstituted, igF spectral parameters 930 tetrafluoride rgF spectrum 531,936 lgF spectral parameters 936 perlkoroalkyl derivatives, rgF spectral parameters 937 Super operators 435 derivation 436 matrix representation 437 Susceptibility bulk diamagnetic corrections 260 diamagnetic 65 pammagnetic 35 volume 3, 262, 605 (compilation) Symmetrical equivalence 284 Symmetrised spin functions 321 notation 335 Symmetry transformation group 298 t values 265 charts 1131 for a variety organic compounds (Tiers’ compilation) 1115 Tl,spin-lattice (or longitudinal) relaxation time 19 contribution from aniso&opy of u 76 effect of anisotropic shielding 27 diamagnetic substances 26 ZH on rH resonance 23 pammagnetic substances 21, 25. 30 pressure 28 quadrupoles 21, 26 viscosity 23 intermolecular contribution 22 intramolecular contribution 22 measurement by direct method 46 measurement by progressive saturation method 47 of electrolyte solutions 527 of ice 19 of liquids 19 of solids 19 T,,spin-spin (or transverse) relaxation time 29, 37 effect of paramagnetic ions 30,505 inter- and intramolecular contributions 30 measurement of 49 measurementsonelectrolytesolutions 527 Tautomerism 549 Temperature
SUBJECT Temperature, control of sample 212 dependence of sgCo chemical shifts 1081 effect on rotational isomerism 569 Tetraborane, rrB spectrum 981 Tetraethyl lead, ‘H spectrum 687 Tetramethyl silane (TMS) 264 as a universal reference 267 Thallium alkyls, H-z05Tl coupling constants 1090 ethoxide structure 1091 203Tl-zo5Tl coupling constant 1092 phenyls, H-20sTI coupling constants 1091 salts, 2osTl chemical shifts 1089 effect of anions 1090 Thiazoles, ‘H spectral parameters 805 Thiophene t-bromo- lH spectrum 281 3-bromo-Zaldehyde, ‘H triple resonance 476 2-bromo-S-chloro-, double resonance 472 deuterated, ‘H spectrum 804 2-methylthio-3-thiophene thiol, ‘H spectrum 311 “Tickling” experiments 471 Tin alkyls rlgSn chemical shifts 1083 llgSn coupling constants 1087 halides 531 llgSn chemical shifts 1083, 1084 1lgSn-H coupling constants 1087 resonance spectra 1082 tetramethyl, llgSn spectrum 1087 205Tl chemical shifts 1088 comparison with those of 1lgHg and 207Pb 1098 of thallium salts 1089 effect of anions 1090 205Tl coupling constants 1090 in thallium phenyls 1091 in thallium trialkyls 1091 signs 1091 205Tl resonance 1088 203Tl and 205Tl resonances of aqueous electrolytes 522 Tl(I)-TI(III) exchange reaction 506 T12Cld, 2osTl spectrum 1089 Toluene, ‘H spectrum 6 Trace of a matrix 431 Transient effects 40 Transition intensities 306,310,315 metal compounds, ‘H spectra 825
INDEX
XliX
Transition, “0 chemical shifts 1044 probabilities 16, 19, 33 Transistory selective irradiation 475, 476 Triaryl carbonium ions, ‘H chemical shifts 774 Trifluoroacetic acid as a solvent 989 r3C I-NDOR spectrum 989 Triple resonance 476 Triplet collapse by chemical exchange 491 Tropylinium cation, rH chemical shifts 779 Twin-T bridge 217 u-mode (dispersion)
42, 210, 218
V-mode (absorption) 41, 210, 218 VO_V(IV) exchange reaction 504 Variation theory 77 Vicinal coupling constants, see Coupling constants Vinyl bromide ‘H spectra for various solvents 443 ‘H spectrum 711 Vinyl chloride ‘H chemical shifts 722 H-l% coupling constants 722 H-H coupling constants 722 Vinyl cyanide and its methyl derivatives, ‘H chemical shifts 719, 734 Vinyl derivatives ‘H chemical shifts 717, 727, 735 anomalous 719 correlation with group dipole moments 718 correlation with Hammett u constants 717 correlation with substituent electronegativity 714, 717 effect of methyl substitution 733 disubstituted ‘H chemical shifts 727 H-H coupling constants 727 metal, H-X coupling constants 743 trisubstituted ‘H chemical shifts 735 H-H coupling constants 735 Vinyl ethers, ‘H chemical shifts 721 Vinyl fluoride, coupling constants 425 Virtual coupling 814 Volume susceptibilities 3,262 compilation 605
Water as a solvent
257
1
SUBJECT INDBX
Water, effect of electrolytes on *H resonance 516 H-*?O coupling constants 1048 proton exchange 509 Wiggle beat method of spectrum calibration 276 Wiggle beats 41
Wiggles
40, 224, 226, 508
Xylenes (substituted) 1H chemical shifts 761 correlation with Hammett a constants 762
