Vibrational spectra: principles and applications with emphasis on optical activity

studies in physical and theoretical chemistry 85

VIBRATIONAL SPECTRA" PRINCIPLES AND APPLICATIONS WITH EMPHASIS ON OPTICAL ACTIVITY

studies in physical and theoretical chemistry Recent titles in this series

53 54 55 56 57 58

Potential Energy Hypersurfaces by P.G. Mezey MathlChemlComp 1987 edited by R.C. Lacher Semiconductor Electrodes edited by H.O. Finklea Computational Chemistry by M.D. Johnston, Jr. Positron and Positronium Chemistry edited by D.M. Schrader and u Jean Ab Initio Calculation of the Structures and Properties of Molecules by C.E. Dykstra 59 Physical Adsorption on Heterogeneous Solids by M. Jaroniec and R. Madey 60 Ignition of Solids by V.N. Vilyunov and V.E. Zarko 61 Nuclear Measurements in Industry by S. Rozsa 62 Quantum Chemistry: Basic Aspects, Actual Trends edited by R. Carb6 63 Math/Chem/Comp 1988 edited by A. Graovac 64 Valence Bond Theory and Chemical Structure edited by D.J. Klein and N. Trinajsti~ 65 Structure and Reactivity in Reverse Micelles edited by M.P. Pileni 66 Applications of Time-Resolved Optical Spectroscopy by V. Br~ckner, K.-H. Feller and U.-W. Grummt 67 Magnetic Resonance and Related Phenomena edited by J. Stankowski, N. Pilewski, S.K. Hoffmann and S. Idziak 68 Atomic and Molecular Clusters edited by E.R. Bernstein 69 Structure and Properties of Molecular Crystals edited by M. Pierrot 70 Self-consistent Field: Theory and Applications edited by R. Carb6 and M. Klobukowski 71 Modelling of Molecular Structures and Properties edited by J.-L. Rivail 72 Nuclear Magnetic Resonance: Principles and Theory by R. Kitamaru 73 Artificial Intelligence in Chemistry: Structure Elucidation and Simulation of Organic Reactions by Z. Hippe 74 Spectroscopy and Relaxation of Molecular Liquids edited by D. Steele and J. Yarwood 75 Molecular Design: Chemical Structure Generation from the Properties of Pure Organic Compounds by A.L. Horvath 76 Coordination and Transport Properties of Macrocyclic Compounds in Solution by B.G. Cox and H. Schneider 77 Computational Chemistry: Structure, Interactions and Reactivity. Part A edited by S. Fraga Computational Chemistry: Structure, Interactions and Reactivity. Part B edited by S. Fraga 78 Electron and Proton Transfer in Chemistry and Biology edited by A. MUller, H. Ratajczak, W. Junge and E. Diemann 79 Structure and Dynamics of Solutions edited by H. Ohtaki and H. u 80 Theoretical Treatment of Liquids and Liquid Mixtures by C. Hoheisel 81 M6ssbauer Studies of Surface Layers by G.N. Belozerski 82 Dynamics of Excited Molecules edited by Kozo Kuchitsu 83 Structure, Fluctuation and Relaxation in Solutions edited by H. Nomura, F. Kawaizuma and J. Yarwood 84 Solid State NMR of Polymers edited by I. Ando and T. Asakura

s t u d i e s in physical a n d t h e o r e t i c a l c h e m i s t r y 85

VIBRATIONAL SPECTRA: PRINCIPLES AND APPLICATIONS WITH EMPHASIS ON OPTICAL ACTIVITY by RL. POLAVARAPU

Department of Chemistry Vanderbilt University 7332 Stevenson Center, Nashville, TN 37235 U.S.A.

1998 Amsterdam - Lausanne - New York - Oxford

- Shannon

ELSEVIER - Singapore - Tokyo

ELSEVIER SCIENCE B.V. Sara Burgerhartstraat 25 P.O. Box 211, 1000 AE Amsterdam, The Netherlands

9 1998 Elsevier Science B.V. All rights reserved.

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To Bharathi, Sruthi and Aaseesh

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vii

Preface This book originated out of a desire to have the vibrational absorption, Raman scattering, vibrational circular dichroism and Raman optical activity combined in one source. The younger generation students practicing vibrational spectroscopy would undoubtedly come across these different areas and would benefit from having these topics covered in a unified manner. Some of the material presented in this book, especially portions of Chapters 2-4, has been taught as a part of Molecular Spectroscopy and Quantum Chemistry courses for graduate students. The topics covered are generally of an advanced level, so this book would be appropriate for graduate students and practicing scientists in vibrational spectroscopy. To keep the book reasonable in length, the contents are limited and several important related topics are not included. For example, rotational structure associated with a vibrational band is not considered. Similarly, in discussing the applications of symmetry, character tables of point groups are not included as they are available in many text books. It is not possible to write a book on vibrational spectroscopy, without being influenced by the pioneering books of G. Herzberg, Infrared and Raman Spectra, E. B. Wilson, J. C. Decius and P. C. Cross, Molecular Vibrations, and of S. Califano, Vibrational States. I have also depenOecl on the books of P. W. Atkins, Molecular Quantum Mechanics, and of L. D. Barron, Molecular Light Scattering and Optical Activity, for some of the material covered in this book. It is a pleasure to acknowledge Professor L. D. Barron, who patiently read most of the chapters and made valuable comments; Professor L. A. Nafie for very useful general comments, Professor T. A. Keiderling for suggestions on Chapters 11-14, Professor P. R. Griffiths for comments on Chapters 11-13, and Professor J. Tellinghuisen for help in scanning the figures. The influence of former mentors is a key factor in the development of one's career; so my sincere appreciation goes out to Professors Laurence A. Nafie, Duane F. Burow, Surjit Singh and A. S. N. Murthy, with whom I have conducted research during my formative years. I would like to thank Ms. Swan Go of Elsevier for her patience in letting me complete this book even after missing many deadlines; and Jewell Carter for an excellent job in typing most of the manuscript. Finally, copyright permissions from John Wiley & Sons, Wiley Chichester, The Royal Society of Chemistry, Munksgaard International, Cambridge University Press, The Society of Applied Spectroscopy, and The American Chemical Society are gratefully acknowledged.

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ix

CONTENTS 1 2 2.1 2.2 2.2.1 2.3 2.3.1 2.3.2 2.3.3 2.3.4 2.4 2.5 2.6 3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 4 4.1 4.2 4.3 4.4 4.5 4.6 5 5.1 5.2 5.3 5.4 5.5 6 6.1 6.2 6.2.1 6.2.2

Introduction Quantum theoretical techniques Interaction with electric field Time independent perturbation theory

Static electric dipole polarizability

1 3 4 7

9

Time dependent perturbation theory

10

Electric dipole-electric dipole polarizability Electric dipole-magnetic dipole polarizability Expressions for general molecular properties Interaction with circularly polarized light

11 14 15 16

Transition rates Spectral intensities Transition polarizabilities Diatomic molecules Vibrational equation Energy levels of harmonic oscillator Energy levels of anharmonic oscillator Energy levels of Morse oscillator Electric dipole transition moments of harmonic oscillator Electric dipole transition moments of anharmonic oscillator Electric dipole transition moments of Morse oscillator Magnetic dipole transition moments of harmonic oscillator Magnetic dipole transition moments of anharmonic oscillator Transition polarizabilities Polyatomic molecules Energy levels of harmonic oscillators Energy levels of anharmonic oscillators Electric dipole transition moments of harmonic oscillators Electric dipole transition moments of anharmonic oscillators Magnetic dipole transition moments of harmonic oscillators Magnetic dipole transition moments of anharmonic oscillators Vibrational analysis Cartesian displacement coordinates Internal coordinates Energy distribution Non-equilibrium geometries Redundant internal coordinates Local modes Hamiltonian in local coordinates Approximations

19 21 23 27 27 30 32 41 43 44 51 53 55 58 61 61 64 73 74 78 78 81 81 86 96 97 99 107 107 110

Independent local oscillators Perturbation treatment of inter-oscillator coupling

110 I 11

7 7.1 7.2 7.2.1 7.2.2 7.2.3 7.2.4 7.2.5 8 8.1 8.1.1 8.1.2

8.1.3 8.1.4 8.2 8.2.1 8.2.2

8.2.3

8.2.4 9 9.1 9.1.1 9.1.2

Vibrational frequencies and force constants Molecular orbital theory Energy derivatives First derivatives of energy Second derivatives of energy Higher derivatives of energy Experimental determination of force constants Sum rules Vibrational absorption and circular dichroism Vibrational absorption spectra Experimental vibrational absorption intensities Quantum mechanical methods (A) Nuclear atomic displacement gradient method 03) Numerical electric field gradient method (C) Analytic wavefunction derivative method (D) Analytic energy derivative method (E) Analytic dipole moment derivative method (F) Analytic nuclear electric shielding tensor method Classical models (A) Atomic charge concepts (B) Bond charge and bond moment concepts Sum rules Vibrational circular dichroism spectra Experimental VCD intensities Quantum mechanical methods (A) Localized molecular orbital method 03) Analytic magnetic fieM perturbation method (C) Analytic nuclear electric shielding tensor method (D) Vibronic coupling method (E) Localized orbital-local origin method Classical models (A) Atomic charge concepts (B) Bond charge and bond moment concepts (C) Ab initio bond charge concepts Sum rules Vibrational Raman and Raman optical activity Vibrational Raman spectra Vibrational Raman scattering cross sections Quantum mechanical methods (A) Numerical atomic displacement gradient method 03) Numerical electric fieM gradient method (C)Combined atomic displacement-electric field gradient method (D) Analytic energy derivative method (E) Analytic electric dipole moment derivative method

117 118 124 124 126 130 131 137 143 143 143 146 147 148 149 152 152 153 155 155 157 159 161 162 163 165 167 169 171 173 173 173 173 176 177 183 183 183 188 188 189 189 190 190

xi 9.1.3 9.1.4 9.2 9.2.1 9.2.2 9.2.3 9.2.4 10 10.1 10.2 10.3 10.4 10.4.1 10.4.2 10.4.3 11 11.1 11.2 11.2.1 11.2.2 11.3 11.4 11.5 11.5.1 11.5.2 11.5.3 11.5.4 11.5.5 12 12.1 12.2

Classical models (A) Atom-dipole interaction concepts 03) Bond polarizability concepts Sum rules Vibrational Raman optical activity spectra Vibrational Raman optical activities Quantum mechanical methods (A) Nuclear displacement method 03) Orbital polarizability model Classical methods (A) Atom and bond polarizability concepts (B) Ab initio bond polarizability concepts Sum rules Applications of symmetry Introduction Symmetry operations and point groups Representations Vibrational properties Fundamental vibrations and symmetry species Symmetry of vibrational states Selection rules Principles of spectral measurements Introduction Polarization modulation using a photoelastic modulator Circular dichro ism (A) Sample configuration (B) Calibration configuration Linear dichroism (A) Sample configuration (B) Calibration configuration Polarization modulation using a quarter-wave retarder Double polarization modulation Dispersive spectrometers Dichroism measurements Raman optical activity measurements Time resolved infrared measurements (A) Measurements using boxcar integrator (B) Measurements using lock-in amplifier Dynamic infrared linear dichroism Two dimensional spectroscopy Amplitude division interferometry Introduction Principles of measurements

190 191 193 194 196 196 200 200 201 202 202 203 205 211 211 211 213 221 221 227 231 235 235 237 239 239 241 243 243 245 246 248 249 251 253 256 256 257 258 260 263 263 263

~176

Xll

12.2.1

Transmission configuration (A) Background interferogram (B) Transmission interferogram (C) Dichroisrn interferograms

12.2.2 12.2.3 12.2.4 12.2.5

12.2.6 13 13.1 13.2 13.3 13.3.1 13.3.2 13.4 13.4.1

Reflection-absorption measurements Real time sampling Raman scattering (A) Circular Intensity difference (B) Linear intensity difference Time resolved measurements (A) Rapid sweep method (B) Stroboscopic or synchronous perturbation method (C) Asynchronous perturbation method Further developments Polarization division interferometry Introduction PDIs based on Wollaston prism PDIs based on wire-grid beamsplitters

PDIs with roof-top mirrors PDIs with flat mirrors Principles of measurements

Transmission configuration (A) Residual interferogram (B) Background interferogram (C) Sample transmission interferogram

13.4.2 13.4.3 13.4.4 13.5 13.6 13.7 13.7.1

14 14.1 14.2 14.2.1 14.2.2

(D) Dichroism interferograms (E) Calibration interferograms Reflection configuration (A) Differential polarized reflectance interferograms 03) Polarized reflectance interferograms Raman scattering Emission spectra Additional modulations Time resolved measurements Double polarization modulation interferometers

Transmission configuration (A) Residual interferogram (B) Background interferogram (C) Sample transmission interferogram (D) Dichroism interferograms Applications and molecular structure Vibrational frequencies and absorption and Raman spectra Vibrational circular dichroism spectra

Ab initio theoretical predictions Empirical predictions

265 265 265 265 277 278 279 281 282 282 284 284 286 286 289 289 289 291 291 295 295 296 296 296 297 298 302 304 304 306 307 309 309 311 312 313 313 314 315 316 321 321 333 333 336

xiii 14.2.3 14.2.4 14.2.5 14.2.6 14.2.7 14.2.8 14.2.9 14.2.10 14.2.11 14.3 14.3.1 14.3.2 14.3.3 14.3.4 14.3.5 14.3.6 14.3.7

Amino acids, peptides and proteins Nucleic acids Carbohydrates Transition metal complexes Other Molecules Heine proteins Enantiomeric Excess Kinetics Chiral detection Vibrational Raman optical activity spectra Ab initio theoretical predictions Emp irical predictions Amino acids, peptides and proteins Carbohydrates Nucleic acids Other molecules Enantiome ric Excess

340 353 357 357 358 359 359 360 360 360 360 362 363 370 373 373 374

Appendices Summation convention for vector products Harmonic oscillator integrals with dimensionless p and q operators Some relations among dimensionless p and q operators Polarized light Summation convention for tensor products Vibrational Raman optical activities Classical expressions for Aa~7 and G'ec~ Index

383 384 387 388 394 396 404 407

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Chapter 1 INTRODUCTION Vibrational spectroscopy has contributed an enormous amount of information on molecular structures and chemical compositions; it also has served as an important analytical tool. In most of the early vibrational spectral studies, the emphasis has been on vibrational absorption and Raman scattering processes. Excellent books 1-3 are available on these topics and they continue to be invaluable sources on the subject. The presence and identification of characteristic vibrational frequencies associated with chemical groups has popularized the role of vibrational spectroscopy as an important laboratory tool. In another direction, vibrational spectral data have been used for deducing the force constants, and those parameters that reflect atomic and bond properties, although this outlook has changed significantly in the last decade owing in part to the development of quantum mechanical methods. Vibrational circular dichroism (VCD) and Raman optical activity (VROA) have emerged in the last two decades as two new areas of vibrational spectroscopy with emphasis on three dimensional structural characterization of chiral molecules. Two excellent books 4,5 are available on the subject providing different emphases and perspectives. VCD measures the difference in absorption of left versus right circularly polarized light by chiral molecules; VROA measures the difference in Raman scattering for right versus left circulary polarized incident light, again for chiral molecules. Since chirality plays an important role in biology, the implications for the utility of VCD and VROA are significant. The instrumental 6 and theoretical aspects of VCD and VROA have now been developed to the extent that the practice of these areas is becoming almost as routine as the conventional vibrational absorption and Raman spectroscopies. The recent developments, mentioned above, provided the primary impetus for this book. The underlining theories of vibrational absorption, Raman scattering, VCD and VROA are treated here on a common basis. Since VCD and VROA phenomena encompass those of vibrational absorption and Raman scattering, special emphasis has not been placed on the latter two, but they come out naturally during the course of the description of the first two. The same approach is maintained for the instrumentation involved. The outline of the book is as follows. Chapters 2-10 deal with theoretical details, with Chapter 2 providing the needed quantum mechanical machinery. Time independent and time dependent perturbation theories presented in this chapter form the basis for much of

2

Chapter 1

the development of optical activity and anharmonic vibrational properties in later chapters. Chapters 3 deals with vibrational equations and transition moments for diatomic molecules at both harmonic and anharmonic levels. Same aspects are considered for polyatomic molecules in Chapter 4, although the mathematical complexity involved here restricted the length of coverage. Vibrational analysis is presented in Chapter 5. Setting up of vibrational secular equation, various coordinates (normal, internal, and cartesian coordinates) involved, the role of nonequilibrium geometries and that of redundant coordinates are considered in detail in this chapter. Vibrational equations cast in terms of normal coordinates are converted to those in terms of local coordinates in Chapter 6, providing a brief introduction to the local modes. Chapter 7 presents the methods for vibrational frequencies and force constants. In particular, the molecular orbital methods for evaluating the energy derivatives and experimental means of deducing the vibrational properties are summarized. The nature of vibrational absorption and circular dichroism spectral features are presented in Chapter 8 while vibrational Raman and Raman optical activity are presented in Chapter 9. The experimental quantities involved and the quantum mechnical methods to determine them are emphasized in these chapters. The use of molecular symmetry, with group theoretical concepts, is presented in Chapter 10. The next three chapters provide signal analysis for dispersive and FTIR spectrometers. The emphasis in these chapters has been placed on using polarization modulation as a means of measuring the optical activity. Fourier transform IR spectrometers based on Michelson interferometer are discussed in Chapter 12 and those based on polarization division interferometer are discussed in Chapter 13. The final chapter includes a survey of applications with a majority of them pertaining to VCD and VROA. References 1 E . B . Wilson, J. C. Decius and P. C. Cross, Molecular Vibrations, McGraw Hill, New York (1955). 2 G. Herzberg, Infrared and Raman Spectra, van Nostrand Reinhold Co., New York ( 1971). 3 S. Califano, Vibrational States, John Wiley & Sons, New York (1976). 4 L . D . Barron, Molecular Light Scattering and Optical Activity, Cambridge University Press, Cambridge (1982). 5 M. Diem, Introduction to Modern Vibrational Spectroscopy, John Wiley &Sons, New York (1993). 6 Commercial VCD instruments are available from Bomem-BioTools Inc. Commercial VROA instruments are not yet available.

Chapter 2 QUANTUM T H E O R E T I C A L TECHNIQUES For a particle with mass rn and velocity Vx in the x-direction, the kinetic energy in classical mechanics is given by the relation T = (1/2) mv

2

X 9

In terms of the linear momentum, Px - mvx, this relation is T =

p2x/2m. In quantum mechanics1,2, the state of the particles is described by a function, referred to as the wavefunction ~t, and the energy E is determined by solving the Schrtidinger equation H ~ = E ~ where H is called the hamiltonian, or the energy operator (energy written in the operator form). The kinetic energy operator, -(h2/8rt2m) ~2/~x2, is obtained by replacing Px with (h/2r~i)O/Ox. The potential energy operator is simply the potential energy written as a function of the positions of the particles. The total energy of a system of particles is the sum of kinetic and potential energies. For a one-electron system, such as the hydrogen atom, the Schr/Sdinger equation can be solved. The acceptable solutions result with discrete energy levels and wavefunctions. So the Schr6dinger equation is written more transparently as H~n = EnXl/n, where En and ~n are respectively the eigenvalue (or energy) and eigenfunction (or wavefunction) of the system in state n. For multielectron systems the Schr6dinger equation cannot be solved analytically due to the complications originating from the interelectron interaction terms. Nevertheless, approaches are available to obtain approximate solutions and this will be addressed in Chapter 7. For the purposes of this chapter we assume that the eigenvalues and eigenfunctions of the system are known or can be determined, so that the necessary foundation for quantum mechanical applications to spectroscopy can be developed. As a useful example, to illustrate the eigenvalues and eigenfunctions of the Schrtidinger equation, we will consider the familiar case of a particle moving freely in a one-dimensional box of length L. The kinetic energy operator, as mentioned above is (-h2/8rc2m)O2/Ox2. For a freely moving particle the potential energy is zero within the box, but since the particle is confined to the box, the potential energy is infinitely high in the walls. That is, the potential energy V(x) = 0 for x greater than or equal to zero and for x less than or equal to L; V(x) = oo for x less than zero and greater than L. To satisfy this potential energy form we require, in solving the Schrtidinger equation, that the wavefunction goes to zero within the walls. Between the walls the wavefunction is continuous and single valued. The equation to be solved with these conditions is (-h2/8r~2m)(O2~t/Ox2) = E~.

4

Chapter 2

The functions ~n(X) "- A sin(nrtx~) and energies En = n 2 h 2 / 8 m L 2, with n representing any non-zero integer (1,2,3...etc) and A respresenting a constant, satisfy this equation. Thus we have n equations of the form H~n = En~n or -(h2/8~;2m)~)2~n/~)x2 = En~n, each describing the system in a different quantized state. The constant A can be determined by requiring that the wavefunctions be normalized. That is, by setting ~ ~n*(X) ~n(X) dx -- 1 one obtains A 2 -- 2/L; ~n*(X) represents the complex conjugate of ~n(X). The wavefunctions are orthogonal, that is ~ ~n*(X) ~n'(X) dx - 5nn', where Kronecker delta function 5nn" = 1 for n = n' and zero otherwise; and this property can be easily verified using ~n(X) - (2/L) 1/2 sin(nr~x~). In the bracket notation1, 2 the above mentioned orthonormal conditions are written as ~ ~n*(X)~n'(X) dx = <~n(X) I ~n'(X)> = ~nn'In summary, the Schr6dinger equation describes the system in terms of quantized energy levels and wavefunctions. Each state of this system is characterized by its energy and wavefunction. The wavefunctions of the system are orthogonal and normalized. We will use this information as the background for describing the techniques needed for applications to spectroscopy. 2.1 Interaction with electric field Consider a particle with electric charge eq0 under an electrostatic potential V e (V e has the units of energy per unit charge). Expanding V e with respect to distance R~t from the origin, V e - W~ + (c)V e / ~ R ~ ) 0 Ro~ + (1//2) ( ~)2Ve / 3Ra3RI3)0 RaRI3 +..,

(2.1.1)

where t~ or 13 represent the cartesian directions x, y or z; and subscript 0 represents the expansion at Rot = 0. It is to be noted that when products of vector components are involved the summation sign for ct =x, y and z is omitted, and Einstein summation convention (see Appendix 1) is used throughout this book. The gradient (/)ve/ORa)0 represents the negative of force per unit charge along ct th axis, and the electric field F0a is force per unit charge, so the energy of interaction Eint can be written as 3,4, Ein t - eq0IV0 - F0ctRtx

Rc~RI3+... ].

(2.1.2)

For a collection of particles (with charges q0A) in a molecule, the right hand side of the above equation can be summed over all particles A in the molecule. Then the energy of interaction can be written as

Quantum Theoretical Techniques Ein t - E0,in t - l.t0ocF0a - 00a~F0ocfI + . . . , where E0,in t - . V ~ e q 0

(2.1.3)

A is zero for neutral molecules, which satisfy

A

~ eqo A - O; FO~ represents the electric field gradient; the electric dipole A

moment go~ and the electric quadrupole moment e o ~ are given as, /.t0c z - ~ eq0ARAc z ,

(2.1.4)

A

00o~fl _ (1~)~_~ eq0ARAc~RAI3

"

(2.1.5)

A

An alternative traceless definition of the electric quadrupole moment, given as 00o~13 - (1~)~., eq0A (3RAaRAI 3 _ R~Sa13) '

(2.1.6)

A

is often used. ff this definition of 00tx~ is used then a factor of 1/3 should be associated with the third term on the right hand side of Eq. (2.1.3). That is, Eint - E0,int_/.t0czF0o~_ (1~)00tzl3F0czl3+ . . . .

(2.1.7)

The total energy E of the system in the presence of interaction is the sum EO + Eint, where Eo is the energy in the absence of interaction. To obtain the second order influence of the electric field, one can expand the energy as a function of the external electric field in Taylor series at F o a - 0, as E - E 0 + (3E / 3F0cz)0F0c t +(1//2) (32E / 3F0cz~)F0fl)0F0czF0fl+... (2.1.8) Noting (see Eq. (2.1.7)) that, ILt0c~ - -(/)E / OF0cz)0, and with the definition of electric dipole polarizability,

(2.1.9)

6

Chapter 2

(2.1.10)

c~0a[3- -(02E / ~)Fotx~)Fo[3)o

the energy, to second order, in the presence of the external electric field becomes, (2.1.11)

E - E 0 - ~t0aF0a - ( 8 9

Thus to first order, the energy of interaction in the external electric field is given by the terms -l-t0aF0a. The corresponding second order term is -(1/2) 13q3~f~F0aF0~. Note that the subscript 0 on I~ and ot indicates that these are in the absence of external fields, while that on F represents the field F at a chosen molecular origin. These subscripts are often omitted for the sake of convenience, so the interaction terms are simply written as - ~ F a , and-(1/2~a~FaF~. So, (2.1.12)

E - E 0 - ~ a F a _ (1)O~aI3FotFI3+ ...... _(1) 0al3Fc~l3+...

When interaction magnitudes are small, perturbation theory can be used to obtain the quantum mechanical expressions for the energy corrections. Remembering that the energy derivative (0E/0RAS) represents the negative of the force at the particle A, and force per unit charge represents the electric field, one can write 5-7 from the above equation, FA8 - -(1 / eqA)(0E / 3RA8 ) = FOB + ( 1 / e q A ) [ ( 0 . a / ~ ) R A s ) F a +(1)(0o~a[3/0RAs)FocFI 3 +...+(1) (~)0a[~, ORAs)Fa~+...]

9

(2. 1.13)

Here FA8 represents the electric field at particle A in the presence of external field. Expanding FA8 in Taylor series at Fc~ = 0 one obtains,

+ (3F A8 ] 3F ~13) F oq3+ . . . .

(2.1.14)

Equating the coefficients of like terms in Fa, one finds from the above two equations that the molecular property derivatives (1/eqA)(0~/0RAS), 1/eqA(0~a~0RAS) and (1/eqA)(00a~0RAS) can also be interpreted to represent the variation of the electric field at the particles resulting from

Quantum Theoretical Techniques the influence of the external electric field. In the absence of the external electric field (Fcz = 0), if the particles are in an equilibrium configuration the forces on the particles vanish, so l~A5- 0. Upon the influence from an external electric field the particles move to a newer set of positions so as to make the net forces vanish. These compensating forces depend on how the molecular properties (Pcz, o~zp, 0czp etc) change with respect to nuclear displacements. We can make use of this observation to interpret the vibrational spectral intensities.

2.2 Time independent perturbation theory Consider a system characterized by states with energies E 0 and wavefunctions ~n" The Schr6dinger equation for state s of this system is Ho~O - E s0~ s0 ,

(2.2.1)

where H0 is the hamiltonian for the system. The subscript zero for H, and superscript zero for E and ~, indicate that these quantities are in the absence of any external perturbation. The energy of the system in these states is calculated as o o o E s - < ~lSslH01~s > .

(2.2.2)

Now consider the influence of an external perturbation that is weak and time independent. The total hamiltonian can be written as, H=Ho+HI+H2+...

,

(2.2.3)

where H 1 and H2 represent the first and second order perturbations added to the unperturbed hamiltonian H0. The resulting energies and wavefunctions for each state can be written similarly. That is, 0

f

H

E s - E s + E s + E s +... , f

H

9 s - ~ 0 + ~s + ~s+... ,

(2.2.4) (2.2.5)

where the superscripts ' a n d " on E and ~t represent the first and second order corrections. When the above definitions of H, ~s and Es are substituted into the Schr6dinger equation for the perturbed system, H~s = Es~s ,

(2.2.6)

8

Chapter 2

one finds terms of different orders. These include two zero order terms, H0~s, and E~ four first order terms H0~s ' H I ~ s, Es~ s and Es~s; six second order terms (vide infra); and so on. Equating the terms of like order in Eq. (2.2.6), one gets an equation for each order. The zero order equation is same as Eq. (2.2.1) given above; the first order equation is (H0 - E0)Xl/s = (Es - H1)~s .

(2.2.7)

We assume that the first order correction to the wavefunction ~s can be written as a linear combination of unperturbed wavefunctions, ~s = Z c n ~ n '

(2.2.8)

where c'n are the unknown coefficients and the summation is over all states of the system that satisfy Eq. (2.2.1). Upon substituting Eq. (2.2.8) into Eq. (2.2.7), left multiplying with @s (or its complex conjugate in the case of complex wavefunctions) and integrating, one finds that the orthogonality of zero order wavefunctions, namely (2.2.9) leads to the expression for first order correction to the energy of the unperturbed state s as, E'=<~slHll~s>'s

(2.2.10)

Instead of left multiplying with ~s if we had used ~ o with k = n, but k r s, then the above procedure would lead to the result that c n= < ~nlH1 I xl~s> / ( E 0- E0)-

(2.2.11)

Then from Eqs. (2.2.8) and (2.2.5), wavefunction correct to first order is obtained as, VS=~s + E

[< ~(n)nI H11 ~)s > / (g~s - E(n)n)]~n 9

n:g:s

Similarly let us consider the second order equation,

(2.2.12)

Quantum Theoretical Techniques ( n 0 - E 0 ) ~ s ' = ( E s ' - n 2 ) ~ s + (E~-nl)k~f's ,

(2.2.13)

and express the second order correction to the wavefunction as a linear combination of unperturbed wavefunctions,

Igs'= E C"n~n"

(2.2.14)

Here c"n are a different set of coefficients than the c'n used in the expansion for first order correction. Substituting Eqs. (2.2.14) and (2.2.8) into Eq. (2.2.13), left multiplying with ~s (or its complex conjugate in the case of complex wavefunctions) and integrating, one finds that the second order correction to the energy is given as E"=s

<~s IH2 1~g0 > +

E

n~:s

[< ~n'Hl 'V0 > < ~ s l H 1 'V0 > / (~ss- ~nn)] "

(2.2.15) pp

To second order, the total energy of the system is Es - E 0s + E's + E s, and Eqs. (2.2.1), (2.2.10) and (2.2.15) permit the evaluation of this energy. The expressions for the first and second order corrections to the energy and for the first order correction to the wavefunction are of vital importance as these quantities help understand how the system behaves under the influence of a weak external perturbation. One can proceed in the same manner to obtain higher order corrections to both energies and wavefunctions. 2.2.1 Static electric dipole polarizability To show the applications of time independent perturbation theory, let us evaluate the electric dipole moment in state s using the wavefunction corrected to first order. Considering only one component of the perturbing electric field, say Fx', and writing H1 as-t-tx' Fx', the x-component of the electric dipole moment of the system can be obtained using Eq. (2.2.12). i.e., < IXx > = < ~gsl ~ ~gs > =

<

}.tx >0-~.~\' (la.x,snl-tx',as + ].tx,nsl.tx',sn) F x ' / ( E 0 - ~ n n ) '

(2.2.16)

n~s

where ~x,sn = < ~ s l ~ x [~n > and < ~x >0 = < ~ s I ~x I ~ s >; [note that x',

10

Chapter 2

representing a component of the field perturbing the system, is distinguished from x, the component of the electric dipole moment being evaluated]. The second order terms in Fx' are ignored in arriving at this equation. For real wavefunctions ~tx,ns - t.tx,sn. We can generalize the above equations by considering all components of electric field, so the above equation can be written using the Einstein summation convention (Appendix 1) as < Ixot>- < ~ >0 + o~aa'Fot' ,

(2.2.17)

where CZczcz' - - 2 Z

< ~s I btcz I ~n > < ~n I I.t~' I ~ts0>/(E~s- E~n) ,

(2.2.18)

ng:s

is the general expression for the static electric dipole polarizability of the system. A second application is to calculate the energy to second order in the presence of the electric field perturbation given as H1 = -l.totFcz. From Eqs. (2.2.4), (2.2.10) and (2.2.15), the energy can be written as Es - E ~S - < bt~ >o Fc, - (1/2) ~otocF2

(2.2.19)

where < l.tcx>0 and cxcxcxare defined above.

2.3 Time dependent perturbation theory The time dependent Schr6dinger equation is written in terms of the wavefunction that depends on time. This wavefunction ~ ~ can be written as the product of two terms as, ~n (t) - ~n e-27tiEn0t/h

(2.3.1)

where h is the Planck's constant and t the time; e -2niE0t/h describes the time dependence. The time dependent Schr/Sdinger equation becomes, H0~n(t) = En0~n(t) = i(h/2rt)(~)~n(t)/~)t) ,

(2.3.2)

where H0 is the unperturbed, time independent hamiltonian of the system. The inclusion of time dependent perturbation Hi(t) modifies the hamiltoinan to H(t). That is,

Quantum Theoretical Techniques H(t) - Ho + Hi(t).

11 (2.3.3)

Assuming that the new wavefunction for state k can be written in terms of the original wavefunctions ~n(t) using time dependent coefficients Cn(t), as

yk(t) = ~ Cn(t) ~n(t),

(2.3.4)

the Schr6dinger equation for the perturbed system becomes, [H0 + H I ( t ) ] [ Z

Cn(t)~n(t)] = i ( h / 2 r : ) ~ [ ~ Cn(t)~0(t)]/0t.

(2.3.5)

The summation index in Eqs. (2.3.4) and (2.3.5) runs over all states of the system that satisfy Eq. (2.3.2). Expanding both sides of this equation, making use of the unperturbed equation, Eq. (2.3.2), left multiplying with @k* (* represents complex conjugate), and using the orthonormal property of @n' one obtains the expression for the time dependent coefficients as, ~)Ck(t)/& = (2rffih)Z Cn(t) < V01Hl(t) I ~n > e-2ni(E0- l~)t/h

(2.3.6)

Now we integrate this equation from time zero to a certain time, noting that at time zero the system is in the initial state s, so Cs(0) = 1 and Cn(0) = 0 for n ~ s. Assuming that the coefficient Cs(t) deviates from 1 only slightly at time t, while other coefficients Cn(t) for n ~ s are very small, the expression for the time dependent coefficient Cn(t) associated with a state n is given as, Cn(t) - (2~;/ih) f < ~n I Hl(t)I ~ )s > 0 c~ dt,

(2.3.7)

where COns = 2rl; ( ~ - E0)/h. This is the most important equation for understanding the temporal behaviour of the system under a time dependent perturbation. 2.3.1 Electric dipole-electric dipole polarizability Let us consider the influence 1,2,8 of a dynamic field perturbation on the system, and assume that the perturbation is weak. The time dependent electric dipole moment is given as ~toc(t) = < ~s(t) I ].tc~I ~gs(t) >,

(2.3.8)

Chapter 2

12

with wavefunction ~s(t) given by the Eq. (2.3.4). In the case of electric field perturbation, the time dependent coefficient is evlauted with H 1 (t) = -g~,Fa,(t) [here again o(, representing the component of the field perturbing the system, is distinguished from ~, the component of the electric dipole moment being evaluated] as Cn(t) - -(2rc/ih) I.ta',ns ~ Fo((t) e k~

(2.3.9)

dt ,

where g~',ns = < @n I g~' I @s >" For the real electric field given as Fa'(t) = Fa' [e ic~ + e -i~

(2.3.10)

= 2 Fa, cos cot ,

where co is the frequency of the applied electric field, the time dependent coefficient becomes,

I'_ .-, iO~nst -iO~nst ] c n (t) - [4~ga,,nst~a,e (COnscos C0t - ic0 sin cot - C0nse )

x [1/h(COZns- co2)]

(2.3.11)

These coefficients, their complex conjugates c*(t) and Eq. (2.3.4) can now be used in evaluating ~s(t) and hence ~ta(t). The resulting expression for t.ta(t) is, ~ta(t ) - < Vs01gulv 0 > + ~[4rl;.o~,,nsgo~,snFc~, x (COns cos cot- ico sin cot- COns e-iC~ n~:s

[

+ ~ 4gg~,,snge~,nsF~, x (COns cos cot + im sin cot- runs

e+i~n~t~1

n~:s •

....

In deriving this equation, the hermiticity relation 1,2, < ~~

<

~sOIll~,l~R0n >,

l ~ ~ >* =

was used. The imsinmt term drops out when the product

Quantum Theoretical Techniques <~g01lxallg0><~01t.ta, l v 0 >

is

real,

so

it

13 is

equal

to

; the cosmt term drops out when the product < V~

0 > < ~g~

0 > is imaginary, so it is equal to

- < V~ 0 > < xg~ 0 >. The (COnse-+-i~ terms represent oscillations at runs frequency when the perturbation has frequency m. These oscillations can be eliminated by modifying 2 the electric field perturbation as,

Fo,~t,_ Fo,[eiOt+ e-iOtl[1_ e-t',]

(2.3.13)

For times t >>> "t, and (COns4-m)>> ll'r,, this modification results in an additional term, 2 - O)2) , 4rq't~x',nsFa'mns / h(mns

(2.3.14)

to be added to the right hand side of Eq. (2.3.11). Then the general expression for time dependent electric dipole moment becomes, l.ta (t) - [.ta (0) + 2otcx{x,Fa, c o s m t - 2a~a,F a, sinmt + . . . .

(2.3.15)

The frequency dependent polarizabilities cxaa, and cx~xcx, are given by the expressions

oo = ReI nsOns' 4 .o .,s ~ r

ocaa, - Im - ~ 4ztl.tcx,snt.tcx,,nsm / h m2s k n~s

(2.3.16)

1

(2.3.17)

where Re[...] indicates the real part of the terms in parenthesis and Im[...] indicates the imaginary part. The sinmt term can be modified to that of time derivative of electric field, so Eq. (2.3.15) can also be written as, p

ga(t) = ga(0) + o~aa,Fa,(t ) + ~aa,Foc,(t) / m + ... ,

(2.3.18)

Chapter 2

14

where dot on Fa,(t) indicates time derivative. For O~ns>> o~, Eq. (2.3.1 6) reduces to that of static polarizability (Eq. 2.2.18). 2.3.2 Electric dipole-magnetic dipole polarizability The expressions for the electric dipole moment in the presence of the dynamic electric field perturbation given above provide the necessary framework for considering other perturbations 8. To evaluate the electric dipole moment in the presence of a dynamic magnetic field Ba,(t), the perturbation would be changed to H1 (t) = - n ~ , B a , ( t ) , where m is the magnetic dipole moment; then in Eqs. (2.3.9)-(2.3.14) the symbols associated with lxa' should be replaced by ma, while the symbols associated with index ~ remain the same (since this index represents the electric dipole moment component being evaluated). Then, the electric dipole moment can be obtained from Eq. (2.3 15), by replacing Fa,, aa and o~a, with Ba,, Gaa' and G~a, respectively as, p

~

laa(t) - ~ta(0) + Gaa,Ba,(t) + Gac~,Ba,(t) / 03 + ... ,

(2.3.19)

where the electric dipole-magnetic dipole polarizabilities Gaa' and G~a' are given as,

ooo "e[s4"osr,monsOns'(Ons0 )1 Iml 4 osnmor,sO'(Ons

(2.3.20)

(2.3.21)

L nc:s

If both electric and magnetic fields are considered, then the sum of Eqs. (2.3.18) and (2.3.19) will be involved (noting that g~(0) will be counted only once). Similarly, to evaluate the magnetic dipole moment in the presence of dynamic electric field F~,(t), the symbols containing g~ should be replaced by those containing m~ (since the index a ' represents the electric field component while a represents the magnetic dipole moment component to be evaluated) in Eqs. (2.3.18). That is, p

~

ma(t) - ma(0) + Ga,aFa,(t) - Ga,aFa,(t) / co + . . . .

(2.3.22)

In this manner, one can obtain the general expressions for the dynamic electric dipole, magnetic dipole and electric quadrupole moments.

15

Quantum Theoretical Techniques

2.3.3 Expressionsfor general molecularproperties In the previous two sections we have evaluated the time dependent electric and magnetic dipole moments for the system in a given state. Under the influence of time varying electric and magnetic fields, the expressions obtained can be generalized for any molecular property. This is done by replacing ga or m~ with an appropriate symbol, say Pa representing that property. Then, p

~

Pa(t) = P~(0) + Sec~'F~'(t) + See,Fe,(t) / co + Tee,B~,(t) + T~,Bc((t)/co +... , p

~

(2.3.23)

where, f'~ 1 '

(2.3.24)

1 '

(2.3.25)

~ 1 '

(2.3.26)

T&a,-im[_Z4rcpo~,snmet,,nsm/h(o~2 s n,s L -m2) 1 "

(2.3.27)

See'-ReFZ4gPe,snge,,nsf, Ons/Ln~s

h(f'~

S~~

imI_Z4~;pcx,sngc~,,nsO~/h(co2 s n ~ s

Tcx~

Re[Z4~;Pcx,snmcx,,nsO3ns/Ln~:sh(C~

-~

Note that, P~,sn = < ~0 p~ gOn > with P~ in the integral representing the appropriate operator for the molecular property being evaluated. We will use these expressions in Chapter 7 to evaluate the time dependent fields present at the nuclei in a molecule. As an example, to evaluate the electric quadrupole moment 0ap(t) induced by the electric field Fa,(t), Eq. (2.3.23) gives 0ap (t) - 0ap (0) + A ~ , F ~ , ( t ) ,

(2.3.28)

where Aapa, is the electric dipole-electric quadrupole polarizability,

16

Chapter 2

Ac~pcz'- Re[Ln~#s4~0cxp,snl'tcx''nsO~ns / h(CO2s - (02)1 ' and 0cz~,sn = < 11/0 0czl3 11/0 >.

(2.3.29)

The above two equations can also be

written (by noting that the real values of the products 0 czp,snPa ,ns and l-to(,sn0o~13,nsare equal) as, 0cxp(t) = 0ctp(0) + Act'cxpFcx'(t),

(2.3.28a)

Aa,czp - R e [ E 4~lttec,,snOczp,nsOlns / h(012s - 012)1 9

(2.3.29a)

kn#s

2.3.4 Interaction with circularly polarized light Let us now consider the influence of electric and magnetic fields associated with circularly polarized light incident on the system. Taking the z-axis as the direction of propagation of light, the circularly polarized electric vector, F+(t)(where the superscript represents the handedness, + for left circular and - for right circular) can be written as a sum of the phase shifted electric vector components along the x'- and y'-axes. Thus, F+(t) - 2F0[ficosco(t- z / c) + r

co(t- z / c)],

(2.3.30)

where fiand r162represent unit vectors along x'- and y'-axes and c is the velocity of light. The corresponding components of magnetic field can be obtained from the Maxwelrs relation, (2.3.31)

_(VxF +) - B + .

Finding the x and y components from this vector product and using the relation B0 = Fo/c, B +-(t) is written as B+(t) - ~2coB0[ficosco(t - z / c) + r

co(t- z / c)].

From this expression B---(t) is found to be

(2.3.32)

17

Quantum Theoretical Techniques

(2.3.33)

B + ( t ) - -2Bo [_+fisin co(t - z / c ) - r162 co(t - z / c)].

It is of interest to note that (2.3.34)

B + (t) - ~ ( ~ / c ) F + ( t ) .

Since F+(t) and B+(t), propagating along the z-axis, have only x' and y' components, the perturbing hamiltonian can now be written as +

+

H~ (t) - -J.tx,F+, (t) - lay,Fy, (t) - mx,Bx, (t) - my,By,(t) , where F+,(t) = 2F0 cos m ( t - z/c),

Fy,(t) = + 2F0 sin m ( t -

(2.3.35) z/c), B +'t" xt )

=-T- 2Bo sin m ( t - z/c) and ~,(t) - 2B0 cos m ( t - z/c). Substituting u these x' and y' components into the above equation one obtains, H~(t) - F0{-~x,[e ic~ +e -ic~ _+i~ty,[e ic~ - e-iC~ ]} -B0{__+imx,[e ic~ - e-it~ + my,[e ic~ + e-i~ ,+

9

(2.3.36)

+

The coz/c terms associated with F~a,(t) and B~t,(t) have been omitted here for the sake of economy, and they will not affect the following equations. The time dependent coefficients for the perturbation given by the term [eimt + e -i~t] have been discussed earlier (Eq. (2.3.10)-(2.3.12)). For the case of a perturbation given by the term iFo~'ga'[e imt - e-lint], the time dependent electric dipole moment is given by the relation, l.ta(t) - l.ta(0) + ~ 4r~gtx,sngcz,,nsFcx,[C0ns sin cot + icocoscot]/h(CO2s-co 2) n~:s

+ E 4/I;got,nsgot,,snFtx,[COns sin cot- icocoscot]/h(CO2s- 0)2). n~:s

(2.3.37)

For the perturbation given by the term iBcz'mcx' [e ir176- e -i~ symbols involving g~, in Eq. (2.3.37) are replaced by the corresponding ones with

18

Chapter 2

ma, and Fc~' by Bcz'. Thus all the ingredients to evaluate the electric dipole moment with H l(t) given by Eq. (2.3.36) are now in place. The resulting electric dipole moment is given as, + t "+ + gfi(t) - l.tr (0) + o t ~ , F ~+, (t) + otot~,F~, (t) / 03 + Gcxcx,B~, (t) + p

-q-

(2.3.38)

G~a,B~, (t) / co. The tensor components otaa', ~c~', Gaa', and G~a, are defined by Eqs. 2.3.16, 2.3.17, 2.3.20 and 2.3.21. The ot~a, and Gaa, tensor elements are non-zero only in special situations, and for most common cases only the o~aa' and G~cz' tensors need to be considered. In that event, Eq. (2.3.34) can be used to rewrite the above equation as 9 l.ta (t) - ~a(O) + (~cxa' -Y-(1 / c)Gaa,)F+,(t).

(2.3.39)

This relation makes the nature of interaction of molecular systems, with circularly polarized electric vectors of incident light, quite transparent. Note that the systems that do not support G~cz' cannot discriminate between the left and right circular polarizations. For a general case, the difference in electric dipole moment induced by right and left circularly polarized incident light is proportional to 2 G~a'. As seen from Eq. (2.3.39), the induced electric dipole moment obtained with right circularly polarized light can be different from that with left circularly polarized light in optically active molecules. So the refractive index of the medium can be different for these two polarizations. Since a plane polarized light vector can be written as an appropriate combination of the left and right circular polarizations, and these polarizations experience different refractive indices in a chiral medium, it can be seen that the plane polarized electric wave can be rotated as it passes through the sample. This optical rotation 0 (in radians) is given as 0 - l.too32Ngll(n 2 + 2) / 3,

(2.3.40)

where N is the number density, g is the sample thickness, go is the permittivity, n is the refractive index of the medium and [3 is related to the mean of the G' tensor a s - c o - k 3 ' - 13. Eq. (2.3.40) is applicable when 13 is expressed in SI units. When [3 is expressed in CGS units, the corresponding expression for optical rotation is

19

Quantum Theoretical Techniques 0 - 4rco~2Ng~(n 2 + 2) / 3c 2 .

(2.3.41)

The experimental optical rotations are most commonly reported as specific rotation [ct] in units of deg/[dm (gm/cc)]. The corresponding theoretical quantity is [tx] = 3600t~Vm/2~M where M and Vm are respectively the molar mass and molar volume. Using these definitions, we obtain a convenient expression for the specific rotation as [ct] = 0.1343x 10-3 ~ ~2 (n 2 +2)/3M,

(2.3.42)

with [3 in units of (bohr) 4, M in gm/mol and ~ (wavenumber at which the optical rotation is measured) in cm-1. Sometimes molar rotation, defined as [M] - [a] M/100, is also reported in the literature. Using the quantum mechanical expression for optical rotation [Eqs. (2.3.40)-(2.3.42)], ab initio calculations have been reported 10 only recently. 2.4 Transition rates The evaluation of time dependent coefficients was discussed previously in the context of evaluating the time dependent moments of the system. When the incident light causes the system to transit from the initial low energy (ground) state to a higher energy (excited) state, this process results in transfer of energy from the incident photons to the system and the absorption of the incident light takes place. When the transition is from a higher energy to a lower energy state, the energy transfer is in the opposite direction and stimulated emission results. For these cases the evalution of the time dependent coefficients follows the procedure discussed below. We will only concentrate on the absorption phenomenon in this section. The perturbing hamiltonian (Eq. (2.3.36)) considered earlier from circularly polarized incident light will be used here, as this is needed for determining circular dichroism intensities and the same can be used for absorption intensities and to generalize for a situation when the incident light is linearly polarized. The influence of [e ir176+ e -i~ terms in the hamiltonian (Eq. (2..3.36)) on the time dependent coefficients (Eq. (2.3.7)) is that they will result in the integrals,

f(e ir176+_e-iC~176

_

e i(~ns+co)t - 1 + _ e i(~176

[

i(~ns + CO)

- 1

i(0~ns - co)

1

(2.4.1)

The first term on the right hand side of Eq. (2.4.1) is negligible for absorption process (since COns- co, denominator is large) and the second

Chapter 2

20

term is negligible for stimulated emission (since cons --CO). So, for the absorption process, time dependent coefficients resulting from the circularly polarized perturbation become,

(

c +n (t) - -2rl;V~,ns e i(r176176 - 1) / h(tOns - r

(2.4.2)

where, ns

(2.4.3)

-gx',nsF0 +lgy',nsF0 +imx',nsB0 - my',nsB0 9

The probability P~n+(t) of finding the system in state n (that is transition from ground state s to excited state n) at time t is given by the product +

+

Cn(t)(Cn(t))*, where * represents the complex conjugate. Thus,

P~ (t)

-

16tl;2V~,ns V ~ n s

sin2 (COns - r

] h 2 (cons - 0)) 2

/ 2.

(2.4.4)

The term V +1,ns(VLns) + * can be written as,

+

+

9 [

]

V~n s (V~n s) - -l.tx,,nsF 0 -T-ll.ty,,nsF0 + imx,,nsB0 - my,,nsB 0 • ,

*

*

*

]

-gx,,nsF0 + igy,,nsF 0 ~ imx,,nsB 0 - my,,nsB0 Expansion of this product gives sixteen terms.

9 (2.4.5)

Of these terms,

-T-igx,,nsgy,,n s cancels with +igy,,nsgx,,n s, gx,,nsmy,,ns cancels with

my,,nsgx,,n s, gy,,nsmx,,ns cancels with mx,,nsgy,,n s and T-imx,,nsmy,,ns cancels with +imy,,nsmx,,n s. The surviving terms can be rearranged to the form, V +1,ns(VT, + ns) 9 -

(gx',snl-tx',ns + I-ty',sngy',ns) F2

+ (mx,,snmx,,n s + my,,snmy,,ns)B 2 T- 2i(gx,,snmx,,n s + gy, snmy,,ns)FOB0 .

(2.4.6)

Quantum Theoretical Techniques

21

For an isotropic sample, one needs to average over all orientations, so this equation can be rewritten as, + + )* V~ns(V~n s - (2 / 3)[}.tcx,sn~tcx,nsF2 + mcx,snm~,ns B2 + 2 Im(g~,snmoc,n s)FOB 0]

.

(2.4.7)

The terms F~, B~ and FoB0 can be converted to the energy contained in the electromagnetic wave (remember to use Eqs. (2.3.30) and (2.3.33)). Now we have to consider the density of states in the energy range of the transition, and integrate over all such transitions. With these two considerations2,11, and noting that the transition rate is given as , W + = dP+/dt, one obtains the transition rate for left and right circularly polarized incident excitation as, W + _ (8rt 3 / 3h2)[~c~,snl.ta,ns + ma,snmcx,ns + 2Im(l.ta,snmo~,ns)]P(v) =(8rt 3 / 3h2)[la, sn.l.tns+msn.mns+2 Im(~n.mns)]p(v) - B+p(v)

(2.4.8)

where p(v) is the energy density (energy per volume per Hz). This is the fundamental equation for deriving expressions for absorption and circular dichroism intensities. For lineary polarized incident excitation, Eq. (2.3.30) will be modified to contain only one component (say fi cos c0(t- Z/c)) and the rest of the procedure outlined above is repeated. Then, one would see 2,11 that the cross term, 2 Im(l.ta, snma,ns), will not appear in the transition rate. 2.5 S p e c t r a l i n t e n s i t i e s

Consider a sample, with concentration CO (moles per liter), absorbing a photon of energy hv from the incident light of intensity I(v). Then the change in intensity of the light when it exits the sample is proportional to the sample's thickness, or the optical path length g. Using the relation, I(v) = cp(v) in the expression for transition rate, and assuming that the energy difference between states s and n is sufficiently large to have all molecules initially in the groud state s, one obtains 2 (dI(v) / I(v)) + - (B + / c)hvCoNdg,

(2.5.1)

where N is Avogadro's number and B + is defined in Eq.(2.4.8). For the case of normal absorption, the superscript + can be left out, and only the

Chapter 2

22

/.tcz,sn/.I.cz,n s

term in Eq. (2.4.8) is needed. Then,

ln[I0(v) / I(v)] - (8~3vCoNg / 3hc)Dns,

(2.5.2)

where I0(v) is the intensity of light before passing through the sample and Dns = /.tot,sn/.tcx,ns =/~sno~tns,

(2.5.3)

is called the electric-dipole transition strength. Beer's law of absorption is given as I(v) = I0(v)e -r

,

(2.5.4)

with r representing the absorption coefficient. From Eqs. (2.5.2) and (2.5.4), the absorption coefficient becomes, r

(2.5.5)

= (8~3vN/3hc)Dns .

Taking the spectral band shapes into account and integrating this quantity over the band, it is necessary to evaluate the integral J(cz(v)/v)dv. However, since v does not vary very much over the width of the band, v can be replaced by v0 (the frequency at the center of the spectral band). Then the integrated absorption coefficient is given as A = fcz(v)dv - (8rc3v0N/3hc)Dns.

(2.5.6)

Sometimes Beer's law is written as I ( v ) - I0(v) 10 -~(v)C01 and r is related to ~ (v) as r = 2.303 ~ (v). To facilitate the conversion between the experimentally measurable quantity A and theoretically predictable parameters Dns, the following relation can be obtained from Eq. (2.5.6). A = (100.38/4) v0 Dns x 1040

,

(2.5.7)

where the integrated absorption coefficient A is in commonly expressed units of cm/mol, the band centerV0 is in units of cm -1 and dipole strength Dns is in units of esu 2 cm 2. To obtain the circular dichroism intensities, Eq. (2.5.1) can be used to obtain r and hence A-+-. Then the difference A A - A+ - A-, is obtained from Eqs. (2.4.8) - (2.5.1) as

Quantum Theoretical Techniques

-4(8~3voN/3hc)Rns

23

(2.5.8)

where the rotational strength (some times called rotatory strength) Rns is defined as Rns-Im[l-tsnmns] = Im[~sn.mns].

(2.5.9)

With Rns also expressed in units of esu 2 cm 2, and AA in (cm/mol), the simplified equation, analogous to Eq. (2.5.7), is obtained as AA = 100.38 V0 Rns x 1040 .

(2.5.10)

The ratio & ~ A , referred to as the dissymetry factor can be seen from Eqs. (2.5.7) and (2.5.10) to be AM/A = 4Rns/Dns 9

(2.5.11)

2.6 Transition polarizabilities The expression for time dependent electric dipole moment (see Eq. (2.3.18) obtained in the presence of dynamic electric field indicates that the dipole moment oscillates at the frequency of the electric field (associated with incident light on the sample). An oscillating electric dipole moment emits radiation, so the o~aa'Fc~'(t) term serves as the source for scattered light from a molecule. When the frequency of the scattered electromagnetic wave is the same as that of the incident electromagnetic wave, this light scattering phenomenon is referred to as Rayleigh scattering or elastic scattering. To consider the originl2 o f Raman scattering (or inelastic scattering) where the frequencies of the scattered light components are different from the frequency of the incident light, it is necessary to consider the time dependent electric dipole transition moments, < ~gf(t)llx=l~gs(t ) >. The recipe for evaluating this integral proceeds much the same way as that discussed in Section 2.3.1. To keep the identity of time dependent coefficients associated with the expansion of ~gs(t) and ~gf(t) separate, it would be useful to add an additional subscript to Cn(t). For example Cn(t) in Eq. (2.3.4) can be labelled as Cns(t) when k - s and as Cnf(t) when k = f. Then the transition moment integral will have four contributions. Of these, the contribution which has the products Cns(t)Cnf(t) is of higher order and will be ignored. Then,

24

Chapter 2

< vf(t)ll.ttxlVs(t) > - l.ttx,fselmfst + Z 4gl'ttx,fnbtot',nseiC~

[C~ cos cot - it.osin cotl / h(O~2s - co2)

ng:s

+ Z 4rl;btot,nsbtot',fneimfstFoc' [~ n;ef

cosmt + imsin rot] / h(ol2f - ol 2 ). (2.6.1)

This equation can be rewritten in terms of Fct'(t) and Fa,(t), using Eq. (2.3.10), as

(Ve(t)l t=lvs(t))-

+ [~~']fsFod(t) + [0ta~']fsF~'(t) / f'0}eic~ , (2.6.2)

where [otaa, ]fs - Z 2/1;l'tct,fnl'tod,nst'~ / h(~

- 012 )

ng:s

+ Z 2/1;~ot,nsl'tct',fnOlnf / h(t'O2f - t'O2)' n;ef [0~o~' lfs - Z 2rtil'tcx,fnt'ta',nsm / h(~

(2.6.3)

- c~

ng:s

-

2 m2 Z 2gig~,nsg~',fnm ! h(~nf ) 9 n~f

(2.6.4)

The first term btcx,fse imfst is the oscillation of transition moment between unperturbed states. The terms containing the product e~mfstF~'(t) represent oscillations at co + COfs;that is, the transition moment oscillates at the beat frequencies 03 _+ mrs. As mentioned earlier, an oscillating dipole serves as the source of scattered light, so the scattered light can have higher (co + COfs) or lower ( m - mrs) frequencies than the frequency (03) of the incident light. Thus Eqs. (2.6.1) and (2.6.2) expose the quantum t mechanical origin for Raman scattering. The coefficient of e lmf~ F~,(t) represents the transition polarizability, [ot~c(]fs. Note that when the initial

Quantum Theoretical Techniques

25

and final states are the same (s = f), this equation reduces to the dynamic polarizability [see Eq. (2.3.16)] discussed in Section 2.3. The coefficient of the product e ic~ Fa,(t) " represents the imaginary part of the transition polarizability (similar to ct'aa' in Eq. (2.3.18)). We will use Eq. (2.6.3) to describe vibrational Raman scattering in later Chapters. References 1 H. Eyring, J. Walter and G. E. Kimball, Quantum Chemistry, John Wiley & Sons, New York (1944). 2 P . W . Atkins, Molecular Quantum Mechanics, Oxford University Press, Oxford (1983). 3 L. D. Barron, Molecular Light Scattering and Optical Activity, Cambridge University Press, Cambridge (1982). 4 W.S. Struve, Fundamentals of Molecular Spectroscopy, Wiley, New York (1989). 5 H. Sambe, J. Chem. Phys. 58 (1973)4779. 6 P. Lazzeretti and R. Zanasi, Phys. Rev. A24 (1981) 1696. 7 P.W. Fowler and A. D. Buckingham, Chem. Phys. 98 (1985) 167. 8 A.D. Buckingham, Adv. Chem. Phys., 12 (1967) 107. 9 A.D. Buckingham, P. W. Fowler and P. A. Galwas, Chem. Phys. 112 (1987) 1. 10 P. L. Polavarapu, Mol. Phys. 91 (1997) 551. 11 I. N. Levine, Molecular Spectroscopy, John Wiley & Sons, New York (1975). 12 G. Placzek, The Rayleigh and Raman Scattering, UCRL Translation No. 526(L), translated by Ann Werbin (1959).

This Page Intentionally Left Blank

27

Chapter 3 DIATOMIC MOLECULES 3.1

Vibrational

equation

Consider a diatomic molecule with its constituent atoms designated as A and B. The nuclear kinetic energy operator TN is obtained from the operator equivalent (see Chapter 1) of linear momentum, (h/2rri) (O/OXA + 0/OYA + t)/t)ZA) = (h/2ni)VA, as TN = - ( h 2 / 8 E 2 ) ( V 2 / m A + V~3/mB) 9

(3.1.1)

Here mA and mB represent respectively the masses of A and B. Using the relative coordinates, R = R A - RB, the relation for center of mass coordinates Rcom = (RAmA + RBmB)/M and the relations among partial derivatives, (~)/~)XA)= (~)x]c)XA)(()/t)x) + (~)Xcom/C)XA)(C)/OXcom),the kinetic energy operator can be transformed (remembering that terms such as O2/Ox2 are written as (t)/t)XA)(~)/t)/XA)) to the form, TN = -(h2/8rl;2g)V 2 - (h2/8g2M)V2om 9

(3.1.2)

Note that M = mA + mB, the reduced mass g = mAmB/(mA + mB) ' V 2R -

-

~2/~R2 and V2com - O2/OR2om. The term (-h2/8r~2M)V2om represents the

kinetic energy operator for translational motion of the molecule and can be ignored as it does not couple to, or influence, the vibrational motion that is of interest here. With electronic, vibrational and rotational states designated respectively by subscripts n, a9 and J the Schrtidinger equation is given as H/l/nagj (r,R) = Ena~Jlgn~J (r,R) ,

(3.1.3)

where II/n~ J (r,R) is the total wavefunction that is dependent on electron

coordinates r and relative nuclear coordinates, R. The total hamiltonian is H = TN + VN + He l, with TN defined by Eq. (3.1.2) and Hel and VN defined below" Hel - _(h 2 ] 8~;2m)~ V 2 _ i

i

ZAe2/riA-ZZBe2/riB i

+ZZe2/rij i j,i

28

Chapter 3

(3.1.4) VN = ZAZBe2/RAB 9

(3.1.4a)

Note that ZAe and ZBe respectively represent the charges of nuclei A and B, V 2 is given for electron i as (c)2/~)x~ + ~)2/~)y2 + ~)2/~)z2), riA is the distance from electron i to nucleus A, rii is the distance between electrons i and j, and RAB is the distance between fiuclei A and B. The nuclei and electrons do not move independently, but the relatively large mass difference between them permits certain approximations to be introduced. For any slight perturbation in the nuclear positions, the electrons reorient their positions instantaneously. So it would appear that the total wavefunction can be written as the product ~lnoJ (r,R) = Vn (r,R) ~l/oj (R) . For the sake of simplicity the parentheses showing dependence on r and R will be dropped, so ~l/n (r,R) ~/1JJ (R) - Xi/nXi/~j;

(3.1.5a)

/~/9j describes the motion of slowly moving nuclei in electronic state n, and ~tn describes the motion of fast moving electrons with parametric dependence on nuclear positions. With this assumption, the Schr6dinger equation leads us to the evaluation of a complicated term, TN ~nV~J- This in turn leads us to (-h2/8~2kt) ~l/l)j (02~n//)R 2) and (-2h2/8~2~t)(/)Vn//)R) (t)~j/~)R). Both of these contain (t)~n/t)R) which describes the way the electronic wavefunction changes with nuclear displacement. When these terms are considered small and ignored, the Schr6dinger equation can be rewritten as (-h2/8/1;2~)Xl/n(C)2xl/agj/t)R2) + VnXl/nXl/ajj+ ~ J H e l ~ n - E Xl/n~l/~j . (3.1.6) We further assume that the electronic part of the Schr6dinger equation He 1~n = EnXl/n ,

(3.1.7)

can be solved, where En is the electronic energy of nth state. Then the nuclear part of the Schr6dinger equation is obtained as [(-h2/8~21.t)V2+ V ( R ) ] v v j = EvJXl/vj ,

(3.1.8)

29

Diatomic Molecules

where V(R) = V N + En. In other words, the electronic equation is solved first for a given nuclear geometry and with the effective potential V(R) at that geometry the nuclear equation is to be solved. The above mentioned assumptions leading to the separation of electronic and nuclear motions are referred to as the Born-Oppenheimer approximation. The nuclear Schr6dinger equation describes the rotational and vibrational motions of the molecule as indicated by the subscripts J and respectively. These two motions again can be separated under certain approximations. The nuclear wavefunction 1-3 can be written as ll/agj(R ) II/agYj/R where ~ v describes the vibrational wavefunction; Y j a r e the Legendre polynomials describing the rotational motion. Since our interest here is to consider the vibrational motion only, we can ignore the rotational motion (or assume that rotational quantum number J - 0) and obtain the vibrational equation as

32 ~ ~ +

8n2g{E - V(R)}}Vv - 0 h2

~

(3 1 9) 9

.

.

The expansion of V(R), around the equilibrium geometry Re, in Taylor series gives V(R) = V0 +

(0V) 1 (O2V'}( R _ Re)2 + ... , ~ (R-Re) + ~\~R2 ]

(3.1.10)

where the first term, V0 is a constant representing the effective potential energy at R = Re; in the second term (OV/OR) represents the negative of force at R = Re and is zero for equilibrium geometry. Thus, ignoring the cubic and higher order terms in Eq. (3.1.10), the potential energy is simplified to

(o v)

V(R) = ~ \~)R2/(R - Re) 2 = ~ K(AR) 2 .

(3.1.11)

The potential energy represented by Eq (3.1.11) is called the harmonic oscillator potential. The frequency of oscillation Ve of a particle, with reduced mass g under a harmonic potential, can be shown to be related to K and t-t via Ve = (1/2rg)(K/~) 1/2.

(3.1.12)

Chapter 3

30

The magnitude of K reflects the nature of potential. For large K the potential varies rather steeply around Re and for small K the corresponding variation is slow. 3.2

Energy levels of harmonic oscillator The vibrational equation (Eq. (3.1.9)) can now be written in the mass weighted coordinate Q, which will be referred to later as the normal coordinate whose importance will become obvious when we consider the polyatomic molecules. Introducing the relations, Q - [I, 1/2 ( R - Re) = [tl/2 (AR), and the relations among partial derivates, (~)/OQ) = (OR/~Q) (0/0R) g-1/E(O/t)R), into Eq. (3.1.9), and using Eqs. (3.1.11)-(3.1.12), one obtains f , ~)2 [ 8rt 2E~0 OQ2 + h2

(3.2.1)

where, O~2

4g 2 4g 2 (4rl;2V2)- (_~)2 -h2 (~)-h2 .

(3.2.2)

The superscript on Eu0 and ~ 0 indicates that these quantities are for the harmonic oscillator and represent zero order quantities. Introducing the dimensionless normal coordinate, q = o~l/2Q, and the relation for the partial derivative, (~)/Oq) = (~Q/~)q)(3/bQ) = ot-1/2 3/~Q, the vibrational equation simplifies to 2 0 ~)q2

2E

+

- q

~

= 0

.

(3.2.3)

hv e

If the energy is expressed in cm -1 units, then the above equation becomes ~ ~)q2

+

Ve

- q

gt ~ = 0 ,

(3.2.4)

where, E 0 = EO/hc and Ve = Ve/C. The solution to Eq. (3.2.4) provides the energy,

Diatomic Molecules

--0

E~

31

(3.2.5)

- Ve(l) + I~) ,

and the wavefunction ~0, ~0

=

NvHaj(q) e-q 2/2 9 I

I

(3.2.6)

I

I

t

4000 -

I

I

I

I

S" 3 0 0 0 ~..2.. E -

I

.)

9i

:

O

I

I

l

l

i

m

e

- ~ . . . . . ~t o

m2000-

:

"

..A.-r

,

r

~

;

1000-

....

o . . .0. . <9. . . . . o . . . . . @" ~3""~ - ~ ' ' e ' ' ~ 9

......

1

"~~

tlJ

o

.

.....

,> .....

,

,'~" @ " 0 - . . . . <9. . . . . <9. . . . . c @

~ .

I

-5

I

-4

I

-3

I

-2

I

-1

?

0

I

1

I

2

I

3

I

4

5

Dimensionless normal coordinate, q Fig. 3-1.

The wavefunctions ~ 0 and potential energy for a harmonic oscillator with Ve

= 1000 cm-1. For ~ - 0, 1 and 2, quantized energy levels, E- o are at 500, 1500 and 2500 cm-1 respectively. The wavefunctions are depicted at the corresponding energy value. The harmonic potential, (Ve/2)q 2 is depicted with filled circles connected by dotted lines.

N~ is normalization constant and H~(q) is the Hermite polynomial (vide infra).

The analytic forms of the harmonic oscillator energy ~0 and

wavefunction g o are a consequence of the "harmonic oscillator" approximation made for the potential energy. The energy in v - 0 state is referred to as the zero point energy (ZPE). The energy levels of the harmonic oscillator for v -- 0, 1, 2... are spaced at constant intervals of ~e. The harmonic potential and wavefunctions ~o are depicted in Fig. 3-1, for

32

Chapter 3

an oscillator with Ve - 1000 cm -1. The wavefunctions for a9 = 0, 1 and 2 are depicted at their corresponding energy values. The harmonic wavefunctions for the first few quantum numbers are given in Table 1. TABLE 1 Harmonic oscillator wavefunctions a for the first few quantum numbers.

NO e -q2/2 N 1 (2q) e -q2/2 N2 (4q2-2) e -q2/2 N3 (8q3-12q) e -q2/2 N4 (16q4-48q2+ 12) e -q2/2 N5 (32q 5-160q3+120q) e -q2/2 aThe constant Nv is given as (4

n2mv/hn)l/4(1/2vv!)l/2;

see Ref. 2.

The experimental data on transition frequencies reveal that the spacing between successive levels decreases with increasing quantum number ag. Thus the harmonic oscillator approximation needs to be improved upon by adding correction terms to it. The higher order terms in the potential energy expansion are referred to as anharmonic terms and the methods and consequences of incorporating these terms will be discussed in the next section.

3.3

Energy levels of anharmonic oscillator To improve upon the harmonic oscillator predictions of transition frequencies, the anharmonic terms in the potential energy, i.e. the terms beyond quadratic, need to be considered. If the contributions from these terms are small then perturbation theory can be used as described below. The hamiltonian for vibrational motion was cast earlier in terms of the bond displacement coordinate R and then transformed to the dimensionless coordinate q. In terms of the latter, the hamiltonian for the harmonic oscillator 1210,in units of cm -1, can be seen from Eq. (3.2.4) as,

33

Diatom& Molecules

(3.3.1)

H o - (9e/2) {p2 + q2),

where p, the dimensionless momentum conjugate to the corresponding dimensionless coordinate q, is p = -i(3/Oq) and the operators p and q satisfy the commutation relation pq - qp = [p,q] - -i. Note that the normal coordinate Q and conjugate momentum P satisfy the relation, PQ - QP = -i(h/2rt) and P and p are related through the relation, P - (h/2u)o~ 1/2 p. Implicit in Eq. (3.3.1) is the fact that the potential energy in cm -1 units for the harmonic oscillator is (3.3.2)

V = Vo + V1 q + V2q 2 9

Correspondingly, the potential energy for the anharmonic oscillator is (3.3.3)

V - VO + Vlq + V2 q2 + V3q3 + V4q4..., where,

V2- a2v-

l

m

ve ,

(3.3.3a)

(3.3.3b)

Oq3 _ 1 o4V V 4 - 24 ~q4

--

(3.3.3c)

Then the total hamiltonian is H-

Ho + H1 +H2

+

.....

(3.3.4)

, n

m

where H0 is the zero order hamiltonian given by Eq. (3.3.1) and H 1 and H2 are the perturbing terms given by H1 = V3q 3 ,

(3.3.4a)

H2 = V4q 4 9

(3.3.4b)

Chapter 3

34

The corrections to the harmonic oscillator energy, resulting from these perturbations, can be derived from the perturbation theory (Chapter 2). The first order correction to energy from perturbation H 1 becomes,

Ea~ = < vOIHllXl/0 > - 0

(3.3.5)

.

This correction is zero because the operator q3 will not connect the same states (vide infra). In the second order approximation, the energy correction to state ~) is

0 --0 ~0 E~ -E~,

(3.3.6)

To evaluate the integrals in Eq. (3.3.6), it is necessary to know the effect of q3 on the off-diagonal (v ~ ~') harmonic oscillator integrals and of q4 on the diagonal integrals. The properties of these integrals are essentially those of hermite polynomials Hv(q) and these are summarized in Appendix 2. With the help of the formulae given in Appendix 2, it can be seen that < ~OIH21~~ > is straight forward to evaluate. This is found to be (6~)2 + 6~) + 3) = 3V4{(~) + 1)2 + 1 ) The second term operator serves wavefunction by ~) + 3 will survive (3.3.6) becomes

in Eq. (3.3.6) can be evaluated by noting that the q3 to step-up or step-down the harmonic oscillator +1 and +3 quanta. That is, only the terms a ~ ' - ~ + 1, in the summation in Eq. (3.3.6). The second term in Eq.

0 ~)'~:~)

(3.3.7)

0 --0 --0 E~ -E~,

(3.3.8) 8~e

Thus the total energy of the anharmonic oscillator, considering the potential energy up to quartic terms, to the second order is 1~ = ~o + 1~ + 1 ~ ' , i.e.,

Diatomic Molecules

3032) 8~ e

(~) + 1) 2 +

35

i 7v32) 3V 4

16~e

(3.3.9)

It is common practice to write this equation as

E~ - Ve('o + 1)- VeXe('o + 1) 2 + A,

(3.3.10)

where, -

/30V32

VeXe --

8~ e

3V4) - ~

,

(3.3.10a)

7~32 (3.3.10b) 16~e ' and to refer to VeXe as the anharmonicity constant (this anharmonicity constant is usually represented 3 by symbol x, but we find the designation VeXe more convenient; Xe now becomes a dimensionless number). The term A is independent of the quantum number v and is sometimes omitted due to its relatively small magnitude. The result obtained for the energy correction depends on how the perturbation is treated and how many terms in the potential energy expansion are considered. Higher order perturbations will contribute to A - 3 V4

higher powers in 09 + 1/2) and a general equation for Ev given by Dunham 4 can be written as Eu - Z a i ( a g + l ) X ,

(3.3.11)

where i is integer (0, 1, 2 ... etc.); ao, al and a2 can be, for instance, identified with A, Ve and -VeXe respectively in Eq. (3.3.10) at the level considered. An alternative to the perturbation method discussed above is to transform the hamiltonian using a contact transformation 5,6, as described below.

In the original hamiltonian, now designated as ~(0), with

superscript (0) representing H before transformation,

36

Chapter 3

(3.3.12) ~(0), ~(0) and ~(0) represent respectively the zero order hamiltonian, the first order correction and the second order correction with decreasing order of magnitudes. If this hamiltonian is subjected to a contact transformation and written as

~(1) = ~(1) + ~ 1 ) + ~(21) +...

,

(3.3.13)

then the terms H~'), " ~ 1 ) and H(21)represent respectively the zero order, first order and second order contributions of the once-contact transformed hamiltonian. The superscript indicates the number of times the hamiltonian is subjected to the transformation. The transformation itself is achieved through the relation

~(1) = T(1)~(0)T(1)-I ,

(3.3.14)

where, T(1) _ e i s l = l + i S l +

(i $1) 2 eoo

9

(3.3.14a)

and T (1)- 1

= e

-i S 1

= 1-iS1 +

(-i S 1)2 2 ... ,

(3.3.14b)

with S 1 representing a dimensionless operator that is chosen to satisfy a criterion explained below. Substitution of Eqs. (3.3.12) and (3.3.13) in Eq. (3.3.14) and equating the terms of similar magnitude, one obtains, (3.3.15) (3.3.15a) (3.3.15b) H(21) = fi(~

i (Slfi~0)- n~~

- 1/2 {SI(S1 fi(00)- fi(o~

Diatomic Molecules

37

_ (Sl ~(0) - H(0)S 1)S 1 }

=H(20) + i [S1,H{0)] - ~/2 [s~, [s~, R(0~

(3.3.15c)

The central theme of contact transformation is to choose S1 in such a way that the off-diagonal integrals of ~(1) become zero. That is, 1 < v~

~{'~, v ~ > = o

for v r a9" .

(3.3.16)

The rationale for this becomes transparent when we look at the expression for the second order energy correction given by Eq. (3.3.6). If the transformed hamiltonian ~{1)is placed in Eq. (3.3.6), and if ~{1)has zero off-diagonal integrals then the only correction term surviving would be that in Eq. (3.3.5); i.e., < ~ o1~(~)1, "'1 , v ,0 >- In other words, contact transformation enables one to avoid evaluating the off-diagonal integrals and obtain the energy corrections through diagonal integrals. The condition of satisfying Eq. (3.3.16) leads to the relation, < v~

< ~ , <,~ > =

= o

< V O. I~(0) llltO, + < V O'li [S1 fi(00)] IV 0 > , H 1

(3.3.17)

which can be rearranged, noting that ~0 are the eigenfunctions of ~b,,),m to give 7

= -i

< v ~ i n~0~ i v ~ v > .

(3.3.18)

This equation defines the approach to finding the operator S 1. The transformed hamiltonian H (1) can be subjected to a further transformation such that,

~(2) = T(2) ~(1) T (2)-1 , T (2) - e i s2 . This will yield,

(3.3.19) (3.3.19a)

Chapter 3

38

~(2)

;-;.(2) + fi]2) ;-;,(2) = H0 + 1-12 + ... , ~(2) = n(1) = n(O) 0

(3.3.20a)

fi(2) = ~{1) = ~]0) + i [ S l , fi(0)]

(3.3.20b)

(3.3.20)

1

n(2)

n(1)

= ta 2

+ i [$2,

n(0)]

.

(3.3.20c)

Note that the zero order and first order terms are uneffected by the second contact transformation and that the off-diagonal integrals involving ~1~'(2) ~(2)

(which equals ~]1)) will be maintained zero. The second order terms 1-12 -(1) and H 2 are different and the second contact transformation adds a term i[S2, H(00)] to the previously transformed contribution H(21). Now the function $2 is chosen to make the off-diagonal integrals < ~ o I ta 2~'(2) I ~o> zero. Thus following Eq. (3.3.18), the condition for choosing $2 is

< v O ' I s 2 1 v O> = - i


i

i v ~9 >

(3.3.21)

This procedure can be continued to higher order terms. To nth order contact transformation, ~(n) = T(n) .... T(1) ~(0) T(1)- 1 .... T(n)- 1 _ Tn ~(0) T -1 n =

ei( S1 + .... Sn) ~ ( 0 ) e_i(S1 + .... Sn) .

(3.3.22)

The transformed hamiltonian will have zero off-diagonal integrals to nth order i.e., ~ n ) , ~(n) .... ~(n) n will have only diagonal integral elements, and the energy becomes -(n)

E'~ = < ~ l 0H ( n )

l~ 0 > .

(3.3.23)

Diatomic Molecules

39

Let us now apply the above described contact transformation to a diatomic molecule. The operator S 1 is given as 8,

SI=

- 2V3 { p 33 + 2-~ } "

(33"" 24)

Ve

Substituting Eqs. (3.3.24), (3.3.15) into Eq. (3.3.23), the energy, to =(1) second order, becomes 1~ . ~(1)

0 = < ~1H(00) + ~ 0 ) + ~(20) +i [S 1, ~(0)] +i [S 1, H~0)]

Is1, Is1,

(3.3.25)

Iv ~ >

These integrals can be evaluated by using the relations in Appendices 2 and 3. The integral < ~0 1H(00) I vO > gives the contribution of Eq. (3.2.5); < V~ I H~~ I V O > - 0; < V~ I H(2~ I vO > gives the contribution of Eq. (3.3.7). In addition,

< v01i iS1, H(00)] I V0> = i< V01S1H(00)I ~ 0 > - i < V0 1H(00)S1 I V 0 > =0 ,

(3.3.26)

which can be seen from the fact that the operator Slnb,, )rn or -_rnHb~)S1 is of fifth order (collectively in p and q) and the odd order operators step-up or stepdown the wavefunction by an odd integer. For example, the effect of qpq3 on V O is to generate wavefunctions vO+5, ~1/o+3 and vO+I which are orthogonal to ~0. The remaining two terms in Eq. (3.3.25) give,

_

~2 (30l)2 + 30~ +11), 8Ve (3.3.27)

which is same as the second order energy correction from ~0) (see Eq. (3.3.8)). The evaluation of integrals in Eq. (3.3.27) are simplified by the relations,

40

Chapter 3

i[S1,H~0)] (3.3.27a) and

44)

i[SI'H~~ - - i / ~ 2 ) ( p3q3 - q P3 3 + qPq - q Pq 2 ~Ve)

.

(3.3.27b)

The integrals involving these quantities are given in Appendices 2 and 3. The energy from Eq. (3.3.25) is

--(1) Ev -- Ve(1)+l)+0+3V4{(1)+ 1)2+ 1 } + 0 - V2 (30~)2+30~)+11) 8~ e

(3.3.28)

which is equivalent to the one obtained from second order perturbation theory using the non-transformed hamiltonian (see Eq.3.3.9). The effect of second contact transformation can be investigated by comparing Eqs. (3.3.15) and Eqs. (3.3.20). The second contact transformation changes ~(1)to w.(2)_ ~(1) + i [S2, ~(0)]. Thus the energy t-12 1-12

~)(2) resulting from the second contact transformed hamiltonian is obtained to second order by evaluating < lit~ I i [$2, H(0~ I/I/0 > and adding it to Eq. (3.3.28). The operator $2 is given as 8

$2

=

( 9~)e3 )2 (1/32){ 15 (p3q + qp3)+ 9 (q3p + pq3)} . _ ( V4 )(1/32){ 6 (p3q + qp3)+ 10 (q3p + pq3)} .

Ve Using this expression the commutator i [$2, ~(0)] is found to be 13

(3.3.29)

Diatomic Molecules

v~

41

{_ 27(p2q2 + q2p2)_ 18 + 60p4_ 36q4 _

64v e m

45qp2q+27pq2p}+_ v4 {1 8(p2q+q 2 2p2)+12+ 24p4 - 4 0 q 4 - 18qp2q + 30pq2p I

"

(3.3.30)

Substituting the appropriate relations listed in Appendices 2 and 3, one finds that < v ~ ~ is:, n~o~ ~ v ~ > - o

.

(3.3.31)

Thus using second contact transformed hamiltonian, the energy of diatomic molecule obtained to second order remains the same as that obtained from the first contact transformation to the same order; i.e., second order, where E~) is given by Eq. (3.3.28) E•) The= tzvthirdto and higher order transformations can be similarly written down, but due to the complexity involved they will not be discussed here. 3.4

Energy levels of Morse oscillator

In the case of harmonic and anharmonic oscillators that we have discussed earlier, the potential energy was expanded in Taylor series for small displacements q. However, no analytic form was associated with the potential. Several different analytic forms have been proposed for the potential of a diatomic molecule. The Morse oscillator 9 potential function has been the most widely used one due to its simplicity. This function is given, in cm -1 units, as Vm =

[)e [1 -exp [- ~t (R- Re)]] 2 ,

(3.4.1)

where ~)e is the dissociation energy and a is constant (in cm-1 units). Using the dimensionless coordinate q, this equation becomes

~rm =

[)e [ 1 - e x p [-aqq]] 2 ,

where dimensionless aq is given as

(3.4.2)

Chapter 3

42

aq =

h )1/2 ~t 4r~Vel.t

.

(3.4.3)

If we use this potential in place of the harmonic oscillator potential, the vibrational equation can be solved 9 to yield the energy, Ea~,m = Ve (~1 + 1/2) - VeXe (a) + 1/2)2 ,

(3.4.4)

where,

VeXe = (h~t2/8r~2cg) .

(3.4.4a)

Substituting Eq. (3.4.3) in Eq. (3.4.4a), one finds

VeXe = (Ve/2) a 2 , and

(3.4.4b)

Xe =

(3.4.4c)

1

2/2 q

.

It might seem disturbing to see drastically different forms for the anharmonicity constant, when compared to that obtained earlier. However, the anharmonicity constants given by Eqs. (3.3.10a) and (3.4.4b) can be shown to be related as described below. At the minimum of the Morse potential curve, the first derivative with respect to q is zero. Using this condition, differentiation of Eq. (3.4.2) gives

~r2,m _ 1/2 ( ~ V m )q=0 - aq2 I~)e'

~r3,m-

3

1/6 (~~33m)q=0 = - a q [ ) e -

(3.4.5)

-aqV2,m ,

= (7/12) a~ E)e = (7/12) a 2q V2,m 9

(3.4.6)

(3.4.7)

In Eqs. (3.4.5)-(3.4.7), subscript m has been placed to remind us that these equations are for the Morse potential. Substituting these derivatives in place

Diatomic Molecules

43

of V3 and V 4 in Eq. (3.3.10a), and remembering that Ve = 2 V2, Eq. (3.3.10a) becomes

2m2 aqV2,m [7~a2~ 2VeXe - ( - ~ ) ~ - t 8 ) qv2,m - a q Ve/2 '

(3.4.8)

which is same as Eq. (3.4.4b). Similarly substitution of Eqs. (3.4.6) and (3.4.7) in Eq. (3.3.10b), for obtaining the constant A of anharmonic oscillator, reveals that A = 0 for the Morse oscillator. The main difference between the perturbation and Morse oscillator approaches is that the higher order potential terms V3, V4, ... etc. are treated as independent quantities in the former, but they are dependent quantities [see Eqs. (3.4.5)-(3.4.7)] in the latter.

3.5

Electric dipole transition moments of harmonic oscillator In the approximation of separating the electronic, vibrational and rotational motions (see Section 3.1) one does not have to be concerned about the electronic and rotational wavefunctions, to the extent of considering the influence of anharmonicity on vibrational transition moments. Furthermore, the vibrational states being considered here belong to the same electronic state (unless specifically mentioned otherwise). Once the hamiltonian has been defined to the order of accuracy desired, the vibrational wavefunction can be determined and transition moments evaluated. In the zero order approximation, the vibrational equation is that of the harmonic oscillator and wavefunctions are defined by the Hermite polynomials (Eq. (3.2.6)). The electric dipole transition moments for transition from vibrational state ~) to ~' can be given as - < v ~ i.ivo

(3.5.1)

>

The electric dipole moment operator ~ can be expanded at the reference geometry in Taylor series using dimensionless normal coordinates as ,

1

=

bt0 +

q + 2 x,~)q2)"~ + 6 \~)q3)'

=

go + g lq + g2q 2 + ~3q 3 + ... ,

+ "'" (3.5.2)

where Ix0 is the dipole moment at the reference geometry. In the electrical harmonic approximation this series is truncated to the first power in q, i.e.

Chapter 3

44

~

-

II,0 + ( ~ q ) q

9

(3.5.3)

The use of Eq. (3.5.3) with harmonic oscillator wavefunctions is usually referred to as the double harmonic approximation. Substitution of Eq. (3.5.3) into Eq. (3.5.1) gives (see Appendices 2 and 3),

- ~l'0~agag" (~q) { (1)§ 1/2 1/2 ~',a~-I }, -I2 ~',~+1 + (2) (3.5.4) where 5 1)l)" = 1 for v = ~)" and zero otherwise.

The first term in Eq.

(3.5.4) survives only for ~ ' = ~; the second and third terms survive only for v ' - v + 1 and t) - 1 respectively. Thus, in the approximation of using harmonic oscillator wavefunctions with the harmonic electric dipole moment operator, the electric dipole transition moments for v --->v + 1 transitions (called fundamentals) can be non-zero, but those for a9 --> v + 2, v + 3, etc transitions (called overtones) are zero. To account for the occurence of overtone transitions, and to accurately describe the transition moments in general, it is necessary to use the anharmonic wavefunctions as well as the anharmonic electric dipole moment operator. There are two approaches one can take to evaluate the transition moments in the double-anharmonic approximation. One approach involves evaluating the anharmonic wavefunctions by using perturbation theory and using the resulting wavefunctions to evaluate the transition moment integral with the expansion of g containing higher order terms (l~2q2, 11,3q3 etc). The second approach involves the contact transformation of the electric dipole moment operator and using the resulting operator with harmonic wavefunctions to evaluate the transition moment integrals. The perturbation method is more tedious than the contact transformation method. Both approaches are described below.

3.6 Electric dipole transition m o m e n t s of a n h a r m o n i c oscillator The wavefunction corrected (see Section 2.2) to first order is

Diatomic Molecules

45

where ~g~ represents the anharmonic wavefunction resulting from the first order correction to zero order wavefunction ~0. It becomes transparent from Eq. (3.6.1) that the perturbation causes the harmonic wavefunction of a given state v to mix with the corresponding harmonic wavefunctions of higher energy states. Thus the anharmonic wavefunction for a given state can be viewed as a mixture of appropriately scaled harmonic wavefunctions of several states, which can be demonstrated nicely for a two level system. If we consider a hypothetical case where only a9 = 0 and 1) = 3 states exist, the transition between these two levels are not allowed in the doubleharmonic approximation. One should consider electrical anharmonicity that includes, at least the ],l,3q3 term, to obtain non-vanishing transition moment. The mechanical anharmonicity resulting from the perturbation H 1 = VC3q3 causes a9 = 0 and a9 = 3 states of this two level system to mix. This can be seen from Eq. (3.6.1), as follows" a 0 ~0 = ~ 0 -

V3 ( 6 ) 1'2 0 3 Ve 11/3

0 0 ~o - b ~3 '

(3.6.2)

m

a 11/3 -

0 1[/3 +

V3

3 Ve

(6"~1'2 0 0 0 ',8J ~g0 - ~3 + b ~g0 9

(3.6.3)

a has a small contribution from lg 03 and ~ga3 has a small Note that ~g0 0 contribution from ~0" The transition moment will have only one

contribution, that from J.L3q3 (ignoring higher order terms). ].t30 =

0 0 b2 0 q3 0 <~g3 1~3q3 I~g0 > <~g0 I!"t3 Ixg3 >

6

0q3)'

Ve

(3.6.4) \ 0 q 3)'

Ve

The first term arises from purely electrical anharmonicity (~3q 3 term) and the second term arises from the interference of electrical and mechanical anharmonicities. The second term can be shown to be small; since the

Chapter 3

46

harmonic term V(2 = Ve/2, (Vg319e)2 can be expected to be less than (1/2)2 and the presence of 1/12 makes the second term at least two orders of magnitude smaller than the first term. Now returning to general multilevel case, and using H 1 = V3q 3, the integral on the right hand side of Eq. (3.6.1) can be seen to be non zero only for ~" = l) + 1 and ~ + 3. For the ground state ~ = 0 and therefore ag" in Eq. (3.6.1) can only be 1 and 3. Then the ground state anharmonic wavefunction becomes n

a ~/0

-

0

3V3

xl/0 -

Ve

m

1

0-

~

V3

3 Ve

q/1

(6"~1/2 \8)'

0 q/3

"

(365) " "

Similarly for the first excited state (~ = 1) anharmonic wavefunction becomes, ~la

-

V10 + 3 V 3 Ve

1

0 _ 3 V3

~

/I/0

0 _ V3 ~f~ /!/40 . Xl/2 3 Ve

Ve

(3.6.6)

The transition moment for the 0 ---> 1 transition becomes, a a a a a a ~L10 "- < V1 I ~14) [ V0 > + < V1 I ~1,lq IV0 > + < V1 I la,2q21 V0 > a a

+ < ~1 ! 11'3q3 I ~0 > + . . . .

(3.6.7)

Substituting Eqs. (3.6.5) and (3.6.6) into Eq. (3.6.7), the resulting integrals can be evaluated using the relations in Appendix 2. Thus, a

a

< V 1 I I,ta31VO> ---

3 V3

3 V3

Ve a a =l.tl < Xt/1 I la'lq I qt0 > ~

~ 1+

= 0 ,

(3.6.8)

Ve

(3.6.9)

Ve

~ m

a

a

(3.6.10)

< q/1 I 1"1'2q2 I ~ 0 > = -

~ a a < ~1/1 I ~t3q31~0 > - 3p,3 {1

Ve 347

(~__$3)2 } Ve

(3.6.11)

Diatomic Molecules

10~ Ill0 = ~ +

1

3q + 2~r~ Ve

47

{1 03~1, _ 53_~ (~) 2 jgq3

i9q2

12 3q + 3-44873311} dq3

(3.6.12)

Note that this equation is obtained by considering only V3q 3 terms as the perturbation and the resulting first order wavefunction. One can proceed similarly to higher order perturbations for obtaining the anharmonic wavefunctions. Expressions for transition moments using up to third order perturbation have been reportedl~ 11. The perturbation method for evaluating the transition moments described in the previous paragraphs is laborious due to the need for keeping track of the off-diagonal elements in the anharmonic wavefunction. Contact transformation method described earlier, in the context of anharmonic oscillator energy levels, provides a convenient alternative 7,8,12. This method is described below. The vibrational equation involving the anharmonic terms in the hamiltonian can be written as

~(0)11/~ = (H =(0) -a a 0 + n~0) + H(0) + ...)V; = E~V~,

(3.6.13)

Ea3 is the anharmonic energy for state a3. Using the contact where -a transformation ~(n) = Tn~(0)Tn-1 (Eq. (3.3.22)), and noting that ~(n) is diagonal in the basis of harmonic oscillator wavefunctions, i.e.,

~(n)ll/O = E' -aa311t~ 0,

(3.6.14)

one finds that the vibrational equation modifies to H[(O)(Tn-llltO) = "-a(Tn 1 0 E a J /l/aj) ,

(3.6.15)

and,

1 o

/lt~ = Tn-/lq3. Using this relation, the transition moment integral becomes

(3.6.16)

Chapter 3

48

],Laj'v = - < IltvO I Tnll,(0)Tn-1 I lit0 > = ,

(3.6.17)

where, 11,(n)

_

Tn~L(0)Tn- 1

.

(3.6.18)

The operator ~t(n) represents the n times contact transformed electric dipole moment operator. The attractive feature of Eq. (3.6.17) is that the electric dipole transition moment integral is evaluated with the harmonic oscillator wavefunctions and there is no need to worry about the anharmonic wavefunction (Eq. (3.6.1)). The electric dipole moment operator however must be contact transformed, which is relatively easy. m

Following the contact transformation method for H, one can obtain the

operator ~(n) by simply substituting !~ in place of H. The electric dipole moment operator contact transformed twice becomes, ~(2)

=

1,1,(2)+ I.t]2) + 1,1,(2 2) + ... ,

=

~(o~ = ~o + (O0~q)q ,

(3.6.19)

where

r it?) p,~O) I.I,(2)

=

=

=

= .?

(3.6.19b)

+

(/)2~'~ q2

1/2 \~)q2 )'

(3.6.19a)

'

(1) 2 + i[S2, I,t(oO)] ,

(3.6.19c) (3.6.19d) (3.6.19e)

and g?) =

1/6 (~3~'~ q 3 . ~q31

(3.6.19f)

Diatomic Molecules

49

It is to be noted that the sum go + (~q)q is taken as the zero order term for the electric dipole moment operator. Also there are no restrictions on ~t~1) and I.t2-(2), unlike those on ~1) and ri~(2) 2 It is straightforward to evaluate the commutators involved in the above equations using the definitions for S 1 and $2 [Eqs. (3.3.24), (3.3.29)]. These are found to be, i[Sl, l.t(00)] = -

V3 ( ~ q ) ( 2 p 2 + q 2 )

'

(3.6.20)

Ve

i[S1, I1~0)]

= - -_V3 (~2~'~ (p2q + qp2 + q3) Ve ~q21 '

1/2[$1, [$1, It(~

= (~-~3)2 ( ~ q ) (p2q+ qp2_ q 3 ) ,

(3 6.21) " (3.6.22)

Ve

i[S2, g(o~

{A (3 p2q + 3 qp2) + B (2 q3) } ,

= (~q)

(3.6.23)

where, A = ~-~

_6 Ve

9 (~__33)2 B

=

32

Ve

(~___4_4),

(3.6.23a)

Ve 10

(~r4) Ve

(36.23b) 9

Combining Eqs. (3.6.19)-(3.6.23), the twice transformed electric dipole moment operator becomes, g(2) =

(32~'~q 2 (331-t'~q3 gO + ~ q q + 1/2 \Oq2/ + 1/6 \Oq3~' !

Ve

50

Chapter 3

VC3(bztt'~

qp2 +q3) _ (_V---~3) - ( ~ q2)

_-\~qg),(p2q+

Ve

(p2q + qp2 _q3)

Ve

+ A (~q)(3 p2q+3 qp2) +B (~q)(2 q3) ,

(3.6.24)

Using this operator the transition moments for the fundamental and first two overtones are found to be as follows.

X

[{ (~---~3)2+9 (~--33)2-3(~---4-4)} (0~J~q) Ve

- 5

0 ( 2 ) Ivv> 0
=

0 - [

+

2- Ve

\Oq21

Ve

+ 1/2 \Oq3 ),

(3.6.25)

,

)]1/2 {1/2 (~2~'~+ (V_y~)(~q) } [ (~+2)(a)+l ~ \~)q21

[ 1/6

(~+3)(a)+2)(~+1)] 1/28

/

\(031~'~0q3+

())

(3~q){ 3 (~_~)29 (~_33)2+1 rV4}] --

Ve

2 ?e

(3.6.26)

z

\(021"t'~Oq2 (3.6.27)

"

In the above equations the contributions from the first and second contact transformation are separately listed to identify their origin. The contributions coming from the second contact transformation are:

(~~-~q)[9 (~---~3) 2 3 V(4] ( I ) + l ) 3/2 in Eq. (3.6.25); and Ve - ~/e 2

(~q)[

9(~-~3)2+1--(~r4)] [ (1)+3'(1)+2)(1)+1' ] 1/2 -

Ve

2 Ve

8

in Eq. (3.6.27).

The expression for the average dipole moment in vibrational state ~ is,

Diatomic Molecules

<~l/~176

~l,0 + (a~+l]2)

2\~)q2/

3

51

.

(3.6.28)

In Eq. (3.6.28), as well as in Eq. (3.6.26), there is no contribution from the second contact transformation.

3.7

Electric dipole transition moments of Morse oscillator Although the Morse potential is a convenient model for describing the anharmonic oscillator energy levels, the evaluation of the transition moments using Morse wavefunctions has not been that popular. The reason for this is that the expressions obtained for the transition moments evaluated with Morse wavefunctions are long and complicated. Nevertheless, they can be handled with computer programs or can be simplified to convenient expressions when the anharmonicity constant is small. The transition moments with the Morse wavefunctions and the electric dipole moment operator expanded to include the quadratic term in R have been evaluated. These expressions given by Heap and Herzberg 13 will be considered below for comparison with those obtained from the contact transformation method. It would be interesting to find out what differences, if any, one might encounter in the results predicted by these two approaches. However a direct comparison is not transparent unless some reasonable approximations are made to the expressions resulting from Morse oscillator transition moments, as will be evident below. In evaluating the Morse oscillator transition moments, the dimensionless coordinate x, given as x

=

( R - Re)/Re

,

(3.7.1)

is often used. This quantity is related to the dimensionless coordinate q as q

=

(ge/2Be) 1/2 x ,

(3.7.2)

where Be is the rotational constant (in cm -1 units) given as,

Be =

h/8rc2~R2ec 9

(3.7.3)

Note that Ix in Eq. (3.7.3) is the reduced mass. Using these relations, and noting that we defined the anharmonicity constant as VeXe (so Xe is a unit less number), the transition moments of Morse oscillator are given as

52

Chapter 3

< ~ ,mllxl~l/~> =

(-1

)~+x)'(Be Ve) 1/2 Xe " I

+ (- 1)~+a~'+1 2

Be ~" ~eXe

C~-~ 3~t (~)" ~))Ve Ox

C~'~

32~

! [ 1 - ( ~ ' + 1)Xe] Ha~'a~1/2 0X2

(3.7.4)

The expressions for coefficients C~-~ and H~-v, which have complicated form 13, will not be given here. However, when Xe is very small the following relations hold for the first three transitions, For

Cu-~ = Xe1/2", Hu'u --- 0 Cu'u = 21/2 Xe; Hu-u = 1 Cu-u = (6 Xe3)l/2; Hu-~--- 3

ag" = 1, x) = 0, 9" - 2, x) = 0, x)'= 3, ~ = 0,

(3.7.5) (3.7.5a) (3.7.5b)

With these approximations, we find m -1 - ~

(~q)

m ~ <~2 I ~ t l ~ 0 > --- +

(3.7.6)

(~q) _ 1 (~2~1,'~ 2~ \3q 2/ '

m

<~3 I~1'1~0 > = -

Xe

\ 3q2/

(3.7.7)

9

(3.7.8)

For the same transitions the results obtained by the contact transformation method, starting from harmonic oscillator wavefunctions, were given earlier. Those expressions can be rewritten by using the relations for the potential constants of the Morse oscillator for direct comparison. Substituting Eqs. (3.4.5)-(3.4.7) in Eqs. (3.6.25)-(3.6.27), we obtain


1 ~l,I,

1 {X e ( ~ q ) + 54Xe (~2~1,'~ 1 (~3~'~}

(2)lvO>-q~ 3q + ~ 2 1

(~q) X/2

<~l~t(2)l~~

= _

2 \ 3 q 2 / + ~ 3q 3 /

1 {1 (~31L_'~} +~2

2

~ /)q3/

+ ~22 X,~q2/,

(3.7 9) ' (3.7.10)

Diatomic Molecules

>

1 XeI ( ~ } ) i

(})

53

1/2(c)2~1,'~ 1 (0_~'~ \0q 2/+d3 \0q 3I "

(3.7.11)

When Xe is small, Eqs. (3.7.9) to (3.7.11) can be seen to be equivalent to Eqs. (3.7.6) to (3.7.8) (note that the cubic term was not considered in Eqs. (3.7.6) to (3.7.8)). The sign difference present in the two sets of equations is not important for calculations on transition probabilities since they depend on the square of transition moment. Using the approximations given by Eqs. (3.7.5), the approximate Morse oscillator transition moments can be generalized 14,15 as, < g ~ I II, I ~

> = _

(~q) ;l [_~

Xeag-

111/2

1 (~21,t~ [l)i xe~)-2] 1/2 (1-~11) 2

0q2 )'

"

2

(3.7.12)

This is a simple relationship that can be used to predict transition moments of higher transitions from those of lower transitions.

3.8 Magnetic dipole transition moments of harmonic oscillator Unlike the electric dipole transition moments the magnitudes of magnetic dipole transition moments are considerably smaller and are usually ignored. However, the magnetic dipole transition moments are essential to observe and explain the circular dichroism in vibrational transitions of chiral polyatomic molecules. Therefore we will introduce the treatment 16 for the magnetic dipole transition moment of a single oscillator here. The magnetic dipole moment ma, for a collection of point charges is given (see Appendix 1) by the expression, mot=

~1 ZA

e or

e~A RAI3l~A7 ,

(3.8.1)

where RA and R A are respectively the positional and velocity vectors of a particle A with charge e~A (e being the unit of electron charge); ~ a~7 is the alternating tensor with values of 1,-1 or 0. There are some important properties characteristic to magnetic dipole moments. First, the magnetic dipole moment is origin dependent, as can be seen from Eq. (3.8.1). If the chosen molecular origin is displaced by a constant a, then the positional vector changes from RA to R'A = RA -a, and the new magnetic dipole moment becomes

Chapter 3

54

rn~ = rn~ A ~a- Z

~ al3T e~A I~AT = ma - ~a ~ r

(01a~,/bt) .

(3.8.2)

Second, Eq. (3.8.1) suggests that the magnetic dipole moments are generated by the nuclear velocities and hence are functions of nuclear velocities. Since the time variation of the electric dipole moment can be written as

0).t~/~t = Z (0I.t~/ORAS)(3RAS/0t)=Z A

(0ILtT/0RAS)I~AS,

(3.8.3)

A

it becomes apparent 17 that Eqs. (3.8.2) and (3.8.3) lead to a

0rn~la I~A8 = - ~ e r (O~t~'ORAa). Thus the magnetic dipole moment vector can be expanded as

(3.8.4)

m - m0 +

(3.8.5)

51~A~

I~A~ + . . .

,

where m0, the value of magnetic dipole moment at R A - 0, is zero for a free molecule in non-degenerate electronic state. The higher order terms in the expansion of m~ can be written by noting the properties of mo~ with respect to time reversal; when t is replaced by -t, m~ changes sign (see Eq. 3.8.1) [but its velocity derivative (see Eq. 3.8.4) does not change sign]. Making use of this time asymmetry for m~ one finds that the expansion of rn~ will contain 17 only odd derivatives with respect to I~A6. For the sake of convenience we will now use the dimensionless normal coordinate q, and conjugate momentum p, and write the Taylor expansion of m in q and p as m

03m "~ + ~2\OpOq,/(p q + qp) + 1/4( ~)P0q 2/(q2p + pq2) (03m'~ p3 + + 1/6 \ Op3 J' ....

(3.8.6)

This expansion is somewhat different from the analogous expansion for electric dipole moment in Eq. (3.5.2) and ensures that the magnetic dipole moment for molecules in a non-degenerate electronic state is zero. Further details on this expansion can be found in Chapter 8. The derivatives

Diatomic Molecules

55

involved in this equation are implied to be taken at p = 0 and q = 0. This equation can be succintly written as, (3.8.7)

m - m01 p + m l l (pq+qp) + m21 (q2p+pq2) + m03 p3 + ... , where m01 "- ( i ) m / 3 P )

(3.8.7a)

,

m l 1 = (1/2)(~)2m/~)p~)q) = (1/2)(O2m/~)q/)p) ,

(3.8.7b)

m21 - (1/4)(~)3m/OpOq 2) - (1/4)(/)3m/~)q2Op) ,

(3.8.7c)

m03 - (1/6)(~)3m//)p3).

(3.8.7d)

In these equations, the two subscripts included for m (i.e., m01, etc.) represent the order of differentiation of m in q and p respectively. In the harmonic case, all terms except m01P are set to zero. Thus, the magnetic dipole transition moment in the double harmonic approximation is given as mag'aj = = i(/)m//)p)[((~+ 1)/2)1/2 5a9',~+1 -(x)/2)1/2 5~',a9-1].

(3.8.8) Special attention must be given to the electronic contributions to Om/Op (and higher derivatives) as will be discussed in Chapter 8.

3.9 M a g n e t i c oscillator

dipole

transition

moments

of

anharmonic

In considering the influence of anharmonicity on magnetic dipole transition moments, we defer the discussion on electronic wavefunctions to Chapter 8. For the present purpose, it is sufficient to write the magnetic 0

0

dipole transition moment for the ~ to ~' transition as <~a~,lml~>, with m defined by Eq. (3.8.6). This statement will be elaborated further in Chapter 8. To obtain the transition moments of the anharmonic oscillator, we use the contact transformation method as described earlier for the electric dipole moment operator. For this purpose we introduce the definitions, m (0)

(o (o) r m 6) + m 1 + m 2 ) + ... ,

m(00) - mol P ,

(3.9.1) (3.9.1a)

56

Chapter 3

(Pq + qP) '

(3.9.1b)

m(2o) - m21 (pq2 + q2p) + mo3 p3 ,

(3.9.1c)

m] ~

-

ml 1

and obtain the following commutators:

qp) ,

i[Sl,m(o~

= (V3~e) Ino1 (Pq +

i[S 1,m]~

- (W3/Ve) m 11 (pq2 + q2p _ 4 p3) ,

(3.9.2) (3.9.3)

-1/2 [S1 ' [Sl,m(o0)]] = (V3/Ve)2 toOl {(pq2 2+ q2p) - 2p 3} ,

(3.9.4)

i[S2,m(o0)] - _ 2 A toOl p3 _ 3 B nb 1 (pq2 + q2p) .

(3.9.5)

Using these expressions, the twice contact transformed magnetic dipole moment operator is found to be m (2) = m01 p + ml 1 (pq + qp) + m21 (pq2 + q2p) + m03 p3 + (W3]Ve) m01 (pq + qp) + ('~/3]Ve) mll (pq2 + q2p_ 4 p3)

+ (~3],0e)2 m01 { pq2 + q2p - 2 p3}_ 2 Am01 p3 2 - 3 B m01 (pq2 + q2p),

(3.9.6)

where A and B are given by Eqs. (3.6.23a, 3.6.23b). The magnetic dipole transition moments < ~ -oI m (2) I ~o> are obtained by using the harmonic oscillator wavefunctions, ~o. These transition moments for the first three transitions are as follows: -V(3)2_ V3 2 0 IvO>= i(~)+ 1)1,2[c)m,~ ~+ 1)3/2
57

Diatomic Molecules

3V4 3m 10vr ~2m'~ ~3m] ( 33m )] + - } (--o-p-)(/gp/gq' + 3/6(/9p3 j + 2/4~'~)p/Oq2 '

Ve

(3.9.7)

2re

i~2mx (~e3)(~p__p)] } { [(1)+1)(9+2)] I/2[1/2~,~p~q)+


(3.9.8) V3 2 _ll,8(_~e V3) 2 <_V~+3 0, 'm (2),IV~>= 0 i[(a3+3) (I)+2) (~+)] 1 1/2[(3(_~_e) 8 02m,~ (03m] 03m )] 9 + 3/2 (~e4)} ( ~0m ) + 6/2 Vr3 (0p~qJ-1/6,, 0p3, + 2/4 (~,~p0q2

Ve

(3.9.9)

The average magnetic dipole moment in state a0is
:0 .

(3.9.10)

It is informative to compare these transition moments with the corresponding electric dipole transition moments of a Morse oscillator. Using the relations among the potential constants of the Morse oscillator (see Eqs. (3.4.5)-(3.4.8), the first three transitions originating from the v = 0 state become,

1 ~_~)~+ ~ 12 {_3Xe(~) =i[~2( 02m't+

+

<~g0lm(2)l~~ _ 01 (2), 0 < ~g3tm I ~ 0 >

1 ~2m "~ { ~22 (~P~qJ - ~

=i(-~ ~)

(03m)+ 1/2 ( 03m ) } ] ~m

(-~) }'

(3.9.11) (3.9.12)

[ Xe(~-~)-~ 27 - 3(~-~) 1/2 [~)2m'~k~)p~)qj

(03m~ 1/2 ( 03m "~] - 1/6 ', ~)p3,I + }q2~)pJ "

(3.9.13)

A comparision of these equations with the corresponding equations for electric dipole transition moments reveals that for the fundamental transition

58

Chapter 3

the purely mechanical anharmonicity contribution is larger for the magnetic dipole transition moment than the corresponding contribution for the electric dipole transition moment. The same statement applies for the first and second overtones also.

3.10 Transition polarizabilities As for electric and magnetic dipole transition moments, since we are only interested in evaluating the influence of anharmonicity, there is no need to be concerned about electronic and rotational wavefunctions. More discussion on this aspect is deferred to Chapter 8. So, vibrational transition polarizability integrals can be written as <~,lTcx~lv~> where Ta~ is a general symbol for electric dipole-electric dipole, electric dipole-magnetic dipole and electric dipole-electric quadrupole polarizabilities oralS, G'c~[~and Acxl~, respectively (see section 2.3). For all of these three polarizabilities Tcx[~can be expanded as a Taylor series expansion in dimensionless normal coordinate q; i.e., T ~ = Tcx~,0+ (~Tcx~q)q + 1 (~)2Ttxi3/~)q2)q2 + ...

(3.10.1)

This is identical to the expansion of ~ (see Eq. (3.5.2)), except that the dipole moment vector component ~cx is replaced by the tensor element Tail. As a result, all equations developed in Sections 3.5 and 3.6 for electric dipole transition moments can be converted to transition polarizabilities with the replacement of ~ by T, and they will not be repeated here. The readers may consult Ref.(18) for further details. In the double harmonic approximation, where the second and higher order terms in Eq. (3.10.1) are not included, the transition polarizabilities for vibrational transitions are non-zero only for a9 to a) + 1 transitions.

References 1 I.N. Levine, Molecular Spectroscopy, John Wiley & Sons, New York (1975). 2 J.P. Lowe, Quantum Chemistry, Academic Press, New York (1993). 3 S. Califano, Vibrational States, John Wiley & Sons, New York (1976). 4 J.L. Dunham, Phys. Rev. 41 (1932) 721; R. C. Herman and W. H. Shaffer, J. Chem. Phys. 16 (1948) 453. 5 J.H. van Vleck, Phys. Rev. 33 (1929) 467. 6 G. Amat, H. H. Nielsen and G. Tarrago, Rotation-Vibration of Polyatomic Molecules, M. Decker, Inc., New York (1971). 7 M . R . Aliev and J. K. G. Watson, Higher order effects in the vibration-rotation spectra of semirigid molecules in" Molecular

Diatomic Molecules

59

spectroscopy, ed. K. N. Rao, Vol. III, Academic Press, New York, (1985). 8 D. Berckmans, H. P. Figeys, P. Geerlings, J. Mol. Struct. 148 (1986) 81. 9 P.M. Morse, Phys. Rev. 34 (1929) 57. 10 R. A. Toth, R. H. Hunt and E. K. Plyler, J. Mol. Spectrosc. 32 (1969) 74. 11 J.P. Bouanich and C. Brodbeck, J. Quant. Spectrosc. Rad. Trans. 15 (1975) 873. 12 A. Willetts, N. C. Handy, W. H. Green, Jr. and D. Jayatilaka, J. Phys. Chem., 94 (1990) 5608. 13 H.S. Heaps and G. Herzberg, Zeitschrift fur Phys. 133 (1952) 48. 14 C. Chackerian, Jr., R. H. Tipping, J. Mol. Spectrosc. 93 (1982) 237. 15 K. Fox, Chem. Phys. Lett. 104 (1984) 21. 16 P. L. Polavarapu, Mol. Phys. 89 (1996) 1503. 17 A. D. Buckingham, P. W. Fowler and P. A. Galwas, Chem. Phys. 112 (1987) 1. 18 S. Montero, J. Chem. Phys. 77 (1982) 23; J. M. Fernandez-Sanchez and W. F. Murphy, J. Mol. Spectrosc. 156 (1992) 444.

This Page Intentionally Left Blank

61

Chapter 4

POLYATOMIC MOLECULES Among the 3N degrees of freedom available for a polyatomic molecule with N atoms, the translational motions (i.e. translation of the center of the mass of the molecule in three mutually perpendicular directions) account for three degrees of freedom. For non-linear and linear molecules there are three and two rotational degrees of freedom respectively. Then the number of vibrational degres of freedom is (3N-6) for a non-linear molecule and (3N-5) for a linear molecule. Unless stated otherwise, the discussion in this chapter will be referred to non-linear polyatomic molecules. As in the case of diatomic molecules, it is assumed that the electronic part of the Schr6dinger equation can be solved for a given nuclear geometry and with the resulting potential the nuclear equation is solved. It is further assumed that the vibrational and rotational equations can be separated with the use of Eckart-Sayvetz conditions 1,2. The total wavefunction then is a product of electronic, vibrational and rotational functions. Since rotational transitions are not considered here, the rotational functions will not be considered further (which is equivalent to assuming the rotational quantum number to be zero). The vibrational energy levels and transition moments for harmonic, as well as anharmonic oscillators, are considered below.

4.1 Energy levels of harmonic oscillators The presence of more than one vibrational coordinate in polyatomic molecules necessitates the consideration of interactions among these modes. These interactions can be incorporated as perturbations to the harmonic oscillator (zero order) hamiltonian. As in the diatomic case, the total hamiltonian can be written as H - H 0 + H 1 + H 2+...

(4.1.1)

Here, H 0 is the harmonic oscillator hamiltonian given as

Z (p a

(4.1.2)

a

where Va is the harmonic oscillator frequency, qa the dimensionless normal coordinate and Pa the momentum of the ath vibrational motion. The nature of normal coordinates in polyatomic molecules is discussed in

Chapter 4

62

Chapter 5; for the present purposes it is sufficient to view qa as representing 3N-6 independent coordinates. H 1 is the first order perturbation term arising from the third derivatives of potential energy, with respect to dimensionless normal coordinates, referred to as cubic /

force constants and designated as Vab c . H1-Z

(4.1.3)

Z ZVabcqaqbqc " a b>a c>b /

Similarly H 2 is the second order perturbation term given as H2-X

(4.1.4)

X X XVabcdqaqbqcqd ' a b>a c>b d>c m

where Vabcd is the fourth derivative of potential energy with respect to normal modes and are referred to as quartic force constants. Note that it is common practice to restrict the summation indices in Eqs. (4.1.3) and (4.1.4) so that off-diagonal elements of Vabc and Vabcd are used only once in the expression. This convention makes the definitions of Vab c i

and Vabcd different from the usual expansion terms of a Taylor series. /

Since the zero order hamiltonian H 0 is a sum of the operators of the individual modes, the zero order wavefunction will be a product of the harmonic oscillator functions and the energy a simple sum of harmonic oscillator energies. In other words, each normal mode represents a harmonic oscillator. Therefore, the zero order vibrational energy 1-3 is --0 --0 --0 EaJ - Eaj1 + Eu 2 + . . . - ZVa(Oa + 1~), a

(4.1.5)

where g0~a is the harmonic oscillator energy and ~a the quantum number of ath oscillator. Similarly the zero order wavefunction is V 0 - vO ~ 0 2 . . . - rlVOa, a where ~ 0

(4.1.6)

is the wavefunction of the harmonic oscillator a. When the

quantum numbers ~a are zero for all oscillators, the resulting vibrational energy, ~o _ ~ Va ! 2, represents the zero point energy of the molecule. a

Polyatomic Molecules

63

Let us consider CHFClBr, a simple chiral molecule to illustrate the vibrational states, and transitions between them, in a polyatomic molecule. The experimental vibrational frequencies4 are 3026, 13 11, 1205, 1078, 788, 664, 427, 315 and 226 cm-1. For molecules without symmetry the vibrational modes are numbered in the decreasing order of frequencies. - 1 - ( I 00000000)

7500

- - -_ -

-

f 7000

6

v

)r,

P

6500

a)

c

a)

-m

6000

2-

c

3-

0 .c

d 5500

.p

> 5000

4-

'-1 .-

5- 6-

---I - --I --:

7(000000000)

-- -

--

a-9-

-----

Fig. 4- 1. Vibrational states and energies of bromochlorofluoromethane with harmonic frequencies 3147, 1311, 1205, 1078,788,664,427, 315 and 226 cm-l. States 1 through 9 are respectively 1100000000>, 1010000000>,1001000000>,100010OOOO>,1000010000>, 10oooO1000>, 100OOOOlOO>, 1000000010> and 1000000001>. The states 10 thru 18 are 1200000000>,..., and 1000000002>; 19 thru 27 are 1300000000>,..., and 1000000003>, etc.

The experimentally observed frequencies are not necessarily the harmonic frequencies, due to the presence of anharmonic contributions. After correcting for anharmonicity, the harmonic frequency of the first mode was determined4 to be 3147 cm-1. The remaining observed frequenicies are assumed to be harmonic frequencies. The vibrational state is defined by the quantum numbers of all nine modes as I U ~ W U ~ U ~ U ~ U S U ~ U ~ U ~ > Fundamental transitions are those when only one vibrational mode is excited and its quantum number changes from 0 to 1. Thus transitions from 1000000000> to I1OOOOOOOO>, 1010000000>, 1001000000>, 1000100000>, 1000010000>, 1000001000>, 1000000100>, 1000000010> and 1000000001> states represent the nine fundamentals. These nine states are

Chapter 4

64

numbered 1 through 9 in Fig. 4-1, where the y-axis represents the vibrational energy [Eq. (4.1.5)] and the x-axis gives the fundamental harmonic frequencies. Transitions such as those from 1000000000> to 1000002000> (state no. 15 in Fig. 4-1) and to 1000003000> (state no. 24 in Fig. 4-1) represent respectively the first and second overtone of the sixth vibrational mode. The first overtones of the nine modes are numbered in decreasing energy order as 10 to 18, second overtone states as 19 to 27 etc. in Fig. 4-1. Transitions such as the one from 1000100000> (state no. 4) to 1001000000> (state no. 3) are referred to as difference transitions; here the transition energy is equal to the difference in harmonic frequencies of modes 3 and 4 (which can be read from the x-axis in Fig. 4-1). Similarly transitions such as the one from 1000000000> to 1001010000> are referred to as combination transitions; here the transition energy is equal to the sum of harmonic frequencies of the modes 3 and 5 (which again can be determined from x-axis values in Fig. 4-1).

4.2 Energy levels of anharmonic oscillators As in the case of a single oscillator (see Chapter 3) the correction --0

terms to zero order energy Ev can be evaluated using either perturbation theory 1 or contact transformation 5-8. Both approaches will be described here. Using perturbation theory, the first and second order energy __P

__PP

corrections, E~ and Ea) respectively, are obtained from Eqs (2.2.10) and (2.2.15), with perturbations H 1 and H 2 given respectively by Eqs. (4.1.3) __P

and (4.1.4). It can be easily seen that the first order correction Ev is zero. --

2

This is because H 1 contains terms of the type _q]' qaqb and qaqbqc which are of odd power in at least one normal mode and such operators do not connect the same vibrational states (see Appendices 2 and 3). Therefore it is necessary to use the expression for the second order energy correction --0

for incorporating the correction to E~. In the expression for the second order energy correction, the contribution arising from H 2 is straightforward to evaluate. The only -

-

4

-

-

2

2

terms which contribute here are of the type Vaaaaqa and Vaabbqaqb. From the relation given in Appendices 2 and 3, the contribution from H 2 is found to be

aaaa[/ a

65

Polyatomic Molecules

+s s a b>a

+ 1~)(~)b + 1//2). (4.2.1)

The evaluation of contributions from H 1 is more involved. The terms which contribute here are the products of integrals of the type, (lifO, q3 igO)

x

a

(tltO q3a lltO,)

aqb

(/1tO' qa lily)

)

x

(/g 0 qaqb

,

'

o o o X (V o Iqbqc qaqb:VaJ) X (qt~ qaqb q/~'), (qtaJ' qaqb qtv) and (~O,}qaqbqc]~O) X {~Olqaqbqc~O,).

)

,

Contributions from these

terms are found to be as follows:

V2aa ,Z ~0_ ~0, <~o, Iq~a)~~><~~I~aI~~ > 15V2aa (~)a +1~) 2 - 7V2a ~ 4v a m

(4.2.2)

16v a

m

VaaaVa~ <~,0I,,al~,O><~,O]~a,~.L,~O >_

,,,,,,

~o_~o,

3VaaaVabb (,Oa + l~)(,Ob + 1~),

(4.2.3)

2v a u

x VaabVbb~ > ~o_~o, <,~Olqa~qbl,O><,~OIq~l~,O

,,,,,,

m

I

3VaabVbbb (X)a + 1//2)(~3b+ 1~), 2~ b

Va~a~
- V2ab 4V b (4V 2 - V~)

(,Oa + 1//2)2

(4.2.4)

66

Chapter 4

3vb + 4~22V_a ~2 (tIa + 1//2)('Ob+ 1//2) 16(4+ 2 -V 2)

(4.2.5)

<+~ lqaqP,lvO><+0lqaq~I+0'>3Va --2 [ 8'V'~,-3'~2 1 2 ('Ob + ~) 16(4~ 2 -Va2) [4Va(4V b -V a )

- Vabb <~--'-77--2-=2.

+---~22!b~.--'~'. (~)a + 1//2)('Db+ 1//2)t , (4~ b - V a ) )

(4.2.6)

v , ~ VaabVbcc ~'-d-~'-O-E~_g~), (vOlqbq21vO')VaabVbcc (,Da + 1//2)(~)c + 1//2),

(4.2.7)

2~ b v~,

(vO]qaq2]~O, ) -

VabbVacc ~.--_-6 - ~.---O-E ~ - Ev,

VabbVacc (a)b + 1//2)(~)c + 1//2), 2v a 2

VaacVbbc gO ~0, (vO'lq2qcl vO)(vO

19'~19

(4.2.8)

q2qcl@ >-

VaacVbbc (1)a + 1//2)(l)b + 1//2),

(4.2.9)

2Vc --2 Vabc

(~ ~ ]q aq bq c ]~0 )(~ 0 1q aq bq c ]~0' ) -

_

--2 Vabc(T 1 + T 2 + T 3 + T 4) , where,

(4.2.10)

Polyatomic Molecules

TI=

67

l)aa)ba) c -(a) a + 1)(a)b + 1)(a9c + 1) 8(Va + Vb + Vc)

T2-

1)al)b(D c + 1 ) - (~)a + 1)(l) b + 1)a9c 8(Va + Vb - Vc)

T3=

T4 =

a)a (agb + 1)agc - (~)a + 1)Db (~c + 1) 8(Va - Vb + Vc) 1)a (1) b + 1)(~ c + 1 ) - (1) a + 1)1)b1) c 8(Va - Vb - Vc) ~PP

The total second order energy correction, E~ is obtained by summing Eqs (4.2.2)-(4.2.10). Let us now consider the contact transformation method 5-8 for evaluating the energy corrections. For this purpose the original nontransformed hamiltonian H given by Eq. (4.1.1) will be redesignated as ~(0); that is,

~(o) _ ~ _ ~o o) + ~ o ) + ~ o ) + . . .

_ no + n~ + H2 +...,

(4.2.11)

and use made of Eqs (3.3.12)-(3.3.20) given for the diatomic case to evaluate the total energy to second order. The operator S 1, for first contact transformation, is 5-7

s -E Z ESabc,

(4.2.12)

a b>a c>b

where the expressions for individual terms of the type Saaa, Saab and ~Sabc are given as follows: Saaa - -Vaaa Va ( } Pa3 + qaPaqa ) ,

Saab =

-Vaab Vb(4V 2 - V 2)

{(2V 2 - V~)qa2pb + VaVb(paqaqb

(4.2.13)

Chapter 4

68

-22

}

(4.2.14)

+qaPaPb) + 2VaPaPb ,

Sabc = -Dabc Vabc {~a(~2 _ ~2 _ ~c2)paqbqc -

- Vc )qaPbqc + Vc

-

V~ -2

- Va )qaqbPc (4.2.15)

-2VaVbVcPaPbPc} 9

The first of the above three terms is same as S 1 for a diatomic molecule. This term makes the off-diagonal integrals of q3a vanish in the contact transformed hamiltonian. Similarly the second term Saab makes the off-

2

diagonal integrals involving qaqb vanish. Note that the operator Saab can be obtained by interchanging labels a and b, and this would make the off-

2

diagonal integrals involving qaqb vanish. The third term Sabc serves the

2

same purpose for qabc. In the expression for Sabc, the definition for Dabc is as follows 1.

Dabc - (Va + Vb + Vc )(Va - Vb - Vc )(Va - Vb + Vc )(Va + Vb - Vc) (4.2.16) In the case of a single oscillator we noted that the non-zero contributions

i[

for the energy correction arise from the terms ~(0) and ~- S 1,

of the

contact transformed hamiltonian. The correction terms ( ~ 0 ~(20)~o) are given by Eq (4.2.1).

For the correction terms {q/OI 89

0) q/~

examining the form of S1 [Eq (4.2.12)] indicates that the non-zero contributions come from terms of the type [Saaa, ~(0)~qaaa]' [Saaa, ~(0)~qabb]' --(o)

[Sabb, Haaa], [Saab, nlbbb], [Sbbb, nlaab], [Saab, ~(0)..maab]' [Sabb, ~(0).qabb]' [Saab, -Hbcc], [Sbcc, ~(0) -llaab], [Saac, ~(0) "'bbc]' [Sbbc, ~(0) "laac ], [Sabb Hlacc], --

~(o)

[Sacc, Hlabb] and [Sabc, **labc ]" The individual contributions from these terms are found to be as follows"

(11/0 I~i [Saaa,~(0)nlaaa]lll/O) - - 15V2aa4~ a ( 1)a+ 1~) 2 - 7VLa 16v a

(4.2.17)

Polyatomic Molecules

m

69

m

(~01.~.i [Saaa , ~(0) labb][~O)__ 3VaaaVbbb ~Va7 (,Oa + 1/2)(,Ob + 1/2)

= {11/0 I~"i[ ab b , laaa

Hlbbb]]~~

- 3VaabVbbb (1,)a + ]~)(1,)b + 1~)

(4.2.18)

(4.2.18a)

-(~,~ i[Sbb b , ~(0) laab]1~,o),

(~o ].~i [Saab,~(O)laab l I~tO) -

4Vb

--2 3VaabVb 16(4V 2 -V 2)

2Va2abVa

'2i[Sab " b ' ~(0) labb]}~0)_ +

4V2 _ Vb2 (~a + 1//2)2

3V2bbVa

(4V~-V~)(~a + ~/(~u + 1,4),

(4.2.19)

Vabb (l)b + ~ ) 2 4Va 4 v ~ - va-2 --2 2VabbVb (,Oa + 1~)(~)b + 1//2)

(4.2.20)

(4v~ -Va) -~

16(4~b~ -Va) -~ i ] ~o)_ (~o I.~[Saab ' ~(o) lbccj

-

VaabVbcc 2Vb

( l)a + 1//2)('0c + 1/2)

i ,cc , laab]1xl/O), - (,,,o i:[s

(4.2.21)

u

(v~

i [Saac ~(0)

' lbbc]lV~

VaacVbbc

2re

('Oa+ 1//2)('Ob+ 1//2)

= (vO I'~i[Sbb " c ' ~(0)]1~0), laac

(4.2.22)

70

Chapter 4 - - - - .

i[sabb"'lacc

(~/012"

-

m

VabbVacc 2V a (~)b + 1/2)(~)c + 1//2)

= (~1/01~i[Sac " c , ~(0) labb]lllt0),

(v~ 189 ~(~

(4.2.23)

ll ~ - - 2-~abc v2bc { Va (~2 _ ~2 _ ~2)(~) b + 1//2)(1)c + 1//2) + ~b(~) _ ~2 _ ~2)(1) a + 1//2)(1)c + 1//2) + Vc(V2 - V2 _ Va2)(1)a + 1//2)(1)b + 1//2) +VaVbVc !2 } ,

(4.2.24)

Comparing Eqs. (4.2.2)-(4.2.10) obtained from perturbation theory with Eqs. (4.2.17)-(4.2.24) from the contact transformation method, a one to one correspondance can be seen between these two sets. The total energy, to second order, is obtained by combining the zero order contribution with the correction terms as mentioned earlier. Noting the restriction on summation indices in Eq. (4.2.12) the total energy, to second order, obtained from the first contact transformation is written as 1

El) - g~)a'Ob...- Z Va(1)a + 1/2) + Z Z X a b ( 1 ) a

+ ,~'~)(1)b + 1/2) + i 2 ,

a b_>a

(4.2.25)

where, Xaa -

3Vaaaa 2

Xab - Vaabb-

15V2aa _ Z V2ab 8v-2a - 3v-2b ) 4~ a br b 4V-----a2-" V---~ ' 3VaaaVabb

3VaabVbbb

Va

V~

Z c~a,b

f

2VaV2ab

+

2VbV2bb

(4Va~-v~) (4v~,-Va~)

VaacVbb~ --Va~c~c(~c-~-~a ~)t -2 -2 Vc

(4.2.25a)

i-62;

'

(4.2.25b)

71

Polyatomic Molecules 7V2aa ) 16Va

+

m2 _ ZZ16(34V~bVb ) b#aa -

_Z Z Z

V2bcVaVbVc .

a b>a c>b

(4.2.25c)

4Dabc

The quantities Xaa and Xab are anharmonicity constants (in the same units as Va, Vb etc). There are two differences in notation for anharmonicity constants of polyatomic molecules compared to the one we used in diatomic molecules. First we designated the anharmonicity constant in diatomic molecules as VeXe (Xe being a number without units). Second, this anharmonicity contribution is added to the harmonic contribution with negative sign, so VeXe is positive (see Eq. 3.3.10). For polyatomic molecules Xaa and Xab are generally negative. Until now we have concentrated only on the first contact transformation to obtain the energy to second order. Since the second contact transformation also gives a second order energy term, i[S 2, H(O~ it is necessary to consider this term for completely determining the energy to second order. The operator for the second contact transformation 5-7 is $2 - Z Z Z Z

1[s~bC(PaPbPcqd + qdPaPbPc) + 1 2 d a b c d LSabc(qaqbqcPcl +pdqaqbqc)

(4.2.26)

with restrictions on the summation given as a
S~bc and Sabc depend 5 on the potential constants and harmonic vibrational frequencies.

The zero order hamiltonian H(O ~

contains

operators p2 and q2 for each oscillator. Depending on the values of indices for a, b, c and d, in Eq. (4.2.26) for $2, the commutator i[S 2, H(00)] may have even or odd powers, collectively, in p and q. In I

either case the diagonal integrals

(v~

H(00']IVO) vanish as It

.Jl!

demonstrated below for the special case of two oscillators. Let us consider the index a for one oscillator and the indices b, c, and d for the second oscillator [in Eq. (4.2.26)]. Then

72

Chapter 4

$2- 1s~bbpa(P~qb + qbP'~)+ 1Sbbbqa(q~Pb+ Pbq~), ~(00) _ 9a (p2 +q2)+ -

2

(4.2.27)

,,b (p~, + q,~) -~

(4.2.28) '

and i[S2, H(0)]- ~-i{sgbb [Va (p~qb + qbp2)(paq2

(

q2pa)

3~

+ VbPa p~,q3 _ qbPb + qbP 4 - P4qb + qbP~q~,

upuqu)] q2 2

+ Sabbb[Va{q2pb + Pbq2)(qaP 2 - pa2qa)

--(23

32

22

22)}

+ Vbqa qbPb - Pbqb + Pbq~ -q4pb + PbqbPb - PbqbPb 9 (4.2.29) Since each term on the right hand side of this last equation has odd power,

ils2, ~c0~

collectively, in p and q the diagonal integrals of are zero. Another choice for indices a, b, c, and d in the expression for $2 is to have the indices a and b for one oscillator and indices c and d for another. Then, Ib

~ _ } ~aaC(pa~Pcqc+ qcPcPa~)+ l~aac (qa~qcPc+ Pcqcqa~), m

Vc/ 2

2

~o~ "a 2 (Pa~+ qa~)+-StPc +net

(4.2.30) (4.2.31)

and

i[s~ , ~o~

~i{/~,c,~c+qcpc /[saaC~c(,~a~,a~P a~q a

!

+Saac~ap~q ~ -qaPa 2 2 + t~ c/,~aac2pa + SaacqaC 2)[p3qc - qcP 3 + pc2qcPc - PcqcPc2 +n~pcnc - ncpcn~ + nqpc - pcqc~]} 9

(4.2.32)

Polyatomic Molecules

73

In this case, the terms on the right hand side of the last equation have even powers, collectively, in p and q. However, the diagonal integrals of 3 3 2 operator (Pcqc + qcPc) are zero, as are those of (Pcqc- qcPc + Pc qcPc 2 2 2 3 3 -PcqcPc +qcPcqc-qcPcqc + q c P c - P c q c ) (see Appendices 2 and 3). Thus i[S 2, H(o~ gives no contribution to the energy, as was the case in K

~

J

the case of a single oscillator (see Eq. 3.3.31). As a result, the energy expression given by Eq. (4.2.25) is complete to the second order and is not affected by further contact transformations unless the third and higher order contributions are considered.

4.3 Electricdipoletransitionmomentsof harmonicoscillators

The procedure to evaluate the transition moments with the harmonic oscillator wavefunctions is similar to the one given in Chapter 3 for diatomic molecules. The differences in the present case being" (a) the wavefunction is the product of (3N-6) harmonic oscillator functions; and (b) the expansion of the electric dipole moment operator contains all vibrational coordinates. In terms of dimensionless normal coordinates, the expansion of the electric dipole moment operator is

It-!10+Z(~)~q~ a

/

qa + I Z Z

c321tl'

a b / ~qaOq b/ qaq b-I-....

(4.3.1)

Limiting this expansion to the first derivatives and evaluating the transition moments with harmonic oscillator wavefunctions, would constitute a double-harmonic approximation. In that approximation, the transition moment becomes ~aJ'~ -(vO'IIII~ O) -(~l/O~O~"'[~t0 + Z

( c)qa c)~t~alVOl~l/02...).

(4.3.2)

a

Making use of the orthogonality of harmonic oscillator wavefunctions it would become clear that the only non-zero terms of this equation are of

0 1Iqa the type (~t~a_+

0 )" ~ga

Therefore, only one quantum transitions

become allowed in this approximation and the transition moment becomes

Chapter 4

74

~a)aa)a-

~qa

3q a

2

5Va'Da-

1

"

(4.3.3)

If the electrical anharmonicity is introduced by incorporating the second and higher derivatives of the expansion in Eq. (4.3.1), one can see that multiple quantum transitions become allowed. However, for such transitions the use of harmonic oscillator wavefunctions may not provide sufficient accuracy.

4.4 Electric dipole transition moments of anharmonic oscillators As in the diatomic case, transition moments of anharmonic oscillators can be evaluated in two different ways. One is to obtain the anharmonic wavefunction using perturbation theory by incorporating the corrections to the zero order hamiltonian. The resulting wavefunction then replaces ~0 in Eq. (4.3.2). The second method involves transforming the electric dipole moment operator using a contact transformation. This transformed electric dipole moment operator then replaces the operator g in Eq. (4.3.2). The second method is much more convenient and therefore the details of this method are presented. For this purpose, the original non-transformed operator g [Eq. (4.3.1)] will be redesignated as I1,(0) 9 It (O) -p,(OO) +It} O) +It(O)+...,

(4.4.1)

where, g ( o O ) - i t O + E c3q'a qa,

(4.4.1a)

a

i1.~O) = g

~)qa~)qb

a b

11(20)= l E E a

x b

)qaqb

/)311" I qaqbqc 9 c3qao3qbc3qc

(4.4.1b)

(4.4.1c)

c

Let us evaluate the effect of first contact transformation, which yields, (4.4.2)

75

Polyatomic Molecules

The zero order term remains unaffected by this or any further transformation, that is,

r

r

r ....

(4.4.3)

The first order term becomes (4.4.4)

+

Further contact transformations will not influence ~I 1). That is,

~ _ ~?~-....

(4.4.5)

Therefore let us first evaluate the consequences of first order contributions and we will come back later to the second order contributions. The operator S 1 for the first contact transformation is given by Eq. (4.2.12). As noted in Section 4.2, S 1 is composed of three different types of terms Saaa, Saab and Sabc. As a consequence i[Sl,ll,(00)] will also have three types of terms which are given below: i[Saaa']l'(0)]-~aaa(~)~/(Va c)qa 2p2+qa)-~, 2 (OSaaa ~-~-a / '

(4.4.6)

{( /

Og (2VaVbqaqb + 4~2 PaPb) i[Saab'~(0)] - - ~b(4;i ab_~2) Oqa

+(Oqu = (OSaab OSaab/~-I-opa OPb

i[Sabc' ~t(0)] + ~

- (( / Vabc Dabc

b~t ~)qa (clqbqc + c4pbpc)

(C2qaqc +C4PaPc) + ~qc (C3qaqb +c4papb)

(4.4.7)

76

Chapter 4

- (c)Sabc c)Sabc c)Sabc/ ~gPa q ~Pb ~" OPc) "

(4.4.8)

In Eq. (4.4.8), c 1, c2, c3 and c4 are defined as follows : Cl_~a(~a2 -- ~ - 2--Vc) c2

-

~ b ( ~ 2--V2c - 2-)V

(4.4.8a)

,

a

(4.4.8b)

,

c3_~c(~2 _Va-2 _~2),

(4.4.8c)

c 4 =-2VaVbV c .

(4.4.8d)

Thus the contact transformed electric dipole moment operator with contributions up to first order becomes II(1) - ~t0 + Z a

~)qa a + I Z Z b ~)qa0qb aqb + Z a

t)Pa

a

+Za b~>i/~)Saab )c)pa + / ~)Saab l ~ ) p /b +ZZ Z ~ ( Oqsabc )C)pa + ( Oqsabc) + (~Sabc)l a b>a c> c)Pb C)Pc

"

(4.4.9)

Noting that the transition moments are to be evaluated with harmonic oscillator wavefunctions, and the expression for i[Sl,l.t(00)] contains only second order operators, such as q2, p2, qaqb, PaPb, it becomes clear that the first order contribution from the first contact transformation introduces anharmonic contributions for two quantum transitions. However, the one quantum transitions remain unaffected. To evaluate the anharmonic contributions for one quantum transitions it is necessary to consider at least the second order contribution resulting from the first contact transformation. The second order contribution resulting from the first contact transformation is given by It(21),

Polyatomic Molecules

g(21)- g(2~ i[Sl,la~~ - }[SI,[SI,I.t(o~

77

(4.4.10)

This second order contribution will be modified by the second contact transformation to yield g(2), !1(22) - 11.(21)+i[S2,~l.(0~ .

(4.4.1 1)

Any further transformations will not affect g(22). That is, 11.(22)-11.(23)=. . . .

(4.4.12)

To incorporate the second order contribution completely one must then include a second contact transformation. In practice, the complete second order contribution 6-8 is seldom incorporated due to the complexity involved. Noting the meaning of individual terms, one may incorporate only those appropriate for the situation under consideration. ILt(20) introduces the second order electrical anharmonicity; i[Sl,g~ 0)] introduces the interference between the first order mechanical and electrical anharmonic terms; [SI,[SI,~t(0)]] introduces the interference II.

dl

between first order mechanical anharmonic terms; and i[S2,g(o0)] introduces the second order mechanical anharmonicity. Of these four second order contributions, that from g(20) [given by Eq. (4.4.1c)] is easiest to evaluate. Since S1 contains three types of components, as mentioned earlier, contributions from i[Sl,g~ 0) ] can be written in terms of these three components, i.e. in terms of i[Saaa, Ill0)], i[Saab, [tl,l0)] and i[Sabc, !1~0)]. Only a few terms in the resulting expressions will contribute to one quantum transitions, while the remaining terms are appropriate for multiple quantum transitions. The expressions for these contributions and those for contributions from }[SI,[SI,g(0~ here for space considerations.

and i[S2,g(0~

are not given

Chapter 4

78

4.5 Magnetic dipole transition moments of harmonic oscillators The evaluation of magnetic dipole transition moments of a single oscillator was discussed in Chapter 3. In the double harmonic approximation, the expansion of magnetic dipole moment is restricted to first order terms, i.e.,

in - ~

C)Pa a,

(4.5.1)

a

and the harmonic oscillator product wavefunction (Eq. (4.1.6)) is used. As a result, only one quantum transitions become allowed, i.e.,

m~,~ _ ( ~ o ~ o . . l ~ ( ~ m ~ " a

=

i~

~

t)Pa

aIvOIVOu'")

1

~'Da,'Da+l --

~a,aga_ 1 -- maga~a. (4.5.2)

4.6 Magnetic dipole transition moments of anharmonic oscillators In order to apply the contact transformation method, the magnetic dipole moment operator is expanded in terms of dimensionless normal coordinate q and momentum p, as discussed for the single oscillator in Chapter 3 [see Eqs. (3.8.6) and (3.8.7)]. The only difference here is that this expansion contains the summation over all (3N-6) modes. The resulting contact transformed magnetic dipole moment operator, to first order is found to be,

m(1)-E

C)Pa a

a

~2mq +lE3paa ~(Paqb a,b

b

+ qbPa)+ E

(Vaaa) am Va C)Pa(Paqa + qaPa)

a

aab {(m)[2a )2qaPb2Va ] )

+ ~a b>aE~b (4-~a2 - V~)

~

+ ~

+

[VaVb(Paqa +qaPa)]

VbPaqb

79

Polyatomic Molecules

+ a~ b~>ac>~bDVab:{(~~a limb ( ~ - ~2_ ~2)pbqc +Vc(Vc2 -Va2 - v~)qbPc] ~m' (~a2 ~ 2 - 2 + (~-~b)[Va - Vc)paqc +~c (~2 _ ~ 2 - v~)qaPc] (~m ][~a(~2 _ ~ 2 - ~2)paq b + ~Pc - V a - V c qaPb

9

(4.5.3)

The magnetic dipole transition moment m~,v for transition from a0 to ~)' is

).

then obtained a s . .(W~176176, , The second order contributions resulting from the first contact transformation and those from the second contact transformation can be found in Ref. 9. References

1 2 3 4 5 6

8 9

S. Califano, Vibrational States, John Wiley & Sons, New York (1976). E.B. Wilson, J. C. Decius and P. C. Cross, Molecular Vibrations, McGraw Hill, New York (1955). M. Diem, Introduction to Modern Vibrational Spectroscopy, John Wiley & Sons, New York (1993). M. Diem and D. F. Burow, J. Phys. Chem. 81 (1977) 476. G. Amat, H. H. Nielsen and G. Tarrago, Rotation-Vibration of Polyatomic Molecules, M. Decker, Inc., New York (1971). M . R . Aliev and J. K. G. Watson, Higher order effects in the vibration-rotation spectra of semirigid molecules in" Molecular spectroscopy, ed. K. N. Rao, Vol. III, Academic Press, New York (1985). D. Berckmans, H. P. Figeys, P. Geerlings, J. Mol. Struct. 148 (1986) 81. A. Willetts, N. C. Handy, W. H. Green, Jr. and D. Jayatilaka, J. Phys. Chem., 94 (1990) 5608; O. B ludsky, K. L. B ak, P. Jorgensen and V. Spirko, J. Chem. Phys. 103 (1995) 10110. K.L. Bak, O. Bludsky and P. Jorgensen, J. Chem. Phys. 103 (1996) 10548.

Chapter 4 This Page Intentionally Left Blank

81

Chapter 5 VIBRATIONAL ANALYSIS 5.1 Cartesian displacement coordinates The vibrational, rotational and translational motions of a molecule can be described by the displacements of the constituent atoms in cartesian coordinates. A linear molecule with N atoms has two rotational motions, one each around the two axes perpendicular to the linear molecular axis, and 3N-5 vibrations. A non-linear molecule has three rotations (one each around three mutually perpendicular principal axes of inertia), and 3N-6 vibrations. Unless specifically mentioned otherwise, the discussion here will pertain to non-linear molecules. The 3N cartesian atomic displacements contain the information on all of the motions, namely 3N-6 vibrational motions, 3 rotational motions and 3 translational motions. In setting up the equations for each of these types of motions, it is first useful to note that these three categories can be separated and treated independently under certain conditions, referred to as Eckart-Sayvetz conditions.I, 2 These conditions specify that during vibrational motions, the atomic displacements are such that the center of the mass of the molecule does not move and no net rotational angular momentum is generated. A mathematical description of the separation of these motions can be found in several standard references. 1-3 Let us consider a molecule whose geometry is specified by the atomic cartesian coordinates

x O, yO and z O. These coordinates are collectively designated (a = x, y or z) and their displacements from X ~

as

0 XAc x

as AXAcx = ( X A a -

xOa). The kinetic energy in atomic cartesian displacement coordinates is given as

2T = EmA(A,

, + Ay2A +

A

EmAA: 2A,

,

(5.1.1)

A

with mA representing the mass of atom A. Using the definition of mass weighted cartesian displacement coordinates as, qAcx - m~AXAcx,

(5.1.2)

the kinetic energy is conveniently expressed in the summation or matrix notation as

Chapter 5

82

2T - ~ 1 2 c t

or

2T - ~Cl 9

A

(5.1.3) ,.,.,

Here, /1 is a column vector of 3N quantities ClAct and q (transpose o f / l ) represents the corresponding row vector. The potential energy, whose functional form need not be specified at this point, can be expanded around a reference geometry as,

/ ~~~ ~+-~~ 2 ~x~x~ ~~~+' A B

A

(5.1.4)

with the derivatives taken at the working geometry. The second derivatives (02V/0XAcc0XBI3) are referred to as cartesian force constants. If the working geometry represents the minimum energy equilibrium geometry then the first order derivatives (0V/0XA(x) are zero, since they represent forces (multiplied by -1) on the atoms and these forces are zero for atoms at the minimum energy equilibrium geometry. V0 is the energy at the reference geometry and this is set arbitrarily to zero since it is a constant. For the sake of convenience let us introduce the abbreviations,

i)V

- -OAo~

c)XA{t )

I o~v

,

(5.1.5)

- fAtx,Bi3 ,

~ A(:zOx B13/

v ) = -OAt ~ , ~)2V

/

t)qActc)qBI3

(5.1.6)

(5.1.7)

- tA(x,Bi3 .

(5.1.8)

Then the instantaneous derivatives of the potential energy are given as -

~q

0

+ ~ tA~,B~ qB~" B

(5.1.9)

Vibrational Analysis

83

The equation of motion in mass weighted cartesian coordinates can be derived from the Newton's law, force equals mass times accelaration, as

d-t ~)dlAa +

~qAa

- 0.

(5.1.10)

For a molecule at its equilibrium geometry, the substitution of Eqs. (5.1.3) and (5.1.9) into the above equation gives, ddlAa + Z t A a , B ~ qB9 - 0. dt B

(5.1.11)

A solution to this differential equation is given as (5.1.12)

qAa - CAa cos(0)t + 5),

where co = 2xv, v is the frequency (in cyc/sec), 5 is an arbitrary phase factor and CAcx is the amplitude of atom A. Differentiating the above equation with respect to time, it can be seen that d/tAa - - ~ 2 q A a. dt

(5.1.13)

Then the equation of motion becomes (5.1.14)

,~(tAa,BI3 - C025Aa,Bfl)qBI3 - 0, B

where the Kronecker delta function (as defined in Chapter 2) ~ A a , B [ 3 - 1 for A - B and a = 13 and zero otherwise. Diagonalization of the t matrix, tlx,lx

tlx,ly

tlx,lz

tlx,2x

.........

tlx,Nz

tly,lx

tly,ly

tly,lz

tly,2x

.........

tly,Nz

t

0OQOOOOOO

tNz,lx

tNz,ly

tNz,lz

tNz,2x

......

tNz,Nz

(5.1.14a)

84

Chapter 5

yields 3N eigenvalues of o~ and 3N eigenvectors with each eigenvector containing 3N elements fA~,k. The eigenvector matrix represents the transformations between mass weighted cartesian displacements qAo~and a set of normal coordinates, Qk. These transformations are given in summation or matrix representation as, q A t x - Z gAtx,kQk

or q - g Q ,

(5.1.15)

k

Qk = Zgk,Acx qAtx or Q - gq.

(5.1.16)

A

Here q and Q are column vectors with elements qAa and Qk respectively. The eigenvectors are orthogonal and normalized, so that gk,Ao~ gAoc,k' - ~kk'

or gg- E .

(5.1.17)

A

Additional properties of the t matrix are given by the relations,4 ZgAtx,km~ - 0 ,

(5.1.18)

A 0 Z % q ] T XAI]gAT, k m A - 0 , A

(5.1 19)

and

~8 gAa'kgB~'kA B S a ~ - k

m~m~6a~ , / ~mA/A

(5.1.20)

- ~a~el3rx; m~m/~X~176 where I ~ is the moment of inertia tensor element. Utilizing Eq. (5.1.2), (5.1.15) and (5.1.16) the corresponding transformations between cartesian displacements and normal coordinates become,

Vibrational Analysis AXAa - Z S A a , k Q k

or

Qk -~Sk~AcxZ~Aa A

or Q - S - I x ,

85

X- SQ,

(5.1.21)

(5.1.22)

where S - M-1/2s

(5.1.23)

S -1 - gM 1~,

(5.1.24) 1/

and M1/2 is 3N x 3N diagonal matrix with m ~ as its elements. The column vector X represents the displacements AXAa. From Eqs. (5.1.21) and (5.1.22), the normal vibration can be visualized as a molecular vibration in which all atoms are displaced from their equilibrium positions. The relative magnitudes and signs of the atomic displacements CAa (Eq. 5.1.12) are reflected by the magnitudes and signs of the g matrix elements corresponding to that normal mode. Since all of the elements of gAa,k in a given kth column correspond to the same eigenvalue co2, all atomic displacements in that normal vibration take place at the same frequency. As the working molecular geometry was assumed (in Eq. 5.1.11) to represent the equilibrium atomic positions (i.e. no residual forces on atoms at that geometry), six of the 3N eigenvalues should be zero for a nonlinear molecule. This is because these six zero frequency modes represent the translational and rotational motions (three each) of the molecule and there is no potential energy associated with the translational and rotational motions of a free molecule (under no extemal perturbation). On the other hand, if the working geometry happens to be a non-equilibrium geometry (with residual forces present on atoms even in the absence of external perturbation) then one need not get six zero frequencies. The reason for this can be visualized from the dependency of cartesian force constants tAa, B[~ on the forces as follows. 5 Consider, for example, the term (~)2V/~q2a), which can be written as,

/ [

( O 2 V / ~ q 2 a ) - ~qAa "OqAa - ~qAa ~k ~Qk

~-q-A-Aa

Chapter 5

86

- j~. OQj

9

OQk

OqAa

OqAa

OqAa

0q2A~

(5.1.25) From the second term on the right hand side of Eq. (5.1.25), it is evident that the second order geometric terms (O2Qk/Oq2Aa) and the residual forces,-(3V/OQk), contribute to the cartesian force constants. Such effects can be eliminated by using the projection or correction methods (vide infra), both of which can be better understood with the introduction of internal coordinates, in the next section. It is useful to note that these effects have been addressed more commonly after the development of quantum mechanical methods for determining the cartesian force constants. In the older literature, where vibrational analyses were carried out by assigning force constants in internal coordinate space (vide infra), the issue of residual forces and their consequences did not arise. At this point it is useful to emphasize that if the objective is to determine the vibrational frequencies and normal modes (in terms of cartesian displacements of atoms) at the equilibrium geometry then the equations given above are all what is needed. The needed force constants (~2V/~XAoc~XBI3) in cartesian space are most conveniently obtained from quantum mechanical calculations. When the working geometry is away from equilibrium, or an analysis in terms of more physically transparent coordinates involving chemical bonds is required, further considerations as described in the following sections are needed. 5.2

Internal

coordinates

The use of internal coordinates in vibrational analyses is both important and complicated. It is important because the quadratic potential energy derivatives in internal coordinate space, referred to as the internal coordinate force constants, can be related directly to the properties of chemical structures as viewed in terms of the chemical bonds and angles. The magnitudes of diagonal force constants (that is of terms such as (~2V/~R2) where R:1 represents the jth internal coordinate), and signs and 9 magmtuc~es of off-diagonal force constants (that is of terms such as (~2V/~Rj~Rk) for two different internal coordinates j and k) provide much insight Into the nature of chemical bonding. A significant amount of discussion on this topic can be found in the literature. 6 The consideration of internal coordinates is also important to understand how the energy is

87

Vibrational Analysis

distributed among local portions of the molecule during a vibration, and how the charges or polarizabilites respond to the perturbation of localized structures. At the same time, the use of internal coordinates introduces some additional complexities, due to the nature of the coordinate transformations involved and also due to the interdependancy of some internal coordinates. Four different types of internal coordinates are commonly used. 1,2 These include: (a) stretching coordinate which represents change in length between the atoms A and B forming a chemical bond; (b) angle bending coordinate which represents the change in angle A - B - C , between the chemical bonds A-B and B-C; (c) torsional coordinate which represents the change in the dihedral angle formed by the planes defined by atoms ABC and BCD; (d) out-of-plane bending coordinate which represents the out-of-plane bending of the bond A-B from the plane containing all of the atoms A, B, C and D. Hi

I I

"~.L

Arl

~

,

~

O

..n2

Ar 3

--

I'

IA

&'2

'

Ar 1

Aot2

Ar2

Ar3 02 (a)

(b)

Fig. 5-1. Internal coordinate definitions for (a) non-planar H202 and (b) planar H2CO.

With the non-planar H202 molecule (see Fig. 5-1), as an example, one finds a total of six internal coordinates, with two O-H stretching coordinates (Arl, Ar2), one O-O stretching coordinate (kr3), two H-O-O bending coordinates (AOtl, Aot2) and one H-O-O-H torsional coordinates (A'0. The number of internal coordinates in this case happens to be the same as 3N-6, the number of molecular vibrations. However, this coincidence is not a general situation and one often finds that more than 3N-6 internal coordinates can be defined for most molecules. In such cases, some of the internal coordinates will exhibit interdependencies, and such a set of internal coordinates is referred to as a redundant set. For example, for the planar formaldehyde molecule (see Fig. 5-1) one can

88

Chapter 5

define two C-H stretching (Arl, Ar2), one C=O stretching (Ar3), one H-CH angle bending (Act), two H-C=O bending (All, A~2) and one O=C outof-plane bending (not shown) coordinate yielding a total of seven internal coordinates - one more than the number of vibrations. The changes in three in-plane angles within this molecule can be seen to be interdependent because, At~ can also be represented as -(A~I + A~2). So one may specify only the two H-C=O angle bending coordinates as a part of the 3N-6 internal coordinates. That will eliminate the redundancy problem, but the choice to eliminate Atx is not unique. One could have chosen Atx and A~I-A[~2 as the two in-plane angle bending coordinates, with the former representing the symmetric in-plane H-C=O bending and the latter representing the asymmetric in-plane H-C=O bending. Both approaches are equally correct and there are other equally valid choices. Such ambiguities are quite common for most molecules. In such cases, one can select 3N-6 internal coordinates avoiding those that will lead to redundant relations (such as eliminating Ao~ for H2CO). Alternately, one can make use of the molecular symmetry and formulate linear combinations of the internal coordinates to arrive at the 3N-6 independent symmetrized internal coordinates (see Chapter 10). The relations between internal and cartesian displacement coordinates can be derived in a straight forward manner. 1,2 Consider the bond between atoms A and B with bond length rAB. In terms of the cartesian coordinates of these two atoms, in an arbitrary set of cartesian coordinate system, the bond length is given as r2B _ (x O _xO)2 +(yO _yO)2 +(z O _ zO)2.

(5.2.1)

For an infinitesimal change in the bond length, one may write rABArAB - (xOtx - xOtx)(AXAet - AXBct).

(5.2.2)

Since ArAB represents the stretching internal coordinate, say designated as Ri, the gradients of this internal coordinate with respect to the cartesian displacement coordinates become, 0 _ X B~ 0 )/rAB = Bi,A~, (c)Ri/t)XAt~) = (X Ao~

(5.2.3)

0 0 (t)Ri/t)XBa) - - (XAa - XBa)/rAB - Bi,Ba.

(5.2.4)

Such relations can be formulated for all types of internal coordinates described above. Since excellent descriptions 1,2 to obtain these relations,

Vibrational Analysis

89

and standard programs incorporating these relations, are available they will not be rederived here. For the present purposes, we will assume that a set of 3N-6 independent internal coordinates have been selected. From the above equations one can see that the transformation from cartesian displacements to internal coordinates can be expressed as, R i - Z Bi,AsAXA5 A

or

R- BX.

(5.2.5)

Here R is a column vector of 3N-6 internal coordinates Ri, X is the column vector of 3N cartesian displacements AXAc~ and B is the transformation matrix with 3N-6 rows and 3N columns. These internal coordinates with linear relationship (Eq. 5.2.5) to the cartesian displacements are referred to as the rectilinear coordinates. The B matrix elements possess some interesting properties. The sum of B matrix elements, corresponding to x, y, or z component, in any given row add up to zero (see Eqs. (5.2.3) and (5.2.4) for the stretching coordinate considered). That is, Z Bi,A~5 - 0, for 5 - x, y or z. A

(5.2.6)

Similarly the sum over cross products of the positional vectors of atoms, 0 X A, with the B matrix elements for the corresponding atoms, in any given row, is zero. That is, 0

Z etxlS), XA~Bi,Av - 0. A

(5.2.7)

These two properties are equivalent 7 to Eckart-Sayvetz conditions since they ensure the absence of kinetic energy coupling between internal and external coordinates (vide infra). The rotational and translational coordinates, p~r and p~t (three each with tx = x, y and z), are referred to as the external coordinates p, and are defined as follows:

A

Chapter 5

90

-

I1 /1 /

mAAXAa.

m A

(5.2.9)

They can be expressed in terms of the cartesian displacement coordinates X, in analogy to Eq. (5.2.5) as pj

-

EI3j,AS~A8

or

p - 13X,

(5.2.10)

A

where p is a column vector of three rotational P~t and three translational ptcoordinates and 13 is the transformation matrix with six rows and 3N columns. It is convenient to include the internal and external coordinates together and write the transformation from cartesian displacement coordinates as X

.

(5.2.11)

A line between R and p or B and [3 indicates the stacking order of their elements. Since cartesian displacements are dependent on both R and p, the inverse transformation is given as, AXAa - E AAS,i Ri + E ~162 Pj i

(5.2.12a)

j

or

x-

/

mb,

To ensure that the coordinates R and p are independent of each other, it must

be

that 8 c)Ri/~)pj - E(~)Ri/~)XAs)(~)XA8/c)pj)- 0, A

c)pj/~)R i - E(~)pj/C)XA~5)(C)XA~5/~)Ri)- 0. That is, A

and

Vibrational Analysis Z Bi,A~St~A~5,j - 0

91

or

B a - 0,

(5.2.13a)

or

[~A- 0,

(5.2.13b)

A Z~j,ASAAs,i - 0 A

where a and A matrices are defined in Eq. (5.2.12). Furthermore, since each of the internal and external coordinate spaces are complete, - ~ii'

or

BA - E 3 N _ 6 ,

(5.2.14a)

Z~j,AS0~A~5,j, - 5jj,

or

~a-

(5.2.14b)

ZBi,ASAAs,i' A

E6,

A

where E is the unit matrix (diagonal elements equal one and off-diagonal elements zero) with its dimension indicated by the subscript. The A matrix is referred to as the inverse of the B matrix, and a matrix as the inverse of the 13 matrix. Note that the products al~ and AB are not unit matrices and obey the relation, AB + a ~ = E 3 N .

(5.2.15)

From Eqs. (5.2.12) - (5.2.15) the A matrices of isotopic molecules can be seen to be related as 8 A2 = A1 + @1~1A2 ,

(5.2.16)

where the subscripts 1 and 2 represent isotopically substituted molecules. These relations play a critical role in transforming the molecular properties such as dipole and polarizability gradients between internal and cartesian displacement coordinates. The a matrix can be constructed from the following relations. (5.2.17) AXAa = [1/~mA]l/2p t = (1/M)I/2p t

(5.2.17a)

A

Using the Eqs. (5.2.12) and (5.1.1) the kinetic energy can be written in matrix form as

Chapter 5

92

-

(/ MA l XMa

(5.2.18)

The matrix, G -1 = 7tMA,

(5.2.19)

is called the kinetic energy matrix in internal coordinates. Similarly the product @Ma represents the kinetic energy in external coordinates. The products 7tMa and @MA, representing the cross terms of kinetic energy between the internal and external coordinates, vanish as shown below. Let us consider the inverse kinetic energy matrix, which contains as the diagonal blocks G = BM -1 B,

(5.2.20)

inverse kinetic energy matrix in internal coordinates, and 13M-11~, the corresponding term in external coordinates; ~M-II3 and BM- 1~ are the off-diagonal blocks. The off-diagonal blocks can be written as, B M "1 ~ - s

(1 / m A)Bi,Aa 13j,Aa

(5.2.21)

A

Substituting the expressions given for the 13 matrix (see Eqs. (5.2.8) to (5.2.10)) into Eq. (5.2.21) one finds that these cross terms are zero when Eqs. (5.2.6) and (5.2.7) are satisfied. These relations were already mentioned above to be the characteristic of the B matrix. Therefore, the inverse kinetic energy matrix does not have cross terms between the internal and external coordinates. Then it must be that the kinetic energy matrix is also in block diagonal form, so AMa = ~MA = 0. Thus the kinetic energy given by Eq. (5.2.18) becomes,

(5.2.22)

Vibrational Analysis 2T - R G ' I R + b&Moq5 .

93

(5.2.23)

Until now there is no restriction placed on the molecular geometry, and the above equations apply whether or not the working geometry is an equilibrium geometry. In considering the potential energy, however, it is necessary to specify whether or not the working geometry is an equilibrium geometry. Let us consider the case of equilibrium geometry, where by definition there are no residual forces on atoms. Then, the potential energy can be written from Eqs. (5.1.4) and (5.2.12) as, 2 V - (Ri 9)

AfxA IXfxA

Offxa

(5.2.24)

Here fx is the cartesian force constant matrix with elements fAa,Bl3 (see Eq. (5.1.6)). The internal coordinate force constant matrix F is given as F = AfxA.

(5.2.25)

The products ~fxa, Afxot and ~fxA should be zero because they represent respectively the force constants associated with external coordinates and interaction force constants between internal and external coordinates. The potential energy of a free molecule does not change with molecular translations and rotations. That is, ~,fxOt- O,

(5.2.26)

A f x a - ~fxA = 0 .

(5.2.27)

One can substitute the analytic formulae (Eqs. 5.2.17) for ot into Eq. (5.2.26) and derive the translational and rotational invariance conditions for fx. There are situations where the quantum mechanical force constants are calculated at non-equilibrium geometries which result in non-zero forces at the atoms. Then the above equation may not hold and one has to give more careful attention to these terms (vide infra). In order to solve the vibrational secular equation in internal coordinates, let us consider Eqs. (5.2.23) and (5.2.24). The external coordinates can be dropped here after, since the kinetic energy matrix was shown to be block diagonal and the potential energy at the equilibrium geometry is independent of external coordinates. The transformation from normal coordinates to internal coordinates can be written using Eqs.

Chapter 5

94 (5.2.5), (5.1.21) and (5.1.23) as R = LQ,

(5.2.28)

where, L - BM -1/2 t .

(5.2.29)

Then the kinetic and potential energies can be cast in normal coordinates using Eqs. (5.2.23), (5.2.24)-(5.2.27) and (5.2.29) as 2T - ( ) L G ' I L Q ,

(5.2.30)

2V - 0 L F L Q .

(5.2.31)

Noting that kinetic and potential energies are diagonal in normal coordinate space, one obtains the relations [,G -1 L - E3N-6 or G = L [ , ,

(5.2.32)

[,FL = A

(5.2.33)

or F = ~-1 A L - 1 ,

where A is the diagonal matrix with eigenvalues representing the vibrational frequencies, Ok 2. From Eqs. (5.2.32) and (5.2.33) one finds that GFL = L A ,

(5.2.34)

which is called the secular equation in internal coordinates. The G matrix required in Eq. (5.2.34) can be obtained from Eq. (5.2.20). The construction of the G matrix is straightforward, once the B matrix has been calculated using the well known relations 1,2. If the F matrix was obtained from quantum mechanical calculations, it is likely that it was transformed from fx. In such cases, one should take note of the corrections needed if the geometry at which the calculations were done contained residual forces on atoms. The product G F is not a symmetric matrix and therefore a double diagonalization method 9 is commonly used. The symmetric G matrix will have 3N-6 non-zero eigenvalues (provided the chosen 3N-6 internal coordinates are independent), so one can find an orthonormal matrix V such that

Vibrational Analysis ~rGV = '1:,

95

(5.2.35)

where x is the diagonal matrix with 3N-6 eigenvalues of G. Then G can be written as G- WW,

(5.2.36)

where W-

VX 1/2 .

(5.2.37)

Then the secular equation becomes, (WWF) L - L A ,

(5.2.38)

or

~fF L = W-1L A .

(5.2.39)

This can be rewritten as

OYCFW)w - 1 L = W- 1 L A ,

(5.2.40)

and using the abbreviations W -1 L = D ,

(5.2.41)

~'FW = H,

(5.2.42)

one has the equation, H D- D A.

(5.2.43)

The diagonalization of the H matrix provides the needed eigenvalues. The required L matrix is obtained as L- WD,

(5.2.44)

where W was obtained from the eigenvalues and eigenvectors of the G matrix. An alternative to the secular equation (Eq. 5.2.34) is to set up the secular equation in terms of the inverse force constant matrix. From Eqs. (5.2.32) and (5.2.33) one can obtain,

Chapter 5

96

F-1G-1L = L A - 1 .

(5.2.45)

The inverse force constants are known as compliance constants. 1~

5.3 E n e r g y distribution The vibrational eigenvalues, intemal coordinate force constants and the L matrix elements can be seen from Eq. (5.2.33) to be normalized as

i

~ LikLjkFij / A k - 1. j

(5.3.1)

In this double summation, the portion defined as Dik, Dik - ~ L i k L j k F i j / A k , J

(5.3.2)

represents the fractional energy contribution from the ith internal coordinate to the kth normal vibration. This distribution of energy among 3N-6 internal coordinates is referred to as the potential energy distribution. From the kinetic energy expression in normal coordinate space (Eq. 5.2.30), one finds that

i

~ L i k L j k G i j 1 - 1. j

(5.3.3)

Since this expression is also normalized, here again, one can define the energy distribution as Dik - ~ LikLjkGij 1 . J

(5.3.4)

However, it is important to note that these two apparently different definitions can be seen to be equivalent to each other. 11 In view of Eq. (5.2.32) and (5.2.33), the energy distribution given by Eqs. (5.3.2) and (5.3.4) can as well be written as -1 Dik - LkiLik ,

(5.3.5)

-1 since L -1 L = E3N-6. Then the product LkiLik is same as the "kinetic energy distribution" of Eq. (5.3.4) or the "potential energy distribution" of

Vibrational Analysis

97

Eq. (5.3.2). The energy distribution is most commonly calculated from Eq. (5.3.2) because the need for additional computation of G -1 or L -1 is avoided.

5.4 Non-equilibrium geometries As mentioned earlier, the potential energy has to be handled carefully when the working geometry is not an equilibrium geometry because such geometries will impart residual forces on atoms which in turn contribute to the force constants through higher order terms (see Eq. (5.1.25)). As a result, the fx matrix need not satisfy the invariance to rotational motions which results in non-zero frequencies corresponding to the rotational motions. To see this further, let us assume that we have a set of non-zero cartesian forces on atoms, i.e. (~V/~XAa) are not zero. Transformation of these cartesian forces to translational and rotational coordinates using Eqs. (5.2.17) gives,

(ov,op )-

2(ov oxa /5 ,

(5.4.1)

A and ( c ) V / ~ p ~ ) - 2 %t[~,(c)V/c)XAa)XO~,/I~. A

(5.4.2)

The translational forces are zero as long as the sum of forces on atoms, along each car-tesian axis, add up to zero. For a free molecule this is true so the translational motions will still give zero frequencies. The presence of non-zero rotational forces will mean that rotational harmonic force constants are not zero, since rotational forces compensate these force constants at a higher level (vide infra). Two different approaches have been used in the literature to deal with this situation. One will be referred to as the correction method and the other as the projection method. In the correction method, 12 the expansion of intemal coordinates in terms of atomic cartesian displacement coordinates is considered up to the second order. That is, R i - Z B i , A c ~ ~ A a + 1 Z Z B iA(x,B~ ~ A a A X B A A B where,

(5.4.3)

Chapter 5

98

BiAa,BI3 -(a2Ri/aXAczaXBI3)

9

(5.4.4)

The potential energy expansion to second order in internal coordinates is given as,

V - V 0 + Z(~)V, ~)Ri)R i + (1)ZZ(c)2v/~)Ri~)Rj)RiR i i j

j 9

(5.4.5)

Setting the constant V0 arbitrarily as zero and substitution of Eq. (5.4.3) in the above equation gives, to second order in AXA~, V - Z AXAe Z (OV / 3Ri)Bi,Acz

A

i

+ (89 E aXA aXB E (aV / ORi)BiA,B A B

i

+(1)Z ZAXAaAXB[3Z(t )2V/c)Ric)Rj)Bi,AaBj,B 13 A B i,j

(5.4.6)

Comparison of this expression with the corresponding expression to second order in the cartesian displacement coordinates (Eq. 5.1.4) gives,

( ~)2V/~XAac)XBI] ) - Z(~)v/c)Ri) B iAa,B[$ i + Z(O2V/()Ri~)Rj)Bi,AaBj,B~ i,j

.

(5.4.7)

Note that this equation is similar to Eq. (5.1.25) given earlier in normal coordinates. Now defining the 3Nx3N matrices B i with elements BiAoc,BI3 for each internal coordinate i, the above relation in matrix notation becomes, fx + E ~ R B ' -

BFB,

(5.4.8)

where ~ R - (0V/0Ri). So the corrected internal coordinate force constant matrix is obtained as

Vibrational Analysis

( /

V-,g, fx+~r

B1 A ,

99

(5.4.9)

i

where A, the inverse of B, can be obtained from Eq. (5.2.20) as, A - M-1BG -1 .

(5.4.10)

The second order derivatives, (~)2Rj/c)XAa~)XB[~), although not known as commonly as the first order B matrix elements, can be evaluated 13 from the geometrical parameters. To consider the projection method, 14 let us go back to the transformation from internal and external coordinates to the cartesian displacement coordinates (Eq. 5.2.12). From this equation, it can be seen that the product A R gives the projection of the vibrational part of the cartesian diplacements and a 9 gives the projection of the external (rotational and translational) part of the cartesian displacements to the total atomic cartesian displacements X. The product AB, given as AB = E3N - ot~,

(5.4.11)

is called the projection operator (or projector). To project the external contributions out of the fx matrix a projection operator was defined 14 as P = (A'B)(AB)=fi(Bfi) - 1 B .

(5.4.12)

Then the product PfxP = BFB gives internal coordinate force constants matrix F with external contributions projected out. That is, F = ,g,PfxPA.

(5.4.13)

5.5 R e d u n d a n t i n t e r n a l c o o r d i n a t e s

As mentioned earlier, the number of internal coordinates that one can choose in a given molecule usually exceeds the number of vibrational degrees of freedom. Let the total number of internal coordinates chosen for a non-linear molecule be given as 3N-6+m. In such cases two different problems will be encountered. Due to the interdependency of these internal coordinates, one finds m redundant relations among these internal coordinates given as follows.

1O0

Chapter 5

2ajRj - 0 .

(5.5.1)

Substituting Eq. (5.2.5) into (5.5.1), one finds that 2aj2Bj,AaAXAa j A

- 0.

(5.5.2)

Rearranging the summation indices this equation can be written as 2 AXAtx2 ajBj,Aa - 0 . A j

(5.5.3)

This relation indicates that certain combinations of elements in a column of B matrix become zero. Since the inverse kinetic energy matrix is given

as Gij - Z B i , A a B j , A a / m A ,

(5.5.4)

A the sum of Gij weighted by the coefficients ai becomes,

i ~ a i G i j - ~A Bj,Aa( i~aiBi,Aa/ / m A - 0 ,

(5.5.5)

indicating that certain combinations of G matrix elements in m rows (or columns) will also be zero. In other words, the G matrix becomes singular in a redundant set of internal coordinates. As a result when the G matrix is diagonalized one would find 3N-6 non-zero eigenvalues and m zero eigenvalues. So eq. (5.2.35), written equivalently as GV = Vx becomes, G ( N i Z ) - (N!Z)(t~-~O) ,

(5.5.6)

where N is that portion of V representing the 3N-6 eigenvectors of the non-zero eigenvalues; Z represents the m eigenvectors of the zeroeigenvalues; and t is the diagonal matrix of 3N-6 non-zero eigenvalues. The m columns of Z give the coefficients aj defining the redundant relations. The 3N-6 columns of N give the coefficients of transformation between a redundant set of 3N-6+m coordinates Ri and 3N-6 independent

Vibrational Analysis

101

internal coordinates Sj as, Sj - ~ N i j R i

0 - Z ZijRi i

j - 1 , 2 . . . 3 N - 6,

J - (3N - 6 + 1)...(3N - 6 + m ) .

(5.5.7)

(5.5.8)

The summation index i runs from 1 to 3N-6 + m. The independent set of internal combination coordinates Si, can also be derived using the molecular symmetry (see Chapter 1(0) in which case the symbol N is usually replaced by U and referred to as the internal symmetry coordinates. The relations derived from diagonalization of the G matrix can be conveniently handled in automated computer calculations. 15 The transformations between internal combination coordinates and cartesian displacements coordinates become, S - NBX - BsX, X - AsS + a p .

(5.5.9) (5.5.10)

The G matrix in a non-redundant set of internal combination coordinates becomes, G s - BsM-l~s

,

(5.5.11)

where Bs = NB as defined in Eq. (5.5.9). The cartesian force constants can be transformed to non-redundant internal coordinates (assuming that we are working with equilibrium molecular geometry) as, F s - / i , sfxA s .

(5.5.12)

So the vibrational analysis procedure becomes identical to the one we discussed earlier except for the replacement of R, B, G and F by S, Bs, G s and Fs. The matrix As is obtained as As - M-l~sGs 1 .

(5.5.13)

The above procedure is adequate if the intention is to solve the secular equation in the presence of redundant internal coordinates. The internal combination coordinates as determined from the diagonalization of the G

Chapter 5

102

matrix (see Eq. (5.5.9)) will however have a complicated nature due to the presence of several internal coordinates in a given combination coordinate. So the physical meaning of force constants in the S coordinate set is not as transparent as that in the R coordinate set, and the issue of transferability of force constants among different molecules in the S coordinate set becomes complicated. The second inconvenience originating from redundant internal coordinates is the presence of non-zero linear potential energy terms 16 even at the equilibrium geometry. Recall that for the equilibrium geometry, the linear terms (aV/aRj) in the potential energy expansion are set to zero and this would not hold in the presence of redundant coordinates. This requires special considerations, especially in the quantum mechanical calculations of optimized geometries and force constants. To gain insight into the issues here, let us consider the case when there is only one redundant internal coordinate, that is the case when rn = 1. Instead of expanding the potential energy in terms of 3N-6 independent intemal coordinates, let us now expand it in terms of 3N-6+m coordinates (with m = 1): ~)V ~)2V RiRj V - Z. ORi Ri + 1Z.. aRiaRj 1

l,J

OV R m + l a2V 2 + aRm 2 aR___~mRm + I s

a2V RmR i+... aRm0Ri

,

(5.5.14)

1

where the indices i and j run over 3N-6 independent coordinates and m is the extra coordinate introduced leading to one redundancy. If we now write Rm as a Taylor series expansion in terms of independent coordinates Ri, then at minimum energy geometry it should follow that, aV aV aR m ) R i _ 0 . Z1. nqR-====4~, aR m ~)Ri

(5.5.15)

So the linear terms aV/aRi or aV/aRm are no longer zero. As a consequence the meaning of force constants also becomes a bit more complex, because the quadratic force constant Fi"J involving coordinates Ri and Rj, appearing as coefficient terms of RiRj now become, Fij _ aV a2Rm a2V t)2V aR m aR m aR m aRiaR j + aRiaRj ~ aR 2 aR i ~)Rj "

(5.5.16)

103

Vibrational Analysis

This expression indicates that the linear term OV/ORm also contributes to the internal coordinate force constant, making the visualization of force constants difficult. The choice of 3N-6 independent internal coordinates for a given molecule is not always uniquely defined. It may be convenient to define all possible internal coordinates for a given molecule, which then leads to several redundant relations and the above mentioned uncertainties in interpretation. It is useful to find a way of defining the force constants in terms of the complete internal coordinate basis set, and projecting out the contributions from redundancies. From the eigenvector matrix N associated with 3N-6 eigenvalues of the G (see Eq. (5.5.6)), one finds that NN

= E -

ZZ,

(5.5.17)

where E is the diagonal unit matrix with dimension 3N-6+m. product, P = N~I ,

The

(5.5.18)

is called the projection operator, projecting out the contribution of redundant coordinates. The force constants in 3N-6+m internal coordinates, but with projected out redundant contributions, is given as, 17 F = NFs~I.

(5.5.19)

An alternate approach involves the use of a generalized inverse method for inverting the singular matrix G in 3N-6+m basis. The generalized inverse 18 of G is given as,

G'I-(N'Z)~o

I0

'

(5.5.20)

where t -1 is the inverse of the diagonal matrix t containing the non-zero eigenvalues of G. The conversion of cartesian force constants to internal coordinate force constants in a 3N-6+m size basis can be achieved 19 from F = AfxA where A is now obtained from Eq. (5.4.10) with G -1 given by the Eq. (5.5.20). For interpretational purposes the F matrix is now in the original 3N-6+m basis. Quantum mechanical calculations of geometry optimizations 19 are often performed in the internal coordinates. In such calculations, the 3N cartesian forces are converted to the internal coordinate basis using the relation,

104

CR - CxM-1BG -1 ,

Chapter 5

(5.5.21)

where Ox and OR a r e the vectors of forces respectively in cartesian and intemal coordinate bases. During the geometry optimization cycles, the bond lengths and angles have to be corrected to minimize the forces. These corrections are determined from the inverse of OR - - FR, so one needs to determine the compliance matrix F -1. This is accomplished f r o m l 9 the relation F -1 - P ( P F p ) - I P ,

(5.5.22)

where P was defined above as N~I and the inverse (PFP) -1 is the generalized inverse, similar to the generalized inverse of G given in Eq. (5.5.20). References 1 E . B . Wilson, J. C. Decius and P. C. Cross, Molecular Vibrations, McGraw Hill, New York (1955). 2 S. Califano, Vibrational States, John Wiley & Sons, New York (1976). 3 D. Papousek and M. R. Aliev, Molecular Vibrational-Rotational Spectra, Elsevier, New York (1982). 4 J.K.G. Watson, Mol. Phys. 5 (1968) 479. 5 W.T. King and A. J. Zelano, J. Chem. Phys. 47 (1967) 3197. 6 M. Diem, Introduction to Modem Vibrational Spectroscopy, John Wiley & Sons, New York (1993). 7 R. Malhiot and S. Ferigle, J. Chem. Phys. 22 (1954) 717. 8 B. Crawford, J. Chem. Phys. 20 (1952) 977. 9 T. Miyazawa, J. Chem. Phys. 29 (1958) 246. 10 L. H. Jones, C. Kennedy and S. Ekberg, J. Chem. Phys. 69 (1978) 833. 11 J.C. Whitmer, J. Mol. Spectrosc. 68, 326 (1977). 12 P. Pulay, Direct Use of the Gradient for Investigating Molecular Energy Surfaces, in: Modem Theoretical Chemistry, ed. H. F. Schaefer HI, Plenum, New York (1977). 13 I. Suzuki, Appl. Spec. Rev. 9 (1975) 249. 14 I.H. Williams, J. Mol. Struct. 94 (1983) 275. 15 M. Gussoni and G. Zerbi, Atti. Acc. Naz. Lincei. 40 (1966) 1032 (as quoted in ref. 2). 16 B. Crawford and J. Overend, J. Mol. Spectrosc. 12 (1964) 307; M. Gussoni and G. Zerbi, Chem. Phys. Lett. 2 (1968) 145; I. M. Mills, Chem. Phys. Lett. 3 (1969) 267; V. I. Pupyshev, S. V.

Vibrational Analysis

105

Krasnoshchiokov and YU. N. Panchenko, J. Mol. Struct. 131 (1985) 347; H. H. Eysel and B. M. Bussian, Appl. Spectrsoc. 35 (1981) 205; N. Neto, Chem. Phys. 87 (1984) 43. 17 K. Kuczera, J. Mol. Struct. 160 (1987) 159. 18 M. Gussoni, G. Dellepiane and S. Abbate, J. Mol. Spectrosc. 57 (1975) 323. 19 P. Pulay and G. Fogarasi, J. Chem. Phys. 96 (1992) 2856.

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107

Chapter 6 LOCAL MODES

6.1 Hamiltonian in local coordinates To understand the local modes and to make a connection with the normal modes of polyatomic molecules we will start with the zero order hamiltonian H 0 that was considered for polyatomic molecules. As discussed earlier, with this zero order hamiltonian the vibrational problem can be treated as a sum of 3N-6 independent equations and the vibrational wavefunction becomes the product of 3N-6 harmonic oscillator wavefunctions. In terms of dimensionless normal coordinates qa and conjugate momenta Pa, (6.1.1) 2 where the summation index a runs over 3N-6 harmonic oscillators. Let us consider each of the two terms on the right hand side of Eq. (6.1.1) separately. The relations p = -i~)/~)q and q = (4n 2 v/h)l/2Q (see Chapter 3) can be used to derive the relation

Va 2 _ ,h a ~ 2 "-pa

~2

(6.1.2)

8/i;2c~a OQa2 '

where Qa is the normal coordinate for harmonic oscillator a. Now using the relation between the normal coordinates Qa and internal coordinates Ri, (see Chapter 5) given as Ri - ~LiaQa ,

(6.1.3)

the operator ~9/,gQabecomes

b

b

= ~_~"Lia . ~)Qa i 3Ri Substituting Eq. (6.1.4) into (6.1.2), one obtains

(6.1.4)

Chapter 6

108

~_~p2 _ "~1 ZL2a a 8~2c " a

t)R2

+

ZLiaLja "" a

~)Ri ~)Rj

" (6.1.5)

Using the relation of the L matrix (see Chapter 5) with the inverse kinetic energy matrix G, LL - G ,

(6.1.6)

Eq. (6.1.5) becomes

~._~_p2_ 8/1;2C __ ~i Gii /)R 2 + . . G IJ~)Ri c)Rj

(6.1.7)

9

Our interest here is to describe the vibrational motion in terms of the localized coordinates such as the internal coordinates Ri. To convert the internal coordinates Ri to dimensionless internal coordinates qi and conjugate momenta Pi we use the relations 1 analogous to those for normal coordinates given in Chapter 3, i.e.,

-

-

4/I;2Vi[.I,i

(6.1.8)

qi,

(6.1.9)

Here vi is the frequency ofthe local oscillator i, given as 2-2y/Fii) 1/2 with

\~ti

reduced mass ~i and force constant Fii (vide infra). Then Eq. (6.1.7) becomes,

~.~TPa2 -

8 9i ~

viktiGiip2+j~i(vivj)l~(kti~j)l~GijPiPJ

'

(6.1.10)

a

where Vi

= Vi/c.

Note that the summation index a on the left hand side of

109

Local Modes

this equation runs over 3N-6 normal coordinates while the index i on the right hand side is over 3N-6 independent internal local coordinates. For the second term in Eq. (6.1.1), we have Va 2 471;2 2 2 aVaQa Za ~ - - q ~ c Za 9

(6.1.11)

Using the relation -1 Qa - ~ L a i R i ,

(6.1.12)

where L -1 is the inverse of L in Eq. (6.1.3), Eq. (6.1.11) becomes Z ~_a qa2 - 2tlc . . Z 4~:2VaLai 2 - 1 -Laj 1 ) RiR j 1 1 i ~. I ( ~ a _4g 2VaLai 2._12)R2+j~i( a

a

(6.1.13) This equation can be simplified by noting (see Chapter 5) that LFL - A,

(6.1.14)

where A is a diagonal matrix with elements A a= 4n2v 2 and Fij are the force constant matrix elements. Using the relation with dimensionless coordinates qi (see Eq. (6.1.8)), Eq. (6.1.13) can be written as Va 2

Fij 9

9

9

]1//2 .

(6.1.15)

Thus the zero order hamiltonian of harmonic oscillators can be written in terms of the local coordinates using Eqs. (6.1.1), (6.1.10) and (6.1.15), as,

[ i{ ioiip i

110

Chapter 6

+1 ~~

(~i~j)l~

(~tigj

)1//2 GijPiPj +

Fij (FiiFjj) 1~ qiqj (6.1.16)

6.2 A p p r o x i m a t i o n s

At this point, we have to consider the individual cases in modifying Eq. (6.1.16). If we restrict ourselves to the bond stretching coordinates then it can be shown that giGii = 1 (see the B matrix elements discussed for stretching coordinates in Chapter 5), and the first term on the right hand side of Eq. (6.1.16) is similar to the harmonic oscillator hamiltonian in normal coordinates. The reduced mass gi is not well defined for nonstretch coordinates, where we may define an effective reduced mass as gi = 1/Gii.. With this approximation, Eq. (6.1.16) can be rewritten as follows: HLO - l ~ v i { p 2

+q2}

+ 1 Z X(~i~j)l~[

Gij

PiPj +

L(GiiGjj)~

i jr - H0,LO + H1,LO

Fij

)1/2 qiqj

(6.2.1)

FiiFjj

9 m

The subscript LO on H indicates that the hamiltonian is for the local oscillators; H0,LO is the harmonic part and H1,LO the coupling part. Further developments of the local mode concept can be made at different levels of approximations: (a) consider only the zero order local oscillator part, H0,LO; (b) evaluate the influence of inter-oscillator coupling, H1,LO on the local oscillator energy levels resulting from H0,LO; and (c) introduce anharmonicity using either the higher order terms of potential (involving q3i, qi4 etc.) or the Morse potential. 6.2.1 Independent local oscillators First let us consider H0,LO. This term is similar to the harmonic oscillator hamiltonian in normal coordinates [see Chapters 3, 4]; the only difference being H0,LO is cast in terms of local coordinates qi and

Local Modes

111

conjugate momenta Pi rather than those of normal coordinates. The eigenvalue problem would then be identical to that of the normal coordinate harmonic oscillator (see Eqs. (4.1.2), (4.1.5) and (4.1.6). That is,

--

HO,LO~LO - ELO~LO,

0

--0

0

(6.2.2)

EOo - E vi (~)i + 89

(6.2.3)

0

vOo - .I/~i/i,LO,

(6.2.4)

1

0 where ~/i,LO represents the local coordinate harmonic oscillator wavefunction with frequency vi (which is different from the normal coordinate harmonic oscillator frequency Va)" 6.2.2 Perturbation treatment of inter-oscillator coupling

The second term H1,LO can be treated as a perturbation using 0 perturbation theory with harmonic oscillator basis functions ~l/i,LO. If we consider all (3N-6) independent internal coordinates, each representing a local oscillator, then the number of non-zero terms in H1,LO will be prohibitively large to provide any qualitative picture. Therefore we will consider only the predominant terms as appropriate for individual systems. Let us consider a X2Y2 non-linear molecule, where the two identical X-Y bonds are coupled through the bond between the two Y-atoms (an example being H202). As this type of molecule has six internal coordinates (two X-Y and one Y-Y stretchings, two X-Y-Y bending and one X-Y-Y-X torsion) one has to consider, in principle, the interaction among these six oscillators. For the sake of brevity we will restrict our consideration to the two X-Y stretches interacting through the Y-Y stretching oscillator. Designating the two X-Y and Y-Y oscillators as 1, 2 and 3 respectively, and noting that Vl - v2, Fll - F22, G11 = G22, G13 - G23, F13 = F23 and G12 = 0, the perturbing hamiltonian becomes

H1,LO- Vl [(~121 ]qlq2 ] + (VlV3)~I (G1

G13 1G33

.1/(PlP3 + P2P3)

112

Chapter 6

+

F13 (qlq3 + )1 (F11F33)1//2 q2q3 (6.2.5)

The dominant contribution in Eq. (6.2.5) depends on the nature of atoms in the molecule. Let us explore the situations separately. First, consider the case when v1F12 / F 11 is negligible compared to the second term in Eq. (6.2.5). Then,

H1,LO - (VlV3)I~I(G1

G13

.~ (PlP3 + P2P3)+

1G33 )

F13 )1~ (qlq3 + q2q3) 1" (F11F33 (6.2.6)

This perturbation causes the excited states within a given 'block' to mix. For example, if the total quanta of excitation is 1, then the excited states that are the members of a 'block' are, (Vl - 1, 1)2 - 0, 1)3 - 0), (a91 - 0 , v2 - 1, v3 - 0 ) and (a91 - 0 , v2 - 0 , v3 - 1), and these states are forced to mix by the perturbation in Eq. (6.2.6). To evaluate the energies of these states after perturbation, we have to diagonalize the following interaction energy matrix which can be derived using the integrals in Appendix 2: lagl = 1 , ~ 2 = 0, xr3 = 0 > <~)1 = 1, X,'2 = 0 , ~ 3 = 0 1


aY2 = 1 , 9 3 =

01

<1)1 = 0 , 1 3 2 = 0 , 1 ) 3 = 11

I~1 = 0 , ~ 2 =

1,~3=0>

I ~ 1 = 0, ~ 2 = 0,a93 = 1>

A

0

B

0

A

B

B

B

C

where, A - ~1 +(v1 +v2 + v 3 ] 2 B - (VlV3) 1/2

G13

(6.2.7)

+

F13

,

(6.2.8)

Local Modes

C-~3+

113

(~1 + ~2 + ~3 ) 2

(6.2.9)

The diagonal elements of the interaction energy matrix are the unperturbed energies obtained with the hamiltonian H0,LO and the off-diagonal elements are the perturbations introduced by H1,LO. The eigen values resulting from diagonalization mentioned above are: 2~1 - A,

(6.2.10)

(A+c)+/A C/El+S/ B 2

~3-

2

A-C

'

(6.2.11)

(A B+C)- 2(A 2- C)[1+8(A-C/21~ (6.2.12)

The quantities A, B and C are defined in Eqs. (6.2.7)-(6.2.9). The wavefunctions corresponding to the energies given by Eqs. (6.2.10)(6.2.12) are obtained as linear combinations of the three functions ~Oo(a) 1 - 1, ~2 - 0 , a~3 - 0 ) , ~Oo(a) 1 - 0 , 1)2 - 0 , 1)3 -1 );

~Oo(a) 1 - 0 , ~2 - 1, ~3 - 0 ) ,

an d

the amount of mixing among these three

functions depends on the magnitude of B/(A-C). If B--- A-C = V l - v3, then the energies become ~2 = 2A-C and ~,3 "--2C-A. Now let us consider the case where the second term in Eq. (6.2.5) is relatively less important than the first term, and

__

[. Fll qlq2

19

(6.2.13)

The perturbation given by Eq. (6.2.13) causes the states of oscillators 1 and 2 in a given block to mix with each other through force constant F12 and oscillator 3 does not play any role. For example, (~1 - 1, l) 2 - 0) and (301 - 0 , 1)2 - 1 ) will be the members of a block when total quanta of excitation is 1. In that case, using the integrals in Appendix 2 the energy matrix to be diagonalized becomes

114

Chapter 6

I~1 = 1,a92 = 0 >

I~1 =0,~2 = 1 >

< ~)1 = 1,1)2 =01

A'

B'

<1)1 = 0 , 1 ) 2 = 11

B p

mp

where A ' - 2~ 1 +

(v3 ]

B'

2) (F12 / F 11)

-

(v1

/

(6.2.14)

2),

(6.2.15)

9

As a consequence of this perturbation, the and

~ o 0 ( ~ 1 - 0 , v 2 - 1)

[~~

-1, ~2 - 0)-Y-~~

are

states xl/Oo(~)l -

mixed

- 0, ~2 - 1 ) ] / ~ / 2 ,

to with

1, 1)2

- 0)

give energies

2Vl + (v3 / 2) + (Vl / 2)(F12 / F11)One point worth noting here is that the kinetic energy coupling term G12 is absent in Eq. (6.2.13). This is applicable when the two X-Y oscillators do not share a common atom. In non-linear molecules such as X2Y (for example, H20), where two X-Y oscillators share common atom Y, G12 is non-zero and is given as 1 / m y cos0, [0 is the angle XYX and my is the mass of central atom Y]. In such cases, H1,LO-Vl[ G12G11PlP2 + F1--!2 qlq2],F11

(6.2.16)

and B' in Eq. (6.2.15) becomes (V1 / 2)((G12 / Gll ) + (F12 / Fll)). The one quantum symmetric stretching vibrational transition will have a frequency Vs - Vl + (Vl / 2)((G12 / G l l ) + (F12 / Fll)) and the corresponding antisymmetric vibration will have a frequency Va = v1 -(v1 / 2)((G12 / Gll) + (F12 / Fll)). For 0 > 90 ~ G12 is negative and if F12 is also negative (or IF12 / Fll I<
Local Modes

115

role in determining which of these transitions have higher frequency. To consider this issue in somewhat more detail let us examine trans-1,2-d2oxirane. Here we have two each of C-H and C-D oscillators sharing a common C-C bond. Since the two C-H oscillators do not have a common atom, G12 = 0. The same is true for the two C-D oscillators. Then H1,LO is given by Eq. (6.2.13) both for the C-H oscillators and the C-D oscillators. F12 is also the same for both the C-H and C-D oscillators. Based on these points one might expect that the sign of F12 will determine whether or not the symmetric vibrational frequency of a one quantum transition is lower than that of the antisymmetric counterpart. The real situation is more complicated. For the C-H oscillators, the symmetric vibrational transition is found 2 to be of lower energy than the antisymmetric counterpart. On the other hand, for the C-D oscillators the opposite trend was found. 2 This difference is due to the fact the C-H and C-D oscillators are coupled to other internal coordinates, especially bending coordinates that involve the C-H/C-D bond [i.e, H-C-C, H-C-O, D-C-C, D-C-O bending coordinates]. Although the magnitude of kinetic energy coupling between stretch and bend coordinates is independent of the C-H/C-D substitution, that between the bending internal coordinates depends on whether the bending coordinates involve C-H or C-D bond (note that the potential energy coupling will not discriminate between the C-H and C-D oscillators). This difference in the kinetic energy coupling between the bending coordinates affects the energy levels of the bond stretching oscillators. Thus ignoring many of the coupling terms in the second term of Eq. (6.2.1) can lead to a severe, sometimes, misleading approximation for fundamental transitions. From the above discussion, it is apparent that unless H1,LO is included with all possible inter-oscillator coupling terms, the results obtained for fundamental transitions will only give approximate estimates. But if H1,LO is included without approximations, then the final result obtained will be same as that for the harmonic oscillator in normal coordinates and no advantage of the local oscillator concept is realized. For higher quantum transitions, however, it has been noted that the results can be satisfactorily interpreted using the local oscillator concept and it is this observation which makes the local oscillator concept 3 important. For these higher quantum transitions, the anharmonicity plays a significant role and therefore approaches incorporating the anharmonicity have to be explored. The Morse oscillator (see Chapter 3) has been widely used in the literature for this purpose. The incorporation of anharmonicity in the normal mode basis has been discussed in Chapters 3 and 4. The procedures for incorporating anharmonicity in the local oscillator basis 3 and the connection with that in the normal mode basis are

116

Chapter 6

the subjects of numerous investigations. 4 A paper by Mills and Robiette 5 is of particular significance to understand the connection between local mode and normal mode concepts, when higher quantum transitions are to be interpreted. Vibrational circular dichroism spectra of C-H overtone transitions have been reported for a limited number of molecules. 6 The role of local modes in interpreting these spectra has been the subject of investigations by Abbate and coworkers. 7 References 1 T. Onishi, I. Suzuki, and J. Overend, Spectrochim. Acta 42A (1986) 467. 2 T.B. Freedman, M. G. Paterlini, N.-S. Lee and L. A. N afie, J. Am. Chem. Soc. 109 (1987) 4727. 3 B . R . Henry, The Local Mode Model, in" Vibrational Spectra and Structure, ed. J. R. Durig, Elsevier, Amsterdam, (1981); M. L. Sage, and J. Jortner, Bond Modes, in" Photoselective Chemistry, ed. J. Jortner, Wiley, New York, (1981); L. A. Findsen, H. L. Fang, R. L. Swofford and R. R. B irge, J. Chem. Phys. 84 (1986) 16. 4 M. S. Child and L. Halonen, Adv. Chem. Phys. 57 (1984) 1; K. Lehmann, Jr. Chem. Phys. 79 (1983) 1098. 5 I.M. Mills and A. G. Robiette, Mol. Phys. 56 (1985) 743. 6 T.A. Keiderling and P. J. Stephens, Chem. Phys. Lett. 41 (1976) 46; S. Abbate, G. Longhi, L. Ricard, C. Bertucci, C. Rosini, P. Salvadori and A. Moscowitz, J. Am. Chem. Soc., 111 (1989) 836; S. Abbate, G. Longhi, J. W. Givens III, S. E. Boiadjiev, D. A. Lightner and A. Moscowitz, Appl. Spectrosc., 50 (1996) 642; M. Miyazawa, Y. Kyogobu and H. Sugeta, Spectrochim. Acta 50A (1994) 1505. 7 S. Abbate, G. Longhi and K. Kwon, Paper No.TC09, 52nd Molecular Spectroscopy Symposium, Columbus (1997).

117

Chapter 7

VIBRATIONAL FREQUENCIES AND FORCE CONSTANTS Vibrational spectra are presented with vibrational frequencies (the difference in the energy levels of the molecular transition involved) along the x-axis and the spectral intensities along the y-axis. The experimental vibrational frequencies and spectral intensities are two individual sources of information for molecular structural determination, as well as for analytical applications. From the expressions for the vibrational energy levels (see Chapters 3, 4), it is clear that they depend on the second and higher derivatives of potential energy with respect to the nuclear coordinates. The second derivatives are called the quadratic (or harmonic) force constants; the third and fourth derivatives are respectively called the cubic and quartic force constants. The fifth and higher order derivatives are not usually encountered, because their magnitudes are small and are also difficult to determine. The cubic and higher order force constants represent the anharmonic force constants. When the molecular geometry being considered does not correspond to the equilibrium positions of nuclei, the net forces at the nuclei are not zero. These forces, sometimes referred to as linear force constants, contribute to the higher order energy derivatives when the atomic displacements do not correspond to the linear changes in bond lengths and angles in a molecule (see Chapter 5). As a consequence, one has to consider the first derivatives of energy as well when they are not zero. The prediction of vibrational frequencies therefore requires a procedure to determine the potential energy derivatives with respect to nuclear displacement coordinates. In the older literature the common procedure was to approximate the force constants, or to transfer them from other molecules where they would have been established from the experimental data on small molecules. In the last decade there has been an explosive development in ab initio molecular orbital (MO) theories, and in the computer technology to implement them. This has so much impact on vibrational spectroscopy that the ab initio MO methods are becoming the first choice to predict force constants and hence vibrational frequencies. Theories at the Hartree-Fock (HF) and post-HF levels have been developed, but we will restrict the treatment here to the HF level. Two developments are of particular significance for the calculation of vibrational frequencies. One is the recognition that the first derivatives of energy can be calculated using the analytic expressions (as opposed to calculating the energy at different nuclear positions and obtaining the numerical derivatives). The second is the realization of a perturbative method for determining the higher order energy derivatives. These advances have made the predictions of vibrational frequencies very convenient.

118

Chapter 7

In this chapter we introduce the molecular orbital theory and its applications to the energy derivatives. The applications to spectral intensities of vibrational absorption, vibrational circular dichroism, vibrational Raman scattering and vibrational Raman optical activity are considered in Chapters 8 and 9. For each case, the experimental quantities involved and the procedures to determine them are discussed. 7.1 Molecular orbital theory 1-5 For a molecule with n electrons and N nuclei the hamiltonian can be written as a sum of two terms, one depending on the electron coordinates and another on nuclear coordinates. These are designated respectively as Hel and Hnuc, so that H-

Hel + Hnuc ,

Hel = -(h2/8/I;2m)Z Vi2 - Z Z ZAe2/riA+ Z Z e2/rij , i j
2 + Z E ZAZBe2/rAB . Hnuc - -(h2/8/I;2 ) Z VA/mA A A B
(7.1.1) (7.1.1a)

(7.1. lb)

In these equations, the summation index i is for electrons, and A for nuclei; ZAe and mA represent the nuclear charge, and mass of nucleus respectively; m represents the mass of the electron; rAB is the distance between nuclei A and B; riA is the distance from electron i to nucleus A; rij is the distance between electrons i and j; V 2 is the differential operator (~2/~x2 + ~2/~y2 + ~2/~z2). In the approximation that the nuclear kinetic energy operator does not influence the electronic wavefunction, -(h2/8x 2)

ZA

VA/mA in Eq. (7.1.1b) can be omitted and the Schr6dinger

equation can be solved at fixed nuclear positions. Then Hnuc is equal to the nuclear potential energy term, i.e.,

Hnuc = X ZZAZBe2/rAB, A B
(7.1.2)

and is a constant for a given nuclear configuration. For this reason, we do not have to explicitly write down Hnuc in all of the expressions, but can add this to the final expressions as needed. Focusing on He l, if this term can be written as a sum of independent one-electron terms (that is, ignore the electron repulsion terms e2/rii) then the total wavefunction can be written as a product of one-electron wavefunctions. In the presence of

Vibrational Frequencies and Force Constants

119

inter-electron interactions one may still view He l as a sum of one-electron terms (but with each electron experiencing some effective potential) and the total wavefunction as a product of one-electron wavefunctions. The one-electron wavefunctions in a molecule are called molecular orbitals. The Pauli exclusion principle requires that the total wavefunction, including the spatial and spin parts, should be antisymmetric with respect to exchange of two electrons. The one-electron function written as a product of the spatial and spin functions is called a spin orbital, gi. The electron spin function is represented by t~ or ~l (for +1/2 and-1/2 spins). For the two electrons to be associated with a spatial function tpl, the Pauli principle requires them to have opposite spins. So the total wavefunction ~ , for two electrons in one molecular orbital, is given as tPl(1)tPl(2)[o~(1)]3(2) - I](1)ct(2)]. Note that the spatial part tPl(l~l(2) is symmetric, but the spin part [ct(1)13(2 ) - 13(1)ct(2)], (and hence the total function) is antisymmetric with respect to interchange of electrons. This total function can also be represented by a determinant

] q~l(1)tx(1) =

q~l(2)tx(2 )

q~(1)~(1) q~1(2)~(2)

(7.1.3)

or as ~t - 2 - , (-1) P Pk{ q~l(1)ct(1)q~l(2)~(2) },

(7.1.3a)

k

where Pk is the permutation operator that permutes the electron indices n! times (n being the number of electrons). (-1)P is +1 for an even permutation and -1 for an odd permutation of electron indices. An nelectron system with all electrons paired (that is each spatial function is associated with two electrons of opposite spin) is referred to as a closed shell system. Noting that the total wavefunction is normalized [i.e. *~dx = < ~ t l ~ 1] and spatial and spin functions are orthonormal [i.e., = 8ij, = <~(i)l~(j)> = 8ij], the determinantal wavefunction for a closed shell n-electron system can be written as, q~l(1)cz(1) t91(1)~(1) q)2(1)cz(1) t92(1)[3(1) ... q~n/2(1)[3(1) ~ = (n!)-I/2 q~1(2)ot(2) qo1(2)13(2) q~2(2)0~(2) q~2(2)[3(2) ... qOn/2(2)13(2)

q~l(n)a(n) q~](n)13(n) q~2(n)a(n) ~2(n)~(n) ... q)n/2(n)13(n)

Chapter 7

120

= ( n ! ) - l / 2 ~ (-1) P Pk{qOl(1)o~(1)tPl(2)~(2 ) .... tPn/2(n)o~(n)qOn/2(n)~(n)}, k (7.1.4) where (n!) -1/2 is the normalization constant included to satisfy I ~*vd1: = 1. To save space, one can write the product ~(1)o~(1) as spin orbital g l, q~1(2)13(2) as ~2 etc., so that the total wavefunction written in terms of spin orbitals becomes, = (n!)-l/2~a (-1) P Pk{~I~Z2 ... ~Zn} 9 k

(7.1.5)

Using this wavefunction, the electronic energy is determined from, (7.1.6)

J'~*Hel~ d'l: = Eel.

To make the energy evaluation more transparent, consider a twoelectron closed shell system (for example, the H2 molecule). The normalized wavefunction is V =

(1/~/~) { [q~1(1)~(1)] [q~1(2)13(2)] _ [q~1(1)13(1)][q~1(2)o~(2)] } (11~-2) q)l(1)(Pl (2) [o~(1)[3(2)- 13(1)o~(2)] .

(7.1.7)

Let us rewrite the electronic part of the hamiltonian, Hel, as He l

=

Hcore = Hint

=

Hcore + H i n t , (-h2/8x2m)~.~~ V 2 - ~ ~ ZAe2/riA, i i A e2/r12,

(7.1.8) (7.1.8a) (7.1.8b)

and evaluate the energy contribution separately from Hcore and Hint. Remembering that the spatial and the spin functions are orthonormal one finds that the only non-vanishing terms associated with Hcore are and <~Pl(2)lH(2)lq~1(2)>, where H(i) is a part of Hcore that belongs to electron i. Thus,
Hii . i

(7.1.9)

121

Vibrational Frequencies and Force Constants

where Hii = s

<~Pi(1)lHcorekPi(1)>. Similarly the non-vanishing term

i associated

with

e2/r12

is

=

. This is called the (diagonal) coulomb integral Jii, so <Wle2/rl2lW> - Jii = <~i( 1)~i( 1)le2/rl2ltpi(2)tpi(2)>.

(7.1.10)

However, the above result is for the specific case of two electrons associated with one spatial function ~1. In a multielectron case, the two electrons under consideration can be associated with two different spatial functions. Then the permutation operator would have lead 5 to products such as tpl( 1)t~(1)tPl(2)13(2), q~l(1)~(1)tpl(2)a(2), tPl(1)ct(1)q~2(2)~(2), tpl(1)t~(1)~E(2)t~(2), q~1( 1 )13(1)q~2(2)t~(2), tpl(1)l~(1)tp2(2)13(2) in the wavefunction. Of these products the first two would lead to two diagonal coulomb integrals Jii and the last four to four off-diagonal coulomb integrals Jij, where

Jij =

.

(7.1.11)

Since the permutation operators involved for ~ and ~* (the right and left hand sides of operator in the integral <~lHintl~>) are independent, one would also find the integrals,-Kij, (Kij are called exchange integrals), given as Kij = <tpi(1)tpj(2)le2/rl21q)i(2)tpj(1)>,

(7.1.12)

but only one-half as many as the Jij integrals (because the integrals offdiagonal in spin, namely < tp 1( 1)tx( 1)tp2(2)13(2) le2/r 12Iq)1(2)tx (2) cp2(1)13(1)> are zero 5 by virtue of the orthogonality of spin functions). The exchange integrals appear in the energy expression with a minus sign because the permutation of ~ and ~* would differ by an odd number. Then for a general n-electron closed shell case, the energy expression becomes 5,

Eel = 2 s

Hii + s i

i

s (2 Jij - Kij), j

(7.1.13)

where the summation indices run over n/2 spatial functions (or occupied molecular orbitals). This energy expression can be evaluated when the

Chapter 7

122

analytic form of the spatial functions q~i are known, which is not the case in general. Since the spatial functions for the hydrogen atom are known (from the solution of Schr6dinger equation) one can use these hydrogen atomic orbitals, or some variations of these, to formulate the molecular orbitals. One such approach approximates the molecular orbital as a linear combination of atomic orbitals (LCAO) or of suitably modified atomic functions called basis functions. Thus each spatial function q~i can be written as q~i = E Cpifp, (7.1.14) P where the atomic basis function are denoted as fp. In terms of these basis functions, the energy expression becomes 5, Eel - E 2 P p q H p q + EPpqPrsIpqrs" p,q p,q,r,s

(7.1.15)

In the above equation, Ppq = E CpiCqi ,

(7.1.15a)

Hpq

=

,

(7.1.15b)

Ipqrs

-

2Jpqrs- Kprqs,

(7.1.15c)

Jpqrs

=

,

(7.1.15d)

9

(7.1.15e)

Kprqs =

where we assumed all functions and coefficients Cpi to be real. The summation index i in Eq. (7.1.15a) runs over occupiedmolecular orbitals and the matrix with elements as 2Ppq is called the density matrix. The orthonormality condition associated with the molecular functions becomes, Sij - 8ij - E CpiCqjSpq p,q

(7.1.16)

where the overlap integral Spq = . The evaluation of Eel via Eq. (7.1.15) is now subject to the condition that the coefficients Cpi can be determined. This is achieved using the

Vibrational Frequencies and Force Constants

123

variation principle. According to the variation theorem, the energy obtained using an approximate wavefunction can only be greater than or equal to the true energy (it cannot be less than the true energy). This can be easily seen by writing the approximate wavefunction as a linear combination of the true eigenfunctions of the hamiltonian. If ~n are the true eigenfunctions of H~n = En~n, and the approximate wavefunction Wapp is ~ Cn~n, then, n

(7.1.17)

<~applHkl/app>/<Xl/applXl/app> = Z Cn2En~ EO,

where E0 is the lowest true eigenvalue of the system. Since En are the true energies (with eigenfunctions ~n) and C2n is positive, the weighted average

~

Cn2En can

never

be

lower

than

the

smallest

n

value of En. That means the energy calculated with an approximate wavefunction should be minimized to reach towards the true energy of the system, although that does not guarantee that the true energy will be obtained. Among different approximate wavefunctions, the one which results in lowest energy can be considered to be closest to the true wavefunction. The coefficients coi can be optimized to achieve this energy minimization by setting (t)Eel/C)Cpi) to zero and solving the resulting equations for the coefficients; i.e., the condition O/OCpi[<~applHell~app>/<Xltappl~app>] = 0 leads to the set of equations,

E (Fpq - EiSpq)Cqi = 0,

(7.1.18)

where, Fpq = Hpq + X PrsIpqrs ,

(7.1.18a)

r,s

and Ei are the orbital energies. These are called Roothan's equations and Fpq are called the Fock matrix elements. The secular equation is of the form FC = S Cs (~ is diagonal matrix with elements ei) which can be rewritten as (S-1/2FS-1/2)(Sl/2C) = (Sl/2C)g. By diagonalizing the (S-1/2FS -1/2) matrix, the energies of the molecular orbitals ei are obtained as the eigenvalues. The eigenvectors are equal to $1/2C from which the matrix of coefficients C can be extracted. But, since the Fock matrix

124

Chapter 7

elements Fpq themselves depend on the coefficients one needs these coefficients even before the diagonalization of the (S-1/2FS -1/2) matrix. Usually one approximates the FI~q as Hpq, diagonalizes the (S-1/2FS -1/2) matrix for coefficients and uses the resultant coefficients to construct the full Fock matrix (see Eq. 7.1.18a). The diagonalization procedure and the construction of Fpq is repeated until the energy and coefficients obtained in the current step are consistent, within a certain tolerance, with those obtained in the previous step. This cyclic procedure to determine the coefficients and energy is referred to as the self-consistent field (SCF) Hartree-Fock (HF) method. The lack of correlation in the motion of electrons with opposite spins is considered to be a serious deficiency in the HF procedure described above. Different methods are available to correct for this deficiency, but they are beyond the scope of this book. Interested readers are encouraged to refer to the original references on this subject. 6

7.2 Energy derivatives The derivatives of energy are important quantities in vibrational spectroscopy, as reflected by their attributes given below. The first derivative of energy with respect to a nuclear displacement gives the negative of the force at that nucleus. These forces are needed for the geometry optimization process. Variation of these forces with respect to the electric field can be used in deriving the nuclear displacement derivatives of the electric dipole moment. The second derivative of energy with respect to electric field is related to the electric dipole polarizability. The variation of forces with respect to the magnetic field can be used to formulate the nuclear displacement derivatives of the magnetic dipole moment. The second derivative of energy with respect to magnetic and electric fields is related to the electric dipole-magnetic dipole polarizability. The second and higher derivatives of energy with respect to nuclear displacements represent the force constants, which are required for vibrational frequencies. The procedures to evaluate the energy derivatives using the molecular orbital theory are outlined below. 7.2.1 First derivatives of energy 7-14 Direct differentiation of the energy expression (see Eq. 7.1.15) for Eel with respect to a parameter b results in two different types of terms. One involves the derivatives of integrals, i.e., (~Hpq/Ob) and (OIpqrs/Ob), and these derivatives can be handled by directly dit~ferentiating the analytic expressions of integrals involving the basis functions. The second type of terms contain the derivatives of coefficients, i.e., (OCpi/Ob). These are not known a priori, but they can be avoided by using the orthonormality condition among the molecular orbitals as follows. From Eq. (7.1.15), the

Vibrational Frequencies and Force Constants

125

differential of Eel with respect to parameter b is, (7.2.1)

(OEel/~)b) = (OEel/Ob)I + (OEel/Ob)II, where, (OEel/ Ob)I - Z 2Ppq (OHpq / Ob) + Z Ppq Prs (~Ipqrs / Ob) , p,q p,q,r,s

(7.2.1a)

(OEel/ Ob)ii - 22(~)Ppq /Ob)Hpq p,q + 2[(clPpq / ~)b)Prs + Ppq (~)Prs / ~b)]Ipqrs. p,q,r,s

(7.2. lb)

Using the expressions (see Eq. 7.1.18) for Fock matrix elements and secular equation (~Eel/3b)II can be simplified l0,13 to (OEel / c)b)II - Z ~i 2 2[(OCpi / c)b)Spqcqi + (OCqi / Ob)Spqcpi]" (7.2.2) i p,q From the orthonormality relation (Eq. 7.1.16) for molecular orbitals, one obtains Z[(C)CP i / c)b)cqiSpq + (C)Cqi / c)Sb)CpiSpq ] - - Z CpiCqi (c)Spq / c)b). P,q p,q (7.2.3) Substituting Eq. (7.2.3) in Eq. (7.2.2) one obtains, (7.2.4)

(OEe' / c)b)II - - 2 [ Z 2CpiCqil3i](~)Spq / c)b). p,q i

From the above equations, it can be seen that the first derivative of energy with respect to a parameter b can be calculated if the derivatives of integrals, i e, (OHoq/~b), (OJpqrs/Ob), (OKprqs/Ob) and (OSp/3b) are known When b represents a nuclear i:lisplaceme-nt~ these derivative integrals can be evaluated using analytic expressions .Iat may be noted that to obtain the total energy derivative, one should a,~,~ (~Enuc/~b) to (3Eel/~b); Enuc for fixed nuclei is given by Eq. (7.1.2), which can be differentiated with respect to nuclear displacements. 9

9

.

.

.

.

fl

9

Chapter 7

126

7.2.2 Second derivatives of energy7-14 Using the expression for the first derivative of energy, differentiating again with respect to a parameter b' gives, (~)2Eel/~)b~)b') =

2[Ppq (~)2Hpq/~)b~)b') + (~)Ppq/~)b')(~)Hpq/~)b)] p,q {Ppq Prs(()2Ipqrs/()b()b') p,q,r,s + (~)Ipqrs/~)b)[Ppq(~)Prs/~)b') + Prs(/)Ppq//)b')] }

-Z P,q

[(~(Z

I~iCpiCqi)/()b')(()Spq/()b) i

EiCpiCqi)(()2Spq/~)b~)b') ].

- (Z

(7.2.5)

i

The second derivatives of the integrals (~)2Hpq/~)b~)b'), (~92Spq/~Ob~Ob') and (()2Ipqrs/()b()b') can be evaluated using the analytic expressions. But the derivatives of coefficients and orbital energies, i.e., (~coi/~gb') and (~)ei/~)b') are not known. A procedure which enables us to calcialate the first order corrections to Cpi and I~i due to a perturbation from b', will allow us to determine the second derivatives of energy. This procedure, called coupled perturbed Hartree-Fock (CPHF) method is summarized here. For this purpose, we follow the work of Pople and coworkers. 13 The secular equation, (Eq. 7.1.18), that we discussed earlier to obtain the orbital energies, has resulted from using a hamiltonian for the free molecule. If the molecule is under a perturbation, represented by parameter b', then the secular equation would have the same form, but each of the quantities, namely Fpq, Spq, Cpi and ~i would have been a function of b'. That is, the secular equation now can be written with functional dependence on b' as F(b')C(b') = S(b')C(b')E(b').

(7.2.6)

Here s(b') is a diagonal matrix with elements ei(b'), C(b') is a matrix of coefficients with elements Cpi(b'), S(b') is an overlap matrix with elements Spq(b') and F(b') is the Fock matrix with elements Fpq(b'). The goal is to determine Cpi(b') without having to solve the secular equation containing the perturbation b' (because, when b' represents a nuclear displacement solving the secular equation for each nuclear displacement in a molecule is

Vibrational Frequencies and Force Constants

127

computationally prohibitive). Instead, we can write Cpi(b') in terms of Cpi(0), which are obtained in the absence of the perturbation (b' = 0), as Cpi(b') = Z Cpg(O)ugi(b') or C ( b ' ) = C(O)U(b'). g

(7.2.7)

Note that the summation over g contains not only the occupied molecular orbitals, but also the unoccupied molecular orbitals (commonly referred to as virtual orbitals). In order for this equation to be valid at b' - 0, it is necessary that uei(0) - g~i, i.e., the matrix U(0) containing the elements uei(0) is a unit matrix. Then our goal is to find the first order correction to uei(0) in the presence of perturbation b'. Substituting Eq. (7.2.7) into Eq. (7.2.6) gives, (7.2.8)

F(b')C(0)U(b') = S(b')C(0)U(b')g(b') . Multiplying this equation with C(0) and designating Ft(b') = C(0)F(b')C(0),

(7.2.9)

St(b') = C(O)S(b')C(O),

(7.2.10)

one obtains a transformed equation, Ft(b')U (b') = St(b')U (b')t~(b').

(7.2.11)

Note that, (7.2.12)

St(O) = C(O)S(O)C(O) = E,

where E is the unit matrix. This is because St(0) is the overlap matrix in the molecular orbital basis and the molecular orbitals are orthornormal. Considering only the first order changes, one can write, Ft(b') = Ft(0) + Ft 9

P

= E(0) + Ft; U(b') = U ( 0 ) + ~ = E + U " ,

St(b') - St(0) + St =

E + S t and E(b') = ~(0) + E'. Then Eq. (7.2.11) for first order changes becomes, P

Ft + s

I

P

= I~'+ U'g(0) + Stir(0).

(7.2.13)

128

Chapter

7

Diagonal elements of this equation provide the first order corrections to energy as, 9

p

(7.2.14)

e"e - Ft,gg - St,ggEg(O ) 9 p

The off-diagonal elements provide the first order corrections Ugm as, p

p

p

(7.2.15)

[gm (0) - Eg(0)]ug m - Ft,gm - St,gmg m (0). p

In Eqs. (7.2.14) and (7.2.15), St,gm is obtained from the first order contribution of Eq. (7.2.10) as p

I

St,gm = E Cpg(O)Spqcqm (0), (7.2.16) P,q where S' - (OSpq/Ob') can be obtained via analytic differentiation of Spq. pq p

The evaluation of .Ft,em is more involved. transformed Fock matrix is written as,

From Eq. (7.2.9), the

Ft,gm = E Cpg(~ (b')cqm (0) , (7.2.17) p,q where Fpq(b'), in analogy to the Fock matrix at b ' - 0 (see Eq. 7.1.18a), is written as

Fpq(b')- Hpq(b') +EPrs(b')(2Jpqrs-Kprqs)b' 9 r,s

(7.2.18)

The subscript b' in the second term on the right hand side of Eq. (7.2.18) indicates the integral dependence on b'.

Noting that Prs(b')

=

Cri(b')csi(b') and Cri(b') is given by the expansion in terms of Cri(0) (see 1

Eq. (7.2.7)), Ft,trn is obtained from Eqs. (7.2.17) and (7.2.18), as Ft,gm = E Cpg (0) Hpq(b') Cqm(O) p,q

Vibrational Frequencies and Force Constants

129

+Z Cpg (0)Cqm(0)Z Z Z Uni (b')uoi(b')cm(0)Cso(0) p,q

r,s i n,o • (2Jpqrs- Kprqs)b'.

(7.2.19)

To extract the first order contribution from this equation, we need to remember that Uni(b'), Uoi(b') and the integrals have to be expanded at b' = 0 and that the summations over n and o run over all molecular orbitals (occupied and virtual). Thus, F't,gm) - ZCpg (0)Hpq(b')cqm(0) ' p,q + ZCpg(O)cqm (0)Prs(0)[c)(ZJpqrs - Kprqs ) [ 3b'] p,q,r,s + ~ Uni Z Cpg(0)Cqm(0)Crn (0)Csi (0)(2Jpqrs - Kprq s) i,n p,q,r,s + ~ Uoi Z Cpg(0)Cqm(0)Cri (0)Cso (0)(2Jpqrs - Kprq s)" i,o p,q,r,s

(7.2.20)

Substituting this expression into that for U~m [Eq. (7.2.15)], one finds that the off-diagonal elements of U' are present on both sides of the equation, so one has to use some kind of iteration procedure to determine the offdiagonal elements of U matrix. The readers may consult Ref. 13 for further details. The diagonal elements u~e, however, can be determined from the orthonormality condition associated with the molecular orbitals. This condition can be written for the molecular orbitals obtained in the presence of the perturbation as (7.2.21)

C(b')S(b')C(b') = E . Using the transformation in terms of

Cpi(0),this equation becomes

[I(b') C(O) S(b') C(O) U(b') = [l(b') St(b') U(b) = E .

(7.2.22)

The first order contribution from this equation for diagonal elements (noting that St(0) - U(0) = E), is given as s ,ee = - 2u e. Since S~,tt can be determined as described earlier (see Eq. 7.2.16), the diagonal elements of the U' matrix can be determined. Once the U' matrix elements are determined, the derivatives ~Cpi/3b' can be obtained, and the second

Chapter 7

130

derivative of energy O2Eel/Ob/)b' calculated. The ab initio force constants for CHFC1Br are listed in Table 1; the vibrational frequencies derived therefrom are given in Table 2.

7.2.3 Higher derivatives of energy The third derivatives of energy such as ~)3Eel/~)b~)b'~b" can be obtained by differentiating the expression for O2Eel/~)bOb' with respect to b". The resultant expression contains coefficient derivatives OCpi/Ob" and O2Cpi/Ob'bb". Readers may consult Ref. (14) for further details. TABLE 1 Force constants(mdyn/A) a in symmetry coordinates for (S)-CHFC1Br

S1 $2 $3 $4 $5 $6 $7 $8 $9

S1

$2

$3

5.63

0.15 6.40

0.05 0.60 3.69

$4

$5

$6

$7

$8

0.03 0.54 0.39 2.74

0.02 -0.10 -0.14 -0.09 0.71

0.04 0.42 -0.15 -0.14 0.06 0.68

0.01 0.00 0.29 -0.27 0.01 -0.01 0.58

0.00 0.16 0.15 -0.21 -0.16 0.03 0.06 0.57

$9 -0.00 -0.18 0.16 0.00 -0.03 -0.04 0.02 -0.02 0.32

aObtained using DZP basis set with electron correlation incorporated via MP2 method. The atomic cartesian coordinates (in Angstrom units), representing the optimized geometry, are" C(-0.4320, 0.4612, -0.6383); H(-1.5179, 0.5752, -0.6709); F(0.1574, 1.6482, -0.8538); C1(0.0564, -0.6854, - 1.8782); Br(0.0216, -0.1643, 1.1408). The symmetry coordinate definitons are as follows: S 1 = ArC-H; $2 - ArC-F; $3 = Arc-c1; $4 = Arc_ Br; $5 - (1/~/-6) (A0~HCF + A0~HCC1 + A~HCBr - A 0~FCCl - A (/C1CBr A0~BrCF); $6 - (1A/-6) (2A0~HCF - A (~HCC1 - A ~nCar); $7 - (1/~/-2) (At~HCC1 - At~HCBr); $8 - (1/~/6) (2AtXFCC1 - A (R21Car - A~BrCF); $9 = (1/~-2) (At~lCar- ACtBrCF). In the construction of B and G matrices, the H-C-X angle coordinates were scaled by C-H bond length, while the F-CC1, C1-C-Br, and Br-C-F angle coordinates were scaled by C-F, C-C1 and C-Br bond lengths respectively.

Vibrational Frequencies and Force Constants

131

7.2.4 Experimental determination of force constants The force constants can be defined in any of the coordinate systems (i.e., cartesian displacements, internal and normal coordinates) discussed in Chapter 5. However, it is convenient, and intuitively transparent, when the force constants are written in internal coordinates. For this reason we restrict the discussion to these internal coordinate force constants. The simplest example to illustrate the application of experimental data in determining the force constants would be a diatomic molecule. The TABLE 2 Predicted vibrational properties for (S)-CHFC1Br

~a 3209 1372 1266 1151 822 678 439 321 233

Ab

Rc

Sd

6.0 0.5 23.4 -8.1 82.6 7.1 189.7 1.0 174.8 15.1 60.6 - 15.5 1.6 0.7 0.1 -0.4 0.1 0.2

80.7 4.9 3.6 1.3 4.0 9.6 4.0 3.1 4.6

pe ( t ~ • 103)f 0.24 0.75 0.65 0.74 0.63 0.16 0.28 0.52 0.52

-0.16 -0.00 -1.8 0.06 0.72 -2.1 -0.64 0.93 -0.57

(72/f.o)f

(52/o~)f

-0.0264 -0.0141 -0.0074 -0.0036 -0.0725 0.0627 0.0077 0.0054 -0.0104

0.0198 0.0006 -0.0231 -0.0048 0.0186 -0.0107 0.0018 0.0025 -0.0017

aFrequency (cm -1). The major contributions to potential energy ditribution (PED) for each of the vibrational modes are as follows: 3209 cm -1" S1(100); 1372 cm -1" $6(100); 1266 cm -1" $7(100); 1151 cm -1" $2(100); 822 cm -1" $3(85), $8(18); 678 cm -1" $5(59); $4(52); 439 cm-l: S5(27), $8(23), $3(18); 321 cm-l" $9(38), $8(30), $4(27); 233 cm-l" $9(54), $8(21). bAbsorption intensity (km/mol); Crotational strength (10 -44 esu2cm2); dRaman activity (A4/amu); edepolarization ratio; fROA o

o

parameters (A5/amu). The frequencies, absorption intensities, Raman activities and depolarization ratios were obtained with DZP basis set and electron correlation using MP2 method. The remaining properties were obtained at the Hartre-Fock level. vibrational frequency of its fundamental transition can be obtained from the vibrational absorption or Raman spectrum. If we assume that the potential energy is given in the form of that for the harmonic oscillator (i.e., third and higher derivatives of energy in the potential energy

Chapter 7

132

expansion are ignored; see Eq. (3.1.11)), the vibrational frequency Ve is related to the quadratic force constant K = O2V/OR2, as Ve -- (| [ 2gc)(K / g)l/2,

(7.2.23)

where tx is the reduced mass (see Eq. 3.1.12). For the harmonic oscillator the vibrational energy levels are given by Eq. (3.2.5). With ~?e, K and expressed respectively in cm -1, mdyn//~ and atomic mass units, Eq. (7.2.23) can be conveniently written as Ve -- 1302.8(K / it) 1/2.

(7.2.24)

Although it may seem straightforward to determine the quadratic force constant from the experimental vibrational frequencies using Eqs. (7.2.23) and (7.2.24), the underlying harmonic approximation is not necessarily valid for most cases. If the harmonic approximation is valid, then the experimental vibrational frequencies of overtone transitions 0 ---) 2, 0 ~ 3 etc. should respectively be at 20e, 3Ve etc. The observed overtone frequencies are usually found at frequencies lower than these values. So one must consider anharmonicity, i.e. higher energy derivatives in the potential energy expansion. When the cubic and quartic terms are included (see Eq. 3.3.3), the vibrational energy levels are given by Eq. (3.3.10). The frequencies of 0 ~ 1, 0 ---) 2, 0 ---) 3 etc. transitions can then be seen to be, respectively, V e - 2VeXe, 2(Ve- 3VeXe)' 3(Ve- 4VeXe)' etc. Note that for diatomic molecules we defined the anharmonicity constant as VeXe (see Eq. (3.3.10a)). A general expression for the vibrational frequency v 0 ~ v of the 0 ---) ~) transition can be found from Eq. (3.3.10) as V0---~ -- Ve l) -- VeXe

(v2 + v).

(7.2.25)

A plot of ( v 0 ~ / ~) vs (~) + 1) should be a straight line with the slope given as -VeXe and intercept as re. The anharmonicity term VeXe is related to the cubic and quartic force constants via Eq. (3.3.10a), and ve is related to the quadratic force constant via Eq. (7.2.23). From the above discussion it is apparent that the determination of force constants requires the availability of experimental data for several overtone transitions, along with that for fundamental transitions. One may include higher order anharmonicity contributions also and in such cases

Vibrational Frequencies and Force Constants

133

the energy levels are given by Eq. (3.3.11). These higher order contributions are rarely included. For a polyatomic molecule the situation is more complicated. First let us assume that the experimental fundamental frequencies either represent, or can be converted (vide infra) to, the harmonic frequencies. For small symmetric polyatomic molecules one may confidently use the harmonic frequency data to determine the quadratic force constants, but for the rest one may not be able to obtain a unique solution. To explore this further, first let us recall that the vibrational secular equation (see Chapter 5) in the harmonic approximation is solved from G F L = LA, where G is the inverse kinetic energy matrix (determined from atomic masses and geometry), F is the force constant matrix, L is the eigenvector matrix and A is the diagonal matrix of eigenvalues (or vibrational frequencies). It is useful to note that in the older literature the F matrix based on different force fields (namely central force field, Urey-Bradley force field and valence force field) have been employed. These force fileds will not be discussed here as they are not used frequently (as reflected in the recent literature). For a non-linear molecule with 3N-6 independent internal coordinates, the F matrix would have 1/2(3N - 6)(3N - 6 + 1) elements (note that F is a symmetric matrix). For example, for the non-linear H-OF molecule the three internal coordinates (O-H stretch, O-F stretch and HO-F bend) would lead to six independent internal coordinate force constants. For the non-linear and non-planar HOOF molecule, the six internal coordinates (O-H stretch, O-F stretch, O-O stretch, H-O-O bend F-O-O bend and H-O-O-F torsion) lead to 21 independent internal coordinate force constants. Since HOF and HOOF molecules have only three and six vibrations, respectively, one cannot uniquely determine all of the internal coordinate force constants from the experimental vibrational frequencies. For small symmetric polyatomic molecules such as H20 and H202, however the situation is more favorable. For H 2 0 there are four independent internal coordinate force constants, namely

O2V/OR2

~)2V/OR2, ~)2V/~)RI~)R2, ~2V/OR 1~o~ = O2V/OR2~)~ and ~2V/~)~2 (where R1, R2 and a are internal coordinates representing the changes in two O-H bond lengths and angle). The three fundamental vibrational frequencies of H20 are not enough to determine the four force constants, but one can use the experimental data for isotopic molecules such as D20, HOD, H2180. Recall that the F matrix is invariant to isotopic substitution (but G matrix and hence vibrational frequencies change upon isotopic substitution). With the experimental data for the parent molecule and any one of the isotopic molecules, the total number of available vibrational frequencies increases to six (two greater than the number of unknowns). But the vibrational frequencies of isotopic molecules are related through

134

Chapter 7

the product rules (see next section). In the case of H20 and D20 (or H2180) there are two product rules (in the case of H20 and HOD there is one product rule). Thus there are four independent vibrational frequencies for H20 and D20, which are adequate to determine the four independent force constants. In the H202 case there are 12 independent internal coordinate force constants (the reader can easily verify this by using the equivalence of two O-H groups and the internal coordinates mentioned earlier for HOOF), but only six vibrational frequencies. The six vibrational frequencies of D202 will bring the total number of frequencies to 12, but there are two product rules among these frequencies making the number of independent frequencies to be only 10. So one must have the vibrational frequencies of HOOD as well. The eighteen vibrational frequencies of the three molecules H202, D202 and HOOD, must satisfy one sum rule (see next section) and four product rules (two product rules for H202 and D202, one for H202 and HOOD and one for D202 and HOOD). Then one has 13 independent vibrational frequencies from three molecules H202, D202 and HOOD, which can be used to determine the 12 independent internal coordinate force constants. For larger polyatomic molecules (even with some symmetry) the number of independent internal coordinate force constants would, in general, be so high that it would be impossible to obtain enough independent vibrational frequencies from the experimental data. For this reason, one has to make severe approximations (for example, approximating some force constants to be zero and some to be constrained to certain values guided by chemical intuition etc). In these cases there are no unique solutions for determining the force constants from the experimental vibrational frequency data. The coriolis interaction constants and centrifugal distortion constants are also related to the force constants, so one can augment the experimental vibrational frequency data with this additional information in determining the force constants. But the experimental data on coriolis and centrifugal distortion constants are not available for many polyatomic molecules, in general, and therefore their use is limited. The reader interested in these aspects may refer to a detailed discussion in Ref. (15). Coming to the cubic and higher order force constants in polyatomic molecules, one may be able to determine them only for selected small molecules. The reason for this would be clear when we analyze the needed experimental data for determining these higher order force constants. The vibrational energy levels with anharmonic contributions are given by Eq. (4.2.25). For transitions restricted to the ath vibration (for all other vibrations ~b = 0), the transition frequencies can be given by the general expression,

Vibrational Frequencies and Force Constants

V0"->aga -- Vaaga + Xaa (1)2 + a)a) + E l)aXab / 2 .

135

(7.2.26)

b,a A plot of (v0-->u~ / ~)a) vs (l)a + 1) gives a straight line with slope given as Xaa and intercept as (Va + ~ Xab / 2). Thus the diagonal anharmonicity b,a

constant Xaa can be determined if the fundamental and at least one of the overtone frequencies of the ath vibration can be measured. The offdiagonal anharmonicity constant Xab and the harmonic frequency Va are lumped together in the intercept of the above mentioned plot, so additional experimental information would be needed to separately determine these quantities. In a combination transition two different vibrations a and b are simultaneneously excited by different quanta 9a and agb. Then from Eq. (4.2.25) the combination frequency is given as Vo,o__>aga,~b -- ~al)a + Vba)b + Xaa(a)2 + I)a) + Xbb(1)~ + I)b) + Xab[(1)a / 2)+(I) b / 2)+ 1)al)b]+ ~(1)aXac + ~)bXbc) / 2. c~a,b

(7.2.27)

Subtracting Eq. (7.2.26), and the analogous equation for V0.._>l)b, from Eq. (7.2.27), one obtains (Vo,o....>,Oa,,Ob -- Vo...~,Oa -- Vo...f,Ob ) -- Xab'Oa'O b .

(7.2.28)

Thus to determine Xab one requires the combination frequency V0,0--->~a,~b and the corresponding individual fundamental frequencies for transitions v0-->~a and V0-->~b" Such data are not available for most polyatomic molecules. Once Xab is determined from Eq. (7.2.28), the harmonic frequency Va of the ath vibration can be extracted from the intercept of the plot (V0__>ua / ~ a ) vs (a)a + 1) mentioned above (see Eq. 7.2.26). Remember that Va is related to the quadratic force constant (see Eq. 3.3.3a) and Xaa and Xab are related to the cubic and quartic force constants (see Eq. 4.2.25), all in normal coordinate space (not in internal coordinate space that we discussed earlier). Once the harmonic frequencies Va are determined for all fundamental vibrational transitions, we go back to the discussion given earlier (for H20 and H202) to determine the quadratic force constants in internal coordinate space (where

Chapter 7

136

the use of isotopic information is needed). In the determination of the anharmonicity constants Xaa and Xab discussed above one has to address the possible resonance interactions between different transitions. Looking at the expressions for Xaa and Xab given by Eqs. (4.2.25a) and (4.4.25b), it can be noted that the magnitudes of Xaa and Xab can become very large in two situations. (a) the first overtone transition frequency 2 Va of the ath vibration is approximately equal to the fundamental transition frequency Vb of vibration b; (b) the combination frequency Va+ Vb of vibrations a and b is approximately equal to the fundamental transition frequency Vc of vibration c. The first case was observed in CO2 by Fermi, 16 so this type of resonance is known as Fermi resonance. In such resonance situations, the second order perturbation theory used to obtain the expressions for Xaa and Xab is no longer valid. This is because the perturbation theory is based on the premise that the second order corrections are small, but Xaa and Xab acquire larger magnitudes in resonance situations. So one has to identify the transitions that are in resonance, remove the corresponding contribution from the expressions for Xaa and Xab, and separately correct the observed frequencies for resonance effects. Let us consider the situation where resonance is occuring from 2V a - Vb = 0, i.e., 2 Va " Vb" That means the states laga = 2, 1)b = 0> and I~a = 0, 1)b = 1> are mixing -

2

each other and the perturbation comes from the cubic term Vaabqaqb (see Eq. 4.2.25). From the integrals in Appendix 2, it can be seen that, = 112. I~a=2,~b=0> (Va+3 Vb)/2. be written as To obtain the

The harmonic energy of the

state is (5Va+Vb)/2 and that of I~a-0,a3b=l> state is Since 2 Va-Vb = A " 0, the above mentioned energies can (5~a+ Vb)/2 = (7 Va- A)/2 and (Va+3 Vb)/2 = (7 Va- 3A)/2. corrected energy levels, we diagonalize the 2x2 matrix

(7 Va - A)/2

Vaab/2

Vaab/2

(7V a - 3A)

- 2 1/2]/2 as the new energies of the two yielding, [(7 Va- 2A) + (A2 + Vaab) levels.

Since the zero point energy is (Va+Vb)/2 = (3V a - A)/2, the

transition frequencies become 2 Va- A/2 _+(A2 + V 2aab)1/212.

137

Vibrational Frequencies and Force Constants

The energy levels before and after perturbation are sketched in Fig. 71, for the case 2 Va=Vb . In this case the difference between the two observed frequencies of the above mentioned transitions gives the magnitude of the cubic force constant Vgaab, and the contribution from this term should be dropped in the expressions for Xaa and Xab. For the second case, where the resonance occurs from Va + V b - V c = 0, the two states with quantum numbers II)a - 1, ~ b - 1, l)c = 0 > and I~a - 0, lkb -

0 , 1)c =

=0)

1> will be mixed by the perturbing term Vabcqaqbqc- The

. . . . .

( ~ a = O, ~b = 1)

,,,

....

I

, 1

abc 9

1)J

4,.

(~a = O , ~ b = 0 )

~J

(ga = 1,X)b = 1,X)c = 0 ) ' ~ (ga = 0,19b = 0 , 9 c =

' ~%

4| 4

-I ( ~ a = 0, ~b = 0, be = 0)

Fig. 7-1. E n e r g y levels depicting the situation for 2 V a = V b , and V a + V b = V c .

procedure to set up and diagonalize the energy matrix is similar to the one discussed above. For the situation where Va+ Vb=V c the two energy levels will be separated by 2Vabc/ff2, as sketched in Fig. 7-1. So the difference in the observed frequencies of the two transitions can be used to deduce the magnitude of V abc.

7.2.5 Sum rules The vibrational secular equation in the cartesian displacement coordinates (see Eq. 5.1.14) reveals a sum rule relating the vibrational frequencies to the diagonal cartesian force constants. The trace of the t matrix (see Eq. 5.1.14) is equal to the sum of the vibrational frequencies (provided the atoms are at equilibrium geometry, i.e. aV/aXAtx = 0), so one can write the sum rule as,

[(a2v / ax2)+ (a2v / ay2a,,)+(a2v / az2)1/mA A

E492c2V2" a

(7.2.29) This relation can also be obtained from Eq. (5.1.25) (note that the normal coordinates are orthonormal and (aV/aQa) = 0 at the equilibrium

Chapter 7

138

geometry). Since c)2V/c)X2Acxare independent of isotopic substitution one can estimate the vibrational frequency sums for isotopically related A

molecules from a knowledge of cartesian force constants ~)2V/~)XAo~. One can define an effective atomic cartesian force constant as, f 2 _ ~)2V / ~9x2 + ~)2V/~9y2 + ~)2V / ~)z2 '

(7.2.30)

and simplify the sum rule given above as, 17 X f 2 / mA _ X 4rl:2c2v2 A

9

(7.2.31)

a

If the effective atomic cartesian force constants fA are transferable for atoms in similar chemical environments, the vibrational frequency sums can be estimated from the above equation for a molecule of interest. Alternately, for molecules where the experimental vibrational frequency data are available, but the atomic compositions are not certain, the above mentioned sum rule provides a criterion to suggest the atomic compositions. The sum rule given by Eq. (7.2.31) leads to additional sum rules 18 for isotopically substituted molecules. As an example, let us consider Eq. (7.2.31) for H202, D202 and HOOD. These are,

= (1/4rC2c2){2fH/m H +

Va

2fo/mo},

(7.2.32)

H202

{Xa} {Xa} Va

= (1/4x2c2){2fH/m D + 2fo/mo},

(7.2.33)

D202

Va

(1,4.~c~)~f. ,m. + f . , mo + ~o 'mo~

HOOD

Substracting the latter two equations from the first, one obtains

(7.2.34)

Vibrational Frequencies and Force Constants

Va

-

12fH(1 / m H - 1 / m D),

Va

-

139

H202

D202

(7.2.35)

Va

-

=(1/4n2c2)fH(1/m H-1/mD).

Va

H202

HOOD

(7.2.36) From Eqs. (7.2.35) and (7.2.36) it is easy to see that,

Va

-2 H202

{a

Va

+ HOOD

= 0.

Va

(7.2.37)

D20 2

There is no need to restrict ourselves to H/D substitution, and one could have written similar equations by replacing 16 0 with 18 0 . Experimentally, however, deuterium substitution is much easier than 180 substitution. The product of vibrational frequencies also provides a useful relation, which is commonly referred to as the Teller-Redlich product rule. 19 Although this product rule should not be classified as a sum rule, it is included here due to its importance. The product rule can also be obtained from Eq. (5.1.14) as follows. The determinant of the t matrix (see Eq. 5.1.14) is equal to the determinant of the eigenvalue matrix. The eigenvalue matrix is diagonal, so its determinant is equal to the product f.0102...f.03N . 22 2

The t matrix itself can be written as t = M-1/2fM-1/2 [see Eqs.

(5.1.2), (5.1.6) and (5.1.8)], where M -1/2 is the diagonal matrix with its -1/2 and f is the matrix of elements b2V/bXAaOXB~. elements being m A Then the determinant of t is equal to the determinant of f times the product of inverse atomic masses, m -1 A. Since f is independent of isotopic substitution, the ratio of the product of 3N eigenvalues for two isotopic molecules becomes,

nv~/~vf~2 - nmk/rlmA, k

A

A

(7.2.38)

Chapter 7

140

where primes identify the quantities for an isotopically substituted molecule. In Eq. (7.2.38) the index k runs from 1 to 3N, so the products of eigenvalues include not only those of (3N-6) vibrational modes, but also of three rotational and three translational modes. The above equation can be rewritten as, 17v a / H a

-

a

A

A

j

-

,

(7.2.39)

j

where the index a is for 3N-6 vibrational eigenvalues and j for six rotranslational eigenvalues. If we assume that the translational and rotational eigenvalues have very small magnitudes (as in the presence of a weak external force), they can be related for isotopic molecules as 18-20 . . . . vi 2 / v 2 - M / M'for translations and as vi 2 / vj - Ij / Ij for rotations; here M is the molecular mass and Iithe principal moment of inertia along jth principal axis. Then, Eq. (7.2.3q) can be rewritten as

6

V{V2. V3N-6.

mlm2 .

mN .

\~--Tj

T'x~y-z T' T'

/

"

(7.2.40)

This product rule has been widely used 20,21 in the literature. Note that the product rule equation [Eq.(7.2.40)] can be written down separately for vibrations belonging to different symmetry species (Chapter 10). For example, the vibrations of H 2 0 (and also D20) belong to A1 and B1 symmetry species, so a product rule each for vibrations of A1 and B1 symmetry species can be written down for H20 and D20. Only those rotations and translations belonging to the appropriate symmetry species appear in the product rule equations. References

1 2 3 4 5 6

D.R. Hartree, Proc. Cambridge Phil. Soc. 24 (1928) 89. V. Fock, Z. Physik. 61 (1930) 126. G.G. Hall, Proc. Roy. Soc. London A205 (1951) 541. C . C . J . Roothan, Rev. Mod. Phys. 23 (1951) 69. J . A . Pople and D. L. Beveridge, Approximate Molecular Orbital Theory, McGraw Hill, New York (1970). (a). Electron correlation can be treated at different theoretical levels. A review of these methods can be found in: K. Raghavachari, Ann. Rev. Phys. Chem. 42 (1991) 615; (b). Density functional methods are now becoming popular for treating the electron correlation. Reviews on the subject can be found in" B. B. Laird, R. B. Ross, and T.

Vibrational Frequencies and Force Constants

10 11 12 13 14

15 16 17 18 19 20 21

141

Ziegler, Chemical Applications of Density-Functional Theory, ACS Symposium Series 629 (1996). R. M. Stevens, R. Pitzer and W. N. Lipscomb, J. Chem. Phys. 38 (1963) 550. D. M. Bishop and M. Randic, J. Chem. Phys. 44, 2480 (1966). J. Gerratt and I. M. Mills, J. Chem. 49 (1968) 1719; 49 (1968) 1730. P. Pulay, Mol. Phys. 17 (1969) 197. T. C. Caves and M. Karplus, J. Chem. Phys. 50 (1969) 3649. R. Moccia, Chem. Phys. Lett. 5 (1970) 260. J. A. Pople, R. Krishnan, H. B. Schelgel and J. S. Binkley, Int. J. Quant. Chem. 13 (1979) 225. Y. Yamaguchi, Y. Osamura, J. D. Goddard and H. F. Schaefer III, A New Dimension to Quantum Chemistry: Analytic Derivative Methods in Ab Initio Molecular Electronic Structure Theory, Oxford Univ. Press, New York (1994). S. Califano, Vibrational States, John Wiley & Sons, New York (1976). E. Fermi, Z. Physik. 71 (1931) 251. W. T. King and A. J. Zelano, J. Chem. Phys. 47 (1967) 3197. E. B. Wilson, J. C. Decius and P. C. Cross, Molecular Vibrations, McGraw Hill, New York (1955). O. Reidlich, Z. Phys. Chem. B28 (1935) 371. G. Herzberg, Infrared and Raman Spectra, van Nostrand Reinhold Co., New York ( 1971). N. Mohan and A. Muller, J. Mol. Spectrosc. 80 (1980) 455.

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143

Chapter 8 VIBRATIONAL ABSORPTION AND CIRCULAR DICHROISM In the previous chapter, we discussed the methods to determine vibrational frequencies, the x-axis of vibrational spectra, and force constants. Coming to the spectral intensities it is important to note that they are not independent of the force constants which determine vibrational frequencies. The spectral intensities are related to the molecular property (dipole moment and polarizability) derivatives in normal coordinate space, and the normal coordinates are composed of individual atomic displacements in a molecule. The composition of these atomic displacements in a given normal mode, which is important for determining or interpreting the vibrational intensities, is also influenced by the force constants. Equally important are the molecular property derivatives with respect to the individual atomic displacements. Without a reliable description of these properties the prediction or interpretation of vibrational intensities would not be successful. Here also the ab initio molecular orbital theories are becoming the first choice.

8.1 Vibrational absorption spectra The vibrational absorption spectral intensities may be used to derive molecular properties, which include electric dipole moment derivatives, and atomic or bond properties, which include effective charges and their changes during nuclear motions. Alternately, absorption spectral pattern may be predicted by developing methods to predict the above mentioned properties. These approaches are considered below.

8.1.1 Experimental vibrational absorption intensities The relation between the integrated absorption coefficient and the electric dipole strength is given in Section 2.5. For the specific case of vibrational transitions two additional aspects have to be taken into account. In the approximation of separating the electronic, vibrational and rotational motions, the total wavefunction is written as a product of the electronic, vibrational and rotational wavefunctions (see Chapter 3). Since the vibrational transitions considered here are in the same electronic state, we need not be concerned about the electronic wavefunction for now (see next Section). Then the states s and n considered in Section 2.5 can be represented by the products ~ v ~ j and ~ , ~ j , , and the vibrational transition moment integral written as < ~ v , ~ j , l ~ l ~ ~ j > . Converting l.tc~ from the space fixed (non-rotating) axes to the molecule fixed (rotating) axes as laa = uo~a'~', with uaa, representing the transformation matrix element, the

144

Chapter 8

transition moment integral becomes ~l,(t',9'9Ut~',J'J, where kta,,a~,v =

<~v,l~t~,l~> and U ~ ' , J ' J -

< ~ j , lucto~, I ~ j >. The rotational integrals can

be replaced 1 by the averages of direction cosines, in which case the dipole strength becomes l.ttx,,~%kt *~, , ~ ) ' " 0 - I.t2, , ~,~ " The energy difference between vibrational levels is such that at room temperature the population in higher vibrational levels is usually non-zero. Then the transition rate per molecule given in Section 2.5 needs to be multiplied 1 by nv- nv, where n~ and nv, are the number of molecules per unit volume in vibrational states ~ and v'. Then Eq. (2.5.1) becomes, (dI(v)/I(v)) + = (B+/c)hv(nv - nv,)dl .

(8.1.1)

Proceeding further as in Section 2.5, the integrated absorption coefficient for a fundamental transition from l)a to ~)a+l (associated with normal mode a and observed band center V0a) becomes Aa - (4~3V0a/3hcCo)(nv a - naJa+1)(~a + 1)(c)kta/~)qa))2 .

(8.1.2)

In the harmonic oscillator approximation the energy difference between successive vibrational levels is the same, so all hot transitions such as 1---)2, 2---)3, etc., appear at the same location as the first fundamental transition 0---~1. Then Eq. (8.1.2) needs to be summed over all vibrational quantum numbers. Using the Boltzman distribution, n~ a can be expressed

as, nva - n e -hc~a (~)a+l/2)/kT / Z

e-hcva (aga+l/2)/kT,

(8.1 .3)

~a where n is the total number of molecules per unit volume and Va is the harmonic vibrational frequency of normal mode qa. Then the sum ~ ( n ~ a - n~a+l)(~a + 1), with the summation going from ~)a - 0 to co, can be shown 1 to be equal to n or NCo (where Nis Avogadro's number and Co is concentration). As a consequence, the absorption intensities of fundamental transitions, in the harmonic approximation, are expected to be independent of temperature (assuming that the temperature changes do not lead to conformational changes). Using the conversion from the dimensionless normal coordinate qa to the normal coordinate Qa, q2 _ (4rt2va/h)Q2 and taking the observed band center V0a as equal to the harmonic frequency Va, Eq. (8.1.2) can be

Vibrational Absorption and Circular Dichroism

145

written as - (~/3c)(~)jacflOQa)2) .

(8.1.4)

This equation gives Aa in units of cm2/mol.sec, with ~ta in units of Debyes (D). Most of the current infrared spectrometers provide the spectra on the wavenumber (cm-1) axis and integrated absorption coefficient A in cm/mol or km/mol. In such cases the right hand side of Eq. (8.1.4) should be multiplied by (l/c). Substituting the standard values for the constants, one obtains the useful relation, N,a = 42.28(~)gJ~)Qa)2) ,

(8.1.4a)

where Aa is in units of km/mol and (~g~x/OQa) in units of D/(A amul/2). For a diatomic molecule A-B, with internuclear axis taken as the z-akis, only (Ogz/OQ)2 component can be non-zero. Furthermore (see Section 3.2), Q = txl/2ARAB, where g is the reduced mass, mAmB/(mA + mB), and RAB is the bond length of A-B; ~)gz/~)RAB is the same as ~)gAB/~RAB where gA-B is the bond moment (for +eq charge on atom A and -eq charge on atom B, the bond moment gA-B = eqRAB). With these considerations, Eq. (8.1.4a) for a diatomic molecule becomes, A - [42.28(mA + mB)/mAmB](~)gA_B/~)RAB)2 .

(8.1.5)

The derivatives ~)ILtA_B/ORAB are to a good approximation independent of isotopic substitution. Nevertheless, it is apparent from Eq. (8.1.5) that the absorption intensities would be influenced by isotopic substitution (for example, absorption intensities of H-F and D-F vibrational bands would be different). The experimental integrated absorption coefficients enable the determination of the magnitude of (~A-B/c)RA-B), which in turn reveal the magnitude of charge movement (or charge flow) during the vibrational motion of the bond A-B. The direction of charge flow however cannot be determined because the sign of (O~tA-B/ORAB) cannot be determined from Eq. (8.1.5). For polyatomic molecules with symmetry, vibrations can be classified according to the irreducible representation they belong to (see Chapter 10). Depending on the point group and the associated irreducible representation, some vibrations can have only one of the three components (~)l.tj~Qa) non-zero. In such favorable cases individual components of (~)l.tot/~)Qa)2 can be determined from Eq. (8.1.4). Unlike in diatomic molecules, Qa is now dependent on several internal coordinates and therefore (O~ta/OQa) does not have as simple a meaning as (O~tA-B/ORAB) of a diatomic molecule. As a result, (3~a/OQa) has to be reduced to

Chapter 8

146

internal coordinates Rj or cartesian displacement coordinates XAor Such a reduction is complicated by the uncertainty in the signs of (~)l.tcfl3Qa), because this process involves the relations,

(o~l.ta/c3Rj)--- ~_.,(o~laoJc3Qa)(0Qa/c3Rj=)

~(~)l.ta/~)Qa)La] ,

(8.1.6)

a

(~)l-ta/~)XBI3) = ~(~)l.tcfl~)Qa)(~)Qa/~)XBI3) = ~(~l.tcx / ~gQa)Sa, ll3 9 (8.1.6a) a

a

aj and S-1 a,B[~ are discussed in The transformation matrix elements L -1 Chapter 5. The nine elements 691Ja/~OXB[~),with o~ = x, y or z and 13= x, y, or z, are referred to as the atomic polar tensor (APT) elements of atom B. Since the sign associated with each of the (~)l.ta/3Qa) is uncertain, Eq. (8.1.6) has to be evaluated for 2 n possible sign combinations of (~)l-tj3Qa); n is the number of normal modes of the molecule that have non-zero 091.tcd~OQa). In some special cases, the relative signs of (c)gcx/c)Qa) can be determined 2 from the analysis of coriolis effects in vibrational-rotational spectra; but for general cases this sign ambiguity associated with 691.ta/~)Qa) presents a serious problem in further analysis. Quantum mechanical calculations of electric dipole moment derivatives (especially of signs) helped overcome this problem. 3 With the sign ambiguity resolved, the derivatives (c)l.ta/~kRj) or (c)goc/c)XA~) can be further reduced to bond moment parameters using the bond moment model or atomic charge parameters using the atomic charge model (see Section 8.1.3).

8.1.2 Quantum mechanical methods Different quantum theoretical approaches can be identified for evaluating the electric dipole moment derivatives ~)l.tjOQa and hence vibrational absorption intensities using Eq. (8.1.4). In all these approaches it is convenient to separate the electric dipole moment into nuclear and electronic contributions. That is, n u c + go~ el = ~ eZAXAo~-e~ Xio~. goc = goc A i

(8.1.7)

Here eZA is the bare nuclear charge, e is the unit of electron charge, XAo~ is the positional coordinate of atom A and Xia is the positional coordinate of electron i. To evaluate the electric dipole moment of the system in state s, the dipole moment integral <~slg~l~~ needs to be evaluated. Since

Vibrational Absorption and Circular Dichroism

147

the nuclear part ganuc has no influence on i1/0 , this integral becomes ~1.0~ = _ + 0 = 2 eZAXAa- e. A i

(8.1.8)

With the electronic wavefunction represented by a single determinant, the integral <~t0l~. Xialg0> becomes 2~, where the summation 1

k

k is over all occupied molecular orbitals and it is assumed that each occupied molecular orbital has two electrons of opposite spin (closed shell system). The integral is called the orbital centroid, and can be viewed as describing the average position of the paired electrons in the kth MO. If the molecular orbitals are localized then these orbital centroids can be identified with the average positions of bonding electrons in a chemical bond or lone pairs of electrons located on atoms. Although such localization is not a requirement, this process helps to visualize the origins for signs and magnitudes of the electric dipole moment (and its gradients) m terms of bonding and lone pair electrons. Denoting the orbital centroid , for short, as Xk0,a the above equation can be written as _nuc

g0a = ga

el + l.ta = 2 eZAXAa - 2e2 Xk0,a. A k

(8.1.9)

The dipole moment derivatives can be obtained by differentiating this equation with respect to normal coordinates, internal coordinates or . hue cartesian displacement coordinates. The gradient of ga is trivial, i.e.,

(c}~xuc/c}XB~) = eZASABSa[3 .

(8.1.10)

It is the electronic part which requires careful consideration. Note also that in the LCAO-MO approach (see Eq. 7.1.14), each molecular orbital q0k is expressed as a linear combination of atomic orbitals. Different approaches for calculating this contribution are described below. (A) Numerical atomic displacement gradient method In this method g0a is evaluated at slightly distorted geometries and the gradients of the electric dipole moment are obtained as numerical derivatives. 4 The distortions of geometry should be as small as possible

Chapter 8

148

yet should be large enough to overcome the numerical errors. The magnitudes of geometrical distortions used have been in the range of 0.005-0.01A. Improved numerical accuracy is obtained when distortions m both positive and negative directions are used. Then the derviatives in cartesian space are obtained as, --

-

+

(.o

where g0a is the value of g0a when XBfi is increased to XBfl; l.toa is the corresponding value when XB[3 is decreased to X~I3. For a molecule with N atoms, Eq. (8.1.11) has to be evaluated 2x3N times, which requires as many SCF calculations. Therefore this approach is inefficient. For second and higher derivatives this approach is even more tedious. In such cases, the dipole moment is evaluated at several displaced geometries and the resulting values fit to a polynomial. The coefficients obtained from the polynomial fit give the values of higher order dipole moment derivatives. (B) Numerical electric field gradient method As discussed in Chapter 2, the interaction energy to first order in the presence of an electric field is given as -gaFa. So the differential of energy with respect to electric field gives -ga, i.e., (3E/OFa) - -gct. Differentiating this expression with respect to a nuclear displacement gives, o

(31.ta/OXBI3)=-(O/OXBfI(OE//)Fo0)=(-O/OFa(OE//)XBO))=OqOBf~OFa. (8.1.12) The quantity -(t)E/t)XBI3) = qIBI3 represents the force at atom B. When analytic methods to evaluate tpBI3 (see Section 7.2.1) became available 5, Eq. (8.1.12) provided an economical approach to evaluate the electric dipole moment derivatives. Since qlBI3 are obtained using analytic expressions, all of the required 3x3N dipole derivatives (3g~/OXB~) can be obtained from six calculations. That is, (3l.toJ~)XB~) = (~)qOB~)Fot)- (q0B~ -

(8.1.13)

+

where qIB[3 represents the force in the presence of electric field +

F~ and q ~ represents the same in the presence of electric field F~. This procedure 6 requires the incorporation of the electric field perturbation term -gaFa into the hamiltonian while carrying out the SCF, and subsequently

149

Vibrational Absorption and Circular Dichroism

the force, calculations. A significant advantage is realized in this approach because regardless of the number of atoms present in a given molecule only six electric field perturbation calculations are required to obtained all of the 3x3N derivatives Oga/OXBB, while in the numerical method discussed previously the number of calculations required is prohibitively large. (C) Analytic wavefunction derivative method The electric dipole moment derivatives can be obtained using analytic methods in a straight forward manner. This is facilitated by the development of CPHF method (Section 7.2.2). In this section we present the basic theory which also provides the foundation needed to formulate the magnetic dipole moment derivatives in a later section. For this purpose we follow Buckingham and coworkers. 7 The separation of the Schr6dinger equation into electronic and vibrational parts has been discussed in Chapter 3. This separation and solution of the vibrational equation are based on the following assumptions: (a) The wavefunction of the system can be written as a product of electronic wavefunction ~gs0 (which has a parametric 0 dependence on nuclear positions) and the vibrational wavefunction ~v. That is, iltsO = VsVv 0 0 9

(8.1.14)

The superscript 0 indicates that these are unpertubed wavefunctions. In Eq. (8.1.14) it is implied that the vibrational wavefunction ~0 belongs to the electronic state s; (b) the nuclear kinetic energy operator does not influence the electronic wavefunction, i.e., terms of the

type ~2/l/s~

and O~s/OXAa are ignored; (c) the electronic Schr6dinger equation can be solved to obtain the electronic energy E 0s and wavefunction gt~ and (d) the vibrational equation can be solved with the potential (given by the sum of E s0 and nuclear repulsion energy VN) expanded in a Taylor series and this series truncated at the second order term (harmonic approximation). Let us designate the electronic part of the hamiltonian and nuclear repulsion energy terms together as H0 and expand this H0 in dimensionless normal coordinates qa, as

Chapter 8

150

H = H0 + ~ (c)H/c)qa)qa + ... a = Ho + H1 + . . . .

(8.1.15)

Using the first order term H 1 as a perturbation, we can use the first order perturbation theory (see Chapter 2) to obtain the corrected electronic wavefunction. The corrected wavefunction (see Eq. 2.2.12) can be written as,

Vs~

--

o o IIIs lII ~ + Z Z

o o (~nkl/u,,

I 0 0 0 H1 Vs Vv)~t0V 0" / E s n ~ " ,

(8.1.16)

n~s~ H

where E 0s n ~ ,, = E 0s + E 0~) wavefunctions ~0 and xr

E on -

0 In Eq. (8.1.16), vibrational Eu,.

belong to the electronic states s and n,

respectively. With this new wavefunction, the electronic part of electric dipole transition moment integral for a transition from state sv to s v ' is given by the following expression: el

/ so, 0 0 +~~(VnVa~,,IH11

0 0

0 0

9 n~" ) / Esn~"

n:g:s ~)t'

0 0

0 0

lll s I11~) 1 E sn~ % ,,

n C s "Opp

n:Cs m:Cs

~tp ~)m

I, 1

0 0 >, Esnag'~"Esma~" 9

(8.1.17)

The vibrational wavefunctions are orthogonal, so the first term on the right hand side of Eq. (8.1.17) vanishes. For the same reason, if we assume that the vibrational levels of ground and excited electronic states are represented by the same vibrational wavefunctions, then the only surviving terms are those for ~' - ~" in the second term, a9 = ~" in the third term and u " - ~'" in the fourth term. Furthermore, the fourth term is much smaller than second and third terms (notice that the fourth term is a product of two integrals each with H1 operator and the energy denominator is approximately square of that in the second and third terms). So the fourth term can be ignored. Then we are left with the

Vibrational Absorption and Circular Dichroism

151

second and third terms. The denominators of the second and third terms ~ = E s~ - E ~n 1 7 6 v v ' -_ can be written as ~n (l+x) where x = ~ v'/E O sn' Esn

0 For x<
X(~o~o, Inll~~176176I~l ~~ E~

el

n~:s

+X(v~ ~ H1

0

0

~0

)(l+Ea~ov,/Esno) /Esn.o

(8.1.18)

n:#:s

For systems with
0

= <~0nlgetlllt0>, the terms that contain Evv,

cancel. For systems with

the terms that

contain E~ survive. The first situation is true if the integrals are all real and the second situation is true if the integrals are imaginary. Writing the 0 0 , H l]~s0~l/a~> o as ~ 0 0 integral and a

noting that the integrals
el

(see Appendix 2), Eq.

I

g0c,X)aaga-- 2Re E(/I/0 I~)n0/OqaIV0)(V0I~IV o)/(E o -E o n:#s

•

+ 1)[ 2)1~ ~13a,~3a+l+ (~)a [ 2)1~ ~a,19a-1 ].

(8.1.19)

Making use of Eq. (2.2.12), the above equation is rewritten as,

gO~,lga19 a -[2 el

el Re(v s0irt=l~Vs //)qa )]

• (~)a + 1)/2) 1//2~X)a,~a+l + ('Oa ! 2) ~ ~)a,aOa-1

(8.1.20)

The CPHF method (see Section 7.2.2) is used to determine (3~s/3qa) needed in Eq. (8.1.20). The electronic part of the electric dipole moment

Chapter 8

152 derivative is thus given as,

0 el (ag~ /C)qa)- 2Re(Vslg=laVs/c)qa) 9

(8.1.21)

In practice, it is convenient to evaluate these derivatives in cartesian coordinate space and transform them to normal coordinate space, as needed. In that situation qa in Eq. (8.1.21) is replaced by cartesian displacement coordinates. The dipole moment derivatives with nuclear and electronic contributions (see Eq. (8.1.10) for nuclear contribution) are obtained in the cartesian space as,

c)~l,o~[ c)XBi3 - eZA~Aa~oq 3 + 2 Re( Vsl 0 el .lavs0 / ~)XBI3),

(8.1.22)

and in normal coordinate space as

t)"ot ' t)Qa

-

s B

] t)XBl3)(t)XB[3/t)Qa) - Z(()"t~ ' t)XB,13) SBI3,a " B

(8.1.23)

(D) Analytic energy derivative method 8,9 The electric dipole moment derivative (~)goJc)XAS) is equal to -(~)2E/~)XAS~)Fa), as can be deduced from Eq. (2.1.12). The second derivatives of energy can be obtained using the CPHF method as discussed in Section 7.2.2. The parameters b and b' in Eq. (7.2.5) represent, for the present case, XA8 and Fa respectively. (E) Analytic dipole moment derivative method 9 Instead of using the energy differential, one can differentiate the expression for electric dipole moment. For a closed shell system the electric dipole moment can be written, using a single determinantal wavefunction and the LCAO-MO approximation (see Eqs. 7.1.14 and 8.1.9), as laa - ZeZAXAo~ -- Z Z

A

2CpiCqi (fpleXietlfq),

(8.1.24)

,i p,q

where the summation over A is for nuclei, and i is for occupied molecular orbitals with two electrons per orbital. The definitions of other terms in the above equation are as stated in Section 7.2. Differentiation of the above equation gives,

Vibrational Absorption and Circular Dichroism

153

(t)~oJ()XBS) - Z eZA(~ABStx8 A -ZZ2[(C)Cpi i p,q -

/ C)XBS)Cqi + Cpi(~Cqi ! ~)XBS)](fpleXi~lfp)

Z Z 2 C p i C q i [ ( ~ f p / C)XBsleXi~lf q ) + (fpleXi~lc)fql~)XBs)] 9

i p,q (8.1.25) The derivatives c)fo/c)XB8 are obtained from the analytic formulae for the basis functions fp; ("0Cpi/0XBS) are obtained using CPHF method. All three analytic methods discussed so far should give the same result. The ab initio predicted vibrational absorption intensities for CHFC1Br are given in Table 2 of Chapter 7. (F) Analytic nuclear electric shielding tensor method 10 In Chapter 2 we mentioned that the electric dipole moment derivatives can be related to the variation of the electric field sampled by the individual particles of the system. From Eqs. (2.1.13)-(2.1.14), we have (1/eZA)(~)ktJ~)XA8) = ~)FA~/~)Fot,

(8.1.26)

where F A is the electric field at nucleus A and F is the external electric field perturbing the molecule. The electronic contribution to F A can be expressed using the Hellmann-Feynmann theorem 11 as,

el = <Xl/sl_(~)Hel/~)XAS)/eZAiXl/s>' FA8

(8.1.27)

where Hel is the electronic part of the hamiltonian, for a given nuclear configuration. Using the time independent perturbation theory, ~s can be expressed by Eq. (2.2.12) with the perturbing hamiltonian H1 given as -~ta,Fa,. Then, to first order, F~ 8 _ Fel AS,0 + 2 ~ FA8,sn~a,,nsFct, [ (E 0 - E s0 ) ,

(8.1.28)

n4:s

where

FAS, sn = <~s I-(c)Hel/C)XAS)/eZAIV0>,

(8.1.29)

Chapter 8

154

and F el AS,0 is the electronic contribution at Ftx' - 0, given as Fel 0 0 AS,0 - ( ~ s I-(0Hel / ~)XAs) / eZAIXl/s ) 9

(8.1.30)

~nuc ~nuc The nuclear contribution X-A8 can be written as "'A8,0 + F~,8~,8 where nuc F A8,0 is the contribution at Ft~' = 0. If the nuclei are at equilibrium trnuc Fel positions at F~, - 0, then the total contribution, FAS,0 - ~AS,0 + AS,0, adds up to zero; for non-equilibrium positions FA8,0 will be non-zero. The total electric field experienced by the nuclei can be written as,

(8.1.31)

F A 8 - FAS,0 + (8o(8- TA,Sct,)Fct', where TA,8a' - - 2 E FAS,snl.toCns/(E 0 - E 0 ) n:g:s = -2(~0l-(~)nel / c)Xgs) [ eZgl(O~ s / 0Fct, )),

(8.1.32)

is called the nuclear electric shielding (also electric dipole shielding) tensor. From Eqs. (8.1.26) and (8.1.31) one obtains, (8.1.33)

(I/eZA)(t~ILtjt~XAS) = 8ct8- TA,S(~ 9

To obtain the analogous expressions in the presence of a time varying electric field Fct'(t), rather than a static electric field Fa,, it is necessary to use the time dependent perturbation theory. Such expressions were given for general molecular properties in Chapter 2 [see Eqs. (2.3.23)-(2.3.27)]. Replacing P~(t) in Eq. (2.3.23) with FA8(t), the expression for TA,&t' is obtained from the coefficient of Fct'(t). Thus,

TA,8oC - - Re

4rCFAS,sn~tog,nsf.0ns/ h COnsLn~:s

.

(8.1.34)

This equation reduces to Eq. (8.1.32) when (Ons>>O). From the details given above it can be seen that the nuclear electric shielding parameters T A,8o~' can be used to obtain the electric dipole moment derivatives (Eq. (8.1.33)), and hence the absorption intensitites, of

Vibrational Absorption and Circular Dichroism

155

harmonic fundamental vibrational transitions [see Eqs. (8.1.4) and (8.1.6)]. To obtain "~A,Sa' using Eq. (8.1.32), Random phase approximation method 10 has been used. It is important to note that Eq. (8.1.27) is valid only for exact wavefunctions.

8.1.3 Classical models In these models a molecule is viewed to consist of a set of effective atomic charges e~A, bond charges eqAB or bond moments gA-B. The molecular dipole moment can be written in terms of these parameters and appropriate derivatives formulated. These classical conceptual parameters were widely investigated, before the ab initio quantum mechanical programs became commonly available. At the present time very little activity is apparent in this area, mainly due to the overwhelming influence of quantum mechanical developments. It is worth noting that there are no quantum chemical analogues for the above mentioned classical parameters, although quantities equivalent in spirit can be defined from the wavefunctions. Nevertheless, it is useful to know how these classical parameters are used in vibrational spectroscopy. (A) Atomic charge concepts Suppose that each atom in a molecule has an effective charge e~A (these effective atomic charges are not equivalent to the charges determined from quantum mechanical calculations using Mulliken 12 or other 13 population analyses). For a neutral molecule the sum of effective charges is zero. The molecular dipole moment can be written in terms of these charges as ga = ~ e~AXAa 9

(8.1.35)

A

The required electric dipole moment derivative is obtained 14 by differentiating this equation. For the electric dipole moment derivatives in cartesian space, the nine quantities ~ta/~XA~5 (c~, 5 = x, y or z) pertaining to atom A are referred to as atomic polar tensor (APT) elements; ~Ia/~XA is referred to as APT of atom A. At the simplest level, the effective atomic charges are assumed to be 'fixed', that is they are assumed not to change as the atoms are displaced. This is referred to as the fixed partial charge (FPC) approximation. In this FPC model, the atomic polar tensor elements become ~l-ta/3XB~- ~ e~,~SABSc~5, A

(8.1.36)

Chapter 8

156

and the atomic polar tensor associated with atom A becomes diagonal with the three diagonal elements being equal. In terms of intemal or normal coordinates, the dipole moment derivatives in the FPC approximation become Z eg~,AAa,j , A

(8.1.37)

~)].td()Qa = E e~ASAa,a , A

(8.1.38)

~176

where the A and S matrices are discussed in Chapter 5. Thus the required electric dipole moment derivative can be obtained from Eqs. (8.1.36)(8.1.38) by assigning a set of effective charges for atoms in a molecule. The resulting values would be only approximate however. At the next higher level, the effective atomic charges are not considered to be fixed and to vary differently as different atoms are displaced. Then the above mentioned derivatives become, O).ta/OXa8 = Z [e~AIiABISa~+ e(C);A/c)XBS)XAo~], A

(8.1.39)

})~oJ()Rj - Z [e~AAAoqj + e(})~A/})Rj)XAc~], A

(8.1.40)

c)].toJ()Qa = E [e;ASA(x,a + e(()~A/c)Qa)XAo~]. A

(8.1.41)

The quantities e(~)~A/~IXBIS)indicate how the charge of atom A 'flows' as the atom B is moved; e(~)~A/~)Rj) indicates the same when internal coordinate (bond length, bond angle, etc.)j is displaced; and e(~)~A/c)Qa) has the corresponding meaning for the normal coordinates. Of these three quantities e(O~A/~)Rj) is much easier to connect with chemical intuition. Eqs. (8.1.39)-(8.1.4I) constitute the atomic charge-charge flux model (this terminology was originally used 14b for symmetry coordinates). To obtain the required electric dipole moment derivative, one has to assign not only the effective atomic charges e~A, but also the charge flux parameters e(~)~A/~)XBIS), e(~)~A/~)Rj) or e(~)~A/~)Qa). Looking at these equations from a different viewpoint, if the electric dipole moment derivatives can be determined experimentally, then one has a method to determine the effective atomic charges and charge flux parameters. But the number of charge flux parameters, in general, is

Vibrational Absorption and Circular Dichroism

157

larger than the number of independent electric dipole moment derivatives, so one has to resort to approximations in solving Eqs. (8.1.39)-(8.1.41). This is one reason for the lack of wider utility of these models in vibrational spectroscopy. There are however favorable situations, which can be used to illustrate the importance of these models. Consider the planar molecules and let us assume that the molecule is in the x-z plane. The out-of-plane vibrations are not coupled to the in-plane vibrations, so we can discuss the out-of-plane vibrations independent of in-plane vibrations. The absorption intensity associated with the out-of-plane bending vibration will be determined by the derivative Og/OQo , where 9 . Y. Qop is the normal coordinate for out-of-plane bending vibration. ~t is easy to see that there will be no charge flux contribution to /)gy/OQop [differentiation of gy =~e~AYA with respect to Qop will give two A contributions" one from e~AOYA/OQop and another from (Oe~A/OQo)YA p 9 For a planar molecule in the x-z plane, YA = 0 for all atoms and therefore 3l.t/OQo y p does not have a contribution from the charge flow terms]. Thus the out-of-plane vibrational intensity is determined by the atomic charges e~A and the atomic displacements OYA/OQop. The latter can be determined as discussed in Chapter 5, so one can determine the atomic charges from the out-of-plane vibrational intensity. For example, the atomic charge of fluorine in BF3 and of hydrogen in C2H2, C2H4 and H2CO can be determined in this manner. Using the charges determined in this manner, the absorption intensities associated with the in-plane vibrations can be used to determine the charge-flow parameters. Although it is not possible to determine all individual charge flow parameters, certain combinations of these parameters can be found. For example, in BF3 the combinations (~)e~F/~)rB-F - c)e~F/C)rB-F') and (~)e~F/~t~o - ~e~F/~t~a) can be determined; here Oe~F/~)rB-F indicates how the charge on atom F changes as its distance from B is changed; and ~)e~F//)rB-F' indicates the corresponding charge flow when a neighboring bond B-F' is stretched. The parameters Oe~F/OCto and Oe~F/Ot~a are for the corresponding angle changes (t~o is the angle F'BF', opposite to the bond B-F; ffa is the angle FBF' or FBF"). (B) Bond charge and bond moment concepts Since bond charges are defined from atomic charges, the bond charge and atomic charge concepts are related to each other. For example, consider a A-B-C molecule with B as the central atom and A, C as the end atoms. If the effective charges of atoms A and C are e~A and e~c respectively, then the molecular neutrality defines the effective charge on atom B as e~B = - e~A - e~c. The charges e~A on atom A and - e~A on atom B define the magnitude of bond charge eqA-B in bond A-B as equal to e~A. Similarly the atomic charges e~c on atom C and - e~c on atom B

Chapter 8

158

define the magnitude of bond charge eqc-B in bond C-B as equal to e~c. Thus, with a bond charge eqk in bond k, the bond moment is defined as ].tktx = eqkdka = eqk rk U ka ,

(8.1.42)

where dk is the length vector, rk is the bond length and Uk is the unit vector, all for bond k. If bond k is made up of atoms B and C then dk = XC - XB, U k = ( X c - XB)/rk. The molecular electric dipole moment is then written as a sum of bond moments, so ].ta - ~ laka k

9

(8.1.43)

It is useful to note that in the discussion given above we have only defined the magnitude of bond charge. The sign of the bond charge and the direction of the length vector define the direction of the bond moment. One may choose to define the bond moment vector pointing from + charge t o w a r d s - charge, or vice versa. But the same definition should be maintained for all bonds. Thus, one may define all bond charges as positive and the vectors dk and U k directed from the atom with positive charge towards the negative charge counterpart in that bond. The molecular electric dipole moment derivatives can be obtained 15 by differentiating Eq. (8.1.43) with respect to the desired coordinates. The atomic polar tensor elements, for example, are given as

~)l.ta/~)XA5= ~ [eqkAkA~a8 + e(~)qk/~)XAS)dka],

(8.1.44)

k where AKA = +1 when bond k contains atom A and zero when bond k does not contain atom A. The derivatives/)~ct/~)Rj in internal coordinates, and /)l.ta/~)Qa in normal coordinates, can be obtained similarly using the transformation matrices A and S, as discussed before. The first term in Eq. (8.1.44) represents the fixed charge contribution and the second term represents the charge flow contribution to the molecular electric dipole moment derivatives. Thus the FPC approximation requires the omission of charge flow terms in Eq. (8.1.44). Since the number of independent bond charge parameters (eqk and e~qk//)R') again far exceeds the number of ~)l-ta/~)Qa for a given molecule, it ~s not possible to determine these bond parameters from experimental vibrational spectra without making serious approximations. In favorable cases, one can determine the magnitude of bond charge from the out-ofplane vibrational intensity and certain combinations of charge flow parameters from the in-plane vibrational intensities. This procedure is 9

.

J

.

Vibrational Absorption and Circular Dichroism

159

identical to that discussed earlier in the context of the atomic charge model, and no new information is obtained. From that viewpoint no advantage is gained in choosing the atomic charge concept over the bond charge concept or vice versa. However, when we consider the molecular magnetic dipole moment derivatives in Section 8.2.3, the bond moment concept turns out to be more useful. 8.1.4 Sum rules The integrated absorption intensity (see Eq. 8.1.4) is related to (~l.ta/~Qa)2. To facilitate the derivation of the sum rule for absorption intensities let us consider the transformation between cartesian displacement and normal coordinates, as discussed in Chapter 5. Using Eq. (5.1.21), one can write the atomic polar tensor element as

(8.1.45)

c~l'ta / aXA5 - E (~)['ta / c)Qk)Sk~Afi 9 k

The summation index k goes over all 3N normal modes, comprising (3N6) vibrational modes Qa and six rotranslational modes pj. Noting that the S matrix is related to the orthonormal matrix g (see Eq. 5.1.24), one can write the relation, Z[(c3l.t~ / C3XA)2 + (bl.ta / ~yA A

)2

)2 + (~91-ta / ~}ZA ] / mA Y--,(~91-ta / ~)Qk)2" k

(8.1.46)

The right hand side of this equation can be written as two separate sums, one for the vibrational normal modes Qa, and another for the six rotranslational modes 9j. That is,

E[(b~t a ] ~)XA)2 + ( ~ a A

] c)yA)2 + (~ltla / ~)ZA)2] ] mA =

)2 9

E (~)ga / ~Qa)2 + E (c}ga / ~)Pj a

(8.1.47)

j

The contribution from rotranslational modes can be easily calculated. In the case of translational modes p~, (O~ta/3pl) = 0 for neutral molecules. *dr

The derivatives (O~t,/3or) for rotational modes can be determined 16 from

Chapter 8

160

the relation (see the notation in Appendix 1),

/11/2

oalLtoJOaP; - ~:o~13~,l-h,,~l 3

(8.1.48)

9

Substitution of Eq. (8.1.48) into Eq. (8.1.47), gives E(ogc~ /OQa) 2 - E[(ogoc /OXA) 2 + (Ogcz / OYA)2 + (O~to~ / OZA)2 / mA] a A T1/2 (8.1.49)

- %~ g~ /-~f~

9

If the molecule under consideration has no symmetry, then for a given normal mode all three components of 3gtct/~gQa can be non-zero. In such cases, the integrated absorption intensity depends on the sum [(agx/OQa) 2 +(~gy/~Qa)2 +(agz/~Qa)2], and the sum rule becomes, 17 A a - K~ a

(Ogtoc / ~gQa)2 a

=K~[(Oga/Ox A

)2 +(Ogtcz/3YA )2 +(Ogcz/ozA )2 / m A ] -

A K{(gz 2 + gy2) / ix x + (gz 2 + gx 2) / Iyy + (gx 2 + gy2) / Iz z },

(8.1.50) where K - ~ / 3 c 2 (see Eq. 8.1.4). For molecules with enough symmetry, a normal mode can lead to a change in the dipole moment along only one of the three axes, so only one of the three components of/ggoJ~gQa will be non-zero. In such cases Eq. (8.1.50) can be written separately for modes of different symmetry. The atomic polar tensor elements ~ga/~gXA8 are invariant to isotopic substitution. So the isotopic dependence on the sum of vibrational intensities can be seen from Eq. (8.1.49) to arise from the atomic masses mA (which also affect the moment of inertia II~l~). It is of interest to note that the atomic polar tensor elements themselves satisfy some sum rules" 18

E c)/s / ~ A5 - 0 , A

(8.1.51 )

Vibrational Absorption and Circular Dichroism

Z ~/1$8 (~)~r [ ~)XA~)XA~ - %t[ie ~e 9 A

161

(8.1.52)

Eq. (8.1.51) is called the translational sum rule and can be obtained from the relation (Oga/OP;) - 0 = ~ ( O g a / OXA[~)(OXA[~ / OPt) and Eq. A

(5.2.17); Eq. (8.1.52) is called the rotational sum rule and can be obtained from (Oga/~)P~) - ~ ( 3 g a / 3XAfi)(3XAfl / 3p~) and Eqs. (8.1.48) and A

(5.2.17). A second intensity sum rule 16 can be obtained by transforming the normal coordinates Qa to internal coordinates Ri. Using Eq. (5.2.28) and Eq. (5.2.33), one can write Z(~)gtx / OQa) 2 / 4~2c2v2 - Z(~)ga / ORi)(~)gtx / ~)Rj)Fij 1 . a i,j

(8.1.53)

The inverse force constants F.1j1 are invariant to isotopic substitution. The values of ~ga/~Rj for isotopically substituted molecules are related as

(8.1.54/ where the subscripts 1 and 2 identify the isotopically substituted molecules. The matrices A and ~ were discussed in Chapter 5. Eq. (8.1.54) can be obtained by left-multiplying Eq. (5.2.16) with 311/3X. The sum of frequency weighted integrated absorption intensities can be written [using Eq. (8.1.53) and (8.1.4)] for different isotopically substituted molecules and related to that of the parent molecule using Eq. (8.1.54). 8.2. Vibrational circular dichroism (VCD) spectra

The magnetic dipole transition moments associated with vibrational transitions were discussed in Chapters 3 and 4. These and the electric dipole transition moments discussed earlier determine the spectral pattern in VCD spectra. 19 In fact the only experimental data currently available for deducing the magnetic dipole transition moments are the vibrational circular dichroism (VCD) intensities. VCD intensities may be used to deduce some molecular properties. Alternately theoretical approaches for

Chapter 8

162

predicting VCD intensities may be used to determine the molecular conformations and absolute configurations. These approaches are discussed below.

8.2.1 Experimental VCDintensities The circular dichroism intensities are determined by the rotational (some times called as rotatory) strength (see Eq. 2.5.8). For vibrational transitions, we proceed with the approximations discussed in Section 8.1.1. Then for a fundamental vibrational transition associated with a dimensionless normal coordinate qa, the rotational strength is given in the double harmonic approximation as, Ra

=

(1/2)(3~ttff/)qa)(/)mJ~)pa) (h/4~) (/)~tx/OQa)(~)rn~//)Qa), =

(8.2.1)

where the relations q2 = ctQ2, p _ 0 , and p2 - 4/i;202]h2~, have been used to write the derivatives in terms of Qa and Pa (see Appendices 1, 3); ~ttx and met are in molecule fixed axes. Following Section 8.1.1, the integrated circular difference absorption coefficient (see Eq. 2.5.8)) can be written as, A ~ - (8x2V0aN / 3c) (/)~ta //)Qa)(/)mtx / t)0a) 9

(8.2.2)

Expressing (O~ttx/OQa) in units of D/(A.amu 1/2) and (bma/~0a) in D.sec/(A.amu 1/2), this equation gives ANa in cm2/mol.sec. To use the z ~ in familiar units of kngmol, the above equation is divided by c, i.e., A.~ - (8~2V0aN / 3c 2 ) (t)~ttx / ~)Qa)(Oma / t)0a ) = (8/1;2V0aN / 3c) (t)~ta / ~)Qa)(/)ma / ~)0a ) ,

(8.2.3)

where V0a and V0a represent the peak position of the circular dichroism band a in Hz (or cyc/sec) and cm -1, respectively. Substituting the constants, one obtains the useful relation, (A.~ / V 0 a ) - 1063.2 (~)l.tct//)Qa)(/)ma / / ) 0 a ) ,

(8.2.4)

where the units are" AAa in km/mol, V0a in cm-1, (~)kta//)Qa) in D/(]k.amu 1/2) and (/)ma/O 0 a) in D.sec/(A.amul/2). The product (/)ktJ/)Qa)(/)mtx/~)0a) is more transparently written as

163

Vibrational Absorption and Circular Dichroism

(al~x/aQa)(amx/a0a) + (a~ty/aQa)(amy/a0a) + (al~z/aQa)(amz/a0a). The changes in electric dipole and magnetic dipole moments will occur along the same axis (x, y or z) only for chiral molecules (which do not possess plane of symmetry or inversion symmetry; see Chapter 10). For certain chiral molecules, one can find vibrations which have non-zero values for only one of the three components of the derivatives (~tcx/~Qa) and (~rn~/~(~ a). In such cases Eq. (8.2.4) will contain only one product term, say (iglLtz/~gQa)(igmz/ig0a) and the experimental VCD intensity data permits the determination of (Omz/igl~a), provided the information on (Ol.tz/OQa) can be obtained from vibrational absorption data (see Section 8.1.1) or other means. In the vibrational-rotational spectra the intensities of P, Q and R branches are determined by different components of these derivatives, so vibrational-rotational circular dichroism 20 is one unique source to derive the information on individual product components of Eq. (8.2.4). Since all of the three product components (Ol.ta/OQa)(OmoJO(~a) need not have the same sign, the P, Q and R branches of vibrational-rotational circular dichroism need not have same sign and this in fact has been observed 2~ for the enantiomers of methyloxirane. Just like the electric dipole moment derivatives, the magnetic dipole moment derivatives can also be reduced to the internal coordinate or cartesian coordinate space as,

(a

/a %) -

a

(amdaOa)(a Qa/a Rj))

=

-1 , (3moJig(~ a)Laj

(8.2.5)

fi

-1

B

-

a

(a~/aO a)(aQ a/a:~a 19= ~ (amojaOa)S a,Bl3 a

,

(8.2.6) where the L and S matrices are discussed in Chapter 5. The derivatives in cartesian space, (Oma/OX B[~), are referred to as atomic axial tensor (AAT) elements; the nine elements (o~ = x, y and z and [3 = x, y and z) for a given atom being the AAT for that atom. These derivatives in internal and cartesian coordinate space can be further related to the bond or atomic properties (see Section 8.2.3).

8.2.2 Quantum mechanical methods The evaluation of magnetic dipole moment derivatives (and hence VCD intenstivies) faced some problems in the initial stages of development. Different approaches are now available to overcome these

Chapter 8

164

problems, and these approaches are described below. It is convenient to separate the nuclear and electronic parts of magnetic dipole moment as, ma - manuc + mae l _ (1)EeZAE;o~[3~ ' XA[3J~A~' A

2c1 Eeca[~,Xi~Xi ~, i

(8.2.7)

where the summation index A runs over all atoms and i over all electrons. The derivatives of m~uc are straightforward to evaluate, so c)m~uc/~)XB~5 - ( 1 ) ] [ 2 eZAea[~ XA[~SAB~5~,~5 " A

(8.2.8)

The electronic part requires careful consideration. If we write the total wavefunction as a product of the electronic and vibrational wavefunctions (see Chapter 3, and Eq. 8.1.14), the electronic part of magnetic dipole transition moment integral, for vibrational transition from a~ to a~', in electronic state s, becomes 0

The integral

0

el

~s0 mael ~s0) is zero for molecules in non-degenerate

electronic states. A simple way to understand the reason for this is that the hermitian property requires 1(~0 reel/g0)l* = (/lt01mel 11101. Since m el is an imaginary operator, these integrals satisfy the hermiticity by yielding a zero value. As a consequence the numerical method (Section 8.1.2A) used for electric dipole moment derivatives cannot be used for magnetic dipole moment derivatives, (the electronic part of the magnetic dipole moment (~0 mel Xl/s O) calculated at displaced geometries is also identically zero). A second problem associated with the magnetic dipole moment derivatives is the origin dependence of the calculated rotational strength. Since rotational strength is an observable, it should be independent of which molecular origin one uses to calculate the electric and magnetic dipole moment derivatives. The objective then is to find ways that would restore the electronic contribution to the magnetic dipole moment derivatives, and that would yield origin independent rotational

165

Vibrational Absorption and Circular Dichroism

strengths. (A) Localized molecular orbital method 21 In the molecular orbital approach, Eq. (8.2.9) can be rewritten, for a closed shell system, as o o (~sWv'

o /Itso Iliad) - -2(e / 2c)(/lt O,

k

(8.2.10)

where q)k represents a doubly occupied molecular orbital. The instantaneous position of electrons in the kth molecular orbital, Xk~ is written as a sum of their average positions (or orbital centroid Xk0,~) and deviations AXk~ from Xk0,13. Furthermore J~ 1<7 can be replaced by a

)~kO,7 -

~t

~a

5qa

a

dla -

tko,yqa -

"

~3ko,yQa, (8.2.11)

a

where tk0,7 - OXk0,7 / Oqa and ~k0,7 - OXk0,y / OQa" With these two changes, Eq. (8.2.10) becomes, (,r o o

o o )-(-e/c) lits/lta~

a Ea~yXk0,[3Z tk0,y ( lltO' ]Cta[11tO) a

+(~0 i~., k (q~klea137AXk13Xky]~k)l ~0)] "

(8.2.12)

The second term on the right hand side of Eq. (8.2.12), referred to as the orbital rocking contribution, is ignored as small. This approximation can be justified for localized molecular orbitals for which AXk is expected to be small compared to Xk0. Then, ('r176

9 s0 ~ )0 - Z(~ a

where

'~

a

)(~O'lclal~O) ,

(8.2.13)

Chapter 8

166

~)m~ / c)dla - (-e / c)s

.

(8.2.14)

k In terms of the cartesian displacements, the total magnetic dipole moment derivatives are obtained from Eqs. (8.2.8) and (8.2.14) as,

(~mo~/~)J~BS)- ( e / 2 c ) [ ~ k ZA~ot13TXAI38ABSy8

(8.2.15) Using Eqs. (8.2.1), (8.2.11), (8.2.15), and the normal coordinate derivative of Eq.(8.1.9), the rotational strength is expressed in the LMO model as

l'a [

a ]

x Z A ea137 XAI3SAy,a - 2~ea137 Xk0,~Ok0,7 k

9

(8.2.16)

a

It is common practice to express Ra in esu 2 cm 2. With SAcx,a and Ok0,o~in units of 1/amul/2, XAI] and Xk0,13in ]k, substitution of the constants give a useful relation for Ra in esu 2 cm 2 as

Ra- 1230 x 10-44 X[~AZASAo~,a -

2~k~ ]

x[ZA~yXAI3SAy,a_ 2Zt~o~13y " kXk0,13o~0,y]

(8.2.17)

In eqs. (8.2.16) and (8.2.17), the summation over cx=x, y and z is implied (see Appendix 1). The LMO model treats the electrons in localized molecular orbitals, with positional vectors Xk0 and charge -e, similar to the way nuclei are treated. Any contribution from finite space associated with the LMO are ignored. As a result, the LMO model is not a first-principle method and

167

Vibrational Absorption and Circular Dichroism

possesses inherent limitations (although the rotational strength calculated from Eq. (8.2.16) is origin independent). Despite this limitation the LMO model, when used with ab initio wavefunctions, was found to reproduce 22 the observed VCD spectra qualitatively well. (B) Analytic magnetic field perturbation method7,23,24 The procedure for calculation of magnetic dipole moment derivatives in the analytic derivative method begins very much like the one described for electric dipole moment derivatives (see Section 8.1.2C). Starting with the perturbed wavefunction given by Eq. (8.1.16), the magnetic dipole transition from state saJ to s~' can be written analogous to Eq. (8.1.17). Following the modifications made to arrive at Eq. (8.1.18), the electronic contribution to the magnetic dipole transition becomes,

<~o~oIH~i~o~o><~oImp]vo)[a_EO/Eo]/E0n

el m~,v,v -

n:g:s

+ E( o o iH l o o,}( o Im ] o}[a + EO,/E~ n:g:s

(8.2.18) /

nl

,~11

N\

Since ma is imaginary, the hermiticity relation (Ws[m~t'iWn) 9

el

0

(~~ V ~ requires that (V ~ m(x lgn) - consequence, Eq. (8.2.18) becomes,

0

el

0

(~n ma ~ s ) .

As a

el a -- Eajaaj 0 a [(~Oalqa ~/0 a } d- (~/Oa Iqal~Oa )] m~,lJa~ • Z ( V 0 ]~)U/~)qalV0)(V0 ImPly0)/E0n 2

n:Cs

(8.2.19)

0 a - (Eva 0 - E 13 For aJ'a = 1Ja + 1, the energy difference Eaga~ ~ ) - hva. From the relations for integrals involving operators q and p (see Appendix 2 and 3), one can replace (VOa

i0V~a+1 qa ~ 0) With -1(0~ a + l

o)

Pa V~a

and

qa ~Oa+ 1) with +i(~0a Ipa ]q/1)a+10/. Furthermore, the summation /

term can be simplified by noting that the wavefunctions perturbed by (0H/0qa)qa and maBa can each be written using first order perturbation theory (see Eq. 2.2.12), i.e.,

168

Chapter 8

(c)Xl/s/c)qa)- E ( q t0 ~H / c)qa q/0)q/0/(E 0 _En)o '

(8.2.20)

n#s

(~Vs/~B~--~(V~Lm~lV~ ~

_Eo)

n~:s

= Z(v~ Im=lv~176 ~ E~

(8.2.21)

n~:s

Using these relations, Eq. (8.2.19) can be written for ~3a to ~3'a = X)a + 1 transition as, m~'agaa~a - -ihva[(qtOa PaIVO,/-(q/Oa [Palxl/Oa)]( ~)q/s / c)qal~s/~)B~ = 2hVa(q/O a IPalVOa)Im(~)~s / ~)qalc)q/s/c)gcz ) .

(8.2.22)

If we had used the normal coordinates Qa and momenta Pa, then ~)~s(~)q becomes o( 1/2 ~)q/s/~)Qand p becomes cz 1/2 (2r~/h)P (see Appendix 3). Then the derivative of the magnetic dipole moment can be written as follows:

(c)m~/c)Pa)- (h/rl;)Im(c)q/s/~Qal~Vs/OB=) = -(h/rt)Im(aVs/aBaI~)V s/~Qa)

(8.2.23)

It is more convenient to evaluate these derivatives in cartesian displacements, (3m~/3XA5 ) --(h/n)Im(3q/s/3Bcxl3~s/3XA~5),

(8.2.24)

el and convert them to normal coordinates using the relation, ~)mcc/~)(~a = X (3m.loq}(AS)SAS, el a. The total contribution to atomic axial tensors is A obtained by combining Eq. (8.2.8) and (8.2.24). As mentioned earlier, the derivatives O~s/OXA~i and O~s/OB~ determine the corrections to the

Vibrational Absorption and Circular Dichroism

169

0 wavefunction ~s in the presence of nuclear displacement and magnetic field perturbations respectively. They can be determined using the CPHF method as discussed in Section 7.2. The rotational strengths calculated using the magnetic dipole transition moment from Eq. (8.2.22) are gauge origin dependent, i.e. they depend on where the magnetic field is placed in calculating ~s/~Ba. A distributed gauge origin method has been proposed as one solution 25 to this problem. The rotational strengths predicted for CHFC1Br using this approach are listed in Table 2 of Chapter 7. A better approach 26 is to use the London atomic orbitals, also known as gauge invariant atomic orbitals. These orbitals overcome the gauge origin dependency problem with an added advantage that the calculated magnetic dipole moment derivatives converge faster with increasing basis set size. Eqs. (8.2.19) and (8.2.22) also suggest an approach for formulating the magnetic dipole moment operator expansion when anharmonic effects are to be considered. Note that <wU'--a0I~

III/~ a > _ <1~/~a0 [0 p a q[Ilia9 a a> =

and this relation holds for higher order operator products such as Paq~ (see Appendix 2). As a result, one can evaluate the anharmonic vibrational magnetic dipole transition moment, for ~a to ~ a transition, as <~Oalm]~0~a > when m is expanded as in Eq. (3.8.7). (C) Analytic nuclear electric shielding tensor method 27 The situation here is somewhat different from the one encountered for electric dipole moment derivatives. In the presence of a magnetic field the energy of interaction, Eint = - ma'Btx'. Expanding the magnetic dipole moment as a function of nuclear velocities, Eint can be written as Ein t - - ~ (~m a, / C}XAs):~AsBa,. A

(8.2.25)

The time dependence of XA~5and Ba' can be written respectively as X0,A5 cos cot and B0,a' cos cot and the sum J~ AsBa' + XASI3~' can be seen to be -mXO,ASBO,a' sin 2rot. In the static limit, m--,0 so it follows that J~ AsBa' - - XASBa'. Then Eq. (8.2.25) can be written as

Chapter 8

170

(8.2.26)

Eint - Z (t)mtx / t))( AS ) X As /3 oc', A

which suggests that potential energy derivatives with respect to nuclear displacements can be written as a function of 13a,. That is, (8.2.27)

(o~V ] O~XA(5) - (c~V ] O~XA(5)0 + (bm a, / c)]~AS)]3oc,+... ,

where (c)V/t)XAS)0 is the value of (c)V/t)XAS) at ]~oc' = 0. The electric field FA8 at nucleus A is equal to the force at that nucleus per unit charge, so Eq. (8.2.27) can be used to write the expansion of FA8 as, 7'10'27 (8.2.27a)

FAIl- FA8,0 + (0FAS/~)13c() ]3ct' + ... , where FA8,0 is the value of FA8 at Btx' = 0 and

(8.2.27b)

~)FAS/~)13ct' - - (1/eZA)~)rn~'/~)X A8.

We can write F A8 as the sum of nuclear and electronic contributions, Fnuc el respectively, as in the case of an electric field perturbation. A8 and FA8 Using the expressions for general molecular properties, obtained using the time dependent perturbation theory [see Eqs. (2.3.23)-(2.3.27)], the coefficient of B~, can be obtained. For the present case, the symbols containing P~ in Eqs. (2.3.23)-(2.3.27) are replaced by l~Ala. Then the .

coefficient of B~, is given as (3F~8/313ct, )- Kk,8o ~, - Im I - ~ 4rCFA8,snmct',ns /h(CO2s-o)2) k n~:s

where FA8,sn is given by Eq. (8.1.29). equation reduces to

1,

(8.2.28)

In the static limit the above

22]

K'A,8a, - Im[-(h / 11;)n#sZ4g2FA8,snma',ns / h tons

.

(8.2.29)

This equation can be written using Eqs. (2.2.12), (8.2.20) and (8.2.21), as

Vibrational Absorption and Circular Dichroism

171

Kk,8cx, - (-h / g)(1 / eZA)Im[((OVs / OXAs)I(OVs / 3Ba,)) ] = (h / re)(1 / eZA)Im[((~)Vs / 3Bo~,)l(~)gts / 3XA8)>] = -(1 / eZ A )(0m~' / 0J~A8 )"

(8.2.30)

The last equality follows from Eq. (8.2.24). From Eqs. (8.2.28) and (8.2.30), or from Eq. (8.2.27) it follows that, (~)F~5 / ~)13a,) - -(1/eZA)(~)m ~, / ~)XA~5) 9

(8.2.31)

The total contribution is obtained by adding Eq. (8.2.31) to the nuclear contribution. Thus the electric field at the nucleus, in the presence of the external dynamic magnetic field, is given as 7 FA8 - FAS,0 + [-(1 / 2c)eu138XAi3 + K~,Sa,]B a, .

(8.2.32)

The electronic contribution K~,sa, shields the nuclei from external field and therefore K~,sa, is called the nuclear electric shielding tensor. The relation between K~ and (Om/OXA), given by Eq. (8.2.30), allows the calculation of magnetic dipole moment derivatives, and hence VCD intensities, using nuclear shielding tensors. (D) Vibronic coupling method28,29 In Section 3.1, the vibrational equation was obtained by assuming that the nuclear kinetic energy operator does not influence the electronic wavefunction. This influence can be treated as a perturbation, so the nuclear kinetic energy perturbation is given, to first order, in dimensionless normal coordinates as

a

where ( 0~-%a/ elec operates only on the electronic wavefunction, and (~)

on nuclear wavefunction (see the discussion following Eq. nuc (3.1.5a)). Note that the nuclear wavefunction includes the vibrational

Chapter 8

172

wavefunction (see the discussion preceeding Eq. (3.1.9)). In writing Eq. (8.2.33) the second order term

elec

is ignored. The wavefunction

corrected to first order (see Eq. (8.1.16)) due to the perturbation given by Eq. (8.2.33) becomes,

0~sO\/ 0 t)~Oa\

~-~a/~ ltl/~a t)qa / 0 0 (V0 End--0~_EO~-~-0_ Es -~--OEaga ~/n~/aga'"

0 0 ~l/s~-- ~ts ~/~ + a~ hVan~s ~

(8.2.34) Assuming that (E~ EOa - E 0s - E 0~a)Can be replaced by (E0 _ Es), o the electronic part of the magnetic dipole transition moment can be obtained as follows. Ignoring the term that is second order in energy, and noting that ipa O/Oqa, one obtains the following equation for the ~a to v' transition: --

ma,~a~ e, a

a

_ +ihVa ~ [ (~01m~l~0) (Xl/0 3~t~ ng:s

E0 -E0

(W~

+ E~ ~

a ]Pal~Oa)

c)qa

O) (~0 t)~O\ (I[[/Oa IPa[VOa)]" (8.2.35)

qa/

On the right hand side of Eq. (8.2.35), the first term is zero; the second and third terms are equal (remember hermiticity and the integrals in Appendix 2). Thus, m~,ajaa~a -ihva[(~O~ IPal~Oa)-]

X '~

oel0(

(Vs-~--~ ImalVn) n~:s En-Es

~o

~)V0 (8.2.36)

With the use of Eq. (8.2.21), one can show that Eq. (8.2.36) is equal to Eq. (8.2.22). The vibronic coupling model emphasizes the use of Eq. (8.2.36), with an explicit evaluation of the summation over excited electronic states. The rotational strength calculated in the vibronic coupling method is

Vibrational Absorption and Circular Dichroism

173

origin dependent. To overcome this problem, a distributed gauge origin approach has been implemented. 30 (E) Localized orbital-local origin method 31 An elegant approach to the use of localized molecular orbitals in predicting rotational strengths is the localized orbital/local origin (LORG) method 31. The reader should consult the original work for more details on this method.

8.2.3 Classical models Unlike for molecular electric dipole moment derivatives, the consideration of molecular magnetic dipole moment derivatives requires the use of both atomic and bond charge concepts. In formulating these derivatives it is necessary to verify that the resulting molecular magnetic dipole moment derivatives lead to a rotational strength that is independent of molecular as well as bond origins. Some difficulties were encountered in the initial stages, but the formulation is now well understood. (A) Atomic charge concepts With the effective atomic charges the magnetic dipole moment is written as ma - (1/2c)~ ea[~, e~, XAI3X A~. (8.2.37) A In the fixed partial charge approximation, the derivatives of the molecular magnetic dipole moment can be obtained by straightforward differentiation. For example, the atomic axial tensors are given as c)mec/c)]~B8 - (1/2c)~ ec~13~,e~AXAI38AB878. The derivatives in internal A coordinate space ~ma/O l~j and normal coordinate space Ome~/~(~a, can be obtained similarly using the transformation matrices A and S (see Chapter 5). The incorporation of a charge flow contribution into Eq.(8.2.37) is not as straightforward. First we need to switch over to the bond concept to address this issue. (B) Bond charge and bond m o m e n t concepts 32-34 In order to develop an expression for the magnetic dipole moment with these concepts, it is helpful to consider a triatomic molecule A-B-C. Using the discussion presented earlier for bond charges (see Section 8.1.3B) and noting that e~B = -e(~+~C), Eq. (8.2.37) can be written for a molecule A-B-C, as

174

Chapter 8

mix = (e/2c) eal3Y[~AXA~ XAT+ ~BXB~XBy + ~cXcIIXCy) = (e/2c)

et~fiy[qA-B(XAI3XA? - XB~XB?) + qC-B(Xc~Xcy- XB[SXBy)].

(8.2.38)

For a general case, Eq. (8.2.38) becomes, mo~ = (e/2c) Z qk Z AkAetxl3)XAI3]~ Ay, k A

(8.2.39)

where AkA = + 1 when atom A is contained in the bond k and 0 when bond k does not contain atom A. To express Eq. (8.2.39) in terms of bond moments, let us relate the atomic coordinates to bond coordinates. From the definitions in the Fig. 8-1, it can be seen that the bond coordinate X k and the individual atomic coordinates XA (both with respect to a molecular origin) are related as XA = Xk + dA and dA - d B = dk; the vector dA is from a local bond origin (which can be anywhere along the bond) to atom A in that bond. Substituting these relations along with XA = X k + dA into Eq. (8.2.39) will lead us to an expression in terms of bond moments. For the sake of illustration let us consider only the part pertaining to bond A-B in Eq. (8.2.38). Then,

•

local origin for bond A-B

Dc molecular origin Fig. 8-1. Depiction of the molecular and local bond origins. mix= (e/2c)Eoc~iyqA-B[(dAit- dBIt) ]~ky + Xkl3 (dAy - dBy) + (dAI3ClAy- dalgday)] 9

(8.2.40)

Vibrational Absorption and Circular Dichroism

175

Note that the product eqA-B(dAI3- dBI]) is equal to the bond moment gA-B,~ and the product eqA-B(dA~- dB~) is equal (in the fixed partial charge approximation) to ~tA_B,I3. Since the same expression can be obtained for each bond in the molecule, the general expression can be written as 9

o

ma=(1/2c)2 Eo~y[(gk~ Xky +Xkl3 ~tk~,)+ eqkZ Z~kAdAI3aA~,]. k A

(8.2.41)

The term (1/2c)easy eqk~AkAdA~dA~, represents the local magnetic A

moment with respect to the bond origin. In the fixed partial charge approximation, gka is given as ka = eqk clka,

(8.2.42)

and in that approximation Eqs. (8.2.39) and (8.2.41) are equal to each other, so the molecular magnetic dipole moment derivatives are conveniently obtained by differentiating Eq. (8.2.39). Now, from Eq. (8.2.42) the charge flow contribution can be expected to have the form e/1 kdka; then the total contribution to g ka is ~t koc = e q k cl k a + e

dlkdko~

(8.2.43)

9

Therefore the magnetic dipole moment can be written, with charge flow contributions, in a computationally convenient form by adding the charge flow contribution to Eq. (8.2.39) as

mc~= (1/2c) Z eolian,[eqk~ AkA XA[3XA~,+ edlk Xkl3 dk~,] 9 k

(8.2.44)

A

The atomic axial tensor elements now can be written as ~)ma/~)X B~5= (1/2c)~ eoq3~,[eqk~ AkA XAI35AB578 k

A

+ e(0qk/0XB~5) Xkl3 dk,r .

(8.2.45)

The other derivatives, Omd~ll~j and ~)mc~/OQa can be obtained from Eq. (8.2.45) using the transformation matrices A and S (see Chapter 5). These expressions lead to a rotational strength that is independent of the choice

176

Chapter 8

for molecular as well as bond origins. One may transform the bond charge flow contribution in Eq. (8.2.43) into an atomic charge flow contribution, but the resulting expression would not be as simple and convenient as the one obtained with bond charges. Another alternative 35 is to write e~A as a sum of bond currents IAB and hence e~)~A/~)Qa as the sum of C)IAB/C)(~ a. (C) Ab initio bond charge concepts The bond charge parameters represent bond charges and their variations for a given change in the bond length or angle. These parameters are involved in the prediction of absorption and circular dichroism intensities using the bond charge (or bond moment) model. Some estimates for some of these parameters are available 36 in the literature, but they were not optimized to represent (or to reproduce) the vibrational intensities in a general class of molecular systems. Thus a set of generally applicable parameters are lacking at the present time. This deficiency can be overcome by combining the classical concepts and ab initio methods as follows. The molecular geometries and electric dipole moments for numerous molecules have been (or can be) obtained at very high levels of ab initio theories. 37 These data can be transformed in to bond charge parameters with very little extra effort, by noting that the changes in electronic structure from molecule to molecule reflect not only as changes in dipole moments, but also as changes in bond lengths and angles. The molecular dipole moment ~ta can be written in terms of bond charges using Eqs. (8.1.42) and (8.1.43). Expanding bond charges around a reference geometry, e q k - eqk,0 + (Oeqk/~)rk) Ark + ~ (~)eqk/30k) A0k + ... , k

(8.2.46)

where eqk,0 is the bond charge at the reference geometry (at bond length rk,0 and bond angle 0k,0); A r k - rk-rk,0 and A6k = 0k-0k,0 are changes from these reference geometries; the summation in the expansion given above is over angles that contain the bond k. In the above expansion one can also include first order derivatives with respect to neighboring bond lengths and angles that do not contain the bond k; also higher order derivatives can be included. However they can be considered after evaluating how well the first order approxiamtion given by the above expansion performs. Since each single bond C - X will have three angles surrounding it, there will be three parameters (~)eqk/~)0k) involving angles and one parameter (~9eqk/~)rk) involving the length, leading to a total of four parameters. These parameters can be easily optimized from the molecular geometries and dipole moments of a set of related molecules.

177

Vibrational Absorption and Circular Dichroism

The geometries and dipole moments needed here can be obtained at the converged level, using the ab initio methods, 37 perhaps employing electron correlation. Ideally, one should have as many molecules in the data set as possible, so that the resulting parameters are more generally applicable. With the development of a set of bond charge parameters in this manner, the capability of these parameters to reproduce the vibrational absorption intensities can be evaluated. Here the bond charge parameters in normal coordinate space, aeqk/3Q, can be obtained as .~ (aeqk/aR)(aR'/aQ)j j where Rj is jth internal coordinate. In the first ordeJr approximation, only those internal coordinates that represent bond length change Ark and angle changes A0k (involving the bond k under consideration) can be used in this summation; corresponding derivatives aRj/aQ can be obtained from the normal coordinate analysis (see Chapter 5). The need for including additional bond charge parameters can be identified by comparing the predicted spectra with experimental spectra. The development of such standard bond charge parameters is important for estimating the absorption and VCD in large size biomolecules (where ab initio calculations may not be feasible). These procedures would be similar to those involved in developing standard sets of force fields 38 for bio-molecules. Combining these data sets (force fields and bond charge parameters) it may become possible that the VCD spectra for bio-molecules could be predicted conveniently using a personal computer. 8.2.4 Sum rules As discussed in Section 8.2.1, the VCD intensity is related to the product (a~ta/aQa)(arn~/a(~ a). Following the transformations discussed in the context of absorption intensities [see Eq. (8.1.45)-(8.1.47)], one can verify that, 39

(a/-t(z / aQ a)(amcz / a(~ a) - ~ (a/tot / aXAa)(am(x / aJ~Aa) / m A a

A

- ~ (a/.t(z / apj)(amcz / alSj).

(8.2.47)

J

As before, the summation index "a" runs over (3N-6) vibrational normal modes. The rotranslational contributions to the magnetic dipole moment derivatives amJOlSj can be determined from the sum rules 7,18,39,40 associated with the atomic axial tensors ama/a X Aa:

ama / A

c)]~A~ - - e(x[$~, ~t~, /

2

,

(8.2.48)

Chapter 8

178

~_.,~,5 (~gma / ~RAI3)XA'y =(e/2mp)gasI88 .

(8.2.49)

A

In the above equations mp is the mass of proton, gas is the so called g tensor element. The electronic part of gas is related to the paramagnetic magnetizability Xc~8via the relation, el

g ~ - -(4mpm e /

e2

(8.2.50)

))(;~ / 188 ,

where me is the mass of electron.

In terms of the localized orbital

centroids Xk0, g ~ can be approximately writtenl8a as el

gas

_

-(mp

/

188

)~-, ( X 2 ~ k0~5r 8 - X k 0 , o c X k 0 , 8 ) k

(8.2.51)

,

where the summation index runs over all occupied molecular orbitals of a closed shell system. The nuclear part of gas is given as, (8.2.52)

g~C _ (mp / I~5)~., Z A (X28o~8 - XAaXAs). A

Using the translational sum rule given by Eq. (8.2.48), and the relation given by Eq. (5.2.17b), the translational contribution (3mc~/~15~) is obtained as

~) -

(~m a //)15

~ ( 3 m a //gXAs)(3XA8 /

o~p~)- -

~a~,~t~, / 2M 1/2 .

A

(8.2.53)

This contribution, however, is not necessary for VCD intensities, because Eq. (8.2.47) contains the product (~ta//gpt)(~gma//91St) and/9~oc/~9pt is zero for a neutral molecule. The rotational contribution 3ma/alS~ is obtained |

as,

(~ma / ~0~) - ~ ( ~ m a A

/

.

.112

~)(AS)(~XA8 / ~p~) - ( e / 2mp)ga~f~ff , (8.2.54)

Vibrational Absorption and Circular Dichroism

179

where the a matrix elements given by Eq. (5.2.17a), and Eq. (8.2.49) were used. Using Eqs. (8.1.48) and (8.2.54), we can rewrite Eq. (8.2.47) as Z (~)~a / ~)Qa)(bma / ~)(~a) - Z (~)~a / ~XAf)(~ma / ~):KA5) / mA a A -(e / 2mp) ~al3y gal3 g~, 9

(8.2.55)

The sum of rotational strengths (see Eq. (8.2.1)) is obtained by multiplying both sides of Eq. (8.2.55) with h/4~. The sum of integrated VCD intensities is given from Eqs. (8.2.3) and (8.2.55) as, A ~ /V0a - K { ~ (~)~ta / C)XA~5)(c)ma / c)]KA~5)/ mA a A

-(e / 2mp)(gxyl.t z - gxzl.ty + gyzgx - gyx~tz + gzxgy - gzygx)}

,

(8.2.56)

where K = 1063.2 with the units as defined for Eq. (8.2.4). A second sum rule 41 can be obtained by noting that Z (~)~a / ~)Qa)(~ma / ~)(~a) / 4r~2c2v2 - Z (~)~ta / ~)Ri)(~)ma / ~)l~j)Fij 1" a i,j (8.2.57) The magnetic dipole moment derivatives (/gma/Ol~j) for isotopically substituted molecules can be written, in a manner similar to the electric dipole moment derivatives (see Eq. 8.1.54), as (~9ma / c)l~j) 2 = (~9ma / ~)l~j)1

where the subscripts 1 and 2 represent the isotopically substituted molecules, and the summation i runs over six rotranslational coordinates. The 13 and A matrices were discussed in Chapter 5. As in the case of the integrated absorption intensities, the sum of integrated VCD intensities for isotopically substituted molecules can be related to that of the parent

180

Chapter 8

molecules using Eq. (8.2.3), (8.2.57) and (8.2.58). References 1 E.B. Wilson, J. C. Decius and P. C. Cross, Molecular Vibrations, McGraw Hill, New York (1955). 2 C. DiLauro and I. M. Mills, J. Mol. Spectrosc. 21 (1966) 386; J. L. Hylden and J. Overend, J. Phys. Chem. 87 (1983) 103. 3 W . B . Person and G. Zerbi, Vibrational Intensities in Infrared and Raman Spectroscopy, Elsevier, New York (1983). G. A. Segal, R. E. Bruns and W. B. Person, J. Chem. Phys. 50 (1969) 3811. 5 P. Pulay, Mol. Phys. 17 (1969) 197. 6 (a) A. Komornicki and J. W. Mclver, Jr., J. Chem. Phys. 70 (1979) 2014; (b) L. J. Schaad, C. S. Ewig, B. A. Hess, Jr. and D. Michalska, J. Chem. Phys. 83 (1985) 5348. 7 A.D. Buckingham, P. W. Fowler and P. A. Galwas, Chem. Phys. 112 (1987) 1. 8 Y. Yamaguchi, M. Frisch, J. Gaw, H. F. Schaefer III and J. S. B inkley, J. Chem. Phys. 84 (1986) 2262. 9 R.D. Amos, Chem. Phys. Lett 108 (1984) 185. 10 P. Lazzeretti and R. Zanasi, Chem. Phys. Lett. 112 (1984) 103; Phys. Rev. A27 (1983) 1301. 11 P . W . Atkins, Molecular Quantum Mechanics, Oxford University Press, Oxford (1983). 12 R.S. Mulliken, J. Chem. Phys. 23 (1955) 2343. 13 P.-O. Lowdin and B. Pullman, Molecular Orbitals in Chemistry, Physics and Biology, Academic Press (1964). 14 (a). S. KH. Samvelyan, V. T. Aleksanyan and B. V. Lokshin, J. Mol. Spectrosc. 48 (1973) 47; (b). J. C. Decius, J. Mol. Spectrosc. 57 (1975) 348. 15 L. A. Gribov, Intensity Theory of Infrared Spectra of Polyatomic Molecules, Consultants Bureau, New York (1964). 16 B. Crawford, Jr., J. Chem. Phys. 20 (1952) 977. 17 W. T. King, G. B. Mast and P. P. Blanchette, J. Chem. Phys. 56 (1972) 4440. 18 P.L. Polavarapu, Chem. Phys. Lett. 171 (1990) 271; P. J. Stephens, K. H. Jalkanen, R. D. Amos, P. Lazzeretti and R. Zanasi, J. Phys. Chem. 94 (1990) 1811. 19 G. Holzwarth, E. C. Hsu, H. S. Mosher, T. R. Faulkner and A. Moscowitz, J. Am. Chem. Soc. 96 (1974) 251. 20 P.L. Polavarapu, Chem. Phys. Lett. 161 (1989) 485. 21 L.A. Nafie and T. H. Walnut, Chem. Phys. Lett. 49, 441 (1977); T. H. Walnut and L. A. Nafie, J. Chem. Phys. 67 (1977) 1501. 22 P.L. Polavarapu and P. K. Bose, J. Chem. Phys. 93 (1990) 7524; J. Phys. Chem. 95 (1991) 1606.

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23 P. A. Galwas, Ph.D. Thesis, Cambridge University, Cambridge (1983). 24 P. J. Stephens, J. Phys. Chem. 89 (1985) 748. 25 P. J. Stephens, J. Phys. Chem. 91 (1987) 1712. 26 K. L. Bak, P. Jorgensen, T. Helgaker, K. Ruud, H. J. Aa. Hansen, J. Chem. Phys. 98 (1993) 8873. 27 K. L. C. Hunt and R. A. Harris, J. Chem. Phys. 94 (1991) 6995; P. Lazzeretti, M. Malagoli and R. Zanasi, Chem. Phys. Lett. 179 (1991) 297. 28 L. A. Nafie and T. B. Freedman, J. Phys. Chem. 78 (1983) 7108. 29 R. Dutler and A. Rauk, J. Am. Chem. Soc . . . . . . 989) 6957. 30 D. Yang and A. Rauk, J. Chem. Phys. 97 (1992) 6517. 31 A. E. Hansen, P. J. Stephens and T. D. Bouman, J. Phys. Chem. 95 ( 1991) 4255. 32 S. Abbate, L. Laux, J. Overend and A. Moscowitz, J. Chem. Phys. 75 (1981) 3161. 33 J. R. Escribano and L. D. Barron, Mol. Phys. 65 (1988) 327; J. R. Escribano, T. B. Freedman and L. A. Nafie, J. Phys. Chem. 91 (1987) 46. 34 P. L. Polavarapu, Vibrational Optical Activity, in: Vibrational Spectra and Structure, ed. H. D. Bist, J. R. Durig and J. F. Sullivan, Elsevier, Amsterdam (1989). 35 M. Mosokovits and A. Gohin, J. Phys. Chem. 86 (1982) 3947. 36 M. Gussoni, C. Castigloni, M. N. Ramos, R. Rui and G. Zerbi, J. Mol. Struct. 224 (1990) 445. 37 W. Hehre, L. Radom, P. V. R. Schleyer and J. A. Pople, Ab initio Molecular Orbital Thoery, John Wiley & Sons, New York (1985). 38 A. T. Hagler and C. S. Ewig, Comp. Phys. Comm. 84 (1994) 131. 39 P. L. Polavarapu, J. Chem. Phys. 84 (1986) 542. 40 A. Rupprecht, Mol. Phys. 63 (1988) 955. 41 P. L. Polavarapu, J. Chem. Phys. 87 (1987) 6775.

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183 Chapter 9 VIBRATIONAL RAMAN AND RAMAN O P T I C A L ACTIVITY

9.1 Vibrational Raman spectra The time dependent transition moments considered in Section 2.6 provide clues for the origin of dipole moment oscillations at frequencies greater or lower than the frequency of incident electromagnetic wave. Since an oscillating dipole can emit radiation at the frequency of its oscillation and serve as the source for scattered light, the wavelength of the scattered light can be larger or smaller than that of the incident light. Such scattering is referred to as Raman scattering. When the energy difference between the incident and scattered light components corresponds to the energy difference between molecular vibrational levels the process refers to vibrational Raman scattering. Vibrational Raman spectral intensities are related to the molecular electric dipole polarizability derivatives. Quantum mechanical methods to predict these derivatives have been developed, which permit the prediction of Raman spectral patterns and hence structural characterization. The molecular polarizability derivatives may in turn be related to the bond or atomic polarizabilities and their gradients, providing a pathway to obtain the information on local properties in a molecule. These different aspects are considered here. 9.1.1 Vibrational Raman scattering cross sections The vibrational Raman spectral intensities are represented as Raman scattering cross sections. To make a connection between these cross sections and molecular parameters, we will start with the transition polarizabilities given by Eq. (2.6.3). In the approximation that we can write the total wavefunction as a product of electronic, vibrational and rotational wavefunctions, (see Section 3.1), the wavefunctions for state s, f and n in (Eq. 2.6.3) can respectively be written as ll/el]/l)ll/J,/I/ell/ag'll/j' and ~e"~v"gtj", where we imposed the restriction that the initial and final vibrational levels v and v' and rotational states J and J' belong to the same electronic state. As long as the difference between the states II/elt/~ll/J and ~ e ~ ' ~ J ' is much smaller than the energy difference between ~e~v~J and ~e"~"~J", the frequencies COnsand COnfin Eq. (2.6.3) can be considered to be approximately equal and colf-o)2 can be replaced by co Zs-C02. Furthermore, if the energy difference between the states ll/e"/l/9"lltJ" and lllel[/~311/J is much larger than the energy of the incident radiation, i.e., 2).. can be replaced by 1/COns. This is referred to Olns>>O~, then COns/(O~2s-CO as the off-resonance situation. For the present purposes, the rotational

184

Chapter 9

wavefunction need not be explicitly considered, as the rotational substructure associated with vibrational bands will not be pursued. Instead, the rotational part of the integrals can be replaced with classical averaging for all molecular orientations (vide infra). With these considerations the transition polarizability given by Eq. (2.6.3) becomes < ~ , l t ~ , l g t ~ > , where o~aa' is the electric dipole polarizability given by Eq. (2.2.18) [in the present notation, s and n of Eq. (2.2.18) are e and e" respectively]. Writing txaa, as a Taylor series (see Eq. 3.10.1) expansion in dimensionless normal coordinates qa and retaining only the linear term in qa, the transition polarizability becomes [(va+l)/2] 1/2(/)ctatx'/~)qa) for the l)a to 1)a+l transition and (~a/2) 1/2 (/)t~txtx'//)qa) for the ~a to a~a-1 transition. The transition dipole moment given by Eq. (2.6.2) can then be seen to depend on ((~a+l)/2) 1/2 (/)txtxtx'/~)qa)et%avat( elt~ + e_io)t), which is 9

,

.

the source for vibrational Raman scattering. 1,2 The oscillating dipole is known to emit radiation at the frequency of oscillation, so the vibrational Raman scattering occurs at frequencies 03 + (OU'aVaand t.o- o>u'aVa; these are respectively called anti-Stokes and Stokes frequencies. The intensity of light emitted by a dipole oscillating at frequency v(= co/2rt) is given 3 as 2r~3cv4kt2 s i n 2 0 / r 2, where ILtzis the magnitude of the oscillating electric dipole moment oriented along the z-axis, ~? = v/c, 0 is the angle between the direction of dipole (z-axis) and the direction of observation (y-axis) and r is the distance from the point of observation to the location of the dipole. Total intensity of light scattered into area dA is obtained by multiplying the above mentioned intensity with dA. Defining dA/r 2 as the solid angle df~, the intensity scattered into unit solid angle then becomes 2rt3c~41a2 sin 2 0. For simplicity, we can take 0 = 90 ~ In the present case, the oscillating dipole moment is the transition dipole moment given by Eq. (2.6.2) so kt2Z can be replaced by ((aga+l)/2) (~)Ctzaq/)Qa)F2 for the ~atO t)a+ 1transition and (Va/2)(/)OtzaV/)Qa)F2 for ~a to aga-1 transition. Now F 2, can be converted into the intensity of incident light as (8r~/c)I0. The ratio of the intensity of scattered light per unit solid angle to the intensity of incident light is called the differential cross section and designated as d~/df~, where tj is the scattering cross section (scattered intensity divided by the incident intensity). The above considerations have to be augmented with spatial averaging of the polarizability components and the Boltzman distribution of population among different vibrational states. For spatial averaging the polarizability component ctaa' in space fixed axes can be written in

Vibrational Raman and Raman Optical Activity

185

terms of those in the molecule fixed axes as ctcttx' = t~BB' utx~ua'[5' where u is the transformation matrix between the two coordfr/ate systems. Then ct

2

, will contain products such as uo~u~fl'u~'~utx'fl' and the averages of

the elements can be obtained 4 through trigonometrical relations. Then the end result is that the spatial averaging leads to replacing (3txtxa'/3Qa) with (45~ 2 +4[32)/45 for ct= og and with ~2/15 for ct:k tx'; t2a and [32 are the mean and anisotropy (vide infra) of the polarizability derivative tensor (3ot/OQa). To consider the Boltzman distribution, the number of molecules in vibrational state a)a is given by Eq. (8.1.3). In the harmonic oscillator approximation, the energy difference between sucessive vibrational levels is the same, so the s u m ~ na ( ~ + 1), over all vibrational states X)a, gives 3 n(1 - e -hc~a/kT) for X)a to X)a+1 transitions and n(e +hc~a/kT 1) for ~3a to X)a1 transition; n is the total number of scattering molecules. Now we are ready to write down the Raman scattering cross sections, for different geometrical arrangements. Consider the incident light to be propagating along z axis (see Fig. 9-1), with y the scattering observation axis (so, 0 = 90 ~ and the sample placed at O. Then the incident electric field vector can be along x - o r y-axes and the induced dipole moment, that emits radiation in the y-direction, can be along z-axis or along x-axis. Let us consider the individual cases. x

Fx

i -z (a)

m ,~ m a~lR~

v

(b)

Fig. 9-1. Relative orientations of the incident and scattered electric vectors, in relation to the direction of propagation of incident light.

(a) Incident electric vector along x-axis (i.e., Fa, = Fx). Here the incident electric vector is perpendicular to the scattering plane. The intensity of scattered light with its electric vector along the x-axis (that is, scattered light with polarization perpendicular to the scattering plane) can only be originating from the induced dipole moment along x-axis, so l.ta = t-tx. Only the o~ 2 term contributes to the scattering in the y-direction. XX

The

Chapter 9

186

differential Raman cross section becomes 3,4 (after taking averaging and the Boltzman distribution into account),

spatial

(dr3a / d~) - 16~4(v -T-Va )4 (h / 8~2CVa) x [(45~ 2 + 4[~2) / 45] / [+1 -T-e-T-hcva/kT].

(9.1.1)

In Eq. (9.1.1), t~a and [32 are the mean and anisotropy of the polarizability derivative tensor/)t~aa'//)Qa;

1 (~axx 3ayy 3azz ]2 Ct~ = ~ ,, t)ea + ~gea + t)ea" '

=

(9.1.1a)

1 { (3axx

i~ayy~ (baxx 3azz~ + (, _ "~)Qa - ~ Q a " + ~ ~ a - ~Qa" ~)Qa ~ a " [(Oaxy]2 (~ax~2 (~axz]21 } . + 6 t,, OQa j + "OQa j + "OQa j j '

(9.1.1b)

h/4~;2Cga comes from converting qa to Qa and 9 is the wavenumber of the incident light. It is convenient to rewrite Eq. (9.1.1) as IX= K[(9 T- 9a)4/[ga(+l-Y e+-'hc~a/kT)llSa , X

(9.1.2)

where K = (16~4/45)(h/8~:2c), Sa = 45 a- 2 a + 4 [32a is called the Raman activity and Ix is the differential Raman cross section (or as commonly called Raman intensities) for the incident electric vector along the xdirection (superscript) and the scattered electric vector in the x-directon (subscript). Since the incident electric vector and scattered electric vector are parallel to each other (both perpendicular to the scattering plane), this scattering is referred to as polarized Raman scattering. The vibrational Raman bands will appear at Stokes and anti-Stokes frequencies x7 - Va and 9 + Va and are shifted from the incident light frequency 9 by the fundamental vibrational frequency. The ratios of vibrational Raman cross sections appearing at the Stokes and anti-Stokes frequencies can be seen from Eq. (9.1.2) to be

Vibrational Raman and Raman Optical Activity

I x (Stokes) I x (anti-Stokes) x

= (_~_9a)4 e+hc~?a/kT "

187

(9.1.3)

v + Va

Knowing the wavenumber 9 of the incident light, and Va from the Raman spectral positions, the above equation permits the determination of the sample temperature. (b) Incident electric vector perpendicular to the scattering plane (F~, - Fx) but the scattered electric vector parallel to the scattering plane. To have the scattered electric vector parallel to the scattering plane, the transition moment should be along the z-axis, i.e., g~ = gz. Then the a 2z x term contributes to the scattering, and the Raman scattering cross section becomes I x = K[(9 -T-9a)4/ga(+l-T- e+-'hcva/kT)](3 [32) .

(9.1.4)

The Raman activity Sa in this arrangement then is 3132. Since the scattered electric vector is orthogonal to the incident electric vector, this scattering is referred to as depolarized Raman scattering. (c) One may consider other relative orientations of incident and scattered electric vectors. For example, both Ixy and I yz also represent depolarized Raman scattering (remember that the superscript on I is for the direction of incident electric vector and subscript for scattered electric vector); I ux - I xx + I xZ ~ (u represents unpolarized)"~ ~ = I xX + I yX and so on.

From

the above discussion it can be seen that the ratio IX/I x = 3 z x - 2a + 7 132a). These ratios are called [321(45 ~- a2 + 4 ~a) and ~ /i ux = 6 132/(450~ depolarization ratios and can be used to identify the symmetry nature of a vibrational band. For vibrations that span the totally symmetric irreducible representation (see Chapter 10), d~a will be non-zero, so the depolarization ratio will be less than 3/4 or 6/7, depending on the type of scattered intensities measured. A more convenient way to analyze the polarization properties of Raman scattering is to evaluate the Stokes parameters associated with scattered light. The representation of polarized light and Stokes parameters are described in Appendix 4.

Chapter 9

188

9.1.2 Quantum mechanical methods To predict the vibrational Raman scattering cross sections we need to determine the mean and anisotropy of the polarizability derivative tensors. They can be calculated using different quantum mechanical approaches. (A) Numerical atomic displacement gradient method In the off-resonance situation, the electric dipole polarizability can be approximated by the static polarizability given by Eq. (2.2.18) [with s and n representing the ground and excited electronic states respectively]. In the near resonance situation, one needs to consider the frequency dependent polarizability given by Eq. (2.3.16). In either case, the polarizability expression (being dependent on the summation over electronic states) posed serious problems in determining accurate polarizabilities. Nevertheless, with suitable approximations to this sumover-states expression, polarizabilities have been evaluated. 5 Development of the CPHF method (Section 7.2.2) eliminated the need to explicity calculate the sum over all excited states. Time independent perturbation theory gives an expression for the correction to the wavefunction (resulting from a perturbation) in terms of the sum over excited states. Using Eq. (2.2.12), with H1 = -lal?,FI3,the derivative of the wavefunction with respect to Ffi becomes, o

-

o

Vn/(E s - En0).

(9.1.5) nCs These derivatives can be evaluated using the CPHF method, without the need for knowing the wavefunctions of excited states. Substitution of this equation into Eq. (2.2.18) gives (9.1.6)

otafl = 2 <~sl~tal3~s//)Ffl> .

Thus static electric dipole polarizabilities can be calculated without worrying about the excited electronic states using the CPHF method. Once the polarizability has been determined, the polarizability gradients can be obtained numerically by evaluating t~tx[~ at displaced geometries. Thus, -

=

+

XB~5) ,

(9.1.7)

where t~a13 + is the value of o~afl when coordinates XB~i of atom B is increased to X~ 5 and o~tfi is the corresponding value when XB6 is

Vibrational Raman and Raman Optical Activity

189

decreased to XB~5. Derivatives determined in this manner are not only time consuming but also suffer from numerical inaccuracy. (B) Numerical electric field gradient method In discussing the quantum mechanical methods for electric dipole moment derivatives, we mentioned that since the energy of interaction in the presence of an electric field is -kttxFa, the differential of energy with respect to Ftx gives -kt~. So, O~ct/OXB~5can be written as OtPBS/t)Ft~where t~B is the force on atom B (see Eq. 8.1.12). Since the induced electric dipole moment is given as t~ctl~Ffl, it becomes clear that the polarizability derivatives can be written as 6 (t)O~tx~/t)XB~i) = (~)2q~BS/t)FI3t)Foc) .

(9.1.8)

As the forces (PB8 can be calculated using analytic expressions, the polarizability derivatives can be determined by numerical differentiation of forces with respect to the electric field (which has to be incorporated into the hamiltonian in solving the Hartree-Fock equations). Better numerical accuracy is obtained by using both positive and negative electric field perturbations, so t)ctotl3/OXB8 was written for t~ r 13as7 t)O~tx~/~XB8 = [ q0B~5(o~Ffi) F+ + -

- + ) - q~B~5(otF~) F +q~BS(FotFi3

+ ~BS(F~F/3)]/(F +

(9.1.9)

+

Here Fa and F~ represent positive and negative electric field values and +

+

g~B~i(F~FI3), for example, represents the force on atom B in the presence of perturbations F + and F;. This approach is computationally more efficient than the numerical atomic displacement method discussed above, but still suffers from inaccuracy due to the numerical derivatives involved. (C) Combined atomic displacement-electric field gradient method An alternate approach 8 is to incorporate the electric field perturbation into the hamiltonian and calculate the electric dipole moment. Then the polarizability o~o~ is obtained as the numerical derivative of g~ with respect to FI3, i.e., =

(9.1.10)

190

Chapter 9

Repeating this calculation at displaced nuclear positions allows determination of ~)txtx~)XB~i using Eq. (9.1.7). This approach is computationally inefficient. (D) Analytic energy derivative method The development of the CPHF method for multiple perturbations (see Section 7 .2 2. and . . 7 2 3) permits the evaluation of ~9ota.~/I/)XB~5 using all analytic expressions. This approach, being most efficient, is that currently used in most quantum mechanical calculations. 9,10 The polarizability derivative is the third derivative of energy, i.e., J)tlot~t)XBi5 = (i93E/JgFct~)F~JgXB8) .

(9.1.11)

So taking the energy expression in terms of the molecular orbitals and differentiating it with respect to XBS, Ftx, and FI3 leads to the derivatives of molecular orbital coefficients, which can be determined with the CPHF method. (E) Analytic electric dipole moment derivative method The polarizability derivatives can be shown to be related to electric dipole moment derivatives as ~)t~a~/)XA~5 = (~)21.tJ~)XA~5~)FI3)- The expression for/)l.ta//)XA8 (see Eq. 8.1.25) can be differentiated 10 with respect to FI3 to obtain the analytic expression for 3t~ct~/OXA~5. The resulting expression contains the coefficient derivatives and these are determined using the CPHF method. The two analytic methods discussed above should give the same results. The ab initio predicted Raman activites and depolarization ratios for CHFC1Br are listed in Table 2 of Chapter 7.

9.1.3 Classicalmodels In much the same way as the molecular electric dipole moment is considered to arise from atomic or bond charges, the molecular electricdipole polarizability can be considered to arise from atom or bond polarizabilities. It is natural to view the isolated atom polarizabilities as being spherically symmetric. Then to explain the anisotropy of molecular polarizability one has to introduce interactions among these spherical atomic polarizabilities so that anisotropy, reflecting the local environment in a molecule, is introduced into the atomic polarizabilities. This is not generally required for bond polarizabilities, since anisotropy is naturally apparent for bond polarizabilities (polarizability along the bond axis is different form that along the perpendicular axes). This is a major distinction between the atom and bond polarizability models. The molecular polarizability derivatives derived from the experimental data

191

Vibrational Raman and Raman Optical Activity

may be interpreted in terms of these models. Alternately these models may be used to predict the molecular polarizability derivatives, and hence the Raman spectra. (A) Atom-dipole interaction (ADI) concepts In the presence of a uniform electric field Fct associated with the incident light, one may assume that an electric dipole moment is induced at each atom in the molecule. The effective electric field F~c t at an atom A, is then different from the incident field due to the presence of neighboring dipole moments. This effect can be summarized by the equation 11 F~cx = Fa - ~

TABa~ IXBfl .

(9.1.12)

B~A

In this equation, .[-tB~ is the induced electric dipole moment at atom B, TAB is the dipole Interaction tensor, 2 1 - 3 UAB,x

- 3 UAB,x UAB,y

- 3 UAB,x UAB,z

- 3 u AB,y u AB,x

1 - 3 UAB,y2

-- 3 UAB,y UAB,z

- 3 u AB,z u AB,x

- 3 u AB,z uAB,y

2 1 - 3 uAB,z

1

TAB = .

(9.1.13) 9

X B o c ) / R A B and R A B is the distance between atoms A and B. Using the effective fields, the induced moments at the atoms can be written as w h e r e UAB,oc = ( X A t x -

s

~

m

gAa = CXAFAo~ CXA[Fa Z

(9.1.14)

TABcx~l-tBI3] ,

B~:A

where tXA is the isolated atom polarizability. expression as,

One can rewrite this

(9.1.15) B~A

B

-1 5al~. Inversion of this equation gives, where AAAa~ = O~A

Chapter 9

192

I-tAa = ( ~ BABalS)F[5 , B

(9.1.16)

where B is the inverse of A matrix (these matrices are not same as the B and A matrices discussed in Chapter 5). Thus the polarizability of an atom after sampling the influence from the local environment in a molecule is given as, aAal3 - Z BABal3 , B

(9.1.17)

and the molecular polarizability as a~

- Z aAa~ A

(9.1.18)

9

It is important to note that the molecular polarizability obtained in this manner is symmetric, but the individual atom polarizability tensors IXActl3 need not be. The molecular polarizability derivatives ~gaa~gXA8 can. now.be seen to depend on the corresponding derivatives of the B matrix. Since the product BA gives the identity matrix, the derivative of B with respect to a given coordinate (cartesian, internal or normal coordinate) can be written as B' = -BA'B, where prime indicates the derivative. From Eq. (9.1.15), AA~ =-a p

a

(9.1.19)

~,

9

AABal 3 = TABotl3

(for A r B).

(9.1.20)

The derivatives of T can be obtained in a straightforward manner from Eq. (9.1.13). Eqs. (9.1.17)-(9.1.20) represent the atom-dipole interaction (ADI) model for Raman intensities. 11 In the ADI model, the determination of the molecular polarizability derivative ~gCXall/~XAi5requires a knowledge of atomic parameters IXA and bCtA/OXAIS. While etA (being one parameter for each atom) can be estimated, the derivatives OOtA/~XB8 are not easy to predict. If it is assumed that these derivatives O0tA/~gXB5 are zero, then the model represents a fixed atom polarizability hypothesis (similar to fixed partial charge model for absorption intensities).

Vibrational Raman and Raman Optical Activity

193

(B) Bond polarizability concepts Here bonds are associated with local polarizabilties. 12 By transforming these local bond polarizabilities into the molecular coordinate system, the molecular polarizability can be written as a sum of the transformed bond polarizabilities. That is, aaf~

=

~., aka~

(9.1.21)

,

k

(9.1.22)

OCkal3 =OCka'a' Ukaa' Ukl3a' ,

where (Zka'a' represents the diagonal element of the polarizability tensor of bond k in the local bond coordinate system; the subscript a' represents the local axes (oriented along and perpendicular to the bond direction); akal3 is the bond polarizability in the molecular coordinate system. For axially symmetric bonds, aka'a" for the two perpendicular directions of the bond are the same. Then aka[~ can be written as 12b, (9.1.23)

aka[3 = OCks5a~ + (OCkp - OCks)UkotUk[3 ,

where aks and CXkpare the local bond polarizability components (of kth bond), perpendicular and parallel to the bond axis respectively; Uk is the unit vector along the bond k (see below). Restricting ourselves to the axially symmetric bonds, the molecular polarizability derivatives with respect to cartesian displacement coordinates can be written as

c)(toc~/c)XA8 = Z ~)l~kcx~/c)XA8 , k

(9.1.24)

0akalff~)XA~5 = (C]aks/C]XAS)Sttl3 + (t)(akp- (~ks)/c]XAS)UkaUkl3 + (0~kp - ~ks)(Ak/rk){ Uk~(~r

Uk~xUkS) +Uka(5138-Uk~UkS) }.

(9.1.25)

As mentioned before, if bond k with length rk is made up of atoms B and C and Uk = (Xc - XB)/rk, then Ak = 1 for A = C,-1 for A = B and 0 for A representing any other atom. If the local bond polarizability derivatives are ignored in Eq. (9.1.25), then the resulting expression represents the fixed bond polarizability model. The molecular polarizability derivatives in internal and normal coordinates can be obtained by transforming O~xafgOX using A and S matrices (see Chapter 5) as for electric dipole moment derivatives.

194

Chapter 9

For molecules where there is only one totally symmetric stretching vibration, the experimental Raman intensity data permits determination of the absolute value of (~5~/0Q). This can be reduced to O6~/OR, where R represents a stretching internal coordinate. In a large number of molecules the magnitude of 0d~/OR is found to reflect the bond order, demonstrating a useful application 13 of Raman spectral data.

9.1.4 Sum rules The electric dipole-electric dipole polarizability oq~13does not depend on the origin of molecular coordinates. So the derivatives of o~al~ with ~)~~Opt are zero. From respect to translational normal coordinates pt,, / the relation between the cartesian displacement coordinates and translational normal coordinates (see Eq. 5.2.17a), it follows 14,15 that

~., o5~(x[3 / DXA7 - 0. A

(9.1.26)

This relation is referred to as the translational sum rule for ( ( ) ~ / 0 X A T) [the derivative tensors 0~/0XA, 0~/OYA, and 0~)ZA are referred 16 to as the atomic Raman tensors]. Similarly the change of a tensor element due to rotational motions is obtained 15 as,

0o~(~13 / 0pYr - (1 / I w )(Ea~zo~z[3 + el3yZotc~Z),

(9.1.27)

which follows from the relation for a general tensor T, AT = p x T - T x p. From the relation between cartesian displacement coordinates and rotational normal coordinates (see Eq. 5.2.17a), one obtains 14,15

~,~ (()~oc13 / 0XA8)E(sTeXA8 - Eo~YX~X~ + EI3TL~~ A

(9.1.28)

This relation is referred to as the rotational sum rule for (()~al3]()XAy). The polarizability derivatives with respect to cartesian displacement coordinates can be written 16 in terms of those with repsect to 3N-6 normal coordinates and six rotranslational coordinates (just as we have done for the dipole moment derivatives; for example see Eq. 8.1.47). Here it is useful to adopt the tensor summation notation (see Appendix 5), so the 1 anisotropy 13a 2 can be written as ~a = 2 [ 3 ( 0 a ~ / 0 e a ) ( 0 a ~ 1 3 / 0 e a ) -

195

Vibrational Raman and Raman Optical Activity

(3otcxdaQa)(OotBB/OQa)]. Then, following the procedure for dipole moment derivatives (see Eq. 8.1.47) one obtains, 2132 - Z 1 3 2 / m A - 2132 , a

A

(9.1.29)

j 1

where 132 = 132 Ax +132Ay +132 - Az and 13~x for example, is 132x -[3(O~cx~/OXA)(O~o~/OXA)- (O~cxoJOXA)(O~I3~/OXA)].The rotranslational contributions (summation over j terms in Eq. 9.1.29) depend on the equilibrium electric dipole polarizability and moments of inertia. They can be formulated in the principal coordinates of inertia as follows" Z ~ 2 - 3[(O~yy - ~ z z ) 2 / i x x J + (~XX- ~yy

*(~zz -~xx)2/Iyy

/ IZZ + ~2y(1 / IXX + 1 / I y y + 4 / IZZ)

+ IX2Z(1 / IXX + 4 / Iyy + 1/IZZ ) + CX2yz(4/Ixx + 1 / I y y + 1 / I z z ) l .

(9.1.30)

Similar expressions can be written for isotropic contributions -2 o~a = (~/3Qa) 2 but here the rotranslational contributions are zero, as the translational and rotational motions do not change the mean of c~a~. So o~a = A ~ ~/mA.

The atomic Raman parameters ~

and 13~ may

have interesting properties just like the effective atomic force constants that we discussed in Section 7.2.4. The sum rules expressions can be extended to the Raman activity by noting that the coefficients of -~a 2 and [32 a in the expression for Raman activity depend on the experimental arrangement (see Eq. 9.1.2). It is to be noted that the experimental Raman intensities depend on the exciting radiation frequency and vibrational frequency, so the above mentioned sum rules for -2cx a and 13a 2 do not apply to the Raman intensities (see Eq. 9.1.2) unless some approximations are made. The polarizability tensor derivatives transformed to internal coordinates provide another useful sum rule. The transformation (Eq. 5.2.28) from normal coordinates Qa to internal coordinates Rk, and the relation of inverse force constants (Eq. 5.2.45), provide the following

196

Chapter 9

relations" 17 ~132/4n2c2v2 - 2.

(3Cxcxl3'JCXcxl3'k- ctcxcx'JCX[313'k)Fj-l"

(9.1.31)

a

In these equations, cxcxl~j-," - Octcx~OR'j and similar relations, for other terms, with subscripts j and k representing the internal coordinates. The internal coordinate derivatives will differ for different isotopically substituted molecules and are related 18 through the relations that are analogous to the ones derived for electric dipole moment derivatives (for example, see Eq. (8.1.54)).

9.2 Vibrational Raman optical activity (VROA) spectra The Raman optical activities are determined 19 by the changes in molecular electric dipole - electric dipole polarizability. . cxcx,13 electric dipole - magnetic dipole polarizability G~f~, and electric dipole - electric quadrupole polarizability Acxl3~,,during vibrational motions. The changes in cxcx6during vibrational motions also determine the Raman activities as discus'sed in Section 9.1. The changes in G ~ and Aal3~, during molecular vibrations however are unique to vibrational Raman optical activity. In addition to providing these fundamental quantities, VROA also provides information on the absolute configurations and predominat conformations of chiral molecular systems.

9.2.1 Vibrational Raman optical activities Vibrational Raman optical activity measurements, 20 just as normal Raman measurements, can be carried out in different experimental geometries, namely 90 ~ right angle scattering, 180 ~ back scattering and 0 ~ forward scattering. Some measurements 2~ modulate the incident monochromatic laser light between right and left circular polarization states and monitor the synchronous difference in the Raman scattering from optically active samples of interest. This approach is referred to as the incident circular polarization (ICP) modulation method. In the ICP modulation method with 90 ~ scattering geometry the depolarized and polarized VROA measurements consist of determining the above mentioned synchronous difference for the scattered light with polarization parallel and perpendicular to the scattering plane (the yz plane in Fig. 9-1), respectively. The magic angle VROA measurement is similar to these two except that the scattered light with polarization vector at 35.2 ~ from the xaxis (which is perpendicular to the scattering plane; see Fig. 9-1) is monitored. In the ICP modulation method, the back and forward scattered VROA measurements are performed without an analyzer in the scattered

Vibrational Raman and Raman Optical Activity

197

light. When the intensity difference associated with right and left circular polarization states of the scattered Raman light are measured, with a fixed linear polarization state for the incident radiation, this approach is referred to as the scattered circular polarization (SCP) modulation method. 21 When the intensity difference associated with right and left circular polarization states of scattered Raman light, with in- (or out-of-) phase circular polarization states for the incident monochromatic radiation, is monitored this approach is referred to as the dual circular polarization DCP! (or DCPII) modulation method. 22 The Raman scattering cross sections with exciting light circularly polarized, can be formulated just as was done for ordinary Raman scattering (Section 9.1.1). But here it is convenient to use the Stokes parameters of scattered light for deriving the scattering cross sections, so this discussion is deferred to Appendix 6. The Raman optical activities can be designated as Pa (analogous to Raman activities Sa), which depend on the experimental arrangement. These activities are summarized in Table 1. The terms involved in these activities for the ath vibration have the following definitions:

2 1 ~~ ~O~yy..3G'xx ~9G'yy. ~O~xx ~9O~zz](OG'xx_~G'zz] Y a = 2 { ( ~ a -~Qa)("~a - 3Qa )+('3Qa - ~ a J ~ " ~ a 3Qa" -~yy ~O~zz~(~G'yy_ .~G'zz~ + ("~9Qa - ~9Qa"" ~Qa - ~9Qa~ [0~xy ~3'xy 0G'yx~ 3O~xz(3G'xz OG'zx) +3t~Qa (~Qa + ~Qa" +~Qa "~Qa + 0Qa" 0ayz .~3'yz ~G'zy~] }. + ~9Qa (~gQa + 0Qa ~'J '

(9.2.1)

2 1 co{ (OQa .Oayy - Oaxx~ Oazz~ ~)Ayzx 8a=~ OQa" OAzxy OQa + (.CkXxx ~ a -3Qa" OQa ,~9O~zz ~9~yy.~Axyz ~OCxy.~9Ayyz Azyy OAzxx OAxxz~ + ('~Qa- ~Qa ) ~)Qa + ~Qa (" ~Qa " ~Qa + ~9Qa - ff-Q~a) O0Cxz-OAyzz OAzzy OAxxy OAyxx+~gQa (~Qa - 3Qa + oqQa " 3Qa ) O~yz ,OAzzx OAxzz OAxyy OAyyx. + ohQa t o3Qa r + o~Qa o3Qa ) }; -

(9.2.2)

-

-' 1 (~O~xx OOCyy OO~zz) ,3G'xx ~G'yy ~G'zz) 6~aGa=9 "~Qa + ~gQa+ ~9Qa"( ~gQa + ~Qa + ~Qa ""

(9.2.3)

In the above equations, Qa is the normal coordinate for the ath vibration and r is the angular frequency of the exciting radiation. The definitions

Chapter 9

198

for ~2a

and ~2a are given by Eqs. (9.1. la) and (9.1. lb).

The normal

coordinate derivatives involved in these expressions can be transformed from the derivatives (ao~ai]/aXA~,), o]-l(aC~g,/0XA~) and (aAal]~aXas) using the coordinate transformation discussed in Chapter 5. The expressions given above for the tensor invariants can be written compactly in the tensor summation convention (see Appendix 5). For example, TABLE 1 Relevant Expressionsa for Raman and Raman optical activites i

measurement type

scattering geometry

Raman activity, Sa

Raman optical activity, Pa

i

L I~- Iz' IY- IY

90~

2401 ~ c

12132

IR R -I L L

90~

4[~&aGa - ' + ~-Ya 13 2 "~1 ~ ] ~401 c

2(45(X~+13~2)

90~

401 4[~_ - ' 7 2 1 ~ ] ~c &aGa+ ~ Ta +

2 ( 4 5 ~ + 7 ~ 2)

4001 3c

w 3

I~

[~" 31 -~]2

-

L x x I x, IR - IL

R I, -

I,, IR - IL

90 ~

L u u I~-I u, IR-I L

180~

9601 I'1 2 1 !.~ Ya + ~]

4145~ + 7132]

180~

4801 [~ 2 1 ~ ] c Ta +

24~2

0o

-,, 1601 45&aGa+ c ~Ya g

4145~+7~

0o

16Olc 45 &aGa + ~ Ta-

414 + a

iR -

L

*

*

L IL

L u u - Iu' IR- IL iR _ iLL

&aGa+

~ Ya

c

,

i

+2

a

i,

,,,

a. The definitions for the quantities involved are as follows: Superscripts on the symbol I identify the polarization state of the incident radiation; subscripts identify the same for scattered radiation. R: fight CPL; L" left CPL, *" magic angle; u: unpolarized; x,y,z: linear polarization along these axes with direction of incident light propagating along z-axis, and scattered light propagating along y-axis.

2 ~'a=

1/2

[3 ( a ~ c x l ] ) ( ~G~13 falXo~ ( ~ ) ] aea d e a ) - ~" a e a ) aQa

,

(9.2.4)

199

Vibrational Raman and Raman Optical Activity

-

~' ~)ea

~" c)Qa

(9.2.5)

9

2 was used in some An alternate definition 8a=3C0/2 eaT8(~O~a )kt'OA78~) OQa 2

literature papers and when that definition is used 8a in Table 1 should be replaced by 82/3. The experimental VROA intensities, just as Raman intensities (see Section 9.1.1), depend on the exciting frequency through the ~4 dependence for scattering, and the incident laser power. In addition, the instrumental response needs to be taken into account. To avoid these difficulties it is customary to express the experimental quantities as a normalized circular intensity differential (C1D) given as A = ( I ~ - ~ ) [ ( 1 7 + I~) ,

(9.2.6)

where the subscripts and superscripts identify the polarization state of scattered and incident radiation, respectively. The denominator (I~' + I~) II

depends on the mean and anisotropy of (Oc~a~/OQ) tensor, while the numerator (I~~' - 18~) depends on the corresponding invariants

of

inter-

ference products of (Oaaf~/OQ) and o~-l(OC~/OQ) and of (Oaa~/OQ) and (OAa.By/OQ) [see Table 1]. The expressions for A corresponding to different experimental arrangements are given in Appendix 6. As an example, for the 180 ~ back scattering arrangement A is given as

A =

I R- IL u u = iR u + i Lu

24 co ( ~ - l ~ a .

~-1~/3)

c(45 ~2a + 7 ~2a )

"

(9.2.7)

It is to be noted, however, that the CIDs are not usually reported, except for those bands that are free from the overlap of neighboring bands. For simulating the theoretical ROA spectra the ROA intensity is calculated with the expression, (I~ - I~) o~ .

(v 9 Va) 4 h p [ hcval8/~2CVa a ' +l-T-exp -T- kT

(9.2.8)

200

Chapter 9

and the corresponding Raman intensity is calculated by replacing Pa with the corresponding Sa. 9.2.2 Quantum mechanical methods From the above discussion, it becomes apparent that the evaluation of VROA intensities requires the knowledge of the derivatives (~o~a~)XAT), c0-1(3G~~XAT) and (~9Aa~/~)XA~5). The methods to determine the former are discussed in Section 9.1.2, and those for the latter two are presented here.

(A) Nuclear displacement method The expression for Ga~, given by Eq. (2.3.21), can be simplified in the non-resonance case (Ons>>O) to p

~-1 Gc~l3 - _ (h/g) Z [1/( E0- E0) 2] Im {<~sllaalll/~176176 n:g:s

} 9 (9.2.9)

For a perturbation H1 - - ml3BJ~, where BI3 is the external magnetic field component, one obtains from the first order perturbation theory (see Eq. 2.2.12)

3Vs/~gBI3= ~ n~s

m IV0> Vn01 E 0 E0 ~ n " 0

n

-

(9.2.10)

s

The analogous expression for ~lts/~)Fa (using H1 = - ~ F a ) is,

~ g~ Illts % 0 = '~"
(9.2.10a)

Taking the complex conjugate of Eq. (9.2.10a), multiplying with Eq. (9.2.10) and substituting in Eq. (9.2.9) gives, co-1 Gdf~ = -(h/n) Im { } 9

(9.2.11)

The derivatives (O~s/~)Fa) and (~)tlts/~)B~) can be obtained using the CPHF method (see section 7.2.2). So Eq. (9.2.11) provides an analytic approach 23 for evaluating o- 1 G ~, as the overlap integral of (O~s/OF~) and

201

Vibrational Raman and Raman Optical Activity (3~slaB~). The

0) -1

Gd~ determined in this manner have static limit p

values. The analytic methods for evaluating m-I(3G~/OXA~,), which are needed for determining the ROA intensities, are not yet developed. As a result these derivatives are at present obtained 24 numerically by computing o~-IG~I3 at the equilibrium and displaced geometries, in the static limit. The electric dipole-electric quadrupole polarizability A ~ , is given by Eq. (2.3.29a). Again in the situation when runs > m, this expression can be simplified 23 to t

Aoq3~, - 2 s

< ~l/sOIl't~ ~l/nO><~l/nOI013~'lvsO>

n#s

(9.2.12)

E n~ E s~

This equation can be reduced using the expression for (3~s/OFa) to, Aal3~, - 2 <

(aVs/aF )1013vl s>.

(9.2.13)

Here again (3~s/OFa) are obtained using the CPHF procedure. Analytic methods for evaluating the derivatives (3AaI3~/3XAS) are not yet available. As a result these derivatives are also obtained 24 numerically by computing Aa~y at the equilibrium and displaced geometries. Although the procedures for evaluating the m-lC~ and A~,/tensors were known 23 by 1982, their use for VROA predictions was not recognized at that time. The numerical displacement procedure presented here was first conceived and developed 24 by the author in 1987. The ab initio predicted ROA parameters for CHFC1Br are listed in Table 2 of Chapter 7. In practice, the values of m-lC~ calculated with limited basis sets are p

gauge origin dependent. Although the derivatives m-I(3G~I3/3XAY) were found 24 to have only a mild origin dependence, they can be calculated 25 without origin dependence when London orbitals are used. In addition, the frequency dependent values of m-I(OG~/OXA~,) can be determined 25 without having to restrict them to the static limit. (B) Orbital polarizability model The molecular electric dipole moment can be written in terms of the orbital centroids as discussed in Section 8.1.2 [see Eq. (8.1.9)]. Noting that the electric dipole polarizability aal3 can be derived from the molecular

202

Chapter 9

dipole moments evaluated as a function of the applied electric field [see Eq. (9.1.10)], it can be seen that (xtxl3can be written as a sum of the orbital polarizabilites. For a closed shell system with paired electrons, the electric dipole polarizability can be obained as, (tall = (/)gtx/~)FfI) = - 2e Ek(~)Xk0,tx/~)FfI) = Ek CtktxB 9

(9.2.14)

Here.. O~k(xf) is the polarizability associated with the orbital k and it is implied that each orbital k has two electrons with oppoiste spins, and the summation index in Eq. (9.2.14) runs over all occupied orbitals. t The G~B and A ~ y tensors can be written7, 26 in terms of these orbital polarizabilites ~ k ~ (see Appendix 7). The derivatives of C ~ and A~B7 with respect to the cartesian displacement coordinates would contain the derivatives (OXk0,dOXAS) and (Ol~k~g/OXAS) which indicate respectively how the orbital centroids and the orbital polarizabilities change for a given nuclear displacement. These quantities can be determined by evaluating the Xk0,~ and ak~B at displaced geometries 7 Alternately, from Eqs. (8.1.9) and i8.1.12) it can be seen that k

(t)Xk0,dc)XAS) = -

(1/2e)(/)~8/i)F(x) where (I)~8 is the electronic

el8 contribution to the force at atom A. If it is assumed that the forces t~A can be written as a sum of the independent contributions from the orbitals, i.e. (h~8 = E k (hkAS, then one can obtain (c)Xk0,ot/c)XA8) = -

(1/2e)

(O~kA~OF~). Similarly, from Eq. (9.2.14) given above, it can be seen that (})O~kot~/c)XAS) = (32t~kA6/OF(xOF~). It is to be remembered that the approximation mentioned above, i.e. (h~8 = Ev (hkAS, is not strictly valid, and the reliability of this approximation has not been established.

9. 2.3 Classical models (A) Atom and bond polarizability concepts The atom and bond polarizability concepts used for Raman activities (see Section 9.1.3) can be extended to the present case. In all these cases the molecular electric dipole polarizability is expressed as a sum of the polarizabilities of the constituent units (units being atoms or bonds). These constituent polarizabilities can then used to write down the expressions for G(~ and A~lly (see Appendix 7). The polarizabilities of the atoms and bonds can be derived from Eq. (9.1.17) and Eq. (9.1.23), using the atom dipole interaction and bond polarizability models respectively.

Vibrational Raman and Raman Optical Activity

203

The derivatives of C ~ and Act[37 with respect to the cartesian displacement coordinates, which can be formulated by differentiating the expressions for G~fl and A~flT, contain the differentials of atom or bond polarizabilities, which are the same as those discussed in connection with Raman activity [see Eqs. (9.1.19-9.1.25)]. (B) Ab initio bond polarizability concepts The main problem in the practical implementation of the atom and bond polarizability concepts is the presence of a large number of atom or bond parameters required. Without making severe approximations it is not feasible to use these methods for predicting or interpreting the VROA spectra. On the other hand, the quantum mechanical calculations are so computer intensive that it is not yet feasible to apply the methods discussed earlier to large size bio-molecules. In fact, an ideal situation would be the one where the results obtained for smaller molecules can provide interpretation of the experimental data for larger molecules (without having to undertake ab initio calculations on larger molecules). This is particularly relevant for vibrational Raman and ROA spectra, which are time consuming to calculate reliably even for small molecules. This section describes a scheme to develop a set of standard ab initio local bond polarizability parameters that can be used for predicting the Raman and ROA spectra of a general molecule. In the bond polarizability hypothesis, the molecular electric dipole polarizability tensor t~tx[~can be written as a sum of the bond polarizability tensors t~k~ [see Eq. (9.1.21)] which (in the molecular coordinate system) can be obtained from the local bond axes via Eq. (9.1.22). For single bonds such as C-X (X-H, N, F and O), it is reasonable to assume cylindrical symmetry so one can define the bond polarizability in terms of only two parameters (local bond polarizabilities t~ks and t~kp perpendicular and parallel to the kth bond axis; see Eq. (9.1.23)). The molecular geometries and polarizabilites. ~[3 can be calculated ab initio at very high levels of theory (see section 9.1.2A). Using these geometries and polarizabilities for a series of related molecules, the data can be fit to the local bond polarizabilites Ctks and ~kp and the bond polarizability derivatives, by expanding Ctks and ~kp around a reference geometry. For illustration purposes, let us consider hydrocarbons CH4, C2H6, C3H8 etc., with only single valence bonds. Each C-H bond would have three valence angles9 0-J (that contain the C-H under consideration) so the expansion mentioned above should contain that many valence angles. Considering the Ctks component (k=C-H in this case), the desired expansion to a first order approximation is

204

Chapter 9

aks = Ctks,O+ (t)Ctks/t)rk)(rk-rkO) + E(C)~ks/C)Oj) (Oj-Ojo) J + higher derivatives ,

(9.2.15)

where the summation index j is over angles 0, with j = 1 to 3; ~ks,0 is the bond polarizability at reference geometry (bond length rk0 and bond angles 0jo etc). If we take the C - H bond length and tetrahedral bond angles of CH4 to represent the reference geometry, then the local bond polarizability Ctks in C2H6 (for example) would be modified according to the above equation. This, in a way, is equivalent to saying that a change in the electronic structure among related molecules would reflect as corresponding changes in bond lengths and bond angles; thus the corresponding changes in the bond polarizabilites are accounted for by the expansion given in Eq. (9.2.15). There are five unknown parameters [aks,0; (/)aks//)rk); and (/)t~s//)0j), j = 1-3] for k = C - H in the above equation. Similarly, there are another five parameters for the parallel component t~kp. Analogous parameters can be defined for the C-C bond in saturated hydrocarbons. Although certain symmetry aspects can reduce the number of parameters, we will not elaborate on this aspect for space considerations. The optimization of the above mentioned parameters would require that geometries and polarizabilites would need to be evaluated accurately for a series of molecules. In the higher order approximations, one would include the second derivatives (for example, terms such as c)2~ks/~rk2) and also the first derivatives with respect to the changes in adjacent bond lengths (for example, C-C bonds and other C-H bonds) and adjacent angles in the above expansion. But that would increase the number of parameters to be determined. The same procedure can be extended to other series of molecules (containing different types of bonds, namely C=C, C-H at sp2 hybridized carbon etc.). Once the bond polarizability parameters [Ctks,0, ~kp,0, (c)~ks/~)rk), (/)t~k~rk), (~)0~ks//)0j) and (/)tXkp/~)0j)] have been optimized for different chemical bonds, these parameters can be used for predicting the Raman and ROA intensities. Raman intensities are determined by the invariants of (/)o~afl//)Q) tensor where Q is the normal coordinate for a vibration of interest and these derivatives can be obtained by differentiating Eq. (9.1.22). The bond polarizability derivatives in normal coordinates (/)tXks//)Q) can be determined from the above discussed library parameters (OtXks/Ork) and (c)O~ks/t)0j) using the transformation, (t) ffks/t)Q) = ~ . (/)aks/ORj)(/)Rj//)Q) where Rj is jth internal coordinate; in the first order J approximation only the internal coordinates representing the bond length

Vibrational Raman and Raman Optical Activity

205

and angles of bond k (rk and 0j as discussed above) will be contributing to this summation; (0Rj/0Q) are determined (as the familiar L matrix elements discussed in Chapter 5) from the normal coordinate analysis. VROA intensities would require, in addition, m-I(OG~/0Q) and ( O A ~ 0 Q ) . These tensor derivatives can also be obtained from the bond polarizability library parameters, discussed above, by noting that m - l G ~ and Aal3~, can be written in terms of the bond polarizabilities (see Appendix 7). Procedure to differentiate these expressions with respect to normal coordinate Q, and writing the resulting expressions, using Eq. (9.2.19), in terms of library parameters (0aks/0r) and (0O~ks/O0j) are identical to the procedure just described for molecular derivatives (0o~a~/0e). 9.2.4 Sum rules The polarizability derivative tensors used in the formulation of vibrational Raman optical activities are written in terms of vibrational normal coordinates. These tensors and those in terms of other (cartesian displacement, translational, rotational and internal valence) coordinates are interrelated and possess interesting properties that lead to useful sum rules. The sum rules applicable for electric dipole polarizability tensor derivatives were discussed in Section 9.1.4. Those applicable for electric dipole - magnetic dipole and electric dipole - electric quadrupole polarizability tensors are described in this section. For the c0-1G~ and Aa~, tensors, equations analogous to Eq. (9.1.26) can be written and from the origin dependence 27 of co-IG~[~and A~[3~, it can be shown 15 that (9.2.16)

Z{O-I~)G~xl3 /ohXA T _ _ 1 8~yg~Zor A 3

(9.2.17)

A

These two equations are called translational sum rules for {0-1 (0G(x~/0XA~,) and (0A(~f)yI~)XAS) respectively. To obtain the rotational sum rules for o-I(0G~/OXA~) and (0Ao~OXA8), the same procedure that was used for (Oa~/0XA~,) can be used. These are given 15 as, p

p

~(0G~f) / OXAs) 88~XA8 - 8o~Gz[3 + 8 ~ G ~ , A

(9.2.18)

Chapter 9

206

Z (OAa~ / OXAs)(~8~XA8 = eaTzAzf)~ + ~yzAaz~ + E ~ A a f ) x ' A (9.2.19) The sum rules for the anistropies can be derived, again following the procedure for the anisotropy of electric-dipole polarizability derivatives [see Eq. (9.1.29)], as 17 2

Ey 2 - Ey2/mA-EYj a

A

A

(9.2.20)

2,

(9.2.21)

j

E82 - E82/mA-ZSj a

,

j

2 1 where y2 A - Y2Ax+ y2Ay+ y2Azand y2Axfor example is YAx - ~ [3(~aa~/~XA) 2 is (~)G~[~/C)XA)- (~)0~aa/C)XA) (c)G~[3/C)XA)]; the x components of 8Ax r 82x =~(c)IXa[3 / c)Xn)~a~,8(c)Ay8~ / C)Xn). The rotranslational contributions (summation over j terms in Eqs. (9.2.20), (9.2.21)) can be formulated in the principal coordinates of inertia in a manner similar to that used for Eq. (9.1.30):

2 E 7j - 3[(0~yy - (~ZZ)(G~/y - G~y,) / Ixx J + (~zz - a x x )(Gzz - Gy~X) / I y y + (aXX - a y y ) ( G ~ o r - G~(y ) / I z z ] 3

+ -[O~xy(G~( Y + G~,X)(1 / Ixx + 1 / I y y + 4 / Izz) 2 + a x z ( G ~ z + G~x)(1 / Ixx + 4 / I y y + 1 / Izz) + a y z ( G ~ z + G~y)(4 / Ixx + 111yy + 1 / Izz)] 2 E 8j - C0[(0tyy - 0~Zz)(Azx Y - 2 A x y z + A y z x) / I x x J + (aZZ - 0~Xx)(Axy z - 2 A y z x + AZX Y ) / I y y + ((XXX - 0~yy)(Ayz x - 2 A z x Y + A x y z) / Izz]

(9.2.22)

Vibrational Raman and Raman Optical Activity

207

+---[0~Xy ( A z x X - AXZ X + A y z y - A z y Y) 2

(1 / I x x + 1 / I y y + 4 / I z z ) + axz(AxYx - Azy z + Ayzz - Ayx x) (1 / I x x + 4 / I y y + 1 / I z z ) + a YZ (A z x z - A YXY + A xYY - A x z z ) (4 / I x x + 1 / I y y + 1 / Izz)] .

(9.2.23)

Similar expressions can be written for the isotropic contributions Raft;a = (Oa/OQa)(OCJ'/OQa), but here the rotranslational contributions are zero as the translational and rotational motions do not change the mean of a tensor. The sum rules for polarizability tensor derivatives with respect to internal coordinates provide another useful sum rule. 17 Following Eq. (9.1.31) one can obtain,

'Z

272/4rt2c2V2 - 2.

,

(30~aI3'jGal3'k -~

,

'

(9.2.24)

a

~_Sa2 / 4g2C2Va2 -- 2m. j,~kgaYSaa~'JA75[3'kFj-kl '

(9.2.25)

a

In these equations, a a ~ j - , "- O a a ~ R ' j (and similar relations, for other terms), with subscripts j and k representing the internal coordinates. The internal coordinate derivatives will differ for different isotopically substituted molecules and are related through the relations similar to those given for dipole momemt derivatives (see Eq. 8.1.54). Some important observations 28 emerge from these sum rules. The sum of VROA activities (for all 3N-6 vibrational modes) would be nonzero for those chiral molecules which do not have a spherically symmetric electric dipole polarizability (for example, CHFC1Br molecule). The sum rules predict that rotational motions will lead to non-zero ROA for chiral symmetric top and asymmetric top molecules. In fact, Barron and Johnston 29 independently showed that rotational ROA is supported by chiral symmetric top molecules. Similarly the sum rules for VCD intensities (see Eq. (8.2.47)) lead to the observation 28 that rotational circular dichroism is supported by chiral asymmetric top molecules. Finally, the translational sum rules given by Eq. (9.2.16) and (8.2.48) indicate a connection between VCD and VROA. From these two

208

Chapter 9

equations one can verify that ~o)-l(c)G~xl3 / ~)XAy) =~(/9(/)m~ / ~)XA~,)/ ~)Fct). (9.2.26) A A Based on Eq. (9.2.26) it is tempting to suggest that co-1 (3G~/~)XA~,) can be determined from the electric field dependence of atomic axial tensors, ~9m13/ ~XA~/. References 1 C.V. Raman and K. S. Krishnan, Nature 121 (1928) 501. 2 G. Placzek, The Rayleigh and Raman Scattering, UCRL Translation No. 526(L), translated by Ann Werbin (1959). D. A. Long, Raman Spectroscopy, McGraw Hill, London (1977). E. B. Wilson, J. C. Decius and P. C. Cross, Molecular Vibrations, McGraw Hill, New York (1955). 5 T.C. Jao, N. H. F. Beebe, W. B. Person and J. R. Sabin, Chem. Phys. Lett. 26 (1974) 474. 6 A. Komornicki and J. W. Mclver, Jr., J. Chem. Phys. 70 (1979) 2014. 7 P.L. Polavarapu, J. Chem. Phys. 77 (1982) 2273. 8 W. Kolos and L. Wolniewicz, J. Chem. Phys. 46 (1967) 1426; J. E. Gready, G. B. Bacskay, N. S. Hush, Chem. Phys. 31 (1978) 467. 9 M.J. Frisch, Y. Yamaguchi, J. F. Gaw, H. F. Schaefer III and J. S. Binkley, J. Chem. Phys. 84 (1986) 531. 10 R.D. Amos, Chem. Phys. Lett. 124 (1986) 376. 11 J. Applequist and C. O. Quicksall, J. Chem. Phys. 66 (1977) 3455. 12 (a). M. Eliashevich and M. Wolkenstein, J. Phys. 9 (1945) 101; L. D. Barron and B. P. Clark, Mol. Phys. 46 (1982) 839. 13 B. Fontal and T. G. Spiro, Spectrochim. Acta, 33A (1977) 507. 14 A.D. Buckingham in: Recent Trends in Raman Spectroscopy, Eds. S. B. Banerjee and S. S. Jha, World Scientific, Singapore (1989). 15 P.L. Polavarapu, Chem. Phys. Lett. 174 (1990) 511. 16 P.L. Polavarapu, J. Mol. Spectrosc. 93 (1982) 450. 17 P.L. Polavarapu, Chem. Phys. Lett. 139 (1987) 558. 18 B. Crawford, Jr., J. Chem. Phys. 20 (1952) 977. 19 L.D. Barron and A. D. Buckingham, Mol. Phys. 20, 111 (1971). 20 L.D. Barron, M. P. Bogaard and A. D. Buckingham, J. Am. Chem. Soc. 95 (1973) 603. 21 K. M. Spencer, T. B. Freedman and L. A. Nafie, Chem. Phys. Lett. 149 (1988) 367. 22 L. A. Nafie and T. B. Freedman, Chem. Phys. Lett. 154 (1989) 260. 23 R. D. Amos, Chem. Phys. Lett. 87 (1982) 23. 24 P. L. Polavarapu, J. Phys. Chem. 94 (1990) 8106. 25 T. Helgaker, K. Ruud, K. L. Bak, P. Jorgensen and J. Olsen, Farad.

Vibrational Raman and Raman Optical Activity

209

Discuss. 99 (1994) 165. 26 L. A. Nafie and T. B. Freedman, J. Chem. Phys. 75 (1981) 4847. 27 A. D. Buckingham and H. C. Longet-Higgins, Mol. Phys. 14 (1968) 63. 28 P. L. Polavarapu, J. Chem. Phys. 86 (1987) 1136. 29 L. D. Barron and C. J. Johnston, J. Raman Spectrosc. 16 (1985) 208.

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211

Chapter 10 APPLICATIONS OF SYMMETRY

10.1 Introduction The presence of symmetry in a molecule greatly reduces the effort in predicting the nature of some molecular properties. Although visual inspection can reveal the presence (or lack) of symmetry in a given molecule, a systematic approach to characterize that symmetry would be desirable. The theory of point groups provides one such approach. A point group is a collection of symmetry operations that satisfy, as a group, certain characteristics and leave a point in the molecule unaffected. Based on the type and number of symmetry operations present, a molecule can be classified to belong to a certain point group. The properties of that point group can then be used to deduce the nature of molecular properties.

10.2 Symmetry operations and point groups A symmetry operation is an act upon whose execution the resulting molecular appearance is indistinguishable from the one before that operation was executed. As an example, consider a non-planar A2B2 molecule (for example H202) with A-B-B-A dihedral angle greater than 0 ~ and less than 180 ~ An act of 180 ~ rotation around the z-axis (see Fig. 101) interchanges the two A atoms and the two B atoms. But the molecular appearance is unaffected by this rotation. So this rotation is said to be a symmetry operation. The axis around which this symmetry operation was undertaken is said to be a symmetry element. According to the widely used convention, this symmetry operation is labelled C2 (rotation by 360~ and the axis of symmetry is referred to as the C2-axis. In general, a Cn operation refers to rotation by 360~ around the Cn-axis. In addition to the rotational symmetry, a given molecule may have other types of symmetry, represented by the following symmetry operations. (a) Reflection symmetry: if the reflection plane contains the principal axis of rotation, as in a planar A2B2 molecule, then the reflection in that plane is designated as the symmetry operation 6v (see Fig. 10-1). The reflection plane itself represents the Cv symmetry element. On the otherhand, if the reflection plane is perpendicular to the principal axis of rotation then the reflection operation is designated as Crh. A dihedral reflection plane ~d bisects two C2 axes that lie perpendicular to the principal axis of rotation. (b) Inversion symmetry" In this symmetry operation an atom with coordinates (x, y, z) is transformed into an equivalent one with -x, -y, -z coordinates. This symmetry operation is designated as i. (c) Improper rotation symmetry" This operation

212

Chapter 10

constitutes a rotation of 360~ around Cn axis followed by a reflection in a plane perpendicular to this Cn axis. This operation is designated as Sn.

C2 | , :

A~

:,q) c3

I~V '

,

A2 .t

At

/

oC3

A2

'

+ IjV

.,.B 2 ~v

B!

(c)

(b)

(a)

~e

B3 B2

'

C3

.~v'

Bl (a')

(c')

~ ~ ~ v

Fig. 10-1. Structures showing symmetry elements in C2,C2v and C3v point groups. (a) Non-planar A2B2 belonging to C2 point group; (a') Newman projection of the structure in (a); (b) planar A2B2 molecule belonging to C2v point group; (c) pyrimidal AB3 molecule belonging to C3v point group; (c') top view (looking down the pyramid) of the structure in (c).

Regardless of the number or type of symmetry elements (or operations) a given molecule possesses, every molecule is said to have an identity element (or operation), E. This symmetry operation is no more than doing nothing. That is, every atom in a given molecule transforms into itself upon identity operation. Although may seem trivial, this operation is needed for a collection of symmetry operations to satisfy certain criteria (so that this collection can be classified as a point group), as follows. (a) The product of two symmetry operations (say X and Y of a given point group), is also a symmetry operation of that point group. Here, the product XY simply means operation Y followed by X; and the appearance of the molecule obtained after these consecutive operations could have been obtained by a single symmetry operation of the same group. The product XY need not be same as YX. (b) The symmetry operations of a point group follow the associative law. i.e., X(YZ) = (XY)Z. (c) The inverse of each symmetry operation of a point group is

213

Applications of Symmetry

also a symmetry operation of that point group. The point group to which a molecule belongs is determined by the collection of symmetry operations that a molecule can be subjected to (or the collection of symmetry elements that a molecule has). For example, the molecules which have no symmetry, except the identity, belong to the C1 point group as this group contains E as the only symmetry operation.

10.3 Representations Consider a non-planar A2B2 type molecule (Fig. 10-1), which has E and C2 symmetry elements. The E and C2 symmetry operations comprise the C2 point group, so non-planar A2B2 type molecules shown in Fig. 10-1 are said to belong to this point group. The effect of symmetry operations can be investigated for any molecular property of interest; let us take the two A-B bond lengths r l and r2 as the property basis for this investigation. Under the identity operation r l transforms into itself, as does r2; the C2 operation transforms rl to r2 and vice versa. These observations can be represented as follows.

E (rl,r2)- (rl,r2)[~ ~]- b Db(E) ,

(10.3.1)

C2 (rl,r2)- (rl,r2)[~

(10.3.2)

~1 - b Db(C2).

Here b is the basis vector representating the basis (rl, r2); Db(E) and D b ( C 2 ) are two dimensional matrix representations of the symmetry operations E and C2, respectively, in the (rl, r2) basis. Because of the interconversion of r l and r2 in the C2 operation, the basis (r l, r2) cannot be written individually for r! and r2. In otherwords the two dimensional matrix representations Db(E) and D b(C2) cannot be written as onedimensional matrix representations in the same basis. However, if we had started with symmetrized basis, it would have been possible to obtain onedimensional matrix representations as follows. Suppose that the linear combinations rl+r2 - Sl and rl-r2 - s2 are used as the basis. Then one finds,

E (s l, s2) = (s1, s2)

C2 (s 1, $2) = (s1, $2)

= Sl [1] + S2 [1] = Sl Dsl(E) + S2 Ds2(E),

O]

= S1 [1] + $2 [-1]

(10.3.3)

214

Chapter 10 = Sl Dsl(C2) + s2 Ds2(C2) 9

(10.3.4)

Note that the two-dimensional basis (s 1, S2) can be written individually as one-dimensional ones (Sl) and (s2) because these properties (Sl and s2) are not "mixed" by any of the two symmetry operations. As a result the twodimensional matrix representations in the (s l, s2) basis can be written as one-dimensional matrix representations in the Sl basis and the s2 basis individually. A one-dimensional matrix representation, cannot be reduced any further and therefore it is referred to as an irreducible representation. The trace of a matrix is designated as its character. For one-dimensional matrix representations Dsl a n d Ds2 there is only one number each associated with them, so these numbers provide a character representation. TABLE 1 Character representations of the C2 point group 9

E

C2

I basis ,,

rl

1

1

r2

1

-1

I I

s1I z I s 2 x,y I I

In the present case, Dsl(E) = 1 and Dsl(C2) = 1, so the characters { 1, 1 } form an irreducible representation (labelled F1) of E and C2 operations (in the S l basis). Similarly, Ds2(E) = 1 and O s 2 ( C 2 ) - -1, so the characters {1,-1} form a second irreducible representations (labelled F2) of E and C2 operations (in the s2 basis). The arrangement of these characters in a tabular form provides a character table, as in Table 1. The basis Sl is said to span (or belong to) irreducible representation Ul, while s2 spans U2. The irreducible representations are also referred to as symmetry species. A given irreducible representation may be generated by more than one basis; alternately, several molecular properties may span a given irreducible representation. One could have chosen the atomic orbitals, molecular dipole moment or some other property as the basis (in place of symmetrized bond lengths discussed earlier) to generate the irreducible representations given above. A common choice is to use the cartesian coordinates x, y and z of a point in the molecule. If we had chosen (x, y, z) as the basis, then we would have obtained:

215

Applications of Symmetry 1

0

E (x,y,z) = (x, y, z)

0

1 01

= =x [1] + y [1] +z[1]

(10.3.5)

= x Dx(E)+ y Dy(E)+ z Dz(E), -I C2(x, y, z) = (x,y,z)

0

0

-I

0 = x [-1] + y [-1] + z [1]

0

1 (10.3.6)

= x Dx(C2) + y Dy(C2)+ z Dz(C2) 9

Since there are no off-diagonal elements, the 3x3 matrix representations can be reduced to three one-dimensional representations. The characters of Dz(E) and Dz(C2) are same as those obtained for s 1 basis, so both z and Sl span the same irreducible representation F1. Similarly one finds that x and y span the same irreducible representation F2 (as s2 does). Let us now consider planar A2B2 type molecules with A-B-B-A dihedral angle of 0 ~ These molecules possess E, C2, Gv(ZX) and Ov' (yz) symmetry elements, which comprise the point group C2v. Here Crv(ZX) and ~v' (yz) are respectively the reflections in the xz and yz planes (z axis coinciding with the C2 axis). The procedure for finding the irreducible representations here is same as that discussed earlier for the C2 point group. If we use (Sl,S2) as the basis ( s l - r l + r 2 and s2=rl-r2 as used earlier), then two irreducible representations (labelled F1 and F2) as shown in Table 2 would be obtained. TABLE 2 Some of the character representations of the C2v point group

lq F3

E

C2

Ov (zx) CYv(YZ)[

1

1

1

1

Is 1

1

-1

1

-1

I s21

1

-1

-1

1

I ,

basis

I I

z x

I

Y

216

Chapter 10

But if we use (x,y,z) coordinates of a point as the basis then three irreducible representations (F1, F2, F3; see the Table above) would be obtained. Since the number of irreducible representations obtained depended here on the basis chosen, one has to wonder if there is another basis that would have identified a larger number of irreducible representations. Alternately, what is the maximum number of irreducible representations for a given point group? To answer this question, and find other properties of point groups, let us consider pyrimidal AB3 type molecules. The symmetry elements I

I!

present here, E, C3 +, C3-, Crv, CYv,Cv (see Fig. 10-1) comprise the C3v point group. The rotation operations C3 + and C3- are said to be conjugate and the conjugate operations comprise a class. The conjugacy results from the relation, C3-= Crv-lC3+~v which means that the operations C3 + and C3- are related through another symmetry operation (CYv)of the same point group. The product Cyv-lC3+cv can be depicted by sequential operations CYv,C3 +, CYv-1 as shown in Fig. 10-2. A

A

A

A Ov - 1

B3

132

B3

1

B!'I~2

1

Fig.10-2. Depiction of the effect of CYv,C3 +, CYv-1 and C3- operations on pyramidal AB3 type molecules.

In general terms, two symmetry operations X and X' of a point group belong to one class if another symmetry operation Y of the same point group transforms so that X'=Y-1Xy or YX =XY. The three reflection operations CJv, Cv and ~v can be identified to belong to another class. To find the representations let us consider the basis (sl,s2,s3), where sl=rl+r2+r3, s2-2rl-r2-r3 and s3=r2-r3. For C3 + and C3operations we obtain: !

I!

1

C3 + (s l, s2, s3) - (s l, s2, s3)

0

0

-1/2

-1/2

3/2

-1/2

217

Applications of Symmetry s1

-1/2

[11 + (sz's3) L 3 / 2

-1/2] -1/2

=Sl Dsl(C3 +) + (sz,s3) Ds2,s3(C3+), 1

0

C3- (sl, s2, s3) - (sl, s2, s3) 0

-1/2

0

-3/2

= Sl [11 + (s2,s3)

/2

(10.3.7)

0 1/2 -1/2 1/2] -1/2

=Sl Dsl(C3") + (s2,s3) Ds2,s3(C3") 9

(10.3.8)

The matrix representations Ds2,s3 are now two-dimensional and cannot be reduced any further (as s2 and s3 are mixed among themselves under C3 + and C3-). So the original three-dimensional basis (Sl,S2,S3) is now decomposed into a one-dimensional basis (s l) and a two-dimensional basis (s2,s3). The corresponding matrix representations for the remaining symmetry operations can be obtained as follows: O'v (Sl,S2,S3)= Sl [1] + (s2, s3)I10

?1] =s 1Dsl(Ov ) +(s2,s3) Ds2,s3(O'v), (10.3.9)

Ov' (Sl,S2,S3) - Sl [1] + (s2, s3)

!2

1! 2 J

= S1 Dsl(Ov') + ($2,s3) Ds2,s3(Ov'),

(10.3.10)

r_l/21/~] crv,,(Sl,S2,S3) - Sl (1) + (s2, s3) L 3 / 2 1 / =Sl Dsl(Ov") + (s2,s3) Os2,s3((Yv"),

E (Sl,S2,S3) - Sl [11 + (s2, s3) [~

(10.3.11)

] -" s1 Dsl(E ) + (s2,s3) Ds2,s3(E). (10.3.12)

The characters of matrix representations are obtained by summing the diagonal elements. Then one finds that the characters for symmetry operations in a given class are the same. For this reason, the symmetry

218

Chapter 10

operations in a given class are grouped together in presenting the character table (Table 3). That is, C3 + and C3- operations are collectively written as |

II

2C3; Ov, ~v and ~v are written as 3Ov. The same character representation, as that obtained with (Sl,S2,S3), would have been obtained with (x,y,z) as the basis.; Sl and z basis span F1; (s2,s3) and (x,y) span F2, which originated from the characters of two dimensional irreducible matrix representation. TABLE 3 Some of the character representations of the C3v point group E

2C 3

q

1

1

r2

2

-1

3Crvl

basis

1 I Sll z I I 0 I s2's31 x,y

Let us now return to the question, that we came across earlier, on the maximum number of irreducible representations possible for a point group. To find the remaining (if any), and maximum number of, irreducible representations one can make use of five different properties 1,2 of point groups. (1). The number of irreducible representations of a point group is equal to the number of classes in that point group. For the C2 point group, there are two operations (E,C2), each of which is a class in itself. So there can only be two irreducible representations which we already identified (see Table 1) for non-planar A2B2 type molecules, using the (s 1,s2) or (x,y,z) basis. The C2v point group has four operations; I

E, C2, ~v, and Ov, each of which is a class by itself. So there should be a maximum of four irreducible representations for this point group. For planar A 2 B 2 type molecules (with A-B-B-A dihedral angle of 0 ~ belonging to the C2v point group, we identified (see Table 2) three of these four irreducible representations using the (x,y,z) basis. For the C3v point group, there are three classes of operations, so there should be three irreducible representations. Two of them have been identified (see Table 3) using (Sl,S2,S3) as the basis for AB3 type pyramidal molecules. For both C2v and C3v point groups we need to identify one remaining irreducible representation. (2). The sum of the squares of the dimensions of irreducible representations in a point group is equal to the order (which is equal to the number of symmetry elements) of that group. The order of the C2v point group is four and the three irreducible matrix representations, identified earlier, for this group are all one-dimensional.

Applications of Symmetry

219

So the remaining representation must be one-dimensional. In the case of the C3v point group, the order is 6. The two irreducible representations identified earlier have dimensions of 1 and 2. Since the sum of the squares of their dimensions is 5 (=12 + 22), the remaining representation must be one-dimensional. (3). The irreducible character representations of a point group are orthogonal to each other. This property can be verified for the irreducible representations identified already. For example, in the C3v case the two identified representations (Table 3) are orthogonal because, 1 x2+2x 1 x(-1)+3x 1 x0-0. (4). The sum of the squares of the characters in any irreducible representation of a point group is equal to the order of that group. This property again can be verified for the irreducible representations already identified. For example, for the F2 representation of C3v point group identified earlier, the sums of the squares of the characters gives 22 + 2 x (-1)2 + 3(0)2 - 6. (5). The symmetry operations of a given class have the same characters. This property has been noticed earlier for AB3 type pyramidal molecules belonging to C3v point group. The properties (3) - (5) can be used to identify the characters of remaining irreducible representations, which are: 1-'4={ 1,1,-1,-1 } for the C2v point group and F3={ 1,1,-1 } for the C3v point group. The proof for the five properties mentioned above can be found in standard books 1,2 on group theory. In the previous discussion, the irreducible representations were labelled F1, F2 etc. However, a more widely accepted designation 1,2 is as follows. A one-dimensional irreducible representation is designated as either A (if the character of principal rotation is + 1) or B (if the character of principal rotation is -1). The two representations F1 and F2 of the C2 point group (Table 1) are designated A and B respectively. When more than one such representation is present, they are distinguished by adding subscripts (for example A1, A2... or B1, B2... etc.). Thus the four irreducible representations F1, F2 F3 and F4 of the C2v point group discussed above are designated respectively A1, B1, B2 and A2. Two dimensional irreducible representations are labelled E (and three dimensional ones are labelled T) with subscripts 1,2,... when necessary. So the three irreducible representaions F1, F2 and F3 of the C3v point group discussed above are designated A1, E and A2 respectively. A subscript g (or u) is added when the character of the inversion operation is +1 (or -1). Sometimes a prime (or a double prime) is added when the character of the CYhoperation is + 1 (or - 1). When the procedures for generating the irreducible representations, described earlier, are applied to point groups such as C3 one can gain

220

Chapter 10

some additional insight. Each of the three symmetry operations (E, C3 + and C3-) of the C3 point group is in a class by itself. Then the properties of point groups described earlier suggest that there should be three onedimensional irreducible representations. Using the (x,y,z) basis, as in earlier cases, we obtain Eqs. (10.3.13) and (10.3.14) with 0 = 2rt/3 = 120 ~ It is apparent that x and y are mixed among themselves under C3 + and C3operations. Then we can only obtain one two-dimensional representation (in the x,y basis) and a one-dimensional representation (in z basis). But the two-dimensional representation must be reducible to two onedimensional representations (otherwise the properties of point groups I cos0 C3+(x,y,z) = (x,y,z) - s ; n 0

cosO C3-(x,y,z)-(x,y,z)[si;0

sin0

0 (10.3.13)

cos0001 '

-sinO 0 (10.3.14)

cos0001'

would be violated). This reduction can be obtatined in a basis that has linear combinations of x and y. Using ((x+iy), (x-iy), z) as the new basis one finds:

0 000 e_i001 0 '

e i0

C3+(x+iy, x-iy, z) =(x+iy, x-iy, z)

C3-(x+iy, x-iy, z) - (x+iy, x-iy, z)

I

e -i0

0

0

0

e i0

0

0

0

1

(10.3.15)

(10.3.16)

Since x+iy, x-iy, and z do not mix with each other in any of the three operations, the three dimensional representation can now be seen to be reducible to one-dimensional representations. Then, the resulting character table is shown in Table 4. The last two rows ( F2 and F3) of Table 4 are sometimes added together (thereby getting characters 2, 2

Applications of Symmetry

221

cos0, 2 cos0) and designated as a two-dimensional representation of the (x,y) basis. For the Cn point groups (n=1,2,3,...), referred to as the cyclic groups, it is not necessary to go through a tedious task of finding a basis (as was done for C3 earlier) that leads to irreducible representations. Certain interesting properties 1 make it easy to find the irreducible representations for these point groups. A Cn point group has n symmetry operations given as Cn,Cn2,Cn3...Cn n. Since Cn k refers to a rotation of TABLE 4 Character representations of the C3 point group H

,

H

z

ri r2 r3

i

basis

I

z

1

1

1

1

ei0

e-i0 I x+iy

1

e-i0

e i0 I

I I

I x-iy I

!

k0 (where 0 =2n/n), Cn n can be seen to be equal to E. Each of these operations is in a class by itself, so there should be n irreducible representations. For the a th irreducible representation, where 0
10.4.1 Fundamental vibrations and symmetry species Let us consider non-planar A2B2 molecules, as an example, and address the following question: How many of the (3N-6) fundamental vibrations, N being the number of atoms in the molecule, belong to a given symmetry species of the point group to which these molecules belong ? There are six fundamental vibrations for A2B2 molecules (with N= 4). In

222

Chapter 10

a simple picture, these vibrations can be represented by the changes in two A-B bond lengths, the two A-B-B bond angles, B-B bond length and dihedral angle A-B-B-A. These six independent internal coordinates, designated respectively as Arl, Ar2, Aotl, Aot2, AR, and At: can be considered as the basis for a matrix representations of the symmetry operations. Then, Eb=b Db(E) and C2b=b Db(C2), where b is a vector of six internal coordinates and Db(E) and Db(C2) are given as:

Db(E) -

--

1

0

0

0

0

0-

0

1 0

0

0

0

0

0

0

0

0

1

0

0

0

1

0

0

0

0

0

0

1

0

0

0

0

0

0

1

-0

;

and Db(C2)=

1 0

0

0

0-

1 0

0

0

0

0

0

0

0

1

0

0

0

0

1

0

0

0

0

0

0

0

1

0

0

0

0

0

0

1

_

The characters of Db(E) and D b(C2) are respectively 6 and 2. The character representation obtained from the symmetry operations in the internal coordinate basis, {6,2 }, is a sum of the character representations of six independent vibrations (vide infra) and therefore is reducible. The number of times an irreducible representation appears in the above mentioned reducible representation is determined as n 7 - (1 / h ) ~ ~red~irr, j J 7 9

(10.4.1)

J

In this equation, h is the order of the group, ZJred is the character of J is the reducible representation for jth symmetry operation, ~;irr,7

corresponding character in 3'th irreducible representation. character table (Table 1), one obtains:

From the

nA - ( 1 / 2 ) [ 6 x l + 2 x l ] = 4 ,

(10.4.2)

nB - (1/2)[6 x 1+ 2 x -1] = 2.

(10.4.3)

In other words, the reducible representation {6,2} in the internal coordinate basis is composed of four vibrations that belong to A symmetry species and two vibrations that belong to B symmetry species. Obviously, the next question to ask is which are the vibrations that belong to A vs B symmetry species. In the internal coordinate basis considered, AR and Ax

223

Applications of Symmetry

transform into themselves, under both E and C2 operations, so they belong to A symmetry species. Among the remaining four internal coordinates, Arl is transformed to Ar2 (and vice versa) and ACtl is transformed to Atx2 (and vice versa) under the C2 operation. So

Arl and Ar2 (or AtXl and

Act2) cannot be assigned by themselves to any symmetry species. Here one needs to form symmetry coordinates (or linear combinations of the internal coordinates.) The approach to determine the symmetry coordinates is as follows. If we take Rk as the generating coordinate, the symmetry coordinates are obtained as Sqt = ~ Xirr,), j O j (Rk), (10.4.4) J where S~, is the symmetry coordinate belonging ~h symmetry species, and oJ (Rk) is the jth symmetry operation on the generating coordinate Rk. With At1 as the generating coordinate one obtains S1-SA=lxE(Arl)+lxC2(Arl) S2=SB=lxE(Arl)+(-1)xC2(Arl)

-

Arl + = Arl -

Ar2,

(10.4.5)

Ar2.

(10.4.6)

If we had taken Ar2 as the generating coordinate, we would have obtained (Arl + Ar2) and (Ar2 - Arl) as the symmetry coordinates which are equivalent to S 1 and $2 (except for a phase difference in $2). Similarily choosing Atx 1 as the generating coordinate one would obtain $3 - (Actl + Ate2) and $4 = (AtXl - Atx2) as the symmetry coordinates that belong respectively to A and B symmetry species. After generating the symmetry coordinates, we could have formed (S 1,$2,$3,$4, AR and Aq;) as our new basis and investigated the matrix representations of symmetry operations. Since these coordinates will not be mixed by the E or C2 operations, the six dimensional representation can be broken down into six one-dimensional representations. Then one finds that S 1, $3, AR and Ax form, individually, a basis for A representation; $2 and $4 form (individually) a basis for B representation. From the above discussion it is apparent that are the four vibrational coordinates that belong to the and $2 and $4 are the two vibrational coordinates symmetry species. The procedures discussed above depend

S 1, $3, AR, and A'c A symmetry species; that belong to the B on generating (or

224

Chapter 10

defining) (3N-6) independent symmetry (or internal) coordinates. If the intention is to find only the number of vibrations that belong to a symmetry species (and not their vibrational origin), a more general approach can be formulated from atomic cartesian displacements as the basis. Here one assigns arbitrary displacements x, y, z for each atom in the molecule and these 3N displacements are considered as the basis. It should be noted that the 3N atomic cartesian displacements are related not only to the 3N-6 fundamental vibrational coordinates, but also to the six external coordinates (3 rotations, 3 translations) as discussed in Chapter 5. Thus the cartesian displacements basis provides information on all 3N nuclear (3N-6 vibrational, 3 rotational, and 3 translational) motions. Proceeding with non-planar A2B2 molecules, as an example, one notices that in the C2 operation the displacements of atom A1 get rotated to atom A2 (and vice versa). Similarly, the displacements of B1 and B2 are interchanged in the C2 operation. In matrix representation, C2b bDb(C2), where

Db(C2) =

-0

0

0

-1

0

0

0

0

0

0

0

0-

0

0

0

0

-1

0

0

0

0

0

0

0

0

0

0

0

0

1

0

0

0

0

0

0

-1

0

0

0

0

0

0

0

0

0

0

0

0

-1

0

0

0

0

0

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

-1

0

0

0

0

0

0

0

0

0

0

0

0

-1

0

0

0

0

0

0

0

0

0

0

0

0

1

0

0

0

0

0

0

-1

0

0

0

0

0

0

0

0

0

0

0

0

-1

0

0

0

0

0

0

0

0

0

0

0

0

1

0

0

0

and b is a vector ( AXA1, AyA1, AZA1, AXA2 .... AXB 1 .... AzB2). The character of Db(C2) is zero. In the E operation, the displacements are unaffected so the character of Db(E) is 12. Then the character representation in an atomic displacement basis is { 12,0}. Using Eq. (10.4.1), the number of times an irreducible representation appears in this reducible representation is found to be 6 each for A and B symmetry species. In other words, there are six nuclear motions belonging to A species and another six belonging to B species. But the nuclear motions

225

Applications of Symmetry

include translations and rotations, in addition to 3N-6 vibrations, so we need to identify how many translational and rotational motions belong to each symmetry species. The translational motions, represented by the translation coordinates, are given by Eq. (5.2.9). The displacements AX, A Y, and AZ of the center of mass behave the same way as cartesian coordinates under symmetry operations, so translational motions Tx and Z ~'s

s,,~Y

C2(z) ~'~-

~ J

f~

I S

fX

Rz

S

S

""

.~

C2(z) ",~ ~

f

_

J

Rx

,, ,, t,,j

.-

J

4'

c 2 (z)

"I

Ry Fig.10-3. Depiction of the effect of C2 operation on rotational motions. Ty belong to B and Tz belongs to A symmetry species of the C2 point group. For determining the character representations spanned by rotational motions one can proceed as follows. The sense of rotation

226

Chapter 10

around the z-axis is not affected by the C2 operation (which is also a rotation around the z-axis) but the sense of rotation around x and y axes will be changed upon C2 operation as shown in Fig. 10-3. Thus the rotational motions Rx and Ry span B representation while rotational motion Rz spans A representation. In summary, out of the six nuclear motions that belong to A symmetry species, one is a translation motion Tz and another is a rotational motion Rz. The remaining four nuclear motions of this A species must be the vibrational motions. Similarly among the six nuclear motions that belong B species, two are translational motions Tx and Ty and two are rotational motions Rx, Ry. The remaining two must be the vibrational motions. The procedure described above, with cartesian displacements as the basis, may look tedious at first sight but, in actual practice, there are some useful simplifications. First, it is not necessary to write down the large 3N dimensional matrix representations, becuase only the cartesian displacements of those atoms which are not shifted upon a symmetry operation contribute to the character of Db. In the previous example, all four atoms of A2B2 were interchanged in the C2 operation so the character of Db(C2) was obtained as zero. In the E operation, however, all four atoms are unshifted so the character of Db(E) was obtained as 12. In the case of planar A2B2 molecules belonging to the C2v point group, the characters of D b ( C 2 ) and Db(Ov') are zero, while those of Db(E) and Db(O,,) are 12 and 4 respectively. [In the Ov operation, for example, the displacements of each atom change from Ax, Ay, Az, to Ax,-Ay, Az, so the trace become 4 x (2-1) = 4]. So the reducible character representation in the cartesian displacement basis is { 12,0,4,0}. The number of times an irreducible representation appears (see Eq. 10.4.1) is hal -- 1/4 [12xl + 0xl + 4xl + 0xl] - 4 ,

(10.4.7)

1/4 [12xl + 0xl + 4x(-1) + 0x(-1)] = 2 ,

(10.4.8)

nal = 1/4 [12xl + 0x(-1) + 4xl + 0x(-1)] = 4 ,

(10.4.9)

riB2 = 1/4 [12xl + 0x(-1) + 4x(-1) + 0xl] - 2.

(10.4.10)

hA2 =

Thus we have four nuclear motions each belonging to A~ and B~ species and two nuclear motions each belonging to A2 and B2 species. As mentioned earlier, translational motions transform the same way as x, y, z coordinates of a point, so Tz belongs to A~, Tx belongs to B l, and Ty belongs to B2. The sense of rotation around z-axis will be changed by o,, [=Ov(XZ)] and Ov' [=Ov(yz)] operations (Fig. 10-4), so Rz belongs to A2.

227

Applications of Symmetry

Based on similar arguments Rx and Ry can be seen to belong to B2 and B l species respectively. Thus the number of vibrational motions belonging to A~ is 3 (4 nuclear motions with one translation), to A2 is 1 (2 nuclear motions with one rotational motion), to B~ is 2 (4 nuclear motions with one translation motion and one rotational motion ) and ".o B2 is 0. Z y

~v (xz)

\ R

"X

ov(yz_ club> jr Fig.10-4. Depiction of the effect of 6v(XZ) and Crv(yz) operations on rotational motion Rz.

The two cases (non-planar and planar A2B2 molecules) considered above provide, hopefully, the needed background for applying this procedure to a molecule of interest. For generalized cases (molecules as well as symmetry operations), simplified equations can be developed 3,4, but they will not be discussed here.

10.4.2 Symmetry of Vibrational States The behaviour of vibrational wavefunctions under symmetry operations can be used to assign symmetry species to the corresponding states. The analytic forms of harmonic vibrational wavefunctions are given in Table 1 of Chapter 3. The symmetry assignment resulting from the behaviour of these harmonic wavefunctions would be the same as that obtained with anharmonic wavefunctions. This is because, an anharmonic

228

Chapter 10

wavefunction can be written as a linear combination of harmonic wavefunctions, and only those harmonic wavefunctions that have the same symmetry can be present in that summation. For the ground vibrational state (all vibrational quantum numbers, ~ = 0 ) , the wavefunction is a product of 3N - 6 harmonic oscillator - E q a2 functions (See Chapter 4), so it is proportional to e ~ where qa is the dimensionless normal coordinate of the ath oscillator. Any symmetry operation on a non-degenerate qa will change it to +qa, so qa lS invariant. In the case of degenerate coordinates, a symmetry operation mixes the normal coordinates of the members of a degenerate block, but the sum of squares of the normal coordinates of the members of that block remains invariant.

So the sum

~ q a 2 is always invariant under symmetry a

operations. Then the lowest vibrational state belongs to the totally symmetric species (that is to a representation with characters of all operations being + 1). For a fundamental vibrational state, the quantum number changes for only one vibration, from 0 to 1. The wavefunction in this case is

-Zqa proportional to qb e a

where qb is the normal coordinate of that

vibration whose quantum number agb changed from 0 to 1. This wavefunction transforms the same way as %, so the symmetry of a fundamental vibrational state is the same as that of the normal coordinate representing the fundamental transition. In the case of overtone states, ~a = 2,3... etc., the wavefunction is -~,qa2 ~)b p r o p o r t i o n a l to one or more terms of the type qb e a ~)b - m

-Zq

2

qb e a , where m=2,4 etc (until Vb - m > 0). Let us first consider the case where qb is non-degenerate. In such cases, it can be seen from the arguments made earlier that the terms q ~)b b and q ~)b b -m symmetry operations when

'D b i s

are invariant to

even and transform the same way as qb

when ~)b is odd. Thus the first, third, etc. overtone states (~)b=2, 4, etc.) belong to totally symmetric species while the second, fourth, etc overtone states (agb = 3, 5, etc.) belong to the same symmetry species as that of the normal coordinate representing the overtone transition. When degenerate normal coordinates are involved the procedure for determining the

229

Applications of Symmetry

symmetry is a bit more involved. Let us assume that the vibration we are dealing with is doubly degenerate. Then there are two normal coordinates (now designated as qbi and qbj) which are mixed by symmetry operations. For the first overtone transitions, the two quanta involved can be concentrated in any one of the two components, or be distributed among the components, of the degenerate set. That is, three different overtone states, (lkbi=2, 1,)bj----0),(a)bi=l, ~)bj=l) and (1)bi----0, akbj=2), form a block of states. The wavefunctions of these three states are proportional to qbi2, qbiqbj and qbj2 9 Similarly for second overtone transitions, the components of a block of states are (a)bi=3, 'Obj----0), (l)bi=2, 'Obj'-1), (~)bi=l,l)bj=2), and (a)bi=0,akbj=3). The wavefunctions of these four states would be proportional to qbi 3 (and qbi ), qbiZqbj, qbiqbj2, and qbj3( and qbj), respectively. Now we need to find out how the product terms of the type q~biqbjbj transform under symmetry operations.

Although a

symmetry operation mixes the components of a degenerate set of coordinates, it is always possible to find a set of transformed coordinates (or a combination of original coordinates) such that, O(qbi)=Ri qbi and O(qbj)=Rjqbj 9 Here Ri and P~ are constants and qbi and qbj represent such a transformed coordinate basis. The character of this operation in the transformed basis is Ri+R i which would be same as the character one would have obtained for fhe same operation in the original coordinate basis (before transformation). When an operation is performed n times the character in the transformed basis is g(on)=Ri n + Rj n where O n means the operation O carried out n times. It is to be remembered that O n must be equal to an operator, say P, which is also a member of the same group (see the properties of groups.) Then the character g(O n) =Ri n + Rjn---z(P) and z(P) can be looked up in the group character tables. Similarly when an operation is performed on q~ibi' q~i' and q~ibiq a~jbjseperately, one would get

O(q~ibi)=R~ biqb~ibi,

O(q

)=R

bjq

,

and

9 Dbi O(q~ibiqb;J)-R~biR~bjqb i qb; J.9 For example, for the first overtone 9

2

2

2

states 'Obi + 1)bj --- 2 and the operations of anterest are O(qbi )=Ri qbi , O(qbjZ)=Rj2qbj2 and O(qbiqbj) = RiRjqbiqbj. The sum Ri z + Rj 2 +RiRj, designated as g2(O), represents the character of operation O in (qbi,,qbj) basis for the block containing three overtone state wavefunctions. Similarly, for 'Dbi -I- l.)bj ----- 3 , the character for the block containing four overtone state wavefunctions is given by the sum Ri 3 + Rj 3 + RiZRj + RiRj2,

230

Chapter 10

and designated as %3(0). The discussion in the previous paragraph pertains to any one operation of a point group, for which the transformed coordinates qbi and qb) are not mixed. The same set of coordinates may not behave as a nonmixing set under a different class of operations of the same point group. However one can surely find a different set of transformed coordinates, for different symmetry operations which will satisfy the relations given in the paragraph. Thus the relations for the characters given in the previous paragraph apply for all operations of a point group. The character %2(O), for the first overtone block, can be rewritten as [Ri 2 + Rj 2 +(Ri + Rj)2]/2. Since as mentioned earlier, %(O2) = Ri 2 + Rj 2 and %(0) = Ri + Rj , one can write %2(0) = [%(0 2) + %(0)%(0)]/2. Similarly the character %3(0)= [Ri 3 + Rj 3 +(Ri + Rj)(Ri 2 +RiRj + Rj2)]/2 = [%(03) + %(0)%2(0)]/2. These results can be generalized for any block of overtone states as ~ n ( O ) = [ ~ ( O n) + ~(O)~n_l(O)]/2, where n=(~Jbi + ~)bj). Once the characters for all symmetry operations of a point group are determined in this manner, the resulting character representation can be used with Eq. (10.4.1), along with the irreducible character representations of that point group, to identify the symmetry species of the overtone states in a degenerate block. The symmetry of a vibrational state that involves combination transitions (where more than one vibrational normal coordinate has ~a>0) can be determined as follows. Suppose that two different vibrational coordinates are simultaneously excited so that aga=l and ~b=l. The -~qa 2

wavefunction of this combination state is proportional to qaqb e a

.

Under a symmetry operation O, qa and qb would transform as %~(O)qa and ~b(O)q b respectively so the wavefunction of the combination state

_Xq2

transforms as %a(o)%b(O)qaq b e a . The character of the transformation for the wavefunction is then simply the products of the characters of irreducible representations to which q~ and qb belong. This is true even for the case of degenerate vibrations3, 4, although we will not show this explicitly. The structure of the character representation so obtained for the combination state can be determined, if needed from Eq. (10.4.1). As an example suppose that qa and qb belong to B~ and B2 representations of the Czv point group. Then the symmetry of the combination state with ~)a=l amd agb=l is determined from the representation obtained as the product of the characters of B~ and B2

Applications of Symmetry

231

representations. The latter two representations are { 1 -1 1 -1 } amd { 1 -1 -1 1 }, so their product is { 1 1 -1 -1 } which is the A2 representation of the C2v point group. (This result is written as FB1 x FB2 - FA2 or simply as B1xB2=A2). Thus the combination state under consideration has A2 symmetry.

10.4.3 Selection Rules The symmetry of vibrational states, discussed in the previous section, can now be used to determine whether or not a given vibrational transition supports, or can be seen in, a particular spectral measurement. A vibrational transition supports the absorption spectrum only when the electric dipole transition moment integral (gt~,l~t~igta9) is non-zero (it may be recalled that the absorption intensity is proportional to the square of the transition moment integral.). This integral is invariant to symmetry operations, so it should span the totally symmetric irreducible representation. In other words the representation obtained from the product of characters of representations spanned by gtv,, ga, and ~ v should contain totally symmetric representation, or F~g,~,x Fg~ x F~,~ should contain the totally symmetric representation. Since the ground vibrational state always belongs to the totally symmetric representation, the product Fvv, x Frt a x FV~ will satisfy the above requirement when the representations 1-'V,, and Fga spanned by the excited state and dipole moment are the same. For a fundamental transition, the symmetry of g%, is same as that of the normal coordinate qa. So only those normal modes which belong to the same representation as that of ga will support the absorption spectrum. The representations spanned by the three components of g~ are same as those spanned by x, y and z coordinates (or by translational coordinates Tx , Ty and Tz ) which are commonly identified in character tables 1-4. So a simple inspection of character tables permits one to identify which symmetry modes are active (or inactive) in the absorption spectrum. The arguments applicable to the vibrational Raman spectrum are similar except that here we consider the integral (gt~,lc~l~lgt~). For a fundamental transition to support the Raman spectrum the representation spanned by qa and o~13 should be the same. Since the components axx,, yy, O~zz,axy, O~xz and ayz transform the same way as the products x z y2 z , xy, xz, and yz, the representations spanned by o~al3 can be determined from those of x a , y z , z 2, xy, xz and yz.

Their symmetry can be

232

Chapter 10

determined from the product of the characters of individual representations (for example Fx x Fy yields Fxy) if their individual representations are not degenerate. But if one or both of the individual representations are degenerate, then this recipe is not applicable and a more involved procedure 3 is needed. The character tables commonly identify the irreducible representations spanned by o~al~ (or by appropriate combinations of o:cx[3). So one can identify the normal modes that support (or do not support) Raman intensity, by visual inspection of the character tables. It is of interest to note that the point groups containing an inversion symmetry element possess an important property that a representation spanned by T x, Ty or Tz is not spanned by O~c~13. That means, for molecules possessing inversion symmetry, a normal mode supporting infrared absorption would not support Raman intensity and vice versa. This property, referred to as the mutual exclusion principle, is valuable to identify the molecular symmetry from the vibrational absorption and Raman spectral observations. As discussed in Chapter 9, the depolarization ratio is given as -2a + 7 132a) where ~a and ~2 a are respectively the mean and 6~2a/(45 0~ anisotropy of the polarizability derivative tensor 3oto~/~qa. The mean of this tensor is invariant to coordinate transformation, so ~a is non-zero only for totally symmetric normal modes. In other words, for normal modes belonging to the totally symmetric representation, the depolarization ratio would be less than 6/7. Thus Raman spectra also allow us to determine whether or not an observed spectral band originates from a totally symmetric normal mode. In the case of vibrational circular dichroism (see Chapter 8), the band intensity is proportional to (~g~lgo~l~gu,)(~gv, lmal~gv), where rn~ is the magnetic dipole moment component. So it is the product of two integrals that needs to be considered here to determine whether or not a vibrational transition supports the VCD spectrum. The integral (gtajl~c~l~g~,) is same as the one we considered earlier for vibrational absorption. For a vibrational transition to support the VCD spectrum, the representation spanned by ~g~, should also be spanned by both g~ and ma. For fundamental transitions, this is equivalent to saying that both g a and ma should span the same representation as that spanned by a normal mode. The electric dipole moment is a polar vector and its components transform the same way as translational coordinates Tx, Ty and Tz. On the other hand magnetic dipole moment is an axial vector and its

233

Applications of Symmetry

components transform the same way as rotational coordinates Rx, Ry and Rz. Based on these transformation properties, the same component of ga and mo~ can be seen to span the same representation only for chiral point groups (those groups without reflection or inversion symmetry elements), namely Cn, Dn, T, O and I. As a consequence the VCD spectra are supported by the vibrations of those molecules that belong to the chiral point groups. The vibrational Raman optical activities are determined (Chapter 9) by the products ( ~ 1 c~c~[~l~ , ) ( ~ , l G ~ I~t~ ) and p

~a~6(~vlo~lglv,)(~v, l A ~ l g l v ) , so the selection rule here is derived from the behaviour of these product integrals. The integral ( ~ v l ~ l ~ g ~ , ) is same as that we considered earlier for the Raman spectrum.

#

G~

transforms 5 the same way as ~ , 8

A~,8~, so it is

sufficient to consider the transformation properties of (~gv,IG~l~gv). For a vibrational transition to support VROA intensity, the representation spanned by ~ , (or by a normal mode qa for fundamental transitions) should also be spanned by both a ~ and G ~ .

Although (x~ is a polar

I

tensor and G ~ is an axial tensor, both of them transform in the same way in chiral point groups only. Thus VROA spectra are generated by vibrations of those molecules that belong to the chiral point groups. References 1 F.A. Cotton, Chemical Applications of Group Theory, John Wiley & Sons, New York (1990). 2. D . M . Bishop, Group Theory and Chemistry, Dover, New York (1973). 3 S. Califano, Vibrational States, John Wiley & Sons, New York (1976). 4 E . B . Wilson, J. C. Decius and P. C. Cross, Molecular Vibrations, McGraw Hill, New York (1955). 5 L . D . Barron, Molecular Light Scattering and Optical Activity, Cambridge University Press, Cambridge (1982).

This Page Intentionally Left Blank

235

Chapter 11 PRINCIPLES OF SPECTRAL MEASUREMENTS

11.1 Introduction The appropriate experimental quantity that is related to the transition rate, as discussed in Section 2.5, is the absorption coefficient ix(v). To determine this coefficient, one should measure the incident light intensity I0(v--), the fight intensity I(v) transmitted through a sample of interest (both at a given wavenumber v), the sample concentration CO and the optical pathlength g in the sample. Since these quantities are related via Beer's law, ix(v) is obtained as ln[Io(v)/I(v)]/Cog using natural logarithm. The integrated absorption coefficient A(v-) is obtained by integrating o~(v) over the width of the vibrational band of interest (see Eq. 2.5.6). However, the spectral intensities are commonly displayed as absorbance A(v), where A(v-) = logl0[I0(v)/I(v)] = e(v)Cos - t~(v)C0s - a(v)/2.303. The difference in absorption coefficients of left versus right circularly polarized incident light, referred to as circular dichroism, can be observed for randomly oriented chiral samples; Chiral systems are said to be optically active. Optical activity can be induced in non-chiral systems by an external magnetic field, but this phenomenon is not considered in this book. Oriented samples exhibit different absorption coefficients for polarizations parallel and perpendicular to the sample's unique axis. This difference in linear polarization absorption coefficients is referred to as linear dichroism. The applications of dichroism to molecular structural elucidation are numerous. Prior to the 1980s dispersive infrared spectrometers 1 (where different wavelength components of a polychromatic incident light beam are dispersed using a grating or a prism) have been commonly used for infrared spectral measurements. The advantages of using a Michelson interferometer with Fourier transform techniques for infrared spectroscopic measurements have been well established. In a Michelson interferometer2 the incoming light is divided into two equal amplitude components, so the Michelson interferometer can also be referred to as an amplitude division interferometer (ADI). The multiplex and throughput advantages associated with ADI made the Fourier transform infrared (FTIR) spectrometers very popular 3. Vibrational Raman and Raman optical activity (VROA) measuremems, on the contrary, were commonly performed using dispersive methods 4. Recent developments 5 made it possible to perform the vibrational Raman

236

Chapter 11

measurements with Fourier transform infrared spectrometers. Since Raman scattered intensifies depend on the inverse fourth power of the incident excitation wavelength, there is an intrinsic advantage to using visible laser sources for Raman measurements. Nevertheless different factors, such as the improvements in the quality of near-infrared detectors and the advantage of reduced fluorescence interference, gave impetus for the development of FI'-Raman spectrometers with near-infrared excitation. An extension of this approach for VROA measurements has been suggested 6, but no measurements were undertaken. For VROA measurements, one may modulate the polarization of the incident laser beam between fight and left circular polarization states and measure the synchronous difference in Raman scattered intensities. Alternately, one can use the linearly polarized incident laser light, and measure the difference in the scattered Raman light intensifies with fight versus left circular polarizations. Polarization division interferometers (PDIs) 7 provide an alternate Fourier transform approach for infrared and Raman spectral measurements, including VCD and VROA. Thus, three different instrumental techniques are available for a practicing spectroscopist. Linear dichroism measurements are commonly made by orienting a linear polarizer in two different orthogonal orientations, measuring the absorbance and taking the numerical difference between these absorbance measurements. A combination of linear polarizer and quarter-wave retarder (QWR), with the axis of the linear polarizer oriented at +45 ~ and-45 ~ to the unique axis of the QWR, generates opposite circular polarizations. Thus by orienting the QWR in a fixed orientation and placing the linear polarizer in two orthogonal orientations or by orienting the linear polarizer in a fixed orientation and placing the QWR in two orthogonal orientations, the measured absorbance difference represents circular dichroism. A variation in this theme is to rotate the linear polarizer or QWR continually around the light propagation axis and to use synchronous detection (vide infra). However the introduction of photoelastic modulator (PEM) 8, provided a means for obtaining both better accuracy in polarization modulation and high frequency polarization modulation. The details involved in polarization modulation using PEM, rotating QWR and combination of these two are discussed below. The double polarization modulation concepts presented in this chapter are new, and experiments have yet to be undertaken. Another polarization modulation approach of considerable potential, especially for the far-infrared region, is to use a polarization division interferometer as an achromatic polarization modulator 9. This approach will be discussed in Chapter 13. In this chapter the principles involved in polarization modulation and signal processing are presented first, then the dispersive spectrometers are described. Conventional infrared absorption measurements are automatically involved during the measurement of dichroism, as are the

Principles of Spectral Measurements

237

normal Raman measurements during Raman optical activity measurements. For this reason, the conventional infrared absorption and Raman measurements will not be discussed separately with special emphasis in this chapter. The next two chapters focus on amplitude and polarization division interferometers for the same measurements with a similar emphasis. The readers who are not interested in signal analysis and instrumental details can skip these chapters and go directly to Chapter 14. 11.2

Polarization m o d u l a t i o n using a photoelastic m o d u l a t o r The concept involved in the operation of a PEM is that a periodic stress applied to an isotropic crystal causes synchronous variation of the difference in the refractive indices along two mutually perpendicular axes. One of these two axes is the axis along which the stress is applied. For linearly polarized incident light with the polarization axis at 45 to the stress axis of the crystal, one can resolve the incident electric vector into two components, one parallel and another perpendicular to the stress axis. The periodic variation in the birefringence of the crystal introduces timedependent phase lag into these two electric vector components of the incident light. The phase variation in time t follows the relation

8m = 8 0 i sin COmt ' Vi where tom =2rCVm is the frequency of stress modulation,

(ll.2.1) and

~0 i the

maximum phase shift introduced for wavelength )q or wavenumber vi. When the maximum stress applied corresponds to a phase shift of 8 0 = re/2, then the radiation of wavenumber ~i is said to be circularly polarized. Alternately, one of the two electric vector components incident on and parallel to the stress axis of the PEM is said to have undergone a 90 ~ phase shift or quarter-wave retardation relative to the other. For a given maximum stress on the PEM, precisely X/4 or quarter-wave retardation is achievable at only one wavenumber, which is represented by Vq. For other wavenumbers at the same PEM setting, the maximum phase shift becomes 50 = 5 0 Vi

Vq

Vi _ Zt Vi _ rt)Vq ~q 2 ~q 2 )~i

(11.2.2)

As sincomt goes through +1, 0, and -1, 8~% goes through +rt/2, 0, and -~/2, which means that the ~q component goes through right circularly

Chapter 11

238

polarized, linearly polarized, and left circularly polarized states, respectively. For the intermediate values of sincomt, corresponding intermediate polarization states are achieved. Thus, only for a minor portion of sincomt cycle is the wavenumber component ~q circularly polarized (to be precise, only at sin Ohnt = +1), and for the remaining portion of the cycle the polarization states are those of intermediate nature. Similarly, when 8 0vi - rc the linear polarization would have been rotated by 90 ~ and for one

sincomt cycle there would be two cycles of linear polarization modulation. Again the desired linear polarizations are obtained only at sino)mt = 0 and +1. Due to the presence of different polarization states at intermediate points of the sincomt cycle, it is important to understand the nature of the signal expected at the detector 10. Two different experimental arrangements will be useful to understand the signal processing involved. In one arrangement (see Fig. 11-1), a monochromatic light of wavenumber Vi is passed through a linear polarizer (P), PEM and an optically active sample (S), in that order, finally reaching a suitable detector (D). In the second arrangement (see Fig. 11-2),

9 P

PEM

S

D

Fig. 11-1. Sample transmission configuration. P: linear polarizer; PEM" photoelastic modulator; S" sample; D: detector.

9 P

PEM

D

Fig. 11-2. Calibration configuration. P, PEM and D are as in Fig.ll-1; B: birefringent plate; A: linear polarization analyzer. the optically active sample is replaced by a birefringent plate (B) and linear polarizer (now called analyzer (A)). The former arrangement will be referred to as sample transmission configuration, and the latter arrangement

239

Principles of Spectral Measurements

will be referred to as calibration configuration. Regardless of the variation in the experimental arrangement, the z axis is the direction of light propagation and the optical axes of the PEM concide with x and y axes; the polarization direction of the linear polarizer will be considered to be at 45 ~ to the x and y axes.

11.2.1 Circular dichroism (A) Sample configuration The electric vector of monochromatic light of wavenumber Vi , after passing through the first polarizer, is given as [FS(Vi)/~/-2](u + v), where and v are the unit vectors parallel to x and y axes, respectively, and FS( Vi) is the amplitude of initial electric vector. As the PEM imparts a timedependent relative phase lag Gm at wavenumber Vi, to one of the electric Vi '

vector components, the resulting electric vector, after passing through the PEM, becomes

1/o +vei'm/

(11.2.3)

If the fight and left circularly polarized vectors are denoted 10 as [FS( Vi) / ,(-2)]x(u + iv) and [FS(Vi)/~/-2](u - iv), respectively, it can be seen that Eq. (11.2.3) is equivalent to

Ill

+ +/1+ l/u_ v,l (11.2.4)

When this electric vector passes through an optically active sample, the fight and left circularly polarized components are absorbed to different extents. The resulting electric vector is

240

Chapter 11

/ m/

+ 1 + ie 1~5~i (u - iv)e -aL (~i)/2

1 .

(11.2.5)

The intensity of light exiting the sample is calculated as the product of this electric vector with its complex conjugate. The voltage output of a linearly responding detector can be represented by the intensity falling on it, which in the present case is I ( ~ i ) - [IS(~i)/2] [(e -aR(~i) + e-aL(~i))+ (e -aR(~i) - e-aL(~i))sin8 m ]

Vi ,

(11.2.6) where IS(vi) is the initial intensity of the wavenumber component vi, aL(vi ) = 2.303AL(Vi), and aR(vi) - 2.303AR( vi)- Because ~m vi is time dependent, sin ~i-m vi can be expressed 11 as sin 8mvi --sin(80,kv sin corot) - 2 E J 2 n - 1 (~0 i ) sin[(2n -1)O~mt ] , n--1

(11.2.7)

where n is an integer and Jn(80Vi) are Bessel functions. From Eqs. (11.2.6) and (11.2.7) it can be seen that the detector signal contains a timeindependent (often referred to as dc ) signal, and a time-varying (referred to as ac) signals at frequency ram, 3ram, etc. The signal at the fundamental frequency ram, can be isolated by passing the detector signal through a band pass filter centered at mm and a lock-in amplifier tuned to COm. Similarly, the dc signal can be isolated by eliminating the signals at frequencies ram, 3ram, etc., using appropriate electronic filters. Then the ratio of the signal demodulated at mm to the dc signal becomes

Icom (Vi) Idc(Vi)

eaLi/Ol

e-aR(vi ) + e-aL(vi ) Gf

(11.2.8)

where Gl and Gf are the gains introduced by the electronics of the lock-in amplifier and filters, respectively. Multiplying the numerator and

Principles of Spectral Measurements

241

denominator of Eq. (11.2.8) by e (aR (~i)+aL(~i))/2, and noting that for small values of 13, (e~-e-~)/(e~+e-~) - tanh 13--- 13, one obtains

I0.)m(Vi)/Idc(Vi)- 2Jl(~0 i ){[aL(Vi)-

aR(Vi)]/2}(G//Gf). (11.2.9)

= 2J1 (~i0i ){2"303[AL (Vi)- AR (~i)]/2}(G//Gf) If Vi is equal to Vq, then 80i -re/2 and

J1 (re/2)---0.57.

The value of

Jl(~50vi)G//Gf can be determined from a calibration spectrum. (B) Calibration configuration Here the optically active sample is replaced by a birefringent plate 12 and analyzer together (see Fig. 11-2). These two components can be oriented with respect to the preceding PEM and first polarizer in four different ways. (a). The fast axis of the birefringent plate is parallel to that of the PEM and the direction of the analyzer is parallel to that of the first polarizer. In addition to the time dependent phase difference previously introduced by the PEM, additional time independent phase lag 8 B vi is now introduced by the birefringent plate, which is given as 2r~idAn; here d and An are respectively the thickness and birefringence of the plate. Then the electric vector exiting the analyzer becomes

F(Vi) - [uS (vi) / 2] Ill + vei(~5~m+sBi)/(11 + V) ,

(11.2.10)

from which the intensity can be found to be

I(~i)- [IS(~i)/2] (1+ COS~im COsfiBvi- sin 8mVisin~SBvi)

9

(11.2.11)

Using Eq. (11.2.7) and a similar expression 11 for cos 8~m ,

cos 8~m- J0 (80i)+ 2Z J2n (80i )cos 2nf.Omt, n=l the expression analogous to Eq. (11.2.9) becomes

(11.2.12)

242

Chapter 11

I0~m(~i) = -2Jl(fiOi)sinSB i G l Idc(Vi)

(11.2.13)

1+ J0(~Oi )cos~ BVi Gf

(b). Let the polarization direction of the analyzer be perpendicular to that of the first polarizer, and the fast axis of the birefringent plate be parallel to that of the PEM. Then (u + v) in Eq. (11.2.10) becomes ( u - v), and the expressions analogous to Eqs. (11.2.11) and (11.2.13) become I(vi)-[IS(~i)/2] (1 - cos~5m m sin ~sB vi cos ~5B vi + sin S~i vi ) '

Ic0m(Vi) = Idc(Vi)

2J1 (80i )sin 8 Bvi G l 1-- J0(~Oi)cos 8 Bvi Gf

(11.2.14)

(11.2.15)

(c). Let the polarization axis of analyzer be perpendicular to that of the first polarizer and the fast axis of birefringent plate be perpendicular to that of the

i(8~n]+~sBi)

PEM. Then e and (u + v) terms in Eq. (11.2.10) become ei(8~m -8 -B vi" and ( u - v), respectively, and the expressions analogous to Eqs. (11.2.11) and (11.2.13) become

)

I(~i) - [Is (vi)/2] (1 -- cos 8 m Vi COS8 Bvi--sin 8 mVi sin 8 BVi),

Io~m (Vi) = Idc (Vi)

-2J1 (~0i )sin fiBvi G l 1-J0(~Oi )cos~ BViGf

(11.2.16)

(11.2.17)

(d). Let the polarization direction of the analyzer be parallel to that of the first polarizer, and the fast axis of the birefringent plate be perpendicular to that of PEM. Then the expressions analogous to Eqs. (11.2.11) and (11.2.13) become (11.2.18)

243

Principles of Spectral Measurements IC0m(Vi)= 2Jl(80i)sinSBv i Idc ( V i ) 1

_Gl + Jo(8Oi)cos~i-BV i Gf "

(11.2.19)

Note that Eqs. (11.2.13) and (11.2.19) are equal but of opposite sign, as are Eqs. (11.2.15) and (11.2.17). Also Eqs. (11.2.13) and (11.2.17), like Eqs. (11.2.15) and (11.2.19), are equal to each other with a nonzero magnitude of +2J1(80V i )G//Gf at cos 8 B = 0 and sin gB _ +1 Vi Vi -

-

-

-

This is .

applicable when,

~5B =-v i r t = ( 2 n + l ) rC Vi

VB 2

-~ ,

(11.2.20)

where ~B is the wavenumber for which the birefringent plate introduces a single quarter-wave retardation and n is an integer. Furthermore, Eqs. (11.2.13), (11.2.15), (11.2.17), and (11.2.19) are equal to each other, with zero magnitude, when sin 8 -B - 0. This condition is met when vi m

fib = _vi = nrc. vi VB 2

(11.2.21)

In practical terms, if the maximum stress setting on the PEM corresponds to one quarter-wave retardation for the wavenumber Vq and the light components of various wavenumbers are investigated using the calibration arrangement, then one would get four curves (see Fig. 11-3) represented by Eqs. (11.2.13), (11.2.15), (11.2.17), and (11.2.19). The non-zero crossings of these curves provide the values of 2J1(80. vi )G//Gf. These values can be interpolated to the desired wavenumber and used in Eq. (11.2.9) to determine the CD of a given sample.

11.2.2 Linear dichroism (A) Sample configuration For considering linear dichroism, Eq. (11.2.3) can be rewritten as

F(~ i) - IF s(Vi)/~-8]

I/ / / / 1 1+ e

(u + v)+ 1 - e

(u- v).

(11.2.22)

Chapter 11

244

Denoting the base e absorbance for parallel and perpendicular polarizations as ap(~i ) and as(Vi ) respectively, Eq. (11.2.22) is modified to represent the electric vector after passing through the oriented sample as I

,,

, I ,

,,

I

,,

, I ,

,,

I , ,

,

I,

,,

!

I

1.0

m

0.5 m

m

0.0

-0.5

-1.0

2000

1800

1600

1400 1200 ~aavenumbers

1000

800

Fig. 11-3. Calibration curves represented by Eqs. (11.2.13), (11.2.15), (11.2.17), and (11.2.19). Birefringent plate is assumed to have a thickness of 0.25 cm and birefringence of 0.02; ~q = 1000 cm -1. The values of Jl( ~0 i ) were determined from the analytic formula.

F(vi) - [F s (Vi) / ~

ll/ / 1+ e

(u + v)e -ap (vi)

+ 1-e

~ ( u - v ) e -a~(~i) o

(11.2.23)

Principles of Spectral Measurements

245

The intensity of fight exiting the sample, determined from the product of Eq. (11.2.23) with its complex conjugate, is I ( v i ) - [IS(vi)/2][(e -ap(~i) + e-as(Vi))+ (e-ap (~i)_e-as

(Vi))cosS~ii] (11.2.24)

From the expansion of cos fvm term (see Eq. (11.2.12)), one would notice that the signal detected would contain components oscillating at frequencies 2corn, 4corn etc. The signal oscillating at 2corn, normalized to the dc signal becomes

e-a

I2a~m (Vi) Idc(Vi)

{[e-ap(Vi) +e -a~(~i)

+J0(~Oi

1

]O1

e-ap(Vi)-e -a~(~i)

1}

6f

"

(11.2.25)

When linear dichroism magnitudes are small (which is not necessarily true in general), the second term in the denominator of Eq. (11.2.25) may be ignored in relation to the first term. In such situations,

I2com(Vi)

"- 212 (~0 i )tanh[Aa(vi)/21(G//Gf), Idc (~i) where Aa( ~i)/2 =(as(~i) - ap( ~i))/2 - 2.303 AA(~i )/2.

(11.2.26)

(B) Calibration configuration Again a calibration measurement is needed to determine the value of J2( 80vi)(G//Gf) in Eq. (11.2.26). The calibration measurement can be performed just as in the case for circular dichroism, except that the calibration signal would now be extracted from lock-in amplifier tuned to 2O3m. Then the terms Icom and sin8 B Vi, in Eqs . (11 . .2.13), . (11 2 15) (11.2.17) and (11.2.19), will be replaced by I2com and cos 8 B vi respectively; 2J 1( riovi ) will be replaced by-2J2(~50vi ) in Eqs. (11.2.13) and (11.2.15) and by 2J2( 8 o. vi) in Eqs. (11.2.17) and (11.2.19). The predicted curves are

Chapter 11

246 shown in Fig. 11-4.

The m a x i m u m values of these curves give

+_.2J2(~50i ) / [ 1 + J0(~Oi )] (G//Gf) from which J2(~0i)G//Gf can be found. 1.5 1.0 0.5 0.0 -0.5 -1.0

-1.5 2000

Fig.

11-4.

1800

Calibration

1600

curves

1400 1200 wavenumbers

for

linear

I000

dichroism

800

represented

by

___2J2(~0 i )cos~Bvi /[1 + J0 (~Oi)cos~ vi B ]" Birefringent plate is assumed to have a thickness of 0.25 cm and birefringence of 0.02", Vq = 500 cm-1 . The values of J0( ~i0vi) and J2( ~0vi ) were determined from the analytic formulae. 11.3

Polarization modulation using a quarter-wave retarder The PEM shown in Fig. 11-1 can be replaced by a rotating QWR for circular dichroism measurements. For a broad wavelength coverage, however, one needs an achromatic retarder. Such a retarder can be constructed 13 from two different birefringent plates, with their fast axes in the plane of the plates and at 90 ~ to each other. The retardance of the combined plates is given by the relation, (2~/~)[Anltl-An2t2], where An is the birefringence and t is thickness; the subscripts 1 and 2 identify the two birefringent materials. Such a retarder made from CdS and CdSe plates has

247

Principles of Spectral Measurements

been reported 14 for the mid-infrared region. The signals applicable with rotating QWR, which are different from those applicable with PEM, are given in this section. For the configuration shown in Fig. 11-5, the electric vector exiting the rotating QWR is found to be, F(~i) - [Fs (~i)/x/2] X {(u + iv)cos 2 cot + i(u -iv)sin 2 tot + (u + v)(1- i) sin mt cos rot},

(11.3.1)

where to is the rotating frequency of achromatic QWR. With an optically active sample placed after QWR, the signal at the detector becomes, I(vi) = [IS(vi)/2] [e -aR(~i) cos 2 cot+ e -aL (~i)sin 2 cot] --[IS(~i) / 2] e -(aL (~i)+aR(~i))/2 [1 + (Aa(Vi) / 2)cos2mt].(11.3.2) Note that the circular dichroism signal is modulated at 20) (i.e. at twice the frequency of QWR rotation). It is easy to see from Eq. (11.3.2) that I2dIdc gives circular dichroism of the sample. For calibration purposes, the _

V

P i

QWR

S

D

i

Fig. 11-5. Sampletransmission configuration for measurementsusing a rotating quarterwave retarder(QWR); P, S, D are as defined in Fig.11-1. configuration shown in Fig. 11-6 can be used (where the PEM in Fig. 11-2 is replaced by a rotating QWR) and the signal in this case becomes I(vi ) - [Is (Vi)/2 ]((1 + 0.5cos 8 B vi ) - s i n ~5B vi cos 2cot- 0.5cos 8 B vi cos 4cot). (11.3.3) vi )" Following the The ratio I2JIdc now gives -sin 8Bi/(l+ 0.5cos 813 procedure in Section 11.2.1B, one can easily derive similar relations for the other three orientations (of the bireffingent plate and analyzer); similarly the expressions for the linear dichroism of the sample can be obtained following

Chapter I 1

248

the procedure in Section 11.2.2. Dichroism measurements using a rotating stressed birefringent plate have been reported. 15

9 P

QWR |

B

A

D

m

Fig. 11-6. Calibration configuration for measurements using a rotating quarter-wave retarder(QWR); P, B, A and D are as defined in Fig.11-2.

11.4

Double polarization modulation

One may achieve better dichroism signal quality with double polarization modulation as shown in Fig. 11-7. Here a rotating QWR and PEM are used together. For the configuration shown in Fig. 11-7, the i,

....

|

xx

P u,

QWR

PEM

S

D

Fig. 11-7. Sample transmission configuration for measurements using double polarization modulation; P, PEM, S, D are as defined in Fig.11-1. electric vector exiting the PEM is found to be

F(v i) - IF s (v i) / 2"V~] x [(u + iv) x A + (u - iv) x B],

/ m/ / / t/ m/ / lvi/t

where

1~.

A-

l+e

" ~

~ cos 2 m t + i 1 - e ~Svi sin 2 mt +

1~.

1-e

9 m

1

-i

l+e

sin mt cos mt,

(11.4.1)

Principles of Spectral Measurements

B=

249

/ m/ /ei i/ 1 - e ~8~i cos 2 m t + i 1+

1+

~i

_ i 1- e

" sin 2Cot+

sin cot cos cot.

With an optically active sample placed after the PEM, the signal becomes, I ( v i ) - IS(vi)e-(aL(~i)+aR(~i))/2{1 + (Aa(vi)/2)[J0(80 i )cos 2t.ot + J2 (~0i) cos(o~m + 2to)t +

J2 (~50i )cos(~ m -

2tO)t + J1 (~0i)sin COmt

--0.5(sin(~ m + 4co)t + sin(~ m - 4to)t)]} . (11.4.2) It is important to note that the circular dichroism signal now appears at 2o), tom, tom +2oo and tom +4o). There are two approaches for detecting the signals at 2o), O~m, O~m+2o) and corn+ 4co sirnultaneously. One approach is to use a digital signal processing method 16 and the other is to use mutiple lock-in amplifiers. If all these components can be detected simultaneously (vide infra) then the relations 11 among the Bessel functions (see Chapter 13) indicate that the measured dichroism signal becomes -2 times larger than the signal measured with PEM or rotating QWR alone. The analogous expressions for linear dichroism and calibration signals can be derived in a similar manner.

11.5 Dispersive spectrometers For a given wavelength, the signals resulting from polarization modulation are described in the previous sections. Light sources, other than lasers, are usually polychromatic. Selection of a given wavelength (to be more precise, a narrow bandwidth of wavelengths) is achieved in dispersive spectrometers, which were widely used in the first three quarters of this century. Although they are not as popular in current times (due to a wide spread use of the Fourier transform techniques), the dispersive spectrometers are unavoidable for some applications. The central component in these spectrometers is a spectral dispersing element, such as a prism or diffraction grating. When polychromatic light is directed to a reflection grating, different wavelengths are diffracted at different angles. The relation governing this dispersion is given 17 as d(sin0 + sin q~i) - m~. Here, 0 is the angle of incidence (measured from grating's surface normal), q~i is the angle of diffraction for wavelength ~,i, d is the distance between successive grooves on the grating, and m is a integer (called the order

Chapter 11

250

number). At higher orders, not only does the spatial separation between two wavelength components increase, but a wavelength of one order may fall imo the region of another order. Thus a filter blocking the wavelengths of a different order might be necessary. When single element detectors are used, the dispersed light is passed through an exit slit (whose width determines the spectral resolution) on to the detector. In this case, a scanning mechanism is used to rotate the grating so that a different wavelength component goes through the slit to the detector. Thus dispersive spectrometers using a single element detector are called scanning spectrometers. With the availability of multi-element detectors (also called detector arrays), the monochromator is replaced by a spectrograph, where the exit slit is replaced by a wide opening to hold the detector array (see Fig.

input slit

polychromatic light

mirror

grating exit slit or detector array

Fig. 11-8. Schematic of a monochromator/spectrograph. 11-8). In this case the spectral resolution isdetermined by the width of individual detector elements (referred to as pixels), which is typically ~25 ktm. The grating needs to be rotated to a new orientation only when the spectral region of interest is not covered by the combined width of the detector elements. Various methods centered around these dispersive spectrometers are discussed below.

Principles of Spectral Measurements

251

11.5.1 Dichroism measurements For a given wavenumber vi, or wavelength ~i, selected with a monochromator the signals expected at the detector with appropriate polarization modulation can be found in the previous sections. In dispersive spectrometers it is common practice to use a light chopper to modulate the source intesity. In such cases, the signals described in the previous sections have to be multiplied 1 by (1 +sino~ct)/2, where O~cis the light chopper frequency (~ 100 Hz). A block diagram of a dispersive CD spectrometer using a PEM for polarization modulation and a light chopper for source intensity modulation is shown in Fig. 11-9. Multiplying Eq. (11.2.6) with(1 +sino~ct)/2, one notices that the CD signal varies as both sin~ m and sin~Smsino~c t, while the sample Vi Vi transmission signal varies as sino~ct. The CD signal component can be extractedl by processing the detector signal with a lock-in amplifier tuned to COm. Alternately, one can use 18 two lock-in amplifiers (the output of lockin tuned to ohn is processed by a lock-in tuned to O}c)for extracting the CD signal. The CD component extracted in either manner needs to be normalized with the sample transmission signal, which is extracted using a mono-

chr~176 _ source chopper o)

vi

~

PEM )

~ "~ O~m polarizer

C

o)

I 0)

Lock-in Amplifier -= 100 Hz

m C

0)

detector

Lock-in rn Amplifier

"

-=-30-100 KHz

Fig. 11-9. Schematic of a dispersive circular dichroism spectrometer.

I I ! ! ! I I I I | ! I I I ! ! ! I ! ! ! I ! I

252

Chapter 11

different lock-in amplifier tuned to toc. Fig. 11-10 shows the first reported VCD spectra, for their historical importance. Modem instruments provide spectra with much improved signal quality. For linear dichroism signals, Eq. (11.2.24) is multiplied with (l+sincoct)/2 and the procedure for extracting the signals is similar to the one described above for CD, except that since the linear dichroism signal varies at 2Ohn the ohn lock-in used for CD is replaced by a lock-in tuned to 2o~. I

'

!

L

,

l

30OO

9

~ cm-I Fig. 11-10. The first reported VCD measurements using a dispersive VCD spectrometer. The traces with labels (+), (___) and (-) represent respectively the VCD spectra of the liquid samples of (+)-, racemic- and (-)-2,2,2-trifluoro-l-phenylethanol, shifted from each other for clarity. The top most trace is the infrared absorption spectrum. Reproduced with permission from Ref. lb. Copyright 1974 American Chemical Society.

When polarization modulation is achieved with a QWR rotating at frequency co, Eq. (11.3.2) is multiplied with (1 +sincoct)/2 and the COrnlockin amplifier shown in Fig. 11-9 is replaced with a lock-in amplifier tuned to 2to (remember to keep co>>toc). The recent emergence of digital signal processing techniques 16 eliminates the need for lock-in amplifiers; here the detector signal for a given wavelength is digitized (at time intervals determined by the higher modulation frequency) as a function of time and the Fourier transform of this time domain signal contains the intensity at different modulating frequencies. Such digital signal processing techniques are particularly important for realizing the advantages of double polarization modulation approach described in Section 11.4.

Principles of Spectral Measurements

253

11.5.2 Raman optical activity measurements For Raman spectral measurements, the sample of interest is illuminated with a laser and the intensity of scattered light is analyzed at wavenumbers that are away from the incident laser wavenumber. The differences between the wavenumbers of scattered and incident light correspond to the differences in energies of vibrational levels. The monochromator/spectrograph discussed earlier is placed in the scattered beam for wavelength selection. Although monochromators were widely used in Raman spectroscopy, the availability of multi-element detectors resulted in a wide usage of spectrographs. Due to the nature of multielement detection, signal processing is not done using lock-in amplifiers. So the polarization modulation here takes a different form (vide infra). For vibrational Raman optical activity (VROA) measurements 19-29, the incident laser beam can be modulated between right and left circular polarization states and the synchronous difference in the scattered Raman intensities measured 4 as a function of vibrational Raman shifts. This approach is referred26, 28 to as incident circular polarization (ICP) VROA (see Fig. 1111). VROA can be measured for 90 ~ scattered Raman light (Fig. 11-11A) with an analyzer parallel or perpendicular to the scattering plane; the former i"

D

D SG

/\ 9 "

A

L

L Las

P

M

(A)

S

P

M

(B)

Fig. 11-11. Schematic of a Raman optical activity spectrometer with 90~ scattering (A) and 180~ backscattering (B) geometries using incident circularly polarized light. P: polarizer; M: modulator; S" sample; L: lens; A: analyzer; SG: spectrograph; D: detector. See Refs. 19-23 for details. is referred to as depolarized VROA4,19, while the latter as polarized VROA 20. If the analyzer in the scattered beam is placed 21 with its axis at 54.7 ~ from the scattering plane, the resulting VROA is called magic angle

Chapter 11

254

VROA. One may also choose not to include an analyzer in the scattered beam. In back- (180 ~ (Fig. l l - l l B ) and forward (0 ~ scattered geometries22, 23 ICP-VROA is measured with no analyzer in the scattered light. A brief description of the procedures followed to make a measurement is as follows (see Fig.11-11). The incident laser light is first linearly polarized; polarization modulation between circular polarizations is achieved using an electro-optic modulator (EOM) 4, QWR 24, or a liquid crystal variable retarder 25. In the EOM case, the voltage supplied to the EOM is usually a bipolar square wave, so the positive and negative halves of the voltage correspond to the two circular polarizations. The data collected during two halves of the square wave are stored in two different files. The first reported VROA measurements are shown Fig. 11-12, for their historical importance. Modern instruments provide spectra with much improved signal quality.

H---C--OH

I

CNjL

-

_

_

_ .

~Jd

J

I

+

-~- 1~cl

"~ <]

-,

.,--.. .1 x.~_

,.

.

9

.

.

.

9

9,,

Fig. 11-12. The first reported depolarized VROA spectra (bottom traces), using a dispersive scanning ROA spectrometer, for the enantiomers of a-phenylethanol (left half) and o~-phenylethylamine(right half). The dotted lines are for (+) enantiomers and solid lines for (-) enantiomers. The top most traces are depolarized Raman spectra. Reproduced with permission from Ref. 4. Copyright 1973 American Chemical Society. In the QWR case, the orientation of the QWR is fixed in one position to give one circular polarization and data are collected into one file; then the axis of the QWR is rotated by 90 ~ to generate opposite circular polarization

255

Principles of Spectral Measurements

and the data are collected into another file. In the case of liquid crystal retarders, the birefringence of the liquid crystal depends on the square of applied electric fried" they are driven by a--2 kHz bipolar voltage (which is needed to prevent "run-away" of liquid crystal molecules into one permanent orientation) and by changing the magnitude of this bipolar driving signal, one achieves the switching of one circular polarization to another.

D

D

S

SG

A

A

L Laser

L

p

QWR P S

(A)

(B)

Fig. 11-13. Schematic of a SCP-ROA spectrometer (A) and DCPI-ROA spectrometer (B). P: polarizer; S: sample;QWR: quarter-wave retarder; L: lens; A: analyzer; SG: spectrograph; D: detector. See Refs.24, 26 and 28 for details. An alternative to the measurements described above is to leave the incident laser beam linearly polarized, and measure the difference between fight and left circularly polarized intensities of the scattered Raman fight (see Fig. l l-13A). Such measurements are referred to as scattered circular polarization (SCP) VROA measurements 26 and have been reported for 90 ~ scattering geometry. In principle, polarization division interferometry (see Chapter 13) is an attractive approach for this measurement 27. In another type of VROA measurement, the incident laser beam is modulated between fight and left circular polarizations and the synchronous difference in the intensities of fight and left circular polarization components of the scattered Raman light are measured (see Fig. 11-13B). These measurements are referred to as dual circular polarization (DCP) VROA measurement 28. If the scattered light was analyzed for fight circular polarization when the incident light is fight circularly polarized (and for left circular polarization when the incident light is left circularly polarized) the measurement is referred to as DCPI VROA. On the otherhand, if the scattered light is analyzed for fight circular polarization when the incident laser polarization is

Chapter 11

256

left circularly polarized (and vice versa), the measurement is referred to as DCPII VROA. The DCPI measurements in the 180 ~ backward scattering geometry have been reported 28. VROA measurements are prone to artifacts and careful control 29 of these artifacts is necessary.

11.5.3 Time resolved infrared measurements Various approaches are now available for monitoring the temporal changes using infrared spectroscopy. A summary is provided in Fig. 1114. The most straightforward approach is to use a monochromatic fight Time resolved infrared spectroscopy

ContinuOUSscanI Rapid sweep I

~troboscopic I

~

1Asynchronous ~

IC~176 Asynchronous

Fig.11-14. Variousmethods for time resolved infrared spectroscopic measurements. source (either a laser or polychromatic light dispersed by a grating) with a fixed polarization and monitor the changes in intensity at desired time intervals 30. This method however restricts the observations to a single wavelength and the entire procedure has to be repeated for each of the other wavelengths of interest. Interferometry obviously overcomes this limitation, but some other considerations need to be evaluated before undertaking time resolved measurements using interferometry (see Chapters 12, 13). (A) Measurements using boxcar integrator A block diagram for time resolved infrared spectrometer 31 is given in Fig. 11-15. A desired time dependent stimulus is given to the sample and the intensity of the light component (with fixed polarization at wavelength ~ or wavenumber vi) transmitted through the sample is determined using

Principles of Spectral Measurements

Vi

Sample A --

257

Dectector "V

[ Trigger]---B"[ Box'CarI

V Fig. 11-15.

Schematic diagram for time resolved measurements using a boxcar

integrator. a boxcar integrator. Using a reference signal, synchronized to the stimulus provided, the boxcar integrator can be configured to record the detector signal after a certain delay from the stimulus and the signal integrated for a certain time window. Both delay and integration time windows can be set as desired. The signal recorded by the boxcar represents the temporal change in intensity of light transmitted through the sample, IS(Vi) e-a(t'vi). The sample transmission at t = 0, i.e. in the absence of the external stimulus, can be obtained by collecting the data as before, but without providing the stimulus to the sample (but a reference signal representing the stimulus given in the previous measurement is needed for the operation of boxcar integrator). The signal measured now is IS(Vi)e -a(t=0'~i), but measured at different delay settings of the boxcar integrator. From the logarithmic ratio of these two signals, the time dependent changes in absorbance, for a fixed polarization, can be extracted. (B) Measurements using lock-in amplifier This approach 32 is suitable for cases when the external stimulus given to the sample is oscillatory, for example a sinusoidal perturbation. If this perturbation has a frequency roe, a chosen molecular property in response to this perturbation may also vary at this frequency, although there can be a phase lag between the applied perturbation and the chosen molecular property. Thus the temporal behaviour of that molecular property can be written as o~ = (x0 + (x(t) = o~0 + & sin ({Oet+ 13),

(11.5.1)

258

Chapter 11

where ~0 is the value of the molecular property that does not change with time, and [3 is the phase lag. The time varying portion of the molecular property can be written as o~(t) = ~ip sin ~et + O~opcos O~et,

(11.5.2)

with o~ip and O~oprepresenting respectively the in-phase and out-of-phase (also called quadrature) components of molecular property and are given as ~ i p - ~ cos ~,

(11.5.3)

0~op = & sin [~.

(11.5.4)

If Oqp and O~opcan be measured separately then the phase lag ~ (which is called the dissipation factor) can be determined from Eqs (11.5.3)-(11.5.4) as

C%p/Oqp - tan $.

(11.5.5)

The temporal characteristics of the molecular property can then be extracted through Eq. (11.5.2). The in-phase and out-of-phase components of the molecular property can be determined using a lock-in amplifier. This principle is the basis for two recently developed 32,33 novel spectroscopic methods, namely dynamic infrared linear dichroism (DIRLD) and two dimensional infrared spectroscopy. These two methods are described below.

11.5.4 Dynamic infrared linear dichroism (DIRLD) The inherent linear dichroism associated with oriented samples can be perturbed with a dynamic perturbation such as a mechanical stress or electrical voltage. In the case of oriented polymers the implications of mechanical stress are of importance. So the question to be addressed here is one of molecular reorientation as a function of the external stress modulation. As linear dichroism is related to the molecular orientation, the variation of linear dichroism with stress modulation can be related to molecular reorientational dynamics. A rheometer is commonly used to modulate the applied stress. A block diagram for DIRLD measurements 32 is given in Fig. 11-16. The source light is chopped by a mechanical chopper at frequency tOc and dispersed by a monochromator. At the wavelength component of interest, the light is polarized by a linear polarizer P and modulated between two orthogonal linear polarizations by a photoelastic modulator (PEM) at 2COm. As a result of this polarization modulation, the detector signal contains a component that represents linear

Principles of Spectral Measurements

259

dichroism of the sample. If the sample is subjected to a mechanical stress perturbation at frequency COsthe linear dichroism undergoes changes at that frequency. In order to isolate the desired signal, these perturbation

Source

Chopper H chromator Mono~

PEM ~

I1

0)c

.

| m

|

|

Controller

Otor

Sample

I

!

Controller O~m

Controller

|

!

|

COs

al nl an an

|

i

Lock-in 0~c

t

,,

w,.

o ,.

,.

,,

Lock-in

Lock-in O~m

m. . , , ., , .

. , , ,.

. , , . i ,.,

.

.m , ,. , . .

. u.

. , ,. m

.w . a . m. m. m . . m ..

. e. m w

a

w

~

t

w

u w

w

i . .

COs

~. .m . -.-

i

,-...

_----5 ....... t ............ tai l

Fig.11-16. Schematic for dynamic infrared linear dichroism measurements.

frequencies are to be well separated from each other. The usual values are COs--- 10 Hz, COc---1000 Hz and 2O~m---75 kHz. The signals from each of these perturbations are to be demodulated in the decreasing order of frequency. The detector signal is first demodulated using a lock-in amplifier tuned to 2O~m, with minimal time constant so that the signals varying at O~c and COs are not attenuated. The output of the 2O~m lock-in amplifier (containing signals varying at O3cand COs)is then demodulated by a lock-in amplifier tuned to COc and a time constant low enough not to attenuate the COs signal. The output of the O~c lock-in amplifier is then demodulated by a COslock-in amplifier. When demodulated in-phase with the reference signal from the rheometer, the output of this lock-in is proportional to the component of linear dichroism that varies in-phase with mechanical stress. When demodulated out-of-phase with the reference signal from the rheometer, the output of the COs lock-in amplifier is proportional to the component of linear dichroism that varies out-of-phase with mechanical stress. These measured dichroism signals in conjunction with Eqs (11.5.2) to (11.5.5) provide temporal changes in dichroism. However, it should be noted that temporal resolution is dictated by the time constant of the COslock-in amplifier.

Chapter 11

260

11.5.5 Two dimensional spectroscopy For a system that responds to the time dependent external stimulus, cross correlation function for two different time varying signals can be given 33 as =

l lf

T---)~, a-T/2

~(vl,t) ~(v2,t + x)dt.

(I 1.5.6)

Here x is the time off-set (correlation time) at which time the function x(x ) is evaluated, ct(Vl) and tx(v2) are some appropriate signals monitored at wavenumbers vl and v2. At x - 0 the function x(x ) is simply the one that represents synchronous variation between the signals tx(~ 1) and ct(v2 )- If Eq.(11.5.6) is evaluated for different z values and is found to have maximum at a non-zero x value, then that particular x(z) indicates the magnitude of asynchronous variations between the signals tX(~l) and tx(~2). Then x represents the phase shift between the two signals. Thus if one were able to measure ct(Vl) and t~(v2) as a function of time, with sufficient time resolution, then such data will enable determination of synchronous and asynchronous variations among different molecular properties. An alternative to this approach is as follows. Suppose that the external stimulus is sinusoidal in nature. Then the time dependent variations in molecular properties being probed can be represented by Eq. (11.5.2) and one uses 33 the relations x('C) = t~(V1, V2) cos O~et+ ~(V1, V2) sin O~et,

(11.5.7)

where, (11.5.8) (11.5.9) The function q)(v1, V2 ) represents the synchronous correlation intensity and 9 (~1, ~2) represents the asynchronous correlation intensity. Since the molecular properties ~p(Vi) and tXop(Vi) can be determined using a lock-in amplifier (as explained earlier), the correlation intensities q)(vl, v2),

Principles of Spectral Measurements

261

9 (v1, v2) can be calculated. The plots of these correlation intensifies represent two dimensional spectra, with Vl and v2 represented along x and z axes respectively. The y-axis represents the correlation intensity. Such plots have been reported not only for infrared absorption but also for infrared linear dichroism. References

1 (a). I. Chabay and G. Holzwarth, Appl. Opt. 14 (1975) 454; (b). G. Holzwarth, E. C. Hsu, H. S. Mosher, T. R. Faulkner and A. Moscowitz, J. Am.Chem. Soc. 96 (1974) 251. 2 A.A. Michelson, "Light waves and their uses", The University of Chicago Press (1907). 3 P. R. Griffiths and J. A. de Haseth, "Fourier Transform Infrared Spectrometry", John Wiley & Sons, NY (1988). 4 L.D. Barron, M. P. B0gaard and A. D. Buckingham, J. Am. Chem. Soc. 95 (1973) 603. 5 T. Hirschfeld and D. B. Chase, Appl. Spectrosc. 40 (1986) 133; D. B. Chase, J. Am. Chem. Soc. 108 (1986) 7485. 6 P.L. Polavarapu, Spectrochim Acta 46A (1990) 171. 7 P.L. Polavarapu, Principles and Applications of Polarization Division Interferometry, John Wiley &sons (1997). 8 M. Billardon, J. Badoz, C. R. Acad. Sci. Ser. B 262 (1966) 1672; J. C. Kemp, J. Opt. soc. Am. 59 (1969) 950; L. F. Mollenauer, D. Downie, H. Engstrom and W. B. Grant, Appl. Opt. 8 (1969) 661. 9 P.L. Polavarapu, SPIE, 1166 (1989) 472. 10 K. W. Hipps and G. A. Crosby, J. Phys. Chem. 83 (1979) 555; J. C. Kemp, Polarized Light and Its Interaction With Modulating Devices, Hinds International inc., (1987). 11 A. Erdelyi, W. Magnus, F. Obeshettinger and F. G. Tricomi, Higher Transcendental Functions, McGraw Hill, New York (1953); M. C. Potter, Mathematical Methods in the Physical Sciences, Prentice Hall, Englewood Cliffs, New Jersey, 1978. 12 L. A. Nafie, T. A. Keiderling and P. J. Stephens, J. Am. Chem. Soc. 98 (1976) 2715. 13 M. Francon and S. Mallick, Polarization Interferometers, Wiley Interscience (1971). 14 D. B. Chenault and R. A. Chipman, Appl. Opt. 32 (1993) 4223. 15 P. Malon and T. A. Keiderling, Appl. Spectrosc. 50 (1996) 669; P. Malon and T. A. Keiderling, Appl. Opt. 36 (1997) 6141. 16 C. J. Manning and P. R. Griffiths, Appl. Spectrosc. 47 (1993) 1345. 17 F. A. Jenkins and H. E. White, "Fundamentals of Optics", McGraw Hill, NY (1976). 18 L. A. Nafie and M. Diem, Acc. Chem. Res. 12 (1979) 296; T. A. Keiderling, Appl. Spectrosc. Rev. 17 (1981) 189; M. Diem, G. M.

262

Chapter 11

Roberts, A. Barlow, and O. Lee, Appl. Spectrosc. 42 (1988) 20; P. Xie and M. Diem, Appl. Spectrosc. 50 (1996) 675. 19 L. Hecht and L. D. Barron, J. Raman Spectrosc. 25 (1994) 443. 20 W. Hug and H. Surbeck, Chem. Phys. Lett. 60 (1979) 186. W. Hug in "Raman spectroscopy" J. Lascombe and P. V. Huong, Eds. WileyHeyden, Chichester, UK (1982). 21 L. Hecht and L. D. Barron, Spectrochim Acta, 45A (1989) 671. 22 L. Hecht, L. D. Barron, A. R. Gargaro, Z. Q. Wen and W. Hug, J. Raman Spectrosc. 23 (1992) 401. 23 L. D. Barron, L. hecht, A. R. Gargaro and W. Hug, J. Raman Spectrosc. 21 (1990) 375. 24 L. Hecht, D. Che, and L. A. Nafie, J. Raman Spectrosc. 45 (1991)18. 25 P. L. Polavarapu (unpublished work). 26 K. M. Spencer, T. B. Freedman and L. A. Nafie, Chem. Phys. Lett. 149 (1988) 367. 27 P. L. Polavarapu, Chem. Phys. Lett. 148 (1988) 21. 28 D. Che, L. Hecht and L. A. Nafie, Chem. Phys. Lett. 180 (1991) 182; G.-S. Yu, L. A. Nafie, Chem. Phys. Lett. 222 (1994) 404. 29 W. Hug, Appl. Spectrosc. 35 (1981) 115; L. D. Barron and J. Vrbancich, J. Raman Spectrosc. 15 (1984) 47; J. R. Escribano, Chem. Phys. Lett. 121 (1985) 191; L. Hecht, B. Jordanov, and B. Schrader, Appl. Spectrosc. 41 (1987) 295; L. Hecht and L. D. Barron, Appl. Spectrosc. 44 (1990) 483; D. Che and L. A. Nafie, Appl. Spectrosc. 47 (1993) 544. 30 B. Locke, T. Lian and R. M. Hochstrasser, Chem. Phys. 158 (1991) 409; M. Lim, T. A. Jackson and P. A. Anfinrud, Science 269 (1995) 962. 31 T. Yuzawa, C. Kato, M. W. George and H. Hamaguchi, Appl. Spectrosc. 48 (1994) 684; K. Iwata and H. Yamaguchi, Appl. Spectrosc. 44 (1990) 1431. 32 I. Noda, A. E. Dowrey and C. Marcott, Appl. Spectrosc. 42 (1988) 203; B. Chase and R. Ikeda, Appl. Spectrosc. 47 (1993) 1350. 33 I. Noda, J. Am. Chem. Soc. 111 (1989) 8116; I. Noda, A. E. Dowrey and C. Marcott, Polymer News 18 (1993) 167; I. Noda, A. E. Dowrey and C. Marcott, Appl. Spectrosc. 47 (1993) 1317.

263

Chapter 12 A M P L I T U D E DIVISION I N T E R F E R O M E T R Y

12.1 Introduction Most of the modern infrared spectroscopic measurements use interferometric principles in some form. To introduce these conceptsl, let us consider two monochromatic light waves with their electric vectors given as F1Ei~ and F2 e i~2 propagating along the z-axis; here F 1 and F2 are the amplitude vectors and t~l and t~2 are the phase terms associated with the two waves. The resultant electric vector is the sum of these two vectors and the net intensity is obtained as a product of the resultant vector with its complex conjugate. Then it is easy to see that the net intensity is I1 + I2 + 2 (Ii12) 1/2 cos Ate; I1 =F1 ~ and 12 =F2~ are the intensities of individual light components; At~ is the phase difference between the two waves. The cross term, 2 (1112)1/2 cos A~, is called the interference term. For F I=F2 it is clear that 11 = 12= I, so the total intensity is 21(1 +cos Ate). The intensity contribution resulting from the interference is 21 cos Ate. In order for this interference to occur, the two waves should originate from the same light source and have the same wavelength. This condition is met by dividing the light from a given source into two components and bringing them back to the same direction of propagation. Another important point is that the two light components discussed above should have the amplitude (or a component of that amplitude) in the same direction. In other words, two light waves with their electric vectors along orthogonal axes (even when they originate from the same light source and propagate in the same direction) do not interfere. To achieve interference between such orthogonally polarized waves, a linear polarizer with its axis between the axes of electric vectors, should be placed in their path. These preliminary details underline the interferometric techniques presented in this and the following chapter. There are a variety of interferometric techniques, but two of them that are most pertinent for the vibrational spectroscopic studies are based on the amplitude division and polarization division principles. The concept of amplitude division is the central theme of a Michelson interferometer 2 as discussed below. The polarization division concept is the subject of next chapter.

12.2 Principles of measurements In an amplitude division interferometer (ADI) the incident light is divided into two components by a beamsplitter 2,3 (see Fig. 12-1). One

Chapter 12

264

component is transmitted in the direction of propagation, whereas the other component is reflected at 90 ~ to this direction. These components are brought back to the beamsplitter by a fixed mirror in the path of one component and a movable mirror in the path of the second component. If the mirrors in the two arms of the ADI are not at equal distances from the beamsplitter, the two components would have traveled different distances before returning to the beamsplitter. For a monochromatic light of wavenumber Vi and source intensity I s(~ i), if the electric vector in one

Fixed Mirror

ADI

Moving Mirror

Light Source

~ Beam Splitter

Detector

Fig. 12-1. Schematic of an amplitude division interferometer.

arm has traveled a distance 5 more than the electric vector in the second arm of the interferometer, then the two electric vectors would have acquired a phase difference of 2~5 vi. Note that 8 is equal to twice the mirror displacement from the zero path difference (ZPD) point. In a continuous-scan interferometer, the moving mirror is made to move continuously in both directions at a velocity V, in which case 5 = 2Vt at time t; in a step-scan interferometer the mirror movement is achieved in

265

Amplitude Division Interferometry

fixed step intervals. The intensity of light coming out of an ideal interferometer would be, (12.2.1)

I = [I S(vi )/2](1 + cos 2~8 vi ).

One-half of the source intensity in an ADI returns to the source and reflected by the factor of (1/2) in Eq. (12.2.1). The electrical recorded by a linearly responding detector would be proportional intensity of light falling on the detector. A component of this signal independent of ~i is known as the dc signal. The remaining component that varies with 8 is known as the interferogram.

this is signal to the that is signal

12.2.1 Transmission configuration (A) Background interferogram For polychromatic incident light, the intensity recorded by the detector as a function of the optical path difference in the two arms of interferometer would be Ib (~5) - f Is(~i) 2 cos2~;~Svi dv .

(12.2.2)

Ib(5) is referred to as the background interferogram. (B) Transmission interferogram When a sample with base e absorbance a(vi) is placed in the optical path of the light exiting the ADI the interferogram becomes, I t (5) - ~

[is ] (Vi) 2

~a(vi) cos 2/I;~Vi d v i

.

(12.2.3)

From the ratio of the Fourier transforms of transmission [Eq. (12.2.3)] and background [Eq. (12.2.2)] interferograms the transmission spectrum of the sample is extracted. The sample absorbance spectrum is derived from the negative logarithm of this ratio. (C) Dichroism interferograms For measuring dichroism with an ADI 4,5, the sample transmission configuration (see Figs. 11-1 and 11-5) would correspond to placing a linear polarizer, a polarization modulator [photoelastic modulator (PEM) or a rotating QWR; see Chapter 11], and a sample, in that order, in the path of the light coming out of the interferometer and going towards the

266

Chapter 12

detector. Similarly, the calibration configuration (see Figs. 11-2 and 11-6) would correspond to placing a polarizer, polarization modulator, birefringent plate, and analyzer, in that order, in the same place. Calibration configuration" First consider the situation where polarization modulation is achieved with a PEM. The intensity expected at the detector is obtained by multiplying Eqs. (11.2.11), (11.2.14), (11.2.16) and (11.2.18) with (1 + cos 298 vi)/2. For the experimental arrangement where the fast axis of birefringent plate is parallel to that of the PEM and the polarization direction of analyzer is perpendicular to that of the first polarizer, the intensity as a function of the optical path difference can be seen 6 to consist of four distinct parts: (12.2.4)

I(~i) = Idc + I1 +I2 + I3, where,

(Vi) [1-Jo(~Oi )cos ~sB] d~,

I d c - ~0 Is 4

vi

I 1 - ~0 Is (vi) [ l - J 0 ( V~50 i ) cos ~B vii cos2~SVi dv, 4

(12.2.4a)

(12.2.4b)

_2J2n (80 i ) cos ~iB I2 - [0 ~IS(Vi) vi c~176 4 I,n=l + 2JEn_ 1(~50i )sin 8 -BVisin[(En - 1)~mt ] }dv,

(12.2.4c)

_2J2n (~0 i ) cos~SB I3 - ~0~IS(vi) vi c~176 4 (n=l + 2J2n-1 (~iOvi)sin8Bvisin[(2n - 1)O~mt] ) cos 2r~SVi d~.

(12.2.4d) The intensity represented by Eq. (12.2.4a) is purely a dc term which can be eliminated during signal processing. The intensities represented by Eqs. (12.2.4b) and (12.2.4c) are the signals varying independently due to the mirror movement in the interferometer and polarization modulation by the PEM, respectively. The intensity represented by Eq. (12.2.4d)

Amplitude Division Interferometry

267

contains interference from these two modulations. When the detector signal is passed through the electronic filters that would transmit only the interferometer frequencies and block the dc as well as no~n signals, one obtains an interferogram It(5),

fOo,s,: i:, [, It(8) -

Vi ] Gf(~i)cos2xSv i dv .

4

(12.2.5)

Similarly, if the detector signal is passed through a lock-in amplifier with a minimal time constant the dichroism interferograms can be derived (depending on whether the lock-in amplifier is tuned to COrnor 2C0m). When tuned to Ohn the the output of the lock-in amplifier becomes I 1 = ~0 ~ IS(vi)[2J1(50 i )sin ~iB 4 Vi] (1 + COS2r~Svi)G/(Vi) dv

(12.2.6)

In the previous two equations, Gf(vi) and G/(v i) represent the gain factors associated with the filter and lock-in amplifiers, which can be wavenumber dependent. Note that the output of the lock-in amplifier contains a dc signal superimposed on the interferogram signal. The dc part is usually eliminated by the electronic filters present in the signal electronics of the FTIR spectrometers. The resulting interferogram, denoted as Io~n(5) to indicate the interferogram signal demodulated at O~m, becomes Ie0m (8) -

f0~IS'i'4

)sin

] cos (2rl;SVi)G/(V i )Gf(v i ) dv (12.2.7)

For the case of linear dichroism, the lock-in amplifier would be tuned to 2O~m in which case the terms O~n, Jl(~Oi ) and sinSBvi in Eq. (12.2.7) will be replaced by 2O)m, -J2(80vi ) and cos 8B vi respectively. In a continuousscan interferometer, the Fourier frequencies 2V~ i and the modulation frequency give rise to the beat frequencies [for example, C0m+ 2V Vi and ~ - 2V vi in the case of circular dichroism]. It is necessary to ensure that the lock-in amplifier used has a bandwidth (measured as Q value) that is adequate for passing the beat frequencies centered around the reference frequency. Any signal attenuation resulting from the inherent properties

Chapter 12

268

of the lock-in amplifier, including its time constant, can be considered to be embedded in G/(vi) term. The lock-in amplifier may also introduce wavenumber-dependent phase factors into the cos(2r~5 Vi) term in Eq. (12.2.4). These phase differences may be accounted for in the Fourier transformation of the interferograms. It is useful to understand 6 the shapes of the interferograms It(8), I00m(8) and I2oha(5) shown in Fig.12-2. Consider the O)m interferogram whose shape is determined mainly by the trignometric function in Eq. (12.2.7). If we limit the bandwidth of the experiment, s o ~Ui is chosen to

((50i)remains positive, then it is sufficient to consider

be around ~/2 and J 1 the integral, I~om ( ~ ) -

f vmax

~B

sin Vi COS 2/d5~ i d~,

,t0 , , , , ~ ,

I , , , , i

I

I

I

I

i_~

(12.2.8)

,=

-

-_-

.-

=,

,=

350

' ' ' '

I ' ' ' '

400

I ' ' ' '

450

I ' ' ' '

500 Data poi nts

I ' ' ' '

550

I ' ' ' '

600

650

Fig. 12-2. Predicted shapes for the interferograms It(~5) (bottom), Io)m(~5) (middle) and I2%n(8 ) (top) in a calibration configuration. Birefringent plate with a thickness of 0.25 cm and birefringence of 0.02, PEM with ~q = 1000 cm-1, unit gain factors and uniform source intensity in the 4000-700 cm-1 were assumed in the simulation. ZPD corresponds to 500th data point. The Bessel function values involved in the simulation were obtained from their analytic formulae.

Amplitude Division Interferometry

269

where Vmax is the maximum wavenumber component detected. integral can be seen to be equivalent to

I~0m(~) -- --0"51COS((~B((~B / 4)/4) + 5)2~max+ 5)2rt -

This

1

cos((~, B / 4 ) - 5)2/1;Vmax - 1 ((~B / 4) - 5)2r~

(12.2.9)

where ~B is the wavelength for which the birefringent plate introduces the first quarter-wave retardation. For a positive path difference 5, the contribution of the second term of Eq. (12.2.9) is large, whereas that of the first term is negligible. Similarly, for negative path difference, the contribution from the first term of Eq. (12.2.9) is large, whereas that of the second term is negligible. Unlike normal transmission interferograms, the COrninterferogram will not have maximum intensity at the ZPD point, that is at 5 = 0 (as seen in Fig. 12-2). Instead, there will be two intensity maxima (one with positive value and the other with negative value), one on each side of the point 5 = ~B/4 = 5c. There will be two similar intensity maxima on the negative path difference side also, one on each side of the point,-5 = ~B/4 = -8c. A similar analysis may be undertaken for It(5) and I2com(5) (note that the transmission interferogram It(5) also depends on a Bessel function; see Eq.(12.2.5)). To obtain the desired calibration curves [see Eqs. (11.2.11), (11.2.14), (11.2.16) and (11.2.18)], the interferograms It(8) and Ic0m(5) have to be Fourier transformed and the ratio of the resulting spectra need to be obtained. The phase corrections required for the Fourier transformation of It(5) can be obtained by the usual procedures, because in this case the ZPD is associated with well-defined intensity maximum. The phase corrections in Fourier transformation of Icom(5), however, require special attention because, as mentioned earlier, the intensity maxima do not appear at ZPD. One may transfer the phase obtained in a measurement on stressed optical plate4; transfer the phase 6 that has been calculated for It(5) to Ioha(5) with the ZPD point determined from It(5); or obtain correct phase 7 by realizing that the calculated phase must accommodate both negative and positive spectral features. The circular dichroism calibration curve, analogous to that represented by Eq. (11.2.15), is obtained as

270

Chapter 12

~I{o m (8) cos2nSvid8

,

(12.2.10)

-1--J0(~ii~os~-Bi Gg(vi) "

~ It (8) c o s 2 ~ v i d 8 4.0

2J 1 (80 i )sin 8Bi

_

ILg ~

loO ~

0,0

J

,,-.

It(9) -l.g --

"~~

~

"~o 0

m

-41,0

_

t i l,w m

! 17 1500

I

I l , ~ , i400

I

llU lmO

i,

I-r,u, imO

I'm'' liO0

I ' ' ' ' iO00

// m

NavenuWoe~'

Fig. 12-3. Experimental circular dichroism calibration curves determined with an ADI. The magnitudes at the non-zero crossings of the curves correspond to unit circular dichroism. The non-zero crossings, connected by dashed lines, show the frequency dependence of calibration values. The procedure to obtain the remaining three circular dichroism calibration curves [analogous to those represented by Eqs. (11.2.13), (11.2.17) and (11.2.19)] is identical to the one presented thus far. The experimentally determined calibration curves are shown in Fig. 12-3. For linear dichroism calibration curves, one uses the 2Ohn interferogram in place of Ohn interferogram. When polarization modulation is achieved with a rotating quarterwave retarder (see Section 11.3), intensity at the detector is obtained by multiplying Eq. (11.3.2) with (1 + cos 2 n 8 ~ i)/2. The signal analysis is identical to the one described above. Sample transmission configuration 9 First let us consider the circular dichroism measurement when polarization modulation is achieved with a PEM. The total intensity at the detector can be obtained by multiplying Eq. (11.2.6) with (1 + cos 2n8 vi )/2. Then the interferogram signal is

Amplitude Division Interferometry

271

(12.2.11)

1(~i) = Idc + I1 +I2 + 13 , where,

f VmaxIS(Vi) (e -aR (Vi) +e -aL (Vi)) d r

(12.2.1 la)

I1- fVmaxIS,:~i~(e aR':~i~+e aL':~i~)cos2,~i

(12.2.1 lb)

Idc-

J0

4

a0

4

i2 __fVmaxIS(vi)(e-alt(~i)- e-aL(~i)) Jo

4

• 13_

2J2n_ 1(;50i)sin[(2n

fVmaxIS(vi)(e-alt(~i)Jo

1)mint] d~

(12.2.11c)

e-aL(~i))

4

• kn=l

2J2n_ 1(~0i)sin[(2n 1)COmt] cos2rl;SVi

d~

(12.2.1 ld)

Following the procedure used for the interferograms in calibration configuration, the normal transmission interferogram and Olm interferogram can be extracted from the Eqs. (12.2.11) as

it(~ ) _

[ ]

f~max IS(~i) ao 4

(e_alt(~i) +e_aL(~i) ) (cos2/1;5~i)Gf(~i)dv, (12.2.12)

r m [IS4 i l eaR i'eaL i o • (cos2/1;~ivi)[2Jl(50vi )G/(vi )Gf (vi )] dv 9 Eq. (12.2.13) is equivalent to

(12.2.13)

272

Chapter 12

IS(vi)(eIaR(~i) + e-aL(~i))(cos2//;~i){tanh[Aa(~i) / 2} Icom ( 8 ) - fVmax J0 4 x [2J1 (50i)Gg (~i)Gf (~i)] d~ ,

(12.2.14)

where Aa(vi) = aL(vi ) - aR(Vi )" Besides the difference in gain factors, the transmission interferogram [Eq. (12.2.12)] and mm interferogram [Eq. Vi) and tanh[Aa( Vi)/2] terms (12.2.14)] differ only in the presence of J1(8 o. in the latter. Even though

J l(~O i)

can be maintained positive within a

bandwidth for CD experiment, Aa(vi ) can have both positive and negative values. Hence although It(8) would resemble the routine transmission interferogram of a sample with the maximum intensity associated with the ZPD, the mm interferogram is expected to be different. The shape of the ,,

,,

i,,

,,I

,,,

,i

,,,

,I,

,,,

i,

,,,

I,,

,,

i,,

,,I

,,,

,i

,,,,

I u

m

= m

"I''

0

''

I'"''

I ....

200

I'''

'I"'

400

'I

....

600 Data point

i,,

''I

'''

800

,i

''''

I000

Fig. 12-4. Simulated VCD interferogram for (S)-CHFC1Br approximated as Imm(8) = I

f Vmax(COS2~5V i){Aa(Vi)

/ 2 } [ 2 J 1(~0 i )]dv. Vibrational frequencies and VCD

.*0

intensities were determined from MP2/DZP calculation (see Table 2 of Chapter 7). ZPD corresponds to 500th data point. The values of J 1(~~vi ) were obtained from its analytic formula with quarter-wave retardation at 1000 cm-1.

Amplitude Division Interferometry

273

Ohn interferogram of the sample can be expected to be closer to the shape of that in a calibration experiment, because the Aa( Vi ) contribution to the former and the sin ;50 contribution to the latter are bisignate Vi

The major

difference is that in contrast to sin5 -B Aa(vi) varies from molecule to Vi ' molecule and is not a periodic function. As a consequence, it does not appear to be possible to predict where the maximum intensity point would be in the O}m interferogram of a sample. From the simulated interferograms (see Fig. 12-4) using the theoretical values of Aa(Vi), the maximum intensity position is not found to be at the ZPD, but is at a few data points away from the ZPD. In practice, the o~a interferogram of samples can appear to be quite different from the one expected from Eq. (12.2.14) and to be similar to the transmission interferograms. This can be C+) !

N _

=

7

Wl:lVlf NUrIBI"P.~

Fig. 12-5a. The first VCD spectra measured with an FTIR spectrometer. Top two traces with (+) and (-) labels are for the (+)- and (-) enantiomers of camphor in CC14 solution. The bottom trace is the single beam transmission spectrum. Reproduced with permission from Ref. 4a. Copyright 1979 American Chemical Society.

explained by noting that the Aa( Vi )/a( Vi ) values for vibrational transitions are of the order of 10 -4 to 10-5 and that the polarization distortions introduced by the optical components can easily exceed this value. Then an instrument-dependent bias should be added to Aa( Vi ) in Eq. (12.2.14), which can make the resulting values monosignate and the resulting interferograms similar to the normal transmission interferograms. The use of high f-number lenses 8-12 to manipulate the light exiting the interferometer

Chapter 12

274

7.5

8.0

I

I

1300

''''

8.5 I

I''''

1200

Microns 9.5 lO.0

9.0 I

I

I'

1100

I .....

, o ~ I -i

11.0 I

''

'-I''

1000 Wavenumber

900

12.0

13.o

I

"'

I

I'

800

14.0

I....

"' " '

I

700

Fig. 12-5b. Absorption (trace A) and vibrational circular dichroism (traces B, C and D) spectra of camphor in CS2 solution. Trace D was obtained as one-half of the difference between raw VCD spectra of the (+)- and (-) -enantiomers. Trace C was obtained as the difference between the raw VCD spectra of (+)-enantiomer and racemic mixture. Trace B was obtained as the corresponding difference between (-)-enantiomer and racemic mixture. can avoid the addition of a bias to Aa(vi). With such precautions, CD is obtained from Eqs. (12.2.12) and (12.2.14) as ~It.o m (8)COS 2/1;~iVid~5/~ It (8) cos 2/I;SVid~ ~ 2J1(50 i )G/(Vi)[Aa(vi)/21 = 2J1 (~0 i ) G / ( v i )[2.303AA(v i ) / 2 ] .

vi)G/(vi)

The value of 2J1(~i 0

(12.2.15)

is obtained from the non-zero crossings of the calibration curves. The first VCD spectra measured with an FTIR spectrometer are presented in Fig.12-5a for their historical significance.

Amplitude Division Interferometry

275

The VCD spectra measured in our laboratory in the 1350-700 cm -1 region are shown in Fig. 12-5b for camphor solutions. Much improved VCD signal quality can be obtained with modem FTIR spectrometers. When the polarization modulation is achieved with a rotating QWR, intensity at the detector is obtained by multiplying Eq. (11.3.2) with (1 + cos 2rc8 vi )/2. The signal analysis procedure here is similar to the one described above. Note that the lock-in amplifier in this case will be tuned to 2co. Linear dichroism measurements are commonly made using the static difference method, where the polarized absorption spectra for two orthogonal orientations of the linear polarizer are separately measured and subtracted. For measuring small linear dichroism magnitudes the polarization modulation approaches provide better signal-to-noise. When a PEM is used for polarization modulation, intensity at the detector is obtained by multiplying Eq. (11.2.24) with (1 + cos 2rc8 Vi)/2" The signal processing here differs, compared to circular dichroism measurement, in that the lock-in amplifier is tuned to 2O~n (rather than O~m)and the voltage applied to the PEM is chosen to provide half-wave retardation for the frequency range of interest. Just as in the case of circular dichroism, there will be four components in the detector signal. Then the transmission and linear dichroism interferograms are obtained as

+ Eas (Vi) 1

-eas(Vi)l)Gf(~i)cos271;~v i dv i ,

IOm

f[IS( i '

(12.2.16)

_ Eas (Vi) 1

x Gf (Vi)GI (Vi) cos 2~8v i dv i .

(12.2.17)

When the linear dichroism signal is very small, as in the case of circular dichroism, the same features attributed to the shapes of circular dichroism interferograms will apply. For the transmission interferogram however, there is an additional contribution

from J0(~Oi )[e ap(Vi) - e

as(~i)] which

is negligible only when linear dichroism is small. In such cases the linear dichroism is extracted as,

Chapter 12

276

[~120)m (~)COS 2/I;~Vi d~] ] [~lt (~)cos271;~V i d(~]

(12.2.18)

= 2J2(8~ )G/(vi)tanh[Aa(vi)/2]. 5

A

.....

;

-!

I'

-

-I

l

4. 3 2 1 o . . . . . . .

t.N

o <

.

.

.

.

.

.

.

.

.

.

.

.

I

I

s 3

21

-1-

-2_

2000

o 1800

: 1600

-

1400

I

. . . . .

1200

:

1000

800

WAVENUMBER

Fig. 12-6. Vibrational linear dichroism spectra of a oriented isotactic polypropylene film obtained with ADI using a PEM (trace A) and static difference (trace B) method. Reproduced with permission from Ref. 5a. Copyright 1984 Society for Applied Spectroscopy.

For oriented samples linear dichroism can be as large as 1 and in such cases the following equation should be used:

Amplitude Division Interferometry

tan a, i

277

{1+ J0 (12.2.19)

An alternate way of processing the data in such cases is discussed in Chapter 13. The linear dichroism spectra of polypropylene obained with ADI using PEM modulation is shown in Fig. 12-6.

12.2.2 Reflection-absorption measurements The polarization dependence of infrared reflection-absorption from thin films on surfaces has been the basis for wide spread use of ADIs in surface characterizationl3. The difference in polarized reflectionabsorption is a sensitive measure of the molecular orientation on surfaces and therefore these measurements are widely used in characterization of molecules on surfaces. The experimental geometry for these measurements is shown in Fig. 12-7. The angle of incidence is the angle between the incident ray and the surface normal. The plane of reference is the i-r plane which is the plane containing the incident(i) ray, reflected(r) ray and the surface normal. The measurements using the geometry in Fig.

surface normal

i-r~lane / .--\

L_ _ ~

--

_ .A 3

Fig. 12-7. Reflection-absorptionconfiguration 12-7 are categorized as infrared reflection-absorption, grazing incidence reflection (with 0 close to 90 ~ or reflectance spectral measurements. One approach to these measurements can be categorized as the static method and the other two as external polarization modulation methods. In the static method, light exiting the ADI is linearly polarized using a linear polarizer and the polarized reflectance spectrum of the sample was obtained. In external polarization modulation method a rapid polarization modulation is introduced. The detector signal synchronously varying with the modulation is either decoded with a phase sensitive detector (lock-in amplifier, as discussed in previous sections) or sampled in real-time 14

Chapter 12

278

using special electronics (vide infra). The difference in polarized reflection-absorption is analogous to the linear dichroism discussed earlier for transmission configuration. The main difference is that linear polarizations are referenced to the sample's unique axis (such as the stretch axis) in linear dichroism measurements, and to the i-r plane in reflectionabsorption measurements.

12.2.3 Real time sampling For dichroism measurements with a continuous-scan ADI using a PEM, the previously discussed method used a lock-in amplifier for demodulating the detector signal that is oscillating at the PEM frequency. However, the use of lock-in amplifier is not necessarily the best approach, although widely used in the absence of any other practical method until recently. A real time sampling method 14 provides an effective alternative. Although the original development was made for measuring the difference in polarized reflectance from samples on surfaces, the description provided here will be for transmission configuration. Let us first consider linear dichroism. Eqs. (11.2.1) and (11.2.24) indicate that at Ohnt = 0 or rr, the detector signal is simply I(vi ) - IS (vi) e-ap (Vi)

.

(12.2.20)

For a voltage setting on the PEM to provide one half-wave retardation at vi the peak retardation is ~5~V i = rt, so at C0mt = rt/2 or 3~/2 the detector signal is I(vi ) - IS (vi) e-as (Vi)

.

(

12.2.21)

These two equations represent the amount of light transmitted through the sample for two orthogonal linear polarizations. Thus the linear dichroism can be derived from the difference between these two signals (with appropriate normalization by the sum signal). In a continuous scan ADI, however, the moving mirror is continuously changing its position and the above mentioned two signals would not be recorded at the same optical path difference. To see this more clearly, let us assume that the optical path difference in ADI is changing at 1 cm/sec and modulation frequency Ohn = 37.5 kHz. Then these two signals represented by Eqs. (12.2.20) and (12.2.21) would have been recorded at different optical path differences, separated by 0.067 ktm. This can be corrected as follows. At t = 0 the interferogram signal would be,

Amplitude Division Interferometry

ip

e

279

(12.2.22)

cos2/I;~Vi dv i .

At mmt = r~/2, the optical path difference would have been 8 + 8', so the interferogram would be

is s[is,:i,] e a' i' cos 2rr(8 + G')vi d v i

,

(12.2.23)

where 8' is the change in Optical path difference due to the timing difference in recording the signal. Similarly at Ohnt = - rr/2 (and 80vi = rr), the interferogram would be

IS'-- I

Is

(vi) 1e-as(Vi) cos 2r~(G - G')~ i 2

dv i

.

(12.2.24)

At tOrnt = 3rr/2, Is -1 i s - I [';'lea

(Vi)cos 2~(8 + 38')v i dv i

(12.2.25) p

/t

The value of Is at t = 0 is estimated 14, by fitting the three data points I s, I s , and ~s l " to a Taylor expansion. This approach has been used to measure circular dichroism 15 also, although the results are not available in the open literature.

12.2.4 Raman scattering Successful measurements of Raman spectra using amplitude division interferometers have been reported only recently 1 6. A monochromatic laser light is focussed onto the sample of interest and the light scattered is collected, collimated and sent to an ADI. In other words, the light scattered from the sample serves as the light source for ADI. Since Rayleigh scattering is much stronger than the Raman scattering, an important point in these measurements is to filter out the Rayleigh light, with minimal influence on the Raman light. Excellent sources 16 are available for the technical details of these measurements. The Raman spectra obtained with ADI are usually of a particular

Chapter 12

280

polarization property. That is, by setting the linear polarization of incident laser light to be perpendicular to the scattering plane, the 90 ~ scattered Raman light with polarization perpendicular and parallel to the scattering plane are measured separately. Although one may measure 17 the difference between these two linearly polarized intensities and between the left and right circularly polarized Raman intensities as described below, the actual experiments are yet to be undertaken. Consider a linearly polarized monochromatic light source and a 90 ~ scattering geometry. The scattered light, collimated by a lens, passes through a photoelastic modulator, linear polarizer (or analyzer), polarization scrambler, ADI and to an appropriate detector as shown in Fig. 12-8. The polarization state of scattered Raman light component can be represented by a polarization ellipsoid with ellipticity rl and azimuth 0 (see Appendix 4). The angle between the x-axis (perpendicular to the Fixed Mirror

scrambler Moving Mirror

lens

~pE

Bean Splitter

analyzer

polarizer t~

Detector

Fig. 12-8. Schematicof Raman measurements with ADI using PEM scattering plane) and the modulator's stress axis is designated by cx. The transmission axis of the analyzer is set at 45 ~ to the modulator's stress axis. The major axis of the polarization elipsoid is usually along the x-

Amplitude Division Interferometry

281

axis and therefore we assume 0 to be zero in the following analysis. Resolving the complex electric vector of the scattered light along the modulator's axis the intensity exiting the analyser is found 17 to be

IS(~i)- 89

+ S3(vi)sin5 -m Vi ]

9

(12.2.26)

In this equation, S0(vi), Sl(Vi) and S3(vi) represent the Stokes parameters for wavenumber vi. When the incident laser polarization is perpendicular to the scattering plane, S 1 and $3 are given (see Appendix 6) as S1~45~ 2+~2 ,

(12.2.27)

$3 ~ 45 ~G' + 7 ,y2 + t~2 ,

(12.2.28)

m

and ~ and G' are respectively the mean of electric dipole-electric dipole and electric dipole-magnetic dipole vibrational Raman tensors, i]2, 72, 82 are the anisotropies defined in Chapter 9. When the incident polarization is in the scattering plane, then S 1=0 and $3 ,~ 72 _ (1/3)82. Eq. (12.2.26) represents the source intensity that enters ADI. Assuming that there is no intensity loss at the subsequent optical components the interferogram signal recorded by the detector in ADI can be obtained by substituting Eq. (12.2.26) in the relation IS(vi) I(8) = ~ 2 [cOS2~Vi] dv. At this point we need to specify the two types of measurements and discuss them individually. (A) Circular intensity difference The Stokes parameter S3( vi ) represents the difference between the intensities of right circular and left circular polarizations of the scattered light. S3(vi) is non-zero for chiral samples. If the detector signal is processed by a lock-in amplifier tuned to the modulation frequency then the emerging interferogram can be derived as

1

It.0m ( ~ ) - ~ f S 3 ( v i ) 2 J l ( ~ O i ) G s

dv .

(12.2.29)

As the values of Jl(~ 0vi ) and Ge(~ i) can be determined separately, the

Chapter 12

282 Stokes parameter

S3( vi ) can be extracted through the Fourier transform of

Eq. (12.2.29). In order to keep the J l(~O i) near its maximum value, it would be necessary to set the PEM so as to give quarter-wave retardation in the frequency range of interest. Since the scattered wavelengths are only slightly different from the incident wavelength, small variations in ~5~ for a given setting on the PEM would not be of serious concern. (B) Linear intensity difference The Stokes parameter S l(~i~i) represents the difference between the intensities of scattered light with perpendicular and parallel polarizations. From Eqs. (12.2.26) one notices that if the lock-in amplifier is tuned to 2tOm, twice the modulation frequency, the resulting interferogram becomes I2o~m(5)= ~

Sl(Vi)sin2o~ 2J2(~50i)Gs

i dv.

(12.2.30)

Hence the Stokes parameter S 1( Vi ) can be extracted through the Fourier transform of Eq. (12.2.30). In order to keep J2(~5~ near its maximum value, the PEM needs to provide half-wave retardation in the frequency range of interest.

12.2.5 Time resolved measurements Most infrared detectors respond to time varying signals on the microsecond time scale. Some special detectors can be obtained with a response on the nanosecond time scale. Although time resolved measurements have been successfully undertaken on normal transmission spectra, time resolved vibrational circular dichroism measurements have not yet been possible with ADI. The difficulty arises from the fact that when a PEM is used for polarization modulation, its operating frequency (--30-100 kHz) interferes with the temporal dichroism changes one wishes to monitor. One can avoid the PEM and fix the polarization state to be either left circular or right circular (by using an achromatic QWR in a fixed orientation). The difference between separate time resolved measurements with these circularly polarized states may be used to extract the time resolved VCD signals. Such a procedure has been used for time resolved linear dichroism measurements, but similar success for VCD measurements would be challenging (owing to the small magnitudes of VCD). The polarization division interferometers discussed in the next section offer an attractive alternative. In this section we discuss the

Amplitude Division Interferometry

283

procedures used for time resolved ADI measurements for normal transmission and dynamic linear dichroism spectra. In a step-scan interferometer the optical path difference is held at a certain value and the detector signal is integrated for a desired period of time. Then the optical path difference is changed to a new value and this sequence is repeated until all interferogram data points are collected. This procedure (which used to be the common approach two decades ago) has now become popular 18 in light of the advantages present for time resolved and other measurements. For time resolved measurements with a stepscan interferometer 19, either a pulse or continuous perturbation is given to the sample by holding the optical path difference at a certain value. The detector signal is digitized at discrete time intervals (as dictated by the desired time resolution). This procedure is repeated at subsequent optical path differences. All these data points are arranged in a two dimensional array as I(At,~5) where At is the time resolution and ~5 is the optical path difference. For a given At, the Fourier transformation of the corresponding row of I(At,~5) gives the temporal spectrum. In this approach, there is no restriction on the rate of chemical reaction except that: (a). the detector used must be fast enough to follow the intensity changes ensuing from the reaction; (b). the reaction under study should be repeatable exactly the same way with the delivery of the perturbation at each optical path difference point. Due to the absence of Fourier frequencies in step-scan interferometry, one has the freedom to work with a wide range of perturbing frequencies as well 19. That is, in addition to pulse or constant perturbation discussed in the previous paragraph, one can also use repeating (such as sinusoidal) perturbations, at each optical path difference point. In this case, the detector signal is demodulated with phase sensitive detection (such as a lock-in amplifier referenced to the repeating perturbation), and the demodulated signal is digitized at each optical path difference point. By digitizing the in- and out-of-phase signals in two different channels, two interferograms are generated. Fourier transforms of these interferograms give in- and out-of-phase spectra, which can be used to generate two dimensional spectra (see Chapter 11). For interferometers operating in the continuous-scan mode, the detector signal is digitized 'on the fly' using a He-Ne laser interferogram as the reference. Each datum is obtained at a different point in time space, so the collection of the interferogram is itself time dependent, and a careful analysis is needed for time resolved spectroscopic measurements. For monitoring time dependent phenomena that vary much slower than the time it takes to collect the interferogram, a simple procedure 2~ referred to as the rapid sweep method can be adopted. On the other hand for phenomena whose time evolution is faster than the time it takes to collect the interferogram two different approaches referred to as stroboscopic (or

284

Chapter 12

synchronous perturbation) method 21 and asynchronous perturbation method 22 have been used. Note that the use of PEM modulation or any other external modulation is not used in time resolved measurements (for obvious timing difficulities). So, these measurements are restricted to a fixed polarization of the infrared beam. (A) Rapid sweep method Let us consider a chemical reaction that can be repeated an infinite number of times. In other words, when a system undergoes changes under the influence of an external stimulus, we expect the system to return to its initial state after some time so that this cycle can be repeated with the next stimulus. The collection of the interferogram is initiated, either immediately or after a certain time delay from the time the stimulus is provided. Since the next stimulus can only be given after the system returns to its initial state there will be a time gap (or dead-time) between the successive stimuli (and hence interferograms). This can be implemented by rapidly sweeping the moving mirror (and synchronously digitizing the data) in the forward direction, but allowing the mirror to return to the starting point rather slowly. This is in contrast to the normal rapid scan method, where the moving mirror is returned to the starting point rather quickly to minimize the dead-time between the interferogram scans. Using high mirror velocities, the interferogram (to achieve 8 cm -1 resolution) can be collected within -5 msec. So the Fourier transforms of these interferograms provide spectra that represent averages within --5 msecs. Repeating the measurement, with the external stimulus turned off, under identical instrumental conditions and substracting the resulting spectrum from the one made in the presence of external stimulus one obtains the spectral changes caused by the external stimulus. It is obvious that only slow reactions (on millisecond time scale) can be followed 20 with this method. (B) Stroboscopic or synchronous perturbation method The data points that constitute an interferogram represent interference at different optical path differences. But due to the nature of continuous mirror movement they also represent different points in time space. Thus for studying a chemical phenomenon that is initiated by an external stimulus, each data point in the interferogram would have been obtained at a different time after the stimulus was supplied to the sample. In other words, all data points of the interferogram would not have been obtained at the same time interval, and hence the interferogram would not provide time resolved spectrum. To overcome this obvious set back, a stroboscopic or synchronous perturbation method 21 is used. This method is explained with timing diagram shown in Fig. 12-9. Let us assume, for the sake of simplicity, that data points 1 through 10 consititute a normal

285

Amplitude Division Interferometry

interferogram. If the time resolution sought for probing one chemical phenomena is At, then the external stimulus is provided to the sample at a time At before the first data point tl of the interferogram is collected. The subscript of ti represents the data point number (or optical path difference), while the superscript represents the data point number or Data point 0

1

2

fi

3

4

5

6

7

8

9

10

4 t4

5 t5

6 t6

7 t7

8 ‘8

9 ‘9

10 0

3 t4

4 5

5 ‘6

6 t7

7 ‘8

2 t4

3 t5

4 ‘6

5 t7

6 ‘8

7

8 0

2

3 t6

4

5 ‘8

6

7 0

I

p4- At tt

5

‘7

8 t9

t9

5

-5 t2

‘9

6 0

4 t9

tl 0

3 t9

‘1 0

2 t9

5

4 3 0 2 0

-6 ‘2 -7 t2

9 0

-5 ‘4

Fig. 12-9. Depiction of data collection in time resolved measurements with a rapid scan ADI.

time lapse since the stimulus is given, In order to obtain the interferogram point, say $, at a different optical path difference but at the same time

286

Chapter 12

lapse the stimulus must be given at At time before ~ was collected. So one has to synchronously repeat the process of providing external stimulus at At time before each of the optical path difference points that constitute 1 t 13' t41 "'" t 1l0 then should be an interferogram. The data points t 11, t 2, assembled to constitute an interferogram that represents the time resolved interferogram whose Fourier transform gives the time resolved spectrum. (C) Asynchronous perturbation method In clear contrast to the synchronized perturbation scheme discussed above, an approach where such synchronization may be avoided has been developed 22. In this asynchronous method the sample is excited by a series of pulsed perturbations and the digitization of the interferogram is accomplished as in conventional measurements. However, the detector signal is passed through a gated circuit (boxcar integrator, for example), which opens the gate only for a certain time (tg) after a delay (td) from the time each pulse is given to the sample. The delay time is determined by the time resolution and the gate width is determined by the signal-to-noise ratio (and also time resolution) needed for the measured spectra. Furthermore, the frequency of external perturbation given to the sample should satisfy the relation, fe > 2(2VVmax) where V is the velocity of moving mirror in the interferometer and Vmax is the maximum wavenumber of the light detected. With Vmax = 4000 cm -1, mirror speeds of 1 cm/sec and 0.10 crn/sec require fe to be greater than 16 kHz and 1.6 kHz respectively. These relatively high frequency perturbations are appropriate for studying the phenomena where impulse response is completed within microseconds or less. For slow chemical changes (greater than a few milliseconds) the external perturbation has to be repeated at a much slower rate which in turn requires the mirror to be scanned very slowly. Such slow mirror scans (less than 0.01 cm/sec) may not be practical in continuous-scan interferometers, so the asynchronous method is more appropriate for studying the phenomena which can accommodate higher perturbing frequencies.

12.2.6 Further developments Recent developments in the commercial availability of the step-scan ADIs attracted their usage for circular dichroism measurements 12,23. No siginificant improvements in the signal quality, over that obtained on continuous scan interferometers, have yet been uncovered. Nevertheless, step-scan ADIs may permit time resolved circular dichroism measurements on a sub-millisecond time scale (the operating frequencies of PEM restrict time resolved measurements on sub-microsecond time scale). As a result, new applications of circular dichroism for

Amplitude Division Interferometry

287

biomolecular dynamics are likely to emerge. References 1 K.J. Gasvik, Optical Metrology, John Wiley, NY (1987). 2 A.A. Michelson, Light Waves and Their Uses, The University of Chicago Press (1907). 3 P. R. Griffiths and J. A. de Haseth, Fourier Transform Infrared Spectrometry, John Wiley & Sons, NY (1988). 4 (a). L. A. Nafie, M. Diem and D. W. Vidrine, J. Am. Chem. Soc., 101 (1979) 496; (b). L. A. Nafie and D. W. Vidrine in Fourier Transform Infrared Spectroscopy, J. R. Ferraro and L. J. Basile, Eds., Academic Press, New York (1982). 5 (a). C. Marcott, Appl. Spectrosc. 38 (1984) 442; (b). C. Marcott, A. E. Dowrey and I. Noda, Appl. Spectrosc. 47 (1993) 1324. 6 P.L. Polavarapu in Fourier Transform Infrared Spectroscopy, J. R. Ferraro and L. J. Basile, Eds. Academic Press, New York (1985). 7 C.A. McCoy and J. A. de Haseth, Appl. Spectrosc. 42 (1988) 336. 8 P. Malon and T. A. Keiderling, Appl. Spectrosc. 42 (1988) 32; T. A. Keiderling, in Practical Fourier Transform Infrared Spectroscopy, J. R. Ferraro and K. Krishnan, Eds., Academic Press, New York (1990). 9 D. Tsankov, T. Eggiman and H. Weiser, Appl. Spectrosc. 49 (1995) 132. 10 R. W. Bormett and S. A. Asher, Appl. Spectrosc. (to be published). 11 G. -C. Chen, P. L. Polavarapu and S. Weibel, Appl. Spectrosc. 48 (1994) 1218. 12 F. Long, T. B. Freedman, R. Hapanowicz and L. A. Nafie, Appl. Spectrosc. 51 (1997) 505; F. Long, T. B. Freedman, T. J. Tague and L. A. Nafie, Appl. Spectrosc. 51 (1997) 508. 13 J. D. Swalen and J. F. Rabolt, in Fourier Transform Infrared Spectroscopy, J. R. Ferraro and L. J. Basile, Eds., Academic Press, New York (1985); W. G. Golden, in Fourier Transform Infrared Spectroscopy, J. R. Ferraro and L. J. Basile, Eds., Academic Press, New York (1985). 14 M. J. Green, B. J. Bamer and R. M. Corn, Rev. Sci. Instrum. 62 (1991) 1426. 15. S. Weibel (private communication). 16 T. Hirschfeld and D. B. Chase, Appl. Spectrosc. 40 (1986) 133; D. B. Chase, J. Am. Chem. Soc. 108 (1986) 7485. 17 P. L. Polavarapu, Spectrochim Acta, 46A (1990) 171. 18 R. A. Palmer, Spectroscopy 8 (1993) 26. 19 R. A. Palmer, J. L. Chao, R. M. Dittmar, V. G. Grigoriou and S. E. Plunkett, Appl. Spectrosc. 47 (1993) 1297. 20 M. S. Braiman and K. J. Rotschild, Ann. Rev. Biophys. Chem. 17 (1988) 541. 21 A.W. Mantz, Appl. Optics, 17 (1978) 1347. 22 K. Masutani, H. Sugisawa, A. Yokota, Y. Furukawa, and M. Taumi,

288

Chapter 12

Appl. Spectrosc. 46 (1992) 560. 23 C. Marcott, A. E. Dowrey and I. Noda, Appl. Spectrosc. 47 (1993) 1324 ; B. Wang and T. A. Keiderling, Appl. Spectrosc. 49 (1995) 1347; M. Niemeyer, G. G. Hoffmann and B. Schrader, J. Mol. Struct. 349 (1995) 451.

289

Chapter 13 POLARIZATION DIVISION INTERFEROMETRY 13.1

Introduction

Most of the FTIR spectrometers used in research, industrial and teaching laboratories are based on the amplitude division interferometers (ADIs) discussed in the previous chapter. Polarization division interferometry provides an alternate choice which, at least in theory, is superior to amphtude division interferometry for certain applications. Three different types of polarization division inteferometers (PDIs) are known to datel-3; one is based on the division of polarized light by a Wollaston prism and the other two are based on the division of polarized light by wire-grid beamsplitters. Although the concepts of PDIs based on a Wollaston prism have been known 1 for a long time, actual spectroscopic applications have not emerged until recently4, 5. The spectroscopic applications of PDIs based on wire-grid beamsplitters appeared soon after the concepts were presented by Martin and Puplett 2. The thrust of these applications 6 remained mostly in the far-infrared and millimeter wavelength regions until recently. With the realization of the advantages7, 8 that these interferometers offer for polarization difference measurements, these interferometers have recently been developed for the mid-infrared region and new applications are being investigated in both the mid- and far-infrared regions 9-18. The commercialization of these interferometers has been much slower than desired. Nevertheless, numerous advantages and new applications available with polarization division interferometry make it worthy to expediate the development of these interferometers. 13.2

PDIs b a s e d on W o l l a s t o n p r i s m

A schematic diagram for this interferometer 1,4,5 is shown in Fig. 13-1. A Wollaston prism (designated as BS in Fig. 13-1) divides the incoming light (from an appropriate source or target) into two components with orthogonal linear polarizations. Commercially available Wollaston prisms are made from either quartz or calcite crystals. This prism contains two wedges held together such that it forms a plane parallel plate. The optic axes of the two wedges are perpendicular to each other, and are parallel to the respective external faces. If the incident light is linearly polarized (using a polarizer P), with the polarization axis at 45 ~ to the optic axis of the first wedge, then the two waves emerging from the prism would have equal amplitudes (with orthogonal polarizations). The two components of light exiting the prism travel with an angle ~ between their directions. This angle can be obtained from the birefringence of the prism material as, ~ -

Chapter 13

290

2(ne-no)tan0; here ne and no are the refractive indices for ordinary and extraordinary rays and 0 is the angle of the wedge. A linear polarization

, /

P

BS

A

L

D

Fig. 13-1. Polarization division interferometer based on a Wollaston prism. P: linear polarizer; BS: beamsplitter based on a Wollaston prism; A: linear polarization analyzer; L: lens; D: multi-element detector. analyzer (designated as A in Fig. 13-1) placed in their path will select the same polarization component (projected on to the analyzer's axis) for the two rays. An imaging lens (L) focusses these two rays on to a focal plane array detector (D), where they interfere and produce an interferogram. Each pixel of the detector array represents a different point of the interferogram in spatial domain. This interference results from the optical path difference 8 = dt~ between the two rays, which increases with the displacement (d) from the center of the prism along the face of the prism that receives incoming light. The Fourier transform of the interferogram recorded by the array detector yields the multispectral information of the source (that is, emission intensity of the source as a function of wavelength). The shortest wavelength that can be detected (or made to interfere at the detector) is given as ~min - 2 t~ w/M where w is the pixel width of the array detector (usually 25 ~tm) and M is the magnification of the optical system (determined by the focal length of the imaging lens). The spectral resolution, in wavenumbers, is given by the relation A9 - M/ctw. The operating range is determined by the design of the Wollaston prism (angle 0 and transmission properties) and the sensitivity of the detector. Using different diverging angles t~ and a magnification factor of 8 it can be seen that this system can perform anywhere from the visible region (above 300 nm) to the near-infrared region

Polarization Division Interferometry

291

( b e l o w - 2 ktm). Emission, absorption and Raman spectral measurements4, 5 using this interferometer have been demonstrated in the visible region, but can also be undertaken in the near-infrared region using a calcite prism. This interferometer is now commercially available (Photonex Inc, Glasgow, UK) and has the advantage that it has no moving parts and can be miniaturized. Due to the absence of moving parts, this instrument is expected to have much higher stability than the conventional interferometric systems and is suitable for rugged applications in the field. Since multispectral information is obtained from the Fourier transform of the interferogram registered at the detector, no mechanical scanning mechanism (such as that needed in the conventional techniques) is needed. As a result, analytical and quality control applications in an industrial setting can be anticipated for this interferometer. Major limitations of this interferometer are: (a) low spectral resolution; (b) difficulties in extending the measurements to the mid- and far-infrared regions. 13.3 PDIs based on wire-grid beamsplitters An interferometer based on the principle of polarization division using a wire-grid beamsplitter was introduced 2 by Martin and Puplett for the far-infrared region. In this interferometer the linearly polarized input fight is divided by a wire-grid beamsplitter into two polarization components and brought back to the beamsplitter using two roof-top mirrors. In a different design3, plane mirrors were used in place of the roof-top mirrors. These intefferometers represent an ideal choice for investigating the molecular polarization properties.

13.3.1 PDIs with roof-top mirrors The principles of PDI can be illustrated using a collimated ray from a light source (SO), as shown in Fig. 13-2. A linear polarizer, referred to as the input polarizer (P), polarizes the collimated fight. The linearly polarized light falls on a polarizing beamsplitter (BS). For the mid-infrared region, this beamsplitter was constructed 11 as follows. A wire-grid polarizer (metal wires on a suitable substrate (BaF2, KBr or KRS-5) and a matching blank substrate (with thickness matched to 0.001 inch, flamess to L/4 in the visible and zero wedge) were held together, with the grid surface at the interface of the two components. The spacing between them was defined by three shims placed 120 ~ apart on the rim. This assembled beamsplitter is mounted in a 45 ~ beamsplitter mount and the axis of the beamsplitter grids was rotated such that the projection of the grid direction is at 45 ~ to the incoming polarization direction. For the far-infrared region, beamsplitters made of free standing W wires (at 12 ~tm spacing) as well as aluminum wires (at 4 ~tm spacing) deposited on a 6 ktm thick mylar substrate were

292

Chapter 13

investigated 17. Since there is no substrate involved with free standing wire

M PDI/SPM S

O

M2

P2 ~ B S

....PDI !r(8

PD,

Ib 8)A

L1

L1

Itj~

Icd

~ ,L1 =-=SA

L2

~L2 q

(A) Fig. 13-2.

(B)

L1

L1

' SA

L2

'A

L2

q

(C)

(D)

Schematic of a PDI with roof-top mirrors.

configurations (A)-(E).

PO,

(E)

See text for details of the

grids, the issue of matching substrate does not arise. In the case of mylar, the film thickness is so small that a matching substrate is not required. Table 1 summarizes the characteristics of different beamsplitters.

293

Polarization Division Interferometry

TABLE 1 Polarization division beamsplitters for the mid- and far-infrared regions ,,|

Support substrate Blank substrate Substrate thickness Spacing between wires Lower frequency limit Residual

BaF2

KBr

KRS-5

BaF2 BaF2 3 mm 0.25 gm

KBr KBr 5 mm 0.25 gm

KRS-5 KRS-5 4 mm 0.25 gm

-800

-400

--300

-1-3%

--5%

-40%

W wires

mylar

none none 12 gm

mylar none 6 gm 4 gm

-25%

-12% i

Polarization of the incoming light can now be resolved (see Fig. 13-2) into two components, one parallel and the other perpendicular to the grid direction of BS. The parallel component (P1) is reflected to a fixed roof-top mirror (M1) equipped with orientational adjustments, while the perpendicular component ($2) is transmitted to another roof-top mirror (M2) that is movable. The polarization component P1 is rotated 90 ~ upon reflection by M1 to result in perpendicular polarization component (S1) and this component is now transmitted by the beamsplitter. Similarly the polarization component $2 is rotated by the mirror M2 to result in parallel polarization component (P2) and this component is now reflected by the beamsplitter. The components S 1 and P2 exit at 90 ~ to the direction of the input beam. When the mirrors M1 and M2 are at equal distances, there is no optical path difference between S 1 and P2, so this position is referred to as the zero path difference (ZPD) point. At the ZPD point, polarization of the output beam is identical to that of the incoming beam, which is set by the input polarizer P. As the mirror M2 travels away from the ZPD point, the polarization of the output beam changes depending on the length of the travel by the mirror M2. For monochromatic input light of wavelength ~,, the mirror travel distances of ~,/8, 2~,/8, 3~/8, and 4~,/8 correspond, respectively, to phase differences between $1 and P2 of ~,/4, ~,/2, 3~,/4 and ~,, or equivalently to polarization states of left circular, vertical, fight circular and horizontal polarization. This polarization modulation is depicted in Fig. 13-3. As the mirror travels to longer distances this modulation cycle is repeated. For polychromatic light, different wavelengths go through the above mentioned polarization states at different mirror travel positions, since for a longer wavelength correspondingly longer mirror travel is needed to complete one cycle of modulation. Since the mirror travel needed for spectra obtained at

Chapter 13

294

A9 cm -1 resolution is -- 1/(2 x A9 cm -1), the number of full polarization modulation cycles each wavelength component undergoes at this resolution is --1/(~,A9). For an interferometer scan to yield 4 cm -1 resolution spectra in the 3 - 25 Bm region, this varies between -800-100 modulation cycles. Thus this PDI serves to modulate the polarization of the input beam among

12.

H

LCP

8- - - * H

LCP

X(tm)

V

V

RCP

H

4LCP 0

1/2

2/2

RCP

H

LCP

3/2

4/2

5/2

6/2

RCP

H

7/2

8/2

Mirror travel from ZPD 0m) Fig. 13-3. Polarization modulation incorporated by PDI/SPM. all possible polarizations states and in the entire region appropriate for a given beamspiltter. As a result it is also appropriate to call this interferometer as a polarization modulation interferometer. In a later section we will discuss a double polarization modulation interferometer; in order to distinguish these interferometers the present PDI will be labelled as PDI/SPM, where SPM denotes single polarization modulation. While the longer wavelength limit of a mid-infrared PDI is determined by the transmission properties of the beamsplitter substrate, the shorter wavelength limit is determined by the beamsplitter grid spacing and roof angle accuracy. The grid spacing of the beamsplitter should preferably be four times smaller 7 than the shortest wavelength to be measured. This condition is satisfactorily met for the mid-infrared region by the beamsplitters with 0.25 Bm grid-spacing. Nevertheless, the extinction coefficient achievable with commercially available wiregrid polarizers is not

Polarization Division Interfe rometry

295

better than 200:1. Using newer methodologies in photolithography, wiregrid polarizers with extinction coefficients up to 10,000:1 (at 10 ~tm) have been made 19. Although they are not widely available, the use of such polarizers for constructing the beamsplitters will be essential for some applications. The roof angle of the mirrors should be 7 within (L/4D)(360/2~z) degrees of 90 ~ where D is the diameter of the beam. With 45 mm clear aperture of the beamsplitter, s for s 3 ktm would correspond t o - 3 arcsec. Commercially available roof-top mirrors, with 2 arcsec tolerance for the roof angle, set the shorter wavelength limit to be -2 ktm; for the beamsplitters with BaF2, KBr and KRS-5 substrates the longer wavelength cut-off is at--12, 25 and 30 ktm, respectively. The collimated light exiting the PDI is better handled with lenses to avoid polarization distortions that are commonly introduced by the fight focussing mirrors. This is particularly relevant here because light exiting PDI has polarization states that vary with the distance of the moving mirror and these polarization states are to be preserved to a high degree. For the mid-infrared region lenses made from KBr, ZnSe or BaF2 may be used. For the far-infrared region poly-IR lenses have better transmission than the Si lenses. 13.3.2 PDIs with flat mirrors When flat mirrors are used3,9,10 in place of the roof-top mirrors shown in Fig. 13-2, the components P1 and $2 retum to the beamsplitter with the same polarization. As a result, the exit beam traces the path of the input beam in the reverse direction. To separate these two beams, the input beam is made slightly off-axis and focussed before entering the interferometer. The exit beam is made to focus at the same place as the input beam, but since they are slightly off-axis a small prism placed at the focal point could be used to divert the exit beam into the sample compartment. Except for these differences in the optical design, the principles of operation would be the same for the two interferometers employing roof-top vs flat mirrors. 13.4 Principles of measurements Assumption of ideal properties for the optical components involved in PDI makes it convenient to formulate the relations involved. Let the source intensity at a given wavenumber be designated as IS(gi). Assuming that the source light is randomly polarized, the intensity of light exiting the polarizer P can be given as IS(g i)/2. For the light exiting the interferometer, polarization component P2 undergoes a phase change

27~89i relative to the polarization component S1, at a given wavenumber 9i. Here 8 represents the optical path difference (measured with reference to the ZPD position) which for conmuous-scan interferometers is 8=2Vt, with V

296

Chapter 13

representing the velocity of moving mirror and t the time. In step-scan interferometers, g=2x with x representing the distance of mirror travel from the ZPD position. For operating the interferometer in step-scan mode some type of source intensity modulation is needed. This can be achieved with a light chopper, rotating analyzer, or jittering mirror. We will address this aspect separately in a later section. The electric vector of the light propagating through the interferometer can be written following the procedure in Chapter 11 and the intensity of light at the detector determined. The results are summarized below.

13.4.1 Transmission configuration For different configurations shown in Figs. 13.2, the interferograms resulting from these phase changes are described below. (A) Residual interferogram In the configuration shown in Fig. 13-2A, light exiting the interferometer is transferred by two lenses to the detector, and no other components are placed in the beam. Since there is no intensity modulation, although the polarization is modulated, the intensity of light at the detector does not depend on the mirror position, and is equal to the total intensity of light transmitted by the input polarizer P. Thus this configuration should yield no interferogram. However, a small interferogram signal, referred to as the residual interferogram It(8), is seen in practice (see Fig. 13-4) which can be related to the non-ideal characteristics of the optical elements involved. An important point in the alignment of optical components is to minimize It(8). A useful quantity is the ratio [Ir(8)/Ib(8)]x100%, where Ib(8) is the background interferogram (vide infra), and these ratios, referred to as the residual, are summarized in Table 1 for different beamsplitters. (B) Background interferogram Intensity modulation as a function of the moving mirror position can be achieved by placing a linear polarizer, called analyzer (A), in the path of the fight exiting the interferometer (see Fig. 13-2B). The intensity of fight at the detector becomes, I = I [IS(x)i)/4] (1 + cos 2r~ 8x)i) dvi.

(13.4.1)

The + sign in Eq. (13.4.1) corresponds to the orientation of the analyzer axis parallel to that of the input polarizer; the - sign corresponds to the perpendicular orientation. Eliminating the dc component in this equation, a interferogram, referred to as the instrumental background interferogram Ib(8) (see Figs. 13-4 and 13-5) is obtained. For the parallel orientation of polarizer and analyzer, Ib(8) is given as,

297

Polarization Dfi~ision Interfe rometry

(13.4.2)

Ib(~5)- ~[IS(9i)/4] cos 2~:~5~i dgi.

The source intensity distribution IS(gi) can thus be obtained from the cosine Fourier transform of Ib(8). , , , ,

I , , , , .

.

1.5-

.

.

.

I , , .

,,

I , , , ,

"

I

,

,

,

,

(d)

_ ~t~,~

_,.

(c)

1.00"~ ri. 0

z>

0.5-

0.0-

-0.5I

3900

Fig. 13-4.

I

I

3950

4000

4050

Data point

41 O0

4150

Experimental interferograms obtained with PDI/SPM (traces a,c) and

PDI/DPM (traces b,d). a,b: Ib(8); c,d: Ir(~). All interferograms are on the same y-axis scale, but displaced from each other for clarity.

(C) Sample transmission interferogram The transmission spectra of samples are obtained by keeping the sample (SA) subsequent to analyzer A (see Fig. 13-2C). In this case the transmission interferogram becomes, It(8) - ~ [IS(x~i)/4] e "a(~'i) cos 2rc~Sx~idgi.

(13.4.3)

Chapter 13

298

Here a(gi) is the base e absorbance of the sample at wavenumber 9i [and is related to decadic absorbance A(x~i) as a(gi)= 2.303 A(gi)]. The ratio ,,

j

l , ,

-..,m.

-

I

. ~ .,ae,-F

, , , ,

--

-

I,

-- ,.s..ar,.m.

n

--

-

I , , , ,

I

,

,

I

.

.

.

.

(c)

mm , m . r . . . ~

I

,

I

,

,

,

,

I

-A'~,.~-,_...

'ue,,,.V~..m.me ,,m..-ue.,...-.me'.,~-.'tL-f.,.,.~'-'~

......

,

.|

"

% .ql~ i' .u%,,,r

(_ b_)

.

.

.

.

.

.

.

.

.

o~ .4.e e-,o I'l

-='4=.-~

-~--,=.'~-,..~.,"~

I

.D_..._.~,.-~

~-~--~..~-~..,,.,L.-

-

-1-" '

I''

5960

I

''

''

I''

4000

I ' ' ' '

I''

4040

I ' ' ' '

I''

4080

I'

' '' I 41 20

Data Poi nt

Fig. 13-5. Interferograms obtained in different configurations with PDI/SPM. (a). Ib(~) with the analyzer's axis parallel (dashed line) and perpendicular (full line) to that of the input polarizer; (b). Ic(8) obtained with the axes of birefringent plate and beamsplitter parallel to each other and with the analyzer's axis parallel (full line) and perpendicular (dashed line) to that of the input polarizer; (c). Ic(8) obtained with the axes of birefringent plate and beamsplitter perpendicular to each other and with the analyzer's axis perpendicular (full line) and parallel (dashed line) to that of the input polarizer.

of the cosine Fourier transforms of It(8) and Ib(8) gives the polarized transmission spectra of samples. Note that the spectra obtained with ADI using Eq. (12.2.3) are unpolarized transmission spectra. To obtain polarized transmission spectra with an ADI, a polarizer has to be introduced into an ADI, in which case the efficiencies of ADI and PDI are the same. For polarization difference studies, however, PDI has the advantage as will be seen shortly. (D) Dichroism interferograms In the configuration shown in 13-2D, analyzer A is not present and the light exiting PDI passes through the sample to the detector. For chiral systems in an isotropic phase, the absorption of left versus fight circularly polarized radiation can be different. When a chiral sample is placed in the optical path the detector signal is given as

Polarization Division Interfe rometry

299

+ [e aR(9i) I - I [IS(vi)/4] [e-aR(gi) eaL(vi)]{ l + t ~ ~ - eaL(gi)]sin 2rd59i}dgi + eaL (vi) (13.4.4)

where aL(vi) and aR(vi) are the absorbances for left and fight circular polarizations. For small differences in aL(gi) and aR(gi) values, following Chapter 11, one can write the ratio [e-aR(gi) - e-aL(gi)]/[e -aR(gi) + eaL(~'i )] as Aa(gi)/2 - [aL(Vi) - aR(gi)]/2. Also [e aL(gi) + e aR(gi)] can be replaced by 2e-a0?i) with a(gi) representing the average of aL(gi) and aR(gi). Then the circular dichroism interferogram extracted from Eq. (13.4.4) becomes, Icd(~5) - I [IS(vi)/2] e -a0?i) [Aa(gi)/2] sin 2~:~59i dgi.

(13.4.5)

The circular dichroism signal, Aa(gi) is extracted as the sine Fourier transform of Eq. (13.4.5) divided by the cosine Fourier transform of It(~5), as follows: Ilcd(8) sin 2n89i d~5 /IIt(~5) cos 2rc~59i d8 - Aa(gi).

(13.4.6)

A comparison of Eq. (13.4.6) with (12.2.15) reveals that PDI is two times more efficient than ADI for circular dichroism measurements (note that 2J1(8 0vi) in Eq. (12.2.15) is ~1 for quarter-wave retardation). The representative CD spectra measured with PDI are shown in Fig. 13-6. For oriented non-chiral samples, the absorption of radiation with two orthogonal linear polarizations can be different. In this case the signal at the detector can be written as,

I -

I IS(vi)/4] [eap (9i)

+

e "as0?i)] { 1+

[e -ap0?i)

e----|cosaS0?i)-! 2rC89i} dvi t" e-ap(Vi ) + e as(gi)J (13.4.7)

where aP(vi) and as(gi) are the absorbances res Pectivel Y for P olarizations . parallel and perpendicular to the sample's axis. For small differences, one can follow the recipe used in obtaining Eq. (13.4.5) from Eq. (13.4.4). Then the linear dichroism interferogram obtained from Eq. (13.4.7)

Chapter 13

300 becomes, . . . .

.==

I,,

,,

I

. . . .

I

. . . .

1

''

I,

,,

I

. . . .

,I

....

,I

,,

,,

''

''

I,

,_

I'

'

~l

_

o

(b)

r r

q. 03 r

,r 0.5

~ 1 O0

I'

1:550

''

'1

. . . .

13;00

I

. . . .

1250

1200

''

1150

I

. . . .

1100

I

1050

1000

wavenumbera

Fig. 13-6. Vibrational absorption (bottom trace) and circular dichroism (top three traces) of (-)-a-pinene. Trace (b) was obtained with PDI/SPM; traces (c) and (d) were obtained PDI/DPM at O~mand 2ram respectively. Ild(5) - I [IS(vi)/41 [e "ap(9i) + e "as(9i)] [Aa(vi)/2] cos 2~89i dvi ,

(13.4.8)

where Aa(gi) - as(gi)-ap(gi). The linear dichroism can be extracted as, 2 111d(8)cos 2~89i dS/[I IP(8) cos 2~;8~i d8 + I I~(8) cos 2~89i dS] =Aa(x?i).

(13.4.9)

In other words, the cosine Fourier transformation of Ild(8) divided by the average of the cosine Fourier transformations of IPt(8) and I~(8) will yield the linear dichroism; IPt(8) and I~(8) represent, respectively, the It(8) obtained with parallel and perpendicular orientations of analyzer with respect to the sample's axis. For the two orthogonal orientations of the input polarizer, or of the sample's axis, Eqs. (13.4.8) and (13.4.9) will have opposite signs. A comparison of Eqs. (12.2.19) and (13.4.9) indicates, as in the case of circular dichroism, a factor of two gain in efficiency for PDI over ADI for linear dichroism measurements. The

Polarization Division Interferometry

301

experimental linear dichroism spectra measured with PDI are shown in Fig. 13-7. It is important to remember that the linear dichroism magnitudes determined from Eq. (13.4.9) reflect true values only when Aa(gi) is small. This is not usually the case and sometimes Aa(gi) can be as large as 1.0. I,, ,,I,,

,,I,,

,,I,,

,,I,,

,,I,,

,,I,,

,,I,,

,,I,,

,,I,,

,,I,,

,,I,,

,,I,,

,,I

-2

:~ 2r O IO

O"

1500

~

1400

1:500

1200

~ 1100

1000

"

l~]

1F!

91

900

~r

Fig. 13-7. Linear dichroism spectra of polypropylene obtained with PDI/SPM and as the static difference between the polarized absorption spectra are shown as traces (d) and (c) respectively. Traces (a,b) are the two polarized absorption spectra. The dichroism spectra obtained with PDI/DPM as 2 J1 ( 80 ) ix A A ( g i )

at O~m (n=l; trace g), 2 J 2 ( 8 0i) x

AA($i) at 2ram (n=2; trace f) and AA($i) at n=0 (trace e) are also shown for comprison. An alternate way to derive the linear dichroism is by normalizing with the individual transmission spectra. Starting from Eq. (13.4.7), normalizing the cosine Fourier transform of Ild(8) with the cosine Fourier transform of ItP(8) gives,

302

Chapter 13

Ild(8) cos 2n 8~)i d8 / ~ I p (8) cos 2rc89i d8 _ [e-ap(vi) _ e-as(vi)] / eap(vi ) , -

[Aa(vi)] 3

Aa(vi) - [Aa(Vi)]2 2

(13.4.10)

Similarly normalizing with cosine Fourier transform of I[(5) gives, Ild(8) cos 2rt89i d 8 / ~ IS(8)t cos 2~:89i d8 _ [e-ap(Vi) _ e-as(Vi)] / eas@i ) ,

-

Aa(Vi) +

[ha(Vi)]2 2

+

[Aa(vi)]3 6

+ ....

(13.4.11)

Adding Eqs. (13.4.10) and (13.4.11) and dividing by 2 leads to, [~2]{[~ Ild (~) c~ 2 ~ i d 8 ] ~ IP (~) cos 2~SvidS] +

= Aa(~i)

+ [Aa(Vi)]3 0 0 0

6

(13.4.12)

For measuring Aa(gi) of-1, processing the data with Eq. (13.4.12) gives -16% higher values while processing the data with Eq. (13.4.9) gives --10% lower values. For better accuracy in the measured linear dichroism values, one may average the results obtained with Eqs. (13.4.9) and (13.4.12). (E) Calibration interferograms In order to determine the absolute magnitudes of measured linear dichroism and circular dichroism, it is necessary to calibrate the magnitudes. This can be achieved with a waveplate that imparts multiple wave retardations (see Fig. 13-2E). Here, the light exiting PDI passes through a birefringent waveplate (W) with its fast axis either parallel or perpendicular

303

Polarization Division Inte rfe rome try

to the grid axis of the beamsplitter; then passes through analyzer (A) whose polarization axis is either parallel or perpendicular to that of the input polarizer (P). With the axes of input polarizer and analyzer being parallel, the detector signal is given as, I

_ f

B

(13.4.13)

J [IS(~'i)/4] [1 + cos (89i_ 2~:~ix)i)] dvi,

where ~Sviis the retardation introduced by the waveplate at v i and the + signs correspond to the orientation of waveplate's fast axis parallel or perpendicular to the grid axis of the beamsplitter. For one of these orientations, the interferogram Ic(8) derived from Eq. (13.4.13) is Ic(~5)- I [IS(gi)/4] [cos ~SvB-icos 2 ~ 5 9 i - sin ~ivB-i sin 2~;59i] dgi. (13.4.14) The interferograms obtained in calibration configuration are shown in Fig. 13-5. Thus the separate cosine and sine Fourier transforms of Ic(~) divided by the cosine Fourier transform of Ib(8) yield, respectively, tJ Ic(8) cos 2rc~gi d5 / rJ Ib(8) cos 2~:59i d8 - cos ~BV i

'

I Ic(~5) sin 2n 89i d8 / I Ib(~i) cos 2~;~vi d~5 - - sin 5Bi .

(13.4.15) (13.4.16)

Noting that 8 B vi - 2rcgidAn, where d is the thickness of the waveplate and An is the difference in refractive indices along the two axes (i.e. birefringence) of the waveplate, it can be seen that Eq. (13.4.15) gives _+1 for those wavelengths at which the waveplate introduces integral multiple of L/2 retardation and these maxima represent unit linear dichroism. Similarly, Eq. (13.4.16) gives _+1 at those wavelengths for which retardation introduced by the waveplate is odd integral multiple of ~/4 and these maxima correspond to unit circular dichroism. The calibration curves are shown in Fig. 13-8. In addition to providing a means to calibrate the absolute magnitudes of the measured linear and circular dichroism signals, this measurement also provides a means of verifying the correct behaviour of the instrument. Integrating Eq. (13.4.13) or (13.4.14) in the frequency range of the instrument, it can be seen that the maximum intensity point in the interferogram Ic(8) will shift 12 from the ZPD point (see Fig. 13-5) and appears at a mirror travel from the ZPD, given as

Chapter 13

304 8 - + dAn.

(13.4.17)

Here it is assumed that An is constant in the frequency range of interest. The two signs in Eq. (13.4.17) correspond to the above mentioned two orientations of the waveplate's fast axis with respect to the grid direction of 1.0

0.5

i

,

J

,

"

i

i

,

/ rl '

I

0.0

I

I

't '

I

I "i

I

I

I

I

I "i

i

i

,

J .'IL '

I I

_

I

I Ji

I

I

i

I

11

I. e,

,

,

,

I "I '

I I

"i

',

i

I "i ~

I I

', I

,

', i.

I

I I "i ~

I "i L

,

J 'i

I

. . . .

I

( '~

'

I "l

.~

I ', I

I

I

I I

I

-0.5

-1.0

i

i

:5000

2500

2000

1 500

i

i

="

1000

wavenumbers

Fig. 13-8. Calibration curves obtained with PDI/SPM, as the cosine (full line) and sine (dashed line) Fourier transforms of a calibration interferogram. the beamsplitter. The utility of Eq. (13.4.17) is not limited to the cases where An is already known. For oriented samples, which exhibit different refractive indices along two mutually perpendicular axes, the birefringence An can be measuredl2; also temporal or field induced variations of birefringence resulting from an external stimulus may be monitored simply by monitoring the shift in the interferogram peak position.

13.4.2 Reflection configuration For the two configurations shown in Figs. 13-9A and 13-9B, the interferograms are described below. (A) Differential polarized reflectance interferograms If the sample is simply a clean metal surface then very small intensity modulation is expected to be seen at the detector, because for a metallic substrate the difference in reflectivities of two orthogonal linear polarizations is expected to be small. For a dielectric substrate reflectivities for different polarizations may not be the same, so the interferogram in this case is expected to contain an intrinsically large differential polarized reflectance signal from the substrate. For the arrangement shown in Fig.

Polarization Division Interfe rometry

305

13-9A, the interferogram is given as, Idb(8)- I [Ipb(Vi) - Isb(Vi)] cos 2 ~ i 9 i d v i ,

(13.4.18)

where Isb(Vi) = IS(vi) Rsb(Vi)/4, and Ipb(Vi) = IS(vi) Rpb(Vi)/4 are the intensifies of reflected light from the bare surface and subscripts s and p

D SA

D SA

Fig. 13-9.Reflectance configurations to obtain: (A) differential polarized reflectance interferograms of bare and sample-deposited surfaces; (B) polarized reflectance inteferograms of bare and sample-deposited surfaces. refer respectively to the incident electric vector oriented perpendicular and parallel to the i-r plane; Rsb(gi) and Rpb(Vi) are the reflectances of a blank surface for s and p polarizations. For samples on surfaces, differential polarized infrared absorption might occur depending on the relative orientation of vibrational transition moments with respect to the incident electric vector polarization. For achiral samples, this differential absorption originates from the interaction of vibrational transition moments with s and p polarized incident electric fields. As a consequence, reflected intensities depend on the polarization state of the incident light. The resulting interferogram is given as Id(8)- I [Ip(gi) - Is(vi)]

cos 2 ~ : 8 9 i d v i ,

(13.4.19)

where Is(vi) = IS(gi) Rs(gi)/4 and Ip (Vi) - IS(vi) Rp(gi)/4 are the intensities of light reflected from the sample-deposited-surfiice for s and p polarized incident light; Rs(vi) and Rp(gi) are the reflectances of sample-depositedsurface for s and p polarized incident light. Cosine Fourier transform of Id(8) gives the difference spectrum, [Ip(gi) - Is(vi)]. It may be noted that

306

Chapter 13

[Ip(gi)-Is(gi)] also contains a contribution from the bare surface, since as mentioned above such difference for dielectric substrates is significant. (B) Polarized reflectance interfe rograms Introduction of analyzer A (see Fig. 13-9B) at the exit port of PDI results in intensity modulation as a function of the moving mirror position, and the function of PDI now becomes identical to that of an ADI with a linear polarizer in the output beam. For a blank substrate the interferogram, referred to as the polarized reflectance interferogram of bare surface Ijb(5), is given as Ijb(8) - Gj ~ Ijb(X)i) cos 271;8X)i dx)i,

(13.4.20)

where the subscript j - s or p, indicating the polarization state; G"J = 1 when the axes of analyzer and input polarizer are parallel, and -1 when they are perpendicular. The polarized reflectance spectrum Ijb(Vi ) of a bare surface can thus be obtained from the cosine Fourier transform of Ijb(5). For a sample-deposited-surface, the polarized reflectance interferogram becomes, Ij(5) - Gj ~ Ij(x)i) cos 2r:~59i dgi,

(13.4.21)

where, j1"(90 represents the intensity of light reflected from the sampledeposited-surface with j - s or p. As mentioned earlier this quantity contains contributions from the sample as well as substrate. The cosine Fourier transform, of 1"(5)j gives the polarized reflectance spectra of the sample-deposited-surface. The differential polarized reflectance spectrum is contained in the Fourier transform of Eq. (13.4.19), but due to the inherent presence of source intensity dependence in this equation it is necessary to normalize the spectrum. This may be achieved from Eqs. (13.4.19) and (13.4.21) as,

~Id(~5) cos 2~;8x)i d~5 / [~Is(gi) cos 2r:Sx)i d~5 + ~Ip(gi)cos 2~;89i d~5] = [Rp(gi)-Rs(vi)] / [Rp(vi) + Rs(x)i)] - (AR]R).

(13.4.22)

If the bare surface exhibits differential polarized reflectance, then it is necessary to write an analogous equation for the bare surface and subtract it from Eq. (13.4.22). Using Eqs. (13.4.18) and (13.4.20), one obtains

307

Polarization Division Interfe rometry

Ildb(8) cos 2rc89i d8 / [Ilsb(Vi) cos 2rr

d8 + IIpb(gi) cos 2rc89i dS]

= [Rpb(gi)-Rsb(X?i)] / [Rpb(Vi) + Rsb(Vi)] - (AR/R)b,

(13.4.23)

and from Eqs. (13.4.22) and (13.4.23), (aR/R)

- (aR/R)b

=

(aR/R)corr

9

(13.4.24)

Differential reflectance measurements with PDI have been demonstrated for monolayers on water surface and gold substrate. 13.4.3 Raman scattering

Raman scattering experiments with PDI have not yet been reported. An experimental configuration for the Raman scattering measurements 20 is shown in Fig. 13-10. The sample is illuminated with linearly polarized monochromatic radiation with its electric vector either perpendicular or parallel to the scattering plane. The 90 ~ scattered Raman fight collimated by a lens serves as the incoming light source for PDI. When chiral molecules are illuminated with plane polarized monochromatic light the 90 ~ scattered Raman light can have linear as well as circular polarization components (see Chapter 9). First consider the scattered light with polarization parallel to the scattering plane and intensity Ih(gi). In this case, the detector signal is given as I - I Ih (vi)(1 + cos 2rt~iS)dv i . 2

(13.4.25)

For the scattered Raman light with polarization perpendicular to the scattering plane and intensity IV(gi), one gets IV

I=I

(~i) 2 (1-cos2rtV iS)dvi

9

(13.4.26)

Similarly for the scattered Raman light with circular polarization and intensity 1+(90, I- I

i+

(Vi)(l+sin2wViS) dVi ' 2

(13.4.27)

Chapter 13

308

where + and - correspond, respectively, to fight and left circularly polarized Raman light. If all four polarization components discussed above are present in the scattered Raman light, then the signal at the detector is simply

M PDI/SPM S~, M~~,

S

D

b..........O

P2 '~---/JBS

A

Collimating lens

P Sample Fig. 13-10. Experimental arrangement for measuring Raman scattering with PDI. the sum of the above three equations. As a result the Raman difference spectrum, (Ih((ti) - Iv(g i)), is obtained directly as the cosine Fourier transform; the Raman optical activity spectrum, (I+(('i) - I-(9i)), is obtained as the sine Fourier transform of the same interferogram. The aforementioned points can be considered rigorously by representing the scattered Raman light with a complex electric vector. Some important conclusions can be summarized 20 as follows. If the objective is to measure the depolarized ROA, then the incident laser polarization is to be oriented in

Polarization Division Interferometry

309

the scattering plane. For polarized ROA the incident laser polarization is to be oriented perpendicular to the scattering plane. When the objective is to measure the depolarization ratios of Raman bands, the incident laser polarization needs to be perpendicular to the scattering plane and an analyzer needs to be incorporated at the collecting lens. Two measurements with this analyzer oriented parallel and perpendicular to the scattering plane give, respectively, the signals represented by Eqs. (13.4.25) and (13.4.26). The absolute ratios of the intensities obtained from the cosine Fourier transforms of the corresponding interferograms provide the depolarization ratio. If the negative sign present in Eq. (13.4.26) is of concern, the axis of analyzer at the output end of PDI can be set parallel to that of the analyzer at the input end and both rotated simultaneously. Then the signs of cosine terms will remain positive in both Eqs. (13.4.25) and (13.4.26).

13.4.4 Emission spectra For samples which support differential polarized emission the methodology is identical to the one for ROA measurements. The laser source is replaced by a suitable exciting light source. The terms representing the Raman scattering intensities in Eqs. (13.3.27)-(13.3.29) are to be replaced by emission intensities. 13.5 Additional modulations In step-scan interferometers, where the movable mirror is advanced in step increments, the mirror is held fixed at each increment and the detector signal digitized. In this type of operation it is necessary to introduce some kind of intensity modulation to avoid the problems associated with source intensity variations during a particular scan. One choice is to use a mechanical chopper, in which case the source intensity is modulated at the frequency of the chopper. Demodulating the detector signal using a lock-in amplifier tuned to the chopper frequency provides the appropriate signal needed depending on the configuration used. However, one would lose a factor of two in efficiency, since light is blocked by the chopper 50% of the time. A second choice is to use a rotating analyzer 7 in which case the detector signal oscillates at twice the frequency of rotating analyzer. Since light exiting the rotating analyzer no longer contains circular polarization states, this arrangement is not suitable for circular dichroism measurements 21. Instead, one can use a rotating achromatic quarter-wave retarder (QWR) in combination with a static linear polarization analyzer; this results in modulating the source intensity at 2co and 4o~ where co is the rotating frequency of QWR. In the absence of a static analyzer, a rotating QWR will modulate 22 the circular dichroism signal at 2co. In a fourth type of modulation the fixed mirror is jittered sinusoidally at frequency o~f, and this type modulation is referred to as phase modulation 23. This phase

Chapter 13

310 modulation is described here. difference changes as

In phase modulation, the optical path (13.5.1)

- 80 + 2D sin o~ft,

where D is the maximum displacement of the jittering mirror and 80 is the optical path difference at zero displacement position of the jittering mirror. Substituting Eq. (13.5.1) for 8 in Eqs. (13.4.1)-(13.4.14), one can derive the relations required for individual configurations. Additional information needed here is to write cos2rcSvi - cos(2rcv80 + 4gDv i sino~ft) and sin2rcSV i = sin(2rp780 +4nDVisino~ft); and use the relations 24, sin[4r~DVi sino~ft ] - 2Jl(a~DVi)sino~ft+... and cos[arcDVi sin raft] = J0(4rcDvi) + 2J2(4rcDVi)cos2o~ft+ . . . . Here J0 (4nDVi), J1 (4r~DVi) and J2 (4rcDvi) are Bessel functions. When the detector signal is demodulated by a lock-in amplifier tuned to the jittering frequency, the relations for background, transmission and linear dichroism and circular dichroism interferograms become 21, Ib (80) - - I [ Is (Vi)/412J1 (4rcDvi) sin 2rcS0vi

,

It (80) - - I [Is (vi)/4] e-a(~i )2J1 (4~Dvi)sin2 rcS0vi I l d ( 8 0 ) - - I [ I S ( V i ) / 4 l i e -ap(Vi) + e -a~ x 2J 1(4rcD~i)sin2r~0~ i

(13.5.2)

dVi ,

(13.5.3)

(Vi)][Aa(Vi) / 2]

d~ i ,

(13.5.4)

Icd(~i0)- I[IS(vi)/4][e -aR(~i) + e-aL(~i)][Aa(vi)/2] x 2J l(4rcD~ i)cos2rc~0~ i

d~ i .

(13.5.5)

It is to be noted that the background, transmission and linear dichroism spectra are obtained from the sine Fourier transforms of the corresponding interferograms; the circular dichroism spectra are obtained from the cosine Fourier transform of the circular dichroism interferogram. The Bessel function J1 (4r~DVi) has maximum value of 0.58 when 4nDgi - 1.832. Therefore for a given D value, the wavelength which is most efficiently modulated is given as 6.86 D. For this wavelength, phase modulation is -2 times more efficient than chopper modulation. However,

Polarization Div&ion Interfe rometry

311

J1 (4~;Dvi) values fall below 0.3 at 4rcDgi - 3.1 and 0.6. As a result, the efficiency of phase modulation becomes poorer than chopper modulation outside the wavelength range of 4D-21D. It would then become necessary to repeat the measurements with different D values, each D value chosen to achieve maximum efficiency in the infrared region of interest. An alternate way to measure Ib(~5), It(~5), I/d(~5) and Icd(~5) is to demodulate the detector signal using a lock-in amplifier tuned to twice the jittering frequency. In this case, the appropriate interferograms become: Ib ([i0)= ~[IS(~i)/412J2 (4rcD~i)cos2~80~i

d~i,

I t ( 8 0 ) - ~[IS(~i)/4]e-a(~i)2J2 (4rcD~i)cos2~50~i

(13.5.6) dVi,

Ild(80) = ~[IS(vi) / 4][e -ap(~i) + e-a~ (vi)l[Aa(vi) / 2] x 2J 2 (4rcD~i) cos 2rc~50~i

dvi

(13.5.7)

(13.5.8)

,

Icd(80)- ~[IS(vi), 4][e -aR(Vi) + e-aL(Vi)][Aa(~i)/2] x 2J2(4rcD~i)sin2~:80V i

d~ i

.

(13.5.9)

The background, transmission and linear dichroism spectra are derived from the cosine Fourier transforms of Eqs. (13.5.6), (13.5.7) and (13.5.8), respectively. The circular dichroism spectra are derived from the sine Fourier transform of Eq. (13.5.9). When this approach is used the maximum displacement of the jittering mirror must be chosen such that J2(4rcDgi) is maximum for the frequency region of interest. A PDI can also be configured to serve as an achromatic polarization modulator 22. Consider a light source that is either tunable in wavelength or dispersed with a monochromator. If the moving mirror is held fixed at the ZPD point [i.e. ~50in Eq. (13.5.1) is zero] and the fixed mirror is jittered as described above, then the desired polarization for a wavelength of interest can be obtained by adjusting the displacement amplitude D of the jittering mirror. Polarization modulation at a frequency (of between two orthogonal linear or circular polarizations can be achieved in this manner. This concept will be particularly useful for the far-infrared region. 13.6

Time resolved measurements

The procedures for obtaining time resolved unpolarized absorption

312

Chapter 13

measurements with continuous-scan ADIs (using stroboscopic and asynchronous perturbation methods) and with step-scan ADIs have been described in the previous chapter. The same procedures can be used with PDI/SPM. However, note that for absorption measurements with PDI/SPM a analyzer is needed before the sample, so these measurements correspond to polarized absorption. A polarizer should also be placed before the sample for polarized absorption measurements with ADI. As a result, the efficiency for polarized absorption measurements is, in principle, the same for PDI and ADI. With regard to obtaining the time resolved dichroism spectra with a continuous-scan ADI, two different time resolved polarized absorption measurements are made and the difference between these polarized absorptions yields time resolved dichroism. Obviously, this approach requires two separate measurements. In a PDI/SPM, however, the interferograms obtained in the absence of analyzer (before the sample) directly represent the difference between polarized absorptions. Since there are no high frequency external polarization modulations involved in a PDI/SPM, the microsecond scale time domain is fully available for monitoring the temporal changes in dichroism interferograms. Fourier transforms of these time dependent interferograms yield time resolved dichroism. Thus PDI/SPM offers 14 a potentially valuable approach for time resolved dichroism measurements. These concepts have been tested 14 using a continuous-scan PDI (with asynchronous perturbation method described in the previous chapter) by monitoring the temporal function of a PEM on microsecond time scale. Also microsecond time resolved linear dichroism of a liquid crystal under external electric field perturbation has been measured. Since vibrational circular dichroism magnitudes are already small, time resolved vibrational circular dichroism measurements are much more challenging and yet to be undertaken.

13.7 Double polarization modulation interferometers Here we discuss a double polarization modulation interferometer 18 which combines the PDI with a photoelastic modulator (PEM). This interferometer is designated as PDI/DPM and depicted in Fig. 13-11. A PDI based on wire-grid beamsplitters also serves as a polarization modulator as discussed in Section 13.3. The polarization of light falling on the PEM (see Fig. 13-11) is continuously changing with the mirror travel. Consider the situation where the unique axis of the PEM and the direction of the wires on BS are parallel to each other. When the light falling on the PEM has linear polarization at 45 ~ to the PEM's axis, the polarization of light exiting the PEM is modulated between two orthogonal circular polarizations (at the characteristic frquency mm of the PEM); for circular input polarizations, the output polarizations are modulated between linear

313

Polarization Division Inte rfe rometry

polarizations. As a result, the PEM is adding a higher frequency polarization modulation on top of the slow polarization modulation resulting from the mirror movement. This double polarization modulation has several consequences as described below.

PDI/DPM P1 S1

M2

$2 P2

,~L~

~

~~'BS

A

D

PEM Fig. 13-11. Schematicof double polarization modulation interferometer.

13. 7.1 Transmission Configuration (A) Residual Interferogram Ir(3) The light exiting the interferometer is transferred by two lenses to the detector (as in Fig. 13-2A), and no other components are placed in the beam. Assuming that these lenses do not distort the polarization, the intensity of light exiting the interferometer does not depend on the mirror position, but is a constant equal to the total intensity of light transmitted by the input polarizer P. However, due to the optical imperfections and due to possible amplitude division interference at the beamsplitter, a small interferogram signal, referred to as the residual interferogram Ir(8), is seen in practice which is identical to that in PDI/SPM. A residual interferogram obtained with PDI/DPM is displayed in Fig. 13-4, where Ir(fi) obtained with PDI/SPM was also included.

314

Chapter 13

(B) Background interferogram Ib(t~) Intensity modulation can be achieved by placing a linear polarization analyzer in the path of the light exiting the interferometer (as in Fig. 132B). The intensity of light at the detector becomes I - ~ [IS(gi)/4] {1 +_ cos [2nfigi_+ ~m]} dgi,

where 6mvi _ ~0i" sin tOmt

and

(13.7.1)

~0i" is the maximum retardation introduced by

PEM at wavenumber x)i. The + sign in Eq. (13.7.1), prior to the cosine term, corresponds to the orientation of the analyzer axis parallel (+ sign) or perpendicular (- sign) to that of the input polarizer; the + sign in the argument of cosine term corresponds to parallel (+ sign) or perpendicular (sign) orientations of the axes of PEM and BS. Eliminating the dc component in this equation, the interferogram for the parallel orientation of the axes of polarizer and analyzer and of PEM and BS, referred to as the insmmlental background interferogram Ib(8), is given as Ib(~) -

I

[IS(x)i)/4] [ Jeven cos21t~ix)i - Jodd sin2~SX)i ] dx?i,

(13.7.2)

where, OO

Jeven - J0(80i)+ 2 Z J 2 n ( ~ O i ) c o s 2n~mt, n=l

(13.7.2a)

Jo -

n=l When the harmonics at tom, 2tOm, etc are eliminated with the use of a low pass filter, the background interferogram Ib(6) becomes,

Ib(~5)- ~ [Is(~?i)/4] J0(80i)cos2ffSvi d~?i.

(13.7.3)

Background interferograms obtained with PDI/DPM and PDI/SPM are compared in Fig. 13-4. Note that the cosine Fourier transform of Eq.(13.7.3) gives the source intensity distribution multiplied with the

Polarization Division Interferometry

315

Bessel function J0(80i). In a PDI/SPM this Bessel function would not be present (see Eq. 13.4.2) and this important difference can be put to good use as follows. The cosine Fourier transform of Eq. (13.7.3) divided by that of Eq. (13.4.2)gives J0(~Oi), which in t u r n can be used to characterize a polarization modulation device such as a PEM. One may also derive Jl(80i) and J2(~iOi) spectra similarly by using ~m and 2~m components of Eq.(13.7.2). The experimental J0 (g0 i ) spectrum derived in this manner is shown in Fig. 13-12. (C) Sample transmission interferogram It(& As in the configuration shown in Fig.13-2C, the transmission spectra of samples are obtained by keeping the sample subsequent to

0.6(d)

0.4

C

J2

0.2 =

0.0

] -0.2

"~

-,~x,

d0,fl e"

-

-0.4-

3500

3000

2500

2000

1500

1000

vavenumbers

Fig. 13-12. Theoretical (traces b-d) Bessel functions in the mid-infrared region with onehalf wave retardation at 2000 cm"1. The J0(~50i) spectrum obtained with a PDI is shown as trace (a). analyzer A. In this case the transmission interferogram becomes,

316

It(8) -

Chapter 13

~ [IS(x)i)/4] ea0?i) {[ Jeven cos2~Sx?i - Jodd sin2~:89i ] dx)i. (13.7.4)

When the harmonics at tom, 2r etc are eliminated with the use of a low pass filter, the ratio of the cosine Fourier transforms of It(5) and Ib(8) gives the polarized transmission spectra of samples. (D) Dichroism interferograms As in the configuration shown in Fig. 13-2D, analyzer A is not needed and the light exiting PDI/DPM passes through the sample to the detector. First let us consider linear dichroism interferogram Ild(5). For oriented samples exhibiting linear dichroism (LD) the signal becomes, I - I [IS(x)i)/4] { [e-ap(~'i)+e"as(9i)] + [e-ap0?i)-e"as0?i)]

x cos [2nfiVi + 5~m ] } dx)i,

(13.7.5)

where ap(X?i) and as(gi) are the absorbances for parallel and perpendicular polarizations. For small differences of A a (gi)=as(gi)-ap(X)i), the interferogram from Eq. (13.7.5) becomes, Ild(5)- ~ [IS(vi)/4] [e-ap(~?i)+eas0?i)][Aa(gi)/2] x [ Jeven cos2g~x?i - Jodd sin2~;[igi ] dvi.

(13.7.6)

Note that the linear dichroism signal Aa(gi) is present at all integer multiples of tom (i.e. at nO)m, with n = 0, 1, 2 etc). Thus one may extract the linear dichroism components at ntOm (n=l, 2 etc) with the use of a lock-in amplifier (tuned to Ohn, 2Ohn, etc) and another component (n=0) without the use of a lock-in amplifier. The linear dichroism spectra of a stretched polypropylene film obtained with PDI/DPM at n=0,1 and 2 are shown in Fig. 13-7, where the linear dichroism measured with PDI/SPM is also shown. For deriving a circular dichroism interferogram, Icd(~i), the detector signal can be written as

317

Polarization Division Interfe rometry

I-I

[IS(gi)/4] {[eaR(~'i)+eaL(vi )] + [eaR6)i)_eaL(~?i)] (13.7.7)

x sin [21r89i+ ~iV m ] }dx)i,

where aL(gi) and aR(gi) are the absorbances for left and right circular polarizations. As before, for small magnitudes of Aa(gi) - aL(vi)- aR(vi), the interferogram from Eq. (13.7.7) becomes, Icd(~i) - I [IS(vi)/4][e "aL(('i)+eaR(r

[Aa(gi)/2]

x [Jeven sin2r~59i -Jodd cos2rc~59i ] dgi.

(13.7.8)

Note that the circular dichroism signal is also present at all integer multiples of O~n (i.e. at nO~m, with n = 0, 1, 2 etc). Thus one may extract the circular dichroism components at nCOm (n=l, 2 etc) with the use of a lock-in amplifier (tuned to O~n, 2O~m,etc) and another component (n=0) without the use of a lock-in amplifier. The circular dichroism spectra of a-pinene obtained with PDI/DPM at O~mand 2O)rn are compared to that obtained with PDI/SPM in Fig. 13-6. Note that VCD spectrum obtained at 2Comwith PDI/DPM would be noisier because for quarter-wave retardation in the region displayed in Fig. 13-6, J2(80i )has lower magnitudes than Jl(80i ) (see Fig. 13-12). But three separate dichroism measurements with n = 0, 1 and 2 increase the experimental time accordingly. If all three components can be measured simultaneously then PDI/DPM becomes two times more efficient over PDI/SPM, and four times more efficient over ADI as shown below. The dichroism signals at n=0, 1 and 2 can be extracted into separate files

o )_+ and one can derive therefrom any of the combinations [ § J0 (8vi

2J2(80i) + 2J1(80i)] x [Aa(gi)/2 ]. Two such combinations [J0(80i) +

2J2(80i) +2J1(80i)1

x [Aa(gi)/2]

and [-J0(80i) + 2J2(~0i)+2Jl(~iOi)]

x [Aa(gi)/2] are useful for the present discussion. functions for n= 0, 1 and 2 obey the relation 24,

Since the Bessel

318

Chapter 13

J1(~0 i ) - [ 80i/2][ J0(80i ) + J2 (80 i ) ],

(13.7.9)

it can be seen that for one quarter-wave retardation [where 80. - n/2, 1

Jl(~50i )--0.57 and J2(~50i )-0.25] the combination, [ J0(~0i )+ 2 J2(~0 i ) + 2 J1 (~0 i )]x[Aa(x~i)/2], of the dichroism signals becomes ~ 2 x [Aa(gi)/2]. Similarly, for one half-wave retardation [where ~0 i Jl(80i ) ' - 0.30 and J2 (50 i ) ~ 0.49] the combination, [-J0(80i ) + 2 J2 (50 i ) + 2 J1 (~0 i )]X --

~,

[Aa(gi)/2], of the dichroism signals becomes ~ 2 x [Aa(gi)/2]. Thus, if all three components of circular or linear dichroism are measured simultaneously, then P D I ~ P M measurements give twice as much signal. In PDI/SPM measurements the dichroism signal measured is proportional to Aa(gi)/2 and the Bessel functions do not appear. In ADI measurements on dichroism, the circular dichroism appears only at nohn with odd values of n (and only n=l is important); the linear dichroism appears only at n~m with even values of n (and only n--0 and 2 are important). This reason, along with the fact that ADI is two times less efficient over PDI/SPM for dichroism measurements (because one-half of the light in ADI goes back to the source), makes PDI/DPM four times more efficiem over ADI. It is also useful to remember that PDI provides larger dynamic range than ADI for dichroism measurements. For a given PEM setting different wavelengths go through different retardations and Jn(80i ) . , values vary across the wavelength region of interest, leading to variations in signal strength as a function of wavelength for a given atom component.

By analyzing the two sums [J0(80i) +

2 J2(~50i)-I-2 Jl(~0i) ] and [-J0(~50i) + 2J2(~50i)-I-2 Jl(~0i) ] (as a function of 80.1 ), it can be seen that with an appropriate PEM setting (for one quarterwave or half-wave retardation) one can realize a gain of at least 1.5 (maximum gain is 2.1) in the whole mid-infrared region (3500-600 cm-1). All of the above discussion is based only on the expected signal strength. The detector will have different sensitivities for the signals at n---0, 1, and 2 and the role of detector noise at different values of norm is not considered. Both PDI/DPM and ADI would have same detector response at tom and at 2O~m, but PDI/DPM would be favorable due to its characteristic larger dynamic range.

Polarization Division Interfe rometry

319

All three components of the PDI/DPM signal (at nOhnwith n = 0, 1 and 2) can be measured simultaneously using three channel digitization. As an alternative one can use digital signal processing techniques. The application of digital signal processing method for step-scan interferometers has been discussed 25. The same approach can be extended to the present case, provided PDIB)PM is operated in the step-scan mode. References 1 M. Francon and S. Mallick, Polarization Interferometers, Wiley Interscience, New York (1971). 2 D.H. Martin and E. Puplett, Infrared Phys. 10 (1969) 105. 3 The Bomem DA3 Based Polarizing Interferometer Accessory, Technical Bulletin, Bomem Inc., (Quebec, Canada, 1987). 4 T. Okamoto, S. Kawata and S. Minami, Appl. Spectrosc. 40 (1986) 691; S. Takahashi, J. S. Ahn, S. Asaka and T. Kitagawa, Appl. Spectrosc. 47 (1993) 863. 5 M.J. Padgett, A. R. Harvey, A. J. Duncan and W. Sibbett, Appl. Opt. 33 (1994) 6035. 6 Principles and Applications of Polarization Division Interferometry, Ed. P. L. Polavarapu, John Wiley &Sons (1997). 7 D.H. Martin in Infrared and Millimeter Waves , K. J. Button Ed., Academic Press, New York (1982). 8 M.J. Dignam and M. D. Baker, Appl. Spectrosc. 35 (1981) 186; J. A. Bardwell and M. J. Dignam in Fourier Transform Characterization of Polymers, H. Ishida, Ed., Plenum, New York (1987). 9 H. Ishida, Y. Ishino, H. Buijs, C. Tripp and M. J. Dignam, Appl. Spectrosc. 41 (1987) 1288. 10 N. Raghunathan, N. S. Lee, T. B. Freedman, L. A. Nafie, C. Tripp and H. Buijs, Appl. Spectrosc. 44 (1990) 5. 11 P. L. Polavarapu, G.-C. Chen and S. Weibel, Appl. Spectrosc. 48, (1994) 1224. 12 P. L. Polavarapu and Z. Deng, Appl. Spectrosc. 48 (1994) 1562. 13 P. L. Polavarapu, G.-C. Chen and Z. Deng, Appl. Spectrosc. 48 (1994) 1403. 14 P. L. Polavarapu, Z. Deng and G.-C. Chen, Appl. Spectrosc. 49 (1995) 229. 15 P. L. Polavarapu, Z. Deng and S. Weibel, Appl. Spectrosc. 50 (1996) 98. 16 P. L. Polavarapu and Z. Deng, Appl. Spectrosc. 50 (1996) 91. 17 P. L. Polavarapu and Z. Deng, Appl. Spectrosc. 50 (1996) 686; P. L. Polavarapu and G. -C. Chen, Appl. Spectrosc.48 (1994) 1410. 18 P. L. Polavarapu, Appl. Spectrosc. 51 (1997) 770. 19 T. Leonard (private communication). 20 P. L. Polavarapu, Chem. Phys. Lett. 148 (1988) 21. 21 P. L. Polavarapu, Infrared Phys. 28 (1988) 109.

320

Chapter 13

22 P. L. Polavarapu (unpublished results); P. L. Polavarapu, SPIE Proc. 1166 (1989) 472. 23 J. Chamberlain, Infrared Phys. 11 (1971) 25. 24 A. Erdelyi, W. Magnus, F. Obeshettinger and F. G. Tricomi, Higher Transcendental Functions, McGraw Hill, New York (1953); M. C. Potter, Mathematical Methods in the Physical Sciences, Prentice Hall, Englewood Cliffs, New Jersey, 1978. 25 C. J. Manning and P. R. Griffiths, Appl. Spectrosc. 47 (1993) 1345.

321

Chapter 14 A P P L I C A T I O N S AND M O L E C U L A R S T R U C T U R E The applications of vibrational spectra are numerous. In particular, the amount of literature on the applications of vibrational absorption (including linear dichroism) and Raman spectra is so much that it would be impossible to cover them in any one source. For this reason, the applications with regard to vibrational frequencies, absorption (linear dichroism as well) spectra and Raman spectra discussed here are only for the purpose of giving the 'flavor'. On the contrary, the applications of vibrational circular dichroism and Raman optical activity are relatively new and thus a review of these applications is appropriate. The material presented in this chapter reflects these points.

14.1 Vibrational frequencies and absorption and Raman spectra Vibrational frequencies are available in the literature for several diatomic molecules. These diatomics are important in different ways. In one direction, the experimentally observed fundamental frequency directly provides a knowledge of the strength of the chemical bond. In the case of heavier diatomic molecules the anharmonic contributions to the observed vibrational frequencies are very likely to be small enough to be ignored. In such cases one can use the harmonic oscillator equation, as discussed in Chapter 2, for deducing the force constant K as 4/1;2c2V2g. With ~ (the observed frequency) in cm -1, g (the reduced mass) in atomic mass units and c (the velocity light) in cm/sec, the force constants in mdyn//k is obtained as K = 58.9666 x 10 -8 gV2. As an example, the observed la fundamental frequency (see Table 1) of 69Ga75As is 208.5 cm -1 which leads to a value of 0.9202 mdyn/A for the Ga-As bond force constant. Using this force constant, and the reduced mass for naturally abundant isotopomer 71Ga75As, the vibrational frequency of 71Ga75As comes out as 207.0 cm -1, which is also the observed value. Similarly for 69Ga31p and 71Ga31p, with observed fundamental vibrational frequencies of 283.6 and 282.5 cm -1 respectively, the force constant of Ga-P bond force constant becomes 1.013 mdyn/~. The corresponding force constant for Ga-Sb bond is 0.7817 mdyn/A. The magnitudes of these force constants decrease from GaP to GaAs and to GaSb, indicating that the strength of the chemical bond decreases in that order. In a different type of application, the characteristic vibrational frequency associated with a diatomic group can be used as a molecular structural probe. For example, the gas phase vibrational frequencieslb of 02, CO and NO are, respectively, 1555, 2143 and 1847 cm- 1. In addition to obtaining

322

Chapter 14

TABLE 1 Observedla fundamental vibrational frequencies and force constants in some diatomic molecules Molecule 69Ga31 p 71Ga31p 69Ga75As 71Ga75As 69Ga121Sb 69Ga123Sb 71Ga121Sb 71Ga123Sb

V(cm -1) 283.6 282.5 208.5 207.0 173.8 173.3 172.8 171.8

Force constant (mdyn/Ai 1.013 1.013 0.9202 0.9202 0.7817 0.7817 0.7817 0.7817

the bond-strength information as described above, these molecules can be used as ligands binding to molecules of biological significance. Then the characteristic vibrational band of a diatomic system serves as the probe of biomolecular structure and dynamics. A very nice illustration of this application can be seen in the following example 2. The hydration of organic nitriles (-CN) to the corresponding amides (-CONH2) is catalyzed by the enzyme nitrile hydratase. This enzyme contains two iron atoms. One of the two iron atoms is oxidized from Fe(II) to Fe(III) upon irradiation with visible light. The structure of the coordination sphere of iron atoms was not fully known, so the acceptor of electron when Fe(II) is oxidized to Fe(III) was not established. The infrared absorption spectra obtained for nitrile hydratase before and after photoactivation showed the presence of bands around 1850 cm -1, which is the characteristic region (vide supra) for the stretching mode of nitric oxide. The infrared difference spectrum obtained as the difference between photoilluminated nitrile hydratase and natural nitrile hydratase showed (see Fig. 14-1) a positive difference band at 1869 cm -1 and negative difference bands at 1855 and 1847 cm -1. These bands were shifted to 1832, 1820 and 1811 cm-1 when nitrile hydratase was labelled with 15N thereby providing further evidence that these bands are most likely due to NO. Two conclusions were derived from these observations. First, the appearance of bands around 1850 cm -1 suggested the presence of NO in nitrile hydratase. Second, since the vibrational bands attributed to NO were perturbed by photoillumination, the participation of NO in the above mentioned oxidation of Fe(II) to Fe(III) is implied. Thus it

Applications and Molecular Structure

323

was suggested 2 that nitrile hydratase intrinsically possesses NO, and bound to the iron center, functioning as the electron acceptor in the photoactivation process. The presence of NO- upon photoactivation however has not yet been confirmed. I

0.00

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-

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-10

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-40 1900

1800

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2000

1800

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Fig. 14-1. The infrared absorption difference (after-minus-beforephotoactivation) spectra. (a) nitrile hydratase (replotted from Ref. 2); (b) carbonmonoxy myoglobin (replotted from Ref. 3). Additional examples include the studies on dynamics of the photodissociation of protein-ligand complexes. The infrared spectrum 3,4 of carbonmonoxy myoglobin (MbCO) shows (see Fig.14-1) an absorption band at 1945 cm -1 which originates from the stretching vibration of the carbonmonoxy group. The presence of a shoulder at 1933 cm -1 suggested that CO either binds at two distinctly different sites or that this shoulder comes from an overtone vibration that is in resonance with the fundamental vibration of CO. Pulse photolysis results in bleaching of the ligand (within picoseconds) and in recombination of the ligand (rather slowly in milliseconds). An infrared investigation 4 using a dispersive spectrometer and 100 ~sec pulses reported the changes in CO stretching absorption intensity in millisecond time scale. Polarized infrared absorption of the CO stretching band in MbCO has also been studied 5 at 10 K. After cooling the sample to 10 K, flash photolysis using the polarized 540 nm light was undertaken in steps. The infrared spectra in the 1900-2100 cm -1 region were then obtained with infrared light polarized both parallel and perpendicular to that of visible light. At 10 K, the stretching absorption band of CO in unphotolyzed MbCO shows three peaks at 1966, 1945 and 1929 cm -1. These bands, labeled as A0, A1 and A3 bands, respectively, are associated with slightly different protein conformations. After photolysis three new bands, labeled as B0, B 1 and B2, attributed to the photolyzed CO were found at 2149,

Chapter 14

324

2131 and 2119 cm -1. From the difference in polarized infrared absorption (that is, linear dichroism) spectra, the tilt angles of CO from the heme plane normal, in the conformations responsible for A0, A1 and A3 bands, were found to be 15 ~ 28 ~ and 33 ~ respectively. No linear dichroism was observed for the bands B0, B 1 and B2 suggesting a random orientation for the photolyzed CO. Amino acids lical segments

O,C R7

I

,R1

J R,~'~----4

R2

R4

I

R3

F-8 (proximal histidine)

Structure showing the location of CO in MbCO Picosecond and femtosecond infrared spectroscopic studies on heme proteins were also reported. Using a 558 nm laser with 25 ps pulses for photolysis, the linear dichroism associated with the photolyzed samples (at ambient temperatures) was measured 6a,6b in the CO stretching region. While MbCO has two absorption bands located at 1944 and 1933 cm -1, HbCO has only one band at 1951 cm -1. From the linear dichroism measured in < 8 nsec after the 25 ps pulses were delivered, the CO tilt angles, from heme plane normal, were determined to be 18 ~ for HbCO and 20 ~ and 35 ~ for the two conformations of MbCO represented by the 1944 and 1930 cm -1 bands. A more recent study 6c however concludes that the the CO tilt angle, from heme plane normal, is <7 ~

Applications and Molecular Structure

325

Several Raman studies were reported on heme proteins with two of the recent studies6d, 6e being those on HbCO and MbCO. In the time resolved resonance Raman spectra obtained with 230 nm excitation, the resonance enhanced vibrational bands of tryptophan and tyrosine residues provided information on the tertiary structural changes in R and T states. In the resonance Raman spectra obtained with 416 nm excitation, the Fe-C stretching vibrational bands (492-496 cm- 1 and 508-512 cm- 1 reflecting two protein conformations) provided the information on the CO binding process. In the above discussion, the focus remained on the vibrational property of a diatomic molecule or group. Convenient relations between vibrational frequencies and force constants can also be derived for small polyatomic molecules, or groups. These relations are available in the literature 7. However, determination of the force constants in polyatomics from the experimental data, as mentioned in Chapter 7, is non-trivial. There is very little activity, at least in recent times, in addressing this issue partly because of quantum theoretical advances in predicting the vibrational spectra. Predictions of any type of vibrational spectra require reliable normal coordinate descriptions as a prerequisite (see Chapters 5, 7, 8 and 9). The normal coordinate descriptions depend on the molecular geometry and on the force constants. In addition to these force constants, reliable prediction of atomic polar tensors and atomic Raman tensors are needed for predicting the infrared absorption and Raman spectra respectively. Satisfactory predictions of these quantities at the Hartree-Fock level require larger basis sets; more frequently, higher level ab initio theories that incorporate electron correlation effects are necessary. A large number of papers are available in the literature on this subject 8 and it is beyond the scope of this book to review them here. In practical applications one would be interested in using the observed vibrational spectra, and trends therein, for deducing the molecular structural information. The discussion below reflects this emphasis. The frequencies of vibrational transitions are sensitive to the conformational differences, although the relations between the observed vibrational frequency differences and the subtle differences in conformational preferences are not unambiguous. Vibrational bands associated with the peptide group ( O = C - N - H ) are often used to estimate I

J

the secondary structure of proteins. The C=O stretching motion, invariably coupled (although to a small extent) to t h e - N - H bending motion, of the peptide group is referred 9 to as the amide I vibration. The C-N stretching and -N-H bending coordinates, usually coupled to each other, give rise to two vibrational modes that are referred to as amide II and III modes. The N-H stretching mode is referred to as the amide A vibration. Proteins usually adopt different secondary structural domains, along with some

Chapter 14

326

loops and turns to maintain the continuity between different structural domains. The variation of the dihedral angle of the O = C - N - H group, and the hydrogen bonding pattern, can influence the frequencies of peptide group vibrations. For this reason, in a given secondary structure one can expect a spread of vibrational frequencies to be associated with the peptide group vibrations. Let us focus on the amide I vibration. A typical protein containing approximately 150 peptide groups should, in principle, give rise to that many amide I vibrational bands, and all of them appear in the 1600-1700 cm -1 region. The spectral resolution in the experimental measurements (in solution phase) is -4 cm-]. So it is not possible to resolve all of the amide I vibrational bands arising from the individual peptide groups. Artificial resolution enhancement methods, such as deconvolution, second derivatives and curve fitting help identification of the band positions to some extent, but they also result in ambiguities or in increased noise level. As a result, one can only hope to establish some empirical observations between the observed bands (within the experimental limitations of spectral resolution) and broadly defined patterns in the molecular structures. Such empirical observations may in turn be used to suggest structural patterns in systems of unknown structure. It is important to realize that the structures deduced in this manner are approximate and can even be in error. When amide I vibrational absorption profiles are measured for proteins in water solution, one encounters some practical problems. Water has an absorption band at 1647 cm -1 (see Fig. 14-2) which overlaps with the amide 10

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I absorption band of interest. The peak absorbance of the water band at

Applications and Molecular Structure

327

1647 cm -1 can be estimated 10 approximately as p/10, where p (in microns) is the optical pathlength in solution. At an absorbance of 1, only 10% of the incident light passes through the sample to the detector, so it is a common practice to keep the maximum absorption, in the region of interest, to less than --1. For these reasons, it is necessary to keep the sample thickness to less than 10 gm, which also translates into requiring higher concentration of the sample in that solution. Commercially available variable pathlength cells can be used 10 to achieve pathlengths as low as --6 lam. Once the absorption profile of proteins in the amide I vibrational region is measured, the interfering water absorption (measured separately at the same pathlength) can be substracted out to enhance the spectral features of proteins. The absorption spectrum obtained 11 for carbonmonoxy hemoglobin in water (see Fig. 14-2) shows a strong absorption band at 1656 cm -1. The corresponding feature for carbonmonoxy myoglobin is at 1654 cm -1. Based on the conclusions from x-ray crystallographic studies, these proteins are thought to be composed of predominantly t~-helical structural components, some unordered components, and no 13-sheet type structural components. Thus the absorption maxima seen for these proteins are associated with or-helical structural components. Deconvolution of the absorption profile of concavalin A, which has a maximum at 1635 cm -1 (see Fig.14- 2), revealed the presence of bands at 1642, 1638, 1632, 1627 and 1624 cm -1. These bands are attributed to the [3-sheet structural components, since this protein is characterized as an all I3-sheet structure protein based on x-ray crystallographic studies. From the deconvoluted absorption spectra of proteins that have both or-helical and [3-sheet type structural components (for example, lysozyme, cytochrome c, t~-chymotrypsin, trypsin, ribonuclease A and alcohol dehydrogenase) the band located at 1650 cm -1 (which appears as a shoulder to the or-helix band mentioned above; and as a separate band in the case of low helical content proteins) was associated with unordered structure. The bands seen at 1688, 1680, 1672 and 1666 cm -1 in the deconvoluted spectrum were associated with various types of 13turn structures. These various band positions and assignments are shown in Table 2. Using these assignments and the integrated areas associated with these bands, the estimated 11 percent composition of secondary structural components in twelve proteins agreed well with the corresponding estimates obtained from other techniques (such as crystallography, and electronic circular dichroism). The problems associated with interfering water absorption is usually avoided by using D20 solvent. Deuterium substitution causes the bending vibrational absorption band of H20 at --1647 cm -1 to be shifted to a lower frequency, so the region around --1650 cm -1 will not have an interfering solvent absorption band. Since the N- H protons exchange with deuterium (in D20 solution), the amide I

Chapter 14

3 28

TABLE 2 Vibrational band positions of amide I component bands 11 Band position Assignment

i624 13-sheet

1627 '" 1632 I~-sheet 13-sheeta

1638 1642 1650 13-sheet [3-sheet unordered

Band position Assignment

1656 or-helix

1666 turn

1680 turn

1672 turn

1688 turn

aalso associated with extended chain. vibrational frequencies of that protein would be slightly shifted. This is because t h e - N - D bending vibrational frequencies are lower than t h e - N H bending frequencies and the mixing of the - N - H bending vibration with the C=O stretching vibration would now be absent. The amide I vibrational bands are now labeled as amide I' to signify this effect. The assumption of complete exchange of N - H protons with deuterium is not strictly valid, because this exchange depends on the accessibility of N - H protons to the D20 solvent, and certain structural features of proteins would not allow solvent molecules to reach the N-H protons located in certain regions of the protein structure. Thus the measured amide I' vibrational band profile may actually contain some amide I vibrational bands, although the extent of their presence is not well established. The analyses of the spectra measured in D20 solutions usually assume or imply complete conversion from N-H to N-D. The amide I' vibrational band positions observed for some proteins can be found in Ref. 12-18. The amide I' absorption profile with peak absorption in the 1650-1658 cm -1 region (as in the case of hemoglobin, for example) is generally associated with or-helical structure. Exceptions to this assignment are found in some cases" for example, epidermal growth factor and basic fibroblast growth factor are found 18 to have vibrational absorption profiles with significant absorption peaks in the 1650-1656 cm-1 region, even though other methods have suggested that these proteins are composed of I]-sheet and loop structures and have no detectable tx-helices. The absorption peaks in the 1620-1640 cm -1 region are generally associated with the 13-sheet structures. The various band components found in this region could not specifically be associated with particular types within such structures. The presence of weak bands around 1630 cm -1 is not uncommon even when a protein is known to have no or small 13-sheet structural component. Such observations lead to the suggestion that these bands can be attributed to short extended segments, which formally may not correspond to 13-sheet structure. The high frequency bands in the 1668-

Applications and Molecular Structure

329

1690 cm -1 region are usually associated with the structures that contain turns and bends. The presence of a broad amide I' absorption band contour with absorption maximum at --1630 cm -1 is considered to be typical of proteins with dominating [3-sheet structure components (see for example the spectra obtained for choleratoxinl4, interleukinl5). Exceptions to this general observation are not uncommon: again in the cases 18 of epidermal growth factor and basic fibroblast growth factor which were suggested to have dominant l-sheet structure components, the peak absorptions with significant absorption are located at 1651 and 1643 cm-1 respectively. When the amide I' absorption profile has two or more maxima (with significant absorption) in the 1630-1650 cm -1 region, (as is the case of lysozyme, for example) then the presence of both ~ and 13 structural segements are implicated. A significant number of vibrational spectroscopic studies 19,2~have been reported on bacteriorhodopsin (BR). In the light adapted form, the retinal chromophore (see the structure below) of BR exists in the all trans form which is characterized by a visible absorption band at 570 nm, and is bound to a lysine residue (Lys-216) of BR570 (the subscript on BR representing the wavelength where visible absorption has a maximum). The absorption of visible photons by the retinal chromophore in this conformation induces a conformational change that triggers a sequence of structural changes leading to the formation of intermediates and ultimately C"

n3

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1 ~I-

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back to BR. This process is collectively labeled as the photocycle. The intermediates identified to date are labeled J, K, L, M, N, and O. The intermediate J decays to K within 6 picoseconds 21. The transition to the K intermediate involves isomerization of the retinal chromophore around the C 13=C 14 bond and the C]4-C 15 single bond. This isomerization brings the Schiff base proton closer to an aspartic acid residue (labeled as Asp 1, for convenience) that is closer to the retinal. The K intermediate can be stabilized by keeping the sample at 80 K. The K to L transition is believed to occur in --1 microsecond and to involve deprotonation of Asp l by transfering a proton to another aspartic acid residue (labeled Asp l' and

330

Chapter 14

believed to be in deprotonated form); Asp l and Asp l' are believed to be located on the opposite sides of the polyene chain of the retinal. The L intermediate can be stabilized at 110 K. The deprotonated Asp 1, being in the vicinity of the Schiff base proton of the retinal, draws the proton away from the Schiff base. At the same time two other aspartic acid residues (labeled Asp2 and Asp3), which existed as counterions to the Schiff base, are believed to accept protons from the cell interior. These changes are believed to take place within ~ 100 microseconds and constitute the L to M transition. The M intermediate can be stabilized at 210 K. Deprotonation of the Schiff base in the M intermediate is believed to cause back isomerization about C14-C15 single bond and draws a proton to the Schiff base. This process also forces the isomerization around the C13=C14 double bond. These changes are believed to occur in the M to N transition. The N to O transition and back to BR involve proton transfer to and from the above mentioned aspartic acid residues. Thus the entire process represents a proton 'pump' mechanism initiated by the photo-absorption. The changes involved in the photocycle can be seen to influence (a) isomerization around C=C and C - C bonds and (b) protonation and deprotonation of carboxylic groups. These structural changes directly affect the vibrational motions, namely: stretching (C=C, C-C, C=O and C-O), bending (angles C=C-C, H-C=C, H - C - C ) , and torsion (dihedral angles, C - C = C - C , H - C = C - H , H - C - C - H ) . Therefore, vibrational spectroscopy in principle should provide information on the type of changes that take place during the photocycle. The sensitivity of detecting these changes however depends on the spectroscopic property being probed. Nuclear magnetic resonance, resonance Raman vibrational spectroscopy and infrared vibrational spectroscopy are three techniques which offer the most potential. Extensive investigations were conducted 20 on the photocycle of BR570 using resonance Raman spectroscopy. Here two lasers were used, one serving to initiate the photocycle and another to probe the status of the sample. The former is referred to as the pump laser and the latter as the probe laser. If the wavelength of the probe laser is chosen to be in the visible region, then only those retinal chromophore vibrations which mimic (or are coupled to) the excited electronic state geometry of the chromophore are resonance enhanced, and the remaining vibrations including those from the other portions of the sample become 'transparent' and are not seen in the resonance Raman spectra. Alternately, if the wavelength of the proble laser is chosen to be in the UV region, then only those vibrations which mimic the excited electronic state geometry of either the peptide group or the aromatic residues are resonance enhanced and seen in the spectrum. The information obtained from resonance Raman technique is usually complementary to that obtained from infrared spectroscopy. As discussed earlier, the strong vibrational absorption band of water solvent at --1650 cm -1 masks the spectral features of the sample being

331

Applications and Molecular Structure

investigated. This problem is generally overcome using concentrated solutions and low (-6 gm) pathlengths or by preparing thin films of the samples (by evaporating the concentrated samples in water). An additional A'+',O.O~ _

-0,(~

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ii

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Fig. 14-3. The difference between the absorption spectra of BR568 and M412 (obtained as BR568-M412). Reproduced with permission from Ref. 19e. Copyright 1985 American Chemical Society.

complication is that the vibrations originating from every residue within the sample will give rise to the vibrational absorption bands, resulting in overly congested spectra. This spectral congestion is removed considerably in the infrared difference spectra, where spectra of different states of the same system are separately recorded and then subtracted (see Fig. 14-3). Utilizing the stability of K, L, and M intermediates at 80, 110 and 210 K, respectively, infrared spectral changes associated with these intermediates were identified 19a-e as follows" the infrared absorption of BR570 before photoinitiation and after photoinitiation at a given temperature (where the intermediate is stabilized) were taken and the difference between these spectra provided the absorption spectral features associated with the intermediate. Polarized infrared difference absorption measurements associated with the C=C and C - C stretching vibrational bands were used 19d,i to determine the orientation of polyene chain of the retinal chromophore with respect to the membrane plane. In a similar manner, polarized infrared difference absorption associated with the Amide A, I and II vibrational bands were used to suggest the helical segment tilt angle of BR in purple membrane (PM). The BR molecules in the native PM are known to exist as trimeric clusters and this structure is affected when one of the associated retinal chromophores isomerizes. This can lead to either a change in secondary structure or a change in the net tilt angle of the helical segment. Both views have been put forward and controversy exists as to the actual mechanism. The polarized infrared measurements (see Fig. 14-4)

Chapter 14

332

were interpreted 19i recently to arise from the net tilting of the helical segment from membrane normal, rather than due to changes in the secondary structure. Utilizing the stability of K, L and M intermediates at 80, 110 and 210 K, the time evolution of these intermediates were also monitored by flash illuminating the BR570 sample and monitoring the changes in spectra

I C.1

A.

4000

3620 3240 2860 2480 21O0 WAVENUMBER

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1:~40

960

Fig. 14-4. Polarized absorption spectra of purple membrane films. AV and AH represent vertical and horizontal polarizations, respectively. Replotted from Ref. 19i.

at 0.7, 8 and 100 microseconds after the illumination. Both step-scan Fourier transform infrared spctrometers and dispersive infrared spectrometers have been used for this purpose 19d-h. PM was used as a model system to study the interaction of general anesthetics with membranes. BR is the only protein found in purple membrane with approximately 80% of the 248 amino acid residues of BR being located within the lipid bilayer of PM. It is known 22 that the commonly used anesthetics, enflurane and halothane shift the absorption maximum of BR from 570 to 480 and 380 nm respectively. The unperturbed and these modified BR are referred to as BR570, BR480 and

Applications and Molecular Structure

333

BR380 respectively. Although the mechanism of this interaction is not yet known, the visible spectral studies 22 indicate that an equilibrium exists among the BR570, BR480 and BR380 forms upon exposure to anesthetics and that B R570 can be fully recovered if the anesthetic vapors interacting with BR570 are removed (via mild vacuum) within -60 min after the exposure. With prolonged exposures, it is thought that BR380 converts to an irreversible form, the structure and mechanism of this being unclear. An infrared absorption study 23 (which appears to be the only infrared spectroscopic study reported to date on the interaction of anesthetics with proteins) revealed that the ratio of absorption intensities of amide I and II bands is sensitive to the structural changes acompanying the conversion from BR570 to BR480 and BR380. This ratio changed from 1 in BR570 to 0.94 in BR480 and to 0.65 in BR380. From the known relations of amide I and II transition dipole moments with respect to a helix axis, it was suggested that the variation in the amide intensity ratio reflected progressive randomizatioin of helices as the membrane is exposed to anesthetics. When BR380 is allowed to become irreversible, a new infrared band at 1632 cm -1 developed, suggesting that the c~-helix structure is being converted to [3sheet. This process of spectral changes was found to be similar to that in thermal bleaching of BR570. The applications discussed so far are a few representative cases and many more applications like these can be found in the literature. It is important to note that when the information derived from ordinary vibrational absorption and Raman studies are supplemented with that from vibrational optical activity spectra (see the following sections), the derived structural details will have greater validity.

14.2 Vibrational circular dichroism spectra 14.2.1 Ab initio theoretical predictions As discussed in Chapter 8, the prediction of vibrational circular dichroism intensities require three different quantities" normal coordinate vectors (which depend on force constants and molecular geometry), atomic polar tcnsors and atomic axial tensors. Several important developments have occurcd in the last few years in calculating the later. The quantum theoretical methods developed for VCD have generally been successful in reproducing the experimental spectra of a variety of molecular systems, when implemented ab initio using moderate to large basis sets. These calculations have been done using Hartrcc-Fock (HF), post-HF and density functional theories (DFT). The atomic axial tensors have been evaluated at the HF level, employing the standard basis sets, with the localized molecular orbital 24 (LMO), magnetic field perturbation 25-28 (MFP), vibronic coupling29(VC)and nuclear electric shielding tensor 30 methods. They were also evaluated with a random phase approximation 31 (RPA), and with the inclusion of electron correlation using the VC method 32. Using the

3 34

Chapter 14

TABLE 3 Molecules with available experimental and ab initio theoretical VCD data isoflurane 37 methyl lactate25e 1,2-trans-dideuteriocyclobutane 25b 6,8-dioxabicyclo [3.2.1 ]octane25j 4-methyl-2-oxetanone 25k 3,4-dideuteriocyclobutane1,2-dione 26b 4-methylazetidin-2-one 29e 1,2-trans-d2-cyclopropane 251 2,3_trans_d2_oxirane 35a 1_13C_ 1,2,3-anti-d3-cyclopropane 251 methyloxirane 251 2,3-trans-dimethyloxirane 251 1,2-trans-dicyanocyclopropane 25c 1,2-trans-dimethylcyclopropane 27a 2,3-trans-dimethyloxirane-2-d 1251 desflurane 38 2,3-trans-dimethyloxirane-2,3-d2251 1-d-ethanol 29c methylthiirane 251 2-methylaziridine 29f 2,3-dimethylaziridine29g methylglycolate-d125f 2-methylthiirane-3,3-d2251 2,3-trans-dimethylthiirane 251 methylglycolate-d425f 3-methylazetidin-2-one 29e 2,3-trans-dimethylthiirane-2,3-d2251 fenchone 35b 2-methyloxirane-3,3-d224g glycido124g 3-methylcyclohexanone24g L-alanine24g isotopomers of L-alanine24g 1-azabicyclo[3.1.0]hexane 29h 1,2-dimethylaziridine29g

2,3-dideuteriobutyrolactone 26a camphor 35b

gauge invariant atomic orbitals (GIAOs), the MFP method has also been used for calculating the atomic axial tensors at the SCF 33, post-SCF 34 and DFT 35 levels. At the time of this writing, the predictions of VCD spectra obtained with the MFP method using GIAOs and DFT appear to be most promising (both interms of practicality and reliability). The anharmonic contributions to the VCD intensities 36a-d of the fundamental bands have also been calculated 36c, but were found to be negligible for trans-2,3-d2-oxirane [nevertheless, circular dichroism in the overtone and combination transitions has been experimentally observed36e-g, so the anharmonic effects cannot be ignored in genera136d; the appearence of combination bands, originating from the fundamentals in the region below 1800 cm -1, in the 4000-2000 cm1 region is quite common, so VCD of combination bands can be found in many of the spectra reported in the C-H stretching region]. A list of molecules which have been subjected to both experimental and ab initio theoretical studies are included in Table 3, with the references quoted being the ones with most recent calculations. The theoretical VCD

335

Applications and Molecular Structure 1600

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,

I

1500

I

1400

'

I

1300

!

'

1200

'

I

1100

,| H I

,

1000

900

wavenumbers Fig. 14-5. Absorption (a,b) and VCD (c,d) spectra of camphor. The experimental spectra (a,c) are for (+)- camphor in a 0.6 M CC14 solution at 4 cm "1 resolution and the theoretical spectra (b,d) were obtained for (1R,4R)-camphor with the 6-31G* basis set using DFT method. The theoretical VCD spectra used the MFP method. Reproduced with permission from Ref. 35b. Copyright 1996 American Chemical Society.

336

Chapter 14

spectra obtained with MFP method for camphor using a 6-31G* basis set and density functional technique are compared with the corresponding experimental spectra in Fig. 14-5. The calculated absorption and VCD bands are in correspondence with the experimental observations (the calculated band positions are some what higher, so they appear shifted to higher frequencies relative to the experimental bands). The absolute configuration and conformations of a molecule for which the predicted VCD spectra match well with the experimental VCD spectra are considered to represent the most plausible molecular structure. The structures determined in this manner were found to compare well with the structures known from other techniques for several molecules (see Table 3). This observation provides the basis for using VCD as a predictive tool of molecular structure. The absolute configurations and predominant conformations of isoflurane and desflurane were first determined 37,38 using the VCD spectral studies. The crystal structures for these two molecules are now available 39.

14.2.2 Empirical predictions While the ab initio studies discussed above represent the preferred approach for determining the molecular structure using VCD data, such calculations are not amenable to derive easily conceivable relations (between the VCD signs and molecular structure) that can be used for general applications. Moreover such studies are not feasible for large molecules (especially those of biological interest). As a result, empirical methods that can provide reliable spectra-structure relations would have useful practical applications. The simplest chiral groups appropriate for developing the empirical relations are: (a). non-planar A-B-B-A [i.e. A2B2 type] molecular segments; (b). a tripodal structure with a central atom attached to three different groups; and (c)" a tetrahedral structure with central atom attached to four different groups. The vibrations associated with A2B2 type molecular segments provide a useful approach for deducing a connection between VCD and three dimensional molecular structure. For this reason let us focus on the A2B2 type molecular segments. Since the two A-B bonds do not share a common atom, there is no kinetic energy coupling between the A-B stretching motions (see Chapters 5 and 6). As long as the mass of atom A is significantly smaller than that of atom B (for example, A representing H and B representing O, N or C) the remaining vibrations of A2B2 segment will be at significantly lower frequencies than the two A-B stretching vibrational frequencies. As a consequence, one can set up the vibrational equations for the two A-B stretching motions independent of other motions. Then the frequencies of the two A-B stretching motions become 40 9+ = (1/2xc) [(kAB +k']/g] 1/2 where kAB and g are, respectively, the force constant and reduced mass of the bond A-B; k' is the interaction

Applications and Molecular Structure

337

force constant; + and - signs are applicable for the symmetric and antisymmetric combinations of the two stretching motions. Expressing k' in mdyn/A and ~ in atomic mass units the frequency separation, A = 9+-9_, becomes A = (k'/I.t) • 0.17 x 107/9 where 9=(9+ + 9_)/2. Experimental determination of this frequency separation permits determination of k'. Alternately, an estimate of k' will give a value for frequency separation, but one has to be careful with the sign of k' which determines the relative frequency order of the two modes. Empirical approaches may sometimes be satisfactory to predict the value of k'. In one such approach the interaction between the two A-B groups can be considered to orginate from the electric dipoles associated with them. The energy of dipolar interaction between two A-B groups can be given as v12 = gA1B1 gA2B2 [UA1B1 " 3 UA2B2- 3 (UA1B1 ~ u12) (UA2B2 ~ U12)] / R12. In this equation, gAiBi and UAiBi are respectively the electric dipole moment magnitude and unit vector associated with bond A i - Bi (i = 1 or 2); u12 is the unit vector for the line joining these two bonds. Since k' = (~2V/~RA1B I~RA2B2) it is necessary to differentiate the above mentioned interaction energy, which then leads to terms of the type ~tAiBi, ~)~tAiBi/~)RAjBj, ~2~AiBi/~)RAiBi~RAjBj" The second derivative 321.tAiBi/3RAiBiORAjBj represents electrical anharmonicity (which is usually much smaller than ~tAiBi and C)~AiBi/c)RAiBi) and can be ignored. In a second approach one assumes that the interaction is between the two transition dipoles moments 41 and the dipolar interaction between these moments leads to a separation between their frequencies. In that case UAiBi referred above represents the unit vector associated with the transition dipole moment [in some cases, one may assume that the line from Ai to Bi represents the direction of transition dipole moment]. This approach, referred to as the coupled oscillator model, is most widely used in dealing o O II II with biological molecules containing repeated units of c... c 9 One subtle, but important, point is that in peptides the C=O stretching vibration is usually coupled to an -N-H bending vibration of the peptide group. By O O II II limiting the consideration to C ... c type segments, it will not be possible to explain the influence of N-H deuteration on C=O stretching vibrational o H II I properties. It is therefore proper to consider the c "" N segment as the repeat unit and include the interaction between the transition dipole moments originating from C=O stretching and-N-H bending vibrational motions.

Chapter 14

338

H

H

Structures

H

H 60 ~

120 ~

120 ~

60 ~

CCW

CCW

CW

CW

I

V

VCD spectra

A

V

Y

A

V

A

Absorption spectra

Vs

A

Va

Vs

Va

Aj/ jnkA ~

Va

Vs

Va

Fig. 14-6. Correlation between VCD signs and molecular structure.

The first qualitative model to be used for VCD was based on the coupled oscillator mechanism 41 that was widely used in interpreting the electronic circular dichroism. Two transition dipole moments (originating from independent oscillators) with a dihedral angle (other than zero or 180 ~ between them result in a net magnetic dipole moment and therefore support circular dichroism. The rotational strengths associated with states representing the symmetric (+) and antisymmetric (-) combinations of the independent transition moments are given as R+--T-(rrv/2c) T12 s(llt 1 x 11,2) where !11 and pt2 are the transition dipole moment vectors, and T12 is the distance vector between these moment vectors. Note that the interaction between IlL1 and ~t2 is assumed to separate the band positions corresponding to R+ and R_ (in the absence of such interaction R+ and R_ would appear at the same vibrational frequency

339

Applications and Molecular Structure

and cancel each other). In deriving this relation no use is made of the theory of molecular vibrations. The principles behind the theory of molecular vibrations can be conveniently applied 42 to A2B2 type molecular segments with C2 symmetry. The principal conclusions are as follows. For a counterclockwise dihedral angle the symmetric and antisymmetric A-B stretching vibrational bands have positive and negative VCD respectively. The same statement applies for A-B-B bending vibrations. These relations are shown in Fig.14-6. In general, the magnitudes of rotational strengths for symmetric and antisymmetric vibrations are not equal. The magnitudes of corresponding absorption intensities are also not equal, but may become equal under some circumstances. When the two Ai-Bi groups are mechanically and electronically independent of each other then the A2B2 molecular segment behaves like a coupled oscillator. These relations between VCD signs and handedness have been used to verify their applicability and to suggest conformational or configurational details in some molecular systems40,43,44. A word of caution in using these empirical approaches is appropriate. If the qualitative predictions obtained for A2B2 segment can be verified with quantum theoretical predictions on a suitable system, then one can gain confidence in using these qualitative predictions. The H202 molecule is clearly an appropriate choice for this purpose. The VCD and absorption TABLE 4 Theoretical vibrational absorption and VCD intensities a for H202 Vibration

Absorption intensity a (km/mol)

VCD intensity (10 -44 esu 2 c m 2) LMOb

~;ym O-H stretch Asym O-H stretch Sym H-O-O bend Asym H-O-O bend O-O stretch Torsion

26.0 88.3 0.5 103.7 0.9 202.4

38.4 -39.2 18.5 5.8 1.4 -234

MFP c -29.6 47.8 14.3 1.8 1.7 -214

VCTd -1.4 7.7 12.8 -20.3 2.7 -190

acalculated at optimized geometries, using 6-31G (ext) basis set with counter clockwise dihedral angle; blocalized molecular orbital method; CMFP method using CADPAC program45;dvibronic coupling method 29b with 631 ~ basis set. intensities predicted for all vibrations of H202 using three different quantum

340

Chapter 14

theoretical methods are given in Table 4. For H202 with a counterclockwise dihedral angle, the coupled oscillator and A2B2 type molecular segment models discussed above predict positive VCD for symmetric O-H stretch and negative VCD for antisymmetric stretch. While these predictions are supported by the LMO results, both MFP and VCT predictions are just the opposite. For the bending modes, bisignate VCD predicted by the empirical models is supported by the VCT calculations, but not by the LMO and MFP calculations. Because the enantiomers cannot be resolved, there are no experimental data for VCD of H202, so it is not possible to judge which models are correct here. All three calculations included in Table 4 predict relatively larger magnitude for VCD associated with the torsional vibration of H202, suggesting the feasibility and importance of VCD in the low frequency vibrations that appear in the farinfrared region. The above discussion indicates that one should not take it for granted that the correlation shown in Fig.14-6 between VCD signs and molecular structure hold in general. Various other empirical models and relations 43,44 have also been formulated, but their use for reliable predictions of molecular structure is not firmly established. Experimental VCD spectral studies were undertaken on different series of molecular systems. Certain correlations could be inferred from these studies which in turn help infer structural details in related molecules as discussed below.

14.2.3 Amino acids, peptides and proteins Alanine, being a simple chiral amino acid, attracted much attention in understanding and interpreting the observed VCD features 46. The infrared and Raman spectra of alanine isotopomers, namely alanine-C*-dl, alanineC-d3, alanine-C-d4 and their N-deuterated analogues, have been helpful in identifying the vibrational band positions of different C-H stretching modes. The methine stretching mode of L-alanine-C-d3-N-d3 is found to have positive VCD and to strongly depend on pD (see Fig. 14-7). At pD 1.5 its VCD intensity was found to decrease 47 by an order of magnitude from that at neutral pD; at pD 13.5 however the VCD intensity remained about the same as that at neutral pD, but the methine stretching frequency shifted (from 2970 cm -1 at neutral pD) to 2936 cm -1 and the associated absorption intensity significantly increased. These changes could be reflecting the influence of change in electronic structure with pD as zwitterionic alanine at neutral pD changes over to monoionic forms at high and low pD conditions. Similar pD dependence studies on a few other simple amino acids and alanyl peptides have also been undertaken. The positive VCD associated with the methine stretching modes in glycyl-L- alanine-N-d4 and L-alanyl-L-alanine-N-d4 was found 47 not to be influenced with pD as dramatically as in L-alanine. The VCD observed 47 for L-alanyl-C-dl-L-

341

Applications and Molecular Structure

alanine-C-dl-N-d4 at neutral pD is very small in magnitude (compared to other dipeptides) which suggests that the interaction of the methine groups among themselves and with other C-H groups plays a significant role. The methine stretching VCD in most of the simple L-amino acids was found to be positive. One exception is (S)-(-)-glycine-C-dl where the methine stretching VCD band at -2990 cm -1 was found 48 to have little VCD. There are two bending modes that can be associated with the methine group and they appear at 1337 and 1291 cm -1 in L-alanine-N-d3, at 1323 and 1268 cm -1 in (S)-glycine-C-dl-N-d3. The higher frequency mode has negative VCD while the lower frequency mode has positive VCD in both molecules.

20-

~,

,

,..~ ,

4

3

~0

,

":"'"'- .-. . . .

o

L:

-I x

.,/

-

<]'10 "

L-ALANYL-L-ALA~INE-N-d4

A

neutral pD

<3-1-2

40-

L-Alanlne-3,5,~- ds- N- d3 ;",,

12

z

:\

z9o

,

6

I

i

,,

9

,.,

30-

.,

/%~

.9. . .

....

pD2 pD13

zo o

3000

z9oo

way E NUMBER

3000 2900 W A V E NU M B E R

Fig. 14-7. VCD (top) and absorption (bottom) spectra in D20 under different pD conditions. Left panel: L-alanine-3,3,3-d3-N-d3[ solid line:neutral pD; dashed line: pD 13.5; dotted line: pD 1.5]; Right Panel: L-alanyl-L-alanine-N-d4. Reproduced with permission from Refs. 47. Copyright 1989 American Chemical Society; 1989 John Wiley & Sons.

Simple L-amino acids attached to the tertiarybutoxycarbonyl [(CH3)3C(O)(CO)] group (labelled t-BOC) or the carbobenzoxy [(C6H5)C(H)(H)(OCO)] group (labelled CBZ) at the N-terminus exhibit 49

Chapter 14

342

bisignate VCD couplets in the C=O stretching region, with positive VCD on the higher frequency side and negative VCD on the low frequency side (see Fig. 14-8). The intensity of this VCD couplet decreases with the concentration in chloroform and dimethylsulphoxide solvents indicating the presence of intermolecular hydrogen bonding. The higher frequency component was suggested to originate from the C=O group participating in intermolecular hydrogen bonding, and the lower frequency component from the free C=O group.

t:} :z ':Z. O ,,...,, a_ bath

u'~

":'7,

i---

_J

oo

,0

N- T-BOI~-L-FLRN

I i~IE

0

|

N-T -BOE-L-i;'ItOL

I tie

0

o

to

[

~

ua / /

-/', "/." ~. ~, ", " " " / ' . - " ' ]

Q

t~so

i~axJ ~

0

HflVO'4LIrSO0~

Fig. 14-8. VCD (top) and absorption (bottom) spectra of N-t-BOC-L-alanine and N-tBOC-L-proline in CHC13 solution at different concentrations: 0.2M (solid lines), 0.02M (dashed line) and 0.002 M (dotted line). Reproducedwith permission from Ref. 49. Copyright 1987 John Wiley & Sons. A simple model system to characterize the VCD associated with the peptide group is alanyl-alanine. The amide I' vibration of L-alanyl-L-alanine appearing at 1665 cm -1 was found not to have any significant VCD. On the other hand L-alanyl-L-alanyl-L-alanine, (L-Ala)3, which has two peptide groups, was found 50 to have a positive-negative bisignate VCD couplet

343

Applications and Molecular Structure

associated with the amide I' vibrations (see Fig. 14-9), indicating that the coupling between the peptide groups is probably important for observing bisignate VCD in the amide I' vibrational bands. A possible solution structure for (L-Ala)3 with e = 120 ~ and ~ = -25 ~ was suggested based on these VCD data. However, this structure is significantly different from the one deduced from VROA spectral data (vide infra). N-acetyl-N'-methyl-Lalaninamide is another model system to study the possible coupling between 2

ZIAxlO5

1

-1 -2 6

AxlO 1

1550

1650 WQvmnumbQr

(I/cm)

17513

Fig. 14-9. VCD (top) and absorption (bottom) spectra of L-alanyl-L-alanyl-L-alanine in D20 solution. Reproduced with permission from Ref. 50. Copyright 1989 John Wiley & Sons.

two peptide groups. The amide I' vibrational bands of this molecule, in contrast to (L-Ala)3 were found to have a simple negative VCD feature. The lack of a bisignate VCD feature here does not necessarily indicate the absence of coupling between the peptide groups because one has to consider the conformational degrees of freedom present in this molecule. The higher oligomers (L-Ala)n, n = 4 and 5 were found to have amide I' VCD features that are similar to the ones found for (L-Ala)3. In chloroform solution, poly(T-benzyl-L-glutamate), abbreviated as PBLG, is considered to adopt a right handed t~-helical structure. The VCD

Chapter 14

344

spectrum of PBLG in the amide A region has a positive-negative bisignate couplet 5] associated with the absorption band--3290 cm -]. The magnitude of the positive VCD component (on the high frequency side of the absorption band) is slightly greater than that of the negativeVCD component (on the lower frequency side of the absorption band). This difference in the magnitudes of the individual components of a bisignate couplet is referred to as 'bias'. A bisignate VCD couplet is also present in the amide I region (see Fig. 14-10), but with opposite sign pattern. That is, negative VCD on the PIX,,Y-~,.,,,m~N'ZYt,.=I,,.-AC~PA.qTATE( - __ =)

PObY--'Y-~r.NZYI.-I.-Q,.L/ThI, UkTE ( - - - ) N - ~ T E O (------)

l

v

11.441

11

A t

(~I~4)

,"a

./,,,,.

9.

~71rali

NF--DEUTERATED ( ~ )

1~11

tB~W~

~'l~O4CY(Cl~-1)

A l , 15

l~)lf

15M

L,? FI~IJE~Y

(~,kl-t )

Fig. 14-10. VCD (top) and absorption (bottom) spectra of poly("~-benzyl-L-glutamate) (left panel) and poly(b-benzyl-L-asparate) in CHC13 solution. Reproduced with permission from Ref. 51a. Copyright 1984 John Wiley & Sons.

high frequency side, and positive VCD on the low frequency side, of the amide I absorption band. In the amide II region, the absorption band at -1550 cm -1 is associated with small negative VCD. But at 1520 cm -1, where the absorption band has only weak intensity, a larger negative VCD signal is present. The VCD features mentioned in the above three regions are considered to be characteristic of right handed-oc-helical structures, as evidenced by similar features for other peptides that are known to have right handed ochelical structures. Poly(D-benzyl-L-asparate), which is known to have a left handed helical structure in chloroform solution, gives 51 opposite signs for VCD (see Fig. 14-10). Although the VCD features seen in the amide II region are not quite mirror images for polypeptides with right and left hand helical structures, a

345

Applications and Molecular Structure clear difference is present between them. n-5 ~-4

li~ Co I B ~

-4. iii rl = iii

ill

41

m

34~

34n

335g

FREQUENCY

~311t~

$~i~

32~

(CM -I)

Fig. 14-11. VCD (top) and absorption (bottom) spectra of Z-(Aib)n-L-Leu-(Aib)2-OMe with n=2 to 5 in CDC13 solution; Z is an abbreviation for benzyloxycarbonyl group. Reproduced with permission from Ref. 52b. Copyright 1986 American Chemical Society. In right handed t~-helix structures the repeating pattern of hydrogen bonds occurs between the carbonyl oxygen atom of nth residue and amide hydrogen of (n + 4) th residue. A variant of this structure is right handed 3 l o-helix where the above mentioned hydrogen bonding is between the residues n and (n + 3). The (q0, ~) angles in t~-helix structures are --(-60 ~ -40 ~ while those in 310-helix structures are-(-70 ~ -5~ Proteins adopting 3 l o-helix structure are less common, but the presence of dialkylated oramino isobutyric acid (Aib) residue in blocked polypetides apparently favors the formation of this structure. Such peptides were studied 52 to establish

346

Chapter 14

the characteristic VCD features associated with 310-helix structures. In the amide A and I regions the differences seen in VCD spectra are maily in the relative magnitudes of the individual components of couplets, but the sign patterns are the same. The amide A couplet is positively biased for tx-helix structures, but is nearly balanced (that is, positive and negative components have nearly equal intensities) for 310-helix structures (see Fig. 14-11). In the amide II region, however, 310-helix structures give negative VCD which is associated with substantial absorption intensity a t - 1 5 2 0 cm -1. This is different from that seen in a-helix structures, where the peak absorption intensity is at -1550 cm -1, and the negative VCD intensity peaks at ~1520 cm-1. Poly (tyrosine) is known to undergo solvent induced structural transitions. In dimethyl sulphoxide solvent, the structure of poly(Ltyrosine) is considered to be a random coil, while that in mixed solvent systems (vide infra) is a right handed o~-helix. A well ordered, but nonrepeating conformation is generally classified as coil structure. Sometimes this is also referred to as random coil (although this does not mean truly random or disordered structure53). Poly(L-tyrosine) in dimethyl sulphoxide solvent shows 51 a negative-positive VCD couplet in the amide A region and a positive-negative VCD couplet in the amide I region. Weak VCD signals are seen in the amide II region with a positive band at 1520 cm -1. These sign patterns are opposite to those seen for right handed or-helix structures, but are similar to those seen for left handed or-helix structures. When the solvent is changed from pure dimethyl sulphoxide to an 80:20 mixture (by volume) of dimethylsulphoxide with dichloroacetic acid, (or with trifluoroacetic acid) the VCD sign patterns in the amide A and I regions are reversed. In addition, the amide II region in this mixed solvent shows two negative VCD bands, one at 1545 cm -1 and another at 1520 cm -1. The VCD spectrum of poly(L-tyrosine) in a 50:50 mixture of dimethyl sulphoxide and trimethyl phosphate is similar to that seen in the dimethyl sulphoxidedichloroacetic acid mixture. These VCD studies are in concurrence with literature conclusions that poly(L-tyrosine) adopts a right handed o~-helical structure in the mixed solvent systems mentioned above. In dimethyl sulphoxide solvent, the structure of poly(L-tyrosine) was suggested to be a random coil, but the similarity of VCD spectra obtained here to those obtained for left handed a-helical structures suggested that the random coil structure here has some left handed local ordering. Although most initial VCD studies were done in non-aqueous solvents, these solvents do not necessarily represent the real situation for biological molecules in aqueous media. Using high concentrations and lower path lengths for water solutions it became possible 54 to measure VCD in the amide I, II and III regions (vide supra). When D20 solvent is used, as mentioned in Section 14.1, the N-H protons exchange with deuterium from

347

Applications and Molecular Structure

the solvent and the coupling of N-H bending vibrations with the C=O stretching band is now eliminated. This, and perhaps other factors, lead to a slight difference 54 in the VCD spectral appearance for o~-helical structures (see Fig. 14-12). The amide I' bands for o~-helical structures show a negative-positive-negative VCD triplet (in place of negative-positive VCD couplet for the amide I bands). This sign pattern was also observed for proteins (such as hemoglobin, myoglobin and albumin) with percent content of ~-helical structure greater than ~60%. The peaks of negative, positive and negative VCD features are at - 1657, 1645 and 1620 cm -1 respectively. For proteins, such as concanavalin A and chymotrypsin, with low percent content of a-helical structure (~20% or less) the amide I' VCD is found to be a positive-negative couplet (weak positive feature at ~ 1645 cm -1, and strong negative feature at -1628 cm -1) with a negative bias (see Fig. 14-12). In the case of proteins with percent ~-helical content in the intermediate range (-20 to 60%; for example, lactoglobulin, cytochrome c etc.) the observed VCD sign pattern is an approximate combination of the previous two, resulting in a 'W' shape, with negative peaks at ~ 1660 and 1620 cm -1 and positive peak at ~1645 cm -1. Based on these VCD sign patterns, the VCD spectra of epidermal growth factor and fibroblast growth factor were interpreted 18 to have very little a-helical content. hemoglob

poly,-.L-lysine

n

._

0

(xto s) -5

1

A

.5

0 1700 1600 1500 w a v e n u m b e r ( e r a - 1)

1700

1600

1500

w a v e n u m b e r (era -1)

17'oo ldoo 1~o w a v e n u m b e r ( c m - 1)

Fig. 14-12. VCD (top) and absorption (bottom) spectra of (a).hemoglobin, (b). achymotrypsin and (c). poly-L-lysine at pH 7.0 in H20 (thick lines) and D20 (thin lines) solutions. Reproduced with permission from Ref. 54. Copyright 1993 American Chemical Society.

Poly(lysine) is known to undergo structural transition from coil to ~-

348

Chapter 14

helix and to [3-sheet structures with changes in pH, temperature or solvent. The VCD spectra of poly(L-lysine) were in fact found 55 to be different under different conditions. In the amide I' region, the VCD spectrum of poly(L-lysine) in D20 (and also in CD3OD + D20 mixture from about 50:50 to 90:10 ratio of CD3OD to D20) shows a positive-negative couplet. This is the same sign pattern seen for amide I bands of poly(L-tyrosine) in dimethyl sulphoxide which was mentioned earlier to have random coil structure. In a solvent mixture of 96% CD3OD and 4% D20, the amide I' bands of poly(Llysine) give a negative-positive-negative VCD sign pattern, which is similar to that seen for right handed o~-helix structures. When the ratio of CD3OD to D20 is between 90 and 95, the amide I' VCD of poly(L-lysine) has a single negative feature, and this is considered to represent an intermediate state between coil and helix structures. At a pD of 11-12 and 65~ significant changes were noted in the absorption and VCD spectra of poly(L-lysine) in the amide I' region. A strong absorption band with associated negative VCD is now the dominant feature, which is considered to be characteristic of [3-sheet structure. A weak negative VCD was also present at 1680 cm-1, but there are some questions about the reproducibility of this VCD feature. Polypeptides with alternating lysine and leucine residues, referred to as (KL)n type polymers, can be changed from coil to ~-sheet type structures using high salt concentration. Polymers of (LKKL)n type on the otherhand are stabilized in cx-helical form under the same conditions. The same effects may also be induced in going from low to high concentration of the poly peptides. At a concentration of -23 mg/mL of D20, the VCD spectrum of poly(LKKL) in the amide I' region has 55 negative-positive-negative triplet, which is characteristic of o~-helical structure. The same pattern, but with higher intensity, is retained in salt solution indicating that the polymer has already stabilized in c~-helical form in the absence of salt. In the case of poly(KL), at a concentration of--23 mg/D20, the amide I' VCD does not uniquely fit anyone of those observed for well defined secondary structures, so mixed conformations are implicated. In salt solutions, amide I' VCD resembles that of ~-sheet structures. The interpretation of the VCD spectra of (KL)8 posed some uncertainties. Because of the presence of a closed ring in proline, this amino acid introduces charateristic effects into protein structure. One effect is that the q~ angle is restricted to -60 ~ Another effect is that the N-H bond, nitrogen being a part of the closed ring, cannot participate in the type of hydrogen bonding network that is common to other amino acids. Finally, both cis and trans peptide linkages are possible with proline. So proline favors the formation of turns in protein structure. Poly (proline) with trans peptide linkages is referred to as poly(proline)-II, while that with cis peptide linkages as poly(proline)-I. Poly(proline) structures are distinctly different

Applications and Molecular Structure

349

from (z-helix and [3-sheet structures, due to the above mentioned effects. Accordingly, the VCD spectrum of poly(L-proline)-II was found 56 to be different from those obtained for right handed ~-helical and 310-helical structures. For example, in the amide I' region, poly(L-proline) in D20 shows a positive-negative VCD couplet (which is similar to those seen for unblocked proline oligomers, (L-proline)n with n = 3 to 7; see Fig. 14-13), whereas o~-helix structures are characterized to have negative-positivenegative triplet in this region. However it resembles the VCD spectra obtained for poly(L-lysine) (see Fig.14-12), and poly(L-glutamic acid) which are suggested to have random coil structures at neutral pH. One difference is that the absorption band maximum in the amide I region is located at --1623 cm -1 for poly(L-proline), while that for poly(L-glutamic acid) is at --1646 cm -1. This difference is attributed to the tertiary amide group present in proline. b) n:

8

6

A 1

0 1750 1650 15,.%0 Frequency ( c m - I)

zSA XIO

4

-2

1750 1650 1550 Frequency ( c m - 1 )

Fig. 14-13. VCD (fight panel, b) and absorption (left panel, a) spectra of (proline)n, n=37 in acidic D20 solutions. Reproduced with permission from Ref. 56b. Copyright 1991 Munksgaard International Publishers.

The absorption spectrum for (L-proline)3 in the amide I' region shows (see Fig. 14-13) two distinct maxima at 1624 and 1641 cm -1, but as n increases the resolution of these two bands becomes poor. For n - 7, the amide I' band has maximum intensity at ~ 1623 cm- 1, with 1641 cm- 1 component appearing as a shoulder. The VCD features in the amide I' region however remain similar as n increases, with some qualitative differences in the width of the positive VCD component. Similar observations were noted for N-terminal blocked proline oligomers with n =

350

Chapter 14

3, 4, 6 and 12. For n = 2 however, only a single negative VCD feature is present in the amide I' region. Thus, it appeared that the on-set of systematic poly(proline) structure begins at n > 3. The similarity of amide I' VCD spectra of (L-proline)3 and higher oligomers to those of polypeptides with random coil structures lead to the suggestion that the local order that is thought to be present in the random coil structures might be at least 3 to 4 residues long. Terminally blocked oligopeptides (L-proline-Aib)n and Aib-(L-prolineAib)n, where Aib stands for ~-aminoisobutyric acid residue, were suggested to have [~-bend ribbon spiral structures which are considered to be a sub-type of 310-helix structure. The VCD spectra obtained for such peptides in chloroform solution indicated 56a-c that the amide I region has a negative-positive couplet, with a positive bias, centered at ~ 1641 cm-1. The somewhat higher amide I frequency here (compared to that in the proline oligomers) is attributed to the presence of both secondary and tertiary amide groups. The amide II region shows a strong negative VCD at 1535 cm -1 which is also the position of strong absorption maximum. These are also the characteristic features associated earlier with 310-helical structures. Then one can support the view that l-bend ribbon spiral structures are related to 310-helical structures. However, this observation may also be viewed as reflecting the inadequacy of amide I VCD features in differentiating between these two types of structures. The chain length dependence on the normalized VCD intensities indicated that the [~-bend ribbon character emerges as early as n = 2 or 3 and develops fully at n = 5 to 6. The amide A VCD of blocked oligopeptides containing proline, in organic solvents, were found 56d to exhibit characteristic features for conformations representing the "t- and l-turns. The amide I' VCD patterns seen 57 for (L-Ala)n, n - 3 to 5 are similar to those seen for (L-Pro)n. But it is difficult to argue that the structures in these two cases are similar, because of the special structural restrictions placed by the proline ring. In the amide I' region, the VCD spectrum of gramicidin-S in dimethylsulfoxide solvent shows 58 a positive-negative couplet (with a negative bias) that closely resembles the amide I' VCD features seen for random coil structures. The positive portion of this VCD couplet loses some intensity when the solvent is 1" 1 mixture of D 2 0 and TFE. It is reported that gramicidin-S, which is a cyclic decapeptide, contains antiparallel l-sheet structure formed by four pairs of residues and ~l-tums formed by two other residues. So the interpretation of VCD here is not as clear as in the well behaved systems discussed earlier. Just as a coil structure represents a non-repeating conformation, a 'turn' also represents a non-repeating conformation with at least four amino acid residues participating in a turn 53. In a common turn (referred to as type I

Applications and Molecular Structure

351

turn), the four o~-carbons make a dihedral angle of ~ +45 ~ In glycine turn (referred to as type II turn), the four t~-carbons and the hydrogen bond (between the C=O of residue 1 and N-H of residue 4) stay approximately in a plane. A y-turn in contrast is stabilized by 1-3 H-bonding. The crystal structure of a cyclic pentapeptide, cyclo-(-Gly-Pro-Gly-D-Ala-Pro) reveals a type II turn and a y-turn. The VCD spectrum in the amide I region showed 59 a negative-positive-positive-negative pattern. These peaks are well separated compared to those seen in VCD spectra associated with t~helix, 13-sheet and coil structures. The VCD spectrum of cyclo-(-Cys-ProGly-Cys-) in dimethyl sulphoxide solvent shows a similar pattern, with much less negative VCD intensity for the low frequency bond. The VCD spectrum of this cyclic peptide and that of cyclo-(Pro-Gly)3 were found to show significant changes depending on the solvent. It was suggested that the conformational mobility of this cyclic peptide can be conveniently probed using VCD. As mentioned earlier VCD spectra in aqueous solutions can be measured 54 if higher sample concentrations and lower pathlength cells (vide supra) can be used. In the amide I region, high t~-helical structural content proteins, such as hemoglobin (see Fig. 14-12), give a negative-positive VCD couplet (with a strong negative bias), while random coil structures, for exaple, poly(L-lysine) at pH7, give the opposite VCD sign pattern (a positive-negative couplet with strong positive bias; see Fig. 14-12). Proteins such as ct-chymotrypsin with a predominant 13-sheet structure give a large negative VCD signal (with a small positive VCD as a high frequency shoulder). In the case of proteins with a mixture of t~-helical and ~-sheet structures, for example ribonuclease S, a 'W' shape VCD spectral pattern was observed. These features 54 are quite similar, with some minor differences, to those discussed earlier for the corresponding amide I' bands. Despite these similarities of VCD features in H20 and D20 solutions, some significant differences in the role of H20 vs D20 solvent were noted in some cases, when the structural changes were monitored as a function of PH (or PD). The VCD spectrum of bovine tx-lactalbumin in D20/phosphate buffer (pH = 7.6) in the amide I' region shows 60 a large negative band at --1650 cm -1 with a small positive band on the high frequency side at --1680 c m -1. This pattern is similar to that seen for concanavalin A and txchymotrypsin which have high 13-sheet content. Under Fourier selfdeconvolution the major absorption band at 1650 cm -1 reveals two components at 1637 and 1649 cm -1 and the corresponding major components in H20 solution are at 1650 and 1660 cm -1. Addition of propanol up to 20% does not affect the shapes of the VCD or absorption bands, but at -33% significant changes were noted. A negative-positivenegative intensity pattern is seen with peak positions at -1675, 1658 and

352

Chapter 14

1628 cm -1. This sign pattern is nearly opposite to that seen without propanol added. At 66% propanol concentrations, the VCD spectrum resembles that of hemoglobin with high or-helical content. The deconvoluted absorption band reveals decreased intensity at 1637 cm -1, increased intensity at 1646 cm -1 (with a shift in peak position to lower frequency) and a new band at ~ 1621 cm -1. The increase in absorption intensity at --1646 cm -1 may also suggest tx-helical contribution. Thus these studies 6~ indicated that the addition of propanol induces conformational change from [3-sheet to o~-helical type secondary structure in cz-lactalbumin. The crystal structures of hen egg white lysozyme and o~-lactalbumin (from baboon) are thought to be similar. But the solution structures appear to be different as revealed by the VCD spectral comparisons 60. The VCD of hen egg white lysozyme in amide-I' region is very similar to that of lactalbumin (with 33% propanol). Addition of 33% propanol to lysozyme does not change the VCD indicating considerable structural stability in this case. The deconvoluted absorption spectrum in D20 shows a peak at ~ 1653 cm -1 with shoulder at 1642 cm-1, while that in water shows a peak at --1657 cm -1 with two principal side lobes at--1672 and--1647 cm -1. Not only are the amide I' VCD spectral signatures of bovine o~lactalbumin different from those of hen egg white lysozyme, there are in fact differences among the spectral signatures of tx-lactalbumin derived from different sources. Human-, bovine-, and goat-lactalbumins are supposed to have high homology yet the amide I' VCD spectral signatures are different in these three cases. Upon changing pH from -7 to -2, however, the amide I' VCD spectral signatures of all these three lactalbumins become similar among themselves and to those of hen egg white lysozyme. These observations indicate that definite structural differences exist for lactalbumins derived from different sources under normal conditions (pH -7) and that they take on similar structures under extreme conditions (pH -2). The structure of hen egg white lysozyme, however, appears rather stable for pH changes f r o m - 7 to -2 as well as to propanol solvent perturbations (because amide I' VCD does not seem to be influenced significantly by these perturbations). The above mentioned changes resulting from the influence of pH are for the amide I' VCD. The amide I VCD however shows a different behaviour. That is, amide ! VCD spectra taken in H20 solutions do not reveal major changes as the pH is changed from -7 to 2. In water solutions, the amide I band originates from the C=O stretching vibrations slightly coupled to the N-H bending motions. In D 2 0 solutions, some (or most) of the N-H protons are changed to N-D and the amide I' band is considered to originate from the pure C=O vibrations. Then the changes in amide I' VCD among lactalbumins and hen egg white lysozyme can originate from two different

Applications and Molecular Structure

353

sources. (a) The extent of deuteration of N-H protons, which depends on the access of N-H protons to D20 solvent and therefore to the type of protein structure; (b) the nature of coupling between C=O stretching and NH bending modes (here one has to consider the extent of the presence of unexchanged N-H protons), which will go beyond the secondary structural details. In either case, the amide I' VCD can be seen to be probing the intricate structural details that are not accessible to normal spectroscopic techniques.

14.2.4 Nucleic acids Two different spectral regions were used to probe the structures of nucleic acids using VCD spectroscopy. One region is 1700-1600 cm -1 where C=O, C=N and C=C stretching modes of the nucleic acid bases are expected to appear. The vibrational bands appearing above --1650 cm -1 are associated with the C=O and those appearing below --1650 cm -1 are associated with C=C stretching modes (with some contribution from C=N stretching as well). Adenosine has no C=O groups but cytosine, inosine, and guanosine have one C=O group each and uridine has two C=O groups. This difference in the basic chemical composition can be expected to influence the nature of VCD of nucleic acids with different base compositions. The second region is 1250-1000 cm -1, where the symmetric and antisymmetric stretching modes of the PO 2 group and some vibrational modes of the sugar moiety are expected to appear. If these modes are not coupled to those originating from the bases, then the VCD appearing in this region can be expected to be independent of the base composition. In the 1700-1600 cm- 1 region, monoribonucleotides were found to have 61 very little VCD. But dimers were found to give a weak negativepositive VCD couplet. The same VCD sign pattern is also present for homopolymers of ribonucleic acids but with the magnitudes of VCD intensities significantly enhanced. These observations lead to the suggestion that the observed negative-positive VCD couplet is primarily due to the base-base coupling. The absorption band positions and associated VCD sign patterns are summarized in Table 5. Among the homopolymers of ribonucleic acids, poly(rI) was found to give two to three times larger VCD magnitudes than poly(rA) and poly(rC), and an order of magnitude larger VCD magnitudes than poly(rG) and poly(rU). While such large differences in magnitudes of VCD intensities should reflect some differences in conformational details, the invariance of VCD sign pattern among these poly ribonucleic acids provided a common factor for structural investigations. Copolymers of ribonucleic acids were also found to give a negativepositive VCD couplet in the 1700-1600 cm -1 region. In the case of poly(rC,rU), the normalized VCD magnitudes were found to be,

Chapter 14

354

TABLE 5 Vibrational absorption band positions and the associated VCD couplets in the 1700-1620 cm -1 region for some ribonucleic acids 61 poly(rC) poly(rA) poly(rG) poly(rI) poly(rU) poly(rG) 9poly(rC) poly(rA) 9poly(rU) poly(rI) 9poly(rC) poly(rC, rU) poly(rA, rC) poly(rI, rC)

1656 (-,+)

1617 (-,+) 1627 (-,+)

1688 (-,+) 1684 (-,+) 1659 1649 1670 1647 1657 1650

1690 (-,+) 1710 (-) 1694 (-,+) 1691 (-,+)

(-,+) (+,-) (-,+) (-) (-,+) (-,+)

1621 (-,+)

1630 (-,+) 1625 (+)

60

,iS BIZI 31ZI

15

15

/xC

'

0

,

XIIZl2

0.

.,,

XIO

~.

-15

I

I

1000

-15.

11211ZII2]

12

500 0 17 50

1650

FREQUENCY

15

(CM -I)

1751ZI IB51ZI 1550 FREQUENCY (CM -I)

Fig. 14-14. VCD (top) and absorption (bottom) spectra of poly (rA).poly (rU). Left panel: at 25~ (solid line) and 52~ (dahsed line); Right panel: at 56~ Reproduced with permission from Ref. 62. Copyright 1993 John Wiley & Sons.

Applications and Molecular Structure

355

approximately, the average of those found for poly(rC) and poly(rU). Such additive feature is not generally applicable, because in the case of poly(A,U) and poly(A,C) the observed VCD magnitudes do not correspond to the averages of the magnitudes found for individual polyribonucleic acids. The VCD spectrum of poly(I,C) was found to be quite similar to that of poly(I), with one difference being that the absorption and VCD band positions of poly(I,C) shifted upwards in frequency position. The VCD spectra of the double strand polymers poly(I) 9 poly(C) and poly(A) 9 poly(U) were found to be similar to those for the corresponding copolymers poly(I,C) and poly(A,U) respectively. The most useful applications of the similarities (and differences) seen among the VCD features of different nucleic acids can be found in deducing the structural transitions of a given nucleic acid under different experimental conditions. One such application was demonstrated 62 by monitoring the VCD of poly(rA)opoly(rU) as a function of temperature. In the 1700-1600 cm -1 region, the VCD spectrum of poly(rA)opoly(rU) in the 25~ range contained (-,-, +,-, +) sign pattern (Fig. 14-14). This pattem changes to (, +, +,-, +,-, +) when the temperature is maintained in the 54-60~ range. The frequency separation among the first three higher frequency bands, considered to originate from the C=O stretching modes of the U residues, increases as the temperature is increased. However, as the temperature is further increased to above 65~ VCD associated with these bands vanishes. The bands appearing in the 1650-1660 cm-1 region (which are considered to originate from the C=C stretching modes of the A residues) however are not affected by this increase in temperature. A plot of VCD intensity changes (in the 1700-1650 cm -1 region) as a function of temperature revealed that abrupt changes occur in the 52-54~ and 60-65~ ranges. Similar temperature dependent VCD studies on triple stranded poly(A) 9poly(rU) 9 poly(rU) revealed no abrupt changes for the bands in 1700-1650 cm -1 region; instead the VCD intensity changes as a function of temperature were found to follow a smooth line. Single stranded poly(rA) and poly(rU) also do not exhibit abrupt changes in VCD intensities as the temperature is increased to 70~ By comparing the VCD intensities observed at different temperatures for poly(rA), poly(rU), poly(rA) 9poly(rU) and poly(rA) 9 poly(rU) 9poly(rU), it was concluded that the two abrupt VCD intensity changes referred to above for poly(rA) 9 poly(rU) correspond to the structural transition from double strand to triple strand (at 52-54~ and to single strand (at 60-65~ forms. The ordinary infrared absorption band positions (or intensities) by themselves were unable to identify the second structural transition (at 60-65~ so the important role of VCD intensities is underscored by these investigations. The B---~Z and B ~ A structural transitions in deoxyribonucleic acids can be induced 63 using high salt concentrations and alcohol concentrations respectively. The B- and A-forms have right handed helical structures, but

Chapter 14

356

the Z-form has a left handed helical structure. The B-form is characterized by the appearance of a negative-positive VCD couplet in the 1700-1650 cm1 region and another couplet of the same sign pattem at 1087 cm -1. The latter couplet is associated with symmetric PO 2 stretching modes (see Table 6). The signs of VCD couplets in these two regions were found to be TABLE 6 Vibrational absorption band positions and the associated VCD couplets of the PO 2 symmetric stretching modes of some deoxyribonucleic acids (dG-dC)n (n = 2,3) poly(dC) poly(dA-dT) 9poly(dA-dT) poly(dA) 9poly(dT) poly(dG-dC) 9poly(dG-dC) Calf thymus DNA

1087 (-,+), 1053 (+) 1087 (-,+), 1060 (+)

1087 1087 1087 1087 /

2 AAx

10 5 0

~ - _

. _.

(-,+), (-,+), (-,+), (-,+), ,~

(+) (+) (+) (+)

%%

//~ ',,,

.,?' \..

, . ,, ,~.,,,'x.,--.-~;

1053 1053 1053 ]o53

"'--...

\~---

9 # ~. I I

-2 -4

15" AxlO 2

10-

,:"".l,/ ! 550

"

". i 1650

Wavenumber (I/cm)

1750

Fig. 14-15. VCD (top) and absorption (bottom) spectra of poly (dG-dC).poly (dG-dC) at low salt (solid line) and 0.7M MgC12 (dahsed line) conditions. Reproduced with permission from Ref. 63a. Copyright 1989 John Wiley & Sons.

Applications and Molecular Structure

357

reversed for the Z-form (see Fig. 14-15). Although the band positions of the C=O stretching modes of the Z-form were found to shift to lower frequencies compared to those of the B-form, the band positions of the symmetric PO 2 stretching modes are essentially the same for both forms.Thus the VCD couplets associated with symmetric PO~ stretching modes of the B- and Z-forms appear more like mirror images of each other and provide a clear-cut approach for identifying this transition. The VCD spectra of the B- and A-forms were found to be qualitatively similar. However, there are differences between them in the C=O stretching region and these differences depend strongly on the sequence or distribution of the bases.

14.2.5 Carbohydrates In the C-H stretching region, 1-O-methyl t~-D-glucopyranoisde was found 64a to give a negative-positive-positive-positive VCD pattern, starting from the high frequency side. The corresponding pattern found for 1-Omethyl ~-D-glucopyranoside was of opposite sign, i.e. positive-negativenegative-negative. Since ct- and 13-anomers differ only in the absolute configuration at C1, and have the same configurations and constituents at other carbon atoms of the pyranose ring, the observed VCD bands were suggested to belong to the CH3 and anomeric C-H groups. These assignments were supported by significant reduction in the observed VCD magnitudes for 1-O-methyl-d3-~-D-glucopyranoside. The fact that a change at one chiral center has so much influence on the observed VCD, lead to the suggestion that VCD may potentially be used in deducing the stereochemical information at individual chiral centers. VCD spectra in the 1650-800 cm-1 region64b, c and in the C-H stretching region 64d have been obtained for simple sugars. A VCD band at --1150 cm -1, considered to originate from the C-O stretching motions, was found to have a correlation to the sequential arrangement of the hydroxyl groups around the carbohydrate ring. The isotopomers D-glucose- 1-dl, D-glucose-2-dl and D-glucose-6,6d2 have also been studied 64d to help locate the positions of individual C-H stretching bands. 14.2.6 Transition metal complexes The conformations of the ring structures formed by the ligands, due to complexation with metal ions, were investigated 65a-d. L-amino acids, ethylene diamine and ~-alanine were used as the ligands for complexation with Cu(II), CO(III) or Cr(III). VCD features observed in the N-H and CH stretching modes and in some bending modes were interpreted in terms of the possible conformations of the complexed ligands. In the C-H stretching region, the complex of Zn(II) with (-)-spartein showed 65e weak VCD, but

Chapter 14

358

those of Co(II) and Ni(II) showed enhanced VCD due to the coupling of vibrational transitions with the underlying d-d electronic transition. 14.2.7 Other molecules In three different series of chiral molecules containing the benzene group, the VCD sign associated with a benzene group band appearing in the

TABLE 7 A correlation 66a between VCD sign and absolute configuration in molecules containing benzene group Band position

Molecule

(R)-(+)-sec-phenethylalcohol (R)-(+)- 1-phenyl- 1-propanol (R)-(+)-2-methyl-l-phenyl- 1-propanol (S)-(+)- 1-methoxy-2-phenylethanol (R)-(+)- 1-phenyl- 1-butanol (S)-( +)- 1-phen yl- 1,2-ethanedi ol

R1

OR1 R2,~Ph

H H H CH 3 H H

R2

CH3 CH3CH2 CH3CH(CH3) CH2OH CH3CH2CH2 CH2OH

VCD sign

(cm l)

(-) (-) (-) (-) (-) (-)

1452 1452 1452 1452 1454 1453

(-) (-) (-) (-) (-)

1456 1452 1455 1455 1455

(+) (+) (+) (+)

1452 1462 1462 1460

COOR 1

(S)-(+)-2-phenylpropionic acid (S)-(+)-2-phenylbutyric acid (S)-(+)- ct-methoxyphenyl acetic acid (S)-(+)-madelic acid (S)-(+)-methylmandelate

H H H H CH3

CH3 CH3CH2

OCH3

OH OH

Phi, R1 P~ (R)-(+)-phenyloxirane (1R,2R)-(+)-phenylpropylene oxide (2R,3R)-(+)-3-phenylglycidol (2R,3R)-(+)-2-methyl-3-phenyglycidol

H H H CH3

H CH 3 CH2OH CH2OH

-- 1450-1460 cm- 1 region, w a s f o u n d 66a to correlate witti the molecular

Applications and Molecular Structure

359

configuration. This correlation (see Table 7) can be viewed as representing an empirical rule for predicting the molecular stereochemistry of chiral molecules containing the benzene group. In phenyl carbinols 66b the sign of a VCD band at ~ 1200 cm-1 was found to correlate with the chirality of the probe group. The absolute configurations of substituted indans 66c appeared to be reflected by the sign of a VCD band orignating from the C*-H bending vibration (* representing the chiral carbon). The VCD spectra in the O-H and N-H stretching regions for a series of ephedra molecules with pharmaceutical importance have been used 67a to deduce the conformational preferences. VCD studies on the H-bonded systems were also undertaken 67b-e, in the O-H stretching region, probing the conformational preferences. The relationships between the VCD observed in the N-H stretching region for 2,2'-diamino-l,l'-binaphthyl, and in the C-H stretching region for 9,10-dihydrodibenzo phenanthrene, and the absolute configurations have been explained67f, g using empirical models. In the case of 2-methyloxetane 67h, the VCD associated with the methine and methyl symmetric bending vibrations are identified as configurational markers. Similarly the VCD spectra in the 1400-1100 cm -1 region of the derivatives of 5-methyl-6,8-dioxabicyclo [3.2.1 ] octane were found 67i to have some common features that reflect the absolute configuration. 14.2.8 Heme proteins The antisymmetric stretching vibration of the azide group covalently bound to low-spin Fe(III) in heme proteins was found 68 to give an intense VCD signal at -2020 cm -1. For the corresponding high spin complex the antisymmetric azide stretching absorption band appears at -2040 cm-1 but shows no VCD. For the azide group bound to non-heme proteins also a significant VCD signal was found. The VCD signals are much larger for the cyanide stretching vibration (at-2120 cm -1) of the cyanide ligand bound to heme proteins. No measurable VCD signals were found for the carbonmonoxy stretching vibration of carbonmonoxymyoglobin. In order to isolate the sources responsible for these large VCD signals, studies were undertaken with substituted (and reconstituted) hemes and with proteins where distal or proximal amino acids were replaced. The VCD signals were found to be more sensitive than the corresponding absorption to proteinligand interactions. The dependence of sign and magnitudes of the observed VCD signal on the protein environment, however, is not yet clearly understood. 14.2.9 Enantiomeric excess The optical purity of chiral samples is commonly measured using optical rotation or circular dichroism in the visible spectral region. In order to

360

Chapter 14

verify whether or not more precise and accurate estimate of enantiomeric excess could be made using VCD spectra, VCD studies were undertaken 69 on trans 1,2-dideuteriocyclopropane. Here VCD spectra were measured in the C-D stretching region for samples with known percent enantiomeric excess (in the range of 30% to --87%). The corresponding values determined from VCD (with reference to the VCD of pure enantiomers) were found to match the known values to within one percent.

14.2.10 Kinetics The rate of racemization of (1R,2R)-cyclobutane-l,2-d2 was determined 70a by measuring VCD in the C-H stretching region as a function of time. The corresponding rate of isomerization was determined by monitoring the absorption changes in the 500-600 cm -1 region. Approximately 15 mg of the sample was used. The rate of racemization of ( 2 S , 3 S ) - c y c l o p r o p a n e - l - 1 3 C - 1 , 2 , 3 - d 3 at 407~ and of (-)-(R,R)cyclopropane-l,2-d2 at 422.5~ were determined 70b,c by monitoring VCD in the C-D stretching region as a function of time. The corresponding rates of isomerization were determined from time dependent changes of vibrational absorption in the 1020-1050 cm-1 region. Approximately 5 mg or less of these samples were used for these measurements. These amounts are much smaller than those used for polarimetric studies. Thus the potential advantages of VCD for kinetic studies were identified. 14.2.11 Chiral detection VCD spectral measurements have been reported 71 for chiral samples eluting from a liquid chromatograph (LC) column. The (R)- and (S)enantiomers of benzoin and of 2,2,2-trifluoro-l-(9-anthryl)ethanol have been separated, on a high performance LC column, from the corresponding racemic mixtures and detected using their VCD features. The limit of detectability was estimated to be a few micrograms.

14.3

Vibrational Raman optical activity spectra

14.3.1 Ab initio theoretical predictions A satisfactory agreement between the theoretical and experimental VROA spectra provides an assurance that the VROA interpretations can be made from quantum mechanical principles rather than at an approximate qualitative level. For molecules studied to date (see Table 8) the ab intio predictions compared satisfactorily with the experimental VROA spectra. As an example, the ab initio depolarized VROA spectra obtained 8l a with the DZP basis set for CHFC1Br (see Table 2 of Chapter 7) are displayed in Fig. 14-16 and compared to the corresponding experimental spectrum. It can be seen that the signs and approximate magnitudes observed for (-)-enantiomer are correctly reproduced (for most of the bands) in the calculation on (R)-

Applications and Molecular Structure

361

CHFC1Br allowing the determination of its absolute configuration. It is possible to determine the absolute configuration and preferred conformations of a small to medium sized molecule in the solution phase using VROA. One of the major limitations in executing these calculations at :. ....

L

. . . .

I

..... ,,

v

i,,,

I, ~

,_~,

,

L__,

CHFCIBr

i 2 i

computed

!

-

expe~ental

~

'

'

'

i

500

'

'

'

'

I

'

1000

'

'

'

I

'

1500

'

'

'

.

I

3,000

wavenumber / cma Fig. 14-16. Depolarized Raman and ROA spectra of CHFC1Br. Theoretical spectra were obtained for (R)-configuration with the DZP basis set. The experimental spectra are for the (-)-enantiomer. Replotted from Ref.81a.

362

Chapter 14

the present time is in the finite nuclear displacement procedure 82 that is used for evaluating the derivatives of optical activity tensors. Thus significant advances can be anticipated in the future for VROA calculations. For large molecules, such as those of biological significance, ab initio predictions might then be feasible. TABLE 8 Molecules with available experimental and ab initio VROA data alanine72,73 L-alanyl-L-alanine72 3-methylcyclohexanone 76 2,3-trans-dimethyloxirane77 tartaric acid-d478 2,3-trans-dimethylthiirane 80 isoflurane81b, c

N-acetyl-N'-methyl-L-alaninamide72, 74 3-methylcyclopentanone 75 methyloxirane 77 tartaric acid78 methylthiirane 79 bromochlorofluoromethane 81a desflurane 81c

14.3.2 Empirical predictions The ab initio studies discussed above provide interpretations on a fundamental basis. Such calculations however do not render themselves to establishing simple predictive relations for general use and it is not prei'erable to undertake ab initio calculations on every single molecule of experimental interest. As a result, reliable empirical approaches that can relate the experimental spectra to molecular structure will have practical applications. A two group model 83a has been widely referenced in the literature for understanding the underlying VROA mechanisms. These concepts have been used to derive convenient expressions 83a,b and to interpret 83b the VROA associated with the methylene scissoring vibrations in some terpenes. If the qualitative predictions obtained with the two group model can be verified with quantum theoretical predictions on a suitable system, then one can gain confidence in using these qualitative predictions. H202, D202 and trans-2,3-d2-thiirane molecules are appropriate choices for this purpose. The normalized VROA intensities predicted ab initio with 631G(ext) basis set are compared with those predicted with the two group model in Table 9. Except for the symmetric bending vibration of H202 the signs predicted by both approaches are in agreement. Thus it appears that, for isolated vibrations where coupling with other internal coordinate motions is not significant and the identity of the vibrations can be discerned, it might be possible to gainfully apply the two-group model. For the cases

363

Applications and Molecular Structure

where the two groups involved in generating the VROA are dissimilar, the chiral influence of the second group was suggested 83c to be imparted on the VROA associated with the first group via Forster-type radiationless energy transfer; and the observed VROA was suggested to depend on the inverse square of the distance between the two groups. This is a very useful observation, because the observed VROA can then be viewed as depending on the immediate environment of a group and long-range influences can be ignored. Other appropriate models for VROA interpretations have focused on using the methyl group as a chiral probe. The degeneracy lifted bending vibrations83d, e and the torsional vibration83f, g associated with the methyl group have been explored. TABLE 9 A comparison 82a of two group and ab initio predictions Vibration

10 4 Az

two-group model

10 4 Ax

6-31G(ext)

two-group model

6-31G(ext)

H202

sym O-H stretch asym O-H stretch sym O-H bend asym O-H bend torsion

-4 2 -2 4 2

sym O-D stretch asym O-D stretch sym O-D bend asym O-D bend torsion

-4 2 -2 4 2

sym C-H stretch asym C-H stretch sym C-D stretch asym C-D stretch

-3 10 3 - 10

-12 7 0.4 3 7

4 -3 7 4

-3 8 0.3 7 8

D202 -12 7 4 -1 -3 3 7 7 4 trans-2,3-d2-thiirane -7 6 17 9 -7 - 17

-3 8 -2 6 8 -3 7 3 -9

14.3.3 Amino acids, peptides and proteins Alanine is one of the first molecules investigated in detail for VROA spectral interpretations72,73, 84a. The Raman and ROA spectra of L-alanine are displayed in Fig. 14-17. For other molecules in this series the

Chapter 14

364

interpretations were done at a qualitative level. Substitution of the methyl group hydrogens of alanine with other chemical groups results in distinct rotamers, whose relative populations are likely to influence the VROA spectral patterns. The populations of these rotamers, as derived from the NMR studies, were used 84 to explain the appearence of experimental VROA spectral patterns in serine, cysteine, valine, threonine and isoleucine. Although serine and cysteine differ only in the replacement of the O-H group in the former with a S-H group in the latter, they exhibit rather different VROA spectra. However, consideration of the differences in vibrational frequencies of C-OH vs. C-SH stretching motions (--1200 cm -1 for the former and 700 cm -1 for the later) and in relative populations of the rotamers of CH2OH vs. CH2SH groups, helped explain the differences. Earlier theoretical anlaysis of VROA spectra of alanine, also helped in understanding the origins of VROA in serine and cysteine. For valine and threonine, the overlap of bands resulting from additional CH3 groups modes restricted the interpretations to the bands that appeared above --1200 cm -1. Interestingly, isoleucine does not exhibit any significant VROA presumably due to the presence of two conformers that may have opposite VROA signs originating from the respective opposite handed H-Ca-CI3-H segments. ]9.0

" 10 6

Raman

__! I==,t

4-

CH3

A A]I

ROA

9

M

I

L-Alanine 700

900

1100

1300

1500

cm -~ 1700

Fig. 14-17. Raman (top) and ROA spectra of L-alanine in aqueous solution obtained in the backscattering geometry. Reproduced with permission from Ref. 84a. Copyright 1992 John Wiley & Sons. The VROA spectrum of L-alanyl-alanine (see Fig. 14-18) can be expected to differ from that of L-alanine itself. The former contains a

Applications and Molecular Structure

365

peptide group ( O = C - N - H ) whose vibrational modes are expected to show I

I

VROA features that may also be found in polypeptides and proteins. The negative VROA band at 1270 cm -1, large positive VROA at 1340 cm -1 and a smaller positive VROA a t - 1325 cm -1 seen 84 in the spectrum of L-alanyl-Lalanine, are also seen in the VROA spectra of proteins. The vibrational analysis 85 suggested that the 1280 cm -1 Raman band is actually composed of three different bands" on the lower frequency side a band originating from a methine deformation (labelled Cc-HI), from the C-H group next to COO- group, is thought to be present; a similar deformation (labeled CN-H I) of the methine group next to the NH3 + group is thought to be present on the high frequency side; the third band, labeled amide III, is supposed to have contributions from the N-H and C-H deformations. Since the negative 1270 cm -] ROA band is on the lower frequency side of the 1280 cm -1 Raman band, this band was associated with a C c - H I deformation. The two positive VROA bands a t - 1 3 2 5 and 1340 cm -1 were associated with amide III modes (the latter mixing between amide N-H bend and Cc-H II bend, while the former with mixing of Cc-H II and CN-H II bending modes). The sum of the VROA spectra of glycyl-L-alanine and L-alanyl-glycine was found 84d to be nearly equal to the VROA spectrum of L-alanyl-L-alanine. 1 . 4 x 10 v

A I[

. ,, u

Hi

Raman

II . . / ~ , , , , / ~ , N II

-3,',.

II

~,.

+

6.3= 10 a

t

0

L - AI anyl - L - al a n in e 700

900

cm -t I I00

1300

1500

1700

Fig. 14-18. Raman (top) and ROA spectra of L-alanyl-L-alanine in aqueous solution obtained in the backscattering geometry. Reproduced with permission from Ref. 84a. Copyright 1992 John Wiley & Sons. The VROA spectral changes seen 84 for L-alanyl-L-alanine on going

Chapter 14

366

from neutral to high or low pH are not dramatic, but pH dependent differences do appear. For example, the strong positive VROA band at --1340 cm -1 appears as two separate positive bands at high and low pH values. The negative VROA band at 1400 cm -1 disappears (almost) at high pH, but remains at low pH. Such changes can result from two different sources" one is that variations in pH may result in a different conformational preference; second, the zwitterionic form present at neutral pH is no longer present at high and low pH. The vibrational mode compositions of CO0- vs. COOH and NH3 + vs. NH2 groups can alter the appearance of VROA. One intriguing observation is that the VROA spectrum of L-alanyl-Lalanine is quite similar to that of higher oligomers, L-alanyl-L-alanyl-Lalanine and L-alanyl-L-alanyl-L-alanyl-L-alanine 84c. Any major differences in the backbone angles of (I) and ~ in these molecules would be expected to seriously influence the appearance of VROA spectra. In the absence of such obvious differences, one is tempted to suggest that the backbone structure in the molecules is fairly consistent. However, based on the pH dependent VCD studies in the amide I region, it was thought 50 that the solution structure of L-alanyl-L-alanyl-L-alanine is stabilized by the interaction between the zwitterionic groups from either end. As the VROA spectra of L-alanyl-L-alanyl-L-alanine in neutral and basic solutions are similar, there is a contradiction between the conclusions emerging from these studies. This discrepancy may have arisen because of the empirical nature of the model used in interpreting the VCD data. ] 2 . 1 9I0 7

'-

?00

900

1100

~

1800

'

'

1004)

ii,

1700

700

900

1100

13~

1~

1700

Fig. 14-19. Raman (top) and ROA spectra of bovine serum albumin (left panel) and lysozyme, from hen egg white, (right panel) in water obtained in the backscattering geometry. Reproduced with permission from Ref. 86b. Copyright 1992 Royal Society of Chemistry.

367

Applications and Molecular Structure

Characteristic vibrational Raman bands associated with tx-helix structures have been suggested86, 87 to appear in the 1645-1680, 1230-1310 and 890-945 cm -1 regions. These three regions are associated respectively with C=O stretching (amide I), N-H and Ctx-H bending (amide HI) and C~C stretching (backbone skeletal stretching) motions respectively. The characteristic modes associated with [3-sheet structures have been suggested to appear in the 1645-1680 cm -1, 1230-1310 cm -1 and 1020-1060 cm -1 regions 86,87. The first two regions have similar origins as in t~-helix structures, and the 1020-1060 cm-1 region is associated with Ca-N stretching motions. Bovine serum albumin, which is thought to have high

.d §

] ~.SxlO ?

ROA

P.OA

..~. i 0

C

___i

n,,, ~

0

700

004)

1100

1300

1500

wav~number/e m -I

1700

700

900

I I00

1300

1500

1700

wawnumber/e m -I

Fig. 14-20. Left pannel: Raman and ROA spectra of o~-helical poly (L-lysine) in NaOH solution at pH 11.0 (top half) and in NaOD solution at pD 11.0 (bottom half) obtained in the backscattering geometry at 2~ Right panel: Raman and ROA spectra of unordered poly (L-lysine) in glycine buffer at pH 3.0 (top half) and in deuterated glycine buffer at pD 3.0 (bottom half) obtained in the backscattering geometry at 20~ Reproduced with permission from Ref. 87b. Copyright 1996 Royal Society of Chemistry.

t~-helical content (and no I]-sheet structure) exhibits a large positive VROA

368

Chapter 14

intensity in the 900-950 cm-1 region (corresponding to the above mentioned Ca-C stretch) and no significant VROA in the 1020-1060 cm -1 region (see Fig. 14-19). Ribonuclease A, lysozyme and oc-lactalbumin, which are thought to have both or-helix and 13-sheet structural components exhibit VROA in both ~-helix and 13-sheet regions. These observations support the hypothesis of characteristic vibrational Raman bands representing ~-helix and 13-sheet structures. Bovine serum albumin also exhibits a large positive VROA band at 1340 cm -1, which was assigned 87b,e,f to a rigid loop structure with local ordering corresponding to that of a 310-helix. Twisting vibrations of CH2 groups in aliphatic side chains and tryptophan residues also contribute around this region. The observation that the VROA band intensity at 1340 cm -1 disappears when bovine serum albumin is dissolved in D20 discards these latter possibilities and suggests the involvement of NH bending vibrations. Lysozyme and o~-lactalbumin also exhibit positive VROA at 1340 cm -1 and their VROA spectra in the 1100-1700 cm -1 region are similar except for some differences in relative intensities of the ROA bands. The sharp features at 1582, 1552, and 757 cm -1 in these molecules are considered to have contributions from the tryptophan residues. The relative intensities of the two positive VROA bands at -~1300 and 1 340 cm-1 are reversed from bovine serum albumin to lysozyme, whereas the relative intensities of these two bands in oc-lactalbumin are approximately equal. The implications of these differences is that, lysozyme and o~-lactalbumin have similar a-helix structures, but a-lactalbumin has more 310-helix loop structure. A comparison of the VROA spectra for poly-L-lysine 87 at pH -- 3 and at pH = 11 (both at 20 ~ revealed differences which can be used to support or discard some of the hypotheses. Poly-L-lysine is thought to exist in random coil structure at pH = 3, and in a-helix structure at pH = 11. At pH = 3, the VROA spectrum (see Fig. 14-20) reveals a positive band at 1318 cm -1 (presumably due to C~-H and N-H bending motions) and a broad negative band in the 1180-1261 cm -1 region (also due to the Co~-H and N-H bending motions). In place of the single positive VROA band, two positive bands appear at pH = 11, with the new one appearing at 1335 cm -1. The relative intensities of these two positive bands are similar to those seen in the VROA spectrum of bovine serum albumin at neutral pH. This observation is in agreement with earlier statements that bovine serum albumin has high oc-helical content (although the assignment of this band to 310-helix loop strcture remains to be independently verified). The negativepositive VROA couplets centered at -1103 cm -1 and 1275 cm -1 (negative on the lower frequency side) in the spectrum of poly-L-lysine at pH - 11 and their similarities to the couplets at about the same position in the case of bovine serum albumin (and insulin as well) suggest that these couplets

Applications and Molecular Structure

369

represent characteristic VROA features of o~-helical structures. The earlier suggestion that the bands in the 900-945 cm -1 region may also serve as signatures of o~-helix structures, can be acommodated by the presence of VROA bands in this region for poly-L-lysine at pH = 3.0 if the random coil structure of poly-L-lysine is considered to be a mixture of well defined c~helix, l-strand etc structures. Concavalin A, o~-chymotrypsin, ribonuclease A and ~-lactoglobulin have the following structural components" (64% 13, 2% o~), (51% 13, 10% o~), (46% 13, 23% ~) and (50% 13, 15% a) respectively. The presence of high 13-sheet and low c~-helical contents make these molecules suitable to discern common VROA features 87 that can be associated with [3-sheet structure (see Fig. 14-21). In all these four cases, a negative VROA band is present in the 1340-1380 cm- 1 region and a positive band is present at 1313 cm-1. The frequency changes in the amide I region, among different structures, are significant. A negative-positive VROA couplet (from the lower frequency side) is seen in the spectra for ribonuclease A, bovine serum albumin (Fig. 14-19), lysozyme (Fig. 14-19), cx-helical poly(Llysine) (Fig. 14-20), a-lactalbumin, (x-chymotrypsin (Fig. 14-21) and 13lactoglobulin (Fig. 14-21). The center of the couplet in l-sheet proteins is at -- 1658 cm- 1 and at ~ 1647 cm- 1 in a-helix proteins. However, poly(Llysine) (at pH = 3; see Fig. 14-20), and insulin show single positive VROA while Pro-Leu-Gly-NH2 (Fig. 14-21) has negligible VROA here. So an unambiguous correlation of VROA with structure is not yet available in this region. Temperature dependent VROA spectra of ~-lactalbumin and lysozyme were used87g, h to address the structure of molten globule state and to uncover a cooperative transition by monitoring the changes in VROA intensities of the bands in amide I, amide II, extended amide III and backbone skeletal stretch regions. From these measurements on cxlactalbumin at pH=2.0, in the temperature range of 2 to 45~ it was concluded87g that the secondary structure present in the native protein persists with only a small gradual decrease with increasing temperature. The tertiary backbone fold however was considered to change dramatically from native like at 2~ to completely disordered at 35~ From the corresponding studies on lysozyme at pH=5.4 it was concluded 87h that a cooperative transition occurs at 12~ involving the loss of mobility in tertiary loop, secondary structure and tryptophan residues. It was further concluded, in both proteins, that no significant conformational change appeared to be involved in the temperature range studied.

Chapter 14

370

o I

A

+

I ....

I

I

,L

I

"'

F

~

'!

IV-

' I

A

I'

/I

.

4-

9

I

I

l

9

I

I

I

9

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i

II

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+

_

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,l~

~

-

.

.

9~

.

.

.

_.

~,

.h.^.,

~.-5.3xlo* I

I

900

em-I I

i

|

9

1300

1

I

I

1

1700

Fig. 14-21. Raman (top) and ROA spectra of o~-chymotrypsin, [3-1actoglobulin and Lpro-L-leu-gly-NH2 in aqueous solution obtained in the backscattering geometry. Reproduced with permission from Ref. 87c. Copyright 1994 Cambridge University Press.

14. 3.4 Carbohydrates The VROA spectra of simple carbohydrates 88 were divided into

371

Applications and Molecular Structure

three regions" anomeric region (600-950 cm-1), fingerprint region (9501200 cm- 1) and the CH2 and C-O-H deformations region (1200-1500 cm1). In the anomeric region, methyl-tx-D-glucoside exhibits VROA bands (although with weak intensity) while the corresponding [3-D-glucoside has no discernable VROA features. The vibrational mode compositions in the 13-anomer are probably such that no VROA is generated. Then, since the 13anomer dominates in aqueous D-glucose solutions no significant VROA would be expected and this in fact is the case. The same conclusion is not applicable for pyranose sugars with some axial hydroxyl groups. For example, both the t~- and 13- anomers of methyl-D-galactoside exhibit relatively strong VROA bands. The VROA band at -860 cm -1 is positive in the tx- and negative in the [3-anomer, correlating with the opposite absolute configurations at the anomeric center in these two anomers. The VROA spectrum of 5-thio-D-glucose in the anomeric region is significantly different from that of D-glucose with significant increase in vibrational Raman and ROA intensities. The VROA spectrum of D-glucosamine hydrochloride however is similar to that of D-glucose itself in the anomeric region. All these observations suggest that vibrational mode compositions and (or) polarizability changes are significantly dependent on the orientation of exocyclic C-O groups, and on the ring C-O (or C-S) bond.

]1.o,lo ~

~

./~o\.

4-

...!

,I

0

" D-glucose 800 800

e m -1

1000

1200

14O0

1600

Fig. 14-22. Raman (top) and ROA spectra of D-glucose in aqueous solution obtained in the backscattering geometry. Reproduced with permission from Ref. 88e. Copyright 1994 Elsevier Science Publishers.

372

Chapter 14

In the fingerprint-region, the VROA spectrum of D-glucose shows (Fig. 14-22) a positive band at 1048 cm -1, negative band at 1111 cm -1 and positive band at 1150 cm-1. This positive-negative-positive pattern is seen to be retained, with minor intensity and frequency variations, in the VROA spectra of D-glucose-l-d, D-glucose-6,6-d2, D-glucose-O-d5, D-allose, 5thio-D-glucose and D-glucosamine hydrochloride. The same pattern is also seen in the VROA spectrum of D-xylose, and methyl-~-D-glucoside, but not in that of methyl a-D-glucoside. It is possible that the observed VROA here is generated from the delocalized coupling of C-C and C-O stretching modes (both exo- and endo-cyclic) with some amount of coupling from the C-H bending modes. A VCD band at 1150 cm -1 was also assigned 64b,c to the delocalized C-O stretching vibrations. In the CH2 and C-O-H deformations region, the VROA spectrum of Dglucose shows a negative band in the 1200-1250 cm -1 region and a positive band in the 1250-1300 cm-1 region. This negative-positive couplet is also present in the VROA spectra of D-glucose-l-d, D-xylose, D-mannose, methyl a-D-mannoside and methyl 13-D-glucoside, at about the same position and appears shifted to lower frequencies by --50 cm-1 in that of Dglucose-6,6-d2. The positive portion of this couplet is missing in the VROA spectrum of methyl a-D-glucoside and D-lyxose, while the negative portion is missing in the VROA spectrum of D-glucose-O-d5, D-galactose, methyl-t~-D-galactoside and methyl-13-D-galactoside. These bands are likely to have contributions from the bending vibrations of the hydroxyl groups (with a small contribution from the CH2 group of exocyclic CH2OH and none from C1-H). The negative band might 88 be due to the gg rotamer, and the positive band to the gt rotamer of the exocyclic CH2OH group. An interesting correlation transpires from the VROA associated with the C1-H bending vibrational band in simple carbohydrates. This band can be identified from the VROA spectra of D-glucose and D-glucose-l-d to be at --1316 cm -1. An earlier analysis based on the FTIR spectra suggested that this band has coupling between C1-H and C5-H bending motions. The negative VROA associated with this band in D-glucose is also present in the VROA spectrum of methyl 13-D-glucoside, but absent in that of methyl a-Dglucoside. The reversal of the sign for an apparently corresponding band at -1350 cm -1 in D-mannose (67% a) and methyl a-D-mannoside suggests that the VROA sign associated with the C1-H bending mode reflects the configuration at the anomeric carbon. The presence of negative VROA at --1320 cm -1 in D-xylose, D-galactose, methyl [~-D-galactoside, D-allose and its absence in methyl-a-D-galactoside support this hypothesis. However, the corresponding band is not apparent in the VROA spectrum of 5-thio-Dglucose. Combining the VROA data obtained for several monosaccharides, as mentioned above, with that for D-fructose, L-sorbose, D-tagatose and L-

Applications and Molecular Structure

373

picose, new vibrational assignments were suggested88g. A VROA couplet at --420 cm -1, in these four sugars, was suggested to reflect the absolute configuration at the anomeric center. All disaccharides studied to date, containing only D-glucose residues, exhibit a bisignate couplet at --430 cm -1. For ~-linked disaccharides the positive portion of the couplet is on the lower wavenumber side (and negative on higher frequency side). For I~-linked disaccharides the opposite pattern is found. The width of the couplet in 1-1, 1-4, and 1-6 linked disaccharides is, respectively, --19, 34 and 39 cm -1. Therefore, based on the sign pattern and width associated with this couplet, it might be possible 88 to use VROA to determine the nature of glycosidic linkage in disaccharides. As mentioned earlier, methyl-13-D-glucoside does not have significant VROA in the anomeric region. In a similar manner, the 13-1inked disaccharides D-cellobiose and D-gentiobiose do not exhibit any significant VROA in the anomeric region; the ~-linked disaccharides D-maltose and Disomaltose exhibit VROA in this region, just as the ~-anomer of methyl Dglucoside. ~-, 13- and y-cyclodextrins show VROA spectral patterns in the 13001400 cm -1 region that are reminiscent of the spectrum of glucose itself. However, a sharp positive-negative couplet centered at --920 cm -1 is present in the VROA spectra of all three cyclodextrins, with a very large magnitude. The corresponding feature in D-glucose is weak. The large magnitude observed for cyclodextrins is probably due to concerted vibrations of the glycosidic link and C1-H bending vibrations. 14.3.5 Nucleic acids The VROA spectra of nucleosides, nucleotides and nucleic acids are now beginning to emerge 87d. The VROA signals for these systems are usually weak, thereby requiring long collection times. Nevertheless, the vibrations appearing in different regions [originating from the base, sugar and phosphate groups] appear to provide information on different structural segments. 14.3.6 Other molecules The VROA spectra obtained in the back scattering geometry with the dual circular polarization (DCPI method, see Chapter 9)) have been reported 89a for selected terpenes (Fig. 14-23), and compared with depolarized ROA spectra obtained in 90 ~ scattering with the incident circular polarization (ICP method, see Chapter 9). The role of VROA associated with the CH2 scissoring modes of the methylene group in deducing the absolute configuration has been pointed out. The presence or lack of VROA spectral correlations among different groups of terpenes was used to provide empirical relationships between the observed VROA

374

Chapter 14

features and molecular structure. & 2

~'

X

,<1

F

E

o -2

tO' 0

+ -10

G~[DV B

C

v

A

10

'T 0

x

,-1,,=1 +

5

i=,=e

BOO

1200

1600

WAVENUMBERS/cm-~ Fig. 14-23. Raman (bottom) and DCPI-ROA (middle) and normalized ROA (top) spectra of (-)-trans-pinane obtained in the backscattering geometry. Reproduced with permission from Ref. 89a. Copyright 1995 John Wiley & Sons.

VROA spectra of four different ephedrine molecules 89b were interpreted in terms of small changes in their molecular structures. A quantitaive comparison of normalized VROA intensities with the corresponding VCD intensities, in the --800-1350 cm -1 region indicated 89c that the former, on the average, are larger by a factor of two to three. 14.3. 7 Enantiomeric excess The analytical applications of VROA spectroscopy in determining the optical purity of chiral samples were also evaluated 90. By measuring the VROA spectra for known mixtures of (+)- and (-)-enantiomers of a-pinene, a calibration equation was developed for optical purity. Using this calibration, the content of (+)-a-pinene, in a mixture containing 83.6% (-) and 16.4% (+)-enantiomers, was determined to be 16.3%.

Applications and Molecular Structure

375

References 1 (a) S. Li, R. J. Van Zee and W. Weltner, J. Phys. Chem. 97 (1993) 11393; (b) H. W. Schrotter and H. W. Klockner in Raman Spectroscopy of Gases and Liquids, A. Weber, Ed., Springer Verlag, New York, 1979. 2 T. Noguchi, J. Honda, T. Nagamune, H. Sasabe, Y. Inoue and I. Endo, FEBS Lett. 358 (1995) 9. 3 R.A. Palmer, J. L. Chao, R. M. Dittmar, V. G. Gregoriou and S. E. Plunkett, Appl. Spectrosc. 47 (1993) 1297. 4 F. Siebert, W. Mantle and W. Kreutz, Biophys. Struct. Mech. 6 (1980) 139. 5 (a) P. Ormos, D. Braunstein, H. Frauenfelder, M. K. Hong, S. L. Lin, T. B. Sauke and R. D. Young, Proc. Natl. Acad. Sci. USA. 85 (1988) 8492; (b) D. Braunstein, A. Ansari, J. Berendzen, B. R. Cowen, K. D. Egeberg, H. Frauenfelder, M. K. Hong, P. Ormos, T. B. Sauke, R. Scholl, A. Schulte, S. G. Sligar, B. A. Springer, P. J. Steinbach and R. D. Young, Proc. Natl. Acad. Sci. USA., 85 (1988) 8497. 6 (a) T. Lian, B. Locke, T. Kitagawa, M. Nagai, and R. M. Hochstrasser, Biochemistry, 32 (1993), 5809; (b) B. Locke, T. Lian, and R. M. Hochstrasser, Chem. Phys. 158 (1991) 409; (c) M. Lim, T. A. Jackson and P. A. Anfinrud, Science 269 (1995) 962; (d) K. R. Rodgers and T. G. Spiro, Science, 265 (1994) 1697; (e) Y. Sakan, T. Ogura, T. Kitagawa, F. A. Fraunfelder, R. Mattera, and M. I. Saito, Biochemistry, 32 (1993) 5815. 7 (a) G. Herzberg, Infrared and Raman Spectra, Van Nostrand Reinholdt Co., New York, 1971; (b) S. Califano, Vibrational States, John Wiley & Sons, New York, 1976; (c) E. B. Wilson, J. C. Decius and P. C. Cross, Molecular Vibrations, Dover, New York (1980). 8 (a) B. A. Hess, Jr., L. J. Schaad, P. Carsky, and R. Zahradnik, Chem. Rev. 86 (1986) 709; (b) T. Ziegler, Chem. Rev. 91 (1991) 65. 9 (a) U.P. Fringeli and Hs. H. Gunthard in Membrane Spectroscopy, E. Grell (ed), Springer Verlag, New York, 1981; (b) S. Krimm and J. Bandekar, Adv. Protein Chem. 38 (1986) 181. 10 The absorption of water measured (in our laboratory) with a calibrated variable pathlength cell indicates that the absorbance (A) of the band at 1647 cm-1, at low pathlengths, follows the relation: A = 0.0969d + 0.11, where d is pathlength in I.tm. 11 A. Dong, P. Huang and W. S. Caughey, B iochem. 29 (1990) 3303. 12 D. M. Byler and H. Susi, B iopolymers, 25 91986) 469. 13 W. K. Surewicz and H. H. Mantsch, Biochim. Biophys. acta 952 (1988) 115. 14 W. K. Surewicz, J. J. Leddy and H. H. Mantsch, B iochem. 29 (1990) 8106. 15 C. L. Wilder, A. D. Friedrich, R. O. Potts, G. O. Daumy and M. L.

376

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383

Appendix 1 Summation convention for vector products In writing the expressions that involve summation over cartesian indices x,y and z, two different notations are often used. The summation sign is included explicitly in one notation and omitted in the other. To clarify these notations, let us consider two vectors a and b. The dot product between these vectors, a . b = axbx + ayby + azbz, can be written as follows: Explicit summation notation"

atxb ~ - a . b , tx=x,y,z Implicit summation notation: atxbtx = a . b .

(1) (2)

The repeating Greek index tx in Eq. (2) is used to imply the summation over x, y and z. Similarly the components of the vector product, a x b = c, can be written as follows: Explicit summation notation" Cx = aybz- byaz, Cy = azbx bzax, Cz = axby - bxay. Implicit summation notation:

ctx= etx~, a~ by,

(3a) (3b) (3c) (4)

where e~fly is the Levi-Civita symbol (also called alternating tensor) with values of 1, 0, or -1. Specifically, exyz = ezxy - eyzx = 1, exzy = eyxz = Ezyx = -1, and zero otherwise. It is apparent, from Eqs. (1)-(4), that the implicit summation notation (also referred to as the Einstein summation convention) provides the most convenient means of writing the vector products and is used throughout this book.

384 Appendix Harmonic operators

2 oscillator

integrals

with

dimensionless

p

and

q

<1)+ 11_1~)>o= i ,,(~+2 1"~ )' 1/2 = i<~+ 11ql~>

< ~ - l l p l ' o > - - i ( 2 ) 1/2 --i<~-llql~> <~+21p21~>-- [(~+1)4(~+2)] 1/2=_ <,0+21q2109> <~lp21~> = (~+1/2) - <~lq21~> <9-21p 210> -

-

t_[9(09-1)]41/2 -

- <1)-_._~2.~)>21ol

<~+31p31~> =- i [(x)+3)(x)82)(~)+l)] 1/2 =- i aJ+l 3/2 _ i<~+llq31~> <~+llp31~>- i 3(--~-)

<9-11p319> - - i 3(2 ) 312 =- i<~)- l lq31~> <~)_31p31~> = i [~(x)-l~(x)-2)] 1/2= i<,o_31q31~> <'o+2,pql~>-i [(~+1)4(~+2)] 1/2= <'o+21qpl~> <~lpql~> = - i/2 = - <~lqpl~> <~-21pqlv> =- i r ~(aJ-1)] 1/2 = <x)-21qpl~> L 4 -- i [(~+3)(a~82)(~+ 1)] 1/2

<~+llqpql~>- i [(a~+l)] 3/2 L 2 <~-llqpql'o>--i [2 ]3/2 <~)-31qpqhJ> - - i [aJ(ag-1)(ag-2)]1/2 L 8 <~)+61p3q3119> - - i [(~+1)(a~+2)(~+3)(a~+4)(~+5)(x)+6)]1/2 = + 64 <,o+4lp3q3l~> = 9 i [(~+1)(09+2)(~+3)(~+4)] 112__ <~+41q3p3109> 64 <~+21p3q31~> - 3 i ('02+3~-3) [ (~+2)(~+1)] 1/2 = + <~+21q3p3h)> L 64

Harmonic Oscillator Integrals

=_ i [18~(~+1)+318 .

=_<9.

385

I o n3 . 3 . ~ )I>

<~-21p3q31~> = - 3 i (~2-~-5) [~(~-1)] 112 - + <~-21q3p31~)> L 64 <~)-41p3q31'U> = 9 i [~('U-1)(642)(~-3) ] 1/2__ <,U_41q3p31,U> <~-61p3q3119> -- i [~(~-l)(a~-2)(~-3)(a~-4)(a~-5)] 1/2 = + <~)_61q3p31~> 64 <'U+61qpqnl'U> -- i [(~+6)(a~+5)(~+n)(~+3)(a~+2)(a)+1)] 112 = + <,U+61qapql,U> 64 <~+41qPq41l)>=i(4~+7) [(aJ+4)(~+3)(~+2)(~+ 1 / 2 6 41)] =I,4~+ { 4a9+7 13)<~+41qnpqll)> "x -' =i(5~2+3~_3)[(~+2)(aJ+l)]l/2-(5~2+3a9-3) 64 -(5~2+27~+33) <~lqpq41~>- - i (18~2+18a9+9) - - 8 Iv(v-l)] 1/2 _ (" 5~2+7v-1 ) <~-21___4l~>qpq = - i (5X)2+7D- 1) k 64 "5V2-17V+ ~ =_i(4,u_3) [~(~-1)(~-2)(~-3)] l/2_[(4,u_3)l(4,u_9)] 64 <~-61qpqnh)> =- i [ag(~-l)(~-2)(ag-3)(~-4)(ag-5)] 1/2_ + 64 = _[('o+4)(ag+3)(ag+2)(ag+l)] 1/2 <~+41q2p21~> 16 = <9+21p2q21~)> = + 4 [(~+2)(v+l)] 112 - _ <~)+21q2p219 > 16 <~lp2q21~> = + (2a92+2'u-1) = <~lq2p2hg> 4 = - 4 [~(~-1)] 112 = _ <,u_21q2p21,u> L 16 <~_4lp2q2ll)>- - [~(~_ 1)(162)(~_3)]1/2_ <~_4lq2p2l~) > <~+3lp2ql~> =- [(~+3)(~82)(a9+1)]1/2= <~+3lqp2l~)> <~+llp2ql~> = (~+3) L---ff-J <~_ 1ip2qlag>

-

= ,,-~,, <~+llqp 21~>

(~_2) [ 8 ] 1/2 = (~--~) ~)-2 <'0-llqp21'O> -

-

< l ) - ._~,_2.~>31anl

386

Appendix 2

- i [(~+3)(~82)(~+ 1)] 1/2 _ <~+31q2pl~ > <'O+llpq21a,)>-i

('o-l)

8 J

<~-llpq21~>- - i ( ~ + 2 ) ( 8 )1/2 = ,~-.(~+2'~<'o- 11q2pl~> <,o_31pq2l,o> = . i [~(~-1~(~-2)] 112= <x)_alq2pl~) > <,t)+41p41a; > = [(~+4)(~+3)('o+2)(x)+1)] 112 = + <~+41q41~> 16 <'0+21p41~> = - (4'0+6) [(~+2)(~+1)] 112 = _ <~+21q41x) > t. 16 <~Ip41~> = 6~2+6~+34 = + <'D..a4A)>IoI <~-21pnl't)> - - (4't)-2) [~)(~-1)] 112 _ _ <~_21q41,t) > t_ 16 <~)-41p41~> = [~(x)-1)(162)(~-3) ] 1/2=+ <,t)_41q41,O> = - [(v+4)(a~+3)(aJ+2)(v+l)] 1/2 - + <~+41pq2pl'o> 16 <~+21qp2ql~> - 0 = + <~)+21pq2pll)> <~lqp2ql~> - 2~2+2~+34 = + <~).r 2r.1,)>lnanl <~-21qp2ql~> - 0 = + <~-21pq2pl~> <~-41qp2ql~> =-[~(~-1)(162)(~-3) ] 1/2 = + <~-41pq2pl~>

387

Appendix 3 Some relations among dimensionless p and q operators q = ~ l / 2 Q , o~ = 4rr2v/h _ ~-1/2 b__~~ at ~Q p=-i~ l/2p P = h ~p=_ih 2n OQ pnq_ qpn - _ i n pn-1 pqn _ qnp - _ i n qn-1 p a q a _ qapa _ _ 2 i (pq + qp) p3q3 _ q3p3 = _ 3 i

[3~ (p2q2 + q2p2) + 1 ]

pq3p2 _ p2q3p = 3 i pq2p q3pq2 _ q2pq3 = i q4 qpqn _ qnpq = _ i (n- 1)qn pqpn _ pnqp = i (n-1)pn qp2 + p 2 q - 2 pqp qpqp - p q p q = i (pq + qp) qpq2p _ pq2pq = i (q2p + pq2) p2qpq _ qpqp2 = _ i 2(p2q + qp2)

388

Appendix 4 Polarized light For monochromatic radiation propagating in the z-direction, the associated electric vector oriented along the x-axis with amplitude Fxo is represented by the expression (1)

Fx = Fxo cos ~(t-z/c) .

It is a common practice to arbitrarily fix the value of z at zero and view the propagation in time. If the radiation under consideration has the electric vector along only one direction (say x as in Eq. (1)), then the radiation is said to be linearly polarized (or plane polarized). If we consider two such electric vectors, one along the x-axis and another along the y-axis, both with the same amplitude Fo then the resulting electric vector is given by the expression,

(2)

F - Fo [ficos(~t + Czx)+ ~ cos(OX + tXy)]/a/-2,

where ~ and ~ are unit vectors along the x and y axes; Ctx and (Zy are arbitrary phase factors associated with individual waves. When the phase factors are equal (Ctx = tx ), the value of phase factor is not important and the resulting F is still linearly polarized, but is oriented at 45 ~ to the x and y axes. In other words, a linearly polarized electric vector can always be resolved into two linearly polarized electric vectors with identical phase. If the amplitudes of x and y components are not equivalent, and are given as Fxo and Fyo, then the angle of resultant vector from the x-axis is given by tx = tan-l'(Fyo/Fxo). On the contrary, a nonzero phase difference in the propagation-of the two electric vectors leads to new polarizations. Defining the phase difference as Cry- (ix = 5, Eq. (2) can be written as F = Fo [fi cos(cot + czx) + "~cos(cot + czx + 8)] / a / 2 , = Fo [ficoscot +~cos(cot + 5 ) ] / a / 2 , where we removed the constant OCx term. 5 = -rt/2 the resulting vector becomes, F + = Fo [ficosoX + ~esino~t] / a/-2,

(3) For a phase difference of

(4)

which is said to be circularly polarized. The handedness of the circular motion can be determined by looking into the z-axis (facing the incoming

Polarized Light

389

radiation) and tracing the orientation of resulting electric vector as a function time (see Fig. 1). In the right handed coordinate system of x, y and z axes, the resulting electric vector is along the positive x-axis at cot = 0, along the positive y-axis at cot = x/2, along the negative x-axis at c o t - x and so on, tracing the counterclockwise direction (left handed helical motion). Therefore Eq. (4) is said to represent a left circularly polarized electric vector. For a phase difference of 5 = x, the resulting electric vector is given as,

5=n' :120 ~5= ~

,.0

0

Q 0

~

right circular

@(3@@@ (D O O (3 0 @@0 0 @ I

0

I

re/2

I

~

I

3~/2

I

linear left circular linear v

2~

COt Fig. 1. Time dependent orientation of resultant electric vector formed from Fx cos tot + Fy cos (tot + ~5),as viewed facing the incoming light (from positive end of z-axis).

F = Fo [fi cos tot - r cos rot ] / a/~ .

(5)

This is again lineraly polarized (between +x and-y axes), but rotated 90 ~ from the situation when 5 - 0. When the phase difference is +x/2, the resulting vector, F" - Fo [ficosoat - r

rot] / a / 2 ,

(6)

has circular motion but with the electric vector rotating in a clockwise direction (or right handed helical motion) in time. Therefore, Eq. (6) is said to represent right circularly polarized electric vector. Note that the sum of left and right circularly polarized electric vectors [see Eqs. (4) and (6)] results in a linearly polarized vector, so a linearly polarized electric

390

Appendix 4

vector can always be resolved into coherent left and right circularly polarized components. For representing a general case, the amplitudes of the x- and ypolarized electric vectors, Fxo and Fyo, are explicitly written down (in place of Fo) with phase difference t~y - O~x - ~5. Then using the relations Fx/Fxo=COS(O~t+O~x) and Fy/Fyo=COS(O~t+t~y) and forming the sum 1, [(Fy/Fyo) sin O~x- (Fx/Fxo) sin O~y]2 + [(Fy/Fyo) cos O~x- (Fx/Fxo) cos O~y]2, one finds that (F x /Fxo) 2 + (Fy /Fyo) 2 - 2(FxFy /FxoFyo) cos~5 - sin 2 5 .

(7)

This is an equation for an ellipse, whose major axis is at an angle from the x-axis. Knowing that the general polarization state of radiation can be represented by an ellipse, we would like to know how to find the intensity of radiation with a certain polarization (for example radiation that is emitted by an oscillating dipole, as in Raman scattering). Alternately, when radiation propagating along the z-axis is specified by a polarization ellipse in terms of its orientation from the x-axis and its ellipticity (see later), we would like to know how to find the intensity of radiation with a certain polarization. This information is obtained from the Stokes parameters (vide infra) which can be conveniently defined using the complex electric vector notation. The left and right circularly polarized electric vectors were given earlier as F+= Fo (fi cos cot + ~ sin tot), where the + sign is for left, and the - sign for right, circularly polarized electric vector. The cos cot and sin oat terms can be replaced by (e it~ + e-it~ and (e i~ - e-i~ With this substitution, and a minor rearrangment of terms, one obtains F+ = Fo e-it~ + F*o eit~ where, Fo = (fi + i~)/2, F*o = (fi ~" i-~)/2 and -- above the symbols represents complex representation. The same representation can also be used for linearly polarized light, but in that case Fo will be either Foil/2 or Fore/2, and Fo = F o Alternately, the electric vector can be also represented as F - Foe -it~ In this representation one takes only the real part of the expression which is written as Re[Fo e-it~ and Fo becomes fiFo or ~Fo for linearly polarized light, (fi + ~)Fo for left and right circularly polarized components. If an electric vector experiences a phase lag ~5 then the exponential argument is replaced by -i(o~t + ~5), which is equivalent to multiplying the original electric vector by e -i8. For ~5- + ~ / 2 , this

Polarized Light

391

multiplicative factor becomes T- i. This is evident in the definition of left or right circularly polarized radiation composed of two orthogonal electric vectors with a phase difference of ~- ~/2; that is, ~ = Fo(fi + i9) e -i~ Now let us consider the linearly polarized complex electric vector components with a general phase difference 8. Fx = Fxo e -i~

(8)

Fy = Fyo

(9)

e -i(~

Using the electric vectors represented by Eqs. (8) and (9), one obtains, ~

-*

~

*

2

2

-

-*

-

*

2

2

FxF x + F y F y - Fxo + F y o - S O ~ I x +Iy , FxF x - FyFy - Fxo - Fy o - S 1 ~: I x - I y , ~

--,

~

(10) (11)

,

- F x F y - FyF x - -2FxoFy o cos5 - S 2 ~: I(~; / 4 ) - I ( - g / 4) ,

(12)

- i ( F x F ; - F y F x ) - 2FxoFyo sinai - $3 oc I- - I + .

(13)

The last expression for $3 can also be written as $3 - Im[FxFy where "Im" means "the imaginary part of". The square of the electric field amplitude can be related to the light intensity, therefore ~ o and F 2yo are proportional to the light intensities Ix and I v respectively. Ix represents the intensity of light with electric vector polarized along the x-axis, Iy represents the corresponding quantity for electric vector polarized along the y-axis. Then the sum ~ o + F2 corresponds to the total intensity yo (Ix+Iy) and is referred to as the Stokes parameter SO. Recalling that two linearly polarized electric vectors with identical phase can be expressed as one resultant linearly polarized electric vector, and that a linearly polarized electric vector can also be written as a sum of left and right circularly polarized electric vectors, the parameter SO can also be written as (I-+I+). Here I- and I+ are the intensities of right and left circularly polarized radiation. The Stokes parameter S 1 represents the difference in intensities of radiation linearly polarized along the x- and y-axes; that is (Ix-Iy). This parameter can be measured by placing a linear polarization anal~czer in the light path with its transmitting axis along x-, and then

Appendix 4

392

along y-axes and taking the difference in the measured intensities. If this analyzer is oriented at +~/4 (between +x a n d - y axes) and -r~/4 (between +x and +y axes) from the x-axis, then the measured difference in the intensities is proportional to the Stokes parameter $2. The Stokes parameter $3 is proportional to (I- - I+). A detailed description of how $2 and $3 represent these difference intensities can be found in the book by Born and Wolf 2. The intensities of monochromatic radiation with a certain polarization can then be extracted from linear combinations of Stokes parameters. From the above discussion it is clear that SO + S 1 o~ Ix and SO- $1 o~ Iy. Note that SO could also have been written as I(x/4) + I(-x/4) or as I++ I-. Then SO + $2 o~ I(rt/4), SO- $2 o~ I(-r~/4), SO + $3 o~ I- and So - $3 o~ I +. Now we want to relate the Stokes parameters to the parameters of a polarization ellipse 3. Referring to Fig. 2, let x' and y' represent the major and minor axes of polarization ellipse and x, y and z represent space fixed (right handed) axes. The angle from x to x' is represented by 0. The ratio

/

x

p

/ #

"-...y,

Fig. 2. Orientation of polarization ellipse for monochromatic light propagating along the z-axis (which is out-of-the plane of paper towards the observer).

of the magnitudes of minor and major axes of the ellipse defines ellipticity, rl = tan-l(F y ,/F x ,) which is the angle of the resultant vector (formed from F x, and Fy,) measured from the major axis. The contribution of F' can be resolved into x and y components. Thus, Fx - Fo[cOsTI c o s 0 - i s i n r l sin0]e -it~ ,

(14)

Fy

(15)

-

Fo[-COSrl s i n 0 - isinTI sin0]e -it~ .

Polarized Light

393

Substituting these expressions for Fx and Fy into Eqs. (10)-(13), the Stokes parameters become 2 SO - F o ,

(16)

2 F o cos 2rl cos 2 0 ,

(17)

2 S 2 - F o cos 2TI sin 2 0 ,

(18)

2 Fo sin2r I .

(19)

S1 -

S3 -

From Eqs. (16) - (19) it is easy to verify that S02 - S 12 + $22 + $32. For right and left circularly polarized radiation rl = +rff4 and-r~/4 respectively. The discussion presented so far is for a monochromatic wave. In practice, one deals with quasi-monochromatic waves. For such cases one has to consider the amplitude and phase factor associated with the quasimonochromatic wave to be also time dependent. We will not go into the details, since the approximation of monochromatic wave is used in all of the considerations that we come across. It is however useful to note that for quasi-monochromatic wave the expressions for S 1, $2 and $3 [Eqs. (17)-(19)] will contain a factor P, called degree of polarization. Then P can be written as [S12 + S22 + 5 3 2 ] 1 / 2 / S 0 . The value of P varies between 1 (for polarized light) and 0 (for unpolarized light).

1D. D. Fitts, Vector Analysis in Chemistry, McGraw Hill, New York (1974). 2M. Born and W. Wolf, Principles of Optics, Pergamon Press, Oxford (1975). 3L. D. Barron and A. D. Buckingham, Ann. Rev. Phys. Chem. 26 (1975) 381.

394 Appendix 5 S u m m a t i o n c o n v e n t i o n for tensor p r o d u c t s

It is convenient to use the cartesian tensor summation notation for writing the anisotropy of a tensor. To explain this convention, let us consider a general tensor Tal3. The product TotaTBf~represents the sum of all possible products with indices o~ and 13 representing any of the possible x, y and z components. Thus, (1)

TaaTi313 - ~TcxcxTi313 - (Txx + Tyy + Tzz) 2 - 9T 2, o~,1~

where T is the mean (Txx + Tyy + Tzz)/3. Similarly the product Tal~Taf~ can be seen to represent the sum, 2

2

2

2

2

2

2

2

2

To~13Tcxl3 - Txx + Txy + Txz + Ty x + Tyy + Ty z + Tzx + Tzy + Tzz.

(2)

From the above two equations, the anisotropy can be represented as [~2 _ (1/2){(Txx _ Tyy )2 + (Txx - Tzz )2 + (Tyy _ Tzz)2 +3(T2y

+

T2z

+

2 Tyz

2 Tyx

+

+

T2x

+

T2y)} ,

(3)

- ( 1/2)(3Tcxl3Tcxl3 - TcxctTI313). Proceeding in this manner, one can verify that 45T 2 + 7[~2 - (3/~2)(7TocllTotl3 + TcxctT[~13) (4) 45T 2 + 13132 - (3/~2)(13Tcxl3Tcxl3-Tcto~Ti313) For products involving two different tensors, the anisotropy becomes, y2 _ (~)(3Ta~T~x~ _ TaciTly) I

p

l

l

= (1/~2) (Txx - Tyy)(Txx - Tyy) + (Txx - Tzz)(Txx - Tzz) I

I

I

l

+ (Tyy - Tzz)(Tyy - Tzz ) + 3(TxyTxy + TyxTy x +TxzTxz + TzxTzx + TyzT~z + TzyTzy)}.

(5)

Summation Convention for tensor Products

395

Using this relation, one can verify that 4 5 T T ' + 7T 2 -

(~)(7T(~I3T~I3 + T(x(~T~I3),

where T' is the mean of T' tensor.

(6)

396

Appendix 6 Vibrational Raman optical activities The electric dipole, magnetic dipole and electric quadrupole moments (~ta, ma and 0~13 respectively) induced by a static electric field Fa, magnetic field, Ba, (their time dependent counterparts are Fa, 13a) and electric field gradient F~B are given in Chapter 2 (see Eq. 2.3.18, 2.3.19, 2.3.22, and 2.3.28)'. For oscillating fields, the properties of light emitted by these induced moments can be analysed from the Stokes parameters SO, S l, $2 and $3. As described in Appendix 4 certain linear combinations of these four Stokes parameters provide the intensities of scattered light with different polarization properties. For example, in 90 ~ scattering with incident light propagating along the z-axis (see Fig. 2 of Appendix 4) the intensity of light scattered in the y-direction with right circular, left circular, linear polarization along the x direction and linear polarization along the z direction are determined respectively by the combination terms (So + $3), (So - $3), (So + $1) and (So- S1). The expressions for the Stokes parameters in right angle scattering geometry were given by Barron and Buckingham 1, and for 180 ~ (or backscttering) and 0 ~ (or forward scattering) geometries were given by Barron 2. The expressions for the non-zero parameters, in terms of the polarizability tensor invariants can be given as follows" S0(90) - (K/2)[(2 / 3)(45~ 2 + 13132)+ ((2 / 3)132 + 30~2)Pcos2rlcos0 +(2 / c)(2 / 3)(45~G' + 13"~2 - ~52)Psin2rl]

,

(1)

SI(90) = (K/2)[((2 / 3)~ 2 + 30~2)(1 + P cos 2rl COS20) + (2 / c)((2 / 3)'y2 + 3 0 ~ ' + 2~52)Psin2rl] ,

(2)

$3(90) = (K/2)(2 / c)[(2 / 3)(45~G' + 13'y2 - 5 2 ) +((2 / 3)'y2 + 30NG' + 282 )P cos 2rlcos 20] SO(180) - K[(2 / 3)(45~ 2 + 7~2) + (8 / c)(2T 2 + (2 / 3)~52)Psin2rl],

(3) (4)

Vibrational Raman Optical Activities

1,180 -

397

(5)

:0],

(6)

(7) S0(0) - K[(2 / 3)(45~ 2 + 7112)+(2 / c)(4 / 3)(45~G' + 72 - 82)Psin 2rl],

(8) S 1(0) - S 1 (180),

$2(0 ) - -$2 (180),

(9)

(10)

S3(0) - K[5(6~ 2 - (2 / 3)132)P sin 21"1 + (2 / c)(4 / 3)(45~G' + 7 2 - {32)]

(11)

In the above expressions, K is a constant; ~ and ~2 are the mean and anisotropy of polarizability derivative tensor (3cxa~0Q); ~UJ' and y2 are the corresponding quantities for interference products of (O~a~/OQ) and (OG~/OQ), while 82 is the anisotropy for interference product of (Ocxa~/OQ) and (3Aa~, //)Q). The angles 0 and 1] are defined in Fig. 2 of Appendix 4. The explicit form of all these quantities for vibrational Raman scattering (in the non-resonance approximation) are given by Eqs. (7.5.1), and (7.6.1)-(7.6.3), for the ath vibration. If the derivatives are replaced by the corresponding equilibrium tensor components, the resulting invariant parameters represent Rayleigh scattering. The master equations given above for the Stokes parameters can now be used to formulate the experimentally measurable intensity differences in different scattering geometries. A normalized quantity, referred to as circular intensity difference (CID), determining the magnitude of observed difference is defined by Eq. (7.6.6). In order to make this definition more transparent, an elaborate notation can be given as,

Appendix 6

398 13~p

{~P

A(~, 0~,0g,[3, [~') - (I~ - II3' ) ' (I~ + II3' ) 9

(12)

Here ~ is the scattering angle, I represents the Raman scattering cross section (see Eq. (7.5.2)); superscripts o~ and o~' on I represent the polarization states of incident light, and subscripts 13 and 13' on I represent the polarization state of scattered light. 1. Right angle scattering This geometry is the most commonly used for vibrational Raman and VROA measurements because of the ease with which the incident and scattered light components can be physically separated. Both incident circular polarization and scattered circular polarization forms of VROA have been measured using this geometry. 1.1. Incident circular polarization modulation The incident laser light is modulated between right circular and left circular polarizations and the corresponding difference in the scattered Raman intensities is measured. From the Stokes parameters associated with this geometry it can be seen that different measurements can be obtained by placing a linear polarization analyzer in the scattered beam at different angles, as described below. (a) Depolarized VROA. Using a linear polarization analyzer with its axis in the scattering plane (that is, along the z-axis), the intensities of scattered light can be determined as A(90, R, L, z, z) - (IR - IL) IR+I L [S0(90 ) - S 1(90)]rl=n/4 -[S0 (90) - S 1(90)]n=_n/4 [So (90) - S 1(90)]rl=~/4 + [S0 (90) - S 1(90)]rl=_~/4

8132

(13)

(b) Polarized VROA. When the linear polarization analyzer is rotated to orient its axis perpendicular to the scattering plane (that is, along the x-axis), the resulting normalized intensity difference in the scattered Raman light can be shown to be (IxR - I L) A(90,R,L,x, x) - ~xR+ ~x L

399

Vibrational Raman Optical Activities

[S0 (90)+ S 1(90)]rl=n/4 -[S0 (90)+ S 1(90)]rl=_g/4 [S0 (90) + S 1(90)]rl=n/4 + [S0 (90) + S 1(90)]rl=_n/4 _ (2 / c)(4 / 3)[45~G' + 7y 2 + ~2] (4/3)[45~ 2 + 7~ 2 ]

(14) "

(c) Unpolarized VROA. In this case the analyzer considered in (a) and (b) is removed. The expression for the normalized intensity difference can be obtained by taking the sum of numerators in Eqs. (13) and (14) and dividing by the corresponding sum of denominators. Then A(90,R.L,u,u)- IR - IL _ (2/c)(4/3)[45~G' + 13T2 - 8 2 ] IR + IL

(4/3)[45~ 2 + 13132]

(15) "

(d) Other forms of VROA. One may also conceive of looking at the difference in x- and z-polarized components of the scattered light, simultaneously with the difference for right and left circular polarizations of the incident light. Then,

(ix (ix -,z ) A(90,R,L,xz,xz)- ilR_ IR)+ (IL_ IL) _ (2 / c)(8 / 3)[4S~G' + y2 + 382]

(16)

(8 / 3)[45~2 +~2] The contributions from the 8 2 term to the observed VROA can be eliminated when the axis of linear polarization analyzer is oriented in the scattered beam at 35.3 ~ from the xy plane (or 54.7 ~ from the yz plane). In this case, the measurement is referred to as magic angle VROA. 1.2. Scattered circular polarization modulation 3. In these measurements the incident laser light remains in one linear polarization state and the scattered light is analyzed with right and left circular polarization analyzers. For incident linear polarization in the scattering plane (along y-axis), the Stokes parameters are evaluated with rl=0 and 0=rr/2 and the normalized intensity difference becomes

Appendix 6

400

A(90, y, y, R, L) - Iy - Iy = 2S3(90 ) _ (2 / c)8172 -(~52 / 3)] I~+IYL 2S0(90) 8~ 2

(17)

For incident laser polarization perpendicular to the scattering plane (along x-axis), substitution of 1"1=0 and 0=0 leads to the normalized intensity difference as A(90, x, x, R, L) - I ~ - I[ = 2S3(90 ) I~ + I[ 2S0(90) (2 / c)(4 / 3)[45~G' + 772 + fi2] (18) (4/3)[45~ 2 + 7132] One can also conceive of measuring the difference in intensities of right and left circularly polarized scattered components, simultaneously with the x x difference for x and y polarized incident light. That is, [(IR-I L) y y x L) x + (IY-I[)] can be obtained by subtracting the numerators (IR-IL)]/[(IR-I of Eqs. (17) and (18) and dividing by the corresponding difference of the denominators. 1.3. Dual circular polarization modulation 3 In these measurements the incident laser light is modulated between right and left circular polarizations (as in ICP-VROA), but the scattered Raman light is also analyzed for the right and left circular polarizations. It is conceivable to use either a right circular analyzer or a left circular analyzer to analyze the scattered light when the polarization state of incident light is right circular; and the same consideration would apply when the incident light is left circularly polarized. A consideration of such possibilities yields the following" (I R - I L)

(2/c)(4/3)[45~'

A(90, R, L, R, L) - (i R + IL ) =

= 0,

+ 137 2 - ~52]

(2/3)[45~ 2 + 13132]

(19)

(20)

Vibrational Raman Optical Activities

(IRR - I L) A(90, R, L, R, R) - (IRR+ IL ) =

(I R -ILR) A(90, R, R, R, L) - ,(iR + IR ) =

401

(2/C)(2/3)[45~G' + 13y 2 -82] (2/3)[45~ 2 + 13132]

(21)

(2 / c)(2/3)[45~G' + 13y 2 -82 ] (2/3)[45~ 2 + 131]2 ]

(22)

2. Back scattering Just as in right angle scattering, different polarization states can be considered for both incident and scattered light components. However, Stokes parameters for back scattering geometry indicate that, for extracting the optical activity tensor invariants, there is no effect of the linear polarization direction either in the incident light or in the scattered light. Thus it is sufficient to consider only the circular polarization states. 2.1. Incident circular polarization modulation The incident light is modulated between opposite circular polarization states and the scattered light is analyzed without analyzer. Then,

A(180,R,L,u,u)- (IuR- IL) - (2/c)(16 / 3)[3y 2 + 5 2 ] (IuR +IuL) - (4/3)[45~ 2 + 7~ 2 ]

(23)

The subscript u indicates that no analyzer was used in the scattered light. 2.2. Scattered circular polarization modulation 3 Here the incident light remains in an arbitrary linear polarization state and the scattered light is analyzed with circular analyzers. Then,

A(180, u, u, R, L) - (I~ -I~) _ (I~ +I~) -

(2 / c)(16 / 3)[3y 2 +82 ] (4/3)[45~ 2 + 7132]

(24)

2.3. Dual circular polarization modulation 3 As mentioned in the consideration of right angle scattering, one needs to consider the possibility of using either a right circular analyzer or a left circular analyzer when the incident light is in a given circular polarization state. Thus different possibilities arise as given below.

402

Appendix 6

(I R -I L) A(180, R, L, R, L) - (IRR+ iL ) (I~ - I L) A(180, R, L, L, R) - (i R + IL )

(2 / c)(32 / 3)[372 + ~2 (4/3)12~ 2

(8/3)[45~ 2 +[~2] '

(25)

(26)

(I~-I~) A(180, R, L, R, R) - (i R + IL ) (4/3)[5132- 45~2] + (2/c)(16/3)[3y2 + 821 (2/c)(16/3)[3y2 + 82]+ (4/3)[45~2 + 7132]

(27)

(... -

(4/3)[5132- 45~2] + (2/c)(16/3)[3y2 + 82 ] (28) The last two equations are obviously different from the rest, since optical activity tensor invariants and normal Raman tensor invariants appear in both numerator and denominator and therefore they are not useful. 3. Forward Scattering The situation here is quite similar to that in back scattering, with one major difference. That is, the optical activity here is dominated by isotropic terms. The optical activity term in back scattering, (6y2+282), is replaced by (45~G'+y2-82) in the forward scattering. The appropriate expressions in different situations are as follows. 3.1. Incident circular polarization modulation

+iu

(29)

Vibrational Raman Optical Activities

403

3.2. Scattered circular polarization modulation 3 (I~ -I~.) (2/c)(8/3)(45~G'+ y 2 -82) A(0'u'u'R'L) - (I~ + I~) = ( 4 ' 3)(45~2 + 7[32)

(30)

3.3. Dual circular polarization modulation 3 (I R - I L) (2/c)(16/3)[45~G' + y2 _ 82] A(0, R, L, R, L) - ~RR + IL ) = (8/3)[45~ 2 +132] (ILR - I L) A(0, R, L, L, R) - (ILR+ IL )

-

0 16"215 "

(31)

(32)

As in the case of back scattering, measurements such as A(0,R,L,R,R) and A(0,R,R,R,L) would not be useful for extracting VROA. 1L. D. Barron and A. D. Buckingham, Ann. Rev. Phys. Chem. 26 (1975) 381. 2L. D. Barron, in Molecular Spectroscopy, Vol.4, Chemical Society, London (1976). 3L. A. Nafie and D. Che, Adv. Chem. Phys., Vol. LXXXV, John Wiley &Sons, New York (1994).

404

Appendix 7 Classical expressions I for Aa~T and G ~ The electric quadrupole moment of a molecule with a collection of charged particles can be written in traceless form (see Eq. 2.1.6) as 2 0[37 - (1/2)E eqi (3Xil3Xi7 - X i 8 ~ , ) , i

(1)

where Xil3 is the positional coordinator of ith unit with charge eqi, and Xi is the distance of that unit from a chosen molecular origin. The 'units' here can represent atoms, bonds or occupied molecular orbitals. In the case of molecular orbitals, the position is defined by the orbital centroid. If the chosen coordinate origin is displaced, then the new coordinates become X i a - a~ and 0137.changes to a new. value. Thus 0137 is. origin dependent. Taking this ongm dependence into account, the expressmn for m~iwdual0" 13. can. beunitsWrittenasin terms of the local electric dipole moments of the

013Y - E0il3y + E[(3/2)Bil3Xi7 + (3/2)~tigXil3 - ~i~5Xi~5~il37] , i i

(2)

where ~ti~ = eqiXi~_ and 0i~7 is the intrinsic electric quadrupole moment of the ith unit. The electric quadrupole moment induced by an electric field Fa is given by Aoti~TFot [see Eq. (2.3.28)] while the induced electric dipole moments of individual units can be written as ~ti13= oq[3o~Fa. Substituting these relations into Eq. (2), one obtains the expression for Aal37 as Aoqh, - ~ Aial3y + E[( i

)oqf axi

+ (3/2)o~iyotXil3 - 0~i~5o~Xi~55137].

(3)

To obtain. . the expression for G&I3, one considers the expression for magneUc dipole moment which is also origin dependent. In terms of the electric dipole moments of the individual units, the expression for ma is given [see Eq. (7.4.38)] m a - ~ m i a + (1//2c)EEal3yOJ, il3Xi7 + Xil3~ti7), i i

(4)

where mia is the intrinsic contribution of ith unit. The induced magnetic

Classical expressions for A afly and G'aft t

405

~

moment by F~ is given as ma =-G~aF~/co (see Eq. 2.3.22) so the expression for G ~ becomes G~x~ - ~ G~al3 - (co / 2 c ) ~ s i

(5)

i

The atom, bond or orbital polarizability models (discussed in Chapter 7) for vibrational Raman optical activity utilize Eqs. (3) and (5) for Aal3y and G~tfl. 1j. R. Escribano and L. D. Barron, Mol. Phys. 65 (1988) 327.

This Page Intentionally Left Blank

407

INDEX AB 3 molecule 212, 216, 218 A2B 2 molecule -planar 215, 226, 212 -non-planar 211,212, 221,224, 336, 339 Absolute configuration 336 -of isoflurane 336 -of desflurane 336 Absorbance 235 Absorption coefficient 22, 235 -intergrated 22, 144, 235 N-Acetyl-N'-methyl-L-alaninamide 343 Adenosine 353 Alanine 340, 363 (L-alanine) n 350 Alanylalanine 340, 362, 364 Alanylalanylalanine 342, 366 Alanylglycine 365 Albumin 347, 367-369 Allose 372 Amide A vibration 325 Amide I vibration 325 Amide r vibration 328 Amide II vibration 325 Amide III vibration 325 Amino acids 340, 363 Anesthetics 332 Anharmonicity 135 -electrical 45, 74 -mechanical 45 Anharmonicity constant 35, 42, 71,135, 136 Anti-stokes frequency 184 Associative law 212 Atomic axial tensor 163, 168, 173, 175 Atomic charge concepts 155 Atomic charge-charge flux model 156 Atomic polar tensor 146, 155, 158 Atomic Raman tensors 194 Azide group 359 Bacteriorhodopsin 329 Basis functions 122 Benzene group 358 Benzoin 360 Bessel functions 240, 315, 317 Birefringence 241,303 Boltzman distribution 144 Bond charge concept 157 Bond current 176 Bond moment concept 157

Born-Oppenheimer approximation 29 Calibration curves -for circular dichroism 244, 270, 304 -for linear dichroism 246, 304 Carbonmonoxy hemoglobin 327 Carbonmonoxy myoglobin 323,327, 359 Cartesian force constants 82, 85, 93 Cellobiose 373 Center of mass 27 Character representation 214 Character table 214 CHFC1Br 63, 130, 131,360-362 Chymotrypsin 351,327, 369, 370 Circular intensity differential 199 Circular dichroism 22, 235 Combination frequency 135, 136 Combination transition 64, 135 Compliance constants 96, 104 Concanavalin A 327, 347, 351,369 Configuration -calibration 239, 241,245,248, 266 -reflection 304 -sample 239,243,247,248, 270 -transmission 265, 270, 296, 313 Conjugate operations 216 Contact transformation 35, 37, 47, 67, 74 -first 67, 74, 76, 50, 70 -second 38, 40, 50, 71, 77 Coordinates -cartesian displacement 81 -dimensionless 30, 41, 51 -external 89, 224 -generating 223 -internal 86, 107 -mass weighted 30, 81 -normal coordinate 30, 107 -redundant 99 -symmetry 223 Correction method 97 Coupled Oscillator 338 Coupled perturbed Hartree Fock(CPHF) method 126 Cyclic groups 221 Cyclic pentapetide 351 Cyclic peptide 351 Cyclobutane-l,2-d2 360 Cyclodextrins 373 Cyclopropane- 1,2-d2 360 Cyclopropane- 1-13C- 1,2,3-d3 360

408 Cysteine 364 Cytochrome C 347 Cytosine 353 D2 02 362 Density matrix 122 Deoxyribonucleic acids 355 Depolarization ratios 187, 309 -for CHFC1Br 131 2,2 -Diamino- 1,1 -binaphthyl 359 trans- 1,2-Dideuteriocyclopropane 360 trans-2,3-Dideuteriothiirane 363 Difference transitions 64 Digital signal processing 252, 319 Dihydro dibenzo phenanthrene 359 Dipole interaction tensor 191 Dipole strength 22, 143 Disaccharides 373 Dispersive spectrometer 249 -for circular dichroism 251 Distributed gauge origin 169, 173 Double diagonalization 94 Double harmonic approximation 44, 58, 73 Dual circular polarization modulation 197, 255 Dynamic infared linear dichroism 258 !

t

Eckart-Sayvetz conditions 61, 81, 89 Electric dipole-electric dipole polarizability

11,183,196 Electric dipole-electric quadrupole polarizability 15, 196 Electric dipole-magnetic dipole polarizability 14, 196 Electric dipole moment 5, 155 -induced 18 -operator 43, 48, 49, 73, 74 -time dependent 17 -time variation 54 Electric dipole moment derivatives 147162 -atomic displacement method 147 -electric field gradient method 148 -energy derivative method 152 -nuclear electric shielding method 153 -quantum mechanical methods 146 -wavefunction derivative method 149 Electric field 4-7 -circularly polarized 16 Electric quadrupole moment 5, 15 Enantiomeric excess 359, 374 Energy -anharmonic oscillator 32-35 -derivatives 124-130

-dipolar interaction 337 -first derivative of 124 -first order correction 8, 64, 128 -harmonic oscillator 30-32 -kinetic 3, 91-93 -levels 61, 64, 136, 137 -local oscillator 110, 111 -of interaction 6 -potential 3, 29, 93 -second derivative of 126 -second order 34, 41, 64 -second order correction 9, 64, 67 -zero order 62 Energy distribution 96 Ephedra molecules 359, 174 Epidermal growth factor 328, 329, 347 Fermi resonance 136 Fibroblast growth factor 329, 347 First overtone 64 Fixed partial charge approximation 155 Fock matrix 123 Force constants -cubic 117 -experimental determination 131 -internal coordinate 133-134 -linear 117 -of CHFC1Br 130 -of diatomic molecules 321-322 -quadratic 117 -quartic 117 Force fields 133 Fructose 372 Fundamental transitions 63 Fundamental vibrations 221 Fundamentals 44 Galactose 372 Gauge invariant atomic orbital 169 Generalized inverse 103, 104 Gentobiose 373 Geometry optimization 103 Glucosamine 371 Glucose 371-373 Glucose-l-dl 357,372 Glucose-2-dl 357 Glucose-6,6-d2 357,372 Glucose-O-d5 372 Glycine 341 Glycyl alanine 340, 365 Gramicidine-S 350 Guanosine 353

409

H202 87, 339, 363 Hamiltonian 27 -contact transformed 36 -harmonic oscillator 32, 61,107 -local oscillator 110 -perturbing 111 -zero order 32, 61, 71,107, 110 Harmonic frequency 135 Hartree-Fock method 124 -coupled perturbed 126 H-bonded systems 359 Heme proteins 324, 359 Hemoglobin 347, 352 Hermite polynomial 31, 34 Hermitian property 164 Hermiticity 164, 167 Identity operation 212 Improper rotation 211 Incident circular polarization modulation 196, 253 Indans 359 Inosine 353 Insuline 369 Integral -Coulomb 121 -exchange 121 Interference 263 interferometer -amplitude-division 237, 263 -polarization- division 237, 289 Interferograms -background 265, 296, 310, 314 -calibration 302 -dichroism 265, 298, 310, 316 -reflectance 304, 306 -residual 296, 313 -shapes of 268 -tramsmission 265,297, 310, 315 -VCD 272 Internal coordinate force constants 86, 93 Internal coordinates 86 -angle bending 87 -out-of-plane bending 87 -stretching 87, 109 -torsional 87 Inverse force constant matrix 96 Inverse kinetic energy matrix 92, 108 Inversion symmetry 211 Irreducible representation 214 -character of 217 Isoleucine 364 Isomaltose 373

Kinetic energy -coupling 114, 115 -operator 27 Lactalbumin 351,352, 368, 369 Lactoglobulin 347, 370 LCAO-MO approach 122, 147 Levi-Civita symbol 383 Linear dichroism 236, 243,299, 316 -of polypropylene 276, 277, 301 Local modes 107 Lysozyme 352, 368, 369 lyxose 372 Magnetic dipole moment -in bond charge model 173 -Taylor expansion 54 -twice contact transformed 56 Magnetic dipole moment derivatives 163175 -atomic charge concepts 173 -bond charge concepts 173 -localized molecular orbital method 165 -localized orbital-local origin method 173 -magnetic field perturbation method 167 -nuclear electric shielding tensor method 169 -quantum mechanical method 163 -vibronic coupling method 171 Magnetic dipole transition moments 164, 167 -anharmonic oscillator 55, 78 -double harmonic approximation 55, 78 -harmonic oscillator 53, 78 Magnetic field -circularly polarized 16, 17 Maltose 373 Mannose 372 Matrix representations 213 5-Methyl-6,8-dioxabicyclo[3.2.1 ]octane 359 Methyl-D-galactoside 371,372 1-O-Methyl -d3-13-D-glucopyranoside 357 1-O-Methyl B-D-glucopyranoside 357 Methyl ~-D-glucoside 371-373 Methyl 13-D-glucoside 371-373 Methyl ~-D-mannoside 372 2-Methyloxetane 359 Michelson interferometer 263 Molar rotation 19 Molecular orbital theory 118 Molecular orbitals 119 Molecular polarizability 190

410 Monoribonucleotides 353 Mutual exclusion principle 232 Newton's law 83 Nitrile hydratase 322 Normal vibration 85 Nuclear electric shielding 153 Nucleic acids 353 Oligopeptides 350 Optical rotation 18, 19 Orbital centroid 147 Orbital rocking contribution 165 Oscillator -anharmonic 32 -harmonic 30, 31 -Morse 41, 43, 51, 115 Overlap integral 122, 127 -matrix 126, 127 Overtones 44 -first 64 -frequencies of 132 -second 64 Pauli principle 119 Perturbation theory -time dependent 10 -time independent 7 Phase modulation 309 Phenylcarbinols 359 Picose 373 ct-Pinene 374 Point group 211 -C 1 213 -C 2 213, 218 -C2v 215 -C 3 219, 220 -C3v 216 -chiral 233 -C n 221 -cyclic 221 -order of 218,222 -properties 218 Polarizability -electric dipole 9, 11, 124 -electric dipole- magnetic dipole 14, 124 -frequency dependent 13 Polarizability derivtive tensors -anisotropy of 185 -mean of 185 Polarization division beam splitters 293 Polarization division interferometer -based on wire grid beamsplitter 289, 291

-based on Wollaston prism 289,290 -with flat mirrors 295 -with roof top mirrors 291 Polarization modulation -achromatic 311 -double 248, 312 -interferometer 294 -using PEM 237 -using QWR 246 Poly [l]-benzyl-L-asparate] 344 Poly [),-benzyl-L-glutamate] 343 Poly [L-glutamic acid] 349 Poly [L-lysine] 347, 348. 349, 351,367, 368, 369 Poly [L-Proline] 348 Poly [L-tyrosine] 346 Potential -harmonic 29, 31 -Morse 41, 42 Potential energy -coupling 114, 115 -distribution 96 -first derivatives of 82, 117 -fourth derivatives of 62, 117 -second derivatives of 82 -third derivatives of 62, 117 Pro-Leu-Gly-NH2 369 Product rule 134, 139 Projection method 99, 103 Proline 348 (L-proline) 3 349 Purple membrane 331,332 Quantum mechanical methods -for electric dipole moment derivatives 146 Quantum number -rotational 29 Raman activities 186, 195 -for CHFC1Br 131 Raman cross sections 183, 185-187 -atom dipole interaction model 191 -atomic displacement method 188, 189 -bond polarizability model 193 -electric dipole moment derivative method 190 -electric field gradient method 189 -energy derivative method 190 Raman optical activities 196 -ab initio bond polarizability concepts 203 -atom and bond polarizability concepts 202

411 -depolarized 196, 253,308 -magic angle 196, 253 -nuclear displacement method 200 -orbital polarizability model 201 -polarized 196, 253,309 Raman scattering -depolarized 187 -polarized 186 Raman spectra 183 Rate of racemization 360 Real time sampling method 278 Rectilinear coordinates 89 Reduced mass 27 Reducible representation 222 -character of 222 Redundant internal coordinates 99 Redundant relations 99 Reflection plane 211 Reflection symmetry 211 Reflection-absorption 277 Relative coordinates 27 Resonance Raman spectra -of bacteriorhodopsin 330 Ribonuclease 351,368, 369 Ribonucleic acids -copolymers 353 -double strand polymers 353 -polymers 353 -triple strand polymers 355 ROA intensities 200 ROA parameters for CHFC1Br 131,201 Rotational circular dichroism 207 Rotational constant 51 Rotational ROA 207 Rotational strength 23, 162, 166 -for CHFC1Br 131,169 Scattered circular polarization modulation 197, 255 Schrodinger equation 3, 4, 28 -electronic 28 -nuclear 28, 29 -time dependent 10 Secular equation 94, 95, 126, 133 Selection rules 231 -for vibrational absorption 231 -for vibrational circular dichroism 232 -for vibrational Raman 231 -for vibrational Raman optical activity 233 Self-consistent field 124 Serine 364 Sorbose 372 Specific rotation 19

Stokes frequency 186 Stokes parameters 187, 281 Sum rules for -absorption intensities 159 -atomic axial tensors 177 -atomic polar tensors 160 -atomic Raman tensors 194 -electric dipole-electric quadrupole polarizability derivative tensors 205 -electric dipole-magnetic dipole polarizability derivative tensors 205 -frequencies 138 -frequency weighted intensities 161 -VCD intensities 177, 179 Symmetry coordinates 101,223 Symmetry element 211 Symmetry operation 211 Symmetry species 214, 211 -of fundamental state 221,228 -of overtone states 228 Tagatose 372 Terpenes 373 5-Thio-D-glucose 371,372 Threonine 364 Time dependent coefficients 11, 12, 20 Time resolved measurements 256, 282, 311 -asynchronous perturbation method 286 -rapid sweep method 284 -synchronous perturbation method 284 Transformation between -cartesian displacement and external coordinates 90 -cartesian displacement and internal coordinates 89 -cartesian displacement and normal coordinates 85 -internal/external and cartesian displacement coordinates 90 -normal and internal coordinates 93 Transition metal complexes 357 Transition moments -anharmonic oscillator 44, 55, 74 -electric dipole 43 -fundamental 50 -harmonic oscillator 43, 53, 73, 78 -Morse oscillator 51 -overtones 50 -0-->1 transition 46 Transition polarizability 23, 58 -double harmonic approximation 58 2,2,2-Trifluoro-l-[9-anthryl] ethanol 360 Two dimensional spectroscopy 258, 260, 283

412 Two group model 362 X2Y 2 molecule 111 Xylose 372 Unit vector 158, 239, 337 Uridine 353

Valine 364 Variation theorem 123 Vibrational absorption -intensities 143-146 -ofCHFC1Br 131,153 -of H20 2 339 -spectra 143 Vibrational circular dichroism 161,333 -experimental intensities 162 -of CHFC1Br 131 -of H20 2 339 Vibrational equation 27, 29 Vibrational frequency 131,132, 134 -of diatomic molecules 321 Vibrational modes of -amide A 325 -amide 1 325 -amide r 328 -amide II 325 -amide III 325 Vibrational properties -for CHFC1Br 131 Vibrational states 227 -combination 230 -fundamental 228 -overtone 228 -symmetry of 228 Vibrational-rotational circular dichroism 163 Wavefunction -anharmonic 44, 45 -corrected 150 -determinantal 119 -first order correction 8 -harmonic 31, 32, 73, 107 -nuclear 29 -of combination states 230 -of fundamental states 228 -of overtone states 228 -perturbed 167, 172 -second order correction 9 -total 28 -vibrational 29, 107 -zero order 62 Wollaston prism 289

Zero frequency modes 85 Zero path difference (zpd) 264, 293,303, 310 Zero point energy 31, 136