Group Theory and Chemistry David M. Bishop OepartmentofChem~uy
University of Ottawa
Dover Publications, Inc. New York
TO IllY TEAOHERS
.J. A. W.,
Copyright © 1973 by David M. Bishop All rights reserved under Pan American and International Copyright Conventions. Published in Canada by General Publishing Company, Ltd., 30 Lesmill Road, Don Mills, 'Ibronto, Ontario. Published in the United Kingdom by Constable and Company, Ltd., 3 The Lanchesters, 162-164 Fulham Palace Road, London W6 9ER. This Dover edition, first published in 1993, is an unabridged and corrected republication of the work first published by The Clarendon Press, Oxford, in 1973. A new section of Answers 1b Selected Problems has been added to this edition. Manufactured in the United States of America Dover Publications, Inc., 31 East 2nd Street, Mineola, N. Y. 11501
Li1n'ary ojCongress Cataloging-in-Publication Data Bishop, David M. Group theory and chemistry I David M. Bishop. p. em. Originally published: Oxford: Clarendon Press, 1973. Includes bibliographical references and index. ISBN 0-486-67355-3 (pbk.) 1. Group theory. 2. Chemistry, Physical and theoreticaL L Title. QD455.3.G75B57 1993 541.2'2'015l22-dc20 92-39688 CIP
D. P. C.,
R. G. P.
Preface THIS book is written for chemistry students who wish to understand how group theory is applied to chemical problems. Usually the major obsta.cle a chemist finds with the subject of this book is the mathematics which is involved; consequently, I have tried to spell out all the relevant mathematics in some detail in appendices to each chapter. The book can then be read either as an introduction, dealing with general concepts (ignoring the appendices), or as a fairly comprehensive description of the subject (including the appendices). The reader is recommended to use the book first without the appendices and then, having grasped the broad outlines, read it a second time with the appendices. The subject materia.l is suitable for a senior undergraduate course or for a first-year graduate course a.nd could be covered in 15 lectures (without the appendices) or in 21 lectures (with the appendices). The best advice about reading a book of this nature was probably that given by George Chrystal in the preface to his book Algebra: Every mathematical book that is worth reading must be read "backwards and forwards", if I may use the expression. I would modify Lagrange's advice a little and say, "Go on, but often return to strengthen your faith". When you come on a hard or dreary paBBage, paBB it over, and come back to it after you have seen its importance or found the need for it further on.
Finally, a word of encouragement to those who are frightened by mathematics. The mathematics involved in actually a.pplying, as opposed to deriving, group theoretical formulae is quite trivial. It involves little more than adding and multiplying. It is in fact pollllible to make the applications, by filling in the necessary formulae in a routine way, without even understanding where the formulae have come from. I do not, however, advocate this practice.
London. November 1972
D.M.B.
Acknowledgements I would like to thank Professor Victor Gold for the hospitality he extended to me while I was on sabba.ticalleave at King's College London. where the major part of this book was written. I also owe a particular debt of gratitude to Dr. P. W. Atkins and Dr. B. A. Morrow who read the final typesoript in its entirety and to Professor A. D. Westland who read Chapter 12. I acknowledge with thanks permission to reproduce the following figures: Fig. 1-2.1 (Trustees of the British Museum (Natural History»). Fig. 1-2.3 (National Monuments Record. Crown Copyright). Fig. 1-2.5 (Victoria and Albert Museum, Crown Copyright), Fig. 12-7.2 and Fig. 12-7.3 (B. N. Figgis. Introduction w ligandfield8. Intersoience Publishers). I am similarly grateful to Dr. D. S. Sohonland for permission to reproduce. in Appendix I, the oharacter tables of his book MolecvlDr symmdry (Van Nostrand Co. Ltd.). Last. I would like to thank Mrs. M. R. Robertson for her immaculate typing.
Contents LIST OF SYMBOLS
XV
1. Symmetry I-I. 1-2. 1-3. 1-4.
Introduction Symmetry and everyday life Symmetry and chemistry Historical sketch
I I 4 5
2. Symmetry oplrltions 2-1. 2-2. 2-3. 2-4. 2-5. 2-6.
Introduction The algebra of operators Symmetry operations The algebra of By1Ilmetry operations Dipole momenta Optical activity Problems
7 8 10 15 19 20 23
3. Point groups 3-1. 3-2. 3-3. 3-4. 3-5. 3·6. 3- 7. A.3-I.
Introduction Definition of a group Some eXBlDples of groups Point groups Some properties of groups Classification of point groups Determination of molecul&r point groups The Rearrangement Theorem Problems
24
24 25 26 31 35 38 40 47
4. Matrices 4-1. 4-2. 4·3. 4·4. 4·5. A.4-1. A.4·2. A.4-3. A.4-4. A.4-5. A.4-6.
Introduction Definitions (matrices and determinants} Matrix algebra The matrix eigenvalue e'Iuation Simila.rity transformations Special matrices Method for detennining the inverse of a matrix Theorems for eigenvectors Theorems for similarity transformations The diagonalization of a matrix or how to find the eigenvalues and eigenvectors of a matrix Proof that det(AB) = det(A)det(B) Problems
48 48 51 55 57 58 61 63 65 67
69 70
5. Matrix raprl..ntBtions 5-1. 5-2. 5-3. 5-4.
Introduction Syrru:netry operations on a position vector Matrix representations for ~2h and ~8.. Matrix representations derived from base vectors
72 73 78 82
Contents
Contentll
lIii
Function spa.GI> Transformation operators O. A ....tisfactory set of transfonnation operators O. A caution An example of determining 0.., and D(R). for the ~.T point group using the d·orbital function space 5-10. Determinants as repre&entations 5-11. SUIDInlPo1'Y A.5-1. Proof that, if T = SR and D(R), DlS). and DlT) are found by consideration of R, S, T on a position vector, then D(T} = 5-1S. 5-6. 5-7. 5-8. 5-11.
A.5-2. A.5-3. A.5.4. A.5-5.
86 88 811 III 92 97 97
D(S)DlR}
98
Proof that the matrices in eqn 5-4.2 form a repreeentation Proof that the matrices derived from a position vector are the same as those derived from a single set of base vectors Proof that the operators O. are (a) linear. (b) homomorphic with R Proof that the matrices derived from O. form a representation of the point group Problems
99 99 100
10l 101
B. Equivelent and reducible repr8H.tetionll 6-1. 6-2. 6-3. 6-4. 6·5. A.6-I. A.6·2. A.6·3.
Introduction Equivalent representation. An example of equivalent representations Unitary representations Reducible representations Proof that the transformation operators O. will produce a unitary representation if orthonormal basis functions are used The Sclunidt ortbogonalization process Proof that any representation i. equivalent, through a similarity tra.neformation, to a unitary representation
103 103 106 108 110 113 113 115
7. Irreducibla rep_ntotio•• lind chlrllctlf tobl.. 7 -1. 7-2. 7-3. 7-4. 7-5. 7·6. 7.7. 7·8. 7 -9. A.7.1. A.7.2. A.7-S.
Introduction The Great Orthogonality Theorem Charaetel'S Number of times an irreducible representation occurs in a reducible one Criterion for irreducibility The reduction of a reducible representation Character tables and their construction Notation for irreducible representations An example of the de~rminationof the irreducible repreeentations to which certain function. belong The Great Orthogonality Theorem Prooftbat. ~ = (/
"
n:
Proof that the number of irreducible representations r equals the number of classes k Problems
117 118 120 123 124 125 128 131 134 138 US 145 all
8. Rapr8l8ntlltions and quantum mechanics 8·1. 8-2. 8-S. 8-4.
Introduction The invariance of He.miltonian operatol'8 under O. Direct product representations within a group Vanishing integrals
151 151 155 158
A.8-1.
Proof of eqns 8-2.12 to 8-2.15 Problems
lIiii 160 163
9. Molecular vibrations 11-1. 9-2. 9·3. 9·4. 9-5. 9-6. 9-7. 9-8. 9-11. 9-10. 11-11. 9-12. 9-13. A.9-I. A.9-2. A.9-3.
In1>roduction Normal coordinates The vibrational equation The T'" (or raN) representation The reduction of ro The cl.....ification of normal coordinates l"urther examples of normal coordinate classification Normal coordinates for linear molecules Classification of the vibrational level. Infra-red spectra Raman spectra The infra-red and Raman epectra ofCH. and CHaD Combination and overton., level. and Fermi resonance Proof of eqns 11-2.17 and 9-2.18 Prooftha.t Dft(R) = o-'D"(R)O Symmetry properties of polarizability functions Problems
164 164 169 172 175 178 182 184 184 J 86 1811 190 192 1113 194 196 195
10. Molecular orbital thlOry 10-1. 10-2. 10-3. 10-4. 10-5. 10-6. 10-7. A.IO-l. A.IO-2. A.IO-3.
Introduction The Hartree-Fock approximation The LCAO MO approxima1>ion The ".-electron approximation Ruckel molecular orbital method Huckel molecular orbital method for benzene Huckel molecular orbital method for the trivinylmetJ>y] radical A1>omic units An alternative notation for the LCAO MO method Proof that the matrix elements of an operator H which commutes with all O. of a group vanish between functions belonging to different irreducible representations Problems
197 1118 201 203 205 206 2]2 217 217 218 218
11. Hybrid orbitals H-I. 11-2. H-3. H-4. U-5. U-6.
Introduction Transformation properties of atomic orbitals Hybrid orbitals for a.bonding systems Hybrid orbitals for ..-bonding systems The mathematical form of hybrid orbitals Relationship between localized and non-localized molecular orbital theory Problems
219 221 225 229 234 241 241
, 2. Transition mlltal chemistry 12-1. ]2-2. 12-3. 12-4. 12·5. 12-6.
Introduction LCAO MOe for octahedral compounds LCAO MOe for tetrahedral compounds LCAO MOs for sandwich compounds Crystal field splitting Order of orbital energy levels in crystal field theory
243 244 251 252 257 260
xiv
Con'"
12.7. Correlation diagrams 12-8. Spectral properties
12-9. Magnetic properties 12.10. Ligand field theory
A.12-1. Spectl'OllCOpio I!tatee and tenn sYJIlbols for many-electron atoms or ions
Problems
Appendix I: Chefed., ubi.. BmLIOGBAFllY
ANSWERS TO SELECTED PROBLEMS INDEX
262
List of symbols
271
273 276 276 278 279 289 290 297
A;;
d
A*
A At A-1 X X; xii
E
o <§
g g, ~
r
r"
D"(R) Df;(R) n" x(R)
X"(R) P"(R)
symmetry element symmetry operation matrix representing R element in the ith row, jth column of D(R) transformation operator corresponding to R matrix, elementa are displayed between two pairs of vertioal lines element in the ith row, jth column of A cofactor of A;; in det(A) determinant of A tra.oeofA conjugate complex of A transpose of A adjoint of A inverse of A matrix ith column of X element in the ith row, jth column of X identity matrix null matrix point group order of a group (number of elementa in a group) number of elements in the ith olass of & group Kronecker delta (equals 0 if i :1= j, equals 1 if i = j) Cartesian coordinates of a point a representation of a point group the pth representation of a point group the matrix representing R in the matrix element in the ith row and jth column of D"(R) the dimension of r ll or the order of D"(R) the character of R in r the character of R in r" the projection operator ~ X"(R)*O.
r"
prieR)
the projection operator ~ Df;(R)*O.
a"
the number of times occurs in the number of classes in & group the matrix representing R in the regular representation r'" any operation of the ith class of a point group the jth operation of the mth class of & point group symbol linking the irreducible components of & reducible representation symbol linking two representations in a direct product representation
k
D·...(R) C,
R; ~
18>
r"
r
llVi
Lin at symllals "th energy level B. wB.vefunction B.ssociated with E y a set of ooordinates for a number of particles B. set of coordinates for B. number of nuclei a set of ooordinates for a number of electrons
(It)
FIG. 1·2.1. (a.) Cymothoe aloatiA; (b) primrose.
(b)
FIG. 1-2.1. (c) ice crystal.
FIG. 1-2.3. The octagonal ceiling in Ely Cathedral.
1. Symmetry 1- 1. Introduction IN everyday language we use the word symmeJ,ry in one of two ways and correspondingly the Oxford English Dictionary gives the following two definitions: (I) Mutual relation of the parts of something in respect of magnitude and position; relative measurement and arrangement of parts; proportion. (2) Due or just proportion; harmony of parts with each other and the whole; fitting, regular, or balanced arrangement and relation of parts or elements; the condition or quality of being well proportioned or well balanced. The first definition of the word has a more scientific ring to it than the second, the second being related to some extent to the rather more nebulous concept of beauty, for example John Bulwer wrote in 1650: 'True and native beauty consists in the just composure and symetrie of the parts of the body'. t It is nonetheless interesting that when we go deeper into the scientific meaning of symmetry we find that the underlying mathematics involved has itself a beauty and elegance which could well be described by the second definition. In this chapter we will first look at symmetry as it occurs in everyday life and then consider its specific role in chemistry. We will end the chapter by giving a historical sketch of the development of the mathematics which is used in making use of symmetry in chemistry. 1-2. Symmltry and 8Veryd.y life The ubiquitous role of symmetry in everyday life has been neatly summarized by James Newman in the following way:
Flo. 1-2.5. A.I1 example of Scottish bookbinding, circa 1750.
Symmetry establishes a ridiculous and wonderful couBinship between objects, phenomena, and theories outwardly unrelated: terrestial magnetism, women's veils, polarized light, natural selection, the theory of groups, invariants and transformations, the work habits of bees in the hive, the structure of space, vase designs, quantum physics, scarabs, flower petals, t This quotation comes from 8 book with the ex.traordina.ry title, Anthropometamor· ph08iB: Man TrYJ4I.8form'd; or the Artijicial OJoangeling. HilJtori<=ally pruemed, in the mad and cruel Gallantry, foolilJh B ....",,"'. ridiculowt Beauty, filthy jinenuee, and loalMome Lovelinue 01 moet NatWne,faehioning and altering their Bod;u from the Mould intended by Nature. With a Vindication 01 the Regular Beauty and Honeety 01 Nature. And an Append"" 01 the Pedigr.. of the EngliIJh Gallam.
.• ,
'··3
I
I I
(b)
,
J
Fro. 1-2.2. (a) Ivy leaf; (b) iris. Dotted lines show plane. of symmetry perpendioular to the page.
PM. world of m<J:Mmatiu, vol. 1, p. 669, Allen and Unwin, London (1960). M. Gardner, The ambide:nrmu ,"niver''", Allen Lane, Penguin Press, London (IVa7).
if
W. A, Mourl
a~J'!I~~ EaiF l Ii f= p.
"'"
#-t4!~JJJ .. J ' l j a
'f
@
;£.
'¥II
@
X-ray interference patterns, cell division in sea urchins, equilibrium positions in crystals, Romanesque cathedrals, snowflakes, music, the theory of relativity·t In nature we find countless examples of symmetry and in Fig. 1-2.1 we show some rather beautiful examples from the animal, vegetable, and mineral kingdoms. Externally, most animals have bilateral symmetry that is to say they contain a single plane of symmetry .. such a plane bisocts every straight line joining a pair of corresponding points. This is the same thing as saying that the plane divides the object into two parts which are mirror imagos of each other. In Fig. 1-2.2 it is seen that the ivy leaf and iris have, perpendicular to the plane of the page, one and three planes of symmetry respectively. Actually, the most frequent number of planes of symmetry in flowers is five. Anyone interested in the predominance of bilateral symmetry in the animal world, with its corollary of left and right handedne88, is recommended to read The ambid~r0'U8 universe.t In the iris we also notice that there is a three-fold axis of symmetry, that is, if we rotate the flower by 2TT/3 radians about the axis perpendicular to the page and running down the centre of the flower, then we cannot tell that it has been moved. Similarly, the ice crystal in Fig. 1-2.1 has a six-fold axis of symmetry: a 2TT/6 rotation leaves it apparently unmoved. Because of its basic aesthetic appeal (regularity, pleasing proportions, periodicity, harmonious arrangement) symmetry has, since time immemorial, been used in art. Probably the first example a child experiences of the beauty of symmetry is in playing with a kaleidoscope. More erudite examples occur in: poetry, for example the abccba rhyming sequence in many poems; architecture, for example the octagonal ceiling in Ely Cathedral (see Fig. 1-2.3); music, perhaps the most astute use of symmetry in art is a two part piece of music which is sometimes
14 J:l1
®
@
t
Table Music for Two
Moderatel)' rasl-wilhspirit (Ke)' of G)
I
t
3
SymlHtry
SymmBtry
2
-
E~
tWLJ I
,~
-".~
-
i I fie
f
~
::I-l~
-
..
~I -
@
II ,~
J!j£?i
@
"
Mml~! J ~
..
~
::I
J.
III: ~~
~ fi~riJJ JIJ~ I'~ ~J3'_J3~JJ J @
"
-
------..... J~ Pi _ _
_
- -
~
J iI
--.....
P!]) 'lilt; ; ,~
.~......-:;..-.o
@
f
01 J 'J! Ju l "lirlQn J I J h j
"
@
p
@
!' - - ~ ~ w L7fDm:J1 $:.ogJI~J'1 i~ @ _.... @ ~!'~B-' ~ I ~ II t U pi Dt¥Fi i
~::I~ rU$J' ~
~§~~r~rr ti t:l?7f?9~~~nn~.~.§i:~f o ._-----------------------------------
•?
OM.L JOJ :l!Snw :llqU.L FIG.
1-2.4. The
800re
of 'Mozart'.' Table Muoio for Two.
attributed to Mozart,t one part being simply the upside down version of the other; consequently a single copy of the score can be used by both players (see Fig. 1-2.4); and painting and design, see for example the specimen of Scottish bookbinding shown in Fig. 1-2.5. tIn KOohel-Einstem'. PM.motk ~ 0/ MtnDrl'. woru this pieoe is m the A"hong of doubtful and spurious pieoes (Anh. 284dd) and EinBtem state.: 'Th""" item. are aurely not by Mozart'.
4
Sym....ry
Symmetry
One thing we notice is that all of these examples involve either a plane, an axis or a centre of symmetry, which in turn define a plane, a line or a point about which the object is symmetric. 1-3. Symmllly end chemistry The involvement of symmetry in chemistry has a long history; in 540 B.O. the Society of Pythagoras held that earth had been produced from the regular hexahedron or cube, fire from the regular tetrahedron, air from the regular octahedron, water from the regular icosahedron, and the heavenly sphere from the regular dodecahedron. Today, the chemist intuitively uses symmetry every time he recognizes which atoms in a molecule are equivalent, for example in pyrene it is easy b
b
to see that there are three Ilets of equivalent hydrogen atoms. The appreciation of the number of equivalent atoms in a molecule leadB to the possibility of determining the number of substituted molecules that can exist e.g. there are only three pOllBible monosubstituted pyrenes. Symmetry also plays an important part in the determination of the structure of molecules. Here, a great deal of the evidence comes from the measurement of crystal structures, infra-red spectra, ultra-violet spectra, dipole moments, and optical activities. All of these are properties which depend on molecular symmetry. In connection with the spectroscopic evidence, it is interesting to note that in the preface to his famous book on group theory, Wigner writes: I like to recall his [M. von Laue's] question as to which results derived in the present volume I considered most important. My answer was tha.t the explanation of Laporte's rule (the concept of parity) and the quantum theory of the vector addition model appeared to me most significant. Since that time, I have come to agree with his a.nswer that the recognition that a.lmost all rules of spectroscopy follow from the symmetry of the problem is the most rema.rkable result. Of course, the basis for our understanding of molecular structure (ra.ther than simply its determination) lies in quantum mechanics and
5
therefore any coIU!ideration of the role of symmetry in chemistry is basically a consideration of its role in quantum mechanics. The link between symmetry and quantum mechanics is provided by that part of mathematics known as group theory.
1-4. Hi"oriClllak..ch In spite of the title, most of the mathematics which occurs in this book is in fact only a small part of the subject known as group theory. We will, however, now briefly sketch the history of this theory. No one person was responsible for the group idea but the figure which loomll largellt ill that of the man who gave the concept its name: Evarillte Galoill (1811-32). Galois had a short if action-packed life, and he was probably the youngellt mathematician ever to make such significant discoveries. He Wlloll born in 1811 at Bourg-la-Reine just outside Paris, and by the age of sixteen he had read and understood the works of the great mathematicians of hill day. However, despite his genius for mathematics, he failed twice in the entrance examinations to the Ecole Polytechnique which Wlloll in tholle daYIl the Mecca for French mathematicianll. Finally, in 1830 he was accepted at the Ecole Normal, only to be expelled the Ilame year for a newspaper letter concerning the actions of the Ilchool'll director during the July Revolution. Galoill had always been a convinced republican and had a strong hatred for all forms of tyranny, so it is not surprilling to find that in 1831 he wall arrested for proposing a toast which was interpreted all a threat on the life of King Louis Philippe. He was at first acquitted but then, shortly afterwardB, he was arrellted again and sentenced to six months in jail for illegally wearing a uniform and carrying weapons. He died on May 31st 1832 when only 20 years old from woundll received from being shot in the intestines during a duel. The duel was fought, under a code of honour, over a 'coquette' but some historians believe it was inlltigated by an agent provocateur on the monarchist side. The night before the duel, Galois with forebodings of death wrote out for posterity notes concerning his most important discoveries, which at that time had not been published. His total work is lesB than sixty pages. The concept of a group had been introduced by Galois in hill work on the theory of equations and this W&ll followed up by Baron Augustin Louis Cauchy (1789-1857) who went on to originate the theory of permutation groups. Other early workers in group theory were: Arthur Cayley (1821-95) who defined the general abstract group as we now
I
Symm..ry
know itt and who at the same time developed the theory of matrices; Camille Marie Ennemond Jordan (1838-1922); Marius Sophus Lie (1842-99) and Ludwig Sylow (1832-1918). For the chemist, however, the most important part of group theory is representaticm theory. This theory and the idea of group characters were developed almost single-handedly at the turn of the century by the German algebraist George Ferdinand Frobenius (1849-1917). Through a decade nearly every volume of the Berliner Sitzungberichte contained one or other of his beautiful papers on this subject. One of the earliest applications of the theory of groups was in the study of crystal structure and with the later development of X-ray analysis this application was revised and elaborated.:j: This is, however, purely a matter of geometrical classification and though useful in cataloguing possible types of crystals, it has no profound physical significance. Of much more importance is the work of Hennann Weyl (1885--1955) and Eugene Paul Wigner (1902-) who in the late twenties of this century developed the relationship between group theory and quantum mechanics. It is interesting that Weyl had a deep conviction that the harmony of nature could be expressed in mathematically beautiful laws and an outstanding characteristic of his work was his ability to unite previously unrelated subjects. He created a general theory of matrix representation of continuous groups and discovered that many of the regularities of quantum mechanics could be best understood by means of group theory. Wigner's greatest contribution was the application of group theory to atomic and nuclear problems; in 1963 he shared the Nobel Prize for physics with J. H. D. Jensen and M. G. Mayer. Finally attention is drawn to a germinal paper on the application of group theory to problems concerning the nature of crystals which was published in 1929 by another Nobel Prize winner, the German physicist Hans Albrecht Bethe (1906-). We will conclude this chapter by noting that it is one of the most extraordinary things in science that something as simple and abstract as the theory of groups should be so useful in the practical and everyday problems of the chemist and it is perhaps worth quoting here the English mathematician and philosopher, A. N. Whitehead (18611947) who said 'It is no paradox that in our most theoretical moods we may be nearest to our most practical applications'. ~
t A. Cayley, Phil. mag., 4th Series, 7, 40 (1864). Artur SchoeD1lies, T"-rie cUr Kmtallnru"'u... Borntraeger, Berlin (1923).
2. Symmetry operations 2-1. Introduc:tion THE purpose of this book is to show how the consideration of molecular symmetry can cut short a lot of the work involved in the quantum mechanical treatment of molecules. Of course, all the problems we will be concemed with could be solved by brute force but the use of symmetry is both more expeditious and more elegant. For example, when we come to consider Huckel molecular orbital theory for the trivinylmethyl radical, we will find that if we take account of the molecule's symmetry, we can reducc the problem of solVing a 7 x 7 determinantal equation to the much easier one of solving one 3 x 3 and two 2 x 2 determinantal equations and this leads to having one cubic and two quadratic equations rather than one seventh-order equation to solve. Symmetry will also allow us immediately to obtain useful qualitative information about the properties of molecules from which their structure can be predicted; for example, we will be able to predict the differences in the infra-red and Raman spectra of methane and monodeuteromethane and thereby distinguish between them. However, to start with we must get a clear idea what it is we mean by the symmetry of a molecule. In the first place it means consideration of the arrangement of the atoms (or, more precisely, the nuclei) in their equilibrium positions. Now, when we look at different nuclear arrangements, it is obvious that we require a much more precise and scientific definition of symmetry than any of those given previously in Chapter I, for clearly there are many different kinds of symmetry, for example the symmetry of benzene is patently different from that of methane, yet both are in some sense symmetric. Only when we have put the concept of symmetry on a. sound basis, will we be able to classif~' molecules into various symmetry types (see the next chapter). The way in which we systematize our notion of symmetry is by introducing the concept of a symmetry operation, which is an action which moves the nuclear framework into a position indistinguishable from the original one. At first sight it would appear that there are very many such operations possible. We will see, however, that each falls into one of five clearly delineated types: identity, rotation, reflection, rotation-reflection, and inversion. Related to the symmetry operation will be the symmetry element. These two terms are not the same and the reader is warned not to
8
Symmetry Operetions
confuse them. The symmetry operation is an action, the symmetry element is a geometrical entity (a point. aline or a. plane) about which an action takes place. It is worth stressing here that one of the problems in the theory of symmetry is the confusion over the meaning of words, some of which have a general meaning in everyday life but a very precise one in the theory of this book. Further confusion &rises from the use of symbols which ha.ve one meaning in arithmetic and another in group theory. The reader is advised to think carefully what the words and symbols in this text rea.lly mean and not to jump to conclusions. With this in mind. we discuss initially in this chapter the algebra of operators. An opera.tor is the symbol for an operation (the words operator and operation are often used interchangeably. and though. semantically. they should not be, no great harm comes from doing so). On the surface this algebra appears to be the sa.me as the algebra of numbers but, in fact. it is not so. At the end of this chapter we will show that knowledge of symmetry can lead us directly to predict whether a molecule can have a dipole moment and whether it can exhibit optical activity. 2-2. The algebra of operators An operator is the symbol for an operation which produces one function from another. Just as a function f assigns to each x in some range a number f(X), so an operator 0 assigns to each function f in a certain class. a new function denoted by Of. (To distinguish operators from other algebraic symbols we will characterize them by bold face italic letters.) An operator, therefore, is a rule or a means of getting one function from another and at the outset it is important to realize their very great generality; they can. for example. be as simple as multiplication by 2 where in mathematical terms we would write:
0= 2 times and if 0 operates on the function f(x) = x' then the new function would be Of(x) = 2 timesf(x) = 2x·. Other examples are: the differential operator 0 = d/dx; if the original function is f(x) = x' +x then 0 produces the new function d Of(x) = dx(xl+x) = 2x+l;
the squaring operator 0 = ( )"; if the original function is f(x) then 0 produces the new function
Of(x) = (x'+x)' = x'+2x"+x";
=
x' +x
I
an operator 0 which changes the sign of x and y; if the original function isf(x. y) = exp(ax)sin(by) then 0 produces the new function
Of(x. y) = exp( -ax)sin( -by); and the logarithm operator 0 = In(); f(x) = exp(ax), then the new function will be
if the original function is
Ofix) = [nfexp(ax)] = ax.
Clearly, there are countless examples of operators. One special kind of operator is the linear operator. An operator 0 will be linear if 0(1, +f.) = Of, + Of., (2-2.1) where f, and f. are functions of one or more variables, and if
O(kf) = kOf
(2-2.2)
whenever k is a constant. From the examples given before, it is clear that d{dx is a linear operator since O (f,+ ! .)
=
d dx (f, +f.)
and
O(kf) =
df, df. dx + dx = Of, + Of.,
=
d~ (kf)
=
k
~~
=
1.: 0 1,
whereas 0 = log( ) is not since and
0(1, + f.) = loge!, +1.) ?,'dog f, + log f. = 0/, + Of.
O(kf)
=
log(kf) 7'= k logf = kOf.
Another special type of operator is the unitary operator, but we will leave discussion of this type until later (§ 5-7). The algebra of linear operators consists of a definition of (1) a sum law. (2) a product law, (3) an associative law. and (4) a distributive law. (I) The sum law. The sum of two linear operators 0, and O. is defined by the equation: e.g.
(2-2.3)
(d/dx+3lx2
=
dx 2 /dx+3x2 •
(2) The product law. The product of two linear operators 0, and O. is defined by the equation 0,0.1 =O,(0J), (2-2.4) that is O. first operates on the function f to produce a new function and then 0, operates on this new function to produce a final function. It should be noticed that the order of the operators in a. product is important (in this respect their algebra is different from the algebra of
10
Symmetry Operations
Symmetry Op• •tions
numbers), and that ° 1 °. is not necessarily the same operator as e.g. if 0 , = d/dx and 0. = x' times, then d
0.0.1 = -dx (x"f) = and O.Od
and symbolically we write:
c.~ x· (~)
0_° 1
df dx
The identical operation
x'~
=
° °.
(2-2.6)
which means that combining O 2 and 0, (sco eqn (2-2.4» first and then this product with 0 , is the same as combining 0 1 and 0. first and then this product with 0" e.g. if 0, = x· times, 0, = d/dx and 0, = d'(dx' then: O,(02 0 ,)f = x'(d'fdx')f
(0,0.>0./
~
d'!
d)d'f ( x· dx, dx.
=
d"f
x' dx'
° 1(° 2 +0,) = ° 1 °.+° 1 °, (0.+0,)° 1 = 0.0 , +°3°1'
This is the operation of doing nothing (leaving the molecule unchanged) and, at first sight, may seem somewhat gratuitous; its inclusion, however, is necessary for the group theory that comes later. The corresponding symmetry element is called the identity and it has the symbol E (from the German word Einheit meaning unity).
The rotation opera-tion This is the operation of rotating a molecule clockwise about an axis. If a rotation by 2-rr/n brings the nuclear framework into coincidence with itself, the molecule is said to have as a symmetry element an n-fold axis of symmetry (other terms are n-fold proper axis and n-fold rotation axis). Necessarily n is an integer. The symbol for this element is C. and for the operator Cn' Ifrotation by 27T/n produces coincidence then clearly so will rotation by k times 27T/n (where k is an integer), such an axis, which will coincide with Cn' is given the symbol C:. The corresponding operator can be interpreted in one of two ways: a rotation by k27T/n or the application of C. k times. It is apparent that C;: = E since a rotation by n27T/n = 27T is equivalent to doing nothing, and hence n must be an integer. It must also be true that for a molecule containing the symmetry element C n an anti-clockwise rotation by k27Tfn must also be a symmetry operation and this is denoted by C;/, from which we see that C~ = C;;-(R-". The axis having the largest n value is called the principal axis. In Fig. 2-3.1 we illustrate these definitions for a square-based pyramid by labelling the four corners of the base. This labelling is merely to enable us to see that an operation has taken place and it has no physical significance: the whole point of the symmetry operation is that the final orientation is indistinguishable from the Original one. Of the operations shown in Fig. 2-3.1, only three, excluding E, are distinct: C., C_, and C:. It is conventional when choosing the symbol for a rotational operation to do so in such a way that n is as small as possible, e.g. C. is used in preference to C;. Finally, it is apparent that quite often symmetry elements will coincide and in such cases we will link the symmetry elements e.g. the C., C., and C: axes in Fig. 2-3.1 will be written as C.-O.-O;.
C:
= x' dx"
and eqn (2-2.6) is satisfied. (4) The distributive law. This law is given by the equations: and
corresponding symmetry element (a point, a line, or a plane) with respect to whieh the operation is carried out. There are five different kinds of symmetry operation.
2xf+x'-
1 of' 0.0" (2-2.5) and we say that 0. and 0. do not commute. Incidentally, the reader must constantly be aware of thc fact that when one writcs a. product of two operators, 0.0. it docs not mean' 0 1 multiplied by 0. although on paper it might appear that way. (3) The associative law. This can be expressed by the equation:
and
11
(2-2.7)
2-3. Symmetry operations A symmetry operation is an operation which 'when applied to a molecule (by which we mean the nuclear framework) moves it in such a way that its final position is physically indistinguishable from its initial position. It should be pointed out that such an operation can have no effect on any physical property of the molecule. Also, in this text, we will establish the convention that the operation is applied to the molecule itself and not to some set of spatial axes. The symbol for such an operation is called a symmetry operator (for which bold-face italic type will be used). J.'or every symmetry operation there is a
IZ
Symmetry Oper.tiDIlS
Symmny Oper.t1DDS
c.-c.-q
~ ~ I I
\
b
I
•
C,
- - -- ... -- ... I,d
c ----
c
a
I,
, I
c;=c.
d
I
•
I I
b ----
c
-- ..!!';
'", d
"
~ I I
,,
C.'
a ----
b r
\b...... - ...
-~-
,
,.
~ ,, \
,
I
d
d
Ii
The reflection operation This is the operation of redection about a plane. If the reflection brings the nuclear framework into coincidence with itself, the molecule is B&id to have a plane of symmetry a.s a symmetry element. The symbol given this element is a (after the German word 8piegel, meaning mirror). If such a plane is perpendicular to the principal axis it is labelled a b (h = horizontal) and if it contains the principal axis a v (v = vertical), if the plane contains the principal axis and bisects the angle between two two-fold axes of symmetry which are perpendicular to the principal axis, it is labelled ad (d = diagonal or dihedral), this latter plane is just a special kind of avo We notice that reflecting a molecule twice in the same plane brings it back to its original position and we can write 0 1 = E. In Fig. 2-3.2 we illustrate these planes for an octahedron and a symmetric tripod. The rotation--reflection operation
d
C.'=E
13
---- -_...!!'!....... _- h (.
This is the operation of clockwise rotation by 2.,/11. about an axis followed by reflection in a plane perpendicular to that axis (or vice versa, the order is not important). If this brings the molecule into coincidence with itself, the molecule is said to have a n-fold alternating axis of symmetry (or improper axis, or rotation-reflection axis) a.s a symmetry element. It is the 'knight's move' of symmetry. It is symbolized by S .. and illustrated for a tetrahedral molecule in Fig. 2-3.3.t It is clear that if a molecule has a 0 .. axis and a plane of symmetry perpendicul&r to that axis, the 0 .. axis is also a 8 .. axis. It is ea.sily seen that the application of S .. twice is the same as the a.pplication of C" twice (the reflection part of S" is simply a.nnulled); this is written a.s
S:
•
=
c:.
In general, k applications of S" will give
b
S: = ObC: if k is odd and
®
S:
I \
I
_~E
(l
FIG. 2·3.1. Rotation.
C:
if k is even.
Consequently can only be interpreted a.s a rotation C: followed by a reflection in the horizontal plane if k is odd; the opposite is also true e.g. a rotation by 2.2.,/3 plus redection is written as and not a.s (which would simply be C:). Furthermore, simple arguments lead to
S:
I I
Ii ---- -
S: =
'","
S:
t In this Figure the tetrahedral atruoture is shown by the oube whioh oiroumaoribea it a.nd the tetrahedral cornera are the alternate oornera of the oube. We shall frequently display tetrahedra in this way.
14
Symmetry Operations
Symmetry Operations
15
....,-----_ _ c
... "
~----+---
Fr G. 2-3.3. Rotation-reflection.
The inverse operation
4 4-
_
__ -
./1
1) ....' ........... /
a
This is the operation of inverting all points in a body about some centre, i.e. if the centre is 0, then any point A is moved to A' on the line AO such that OA' = OA or put another way, if a set of Cartesian axes have their origin at 0, a point with coordinates (x, y, z) is moved to (-x, -y, -z). If this operation brings the nuclear framework into coincidence with itself, the molecule is said to have a centre of symmetry as a symmetry element and this is symbolized by i (no relation to V-I). In Fig. 2-3.4 we show the inversion operation for an octahedral framework.
"
fT-
......
~--
/.
/
"
b
a Fro. 2-3.4. Inversion.
-CT~'
FIG. 2-3.2. Refteotion.
the equations: S,
=
S: =
2-4. The algebra of symmetry operations
(J (Jh
if 11 is odd
E
if 11 is even.
and S::
=
It is apparent that S. and i are equivalent (see Fig. 2-3.5) and that the application of inversion twice is the same as doing nothing, this is written as j" = E. In Table 2-3.1 we summarize the various definitions and symbols which have been discussed.
Symmetry operations like operators can be combined together and when this is done they produce other symmetry operations, e.g. if P and Q are symbols for any two symmetry operations then PQ is the
1.
Sym....ry Oplrations
8ym.. dry Oporations
J-. •
%
%
(x,y,z)
,T -' '4
(-x.-y,z)
To,
c. (z)
-
<7(XlI).
~,'
E
C'jC'
I
x
"
~
(a)
C-l
•
(2-4.1)
," I
/,
r
J.
a~'
<7~
"
C.
-
a
I,
I,
("
b
Identity (E) A ,.·fold axio of rotation (e.)
"
r
"
TABLE 2-3.1 Symmetry operation8 and elements
.... )
•
C,
"
(2.4.2)
(
..
I
"
and say that R is also a symmetry operation. The order of the terms in a product is important since symmetry operations do Dot always commute e.g. for a symmetric tripod (see Fig. 2-4.1)
"
" E
r
PQ =R
"
IJ
h
operation of first applying Q &nd then applying P. Notice the con· vention that the first operation to be carried out is the ODe OD the right. Since both operations, P and Q. leave the molecule coincident with itself, so must their product &nd therefore the combination of two (or, by extension, more) symmetry operations is itself a symmetry operation. Mathematically, we may write
Refleotion in a plane of oymmetry con· taining the principal axio of oymmetry and bioecting the angle between two 2·fold axeo of oymmetry which are per. pendicular to the principal axis (04) Rotation by 2Tr/n about an axis followed by reflection in a plane perpendioular to that axis (S.) InveI'Bion in a centre of oymmetry (I)
-I
•
Y
i (-T, - y, -z)
FIG.2.3.11. The S. opere.tion.
No ohanl!" (E) Rotation by 2"'/0 about an axi. of .ym. metry (C.) Reflection in a plane of .ymmetry per. pendicular to the principal ax;' of oym. metry (Ob) Reflection in a plane of oymmetry con· taining the principal axi. of oymmotry
C
C.
I I
~,
I
y T
17
'"
"
.
<7,_
A plane of oymmetry perpendicular to the prinoipal ax;' of oymmetry (Ob)
A
(h)
C.
0':.
•
plane of oymmetry oontaining the prinoipal axis of oymmetry (Oy)
A plane of oymmetry containing the prinoipal ax;' of oymmetry and bioect. ing the angle between two 2·fold axeo of aymmetry which are perpendioular to the principal axis (04) The n·fold alternating axia (8.l
"
(.
r
b
"
b
"
/,
.
FIG. 2·4.1. (a) Commutation betweenC. andC;l; (b) non·commutation betweenC and , Oy.
The oentre of oymmetry (i)
r
1.
Symmetry Op.l'lItions
Synu..try O.....tlo.
and hence C. and fia' commute (C.c;' = c;lC.) but Cacr~ = a;
and
a~CI =
(combining T on the right of both sidee) but we oannot write
a;
PQT = TR
and hence C. and a~ do not commute (C.a; =F- a~C.). The planes a;, and are defined in Figs. 2-3.2 and 3-4.1. Symmetry operations clearly obey the associative law:
0':,
(1:
(PQ)R
=
P(QR)
=
PQR.
(2-4.3)
If two symmetry operations combine together to give the identical operation E, e.g. eqn (2-4.2), then they are said to be the inverse of each other and the inverse of an operation P is written as p-l (we have, in fact, already been unknowingly using this notation for rotational operations, cf: C. and C;-l), the general situation is written as: PQ = QP = E
1.
(2-4.4)
and we say P is the inverse symmetry operation to 0 (P = Q-1) and is the inverse symmetry operation to P (Q = P-l). A symmetry operation always commutes with its inverse. The inverse of the product of two symmetry operations PQ is Q-lP-1: (PQ)-l = Q-IP-l. (2-4.5)
o
or
TPQ = RT.
2-&. Dipol. mom."
One application of symmetry operations that we can make right away concerns dipole moments. The use of symmetry arguments can tell us whether a molecule has a dipole moment and in many cases along which line it lies. Since a symmetry operation leaves a molecule in a configuration physU:ally indistinguishable from the one before the operation, the direction of the dipole moment vector must also remain unchanged after a symmetry operation. Therefore if a molecule has a n-fold axis of rotation 0" the dipole moment must lie along this axis but if we have two or more non-coincident symmetry axes, the molecule cannot have a dipole moment because it cannot lie on two axes at the same time. Methane CH. has four non-coincident O. axes and therefore has no dipole moment (see Fig. 2-5.1). If there is a plane of symmetry 0", the dipole moment must lie in this plane and if there are several symmetry planes, the dipole moment must lie along their intersection. In ammonia NH. the dipole moment lies along the O.
This is seen to be true by noting that (PQ)(O-lP-l) = P(QQ-l)P-l = P(E)P-l = pp-l = E.
In carrying out the above steps we have made use of the fact that E, the 'do nothing' operation, can always be dropped from any combination. Although there is no process for dividing one operation by another, we can always combine one symmetry operation with the inverse of another symmetry operation and this is essentially equivalent to division, e.g. though we cannot change PQ = R to PQ/R = I, we can combine both sides with the symmetry operation R-l and obtain PQR-l = RR-l = E.
The reader should convince himself that the inverses of E, C", G, S", and i are respectively, E, C;t, a, S;:t, and i. Finally, a word of warning; because the order of operations in a. product is important (the one to the right always being carried out first), one must be careful when manipulating equations involving symmetry operations, for example, if PO = R then we can write TPQ = TR (combining T on the left of both sides) or PQT = RT
FrG. 2·5.1. O. ax... in methane.
20
Symlll8try Operlltl_
axis which is also the intersection of three symmetry planes (see Fig. 2-4.1). A molecule containing a centre of symmetry i cannot have a dipole moment, since inversion reverses the direction of any vector. Our conclusions about dipole moments are all within the context of the Born-Oppenheimer approximation and methane, for example, has in reality a small permanent moment whose magnitude is the order of 10-0 to 10'-' D. This moment is caused by centrifugal distortion effects.t 2-8. Optic.1 .ctivity The origins of the subject of optical activity go back to 1690 when Huygens discovered that light could be polarized by a doubly-refracting crystal of Iceland spar (calcite). Detailed descriptions of light polarization are available in many books and for our purposes it is only necessary to recall that in circular polarization, the electric vector associated with the light describes a right- or a left-handed helix and that if a right and a left circularly polarized ray of the same frequency are superimposed, the resultant electric field vector is a sine wave function along a single direction (plane) in space. Such light is said to be plane polarized. Many substances can rotate the plane of polarization of a ray of plane polarized light. These substances are said to be optically active. The first detailed analysis of this phenomenon was made by Biot, who found not only the rotation of the plane of polarization by various materials (rotatory polarization) but also the variation of the rotation with wavelength (rotatory dispersion). This work was followed up by Pasteur, Biot's student, who separated an optically inactive crystalline m&terial (sodium ammonium tartrate) into two species which were of different crystalline form and were separately optically active. These two species rotated the plane of polarized light equally but in opposite directions and Pasteur recognized that the only difference between them was that the crystal form of one was the mirror image of the other. We know to-day, in molecular terms, that the one necessary and sufficient condition for a substance to exhibit optical activity is that its molecular structure be such that it cannot be superimposed on its image obtained by reflection in a mirror. When this condition is satisfied the molecule exists in two forms, showing equal but opposite optical properties and the two forms are called enantiomers. Whether a molecule is or is not superimposable on its mirror ima.ge is a question of symmetry. A molecule which contains a n-fold alternating axis of symmetry (8,,) is always superimpoaa.ble on its mirror
t I. Ozier, Pn".. B .... LotI.• 'l7, 1829 (1971).
Symmetry O....tiDns
21
image. This is true because the operation 8 .. consists of two parts: a rotation C.. and a reflection CJ. Since a reflection creates the mirror image, the operation 8" is equivalent to rotating in space the mirror image. By definition, a molecule containing a 8" axis is brought into coincidence with itself by the operation 8" and hence its mirror image, after rotation, is superimposable. The reader is reminded that 8. = U and 8. = i, so that a molecule with either a plane or a centre of sym. metry is also optically inactive. However, the most general rule is: a molecule with a 8" axis is optically inactive. Conversely, it can be shown that a molecule without a 8 .. axis is, in principle, optically active. Planes and centres of symmetry are easily identified and molecules having these symmetry elements are readily classified as inactive; alternating axes of symmetry (with n > 2) can be harder to spot. The first example of a molecule with a 8 .. axis (n > 2) to be experimentally studied was 3,4,3',4'-tetramethyl-spirO-(I,I')_bipyrrolidinium ion in 1955 (see Fig. 2-6.1). This ion has a 8. axis and was, as expected, found to be inactive.
S.
H~' : "CH. 1
CR.
~ I I
•
I I
CH.
H
-----
' JC+-
,CH.
1;x~ ,
J.:
,
CH
H
H,
CH.~H CH.
H
- - - - lolirl'or"
"
CH.
H
FIG. 2-6.1. 3.4.3'.4'-tetramethy).opiro-(l,I').bipyrrolidinium ion (inactive).
CH.
s,...etry OlNlr8tiDlIS
22
! /CH = CH -
\ HC ;: C - C ;: C\ \
n
CH = CH - CHI - VOIH
C\= C =iC
"\
HI
\
\\
\
(8) ~------------
II
I
"
H
I,
I
--------------------------~
\('
{·/l' b (a)
------------Mirror The"" two forms are identical (to show this identity the model must
r b
(h)
'~'('
be moved so as to
interchange the left ami right. rings)_
I " (e) FIG.
H"'-----...v FIG. 2.8.2. (a) Mycomycin, .. naturally-ooourring optically-aotiva allena; (b) one form of IBAlti" acid (optically active).
In principle, the lack of a 8ft axis dicta.tes the existence of opticaJ. activity and two examples are given in Fig. 2-6.2, however, in practice, it may not always be powble to actually demonstrate the activity. There are two reasons for this. One is that the opticaJ. activity may be so small that it is virtually undetectable; this is the case for the two
..
..--
CH.CH
.... GH. eH"
I
~---~
I I
I I
I I I
(a)
A----i-~r('H.l.CH.
2·6.4. Lack of activity due to free rotation.
forms of butylethylhexylpropylmethane or for the two forms of the substituted biphenyl in Fig. 2-6.3. The lack of activity in these molecules can be ascribed to the fact that there are only small differences between the substituent groups. The second reason is that free rotation can prevent a distinction being ma.de. For example, the substituted biphenyl molecule in Fig. 2-6.4 has rotation about the central bond restricted by the nitro-groups but nevertheless rotation of both end groups as shown in conformation (a) gives rise to conformation (b) which is completely superimposable on (c) the mirror image of (a). In other words, though (a) and (c) are not superimposable, (a) can be converted to (b) = (c) simply by free rotation. The reader is advised that optical activity is alluded to again at the end of the next chapter in problem 3.7.
I
1
lCH.reH ••
I
CH. [CH,l.
23
r- --- ---- - --------- ------1
,..------------\ \
Symmltry Operations
PROBLEMS
Mirror
2.1. Give all the symmetry elements of H 20, NH. and CH•. For each molecule list the symmetry operations which eommute. 2.2. On the baBis of symmetry. which of the following molecules cannot have a dipole moment: CH.> CHaCl. CHaD•• lIaS. SF.?
2.3. Which of the following molecules cannot be optically active: CHFClBr, lIs0•• Co(en)f+, ci8-Co(enlafNH,.ll+. tnz.....CO(en)2(NH,,):+?
CU.
CD.
}Iirrnr FIG. 2·6_3. (a) Butylethylhexylpropylmethane; (b) .. oubotitllted bipb_yl molecule.
Paint Group.
3. Point groups 3-1. Introduetian
I N this chapter we start on the long path that takes us from the symmetry elements which a molecule possesses to the theorems which will reduce the labour in quantum mechanical calculations. These theorems form a part of the subject called group theory and the connecting link between group theory and the symmetry operations of a molecule is that the latter form what is known as a group. This link opens up the whole wealth of valuable information which is contained in group theory. At first sight, group theory appears so abstra.ct and so unrelated to physica.l reality that it seems amazing that it should be the powerful practiro.l tool which it is. Having defined a group and given some examples, we will consider the specific type of group which is of interest to us: the point group. We will then introduce the notation which must be mastered and which allows us to classify molecules according to the symmetry elements they possess and we will give a simple scheme for determining the point group to which a given molecule belongs. Once this classification of molecules has been made, we will no longer need to consider specific molecules but only bodies having certain symmetry propertiea, i.e. belonging to a particular point group. 3-Z. Definition al • group
A group is any set or collection of elements which together with some well-defined combining operation obey a certain aet of rules. The meaning of the word 'elements' is very general. The elements could, for example, be numbers, matrices, vectors, roots of an equation or symmetry operations. It is important to remember that the definition of a group requires specifying a combining opera.tion. This too is quite general. It could be, depending on the particular group, ordinary addition, ordinary multiplication, matrix multiplication, vector addition, or one operation followed by another. The rules which the elements of a group must obey are as follows. The 'product' or combination of any two elements of the group must produce an element which is also a member of the group, i.e. if P and Q are symbols for two members of the group and PQ = R, then R must be a member of the group. Notice that PQ does not mean P multiplied
Zi
by Q but rather P combined with Q according to thfl defined combining rule. The group must contain the identity element, which is given the symbol E, and is such that when combined with any element in the group R it leaves that element unchanged i.e. RE = ER = R. Notice that E commutes with all elements of the group. The associative law must hold for all elements of the group, i.e. P(QR) = (PQ)R, or, in words, the combining of P with the combination QR must be the same as the combining of the combination PQ with R. Every element R must have an inverse element R-1 which must also be a member of the group. The inverse is defined by the equation: RR-l
=
R-1R
= E.
Those four rules are put into compact form in Table 3-2.1. TA.BLE 3-2.1 Definition of a group 1. PQ = R
R in the group
= ER = R E i n the group P(QR) = (PQ}R For all elementa RR-l = R-1R = Ji1 B-1 in the group
2. RE 3. -i.
3-3. Sam.lll.mpl.. of graupe
The preceding definition of a group seems rather abstract and exceedingly general and in order to bring things down to earth, we give in this section some concrete examples. All positive and negative whole numbers together with zero form a group whose combining rule is algebraic addition. The first rule of a group is obeyed because the addition of any two whole numbers produces another whole number which is, by definition, also an element of the group. If we add zero to any whole number, the number is unc~anged and so for this group the identity element is zero, and this too 18 a member of the group. The associative rule clearly holds and the inverse of any number is simply the negative of that number and this is also an element of the group i.e. a+( -a) = (-a)+a = O. There are an infinite number of elements in this group and for this reason it is called an infinite group. Another infinite group consists of the vectors r = ai +bJ +ck, where i, J, and k are non-coplanar vectors and a, b, and c are positive or negative whole numbers or zero. The combining operation is vector
ZI
Point Groups
Point Groups
addition and the identity element is the null veotor: a = 0, b = 0, c =
o.
Another group consists of the elements 1. -1. i and -i (where i = V-I) and the combining operation of algebraio multiplication. Combination. or multiplication, of any two elements produces one of the four elements; 1 is the identity element; the assooiative rule holds; and the inverse of 1 is 1. of -1 is -1. ofi is -i. of -i is i. This is an example of a finite group. The roots ofthe equation x' = 1 are I, (iV3-1}/2, and -(iv3 + 1)/2 and if the combining rule is algebraio multiplioation. these three numbers form a group for whioh E = 1. The inverse of 1 is 1 and the other two elements are inverses of each other. All powers of two•... 2-.,2- 1 • 2°.2 1 .2 1 •••• form an infinite group if the combining rule is algebraio multiplication. The four matrices 100
0
o
1
0
000
1
o
1
0
o o
0
1
000
001
0
000
1
1
0
0
o
100
1
0
0
10'
000
1
0
0001'
o
1
o
1
000
0
1
0
00'
0
0
form a finite group if the combining rule is matrix multiplication. In this example, the first matrix is the identity element E. The multiplioation of any two matrices produces one of the four. The algebra of matrices is discussed in the next chapter. 3-4. Point groups
From our point of view. the most important type of group is the one which consists of all the symmetry operations (not the symmetry elements) pertaining to a molecular structure. For such a group the combining rule is one operation followed by another. Since the application of any symmetry opera.tion leaves a molecule physically unchanged and with the same orientation in space. its centre of mass must also remain fixed in space under all symmetry operations. From this it follows that all the axes and planes of symmetry of a molecule must intersect at at leut one common point. Such groups are called point groups. (For a crystal of infinite size we can have symmetry operations. e.g. translations, that leave no point fixed in space; these give rise to 8pau groups.) That the symmetry operations of a molecule obey the four rules for a group is easily verified. The combination of any two symmetry operations must produoe another symmetry operation (see eqn (2-4.1», the
Z7
identity element is the 'do nothing' operation E. the assooiative law holds (see eqn (2-4.3» and oorresponding to each symmetry operation there is an inverse operation which annuls its effect and whioh is also a symmetry operation and therefore a member of the group. To illustrate these concepts let us consider the symmetric tripod framework (this has the same symmetry as NH.). In Fig. 3-4.1 we show the following six symmetry operations: (1) (2) (3) (4)
do nothing (E), reflection in the a~ plane (CJ;), reflection in the a; plane (~). relleotion in the plane (CJ:). (5) clockwise rotation by 2fT/3 about the C. axis (C.). (6) clockwise rotation by 4fT/3 about the C. axis or anti-olockwise rotation by 21T/3 (C: C;-1).
a;
==
Notice that the symmetry elements with respect to which the operations are carried out. remain fixed in spaoe; that is. if we introduce a fixed set of laboratory axes (x. Y. z). then the operations can be defined with respect to these axes e.g. a~ is the yz plane. is the plane containing the z axis and 30° clockwise from the xz plane. etc. It is also important to understand that the labels (a. b, c) on the feet of the tripod have no physical significance ; they are only a convenient way of identifying which symmetry operation has been carried out. In the light of what comes later, we will always consider rotations in their clockwise sense e.g. we will interpret C;l as a olockwise rotation of 4fT/3. Furthermore. in all operations it will be the body which is moved not the set of laboratory axes. The reader is cautioned that some text books take the opposite convention and keep the body fixed while moving the laboratory axes; it all comes down to the same thing in the long run but intermediate steps will look different, especially with regard to the signs in the corresponding equations. For this reason. when consulting another book, aJ.ways check the convention being used. The six operations for the symmetric tripod form a partioular point group and the way in which they combine together is conveniently summa.rized. by what is known u a grO'Up table (Table 3-4.1). In this table the operation to be first carried out is given in the first row and the second operation to be carried out in the first column. the combination falls in the body of the table at the intersection of the appropriate row and column. e.g. CJ~CJ; = C•• that is the operation followed by the operation CJ~ is identical to the single operation C. (see Fig. 3-4.2).
a;
CJ;
28
Point amps
Point Group. z
TABLIl 3-4.1 Group table lor /J ."mmetric tripocjf
y
0;.
E (1 )
...
a
E c
b c
b
a
c
a.
C.
G':"'
c.
.....
0;.
G.. G.. G ..
cri G..
E
C.
C;,
C;l
E
C.
.........
C;l
cri G:,
E
G ..
.
• Col
G..
C.
C;l
•
Gil
a
a,_
......
E
t (2)
or; or;
E
C.
b
(3)
II
Gi' -
~
.
Cfy
G..
G..
•
.;
..~
•
C;l
....E.
E
C.
G'
•
c:
EBBentia.lly a group table shows how the first rule for a group is obeyed. Inspection of Ta.ble 3-4.1 shows that each row or column contains ea.oh element once a.nd once only. This is true for all group tables a.nd the proof. the Rearrangement Theorem. is given in Appendix A.3-1. An important concept concerning groups is isomcwphiam. Two groups. <§ with the elements A. B, ... a.nd '6' with the elements A', B'• ...• are said to exhibit isomorphiam if a one to one correspondence
a
A_A'. B_B'•... b
can be established between their elements suoh that
AB=O implies .A'B' = 0'
a, (4)
a.".
a
.
b
(.AB)' = A'B'.
r
"
b
Paraphrasing this definition, we may say that isomorphic groups have group tables of the same struoture or form although they may differ in respect of the notation a.nd nature of their elements a.nd differ in
C'j
(5)
and vice versa. or more briefly if
C,
(/
"
b
C,'
". 0
a
.
c
(6)
a
c~
1
,,; b
b
==C"Jo"l ..
a
a
c
..
C.
b
r
FlO. 1-4.1• .,;.; - C •• b FIG. 3-4.1. Symmetry operations for s symm.etric tripod.
C
31
Paint Oro....
Point 0
their combining rules. The following two groups of order 4 are isomorphic, the combining rule for each being stated in brackets: '§:
'§':
1,
i,
-i
-I,
(ordinary multiplication)
II: ~ 1/·11-: : 11·11-: -: I , I ~ -:(matrix I multiplication).
Indeed if the elements of each set are renamed E, A. B, C (in this order), their common group table is seen to be E
ABC
E
E
ABC
A
ABC
E
B
BeE
A
C
C
B
E
A
Likewise, if the elements E, A, B, C, D, F of some group combine according to Table 3-4.2, which has the same structure as Table 3-4.1,
3-&. SD". prD!*ti_ D' urDU,.
The number of elements in a group is called the order of the group and is usually given the symbol g e.g. for the point group for the symmetrio tripod g is 6 and for an infinite group g is 00. It is clear that the elements of point groups do not necesaarily commute, that is the order in which one combines two symmetry operations can be important (see, for example, Fig. 2-4.1 and Table 3-4.1 where for the symmetrio tripod Cacr~ " cr~Ca). A group for which all the elements do commute is called an Abelian group. One can, of course, combine more than two elements of a group; one simply uses the given combining rule more than once, e.g. PQR is obtained by combining Q with R to obtain some other element of the group and then combining P with this element. If P and Q are elements of a group. then so, by the definition of a group. is the inverse of Q. Q-t, and consequently Q-IPQ. If this latter combination is identical with the element R we can write R
E E A B
a D 11'
A
BaD
A
B
o
D
o
E J!' D
B J!'
F D E
D J!'
B
o
E A B
a8
a
A
B A
D B
a
Table 3-4.1 P J!'
o A B
A J!'
E
B
D
then this group is said to be isomorphic with the group of symmetry operations for the symmetric tripod. In isomorphism each element of one group is uniquely mirrored by an element of the other group, i.e. no two elements have the same image. The more general situation, called homomM'phiBm. still preserves the structure condition (AB)' = A'B' but includes cases in which two different elements of one group '§ have the same imsge in the other group '§'. Thus in lwmomorphi8m structure is retained but individuality may be destroyed. To mention a trivial case, any group '6 is homomorphic with the group '§' whose only element is the number 1; in fact if A -+ 1, B -+ 1. C -+ I, ... a relation of the form AB = C is carried over into 1 xl = 1, whioh is evidently true. In § 5-10 we will see that
31
for a group of square mstrices there is always a homomorphic correspondence between each matrix and its determinant.
TABLB 3-4.2
A gr01J,P table with the 8ame 8tructure
l11li'"
=
Q-IPQ
(3-1i.!)
and we say that R is the transform of P by Q or that P and Rare conjugate to each other. The criterion, therefore, for two elements P and R to be conjugate to each other is that eqn (3-5.1) holds for at least one element Q of the group. We can combine Q and Q-l on the left and right hand sides respectively of both sides of eqn (3-5.1) and produce
QRQ-l
=
QQ-IPQQ-l
=
EPE
=
P.
(3-5.2)
Either of eqns (3-5.1) and (3-5.2) acts as a definition of conjugation between two elements of a group. Furthermore, if P is conjugate to Q and Q is conjugate to R, then P is conjugate to R. This is demonstrated by the following equations:
therefore
P = X-IQX, Q = Y-IRY. P = X-IY-IRYX = (YX)-lR(YX),
(see eqn (2-4.5»
and as YX must be some element of the group say Z we have P
=
Z-lRZ.
(3-5.3)
32
Point GrHps
Pain Gro.p.
The elements of a group which are conjugate to each other are said to form a class and the number of elements in the ith class is given the symbol g,. For the symmetric tripod point group the elements fall into three cls.sses: E; a~, a;, lJ:; C a, c;-l with gl = 1, g. = 3 and g. = 2. For this and all groups E is conjugate with itself: X-lEX = E (all X) and henoe E is always in a claBl3 of its own; we will always consider E to be the:first cl&Bl3. The reader, with the aid of Table 3-4.1, can confirm for himself that if Q = a~ or or then with any of the six elements of the group X the trlLIlSform X-IQX is either a'.,. or a"v or a" 1 v and that if Q = C. or C; then with any element of the group X the transform X-IQX is either C a or c;-l. Any element will always be conjugate with itself as E is always in the group and P = E-IPE. IT P and Q are conjugate, then so are their inverses; this follows using eqn (2-4.5) from
=
(X-IQX)-l
=
(QX)-lX
=
=
(QX)-l(X-l)-l
X-IQ-IX.
(3-5.4)
Hence the inverses of the elements of a. c1&Bl3 belong to a cla.ss (it is sometimes the same one) and the order of a class of elements and the order of the cl&Bl3 of their inverses is the same. In an Abelian group, X-IPX = X-IXP = P, and each element is in a class of its own and the number of cl&8ses is the same as the order of the group g. The technique that has been given for dividing any group into classes can be repla.ced by a simpler one in the ca.se of point groups. This is valuable since the testing of eqn (3-5.1) for all possible combinations of Q and P of a given point group can be quite time consuming, e.g. an equilateral triangle belongs to a point group which has 12 symmetry operations and therefore for this point group we would have to work out Q-IPQ 144 times. That there should be a simpler way is indicated by the fact that in the example of the symmetric tripod point group given above, the c1aBses appear to be a 'natural' sub-division of the point group and to contain operations which are 'similar'. The following simple rules can be established for point groups: (1) The symmetry operations E, i, and a h are each in a olass by themselves. (2) The rotation operation and its inverse c;;~ will belong to the same cl&Bl3 (a different cl&Bl3 for each value of k) provided there is
C:
O.
either a plane of symmetry containing the axis or a axis at right angles to the 0: axis; if not, and c;;" are in classes by themselves. The same is true for the rotation-reflection operations and 8;". (3) Two reflection operations cr and a' will belong to the same c1a.BS provided there is a symmetry operation in the point group which moves all the points on the u' symmetry plane into corresponding positions on the u symmetry plane. A simila.r rule holds true for two rotational operations C: and C:' (or S: and S:') about different rotational axes, i.e. the two operations belong to the same cla.ss provided there is a symmetry operation in the point group whioh moves all the points on the 0:' (or S:') axis to oorresponding positions on the 0: (or S:) axis.
C:
S:
a; a;
p-l
0:
33
The proof of these three rules is based on their rela.tionship with eqn (3-5.1). The symmetry operations E, i, and 0h oommute with all the other symmetry operations of the point group of which they are a part and hence, if P = E, i, or a h and Q is any symmetry operation of the given point group, Q-IPQ = Q-IQP = P and P is in a class by itself (Rule (1». The proof of Rules (2) and (3) depends on the fact that the symmetry operation Q-l PQ must be of the same general type, rotation or reflection, as P (this is easily demonstrated). Now consider the rotations and C;;" for a point group which contains a reflection operation a where the plane of symmetry u contains the axis. The symmetry operation a-lC:lJ, or oC:lJ since 0-1 = a, must be a rotation about the axis since poinb on this axis are unmoved under the three component operations (cr, a-I) and the only question remaining is the magnitude of this rotation. In Fig. 3-5.1 it is shown what happens to the rectangle OABC under the symmetry operation cr-lC:cr if is a clockwise rotation by 8 degrees and u is the ::r:z plane; it is clear that OABC is rotated anti-clockwise by 8 degrees. Hence the rotation cr-1 C:cr must be C;;~ and and c;" must belong to the same c1a.BS. In Fig. 3-5.2 the operation C;-lC:C.( = C.C:C.) is carried out. This operation must a.Iso be a rotation about 0: since points on this axis end up by being unmoved. What happens to the recta.ngle OABC demonstrates that C;-l C:Ca = c;". These results, together with their extension to rotation-reflection operations form the basis of Rule (2). To prove Rule (3) :first consider the reflection operation Q-l cr Q where Q is a symmetry operation which moves aU the points on some symmetry plane u' to corresponding positions on the symmetry plane a. Taking the components of Q-l cr Q one by one: Q will move the
C:
0:
0:
C:,
C:
C:
Point Groups
34
Point Groups
c~
z·
z
(I (J
r
•
=xzplane
x
35
points initially in the (T' plane to the (T plane, .. will leave these points unchanged and 0- 1 will move the points back to the (T' plane. Hence, 0- 1 '10 leaves the points in the (T' plane unmoved and must therefore be a reflection in that plane, i.e. 0- 1 '10 = a' and .. and a' belong to the same class. Now consider the similar rule for rotations: if 0 moves the points on the axis to corresponding positions on the axis, then C~ will leave them unchanged and 0-1 will bring them back to their initial positions on the axis, and hence Q-1C~Q is a rotation
C:'
C:
C:'
n ~:-----x
FIG. 3-6.1. The eifect of .,-lC:-a on So rectangle whioh oontains the oonditions or Rule (2). c~
0: axis, under the z
-
z
z
-(J'-1!!!!0'
FIG. 3-6.3-. The effect of a-tC:a on &. rectangle whioh oonta.ins the conditions of Rule (3).
C:' axis. under the
C:'
z
B
c:
c
~=.,.....:;.;..----x
c
""'=....,=----x
B FIG. 3-5.2. The effect ofCI~=Ct on & rectangle whioh oontains the conditions of Rule (2).
a: axis, under the
about the axis; the magnitude of this rotation can be seen by considering what happens to points in any plane which contains the 0:' axis. If 0 is a"rotation, then 0-IC:O will be a clockwise rotation by k2'1T/n about the C:' axis, that is 0-lC:0 = C:' and and C:' belong to the same claBB. If 0 is a reflection, then 0-lC:0 will be an anti-clockwise rotation by k2'1T/n about the C:' axis and 0-lC:0 = c;.k' (see Fig. 3-5.3). but since for all the point groups which have this property, the symmetry operations c;.k' and fall into the same class, C:, C-;k' and C:' will all be in the same class and Rule (3) will still hold true. A similar proof exists for rotation-reflection operations.
C:
C:'
3-6. Classification of point groups We are now in a position to describe and classify the various point groups that exist. The notation we will use is known as the Schoenflie8
31
PointSr.....
PIIintS,.,. T~BLlI
T~BL:B
8-6.1
'.I.'M poin' gr0UfJ6 MId ,Mit' eum.nal 61/f1Imdry eUmenU
...
.... ..... !II.
!II... !II.d
Y. (.. even)
~'" cPo. oF.. .:r..
3-6.3
The point groupll and all their 61/mmetry elementlt
~~~
PoW,""",
..."1
37
ODe oymmllW'y p1aDe A-'treof~
ODe ...fold azia of .ymmet1'y ODe O. azia plUII .. O• ..... perpendieular to it ODe O. e.zU plua .. vertict&l plaDee a• One O. e.zU plua a horizontal pla>e a,. Th088 of !II. plus a horizontal plane a .. Th088 of !II. plus .. dihedral plaoeo a", One ...fold alterDating e.zU of oymmeW'y Th088 of a regular tetrahedrol1 Th088 of a regular ootabedron or oube Th088 or a regular ioooaheckon Th.- or • .ph8re
t
Theee elem...te .... all in addition to the identity e~t 16 ....hioh is p"""""""d by all point grouptI.
notation and the symbols for the various point groups will be written in script type in order to distinguish them from symmetry elementa or
symmetry operations. Though, strictly speaking, it is the symmetry operations and not the symmetry elements that form the group, it is common to describe each point group by the corresponding elementa. We will continue this practioo with the understanding that when we later use point groups it will be the symmetry operations which we will be dealing with rather than the symmetry elements. In Table 3-6.1 we give the essential symmetry elements for the various point groups. We use the word essential since some of the symmetry elements listed in this Table for a given point group will neoo888rily imply the existenoe of others which are not listed. In Table 3-6.2 some alternative symbols are shown. An ea:kauitle list of
"."1 ..."1
...... ". ". W'. W'.
!II. !II. !II. !II.
!>I.
"",.
"".v "".v
....
"""
",".... ""... ". "" . ..... W'• •
"'h !>I... 9' h !>I... !>I••
9 .. 9 ..
B,
a.
11, "
16 16,0•
16, o.-e: B, o.-e.-et .B, O.-e".4.-o:
.B, o.--e.--e,~.-o: B, a,--e}-d,...d,--e~ 16. a.-e.--d.-e.--C'.-e:--e: E, three a. (mutually perpendicular) E, o.-e:. three a. (perpendioular to 0.) E, O.-e.-d.. four O. (perpEl11dioular to a.): E. a.....c:~-e:. five O. (perpendioul.... to 0.) E. o.-e.--e.~.-e:.six a, (perpendicular to 0.): &aII1e aa 'i'. E,01." two U'Y E, three a. B. four E. a.-e:~-a:.llvea. E. O.--e.--e.~.-e:.six E~ iufinite number of ooinoidental rotational axe_, infinite DlUI1ber of a'Y sa.me as it'.
o.--e:.
a.--o•....o:.
a:
a::
E,C.,'.
Ob
E. a.--e:-s.-.s:.§ a.. E. a.-e.--e"...-s....s:. a 11:. O.--a:--e:-a'~.-B'
;
s:--.s:.§ a.. E. a.--e.-e.--C"...q....s..-s..-s".-.s:.§ a.., ; E. three O. (mutually perpendicular). ;. three a (mutually perpendicular) E. a.--e"~.-.s:.§ three O. (perpendioular to 0.)•.,..., three a~ E. 0.-e..-e:-8....s:. four A. (perpendioular to 0.).:: ;. abo four ..: 16. a.--a:--e:--O:-8.-8'.-.s"o-B:.§ five o. (perpendicular to 0.). a... fl."" a. E. a.--e.-e.-e:-O:-8..-s....s'....s:.§ Ilix O. (perpendicular to 0.1.:: ;. abo liz at E, infinite number of ooinoidental rotational a and alternating S "'08, (8, 0'.. ), iDfiDite number of infinite num.ber of O. ~ E. 0.-8rS':, two O. (perpendicular to each otJ>er and to the other 0.). two ad (through 8.) E. O.--e"..-s.-So. three o. (perpEI11dicular to 0.). ;. three a", E. 0.....0...q..-s....s".-s'.-s:. four a. (perpendicular to 0.). four ad B. 0.--a"...q...q.-B1O~.--s;.-S: •• five O. (perpendiouIu to 0.)••• five a" E. o.-e.-e.-e"o--C'...-s,..-s.~ ....S:.-S'~:. oi::o: O. (perpendicular to 0.). m ad (I."
T~BLJl
3-6.2
AlUmatiw "1Jf1Ibol8 ~. -
ft'LY -
"'1-Y.
..... - Y.
g. --rt
§.Jl-
W'11l -
y.
(.. odd)
t
y.
-.me
v,.,.
E, O...-s.-So." 11:. 0.-e"..-srS':. ;
!II.... --rd sgad •• -_ 9'..,
•
_.....
a.m
r.
From the German
ocdV~.
..
-.me .. 9' ..
aaDle . .
W,•
'"
31
PointG......
PolntG...... TABLB
9'.
.rd Ill..
3-6.3 (00"'.)
'if,. Above and below the pl&ne are different
E. 0.-0.--d.--s,...,s",...,s".--s;
E.
C.-o:.
31
four three JrBeB: (mu~U&1ly perpendioul&r). oiz "d ( _ Fig. 1-11.11) E. four C.-o'rB• three c,-o.--d.--seB: (mutu&lly ~ ) . oiz C•••• three " ... oiz O"d (Bee Fig. 3.11.3)
....s:.
Ax"" which coincide ..... l.i.nked. e.g. C.-o:-S.-S:. : For I'e88OD8 which will beoome oIea1- l&ter on, the a p!&DM in V,y. V .... 16"" and g ... and t.he C. &Dl8 in 16,. 91•• 16 and g ......... oonventiOD&lly lIIlp4IoI'ated into two typM: t.he plane. into ayand 17d pi and the &Dl8 into and 0; &Dl8. on- ctistinctiou ...... &bOWD in Fig. 3-11.1. where i~ io apparent that in V,y and V. y • the p1alMe labelled ad do no~ fulfill the I'e'JWrement of Table 2-3.1. Nonetheleu. the notation gi...... io t.hat .... commended by R. B. Mulliken. (J""""'" of CAemictJl PIt1Iftu IB, 1997 (1966». We will not bot.het' wit.h the differences betw""n the 0; and 0; axeB in 91. and 91• .. t.heoe point
t
d.
~
,
I I
.................
I ;.,.--+--:<
Above and below the plane are different
group. ha_ no ohemical oignifioanoe. IS: io the element ooneoponding to rotation about an axio by lj·2.-{1 (or 2·2.-(1) followed by a reflection in the plane perpendicular to that axio. We oaDDDt UN the oymbol t.hio io iden~ical wit.h 0: (_ I 2-3 pap 13). A oimilaI' a.rgummat holda far
s: ..
a: andS:.
the symmetry element8 for each point group is given in Table 3-6.3 and in Fig. 3-6.4 we show BOme molecular examples. 3-7. Dltel1lli•• tiDn Df ..DI_I.r pDint I"'P.
With experience one comes to recognize the point group to which a molecule belongs simply by analogy with some other known molecule. However, until one builds up a memory file of the point groups of representative molecules, it is best to use some systematic method. A scheme which will enable the reader to do this is shown in Table 3-7.1 and to illustrate how it works. we will consider three typica.l ca.sea. Take, for example, the bent triatomic molecule B-A-B (say. H.O). Following Table 3-7.1, it is not linear, it does not have two or more 0. with 11. ~ 3. it does have a O. axis but there are not 11. 0. axes perpendicular to this axis, it does possess two ff. (vertica.l) planes, it therefore must belong to the 'i'1v point group. Next, take the square planar molecule AD. (say, PtCl:-), it is not linear, it does not have two or more O. with 11. ~ 3 (though it does ha.ve one), its principal axis is and there are four O. axes perpendicular to this axis, the plane of the molecule is a ffla plane and therefore it belongs to the ~&ll point group. Notice that this molecule also possesses O'd pla.nes, but the O'la pla.ne is enough to &8IIocia.te it with the 9J&ll point group.
a.
/c~:
ad
.r-_......,-.:../_....... ---C;,av Above and below the plane are the oame
-O~',ad
:
,.,.",C;,a"
>.:--~' --='o(~ Above and below the plane are the same
" ,
'Gi,O'.,.
FIG. 3-6.1. Some special not&tions for the point. grOUp8
"."1' W".v,
§"'lh
and 9 p .
Finally, consider the puckered octagon (say, S.). it is not linear, does not have two or more aft with 11. ~ 3, its principal axis is 0. and there are four 0. axes perpendicular to this axis, there is no ffla plane, there are. however, four ffd planes, the molecule consequently belongs to the 9J w point group.
41
PolntGraups
Point Graup.
::J
'C.
11
planar
CI.......- " " - r < / H / - v - _ , ....... f'1 H
CI-C-O and H-C-H plano. stan an!!:l.. 4' n x ,,/2 FIG. 3.6.2. Tetrahedron inscribed in a cube. The
0:,
the !J. and the • ax"" are all
O,-S.-s: axe•. The four body diagonala through <>,1>, c, and dare O.~: ax"". The oil< plane. normal to a oube face and pe.ooing through a tetrahedral edge are
a4
plane•.
"
Br
App.ndill
CI
H
\,,/
A. 3-1. The Rurr8n,...ent Thlarsm
Cl---(:-('
This theorem states that in a group table each row or column contains each element once and once only i.e. each row and each column is some permutation of the group elements. The proof is as follows: suppose for a group of elements, E. A. B, 0, D, and F. the element F appeared twice in the column having B as the right member of the combination. We would have, say. AB=F and DB=F
H/'
where A and D are two different elements of the group. Combining each of these equations with B-1 on the right hand side of each side of each equation gives' . AB.n-l = F.n-l DB.n-l = F.n- l
ely"H "
AE = F.n- l A = F.n- l
DE = F.n-l D = FB- l
CI
H
CI
......H
T-Cl~H C.
~Rr'
n...,ither sta.ggered or eclipsed
lrafts·staggered Q
•
fir
"
,
,
H'-
-
\
-~r,1
H-C-H planes at an angle
'+ n x .,./2
Br
Since the combination FB-I is uniquely defined. we have A = D. But we postulated the group elements to be all different, so that A and D cannot
end-on vicw
II
1
/!"", Br/
FIG. 3.6.3. Ootahedron inscribed in a oube. The the !J, and the. ax"" are all O.c.O:-S.-s: axes. The four body diagonals are O.c:-S.-s: axes. The oil< axes through the
F
end-on view
0:,
origin parallel to f...... diagonala are O. axes. The 1l:1J. ft. and 11" planes are a" planeo. The six planes normal to a oube f...... and pe.ooing through a diagonal are a4 plan.eo
FlO.
3·6.4. Molecular example. of the more important point groupo.
41
42
Point GroUPS
PDint GrDups
(,..,.; }'ig. 3-1,.1)
neither 8taggol:'t~(1 or edil'sed
1,!anar
B
H~[
C.:r
~1----t?
li--Cl---t? e'y)
II
HIH cnd·on view
C.
~2h
(mil-on view Gil = molecular plane th.. C. axis is perl'.mdicu!ar to the page
: (}4
I I
H
CI
H
end-on view
NH,,··---....."'C:..O-- XH3 NH. Br
planar (sccFig.3-6.1)
17 h
= molecular plane
t.he C 3 axis is perpendicula.r
FJo. 3-6.4 (Oont.). Molecula.r exarnples of the nlore importa.nt point groups.
to the page
FIG. 3.6.4 (Oont.). Molecular exa.mpl"" of the more important point groups.
43
PaintOnu... Point Groups
~'h
c.~ I
A
Ei's. B
B - -.......::.---B B A (a.,.. Fig. 3-f1'1)
H--e-H pllmes at .,,{2 each other
t<}
H
1
H---
H
.
,,~ ~---H
"'~ C.
pUckered octagon
end-on view
O. top~yiew
.9;. staggered
x, I I
~
~I 1
.l __ ....
I
')\
I'
I
I
, '
(
(_ Jo'ig. 3-6,1)
I'
l'\.
YI •
~ I
x' )
xl/I IIi
• ,'... Y 1 ...._ .... 1
I
..
YI
IX
H-H
puckered octagon
end-on view Fro. 3-6.4 (Oom.). Molecular exampl.s of the more important point groups.
FIa. 3·8.4 (Oonl.). Molecular e"ampl... of the more important point groups.
45
Point Groups
47
be identical. Hence F or any other element cannot appear twice in the same column and because each row or column contains the same number of elements as there are in the group, each element must appear once. TABLllI
3-7.1
.A flow chart/or determining the point group
0/ a
molecule
PROBLEMS 3.1. Determine the point groups of the following: (a) CHoClF; (b) KH.; (0) BCl.; (d) allene; (e) 1,3,5·trichlorobenzene, (/) t.-ana.Pt(N~).Cl. (considered 8B square planar); (g) BFClBr. 3.2. Determine the point groups of the following octahedral compounds: (a) CoN.; (b) CoN1A; (e) ois·CoN.~; (d) trane.CoN.A.; (e) oie-oia-CoNaAa; U) trane-cie.CoNaAa· 3.3. Determine the point groups of the following: (a) chair form of cyclohexane (ignoring the H's); (b) boat form of cyclohexane (ignoring the H's); (e) staggered C.H.; (d) eclipsed C"H.; (e) between staggered and eclipsed C.H•. 3.4. Determine the point group" of the following: (a) ivy leaf; (b) iris; (0) starfish; (d) ice crystal; (e) twin-bladed propellor; (f) rectangular bar; (g) hexagonal bathroom tile; (h) sw8Btika; (i) tennis ball (with seam); (j) Chinese abacus (counters all in their lowest positions); (k) ying-yang.
3.5. Determine the point groups of the following: (a) a square.baaed pyramid; (b) a right cireularcone; (0)" square lamina; (d) a square lamina with the top and bottom sides painted differently; (e) a right circular cylinder; (I) a right circular cylinder with the two ends painted differently; (g) .. right cireular cylinder with a stripe painted parallel to the axis. 3.6. What is the point group for the tris(ethylenediamine)cobaJt(m) ion?
3.7. For which point groups can a molecule (a) have a dipole moment, (b) be optically aotivet
TAllLE 4-1.1
4. Matrices
Special matrices ana term.! Symbol8 (8... al80 LUI oj
4-1. Introduction WE have shown how the symmetry operations for a molecule form a point group and we have introduced the notation which allows us to classify point groups. The next step which leads to further progress is to find sets of matrices which behave in a fashion similar to the symmetry operations; that is, matrices which are homomorphic with the symmetry operations (see § 3-4). These matrices will be said to represent the symmetry operations. Matrix representations of molecular point groups paraphrase in a mathematical way the symmetry of a molecule and are central to all our applications of group theory to chemistry. There are many theorems in mathematics concerning matrix representations and once the link between representations and point groups has been made, these theorems can immediately be invoked for solving chemical problems as different as the molecular orbital theory of benzene and the classification of the vibrational levels of methane. But before we can discuss these matrix representations it is necessary to understand something about the properties of matrices themselves. Therefore in this chapter we define a matrix, describe the algebra of matrices, and introduce the matrix eigenvalue equation and the subject of sinlilarity transformations. The reader who is in awe of the large number of definitions and theorems in this chapter may, if he wishes, go straight to Chapter 5 and then pick up the material on matrices in slow stages as it is required in the succeeding chapters. The various theorems for matrices are proven in the appendices to this chapter but these proofs are not essential for the understanding of future chapters and may be ignored by the reader who finds the mathematics too heavy going. In Appendix A.4-1 certain special matrices and terms are defined and these will crop up in the future; the reader is therefore advised to make himself familiar with them. They are summarized in Table 4-1.1. 4-2. Definitions (matrices and determinants) A matrix is a rectangular array of terms (numbers or symbols) called elements which are written between parentheses or double lines, e.g. al (
u.
or
"
::
:: II-
_boltJ.
p. xv)
.A. - = oonjugate complex of matrix A A = transpoee of matrix .A. .A. t _ adjoint of matrix .A. A-I inV8rIIe of matrix .A det(A) = determinant of matrix .A. Traoe(.A.) = .um of diagonal element. of matrix A "" = Kronecker delta (equal. 0 if. i. equaJa 1 if i = jj ;;0;:
*
D8jinik0n8:
(A-I"
(..4\,
= (A,,)= All
(A t)" ~ (A,,)A-1A = AA-1 = E Sp8Ciol matnc...
Identity matrix
A~j1
Null matrix
A~I
= IJ i1
=- 0 Ail = didifJ d t
Diagonal matrix A real matrix A symmetrio matrix An Hermitian matrix A unitary matrix An orthogonal matrix
-¥- 0
A =A-
A=A
A =A t At = A-' .i = A-1
: Ailor (A)i1 is the element in the -itb row and jth oolumn of matrix A.
There is a definite algebra associated with matrices (see § 4-3). In this book we will only be concerned with square matrices (where the number of rows equals the number of columns) or with single row or single column matrices. We will use the double line notation, e.g.
Au
-1
A.. A •• All A.. A••
2
3
-1
0
An
All
All
Rowand column matrices, e.g.
IIA 1
0
1
a
A. A.II,
b
e
IIx 11
XII
Xli
xull,
Xu
II
X..
II·
ca.n be used to define the components of a vector. In general we will use a capital italic letter to symbolize a matrix and an element in
Mmica
Me.rica
50
the ith row andjth column of matrix A will be written as Au. e.g. if 123
A
4
=
7
8
956
then A.a = 8. The number of rows (or columns) in a square matrix is called the order of the matrix. The reader must not confuse square matrices with determinant8. A determinant is a square array of elements symbolizing the sum of certain products of the elements. Unlike a matrix, a determinant has a definite quantitative value. A single straightla~e~~call line on either side of an array indicates a determinant, e.g.
is a determinant
a.
b.
Hence, we may sucoessively break down any determinant into products involving lower order determinants, until second order determinants are reached, the values of which are given by eqn (4-2.1). The determinant of a square matrix is the determinant obtained by considering the array of elements in the matrix &8 a determinant; if the matrix is A we will write the determinant as det(A) i.e. if Au
All
A,"
Au
A••
A ...
A", A...
A....
A
then
Au All All A••
of second order whose value is alb. -a.bl :
l a
l bll = alba-a.b,
a.
Alft
A ...
det(A) =
(4-2.1)
b.
al
bl
c.
and a.
b.
c. is a determinant of third order whose value is
aa
b.
c.
A...
..1
A
4-3. Metrix elglltre
alb.c. +a.bacI +aab,c. -aob.cl -a.blc. -~ bac•. In general, a determinant is equal to the sum of the products of the elements in any given column (or row) with their corresponding cofa.ctors; the cofactor of an element, .stfu , is the determinant, of next lower order, obtained by striking out the row i and the column j in which the element lies, multiplied by (_1)'+1, e.g. a l bl
a.
b.
:: =
a.
b.
c.+a.x(-l)·x Ib cII +a.X(-I)·x lb. cil
a• .stfu +a• .stf.. +a• .stf81
= al X ( -I)· X
I:: : 1
The algebra of matrices gives rules for (I) equality, (2) addition and subtraction, (3) multiplication, and (4) 'division' as well as (5) an associative and a distributive law. It also includes definitions of (6) a transpose, adjoint and inverse of a matrix. (I) Equality. Two matrices A and B are equal if, and only if, A" = B" for all i and j, e.g. if A
l
b. c. b. c. = al(b.c. -b.c.) -a.(b.c.-baCl) +a.(b.c.-b.cl )
(4-2.2)
=II ~
: II and A
=
B, then B
=" ~ : II·
(2) Addition and subtraction. Matrices may be added and subtracted only if they are of the same dimensions. Under these circumstances, the sum of A and B is given by the matrix 0,
A+B = 0, or
:: :: :: =a• .stfU +b l .stfll +c• .stf,. =a.X(-I).xl:: u.
b.
&1
::1
where 0'1 = A/I+Bil for all i andj, e.g.
c.
II
~
: II + II
~
:" =
Ill: I: II-
52
Matrices
Matrices
Under the same circumstances, the subtraction of B from A produces the matrix 0, A-B = 0, where 0'1 = A'I-BII for a.ll i andj, e.g.
I ~ : II-II: : I I =
-4 -4
=: II·
&3
An easy way of remembering how to carry out matrix multiplication is to reaJize that in doing so, one takes succeeding rOW8 in the jir8t matrix multiplied. vectorially by succeeding colum1l8 in the 8eoond matrix, the ith row and jth column producing the t, j element in the product; or sohematically:
From this, it follows that multiplying a matrix A by a number c produces a matrix B, B = cA, whose elements are given by B'I
e.g.
311
=
cAli' for all i andj,
~ : I II: =
More than two matrices can be multiplied together; one simply uses the multiplication rule more than once, multiplying pairs of matrices at a time (see eqn (4-3.12». For the product of three matrices,
1: I ·
(3) Multiplication. Two matrices A and B may only be multiplied together (ca.lled matrix multiplication) if the number of columns in A, say n, equals the number of rows in B; the product is then defined as the matrix 0, a =AB,
D'I =
=
3
1
14
456
2
32
7
3
50
8
9
2
311
4
5
6
789
= 1130
in the form:
36
4211.
A."B~..,cmJ
A ll111+ A 1I1I.+ A 1I1I. =
123 III
..
(4-3.6)
(4-3.4)
(4-3.2)
=
2
~
AllYl +A ..y.+A.oY. = x. A,"y" +A.,.y.+A.oY. = x.
for a.ll i and j. If the matrix A has m rows and n columns (an m xn matrix) and the matrix B has 11- rows and p oolumns (an n xp matrix,) the matrix a will have m rowa and p columns (an m xp matrix). The following equatioDB are examples of matrix multiplication:
1
• •
LL
(4-3.3)
(4-3.1)
~ : 1111: : 1/ II:: :: II, II: : 1/11 ~ : I II:: :: II,
the general element of the product is given by:
for all i andj, where r is the number of columns in A which must be the same as the number of rows in Band 8 is the number of columns in B whioh must be the same as the number of rows in a (see problem 4.3). Notice the restrictions whioh apply to the number of rows and columns in matrices whioh are to be multiplied together. It is apparent from the examples in eqn (4-3.2) and (4-3.3), that, in general, matrices do not commute, i.e. AB "¢ BA. For this reason care must be taken when matrix equations are being manipulated and one must remember that, like opera.tors, the order in which matrices are multiplied together can be significant. One of the uses of matrices is to express sets of linear equations in a compact form, for example with the above definition of matrix multiplication, it is possible to write the equations:
whose elements are given by the equation:
I
D = ABO,
(4-3.5) or
Xl
(4-3.7)
~:: ~:: ~:: III :: I .4.. I11. II
A.1
A..
AY=X
(4-3.8)
&4
Matrica
where
A
Au Au All All All AI. Au A •• A ••
y=
Y.
This also implies that only square matrices can have an inverse, since only square matrices have a determinant. (5) .As.sooiative and distributive laws. These laws are respectively:
Zl
Yl
x=
Y.
z. z.
Furthermore, if, coupled with these equations, there is also the set of
Buz. +Bl,z,+Buz. = Yl BIlz. +B,,z.+B.,z, = Y. Bllzl + B.,z, + Bllz. = Y.
equations
then
Bn
B= Therefore
B lI
Bn
B Il
B"
B al
Bl i B I I
Zl
and
B ..
z=
=
(AB)O,
(4-3.12)
A(B+O)
=
AB+AO.
(4-3.13)
z,
(AB)'
=
BfAt,
(4-3.15)
z.
(AB)-l
=
B-lA-l.
(4-3.16)
(AB)Z = X.
This result can be expressed in the following way: if the transformation of z's to y's is defined by a matrix B and that of y's to z's by a matrix A, then the transformation of z's to z's is defined. by the matrix AB. (4) 'Division'. As with operators, 'division' can only be accomplished. through an inverse prooess. Every matrix A which has a non-zero determinant, det(A) 0,
*"
is lI&id to be non-singular. For suoh, and only suoh, matrices, an inverse A -1 can be defined. by the following equation: AA-l = A-lA = E,
(4-3.9)
where E is the identity or unit matrix (see Appendix A.4-I(a». The matrix operation which is equivalent to division is matrix multiplication by an inverse, e.g. if (4-3.10) AB=O we may write ABB-l = OR-I (4-3.11) or or
A(BO)
(6) Transpose, adjoint, and inverse of a matrix. The inverse has been defined in (4) above and the transpose and adjoint are defined in Appendix A.4-1 and Table 4-1.1. The reader is left to prove (see problem 4.1) that the transpose, adjoint, and inverse of the produot of two matrices are given by: AR = llA, (4-3.14)
Y =BZ,
where
and
4-4. The matrix eigenvalue 81Juation
For every square matrix A of order n there is an eigenvalue equation
Ax
of the form:
=
(4-4.1)
where x is a column matrix (with dimensions n x I) and 1 is a number or scalar. The solutions of this equation, and in general there will be n whioh are distinct, are the values of 1 (oalled the eigenvalues) and the oorresponding column matrices x (called the eigenvectors). The equation can be expressed in words by saying that matrix A multiplied on the right by the column matrix x produces the same column matrix simply multiplied by a number. We can distinguish between the several solutions ofeqn (4-4.1) with subscripts and write the different eigenvectors which correspond to the different eigenvalues 1.. 1., ... , 1" as X" x., ... , x" or
AE =OB-l
, ... ,
A = OB-l.
Notice that when we multiply eqn (4-3.10) by B-1 to produce eqn (4-3.11), as matrices do not neeessarily commute, we must do 80 on the right-hand side of both sides of the equation. A method for finding the inverse of a non-singular matrix is given in Appendix A.4-2. From this method it is apparent why A-I is only defined when det(A) "# O.
Az,
and we can 'write eqn (4-4.1) as
Az,
=
1,.3:,
i
=
1, 2, ... , n.
(4-4.2)
5&
Matrices
It is customary to require the eigenvectors to be normalized (see
and, if we combine eqn (4-4.6) with eqn (4-4.3), we have
Appendix A.4-I(g», that is:
k = 1,2, ... n
or
x.. ... x:,1I
=11111 1=1,2 ... n
x"'
.
or
~ x:,x~, = 1
~-,
i = I, 2 ...
n.
(4-4.3)
This restriction cuts out those superfluous eigenvectors which differ merely by a constant factor. An alternative form for eqn (4-4.2) is (A-)';E)x,
=
i
=
1,2 ... n
det(A -lE) = O.
0
or
o
A,
o
o
0
An
(4-4.4)
(4-4.5)
The proof of this is given in Appendix A.4-3. Eqn (4-4.5) is often called the characteristic equation of matrix A and it is e8Bentially a polynomial equation in A with n roots: AI, AI, .•. ).... These roots or eigenvalues, which guarantee non-trivial eigenvectors, once found can be used, one by one, in the solution of eqn (4-4.4) coupled with eqn (4-4.3); each eigenvalue Aj leading to a corresponding normalized eigenvector x t (see Appendix A.4-5). There are two important theorems for the eigenvalues and eigenvectors of eqns (4-4.4). If a matrix A is Hermitian (A = At, see Appendix A.4-I(e», its eigenvalues are real and its eigenvectors are orthogonal to each other if they correspond to different eigenvalues i.e. if the eigenvalues are _-degenerate (A~ oF .:t l ). This result can be expressed as ). _ ,. j At 1 = I, 2, ... , n =
provided that A is Hermitian and the roots of eqn (4-4.5) are distinct. Eqns (4-4.7) define a set of normalized orthogonal eigenvectors. The other theorem states that the matrix X formed by using the eigenvectors of a Hermitian matrix as its columns is unitary (for the definition of a unitary matrix, see Appendix A.4-I(g». The proof of these two theorems is given in Appendix A.4-3. The matrix X formed from the eigenvectors of a matrix A may be used to combine the solutions of Ax = Ax into a single equation: AX = XA (4-4.8) where )., 0 o A=
0
where E is the identity matrix and 0 the null matrix (see Appendix A.4-I(f». For this equation to have non-trivial solutions (that is, excluding solutions of the form X t = 0), it is necessary that the eigenvalues At obey the determinantal equation:
xix.
(4-4.7)
l = 1,2, ... n
1=I,2 ... n
IIx~tx:',
67
(4-4.6)
Notice the order is XA not AX; the reader is left to confirm this fact for himself. In eqn (4-4.8), if A is Hermitian then of course X will be unitary and if A is symmetric then X will be orthogonal. 4-6. Similarity transfarmstions If a matrix Q exists such that
Q-IAQ
= B,
(4-5.1)
then the matrices A and B are said to be related by a aimilarity tran8formation. Such transformations will be very important in what follows later. It is immediately apparent that eqn (4-5.1) parallels the relation that exists between two symmetry operations that belong to the same class (see § 3-5). If Q is a unitary matrix (see Appendix A.4-I(g)), then A and B are said to be related by a unitary tran8formation. There are a number of useful theorems concerning matrices which are related by a similarity transformation. If the matrices A and Bare related by a similarity transformation, then their determinants, eigenvalues, and traces (sum of the diagonal elements) will be identical: det(A) = det(B), A's of A = .:t's of B, Trace(A) = Traee(B).
(4-5.2) (4-5.3) (4-5.4)
58
18
If A' = Q-IAQ, B' = Q-IBQ, C' = Q-ICQ etc., then any rela.tionship between A, B, C etc. is also satisfied by A', B', C' etc. If the result of a similarity transformation is to produce a diagonal matrix (see Appendix A.4-I(b», then the process is called diagonalizatian. If the matrices A and B can be diagona.lized by the same matrix, then A and B commute. If X is the matrix formed from the eigenvectoI'8 of a matrix A, then the similarity transformation X-lAX will produce a. diagonal matrix whose elements are the eigenvalues of A. Furthermore, if A is Hermitian, then X will be unitary and therefore we can see that a Hermitian matrix can always be diagonalized by a unitary transformation, and a symmetric matrix by an orthogonal transformation. The final theorem is that a unitary transformation leaves a unitary matrix unitary. These theorems are proven in Appendix A.4-4 and, in Appendix A.4-5, we show, with an example, how a matrix can be diagonalized. Appendices
A.4-1. Special mstricn (a) ldemuy matrix. This is a square matrix whose diagonal elements are all unity and whose off-diagonal elements arc all zero:
o
o E
=
0
i.e. Eij
=
D'I = Oifi ~j, DOl ~ 0 if, =j.
(e) RealmtUrlz. The conjugate complex I- of a number or funotion 1 is obtained by replacing i everywhere inl by - i. TheconjUllate complezof matrix A is written as A - and the elements of A-are the conjugate complexes of the elements of A, i.e. (A-)'I = (A'/)-' For a real matrix all the elements are real and therefore A = A(A.4-I.3) i.e. Au = A'I- for all i andj.
(d) Symmetric matrix. The transp08e of a matrix A is obtained by ohanging rows into columns (or vice versa) and is given the symbol A (or, sometimes, A'), e.g. the transpose of
I
2
3
A= 4
5
6
7
8
9
For a symmetrio matrix: Le. A'I
=
1
7
4
A= 2 5 S
is
3
9
6
A=A,
(A.4-I.4)
Au for all i and j. The matrix
123
A =
2
4
5
356
0
is symmetric. All symmetric matrices must be square.
O. 0
i.e.
I
(A.4-I.l)
"lI = {o I
i
~j
. =J..
(e) Hermitian matrix. The adjoim of a matrix A is ohtained by taking the complex conjugate of the transpose and is given the symbol A I i.e. At = A-. (Mathematicians frequently call A t the Hermitian conjugate and reserve the term adjoint for the transposed matrix of the rofactors of A.) The adjoint of I 3 1 4 A =
~
el
2
-i
3
ell
I
is
At =
4
2
-i
(The symbol ~'I is called Kronecker delta.) Other symbols for the identity matrix are I and 1 and it is sometimes called the unit matrix. It can be of any order and can be inserted anywhere in any matrix equation.
A Hermitian matrix is one which obeys the equation
(b) Diagonal matrix. Any square matrix for whioh all the off-diagonal elements are zero and at least one of the diagonal elements is non-zero is said to be
i.e. Au = Art for all i and j. For example,
diagonal:
d, D =
A=A
1
o o
0
d.
0
o
0
d.
t
A = (A.4-1.2)
e-tll
1
(A.4-1.5)
e-I
-i
2
4
et
4
3
10
11
Metrlcn
is Hermitian. All Hermitian matrices must be square. For real matrioea the criterion for being Hermitian or symmetric is the same. (!) Null (or zero) matnz. A matrix of any dimension whose elements are all zero:
o o
to be orthogonal and normalized, i.e. the rows of A form the orthogonal, normalized vectors .,. All unitary matrices are square.
100
0
A=
0 (A.4-1.6)
0=
0
0
o
w' 0
w
where w = exp(2niJ3) is an example of a unitary matrix. For this matrix the adjoint is I
i.e. 0., = 0 for all i andj. (g) Unilary malriz. A matrix is unitary if its adjoint (see (e) above) is equal to its inverse: t_ A =A 1 • or AtA =E, (A.4-1.7) or t AA = E.
The columns (or rows) of a unitary matrix are related to a set of orthogonal normalized vectors in a general vector space. If, for example,
An An
A."
An A..
.
A n1
A .."
'-1
=
1,2, ...
11.
(A.4-1.8)
j = I, 2 ... n
= 1,2 ... n
~ ALtA s"
=
"If
cos 6
sinO
0
-sin 0
cos 0
0
o
o
I
A.4-2. Mllthod for d8t8n11inillll the inve,.. Df. matrix Consider the fa equations:
+
(A.4-1.9)
onal base vectors. Alternatively, using AA t = E, we have
'-.
For real matrices (see (e) above) the criterion for being orthogonal or unitary is the same. All orthogonal matrices &re square and
+
to be orthogonal and normalized i.e. the columns of A form the orthogonal, normalized vectors r s. In eqn (A.4-1.9) the eJII (k = I, 2 ... n) are unit orthog-
".
0
11. = Auz. AlIz, ... At"z.. 110 = A,.Zl+A002:o ••• +A...z ..
and this is the requirement, by definition, for the general vectors:
j
CUi
is an example of an orthogonal matrix.
Ann
i
= 6.!
o o o
and it is left to the reader to show that AA t = AtA = E. (h) Orthogonal malM. A matrix is orthogonal if its transpose (see (d) above) is equal to its inverse: A = A-I. (A.4-1.12)
A =
is unitary then AtA = E and
.
o
A o"
A=
~ AkjA,,!
At = 0
(A.4-2.1)
Y.. ~ A ...z.+A,,02:o ...+A"..z .. they may be written in matrix notation as
i = 1,2 ...
n
j = 1,2 ... n
(A.4-1.10)
and this is the requirement, by definition, for the general vectors:
i = 1,2 ... n
(A.4-1.11) A ...
Ad
A"..
z ..
13
II or Y = AX. Multiplying both sides of this equation by A-I, the inverse of A, we get X = A-IY or, if we let
A-1
A~l
A;I
Ai"
A;.
A;.
A~tI
Since determinants with two or more identical columns or rows are zero, this equation beoomes or
_
du1l.+dIl1l• ... +d...1I.. = det(A)xI
N.
dud.. d Xl = det(A) 11.+ det(A) 11• ••• + det(A) 11..·
(A.4-2.3) (A.4-2.4)
In the same fashion. with the other cofactors, we can obtain .!JiI11
A.~ft
A~I
A~l
X.
Xl
A~.
A~s
ALI
11.
X.
All
AI.
At..
11.
1
~
(A.4-2.2)
_
d
1"
X.. - det(A) 111 A~l
Z..
.A~I
A~~
All A..
AI" A ...
det(A) = A-I =
A ... A... A ..~ = AU.!Jill l +A I l .!JilI l +···+A"I.!Ji1..1 = Au.!Jil..+ A ••.!JiIn+... + A ...d ..1
.n
d ll det(A}
d.1 det(A)
d ... det(A)
d 1• det(A)
d •• det(A)
.!JiI... det(A)
tYl+dnYl"'+.fJIf...Y. =(Audll+A...fJIfIl+···+A.ld.l}Xl + (AlI.!JiIn +A I ..,rat'1l + ... +A,,"d.1)X I
+... + (A 1 ..d u +AI.,.!JJf1l + ... +A....d ..1)X.. =det(A)zl
All
AI..
A..
All
A..
+
Z ••••
.!JiI...
d .... det(A)
AI..
All
At..
AI..
Au
A ...
+
Z ...
(A- ' )lI = .fJIfu/det(A),
A.I
A ....
A ....
(A.4-2.6)
where (A-l)'i is the element in the ith row andjth column of the inverse of matrix A and d J" the cofactor of All> is (-l}'+i times the determinant of the lower order matrix obtained from A by striking out the jth row and ith column. Clearly, eqn (A.4-2.6), and therefore an inverse, cannot be defined if det(A) is zero (i.e. if A is singular); also. fordet(A) to exist, A must be square. Eqns. (AA-2.4) give the solution of any set of fa equations in fa variables; the method is known as Cramer's rule.
A.C.3. Thlonlms for lilllnlllCton (G) Theorem. Ifl,areohOBeDaetherootsoftheequation: det(A-AE) = 0, then the equations: (A -l,E)x, = 0 i = 1, 2 ... ,. (A.4-3.1) will give non.trivial eigenvectors (oolumn matrices)
A ..I
(A.4-2.4)
So that for any square matrix A
where .!JiI'J is the cofactor of A'i (see § 4-2). If we multiply the first of eqns (A.4-2.1) by d w the second by d ....... the nth by .fJIf", and add, we get:
All
(A.4-2.4)
(AA.2.5)
dIn
= A 1 ...!JiIlN +A...d ... + ... +A .....!JiI....
1
d
+.!JiI d .... det(A) Y•... + det(A) 11..·
I det(A} det(A)
d
d
Comparison of these equations with eqn (A.4-2.2) shows that
Yn
Now the determinant of A can be written as Au Au
.!JiI..
= det(A) Yl+det(A) 11• .•• + det(A) Y..
x,.
85 Proof· (A-l,E)z, = 0 is simply the matrix notation for a set of simultaneous equations which are equivalent to equa (A.4.2.1) with y, = O. Consequently, if the elements of x, are X w zo<' ... x"" we have, by comparison with eqn (A.4-2.3), det(A-l,E)xl , = 0,
det(A-l,E)x.,
Multiplying the second equation on the left hand side by it from eqn (A.4-3.6) gives
(A k
-
zt and 8\lbtraoting
J.,)zkx, = 0 and therefore, if Ak ~ A" xkx,= 0,
(A.4.3.7)
i.e. the eigenvectors are orthogonal.
= 0,
(e) Theorem. The matrix X formed by using the eigenveotors of a Hermitian matrix as its columns is unitary.
det(A-A,E)x",
Proof. The eigenveotors of a Hermitian matrix, if normalized and nondegenerate, obey the relation x4x, = lI~k,11 and consequently we can write:
= O.
x, is to be non-zero, then at least one of its elements,
X~l
say x k " must be non-zero and since det(A-l,E)xk , = 0, this implies that det(A-A,E) = O. Hence, for non-trivial solutions (x, =1= 0) the l, must be the roots of the equation det(A-AE) = O.
X~I
If the column matrix
(b) Theorem. If A is Hermitian, its eigenvalues are real and its eigenvectors or
Proof. Consider eqn (4.4.2):
Ax,
=
l,x,.
(AA-3.2)
taking the adjoint of this equation (see eqn (4-3.15». we have
x,tA f = Atxl or
x,fA
and multiplying by
=
l,·x,t
X:1 X:1
x:J
are orthogonal to each other provided they correspond to non-degenerate
eigenvalues.
%:1 X:1
xtx =
XlI
XII
1
0
XII
x ..
o
1
X..I
E and therefore Xf
=
o 0 x .." X-I and X is unitary. x".
o o
1
Oorollary. The matrix X, formed by using the eigenvectors of a symmetrio matrix &8 its columna is orthogonal. This follows from the fact that the eigenvectors of a symmetric matrix obey the relationship !ii~, = II~k,lI.
A.4-4. TheDrems for similarity trIInsfDrmatictns (a) Theorem. If Q-IAQ = B, then det(A) = det(B).
(sinoe l, is a soalar, A,t = It) (since A is Hermitian, At = A)
Proof. Since det(X Y) = det(X)det( Y) (see Appendix A.4-6) we have;
x, produces
det(B) = det«(T')det(AQ) (A.4-3.3)
But from eqn (A.4-3.2)
= det«(T')det(A)det(Q) = det(Q-l)det(Q)det(A)
x,tAx, = l,xix,
(being a scalar, A,oommutes)
(A.4-3.4)
and subtraction of eqn (A.4.3.4) from eqn (A.4-3.3) gives (l:-A,)xix, = O. Since x~x, =F 0 (soo eqn (4-4.3». (A .4-3.5) or A, is real. Now oonsider the pair of equations AXk = l~k'
Ax,
=
l"",.
Taking the adjoint of the first and multiplying on the right hand side by x,givea t . t xkAx, = J.; xy;, = A~X,
(since Ak is real).
(A.4.3.6)
= det(q-'Q)det(A) = det(E)det(A) = det(A).
(b) Theorem. IfQ-1AQ = B. then the eigenvalues of A and B are identical. Proof. Since, (B-J.E) = (q-'AQ-lE) = q-'(A-lE)Q we have det(B-AE) = det«(Tl)det(A-lE)det(Q) = det«(T'Q)det(A-lE) = det(A-lE). The roots of det(A-AE) = 0 and det(B-lE) = 0 must be identical, sinoe the equations are identical.
&11
Matrica
17
(e) ThWf'em. If Q-IAQ = B, then
Tr&ce(A) = Trace(B). Proof·
Trace(B)
=
.I B"
X-IAX=A.
<
.I .I .I (Q-')'k A 1-RI' < 1 k = .I1 .Ik A.tl .I, Qfi(Q-l)i.t = .I .I Akl(Qq-l)lk =
1
=
where A is the diagonal matrix composed of the eigenvalues of A and multiplying on the left of both sides of this equation by X-I we obtain directly
(see eqn (4-3.6»
.t
If A is Hermitian, then X will be unitary (Appendix A.4-3(c)) and the transformation eqn (A.4-4.1) will be unitary. If A is symmetric, X and the transformation will be orthogonal. (rr) Theorem. A unitary transformation leaves a unitary matrix unitary.
Proof. Since X = U-IAU, where A and U are unitary, we have:
.II .Ik A1-1 "Jk
X-I
= (AU)-lU =
=.IAu =
and
"
Proof. Consider, as an example,
(see eqn (4-3.16))
U-1A-IU
X t = (AU)t(U-1 )t = U t At(U-I)t.
Tr&ce(A).
(d) Theorem. If A' = Q-IAQ, B' = Q-'BQ, 0' = Q-IOQ etc., then any relationship between A, B, 0, etc. is also satisfied by A', B', 0' etc.
(AA-4.1)
(see eqn (4-3.111))
But, by definition, U t = U-1 , At = A-I, and (U-l)t = (ut)t = U. Thus X-I = X' and X is unitary. Likewise, if A and U are orthogonal (real &8 well &8 unitary) then 80 is X.
D=ABO
then,
A.4-5. The di..alUlliutian af a ..atrix ar h_ ta find lb••i••nYalu•• and .igenlltletars of a ..atrix
D' = Q-I DQ
=fr1 ABOQ = Q-IAQq-IBQq-1CQ
Consider the diagonalization of the matrix:
= A'EO'.
=
Do
rr BQ = 1
Db where D", and D. are diagonal matrices, then Q-l(AB)Q = (Q-1 AQ)(Q-1BQ) = DoDb = DbDo = (Q-1BQ)(Q-lAQ) = Q-1(BA)Q
o
-4
0
2
0
2
o
5
A =
(e) Theorem. If A and B are two matrices whioh can be disgonalized by the same matrix Q then A and B commute. Proof. Since, Q-IAQ
-1
This may be accomplished with a matrix X which is composed of the eigenvectors of A. The diagonal matrix created will consist of the eigenvalues of A. The steps involved are: (I) the determination of the eigenvalues AI' As, and As from det(A - AE) = 0, (2) the determination (using these eigenvalues) of the eigenvectors
(diagonal matrices commute) Zat
zit
and therefore AB = BA and A and B commute. (J) Theorem. If X is the matrix formed from the eigenvectors of A, then
X-lAX is a diagonal matrix A composed of the eigenvalues of A.
zli
'I'll
(A -
A,E)
,
z ..
Zu
,
and
from the equations
Z ••
za
z ..
zt.
0
'1'••
0
, and
•
.Iz*~".= 1
It-l
Proof. }'rom eqn (4-4.8) we have AX=XA
z..
0
and (3) the determination of X-I from X.
(l = 1,2,3)
II
••tri_
Mmi_ Therefore,
(1) The determinantal equation is:
Xu
o
-4
o
2-1
o
2
0
5-).
det(A-1E) =
1/"';2,
=
or 1 -6)"+11 ...-6 = O. The roota of this equation are; A,. = I, As = 2, and As = 3. Theee are the eigenvalues. (2) With theee roots, we can now be assured of non·trivial eigenvectors.
X=
(3)
!£i'
=
(X-l)'i = !£u/det(X)
o
1
0
20
2xU+4x31
"':1+X:I+ X:. Therefore, Xn
=
xII
0
4 xOl
0
X-I
= 0 (redundant) = 1 (normalization). 0,
1
0
1/"';5
0
and
"'11
=
i.e.
1
-"';2
0
X-lAX
-.../5
o
-.../5
-1
o -"';2
1
0
o 2
o
-2"';2
1/"';5
-4
XII
0
0
0
0
x..
0
2
0
3 0
"'II
0
",:.+x:.+x:. = x.. = 0,
(normalization).
x.. = I,
and
XII
O.
=
.... =3 -4
0
-4
"'1.
0
0
-1
0
"'II
0
2
0
2
"'II
0
-4:l:•• -4x.. = 0 - X. .
2xu
= 0
+2x.. =
0
x:.+x:a+",~ = 1
= A
0-4
-2/.../5
o
1/.../2
2
o
o
1
0
o
5
1/"';5
o
-1/"';2
100
Note, that had A been Hennitian, step (3) would have been simply X-I = xt, since in such a oase X would have been unitary. The rearrangement of the columns in X with the parallel rearrangement of the ~ in A and the concomitant changes in X-I, or the changing of the sign of every element in a given column of X with the concomitant changes in X-I, all constitute other valid diagonalizations.
(redundant)
1
-2"';2
003
2"'.. +3xII = 0
Therefore,
0
020
0
o= 0
1/"';10 0 -"';5
0
.... =2
-3
-3x.. -4xl • =
-1/"';2
(see Appendix A.4-2)
=
The reader can confirm that
= 0
"'u =
-2/,)5,
1/"';2
0
-"';5
0
-2xu -4:1:'1 = 0 "'II
0
det(X) =
x11
-4
Xu = -1/"';2.
cofactor of X J<
~=I
0
and
-2/"';5
=0
1
-2
x.. =O,
Hence
-1-1
III
(redundant) (normalization).
A.4-I. Proat that det (A8) = det (A) det (8) Necessarily the matrices A and B must be square and of the same order. COllsider matrices of the order 2
AB =
det(AB) =
II A
ll
B l l +A l aBn AnBu+A.aBll
I
AuBn A IIB ll
I
AllBlI AIlB..
AI"B1l
+ An~u
+
AnBlI AlaBIIII A IlB lI +A.aBII
I+ I I+ I
A ..B u A.IBl I
A ll B n
A.aBl l
AnBl I A.aB••
A l aBn AlaEn
I
AlaEu
I
A.aE.. ·
10
11
The first and last determinants vanish sinoe. if the oonatant factor A 11A .. is removed from the first determinant, its two rows are identical and removal of the constant factor from the last determinant leaves it also with two identical rows. Therefore:
A,""'..
4.8. Show that the m ..tri"",,: (a)
(e)
II IfIf vv 22
II
008
-If V211. 11 v 2
6
sin
sin 8
-008
11 v2
(b)
2
If V /1
I/v 2 -If v2
11
'
and
6/1
6
are orthogonal. 4.9. Show that if ~,~•... A" are 1>he eigenvalues of A, then ~-k, ~-k,... A,.-k are the eigenvalues of A -kE. 4.10. Obtain the eigenvalues and normalized eigenvectors of:
The proof oan be extended to matrices of higher order. (a)
AB
=
EA.
(b) (AB)t = BtAt,
A=
4.2. Prove that (a) the product of two unitary matrioes is also unitary and (b) the inverse of a unitary matrix is unitary. 4.3. Show that if four matrices obey the equation D = ABC, then
4.4. (a) Show that Trace (AB) = ~ ~ Ai/Bit. (b) Given two matri""" A and B
4.5. Prove that for any matrix A: (a) AAt and AtA are Hermitian. (b) (A +.4. t ) and itA -At) are Hermitian. 4.6. If AB = BA, show that QAQ and QBQ commute ifQ is orthogonal. 4.7. Find the inverse of:
0
0
0
0
1
0
0
0
0
1
0
0
0
0
1
1
(a)
(e)
(1)
Ila+i~ e+~dll, -c+id a-Ib 0
0
a
0
b
o ,
0
0
0
2
3
1
3
5
2
0
0
2
(b)
(d)
(a)
I
of dimensions n xm and m xn respectively, prove Trace (A B) = Trace (BA).
II:
-:11·
(e)
II~ -~II '
and
8
(b)
10
2
-2
8
-2
11
and
Q=
-If2
- v 3f2
0
- v 3/2
If2
0
0
0
2
008
(e)
6
-sin 6
Ii
10
8
10
2
-2
8
-2
11
(0)
114:i
i 4-14
l·
(b)
If2
v 3/2
0
If2
0
0
0
1
cos 6
sin 8
0
cos IJ
0
0
0
1
, and
0
6
0
0
-v3 f2
-sin 6
sin 6 COB
0
show that Q-IAQ:is diagonal. 4.12. Diagonalize the following matrices:
DiJ = ~ ~ A,..B.,CII' i
~:II'
4.11. If
PROBLEMS 4.1. Show that for two m ..trioes A and B: (a) and (c) (AB)-' = B-IA-l.
II:
10
Ii
Isymmetryl
5. Matrix representations
Imolecular symmetry operations I &-1. Introduction
THE best way to understand how the symmetry operations of a molecule influence its properties is to study the sets of matrices which mirror, by their group table (see § 3-4), those same operations. Such sets of matrices, homomorphic with the point group, are said to be, or to form, a repre8entation of the point group. Essentially, when we introduce a matrix representation, we are repla.cing the geumetry of symmetry operations with the algebra of matrices. Matrix representations are the crucial link between the symmetry of a molecule and the theorems which determine such practical things as to whether a given infra-red band should be present or not. Mastery of the ideas in this chapter is essential to the proper understanding of subsequent chapters. There are several different methods of obtaining sets of matrices which are homomorphic with a given point group and in this chapter we discuss these methods in some detail. One way is to consider the effect that a symmetry operation has on the Cartesian coordinates of some point (or, equiva.lently, on some position vector) in the molecule. Another way is to consider the effect that a symmetry operation has on one or more sets of base vectors (coordinate axes) within the molecule. A third and more complex way is to first find another set of operators 0., which have certain fundamental properties and are homomorphic with the symmetry operations, and then find a set of matrices which are homomorphic with these new operators. This last step is achieved by consideration of the effect that the 0. have on some 'family' of mathematical functions (a so-called Junction space) e.g. a set of five d-orbitals; it will be seen that the choice of the function space and the choice of the O. are bound up with each other. It is important to realize that this method involves two steps as opposed to the first two methods which involve only one. Since it is easy in this subject to 'lose sight of the forest for the trees', the scheme which we are following in these introductory chapters is summarized in Fig. 5-1.1. The reader will probably find it helpful to keep this plan in mind while he pursues the material of this chapter. For the sake of completeneBB, various proofs are given in the appendices but, as in Chapter 4, knowledge of them is not critical for the reader who wants to have only a general understanding of the subject.
[pOint groupsi
!u8trix rep~ntations
from a })()sition
I
I
matrix representation, from sets of baBe
"f"{'tors
vt-'{.1"or
homorphic group of tr&llsfomlation operators Glo
I matrix
representations of transformation operators 0. using different function spares
I
Iofnlatrix poin t grou
repl'e9flntationsl I'll .
I
I
non- equivaleht.
equivalent I representatiollJl
repre....ntations
I
1
lredudble representatiou2i
~irredUeible
representations
I IthN>""ms
1
I "'haracter tabl..,,1 FIG. 5-1.1. Summary.
5-2. Symmetry operations on • poaition vietor
A position vector p is a quantity which defines the location of some point P in three-dimensional physical space (see Fig. 5-2.1). If 0 is the origin of some set of space-fixed axes, the length p of OP and the direction of 0 P with respect to these axes constitute the position vector. If the set of space-fixed axes are mutually perpendicular, the position
74
75
Mmix R.prsnntlltioDS
••trix Reprwent8tions e,
I"~
p
p
.."
~-+---~_
e, FIG. 1)·2.1. A position vector.
F1G. 5.2.3. Effect ofC. on p.
of some point P may also be located by its coordinates Xl' x 2 • and X. with respect to these axes. (Note that for ease of notation later on, Xl> XII and X. will be used in preference to the more familiar X, y, and z.) If, coinciding with these fixed axes. there are three unit vectors (vectors of unit length) e 1 • e 2 • and e., then any position vector p can be expressed as (5-2.1) Corresponding to each point in space :l:u :1:., and x. there is therefore a position vector given by eqn (6-2.1) and we can think interchangeably of a point and the position vector which defines its location (see Fig. 6-2.2). The mutually perpendicular unit vectors e u e•• and e., are called orthogonal base vectors and Xli x., and which double as coordinates, are called the components of the position vector p. We now consider the effect that symmetry operations have on a point or position vector. (1) Rotation. In Fig. 5-2.3 we show the effect on p of a clockwise rotation by fJ (= 2'1f/n) about the direction e. Le. C n . If d is the projection of OP on the plane which contains e 1 and e 2 and c/> the angle it makes with ell then the following relations hold between the components (coordinates) of the initial vector p (point P) x .. x •• and x.
x..
and those of the final vector p' (point p'fx~, x;, an(x;: x~ =
d cos(-fJ) d cos cos fJ+d sin sin (J = d(xJd)cos (J +d(x2 /d)sin fJ = Xl cos (J +x. sin fJ =
d sin(c/>-fJ) = d sin cos fJ -d cos sin fJ = d(x./d)COB (J-d(xl/d)sin (J = - X l sin (J+x. cos (J
Eqns (5-2.2) to (5-2.4) can be combined together (see eqn (4-3.8» to give: COB fJ sin fJ 0 x~ Xl -sin (J
X;
x;
cos (J
0
X.
0
1
x.
0
(5-2.5)
Necessarily, exactly the same set of equations can be obtained from an anti-clockwise rotation of fJ about e. of the base vectors e 1 and e 2 , i.e. moving the point clockwise is the same as moving the laboratory axes anti-clockwise. Eqn (5-2.5) can be used to define a matrix D(Cn ) which corresponds to the operation Cn :
~--+---7"'"-e.
= D(Cn )
sin (J
0
-sin (J
cos fJ
O.
0
0
COB
D(C..) =
(6-2.6)
x. ,
x.
X; and
FIG. 5·2.2. Relation between a point and a position veotor.
(5-2.3) (6-2.4)
x; e,
(?-2.2)
x; =
(J
1
(6-2.7)
71
The invel'lle of D(C,,) is easily found to be (see eqn (A.4-2.6»:
D(C,,)-l =
cos 0
-sin 6
0
sin 6
cos 0
0
o
o
1
Similarly we can also obtain: (5-2.8)
and we 8ee that since D(C,,)-l = D(C,,), the matrix D(C..) is orthogonal. As D(C,,) is real, this implies that it will also be unitary: D(C,,)-l= D(C,,)t. It is apparent that D(C,,)-l corresponds to a clockwise rotation by -6 or an anti-clockwise rotation by 6, i.e.
D(C,,)-l =
17
Metrill RlpnIS8ntltioDS
.atrill R....--tatio...
cos 0
-sin 6
0
cost -6)
sine -6)
0
sin 6
cos 0
0
-sine -6)
cos( -6)
0
0
0
1
0
0 =
1
D(~l).
(2) Reflection. In Fig. 5-2.4 the effect on p of plane containing e l and e.(ul . ) is shown. Clearly, x~ =
-Xl
x~ =
x.
&
reflection in the
-1
0
0
Xl
0
1
0
x. x.
, x. ,
:1:.
0
and D(cr..)
=
0
1
-I
0
0
0
1
0
0
0
1
"'23
.
e.
0
1
0
0
0
-1
:1:; =
0
and
=
D(an )
1
0
0
0
-1
0
0
0
1
(5-2.11)
-XI
and
-1
o
o
-1
o o
o
o
-1
D(i) =
(5-2.12)
(4) Rotation-reflection. Consider a rotation by f) (= 21Tln) about the e. base vector, followed by reflection in the 0'11 plane. The components of the point vector p (or, the coordinates of the point P) will be first transformed by the rotation, as in (1), and then these new components (coordinates) will be transformed by the reflection, as in (2). Using matrix notation, these two transformations can be combined into one step (see § 4-3(3» and we get
D(S,,) =
(5-2.9)
1
0
0
0
1
0
o
0
-1
cos 6 -sin
f)
sinf)
0
cos 6
sinO
0
cos 0
0
-sin 0
cos 6
0
o
1
o
o
o
-1
(5-2.13) (5) Identity. This is the 'do nothing' operation, hence:
100 (5-2.10)
X; = X.
and
D(E)
=
0
1
0
= E,
(5-2.14)
001 e:1
~.
0
where ali is the plane containing e, and e l and Un is the plane containing e 1 and e•. (3) Inversion. The effect of inversion i on a vector will be to invert it, consequently:
~ =X. X~
=
D(a n )
1
where E is the identity matrix. All of the above matrices are orthogonal. To summarize, we have found that the effect of any symmetry operation R on a position vector p = :I:lel +x.e.+x.e. can be expressed as: where
Rp =
R(Xte1 +x.e. +x.e.) =
x~el +x;e.+x;e.
(5-2.Hi)
(5-2.16) Fla. 6-2.4. Effect of .... on p.
78
TABLE 5-3.1
and D(R) is a matrix of order three, characteristic of R. Eqn (5-2.16) can also be written in the form x~ =
a
.I
Dkl(R)XI
k
=
1,2,3
GrrYUp tableJor
(5-2.17)
where D"I(R) is the element in the kth row and jth column of matrix D(R) and XI (j = I, 2, 3) are the coordinates of a point or the components of a position vector.
c,
E
E
C.
i
C. I
ab
Go
C, E ab I
-I
D(C.)D(i) =
Now let us consider two specific point groups: (I) ~ ... and (2) ~3v' (I) ~ Oh' A molecule belonging to this point group is planar transCl
"-C=C / / "-H Cl The point group is composed of four symmetry operations: E, C.. i, and O'h and the group table is given in Table 5-3.1. This table shows the effect of combining one operation with another. Following the discu88ion in § 5-2, the matrices which correspond to the four symmetry operations are
~...
t
I
ah
ah
I C. E
ah
Using matrix multiplication as the combining operation, we can construct a group table for these four matrices (Table 5-3.2) e.g.
5-3. Matrix representations fclr~ah and ~3v
H
E
E C. t The order of combining is AB where A is given at the side of the table and B at the top of the table.
1-1
CaHaCl a
79
Matrix Repr...ntatiollB
Matrix R"l'1IS8ntatiDIIB
o o
o -I
0
0
-I
o
0
o o
-1
o o
o
-1
I
0
o o
I
-1
= D(O'h)'
It is apparent from this table that the four matrices form a group, since (a) the product of any two matrices is one of the four, (b) one matrix, D(E) = E, is such that when combined with the
four, it leaves them unchanged, (e) the associative law holds for matrices, (d) each of the four matrices has an inverse which is one of the four, i.e. D(E)-l = D(E), D(Ca)-l = D(Ca), D(i)-l = D(i), and D(O'h)-l = D(O'b)'
Comparison of Tables 5-3.1 and 5-3.2 shows that they are identical in structure (though the elements and combining rules are different) and consequently the matrix group is homomorphic with the point group; we say that the four matrices form a representation of'if. b • (2) 'if a... Ammonia is an example of a molecule belonging to this point group and it has six symmetry operations which obey the group table introduced in Chapter 3 (Table 3-4.1). If we set up base vectors TABLE 5-3.2 GrrYUp tabk Jor the JrYUr matricu in eqn (5-3.1)t
where the base vectors have been chosen such that e a coincides with aa and e 1 a.nd ealie in the a h plane. D(Ca) has been found by replacing /} by TT in eqn (5-2.7).
0
o o
D(E) D(C.} D(I) D(",,)
DeE)
DeC,)
D(I)
D(ab)
D(E) D(C.) D(I) D(ah}
D(C.) D(E) DC",,) D(I)
Del)
D(ao) D(I) D(C.) D(E)
Deab) DeE) DeC.)
t The order of matrix multiplication is AB where .A. is given at the side of the teble and B at the top of the teble.
80
Matrix Representations
Matrix R"'l'8S8IItatioDB
TABLE 5-3.3 Group table for the six matricea in eqns (5-3.2) to (5-3.4)t
top-yin\\"
D{E) D(E) D(..~) D(..;) D(..;) D{C,) D(C:)
", a,-
a, a,_
Fro. 5.3.1. Axes for the rc:sv point group. The origins of eh e •. and e. are at the centre of m88R; a~, 0;, and are perpendicular to the page.
a;
in accordance with Fig. 5-3.1, then the matrices which correspond to E, a~ (reflection in the planc containing e. and e.), C. (rotation about e. with 8 = 2'fT/3), and C: (rotation about e. with (J = 4'fT/3) are D(E)
D(C.l
=
I
0
0
0
1
0,
0
O· I 0
-1/2
"\13/2
-y3/2
-1/2
0
0
D(er;)
=
D(C.)D(a~) =
D(cr;) D(..;) D(C,) D(Cl)
D{..;)
D{C,l
D(Cl)
D(..~) D(E) D(Cl) D(C,) D(G;) D(";)
D(..:) D{C,) D(E) D(C:) D(G~) D(a;)
D(a;) D(Cl) D(C,) D(E) D(a;) D(~)
D(C.) D(a;) D(G;) D(..~) D{Cl) D(E)
D(c;) D(G.) D(G~) D(..;) D(E) D(C,)
t The order ofmatrix multipJioation is AB where A is given at the side of the table and B at the top of the table. Also C: = 0;'.
-v'3/2
0
-1
0
0'
y3/2
-1/2
0
0
1
0
D(S)D(R) = D( T)
0
0
1
0
0
I
1/2
-y3/2
0
-y3/2
-1/2
0
0
0
0
1
0,
0
1
=
0
, and
D(C:) =
-1/2
-y3/2
0
y3/2
-1/2
0
0
0
and
D(~)
D{..;)
-1/2
D(a~)
-I
1
D(O';) = D(C:)D(er~) =
D(E)
D{a~)
These six matrices form a group for which the combining rule is matrix multiplication and the group table is that in Table 5-3.3 (the reader is left to confirm this for himself). Since this table is identical in structure to Table 3-4.1, we say that the six matrices form a representation of
1 (5-3.2) Using the same technique as we did for Sn in § 5-2(4), we can also obtain the matrices D(a;) and D(a;), namely 0
81
0
is given in Appendix A.5-I.
(5-3.3)
1
0
-1/2
y3/2
0
-1
0
0
-y3/2
-11 2
0
0
1
0
1
0
0
I
0
a;
112
0 y 3/ 2
0
y3!2
-1/ 2
0
0
0
1
(5-3.4) FlO.
~-3.2. The effeot of C. and G~ on a position veotor (or point) in the b...... of a
symmetrio tripod.
82
83
Matrix Repr_ntations
6-4. Matrix rep....entations derived from base vecton It is also possible to construct matrix representations by considering the effect that the symmetry operations of a point group have on one or more sets of base vectors. We will consider two cases, both using the ~.T point group as an example: (1) the set of base vectors e1> e., and e l , introduced in § 5-2; (2) three sets of mutually perpendicular base vectors, each located at the foot of a symmetric tripod.
• ei
a;'
e.
:lO"
e,
e,
ei ei = - (sin 30") e. - (COB 30") e. ei=(cos 3O")e,'- (sin 3O")e.
ei=e.
e-i = - el ei=ej eJ,=ea
ei = (sin 30") e, - (cos 30") e, ei = - (cos 30") e. - (sin 30") e.
ea= eJ
FIG. 11-4.2. Relation between original ....d tra.D8f'onned base veetora for C.. e.....d e~ ..... perpendicular to the page.
1\
1
L-
c.
\
a~ • ....d
..;.
1( • \
lB, li-------....
ez
a;.
FlO. 6-4.1. The effect ofCs and on a set of hue vectors in the base of tripod. e •• e~, and e: are perpendicular to the page.
&
symmetrio
(1) In Fig. 5-4.1 we show, as an example, the effect of the symmetry operations C.. a~, and a: on the three base vectors e e., and e l . The operation C. produces new vectors e;, e~, and and" these are transformed to e~, e;, and e; by a~. The operation will, since = a~CI' transform e e., and e. directly into e~, e;, and e;. Therefore, we " there will be matrices which will link these sets of base anticipate that vectors and which will behave like the symmetry operations. If C. transforms e e., and e. to e;, e;, and e;, then we have (see " Fig. 5-4.2) C ae , = e~ = -eJ2-h/3/2)e. C.e. = e~ = (v3/2)e, - e./2 C.e. = ~ = eo
e;
a:
which can be written as
C.e k
=
e~ =
It is important to notice that the way in which we have formed the matrix D(R) is different from the way we did things when we were considering position vectors (cf. eqn (5-2.17»: the components of the new base vectors are used as the columns of D(R) whereas, before, the components of the new position vector were used as the rOW8 of a matrix. This is what is implied by the order of the subscripts on D in eqn (5-4.2): jk rather than kj. Necessarily
a;
ei ~
does 1Wt equal
D(R)
e.
There is nothing particularly subtle a.bout what we have done, it simply ensures (see the proof in Appendix A.5-2) that the matrices so formed will indeed represent the point group. As a general rule, equations involving coordinates or components of a. position vector will be written in the form:
a
~ Dik(C.)ei ,
k
i-I
where D(C.) =
=
-1/2
.y3/2
0
-.y3/2
-1/2
0
0
0
1
1,2,3
(5-4.1) and those involving functions (see § 5-7) or base vectors, in the form: I~ = ~aiJi i
or
The general equation linking e~ and e k will be given by: Rek = e~ =
•
~ Dik(R)ej,
i_I
k
= 1,2,3.
(5-4.2)
In much of the mathematics whioh follows it is important to bear this rule in mind.
Matrix Repr8ll8ntatioDS
Matrix Rllpruentations
84 If
0': transforms e
1,
e •• and e. to o~el
~,
= e;
0,;, and 0,; (llee Fig. 5-4.2) we find
O'~ea = ~ =
e.
a~ea
e~ =
e.
-1
0
0
0
1
0
=
and D(O'~) =
(By the way, the sets of base vectors need not be parallel.) For example, under C a, we have
Cae. Cae. Cae. Cae. C.e. Cae. Cae. Cae. Cae.
-e1
=
001
If O'~ tr&llBforms e" e., and e. to ~. ~, and
O';e. O';e.
=
=
~
e; (see Fig. 5-4.2) we find and we can write
e./2 -( v3/2)e.
e~ =
=
C.e_
D(a;) =
e~
~
~
= e~ =
-e./2-(v3/2)e. (v3/2)e.-e./2 = e; = e. = e; = -e./2-(y'3/2)e. = e!l = (v3/ 2 )e.-eJ2 = e; = e a
=
= e~ =
ei
L•
). e r, , (eolte.----"re. /
,
e...
e
1
"
(e )'
e.
= v 3/2 -y'3/2 -1/2
o o
0
0
0
0
0
-v3/2
-1/2
0
0
0
0
0
0
0
o
o
1
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
-1/2
v 3/ 2
0
0
0
- v3/ 2 -1/2
0
0
0
1
0
0
o o o o o o
I
.
(et1~e;'
'\
a'
~
\.
e;,,/ e;~ I(e~)----I. e~
(5-4.3)
0
e
--!.....
k = 1,2 ... 9
0
e;~(e;l
e
DI_(C.)ei ,
-V3/2
-1/2
v 3/ 2 2 3 - v / -1/2 0
0
1
0
0
0
0
0
0
0
0
0
or, in general. ReI: = e~ =
0
L" 1-'
,, ,, ,,
=
1/2
The reader may confirm that D(O'~)D(Ca) = D(a:). It is always true that if SR = T and D(S), D(R), and D( T) are defined as in eqn (5-4.2), then D(S)D(R) = D( T); this is proved in Appendix A.5-2. That the matrices found in this way from the base vectors are identical with those found from consideration of the position vector ( § 5-2) is proven in Appendix A.5-3. The base vectors e u e., and e. are said to form a basis for a representation of a point group. (2) If we place sets of base vectors at the feet of a symmetric tripod, these too can produce a matrix representation, as seen in Fig. 5-4.3.
(e·
= -e./2-(-.,/3/2)e. = (v3/2)e. -e./2 = e.
and D(C.)
and
= = =
1-1
-(v3/2)e.-e./2
85
e~
(e.)
/ I
,/
,k..ei'
"
e, \
(e~Ve.'
(e; \~---~: \ el'
es'
FI G. 6·4.3. Symmetry operations 011 baae vectors located at the feet of a symmetric tripod. el' e•• eo ete. are perpendicular to the page.
0
DI_(R)e / ,
-1/2
k = 1,2 ... n.
(5-4.4)
Again, notice the ordering of the subscripts in eqn (5-4.4). The proof that the D(R) constructed in this way form a representation of the point group is the same as the proof given in Appendix A.5-2. It is obvious that there are a large number of different matrix representations which can be found by choosing different sets of base vectors. The base vectors chosen are said to form a basi8 for the representation of the point group. As with the symmetry operations acting on a position vector. the matrices formed by considering base vectors will be orthogonal.
II
M.trill Repres....io..
Me.riJl Rlpreslnt.tion.
combination of these n functions, [in physical space e., e., e. or any three non-coplanar vectors, are linearly i'lUlepe'lUlent and any position vector can be expressed in terms of them]. The n linearly-independent functions are said to span the function space and the space is said to be n-dimensional, [e" e., and e. span physical space which is three-dimensional). Put another way. n is the smallest number of functions from which it is pOBBible to produce all the other functions which belong to the function space. If the n linearly-independent functions are chosen such that the scalar product (f.,II) = "'1' i,j = 1,2 ... n, then they are said to be orthonormal (orthogonal and normalized), [e•• e., and e. are unit orthogonal vectors). It is &1ways possible .to create n orthonormal functions from n linearly-independent functions. Furthermore, orthonormal functions are always linearly independent and therefore, by definition, the maximum number of orthonormal functions in an n-dimensiona.l spaoe is n, [in a general n-dimensional vector space, 80 far not discussed, the maximum number of orthogona.l vectors is n; orthogonality being defined by
5-S Function .,Ice Now we come to a totally different method for producing matrix representations of a point group; B method which involves the concept of B lunction space. The word space is used in this oontext in a mathematical sense and should not be oonfused with the more familiar threedimensional physical spa.ce. A function space is a collection or family of mathematical functions which obeys certain rules. These rules are a generalization of those which apply to the family of position vectors in physical space and in order to help in understanding them, the corresponding vector rule will be put in square brackets after each function rule. Consider the set of functions: I" f., .... If they are to belong to a function space. then the following rules and definitions must hold true.
(1) The addition of any two functions must produce a third. function which is &1so a member of the collection of functions, i.e. belonge to the function space [vector addition of any two position vectors P. and P. produces another position vector P.]. (2) The multiplication of any of the functions by B number must produce a function which is also included in the collection [aP1 is &1so a position vector]. (3) The rules in (1) and (2) can be combined together: any function which is B linear combination ofI.. f., ... (i.e. aJ. +aJ. + ...) must also be B member of the collection [a.p. +a.p. +a.p. is also a position vector). (4) The sc&1ar product of any two functions is defined u: (f.,fl)
=
f f:fl d'T,
where integration is over all the variables off. andfl' [the scalar product of two position vectors p and p' is
p.p'
=
(p, p')
=
pp' cos 8,
where 8 is the angle between the two vectors, and if (e., el) = (J'I' i, j = I, 2, 3 and p = x 1e. +xle.+x.e••
(p, p') = x.xi +x.x~+... x"x~ =
basis lunctions.
Now let us consider two examples of a function space. The solutions of the differential equation
d;C:) = -f(x),
0< x
< 2..
I(x) = I.(x),f.(x) • ... , form a function space, since any oombination of them is also a solution of the equation and therefore a member of the collection of functions, i.e.
dl dxl [/,(X) +11(x»
=
=
dl
(p, p') = :l:'.~+x~+:l:'aXa
(5) If n of the functions f .. /I'" are linearly independent (i.e. aJ1 +aJ.+ ... aJ.. = 0 tmlg if a. = a l = ... a" = 0). then any of the other functions of the space can be expressed &8 lIome linear
OJ.
(6) An n-dimensional function space is defined by specifying n mutually orthogona.l, normalized, linearly-independent functions, [e., e., and e. define physical spa.ce]; they are called orthonormal
p' = ~el+X~I+ziel' then
17
. dx. [al.(x» and
dl d· dxl/.(x) + dx.ll(x)
=
-/.(x) -11(x)
-[/.(x)+ll(x)], dl
=
a dx.I.(:l:') = a[ - f.(x» = -[al.(x)]
dl
dxl [aJ.(x) +aJ.(:t:)+ ...]
=
-[a./1(:t:) +aJ.(x) +...].
88
Matrix Repres.u.tions
Matrix Repr.antations
The general solution of this equation is I,(x) = a. cos x+b. sin x. where a. a.nd b, are arbitrary constants. Since cos x and sin x are linearly independent (otl cos X + ot. sin X is only equal to zero for all x if otl = ot. = 0) and orthogonal
O.(af) = a(O.f)
and
O.(J+g) = 0J+0.u.
(5-6.1)
where a is any number and I and g are any two functions of the function space. (2)
(5-6.2)
where f' also belongs to the function space. (3) the correspondence betwoon Rand 0. retains tl:J.e multiplication rules, e.g. if T = SR. then
Or!
=
0s.1
=
0s(O.f).
° ° _°° _°
such that
and any linear combination of degenerate solutions will also be a solution of the equation. We can therefore say that any degenerate set of solutions (corresponding to a given eigenvalue) of Htp = Etp forms a function space. If n of the solutions are linearly independent, then they can be chosen to be orthonormal and used as basis functions to define an n-dimensional function space. If H is the Hamiltonian operator, then the linearly-independent wavefunctions V' = 'I'~. '/';, '" which correspond to E = E., will be basis functions for an n-dimensional function space. As an example, the three p-orbitals P.., PI" and P. of atomic problems are degenerate and orthogonal and when normalized form an orthonormal basis for the three-dimensional function space of p-orbitals. Any p-orbital can be written, with respect to this basis, as
'f':
V' = a..p.. +a.p.+a.p•.
In § 5-9 we use a function space defined by d-orbitals. The six orbitals of benzene are another example of a function space.
level) and,they will be such that (1) they are linear:
(5-6.3)
Necessarily there will be a unit operator 0E (associated with E)
= Et'l't +E.'I'. = Et(V'l +'1'.) H(a'l',) = aH'I', = aEt'l'l = Et(atpl)
H(V't +'1'.) = H'I't +H'I'. and
89
'IT-
5-6. Transformation operators fDlI} Having defined a function space, we are now in the position of being able to introduce a new group, homomorphic (see § 3-4) with a given point group, in which the elements are transformation operators which operate on the functions of some function space. We will denote the transformation operator which corresponds to the symmetry operation R by 0 •. Every 0. will be defined with respect to some particular function space (e.g. the wavefunctions belonging to a given energy
•
£
E
-
•
-
(5-6.4.)
•
and each operator 0. will have an inverse 0;1:
0iIO. = 0.0,,1 and, since 0.-,0.
=
=
0E
(5-6.5)
0E'
Oil
=
0.-,.
(5-6.6)
This set of transformation operators 0. associated with the symmetry operations of a given point group will therefore have a group table which is structurally the same &II the one for the point group. In the next section we show that, if we introduce a coordinate system into the function space chosen for the 0., we can define explicitly a set of 0. satisfying eqns (5-6.1) to (5-6.6). The reader is warned that sinoe the correspondence between Rand 0. is 80 close. many books (incorrectly) do not distinguish between them. 5-7. Asatisfactory set of transformation oparators fDa)
If an n-dimensional function space is defined by the set of linearlyindependent basis functions II' I., ... f., ... . and f. and if these are functions of three Cartesian coordinates Xl' X.' and x.' then we can define a transformation operator 011 (corresponding to the symmetry operation R) by the equations: (O.f,)(x~. x;, x~) = f.(x l • X., x.),
• = 1,2 ... n
(5-7.1)
where a point in physical space with coordinates Xl' X., and x. is moved by the symmetry operation R to the location x~, x~, and x~. In other words, the new function .f. assigns the value of the old function I. at
°
98
MaUill RepreaentatiDDS
Matrill RepresentatiD. .
z., and x. to the position x;. x;. and z;. Knowledge of f •. x" z., x., x;, x;, and x; allows one to find the form of O.f. (see § 5-9). With this
Xl>
definition, it can be shown that the 0. are linear and that if T = SR, then Or = 0sO. (see Appendix A.5-4). The requirement that .f. produces a function belonging to the given function space (see eqn (5-6.2» will bc met by the proper choice of function space (soo § 5-8). If this is the case, however, we can write. for an n-dimensional function space de£ned by the linearly-independent basis functionsf"f•...• andf..,
°
k
=
1.2 ... n
(5-7.2)
(notice the order of subscripting on D), i.e. the function ORA, if it belongs to the function space, must be some linear combination of that space's basis functions. What is of supreme importance for us, is the fact that the nxn matrices D(R) in eqn (5-7.2) will multiply in the same fashion as the symmetry operations: if T = SR, then D(T) = D(S)D(R). (see Appendix A.5-5). The D(R) so found. therefore form an n-dimensional representation of both the point group and the group of transformation operators 0., and the functions f., f., ... andf.. are said to be a basis for the representation. The operators just described will leave the scalar product of two functions of the function space unchanged: (ORf•• O,l.!i) = (f•• f.)· Suoh operators are said to be unitary and they can always be represented by unitary matrices (see § 6-4). The proof that the 0. are unitary follows from considering
(f.,f.) =
f ft(P)U P )
where P is a general point with coordinates
d'Tp, Xl'
x.,
and x. and
is the volume element at P. Now a symmetry operation R will move P to a point P' with coordinates (x;, x;, and x;) and the volume element to an equal volume element d'T1" = dz; dz; dz; situated at P'. Furthermore, the operator O. is defined such that f.(P) = (O.f.)(P') and f.(P) = (O.f.)(P'). Hence
f ft(P)U P )
d'Tp =
f [(OJ.)(P')]*[(OJ.)(P')] f [(O.f.)(P)]*[(OJ.)(P)] d'Tp'
=
since the range of integration extends over all points P or P'. Therefore,
(f•• f.) = (O.f., OJ.), (5-7.3) that is the scalar product is left unchanged. Our definition of 0. applied to functions of the coordinates Xl' x •• and z. of a point in physical space. but it can be generalized to apply to functions of any number of variables, as long as we know how those variables change under the symmetry operations. For example, if we let X stand for a complete specification of the coordinates of all the electrons (or all the nuclei) of some molecule, i.e. X = zlu , Z~l) X~l») •• ~ zlt'l), x~"), z~·) J
for n electrons (nuclei). and if this specification becomes X' under the symmetry operation R, then we can de£ne 0. by
0J(X') =f(X),
(5-7.4)
where f is a function of all the electronio (nuclear) coordinates. The theorems in Appendices A.5-4 and A.5-5 also hold true for this more general definition.
5-8. A cautiDn We have soon in the previous seotion that the definition of a set of O.s is intimately bound up with some ohoice of function space. The reader is cautioned. however, that not all function spaces can be used to define 0.a appropriate for a given point group. For example. the functions cos x.' sin Zl' cos x., and sin x. do not form a basis for a representation of the ~'T (symmetric tripod) point group; Xl and x. are the coordinates introduced before (see Fig. 5-2.2). The four funotions do define a function space, since the general function is f(z" z.) = a.cos x. +a.sin Z1 +a.cos x. +a~sin x. and addition of any two such functions will produoe a third which belongs to the space. as does a number times any such function. However, if we consider the C. operation of ~.T (clockwise rotation by 21T/3 about e.). then z~ = (-Xl + V3z.)/2 z; = (- V3X1 -z.)/2 z~ = 2:.
or, inverting,
X.
= (-~i -y'3zi)/2
x. = d'Tp
91
x.
(y'3z~ -x~)/2
=~
92
M.trix R.pnll8llt.tioDS
M.trix R.p..... ntlltion.
and selecting the basis function coo Xl' we find 0c,(cos xi)
=
cos x,
= cos
(definition of 0c.)
!l -x~ -
y3x~)
use the d-orbital function space to determine a representation of the 'ifa• point group. We will define the five d-orbitals by the equations:
d, = (x~ -x:)/2. d a = x,x., d. = X,X.' d 4 = xaXo' d s = (3x: -rl )/(2y3),
or, equivalently, 0c,(cos x,)
=
cos i( -x,- y3zl ).
Using the notation 1,= cos X" I. = sin z" I. = cos x •• and I. = sin x. it is clear that • 0c.!, ~ Di,(C.)/i· i
"*
Conoequently, this function space does not provide a basis for a representation of the 'if•• point group. 5-9. An examp'e of determining OilS .nd OrRis for the 'if•• point group using th. d-orbit.' function spO&. A set of five real d-orbitals defines a function space. In opherical polar coordinates r, 0, and ,p. they consist of a common radial function times 0. combination of spherical harmonics Y;"(O, ,p), 1 = 2 and m = 0, ±l, ±2. The combinations of the five spherical harmonics are chosen such that the orbitals are real. It is a well known property of spherical harmonics that if we shift the point r, 0, and rP to r', 0', and "",t the resulting Y;"(O', ,p') can be expressed in terms of a linear combination of aU the Y;"'(O,
')
= =
or or
where a common constant and radial function have been omitted (they are not necessary for the purposes of our discussion). and x" x., and x. are the Cartesian coordinates of a point with respect to the base vectors e" e l , and e. (see Fig. 5-3.1) and r is the distance of the point z" z., and x. from the origin (centre of maS8). The more familiar notation is d, = d,..->", d a = <1"", d. = d,... d 4 = d..., d& = d,•. We will first consider the operation of clockwise rotation by an angle 0 about the e. axio and denote the corresponding transformation operator and matrix by 0, and D(O), respectively. The relations between the coordinates of a point before (z" x •• and z.) and after (x;, x;, and x;) rotation are (see eqn (5-2.5» z~ =
z;
=
(cos O)x, + (sin O)x., -(sin O)x, +(cos O)x.,
zi
=
z.'
or, taking the inverse:
±l, ... ±l, and therefore. 0IlY;"(O',
83
(definition of Oil) a linear combination of Y;"'(O', ,p'). m'
x, x.
Y;"(O,,p)
OilY;" = a linear combination of
=
0, ±l •... ±l
&
i-'
(sin O)zi +(008
O)x~.
From eqn (5-7.1), we have 0, d,(xi,
z., 2'~)
= d,(x" Zo. X.) = (z~-z:)/2
where d" d., ... d. are real d-orbitals. The D(R) will form a representation of the point group. It is apparent that as the above steps do not specify a particular point group, the d-orbitals (or any set of spherical harmonics of the same l value) can be used as the basis for a representation of any point group. With this knowledge, we can now t Note that in general, symmetry operations do not change the distanoe r of a point Crom the origin if the latter is at the centre ofmaas. We can therefore restriot ourselves to shiite for which r is oonsta.nt.
=
~3 = z~.
Yl"',
011 d, = ~ DI«R) d i •
= (cos O)z~ -(sin O)z;,
= [{(COS O)x~ -(sin O)xiP-{(sin O)x~ + (cos 0)zi}"]/2 = (cos 20)(xt -z~·)/2 - (sin 20)z~zi =
(cos 20)
d,(z~. x~.
z;) -(sin 20) d.(x{, xi, ziJ
or. dropping the parameters (they are now the same on both sides of the equation):
O:d,
=
(cos 20) d, -(sin 20) dl'
94
Matrix R.p~tio ...
In the same fashion, we can also obtain
or the inverse
0. d. = (sin 28) d 1 +(cos 20) d. 0. d. = (cos 0) d.-(sin 0) d. 0. d. = (sin 8) d. +(cos 0) d. O.d, = d,.
:1:1
!
D(O)
sin 28
0
0
0
-sin 20
cos 20
0
0
0
o
o o o
cos 0
sin 0
0
-sin 8
cos 0
0
o
0
1
o o
D(C.)
=
and, if 0 = 471/3,
D(C:)
=
V3/2
-v3/ 2 -1/2
0
0
-1/2
v 3/ 2
0
0
0
-V3/2
-1/2
0
0
0
0
0
1
0
0
0
0
0
0
-1/2 - v3/ 2
v 3/ 2
0
0
0
-1/2
0
0
0
0
0
0
0
-1/2 v3/2
0
0
0
Now, let us consider reflection in the have from eqn (5-3.3)
0';
0
o
1
-i(x~'-:l:~')/2-V3/2:1:~:l:i
=
-! dl(X~, z~, :I:~) -V3/2 d.(:I:~, xi, x~)
- v3/ 2 0 -1/2
0
0
1
X 1:1:.
= (!x~ -V3/2xiH -V3/2x~-i:l:~) =
-
Therefore, 0
•
0
and and, likewise;
(5-9.1)
d,(X~, x~,
xi) +! do(x~, x~, xi), etc.
-1 dl(X~, xi, X;)-V3/2 do(x;, x~, x~) -i d 1 -v3/2 d.
=
Oov' d 1 =
O"v' d. = -V3/2 d 1 +1 do, 0"v • d. = ! d.-v3/2 d., 0" v . d. = -v3/2 d.-i d., O"v' d, = d,.
From these equations we obtain
D(a;) =
(5-9.2)
V3/2
dl(:I:~, x~, x;) = d1(x t , x.' x.)
v
(once again, notice the subscripting in Du(O». Hence, if () = 271/3, -1/2
-112
o
=
do(x!> xo' :1:.) =
Du(O) d"
cos 28
=
-V3/2
= {(ix~ -V3/2z~)0-( -V3/2:1:~ -iX~)0}/2
J~l
we get
0
d 1 (Zt. x •• z.) = (x~ - :1::)/2
,
=
-V3/2
Hence,
These equations define the operator 0., that is to say, they specify the new functions created by applying the operator to each of the five basis functions. Introducing the equation 0. d.
1/2
15
-1/2 - v3/ 2
- v3/ 2 1/2
0
0
0
0
0
0
0
0 0 1
0
1/2
0
0
- v3/ 2
- v3/ 2 -112
0
0
0
0
(5-9.3)
The other symmetry operations of 'ifIv give
plane (see Fig. 5-4.1). We D(cr~) =
1
0
0
0
0
0
-1
0
0
0
0
0
-1
0
0
0
0
0
1
0
0
0
0
0
1
(5-9.4)
H
Matrix Representations
D(CJ~) =
Metrix Rep-Utions
-1{2
y'3{2
0
0
0
y'3{2
1{2
0
0
0
°
0
1/ 2
v'3J2
0
0
y'3/2
-1/2
°
0
°
0
0
1
and
1
D(E) =
0
0
0
1
0
0
0
0
0
0
represents a rectangular array of zeros, if D( T) = D(S)D(R), then the matrix multiplication rule leads to the equations
(0-9.5)
0
0
0
1
0
0
0
1
0
°
0
1
and
0
D'(T) = D'(S)D'(R) DW ( T) = DW(S)DW(R)
(5-9.7)
and consequently, the matrices D'(R) etc. and DW(R) etc. form two lower-dimensional representations. Another feature of block form matrices is that if a matrix A is in the block form: [0] [AJ [0]
0
°
117
(5-9.6)
The six 5 X 5 matrices specified by eqns (5-9.1) to (5-9.6) form a five-dimensional representation of '(/a,,; the basis functions for this representation are the Bet of five d-orbitals. It has probably not escaped the reader's notice that each of the above matrices has the same structure, that is to say all their non-vanishing elements occur in the same square blocks along the diagonal:
[0]
[A 0]
[0]
[0]
[0]
[All
then det(A)= det(A1)det(AI)det(A.) ... and if det(A) = 0, then either det(A1) = 0 or or
det(A.) =
°
det(A.) = 0
(5-9.8)
etc.
We will return to the topic of block form matrices in the next chapter; it is the foundation of most of the theorems which follow.
x
X
0
0
0
X
X
0
0
0
0
0
X
}[
0
5-10. Determinants aa representations
0
0
X
X
0
0
0
0
0
X
We might note, in paslling that, since det(AB) = det(A)det(B) (see Appendix A.4-6), the determinants of any set of matrices which form a representation will themselves act as a one-dimensionalrepresentation. That is, if T = SR and D(T) = D(S)D(R) then
What is not 80 obvioU8 is that since the block structure is identical for every symmetry operation, the individual blocks themselves, written as matrices, form lower dimensional representations. Take, for example, the block form matrices D(T) =
II D'(T) [0]
[0]
I
DW(T) , .
D(S) =
I D'(S) [0]
[0] DW(S) ,
II
D(R) =
II D'(R) [0]
[0] DW(R) ,
I
where D'(T), D'(S), D'(R) are square arrays of dimension nxn and DW(T), DW(S), DW(R) are square arrays of dimension mxm and [0]
det{D(T)) = det{D(S)D(R)} = det{D(S)}det{D(R)},
and the group of numbers (or 1 X 1 matrices) det {D(R)} etc. is homomorphic with the group of matrices. 5-11. Summery In this chapter we have shown that there are very many different sets of matrioes which behave like the symmetry operations of a given point group. We have constructed these so-called representations by considering the aotion of the symmetry operations on a position veotor or on any number of base vectors. Alternatively, we have found that we can find tra.naformation operators O. which are homomorphic with the symmetry operations and that from these we can construct
Mattix Repl1lllentlltions
98
99
representations by considering the actions of the O. on functions belonging to some function space. In this way, the p-orbita.Ia, etc. form a basis of representation for any point group, the six .".-orbitals of benzene form a basis of representation for the 9i.b point group, (see Chapter 10) etc. Our next task will be to try and organize, reduce and claaaify the plethora of representations which we can now create. We will try to eliminate from discussion those which are, in a certain sense, equivalent and those which can, in a certain sense, be 'broken down' into simpler (lower order) representations. Where we have now got to and where we are going has a.lready been summarized in Fig. 5-1.1. Appendices
and, because of the matrix multiplication rule, this leads to the desired result D( T) = D(S)D(R). A.&-Z. Proof thllt the mattica constr1lCtlld in eqn 5-4.Z (or 8IIn &-4.4) fonn a repnll8lltlltion of the point group
..
Re~ = ~DI~(R)e,
I: = 1. 2, ... ,71
Sel = ~
j = 1,2 ... ,n
Te~ = ~ D.~(T)e,
I: = 1,2, ... ,71
. Dt/(S)e, ._1 .
1_1
'_1
SRe~ =
A.5-'. ProDf thllt. if tha symmetry Dperlltions R. S. and 10f e point group Dbey the rallltiDn 1=SR. the matrices OrR). O(S). lind 0(1) found by tha considaratiDn of the effect of R. S. lind T Dn a position vector (or point). obey tha relation 0 (1)=0(S)0(R)
Let
p Rp = p'
=
a.nd, if T
xlel+x2e.+x.e.,
=
D/~(R)e/J (the symmetry operators are linear, see eqn (2-2.2»
.
..
I-I
<-I
i
i-I
[iD
I:
~_l
=
.
= SR,
D",,(T) = ~Di/(S)D~(R) f-l
z; = ~• D.i(R)zl
i
zZ
=
= 1,2,3
(A.5-U)
1-1
•
xZ = ~ D~(S}x;
'_I 3
a.]
p' = Rp = zIR.. +x.Re.+ZsRe.. a e~ = Re" = ~ DI,,(R)el 1: = 1,2,
k = 1,2,3
D ..(S)Djf(R)
a
XJ
(A.5-l.2) a
(A.5.3.I)
a
a
p' = Zt~ Dll(R)ef+"'.~ Dla(R)e/+xa~Dla(R)el I-I I-I I-I
1: = 1,2,3.
1'_1
D~f(T) = ~ Dk«S)Djf(R)
a.
I-I
=
Comparing eqns (A.5-l.I) and (A.5.1.2), we have i-I
= X1el+X.e.+x.e.,
1-1
~ [~
1-1
p
•
1-1
=
Proo' that the matrices derived fro.. e position vector an tha seRla 8S those darived fro.. a siogll sat of IIlsa V8CtOrs Consider
I: = 1,2,3
~ D ••(S) ~ D
=
IIG
D(T) = D(S)D(R}.
= 1,2. at A.5-3.
k
(A.5-2.2)
i = 1,2, ...• n I: = 1,2•... ,71
but this is just the matrix multiplication rule,
I-I
• ~ D~f(T)ZI
1,2, ..., n.
If SR = T, eqn (A.5.2.1) and (A.5-2.2) must be identical and therefore
x;e 1 +x;ea+x;e 21
(see Fig. 5.3.2), then, from eqn (5-2.17)
t
f-I
(A.5.2.I)
= ~DI~(R) ~ Di/(S)e.
= z~el+x:ie2+z.e.,
Sp' = p" =
sri
(N.B. SUbscript order)
k = 1,2,3 j = 1.2,3
Di;I (R) 18 tho ma.trix element in tho ith row and .ith ~olu:rnn of matrix D(B}.
a.nd hence
[i
D lI(R)xl]el
1_1
+ [i D.I(R)xl]e.+ [i Dal(R)ZI]" ~-1
1_1
= xiel+z~eJ+z~e.
z; =
a
~ D'I(R)XI
,-1
,=
1,2,3
101
Matrix R.p....ntlItlon.
or changing subscripts x~ =
101
• 2D.s(R)xs
1:: = 1,2.3.
(A.5.3.2)
1_1
Consequently, the same m80trix D(R) defines x~ through eqn (A-5-3.2) (or eqn (5-2.17)) 80nd e~ through eqn (A.5-S.I) (or eqn (5.4.2)).
A.5-5. Praof that the mltriCls derivld from 01/ form I of the point group
We must show that, if T = SR (or, equivalently, Or = OsO., see the previous appendix), then D(T) = D(S)D(R).
A.&-4. Proof thot the oplrBtors D. a... (1) Iin.ar, (2) homolllorphic with the aymmBtry oplII'ltions R (1)
(a) UliB 80 function,
II
Osll =
a number and g = aI, then
...p....entation
i
,-1
Dls(S)!.
(ORl1)(x~, x~, x~) = g(Xt, XI' x.) = a/(x 1 , XI' x.)
= a(OI/f)(x}, XI'
This equation must hold for any point
(x~,
xi)·
x;, ~), hence (Os is linear, see the previous appendix)
(A.5-4.I)
Ol/(a/) = a(OR/).
(b) Consider two functions I and g and let h =/+g, then
= ~lDs.(R){~lD,MJ.~}.
xa, x a) = I(xl' XI' x.) (ORg)(X}, xa. xa) = g(Xl' x.' x.) o RE/(xi, x.;, xaJ+g(zt. XI' xi)] (Ol/f)(x;,
and
= (O.qh)(x~. x';.
xi)
Ths.t ill,
= h(Xt, XI' Xa)
= I(xl' XI' x.)+g(x1 , x •• xa)
= (0./)(x1•
xa, xIJ+(ORl1)(x~,
i {i D,s(S)Df.(R)}/, . DIl,(T) = 2 D /I(S)D .(R)
Or!. =
therefore,
1_1
S-l
S-l
x., XI)'
s
w~ch is the m80trix multiplication rule, therefore D(T) = D(S)D(R), the desrred result.
Since this equation holds for any point xi, xi, xi, we find (A.5-4.2)
(2) Let R transform the point (Xl' X•• x.) to (X;, xi. xi), S the point (x~, xi, xi) to (x~, x;, x;) and T(= SR) the point (x1> XI' x.) to (x~, x;, ",~). Then, if I 80nd g s.re functions belonging to the function space,
PROBLEMS 5.1. Consider the following planar symmetric figure_
(A.5.4.3)
and Now let g =
(OsU)(~,
OJ,
x;, x;) =
g(x~.
Xa, xi).
(A.5-4.4)
then
g(X{,
xa. xi) =
(O./)(x}. ~. XI) = f(x1• XI' x a)
(a) Determine the distinct 8ymmetry operations which take it into itBelf;
construct the group multiplication table for theae operations, and identify the point group to which this figure belonga. (b) Find a aet of two·dimensional matrices which are in one-to-one corre8pondence with the above 8ymmetry operationa. and verify that they have the same group multiplication table &8 the 8ymmetry operation8.
and therefore, going back to eqn (A.5-4.4), (OSORj)(X;. x;. x;) = l(x1 , XI' xa)'
Comparing this lallt equation with eqn (A.5.4.S), we have the desired rellult (A.5-4.5)
5.2. The table below giVeB the effectB of the transformation operatorB 01/ for the 8ymmetry operations R of the point group !JJ. on four functions ft, I., la.
Matrix IlepreuntllliDna
102
andJ•. Construct a four·diInensional representation of St•• E c~. R= C. c: c. C;"
I. I. I. I.
J1 J. I. I.
J. J. I.
II
J. 11 I. I.
J. I. 11 I.
-J. -I. -I. -I.
-J. -11 -I. -I.
C;"
C;.
-J. -I. -I. -I.
-J. -I. -11 -I.
5.3. Consider a set of base vectors located on the nuclei of the molecule 80. in the figure below (ea. e•• eo are perpendicular to the page).
6. Equivalent and reducible representations
88
Construct a nine·diInensional matrix representation for the point group to which 80. belonga. 5.4. For the point group t*.,,: (a) construct a three-diInensional matrix representation using three reaJ p-orbitals 88 basis functions, (b) construct a five-dimensional matrix representation using five reaJ dorbitals as basis functions. 5.5. Consider the planar trivinylmethyl radiea.l with Beven ...-orbitals located shown below:
88
Ulling these ...-orbitals 88 basis functions, construct a Beven-dimensional representation of the 'Ca point group.
6-1. IntrDductiDII HA VING spent a considerable effort in creating many different matrix representations for the point groups we now, ironic as it may seem, devote an equal effort to eliminating many of them from further consideration. We do this in two ways. In the first place, we consider those representations which are produced by the same transformation operators 0. and the same function space but with different choices of basis functions describing that space to be equivalent. We will see that any pair of such equivalent representations have corresponding matrices which are linked by a similarity transformation (see § 4-5). As it will always be possible to find a set of basis functions which produce unitary matrices (a unitary reprflaentation). convenience dictates that we choose such a set for producing a representation which is typical of the other equivalent ones. In the second place. we restrict our discussion to those representations which are composed of matrices which cannot be simultaneously broken down (reduced) by a similarity transformation into block form (e.g. the matrices in § 5-9). It will be for these irreducible representatuma that (in the next chapter) we will be able to prove a number of far reaching theorems. one of the most important of which is the theorem that the number of non-equivalent irreducible representations is equal to the number of classes in the point group. So that, for example. for the point group 'fl'av. which has three classes, rather than dealing with an infinite number of representations we will have only the three which are non-equivalent and irreducible to worry about. Also in the next chapter we will show how to obtain. with the least amount of work, the essential information concerning the non-equivalent irreducible representations which exist for any point group. Irreducible representations are important to the chemist since they provide directly a great deal of information about the nature of vibrational and electronic wavefunctions. 6-2. Equivalent representations
Let ns consider a particular n-dimensional function space, that is one which requires n linearly-independent baaisluncti0'n8 to specify any
1 DC
Equival.nt .nd Reducibl. R.p.....ntetion.
EqulV8l...t end Reducible ..."........io..
function belonging to it. There will be many possible different setll of linearly-independent functions which can act as a basis for this space but we will take just the following two:
and I: = 1,2•... n j
and obtain
/1'/" "'/n e.g. if we were considering the d-orbital function space, the basis functions could be a set of five real d-orbitals or a set of five complex d-orbitals. the latter being just combinations of the former (and vice versa) and both could be used to define any function belonging to the d-orbital space. Since the 9 functions are a basis for the space, any / function can be written &8 a linear combination of 9 functions:
10&
=
1. 2•... n
AB=E.
Therefore the matrices A and B are related to each other in the following way: B = A-I (6-2.4) and A = B-1
Now let us consider the transformation operator OR (corresponding to the symmetry operation R) which is defined by the equation which we have had before: (6-2.6)
1:=1.2....
710
(6-2.1)
(note the subscripting on the coefficient A/k ). and any 9 function can be written as a linear combination ofI functions: j
=
1,2, ... n.
where %1' Z., z. and z~. z~, z; are the Cartesian coordinates of a point before and after the application of R. respectively. Let us assume that 0. in conjunction with our chosen function space produces matrices which represent 0. and R. i.e.
(6-2.2)
"
OJ, = 2.D{..(R)/J:
(6-2.6)
2" Dfi(R)g•.
(6-2.7)
k_1
or
We can combine eqns (6-2.1) and (6-2.2) together to give
i
i
I. = i_I A/k ( 1'_1 B;d.)
0.91 =
i (i B,/A/.)/,
=
1'_1
1-1
J-l
and since the I's are linearly independent, this equation implies that
i = 1,2.... n I: = 1. 2, ...
since then:
I.
=
2" 6 lkl.
1_1
=
(6-2.3)
'/10
A·
The left hand side of eqn (6-2.3) is the element in the ith row and kth column of the matrix formed by multiplying matrix B (composed of the elementll BiJ) by matrix A (composed of the elements A.i ). the right hand side is the i, k element of the identity matrix E. hence
BA We can also write
=
E.
The coefficientll IX.(R) will form the matrix U(R) which represents R in the I basis and the coefficients D~i(R) will form the matrix D-(R) which representll R in the 9 basis. We can readily obtain an alternative equation to eqn (6-2.7) in the following way: 0dl =
oll(i Bill.) ._1
(from eqn (6-2.2)) (0. is linear)
(from eqn (6-2.6)) (from eqn (6-2.1))
1 06
Equivalent Inti Rlducible Reprwentatioa
Equivlllni Inti Reduc:ibll Rllpr_nlltioa
Comparing this result with eqn (6-2.7) and recalling the rule for the product of three matrices eqn (4-3.6), we have: D~(R) =
or
. " 2 2 A~in(R)B<J
A
1/"';2-
-i/"';2
0
1/"';2
i/"';2
0
0
0
I
"'1~_1
Do(R) = B-IDf(R)B
(6-2.8)
Df(R) = A-IDO(R)A.
(6-2.9)
Consequently, the matrices in the g basis representation are related to those in the I basis representation simply by a similarity transformation (see § 4-5); the two representations are said to be equivalent. It is clear that a change of basis for a given function space does not affect the multiplication rules, Le. if Df(SR) = Df(S)Df(R)
then
hence, using the notation of the previous section
DO(R) = ADf(R)B
or, since A = B-1. or
Do(SR) = B-IDf(SR)B = B-1Df(S)Df(R)B
1/"';2
0
i/"';2
-i/"';2
0
0
0
I
where F(r) is a function of the radial distance r, and is common to all three orbitals. If O. is the transformation operator whieh corresponds to a clockwise rotation by an angle 0 about the e. axis (which defines the coordinate z.) and if this rotation takes the point at z .. z •• z. to z;, z;. z;. then (see eqn (5-2.5» z~
= D·(S)D·(R).
z~
D'(R)
B=
1/"';2
and it can be verified that the equation AB = E is obeyed. The three real p-orbitals may be written in terms of a set of Cartesian coordinates Zl' Z•• and Z. as PI = F(r)zl P. = F(r)z. P. = F(r)z.
= B-IDf(S)BB-IDf(R)B
To summarize. the different sets of matrices that can be used to represent the operators OIl in a given space are different realizations of what is really the same representation. Two representations of a point group are equivalent if matrices B and B-1 exist for which
107
+
= (cos O)ZI (sin O)z. = -(sin O)ZI +(C08 O)z.
2:~ = ::1:.
or, taking the inverse Zl = (cos O)zi-(sin O)z; Z.= (sin IJ)z~ + (cos O)z;
= B-IDf(R)B
for every operation R of the point group. Conversely, making the same similarity transformation on all matrices of a given representation is equivalent to simply changing the basis of the chosen function space.
Hence O.Pl(Z~. z~, z;) = Pl(Zl> Z.' z.)
(definition of 0.) F(r):z:1 = F(r){(cos O):z:i -(sin IJ):z:~} =
= (cos 8)Pl(Z~, z~, z;)-(sin IJ)Pa(zi. z~, x;)
B-3. All IxalDple of equivalant reprl..nlltions
As an example of equivalent representations, we will consider the p-orbita.l function space. This space may be described by three real p-orbitals: Pl' P., PI (commonly written as p". P., P.) and we will call this the I basis. Alternatively, we may take three complex p-orbitals: P;. p;, p; (commonly written as Pl' P-l> Po) and we will call this the g basis. These two sets offunctioDs are related by the equations: Pi = (pi+p~)/"';2 P. = -i(p~ -p~)/"';2 p.=pi
Pi
= (Pl+ip.)/"';2 ~ = (Pl-ip.l/"';2 P;=P.
or,
O.Pl = (cos O)Pl -(sin 8)p.
and by carrying out similar steps for P. and P., we obtain
O.P~ =
• D:~(O)Pi
2
i-I
where
cosO Df(O) =
-sinfJ 0
k = I, 2, 3
sin fJ
0
fJ
0
COB
0
I
(6-3.1)
108
Equivelent end Reducible RlpraMtations
Equivelnt Bnd Reducible R.p....ent.tiDIIS
The matrix which corresponds to ])1(8) in the equivalent representation using the g basis p;, p;, p~, D"(O), is given by ~(O)
= B-t])l(O)B = 1/v'2 -i/v'2 1/v'2 i/v'2
A])I(O)B cos
0
-sin
001 1/v'2
-i/v'2
0
i/v'2
0
0
0
I
0
0
0
e-I'
0
0
0
I
0
1/v'2
1/v'2
0
0 cos 8 0
i/v'2
-i/v'2
0
0
o
I
sin 8
0
1/v'2
eI'
°
0
I
0 l
e '/v'2 l
ie '/v'2
I
e- '/v'2
0
-ie-l'1v'2
0
o
1
0
101
TA.BLE 6-3.1 Equivalent representatiDn8 of 'If'v using the p-orbital function spacet
Opera#<m
E
c: (6-3.2)
.
a.
Or, alternatively, we may obtain this same matrix by considering O,p;(x;. x~. x~)
=
p;(xt
•
XI'
x.)
k
=
I, 2, 3
and carrying out the Bame steps for the complex basis functions ~, P;. p; as we did for the real basis functions Pt. P., P.· In Table 6-3.1 we show the matrices for a.II of the operations of the 'If.v point group using both real and complex p-orbitals as basis functions. For the operations C I and C: we have simply replaced by 2'1T/3 and 4'1T/3 respectively in both eqn (6-3.1) and eqn (6-3.2). The matrices for the reflection operations have been obtained in a fashion similar to that used for the rotations. In carrying out these steps it has been assumed that Pt' Pa, and P. lie along the vectors e t • e a, and e., respectively (see Fig. 5-3.1). For obvious reasons the matrix representation in the real basis is identical to the one given in § 5-3(2) and, further, the reader may verify for himself that the matrices using the complex basis obey the 'If.v group table (Table 3-4.1).
°
1-4. Unitery r.pr_tations If we have a number of equivalent representations of a particular point group, it is useful to choose just one of them as a prototype for aU the others. It makes sense that the one we choose for this role has matrices which are unitary, since unitary matrices are much easier to handle and manipulate than non-unita.ry matrices. The reader will recall (Appendix A.4-I(g» that a unitary matrix is defined by A - t = A·. Just as there are two ways of interpreting equivalent representations
to = 000(."./3) =
I, s
~ sin(."./3) =
v'3/2,
£
= exp(2.".i/3).
(change of basis functions or a similarity transformation on the matrices), so there are two ways of proving that it is always possible to find a unitary representation which is equivalent to any given representation. If we choose our basis functions for a particular function space to be orthonormal (orthogonal and normalized) i.e. (f,.fi) = f f:fl d-r = ~/i' then, since the transformation operators are unitary (§ 5-7), the representation created will consist of unitary matrices. This is proved in Appendix A.6-I. It should be stated that it is always possible to find an orthonormal basis and one way, the Schmidt orthogonalization process, is given in Appendix A.6-2. Alternatively, we can prove that there is always a similarity transformation which will transform simultaneously aU of the matrices of a representation into unitary matrices. This is proved in Appendix A.6-3. From now on therefore it will be no restriction to consider. if we wish to, only unitary representations.
110
Equivelent lind Rllducible R.pl'IIS8nt11tions
EquivlIl.t lind Reduciblll Repre8.tlltiou
6-5. Reducible rllp_tations It is convenient at this stage to introduce the symbol commonly used for a matrix representation, namely r. Different representations for a point group can then be distinguished by a superscript on this symbol, for example the representation in the f basis in § 6-2 could be symbolized by r' and that in the g basis by r·. It is important to understand that r is not a symbol for a single matrix but for the whole set of matrices which constitute the representation. Suppose that r is an n-dimensional representation of a group of transformation operators O. acting on the functions of an n-dimensional function space and that we have basis functions 11' f., ..·, f .. with the property that the first m (m < n) are transformed among themselves for all O. (e.g. in § 6-3, the p-orbitals PI and PI were transformed among themselves by all OR and so m == 2 for this case):
and (6-5.3), [Q] must be zero and D(R)
== II
[Q] D'(R)
I
(6-5.1)
where DI(R) is am xm block of elements, D"(R) is a (n-m) X (n-m) block of elements, [Q] is a m x (n-m) block of elements and [0] stands for a (n -m) x m block of zeros. If the basisfl,f.... f .. is chosen to be orthonormal, then the matrices D(R) will be unitary [D(R)-1 == D(R)'] and since D(R-1)D(R)
we have
D(R-l)
==
D(R-IR)
==
D(E)
== D(R)-1 ==
==
E
(6-5.2)
D(R)' DI(R)'
==
II
[Q]'
II
(6-5.4)
[0] D'(R)'
When D(R) is of block form like this, it is said to be fully reducible. If there is a similarity transformation (or, what is the same thing, a change of basis) such that all the matrices in some representation r are brought into identical block form, then r is said to be a reducible representation. If there is not, then r is said to be an irreducible repreBentation. As we have stated before (eqn (5-9.7», the lower dimensional matrices formed from the blocks can themselves form a representation of the point group. If, for example, D(R) are matrices for the representation r and if a matrix A exists such that
==
DI(R)
[0]
[0]
[0]
D'(R)
[0]
[0]
[0]
D3(R)
for all R,
where [0] stands for rectangular arrays of zeros, then the matrices Dl(R), D"(R), and D'(R) form, if they are different and non-equivalent, three new and different representations rt, r', and r" for the point group. We write this symbolically as:
The matrices will then all take the form:
D(R)
== II DI(R) [0]
A-ID(R)A
DI(R) [0]
111
[0] D'(R)"
II
(6-5.3)
As R-1 is one of the operations of the point group. D(R-I) must also have the form ofeqn (6-5.1) and consequently, comparing eqns (6-5.1)
r == r
l
E9
r' E9 r".
This equation is a highly abbreviated version of what we have just done and must be interpreted with care. The symbol fl) does not mean addition and the equation should be read as: 'the representation r can be reduced through a similarity transformation to three representations r l , r', and r",' It is usual to take any reduction that can be carried out as far as possible, that is to reduce r to irreducible representations. Quite often the same or an equivalent irreducible representation will occur more than once, we will then write
r == a1 r 1 6> a.r·... == ~ a.r·
. r'
(6-5.5)
where a. is the number of times or its equivalent occurs in r and the r' are non-equivalent and irreducible representations. Consider the 'WI. point group and the transformation operators O. for the d-orbital function space. If we do not choose our five linearlyindependent basis functions for this space with any particular care, we
112
Equlva'.nt and RHuel.'. R........nt.tlo..
will produce a five-dimensionaJ representation r for 'ifsv • the matrices of which &re not in block form. If. however. we carry out the similarity transformation on each matrix which corresponds to ohanging the basis funotions to those of § 5-9. we will obtain the matrices of eqns (5-9.1) to (5-9.6), whioh are in the same block form and r will have been
TABLE
Equiv,'.nt and RHuei.'. R.p.....ntatione 113 Appondicu A.B-1. Proof that tho transformation op.rator. 0. will prodbco a unitary r.,....lIItation if orthonormal .asis fundions 8re und If an A-dimensional spa.ce is cha.ra.cterized by the A onhonorma.l ba.sis functions fl' f ....· f .., then. by definition, the sca.la.r product is i = 1,2•...• A j = 1,2..... n.
6-5.1
The non-equivalent irreducible representations for 'if:tv U8ing tke d-orbital function space
In §5.7 we showed that the tra.nsformation opera.tors OR are unitary
therefore
R
I~
E
-i
II \1'3/2 c: o'v
-!
II -\1'3/2 1
~II
-\1'3/211 -i
\1'3/211
and hence
i = 1,2,... , n j = 1,2•... , n. Coupling this equation with eqn (6-2.6) and omitting the superscriptf on the ma.trices, we obtain
-!
=
6if
II~ -~II =
f (~lD~(R)Ar fi
k-ll-l
.
@ID,,(R)!I) dT
D~(R)·D,,(R) 6kl *
= ~ Dk,(R) Dkl(R). i-I
reduced to two two-dimensional representations a.nd one one-dimensional representation. Another ohange of basis functions (in fact. simply interoha.nging d. a.nd d., Le. writing d. = ~aX. and d. = ~lZ.) shows tha.t these two two-dimensiona.l representations are equivalent. Clearly the one-dimensional representation is irreducible and. though we have not proved it. so are the two-dimensionaJ ones. We can therefore write
From considera.tion of the formula for the product of two matrices. it is appa.rent that the a.bove relationship leads to D(R)tD(R) = E
or
D(R)-l = D(R)'.
Hence. in an orthonorma.l basis the matrices D(R) which represent unitary opera.tors O. are all unitary. A.B-2. Th. Schmidt orthogon81iz8tion proCUI
where r 1 and r" &re given in Table 6-5.1. Our next task is to discover the relationship between the matrix elements of non-equivalent irreducible representations. the restrictions on the number of such representations. simple criteria for testing for irreducibility and a. method for readily carrying out the reduction of a reducible representation.
Consider the set of linearly.independent functions 'PI' 'P.,'" 'P. where IXI'PI+IXI'PI+'" IX.. 'P. = 0
only if IXI
=
Ota = ...
IX.
= O. The scalar product is defined by
('P" 'PI) =
f"'''''PI dT
114
Equivallllt and Reducibl. lIejI.....nte.i...
Equivalent and R,ducibl, ReprllMnt8tioos
[see § 5-5(4)]. Then the functions equations will be orthogonal:
4>. = -I.
'/'. =
4>.. 4>..... 4>,.
defined by the following
V'1
(.. 'P.) -I. 'PI- (
-I. 't'..
•
"'-
-I. 't'k =
'P/o- "'- (-I.
=
(i'
D'(R) = U- 1D(RW.
(A.6-3.3) then it can be shown by the usual matrix manipulations, that the adjoints of the D'(R) are D'(R)t = U-1D(R)tU. We can now obtain X= U-1HU
= ~ U-1D(R)D(R)tU = ~ U-1D(R)UU-1 D(R)tU
..L )'t',
=
""i' 't'i
and therefore, fonning the scalar product (i'.p,,) where I obtain
<
j
<
k. we
(A.6-3.2)
where X is diagonal and U is unitary. If we define new matrices. equivalent to D(R), aa
t~l (
(i' .)
X = U-1HU
_ "~1 (,.
V''')-I. ,/". ,-, (
,) To prove that these functions are orthogonal we must first show that these definitions have meaning. Le. we must verify that each
1 '1'= O. Assume that, for some lc > 2. 1' 1 ...
1 =F 0, ...• .. =F O. We next show. o.gain by induction, that 1' ... .. form an orthogonal set. Assume that for some k > 2. 1''''
(A.6-3.I)
that is we form a new matrix by adding together the matrices obtained by multiplying each D(R) by its adjoint D(R)t where D(R) is the matrix representing the symmetry operation R or the transformation operator 0 •. The sum in eqn (A.6-3.1) is over all the operations R of the point group. Each product D(R)D(R)t is Hermitian (see Problem 4.5) and hence the sum of them H is also Hermitian. Because of this fact H can be diagonalized by a unitary transformation (see § 4-5)
(1' V'..)-I. (
1{1 •.• (,.-1> .._1{..-1 =
A.I-3. Proof tha. any rep_t8tion is equivalMlt. through a similarity transferma.iOll. to a unit8ry ...,r_nt8.ion To prove this, we first construct the matrix H = ~ D(R)D(R)t.
-I. _ _ (1' 'PI) -I. _ (
.. .) '/'1 (
115
.! D'(R)D'(R)t. It
If we consider a typical diagonal element of X. we have:
V'1o)-1«~'¥'i, ~))(4)i' ,) "PI
XI' = ~ ~ D;i(R)D;i(R)·
i-1
=
(
(
i' i)(,/,i' 't';) = O. ..L
Thus 1 •..• "'" form an orthogonal set and by induction 80 do
... A normalized set of functions can be formed from the
-
p,
X, - ..; (
,
(XI)t = Xi
>'
so that finally (X" XI) = ~ii and X1>'" X,. form an orthonormal set. It is also true tho.t for a general vector space any set oflinearly-independent vectors can be combined in analogous f&Bhion to give a set of orthonormal vectors. In this case the scalar product is defined by (I'. s)
=
and we see that not only is X diagonal but that it has only real positive elements. We can therefore find the square root of X, Xl. simply by taking the square root of each diagonal element: (XI)i' = (Xj,)I. the off-diagonal elements of Xl will, of ooune, be zero. Also, since Xt is diagonal and real
rs cos (j (see § 5.5(4), (5)).
and
(X-i)t = X-i.
Now we define the ms.trices D"(R), equivalent to both D(R) and D'(R) by the equations • D"(R) = X-iD'(R)Xi. (A.6-3.4) It can be shown that the adjoints of these matrices are D"(R)t = XtD'(R)tX-i.
11 &
Equivalent and Reducible RepresentatiDns
We can prove that the matrices DO(R) are all unitary. Consider the partieular operation S, then D"(S)D"(S)t = X-iD'(S)X1XiD'(S}tX-1 = X-iD'(S}U-1HU D'(S)tX-l = X-iD'(SlU- 1 ~ D(R)D(R}tUD'(S)tx-I
•
=
X-iD'(S} ~ U-1D(R)UU-1D(R}tUD'(S}tX-i
=
X-lD'(S).j: D'(R)D'(R}tD'(S)tX-l
=
X-l ~ D'(S}D'(R)[D'(S}D'(R)]tX-l.
If SR = T, thenD(S)D(R} = D(T}, seeAppendixA.5-5,andsineetheprimed matriees are obtained through a similarity transformation, they are equivalent to the unprimed ones and we have D'(S)D'(R) = D'(T}. Furthermore, by the Rearrangement Theorem (Appendix A.3-1), as R runs over the symmetry operations of the point group, so does SR{= T). Hence D"(S)D"(S)t
= X-I ~ D'(T)D'(T}tx-l = X-l ~ U-1D(T)UU-1D{T)tux-i T
= X-lU- 1
j: D(T)D{T}tUX-i
= X-iU-1HUX-l = X-tXX-1
=E. The matrices D"(R) are therefore unitary and the similarity transformation which produees them is (combining eqns (A.6.3.3) and (A.6-3.4» D"(R) = Z-lD{R)Z
(A.6-3.5)
where Z = UX! and U and X are defined by eqn (A.6-3.2}.t
t
The reooer will have noticed th&~ in determining xt we have written (xt).. = (X,,)t rather than (xt)" = - (X"jt which is equally valid. By doing 80 we have merely chosen from the several square roots of X the one whioh is called the poftjit1e "q"UGT6 rool.
7. Irreducible representations and character tables 7-1. Introduction it is the non-equivalent irreducible representations which refleot the essence of a point group, it is upon these which we now focus our attention. Others whioh are equivalent or can through a similarity tra.nsformation be reduced to these are, in a certain sense, superfiuous: they contain no new information about the point group and we can safely ignore them. In this chapter we introduce a theorem which is central to the use we make of irreducible representations in solving quantum mechanical problems. This theorem is called the Great Orthogonality Theorem or the Key Theorem in Representation Theory; both of these names give an indication of the theorem's importance. What it does is to show the relationships which exist between the matrix elements of the nonequivalent irreducible representations. Though the proof of this theorem is fairly complicated, it has an elegance and beauty, the discovery of whioh is the reward for those who master it. The implications of the theorem are many and include such things as the fact that the number of irreducible representations is less than or equal to the number of classes in the point group, and that for a given point group the possible dimensions of its irreducible representations are restricted. It also leads to a formula. for the number of times a given irreducible representation occurs in a reducible representa.tion. We introduce in this chapter the word 'character'. A character is the trace (sum of the diagonal elements) of any matrix which is a part of a representation of a point group. Many of the properties of a point group can be deduced from the characters of its irreducible representations alone rather than from the matrices themselves; this greatly simplifies things. Furthermore, the Great Orthogonality Theorem leads to rules which allow us to construct tables of oharacters of the irreducible representations without explicitly knowing the matrices. We end this chapter with an example of the determination of the irreducible representations produced by certain basis functions using the rules and theorems which we have developed. It is at this point that we are ready, at last, to produce results of genuine chemical interest from the sole knowledge of the point group to which a molecule belongs. SINCE
111
Irreducible Rep,... . .tiORS end Cb.,.ctlIr rebl..
Ineduclble Represeat8tI.1I8 end Cherel:l. rebl..
7-2. The Gra.t Ortholonelity Th..,...
This theorem st&tes that if r" and r' are two non-equivalent irreduoible representations with matrices D"(R) and D'(R) (of dimensions 11... and 11." respeotively) for eaoh operation R of the group '6 then the matrix elements are related by the equation (7-2.1)
Now, the maximum number of g-dimensional veotors whioh can be orthogonal is g. Consider, for example, a two-dimensional veotor spaoe containing the two orthogonal veotors: p = %,e1+x.e. p' = x~e1 +x;e.
where or
where 9 is the order of the group and the sum is over all of the operations R. The proof is given in Appendix A.7-1. If we assume that the irreduoible representations are unitary, and we can do this without any loss of generality (see § 6-4), then
119
(7-2.3)
If there were a third veotor
orthogonal to p and p', then
D'(R-I) = DV(R)-l = D'(R)t
and eqn (7-2.1) becomes:
and
~ D':~(R)D;...(R)· = (9111...)6..,6,,6_.
•
(7-2.2)
n:
For a given unitary irreduoible representation I'" there will be matrix elements corresponding to eaoh R. If the operations are R = R u R., ... R" then the 9 matrix elements of a chosen i andj value,
Solving these last two equations we get:
X, -X;) , Xl". __ 0 (x:! X;ll
and
D'tJ(R 1 ), DMR.), ... .D':J(R,)
can be considered to be the oomponents of a g-dimensional vector. Since there are n~ such sets of matrix elements (corresponding to i = 1,2, ... n" andj = 1,2, ... 11...), there will be 11.: such g-dimensional vectors for each irreducible representation. Eqn (7-2.2) shows that the vectors from a given unitary irreduoible representation are orthogonal to one another and to the vectors formed in & similar way from any other non-equivalent unitary irreducible representation. This is because, as the reader will recaJI (Section 5-5(5», two vectors p = x1e 1+z.e•... +z"e"
p'
(p,p') = ZlX;: +xsX; ... +x..x~ = 0
or, if the vectors are complex, if
x,x;· +xsX~· ... +x..x~·
and
we have
and
= X;:e 1+x~e.... +z~e..
are said to be orthogonal if
(p,p') =
or, since by eqn (7-2.3)
=
O.
Eqn (7-2.2) with iJ =1= " or i =1= j or k =1= m is of the same form as this last equation.
X. +xi)". +X~"* 0 . . X2 = (X: --.(-Xl X. x,x. Xi = . we have x;* = x;· = 0 or x; = x;
Sinoe x~+x: =1= 0, = 0 and hence a two dimensional vector orthogonal to p and p' cannot exist. By extension, this is generally true i.e. the maximum number of orthogonal g-dimensional veotors is g. We therefore have the following result: ~ 11.:
..
< g,
(7-2.4)
120
Irndueibl. R.p....ntation••nd Ch....etsr rlbl..
that is: the sum over all of the non-equivalent irreducible representations of the square of the dimensions of the representations is less than or equal to the order of the group g. This result places restrictions on both the number and size of the irreducible representations of a group. In Appendix A.7-2 we show that, in fact, the equality holds i.e.
in: = g,
Irrsdueibl. R.prnentstions .nd Ch.r.etsr r.bl..
representations will be identical; this is borne out by Table 7-3.1. The reverse proposition: that if the characters of two representations are identical, then the representations are necuBarilg equivalent, will be proved in the next section. If two operations l' and Q of a point group are linked by a third operation X of the point group by the relation P
p-I
where r is the number of non-equivalent irreducible representations. 7-3. ChlIr.etlra
The trace of a matrix is the sum of its diagonal elements: Trace( A) = ~.A II' 4
The trace of a operation of a the symbol z. representation
matrix which represents an element of a group (or an point group) is called a character and is usually given Z(R) is thus the chara.cter of the operation R in the which has matrices D(R), i.e. X(R)
=
Trace{D(R)},
and if we are considering the representation
r·,
c;
0: 0: 0,
t In
e+e·
xl R )
xlR)t
(rflGl bani)
(oompleo: bani)
3 0 0 1 1 1
0 0 1 1 1
X(P) = X(Q)·
This result, too, is borne out by Table 7-3.1, where, for the ~Iv point ' G., • andG"T group E forms one class, C. and C.t form another, an dG., form a third. From eqn (7-2.1) (the Great Orthogonality Theorem) we can obtain for the non-equivalent irreducible representations r· and r':
A;
~ .D:4(R)DiJ(R-I) = (gJn.)6.,6 4 R
=
(gJnJ')6J'A;
and if we sum over i and j, we get:
ft.
n~
~ ~ DMR) ~ DJ;(R-I) =
3
the OOnstruotiOD of this oet of oharacters, the fact that -1 hall beenued,e = exp(2m(3).
=
D(P) = D(X)-ID(Q)D(X)
then
TABLE 7-3.1
E C.
D(X-I) = D(X)-l,
and D(P) and D(Q) are related by a similarity transformation. From this we see that the characters of operations belonging to the same class are identical, i.e. if P = X-IQX
Charac!ers of the representatiem.s giflen in Table 6-3.1 R
(7-3.1)
D(X-I)D(X) = D(E) = E
and
then we write
The complete set of characters for a given representation for the elements of a group is called the character oj the repruentaticm. The characters of the two representations introduced in Table 6-3.1 are shown in Table 7-3.1. We have already seen (eqn (4-5.4» that the traces of two matrices which are related by a similarity transformation are identical. Since equivalent representations have matrices which are linked by a common similarity transformation, the characters of two equivalent
= X-IQX,
then P and Q are conjugate to each other and are said to belong to the same clMB (see § 3-5). If D(P), D(Q), D(X), and D(X-I) are the matrices in BOme representation of the point group representing the operations P, Q, X, and X-I, then necessarily these matrices must mirror eqn (7-3.1), i.e.: D(P) = D(X-I)D(Q)D(X). (7-3.2) Since X-IX = E,
then:
ZP(R) = Trace{D"(R)}.
121
and
• "-1
ft~
"~
(gJnp )6., ~ ~ 6;;
'-1 J-l
1-1
(7-3.3)
If the irreducible representations r p and r' are chosen to be unitary (no loBS of generality), then eqn (7-3.3) becomes ~ XP(R)Z'(R)*
•
= g6.,
(7-3.4)
122
Irr.ducible Repr..entetio... and Charectu T.bl..
since X'(R-') =
Ir.....ucibl. Rapresentlltions and Chareeter Tabl.. we see that the numbers
I D:,(R-') = i D;.(R)* = x'(R)*
i_I
glx"(C,), gjx"(C.), ... glx"(C,,)
.=1
for unitary matrices. If the operations of the group are
can be interpreted lIS the components of a k-dimensional veotor and that similar vectors from the other irreducible representations will be orthogonal to it. Since the maximum number of orthogonal k-dimensional vectors is k, the maximum number of non-equivalent irreduoible representations must also be k, so that if r is the total number of non-equivalent irreducible representations, then
R" R ••... Rif
we can interpret the characters X"(R,), x"(R.) •... X"(RIf) lIS the components of a g-dimensional vector and eqn (7-3.4) then implies that such vectors for different non-equivalent unitary irreducible representations are orthogonal. We will collectively symbolize the operations of the ith clllSs of a point group <§ of order g by C, and we will symbolize the number of operations in the ith class by g, and the number of classes in the group by k. so that 11:
.-,!g,
=
g.
< k.
(7-3.7)
In fact we can show (Appendix A.7-3) that it is the equality which holds. 7-4.
Number of tim.. en irrlducible repr.sentation occurs in a r.ducibl. one
"
...1
C:,
g, = 1,
g. = 2, g. = 3, g = 6, k =3. Since the characters of the operations of the same class are identical, we can write eqn (7-3.4) as:
!" g,X"(C,)X'(C,)*
where r v is the "th irreducible representation, a. is the number of times r' appears in and k is the number of classes in the point group (the number of irreducible representations equals k, see Appendix A.7-3). The characters of the matrices belonging to rOd, x'od(R), are the Bame as the chara.cters of the matrices in their reduced form since only a similarity transformation has been carried out for each one. But the sum of the diagonal elements of the matrices in their reduced form is simply the sum of the diagonal elements of the irreducible matrices x'(R) which occur, multiplied by the number of times that they occur. Consequently, we have
rod
a; 0; or 0;_
,-.
r
Consider the reducible representation rod: we can write (see eqn (6-5.5» rrod = a,I" $ a.r· EEl ••• a.I'· $ ... a"r" = .! a.r·
For example, for the 'fl'av point group we will write: C 1 = E, C.= C. or Ca = or
123
x'od(R) = = g~".
(7-3.5)
where the sum now runs over the different classes and X"(C,) is the character of any operation in the ith olass in the r" irreducible representation. Rearranging eqn (7-3.5) as 11:
,-..! {gtX"(C.)}{gtX·(C.)}*
= g{J....
(7-3.6)
.!" avl
V (
(7-4.1)
R)
_1
for eaoh R, where XV(R) is the charaoter of R in the r v irreduoible representation. If we multiply eqn (7-4.1) on both sides by x"(R)*. the conjugate complex of the oharaoter of R in the r" irreducible representation, and sum over all operations R of the point group, we obtain, by using eqn (7-3.4).
.! Xred(R)X"(R)* = ! .!" a.x·(R)X"(R)* •
•
y_l
=
.!" a.g~". = ga...
"..1
124
Irreducible R.p.....ntation. and Character rebles
Ineducible Reprnentations and Cheracter rabl..
Therefore,
-
a p = g-l ~ X...d(R)XP(R)* = g-l ~ g,x"'d(C,)XP(C,)*
(7-4.2)
i-I
It.
where C, denotes any operation of the ith class and g, is the number of operations in the ith class. Eqn (7-4.2) is an extremely useful formula, which can be easily applied (provided the appropriate characters are available), for determining the number of times "" that the r" irreducible representation occurs in the rred reducible representation. We are now in a position to show that two representations with a one-to-one correspondence in characters for each operation, are necessarily equivalent (see § 7-3). If we consider two different nonequivalent irreducible representations then, since the characters are orthogonal (eqn (7-3.4», there cannot be a one-to-one correspondence. If we consider two different reducible representations r~ and r~ then, by eqn (7-4.2), if the characters are the same, the reduction will also be the same, that is the number of times rIO occurs in f'" (a p ) will, by the formula, be the same as the number of times r" occurs in r~. The reduced matrices can therefore be brought to the same form by reordering the basis functions of either r~ or r~. The reduced matrices are therefore equivalent and necessarily r" and r b from whence the reduced matrices came (via a similarity transformation) must also be equivalent. Hence, we have proved our proposition.
Now, if r~ is irreducible, inspection of eqn (7-5.1) shows that all the a,. are zero except for one which is unity, this one corresponding to the particular irreducible representation which is identical with f"'. So, if f'" is irreducible, the characters must satisfy: ~ X"(R)X"(R)* = g R
or
k
! g,X"(C,)X~(C,)* = ._1
X"(R) =
-
~ apx"(R) ,...1
(7-5.1)
and multiplying both sides by X"(R)* and summing over all the operations R of the point group, we obtain:
~ x"(Rlx"(R)*
= =
~ Li.a"XP(R)}{.t a.x·(R)r
i i a"a.{~ XP(R)x'(R)*}
11_1'1'=1 _ k
=
Jl
~ ~ apa.gd".
I'_ly_1
a
=
9 ~a;. _1
(Note that the numbers a" are necessarily real, i.e. a" = a:.)
9
(7-5.2)
which gives a simple test of whether a representation is irreducible or not. 7-6. The rllduction of a reducible representation When we come to apply the results we have so far discovered to quantum mechanical situations, we will find that the application usually revolves around the reduction of some reducible representation for the point group concerned. We have already seen how to find out which irreducible representations appear in the reduction of a reducible representation, namely if we write
r"'d
-
= ~a.r',
._1
then
-
a. = g-l ~ roo(R)x'(R)* = g-' ~ g,x"'d(C,)X'(C,)*. •
7-5. Critarion for irreducibility From eqns (7-4.1) and (7-3.4) we can discover a simple condition for a representation to be irreducible. Consider the representation r", we can write its characters as
125
1:_1
Now we ask the parallel question-what is the new choice of basis functions for the function space (the one which produced pod) which will produce matrices in their fully reduced form 1 Once again we are looking at the opposite side of the coin whose two faces are a similarity transformation and a change of basis functions. To answer the question we have posed, we will invoke the Great Orthogonality Theorem and carry out a certain amount of straightforward algebra. Suppose that we have found k different function spaces for a given point group, where k is the number of classes or irreducible representations for the point group, and suppose that each function space provides the basis functions for one of the k irreducible representations. If the dimension of the 11th irreducible representation is n•• there will be n. orthonormal basis functions describing the 11th function space. We will write these sets of basis functions as
n,r., ... I:.
,.
=
1, 2, ... k
where the superscript onl shows the function space to which it belongs. By definition, these functions must obey an equation of the same form
126
Irreducible ReprtlMntations and Charact... Tabl..
Irreducible Representations and Charllct8r Tabl..
1;,1:, ···1:,
as eqn (5-7.2), that is, OJ; =
.2 D;.(R)/;
q = 1,2, l ' = 1,2,
11_1
n, k
(7-6.1)
2 (gln p)6 p.6;plJ .t: i
11_1
= (gln p)6py lJi .f/ and, if we define a new operator pri
(7-6.2)
P: by
~
=
i
(7-6.3)
Df;(R)*O"
"
PM; = (gln p)IJ".6i .li.
then
P:
(7-6.4)
i is an operator which is a definite linear combination of the transformation operators 0" with coefficients which are related to the matrices of ['P; it is (for reasons which will be clear later) called a projection operator. If J' = " and q = j, eqn (7-6.4) becomes
p;J; =
(gln,>f;
i--1
•
II
and hence. using eqn (7-6.4),
PPI; or, if J' =F- v.
p"r. =
=
I (gfnp)dpA.n
1_1
0
q = 1.2•... n y (7-6.7)
J' = 1,2•... k 1'=1.2•... k
and, if J' =
JI,
PPI: =
q = 1.2 ,.. = 1. 2
J' = 1. 2, ... k
__ 0
" = 1. 2•... k
and from eqn (7-6.8). if J'
p"I;.o
=
= (glnp)/:'"
J'
= 1.2.... k.
(7-6.10)
We now see why pp is called a projection operator; it annihilates any function which does not belong to the J'th space and projects out (and multiplies by gln p) any function which does. Let us now consider the n-dimensional reducible representation rn d which is produced from the function space whose basis functions are g1' gl' ... g... and let us assume that in the reduction of :rzed no irreducible representation of the point group occurs more than once. One way of looking at the reduction is to see it as a change of basis functions from g1' gl•... gn to where k is the number of irreducible representations. From this it follows that it must be possible to express the g functions as linear combinations of all of the 1 functions: If =
or as
-
g. = ~/:
._1
where
I:
=
8 =
...
~c:Jl
lJ
1,2,... n 1.2•... n
= 1, 2, ... n.
1_1
The functions I: (8 = 1.2•... n) must be functions which belong to the space which produces ['y. since they are simply linear combinations of the basis functions which define that space. If we choose one of the irreducible representations. say r", and apply the corresponding projection operator P" to g•• we obtain from eqns (7-6.9) and (7-6.10). II
nIl
(7-6.9)
JI
(7-6.5)
and Van Vleck has called this equation the basis function generating machine. since from one basis function I; the others can be generated. Furthermore, we can create another projection operator p p by the equation "p "p pp = ~ prj = ~ ~ Dl';(R)*O" = ~ x"(R)*O" (7-6.6) e-1
and we will denote such a general function by 1:'0' Then from eqn (7-6.7) if J' =F- " we obtain. ppraen = pP(a.t!:+a.t!;+ ... rr....,.f'...)
and this equation must be satisfied for every operation R of the point group. Since we will choose the basis functions to be orthonormal. the matrices D'(R) will be unitary (see § 6-4). Now let us multiply eqn (7-6.1) by D:i(R)* and sum over all R. From eqn (7-2.2) we have
=
127
= 1.2.... n
,.. = 1,2, ... k.
(7-6.8)
(7-6.11)
Any function belonging to the function space which has been used to produce r Y can necessarily be written as some linear combination of
Since P" is a linear combination of the operators 0", and O"g, is a known linear combination of gl' gs, ... gn' PPg, must also be a linear
(gln,,)/:
k.
121
Ineducilile Repr-utlons end C1'lrleterTlbl.
Irreducible Repr_ntationl Inti Cblrlchr Tlbl.
oombination of gl' g., ... UK' So. from eqn (7-6.11) we can obta.in a linear combination of g's which are proportional to functions /: which belong to r" and if we apply P" to each g function in turn, we get 11. linear combinations of g's which belong to r" and from which we can find 11." which are linearly independent and. if we wish. orthonormal. Hence we have a method of finding basis functions which belong to a given irreduoible representation, if we are given some function space which produces a reducible representation. Notice that in addition to the 0., the construction of P" (eqn (7-6.6» requires only the knowledge of the characters of the r" representation. IT r" occurs, for example, twice in roo then
=/.1+ .../:+/:' ... +/: and /: and I:' are both funotions belonging to r" and are both linear combinations of g,g, ... ,J:, they differ solely in the coefficients c;;, g.
8 =
1.2, ...
TABLE 7-7.1 The character table Jor the ~h poim groupt WIT
11.
8=1,2, ...
11..
Our method is not capable of separating these two functions; we will always get a com'bination:
P"g. =
three non-equivalent irreducible repreaentationa r"'-J ~ 1'... and r l •
Table 7-7.1. The first row corresponds to r 1 in Table 6-5.1 and the last row to r· in that table. (We have not previously discussed the middle representation.) The headings of the columns in Table 7-7.1 are E, 2C. and v and they imply the identity operation E (one class), the two rotations and (another class), and the three reflections a~, and (a third class). The names of the three representations (AI' A., and E) will be discussed later. It is easy to check that the characters in Table 7-7.1 satisfy the orthogonality relationship (eqn (7-3.5»:
7-7 Cbarlchr lIbl. Ind their construction Since we will continually be requiring the characters of the irreducible representations of the point groups. it is convenient to put them together in tables known as character tables. In the character table of a point group each row refers to a particular irreducible representation and, since the charaoters of operations of the same class are identical, only a single entry X"(C,) is made for all the operations of a given class. The columns are headed by a representative element from ea.oh class preceded by the number of elements or operations in that class g,. For example. the 'ifav point group has three classes (and necessarily three irreducible representations) and its character table is shown in
er;
C.
C:
.,
I
'_I
(U/n,,)(/:+/n.
So that in a case like this we will obtain a mixture of two sets of functions each of which alone would be sufficient to define a function space leading to r". The usefulness of the results of this section will be exemplified by the problem in § 7-9.
I -1 0
t The first oolum.n ohOWB the labels (...... §7-8) of the
er;.
and
E I I -I
3er
i.e.
129
g,x"(C,)x·(C,)· = g6".,
for example, the characters of p-t, are orthogonal to those of poi.: (lxlxl)+(2xlxl)+{3xlx-l) =0,
those of
r.A, are orthogonal to those of rB': (I xl x2)+(2 X 1 X -1)+(3 X 1 xO)
and those of poi. are orthogonal to those of
=
0,
rB':
(lxlx2)+{2xlx-I)+(3x-lxO) =0.
There also exists an orthogonality relationship between the columns of the character table:
.,
I x'{C,)x·(C,)· _1
= (g/g,)6"
(7-7.1)
and the reader may confirm for himself that this equation too is satisfied. The proof of eqn (7-7.1) is part of the proof that the number of irreducible representations is equal to the number of classes of a point group (see eqn (A.7-3.10».
130
Irnduci.'. R.,nantlltiolll .nd Chereetlr rebl.
Though the character tables for a.lI the important point groups are readily available (see, for example, Appendix I at the end of this book), it makes a convenient summary of our results to see how the tables can normally be deduced without explicit knowledge of the matrices themselves. The following four rules can be used: (1) The sum of the squares of the dimensions of the irreducible representations is equal to the order of the point group,
(the proof of this is given in Appendix A.7-2). Since the identity operation is always represented by the unit or identity matrix, the first column of a charaoter table is x"(E) = 11.". Also we have k
1: {Z"(E)}' .._1
= g.
Since the matrices \1111, lilli, •.. form a one-dimensional totally symmetric irreducible representation of any point group, it is customary to put the corresponding charaoters in the first row and 80 X1 (C,) = 1. (2) The number of irreducible representations r is equal to the number of classes k; the proof of this is given in Appendix A. 7-3. (3) The rows must satisfy k
1: g,Z"(C,)X'(C,)*
'_1
= gfJ....
(4) The oolumns must satisfy k
1: X'(C,)X'(C,)* _1
= (g/g,)fJjJ'
From these four rules it is easy. for example, to construot Table 7-7.1. There are three classes for 'ifa• and therefore three irreducible
representations. The only three numbers whose squares add up to six (the order of the group) are 1, I, and 2. We therefore immediately have: E
3cr.
1
1
1
a c
2
1 b
d
131
and have only to determine a, b, c, and d. From rule (3) we have 1+2a+3b = 0 1 +2a"+3b" = 6
and
2+2c+3d
=
0
4+2cl +3d" = 6
hence a
=
I, b
=
-I, c
=
-I, and d
=
O.
There are several general methods for caJculating the characters of the irreducible representations which are more systematio than the method we have given. Their drawback, however, is that they involve long and complex caJculations and are only feasible when use is made of high speed computers (see, for example, John D. Dixon, Numerische MatheflUJli,k 10, 446 (1967». Furthermore, Esko Blokker has described a theory for the oonstruotion of the irreducible representations of the finite groups from their charaoters and though complicated, it can be conveniently programmed for a computer (see, International journal of quantum chemialry VI, 925, (1972». The reader who is interested in the part that computers can play in group theory is recommended to read the article by J. J. Cannon in the Communicati0'n8 of the aBsociation for computing machinery 12, 3 (1969).
7-8. Notation for irreducibl. repr_ntatioas The symbols formulated by R. S. Mulliken are used to distinguish the irreduoible representations of the various point groups. In this section we will outline the general poin~ of the notation and the reader is referred to Mulliken's reportt for the details. One-dUnensional irreducible representations are labeled either A or B according to whether the character of a 2../n (proper or improper) rotation about the symmetry axis of highest order n is + 1 or - I , respectively. For the point groups 'if1 , 'if., and 'ifl whioh have no symmetry axis, a.lI one-dimensional representations are labeled A. For ~. and ~I.h there are three C. axes and the three C. operations fa.lI in different classes; those one-dimensional representations for whioh the
t Thia report WBlI publiahed in T1le JOfM'I14l 0/ chemical ~, 23, 1997 (1966). The reader ohould note that on page 2003 of tbia report the tbird line below Table VI should read: for !I...., fI... =-:. IC•• G'd Ie;; for gl8hJ o. = IC~ Gd == IC•. AbIo, in the diagram for g •• in Fig. I, the d. and do plan... should be interchanged. When theee correotiona ...... made, the definitiona ...... the same ... thoee in Fig. 3·6.1 of this book, with the proviso that our axis is Mulliken'. 0, axis and our 0; &Xi. i. Mulliken'. &Xis. Further information on notation is oont.ained in G. HBBZBBBG'. Molecular .peclra and molocular _noel...... vol. II. Van N Oitrand Reinhold. ::c::II:
do
do
132
Irreducible Representations and Character Tabl..
Irreducibl. R.pr...ntations and ChanctBr Tabl.
characters of all three C. operatioI18 are + 1 are labeled A, while the other one-dimensional representations are labeled B. For !lid' the character of S." determines the label of the one-dimensional representations. Two-dimensional irreducible representatioI18 are labeled E, which should not be confused with the identity element or the identity matrix. Three-dimensional irreducible representatioI18 can be labeled either T or F; usually T is used in electronic problems and F in vibrational problems. If a point group contains the operation of inversion, a subscript 9 (from the German word gerade) or u (from the German word ungerade) is added to the label according to whether the character of i is positive or negative respectively. The inversion operation is always represented by +1 or -1 times the identity matrix; hence the character is either -n where n is the dimension of the representation. Point + n P orwhich " contain i" are ~KIl. (n even), !li groups nh (n even),!li1Od (n odd), l!Jh and !Ii""h and these point groups are often written as ~n®~l (neven), 9 n ®W1(n even), 9n®~t (n odd), l!J0Wt and W "'T®~l respectively (i.e. (J®'tft),t since they contain all the operatioI18 (R) of (J plus all those one can obtain by combining each R with i (i.e. Ri). There are twice as many classes in (J 0~1 as in (J and therefore twice as many irreducible representations. Thus for each irreducible representation of (J there will be represented by + 1 or - 1 times an identity matrix and that there. . are twice as many irreducible representations in '0 @ '€. as in '0. For r'" X"'(R) = XP(R), XP'(iR) = XP(R)
and for [,Po
X"o(R) = XP(R), XPo(iR)
=
-XP(R)
(for a proof of these equations, see Appendix C of Schonland's book Molecular symmetry). If the point group has a ah operation but no i operation (groups ~..h and !!Jfib with n odd) the labels are primed or double primed according to whether the character of ah is positive or negative, respectively. The situation is similar to the one in the previous paragraph in that ah will be represented by + 1 or - 1 times an identity matrix and that there. are twice as many irreducible representations in (J®~. as in (J. For ['P XP'(R) = xP(R), XP'(ahR)
t
=
xP(R)
This notation i. referred to again at the end of § 8·3.
and for
['po
XP'(R)
=
133
XP(R),
XP'(ahR) = -XP(R).
If one can write a point group either as (J0~1 or (J®~. (e.g. !lie.) the former takes precedence. If necessary, numerical subscripts are added to the labels to distinguish the non-equivalent irreducible representatioI18 which are not distinguished by the foregoing rules. Except for the fact that the totally symmetric representation (one-dimeI18ional unit matrices) is numbered and listed first, the numbering is arbitrary and the reader is referred to Appendix I or Mulliken's report for the internationally ll.Ccepted conventioI18. Ifa one-dimeI18ional representation has complex characters a, b, c, ... then there must be another equally acceptable representation with the characters: a*, b*, c*, ... since for a one-dimensional representation the character of an operation equals the single matrix element representing the operation. These pairs are usually bracketed together and labeled E. In fact quite often the reduction which produces the pair of irreducible representatioI18 is not carried out, since no useful information is gained by it and anyway the two always occur together. The two infinite point groups ~ "'.. and 9 ",h (= ~ "'.. ®~l) have their own notation. Because these groups have an infinite number of elements the theorems we have given do not apply and other more elaborate methods are needed to find their irreducible representations. In ~ "".. the pairs of rotations C(tP), C( -,p) through equal and opposite angles, belong to a class of two operations, one class for each ,p value. All the reflections a .. belong to one class. The point group has two one-dimensional irreducible representations and an infinite number of twodimensional ones. In Mulliken's notation these would be labeled A., A., E t , E., ... but a different notation, using Greek letters and which was developed in early spectroscopic work, is usually employed. The character table for 'C"'.. is shown in Table 7-8.1. TA.BLE 7-8.1 Character table for ~ "''' 'if",y
Al = 1;+ A. = 1;E I =IT
E.
=~
E. -= 1/1)
E
2C{+)
"'OOy
1
1
1 2 2 2
1 2 Co. + 2 co. 2+ 200.3+
1 -1
0 0
0
134
136
Irreducible Repr_ntatiollll and CharactarTabl..
Irmucible R8pr_ntatioM and Chanctar Tlbl..
7-9. An Ulmple of thl dltlrmination af the irmucibla
..0 "'0" "C."'O- "'C-
r8p~tatlon.to which cartain functions belong
I
I I
In order to demonstrate BOme of the conclusions of this chapter, we pose the question: for the !lil~h point group. for which irreducible repre-
sentations do three real p-orbitala and five real d-orbitals or their combinations form a basis of representation1 In Fig. 7-9.1 the symmetry elements for the !lilQ point group are shown (see &lao Fig, 3-6.1) as well as our choice of 3:, g and z axes (this choice establishes the orientation of the p- and d-orbitals), In Table 7-9.1 the oorresponding charaoter table is given in full.
.. ----0----0 I I l
e;)
I
I
.
. ----c·----o I i. I I
,;
,;
----~----~ I I I I I
____ 0
I
I
0
I I
"C-..,;..J "Tj-.o
I
I I I
.:!!
~d:C:
I I I
FIG. 7-9.1. Symmetry elemente and axes for ~.b. Exoept for ab. the symmetry planes oontain the • axw and the axw alongside which the plane's label w written, A and B rep.........t two diiferent atoms.
c:~c:
I I I
...
.. .. ..
...
"C"C"'O"'C"'O
I
To find the irreduoible representations which can be produced from the orbitals we need the oharacters r(R) and the effect of O. on the orbitals. Taking the last point first. we find 0.9 for a.l1 R of ~Q and 9 = Pl' PI' PI' d l • d l • d •• d&. d. and the results are given in Table 7-9.2. We have carried out this kind of step before (see § 5-9) and for this partioular point group and axis ohoice. the proOOS8 is particularly simple. for example we have
ex: 3: ex: g PI ex: z d l ex: 3:1 _ , 1 d l ex: :J:1I d l ex: 3:Z d& ex: yz d. ex: 3z"-1'"
(J-
I I
.. .. ... .. ..
----j----i
'"C"'O"C""O"'O
I I
PI
PI
""... ..,;..,;~.. "o..
I I I
.
I~
----cq----cq •
a
_.
».
~
~
"'l~ "'l"~~"l>il'...r...r~rxr ~
...
Ir....ucibl• •p...entation. end Ch.,el:ter Tebl..
Irreducible R.p......tion. encl Cherel:t8r Tebl..
136
and if we consider 0e,' under the C 4 opera.tion we have (800 eqn (5-2.5» ~' 0 ~ 0 -1 ~' ~ 0 1 0 y'
-1
0
0
y
z'
0
0
1
z
or
1/ z
1
0
0
1/'
0
0
1
z'
~
" .:
"
g
and thus
:1
(from the definition of 0.)
"
ex;~
ex; -1/'
~
= - Po(~'. y', z')
or, since the coordinates are now the same on both sides of the equation, Oe,P, = -Po·
0e,P1(Z', y'. z')Oe.Pl(Z', y'. z') - 0c,Po(z'. y'. z')Oc,Po(z'.1/', z') ex; {- Po(z', y', z')}O -{Pl(~" y', z')}O = -d,(~',y', z')
or
Oe.d,
=
-d, .
From Table 7-9.2 and using eqn (5-7.2) we can find the diagonal elements of the matrices which represent the ~4h point group in the p-orbital basis and in the d-orbital basis. From these elements we get the characters of two reducible representations; they are shown in Table 7-9.3. By applying eqn (7-4.2) a ~ -- g-l
we have and
rred (d. basis) =
.
I
x·ed(R)x~(R)·
r..t,• E9 r BI • EEl r B•• EEl r E•.
So that we know that there are p-orbitals or combinations of p-orbitals E which form & basis for the irreducible representations r..t•• and r • and d-orbitals or combinations of d-orbitals which form a basis for the irreducible representations r..t.., r B, •• r B •• and rE·.
5:! .~
'"
'1.;'
"!~
r.;
:l ~ ... .0::
~
ex;
""
1!
Also. using a slightly different but nonetheless straight-forward way. we have Oe,d,(~'. y', z') ex; Oe,(~" -y")
.~
'", ~
~
..:l III
"'!
Eo<
~"'" .. 5
..s'"
~~ " 5:! ~
~J
-of
.J
..;
~~ <..,s ...
~9
....,'"
~J
"
'rS
~ Eo.
I
I
-
'"I
'"
~
I
l;l.,
~
~
-
rS
U
rJ "'J
'" '" "
.~ ,Q
Q.
.:!I
J .,;
137
138
To find out which orbitals or combinations of orbitals produce which representations. we make use of the projection operator P" defined in eqn (7-6.6) &8
and the D(R) are irreducible. then A =
By appiying this operator to each of the orbitals in turn we project out functions belonging to r". In Table 7-9.4 we collect together the results
IJ
p,
P,
P,
d,
d,
d,
d.
d,
A .. A.,
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0
0 0
0 0 0
0 0 0 0
0 0 0 0
16<1,
8d.
Sd.
B ••
0 0 0 0 0 0 0 0 0
E.
8p,
8PI
0 0 0 0 0
0 0 0 0 0
B E," .04>A ••
B lY
16p. 0 0 0
0 0 0 0 0 0 0
16d. 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
Hence. if D(R)A
X-lAX
(all R).
=
X-ID(R)AX-X-IAD(R)X
=
X-I[D(R)A-AD(R)]X.
= AD(R),
D'(R)Z-ZD'(R)
=
0
and Z and D'(R) commute: D'(R)Z = ZD'(R)
and comparing matrix elements on both sides of this last equation. we have:
k_l
k-l
i
=
I, 2,
,n
i
=
1,2,
..
where n is the dimension of the representation. But. since Z is diagonal,
r
Appendicaa A.7-1. The Gnat Orthogonality ThHrem To prove this theorem we first have to prove two theorems (sometimes called Schur's lemmas) concerning those matrices which commute with all the matrices of an irreducible representation. We will call these theorems Theorem I and Theorem II. (1) Theorem I This theorem states that the only matrix which commutes (see § 4-3) with all the matrices of an irreducible representation is a constant matrix. We have to show therefore that if
for all R
~
D'(R)Z-ZD'(R) = X-ID(R)XX-1AX-X-IAXX-ID(R)X
(o/n,.)/:,
AD(R) = D(R)A.
First we diagonalize the commuting matrix A by a matrix X
D'(R) = X-I D(R)X
we see that P, and p. (or P. and P..) belong to pE. and form a ba.sis for that two-dimensional irreducible representation; p. (or p.) belongs to p-t•• ; d , to B, ,; d. to B •• ; d. and d. form a basis for the two-dimensional representation r E,; and d, belongs to I'd".
r
..t
0
Then
of applying the ten projection operators for the ~'h point group to the eight functions. These results are found by simply combining Tables 7-9.1 and 7-9.2. Recalling that (eqn (7-6.11» p"!I. =
o
and define the new matrices
The re8'Ult8 0/ applying P" lor each irreducible representation 0/ ~OJ> to the three p- and jive d-orbitals
16<1,
o ;.
0 0
Z
TABLE 7-9.4
where ;. is a constant. i.e.
).E,
;. 0
A=
B ..
131
InBCIucible Rapresentatlons and Character rablllS
InBCIucible Representations elld Character Tabl_
(A.7-1.1)
Z", = 0
for
k =;I=j
and
Z,,, = 0
for
k =;1= i.
therefore.
D:,(R)ZII = Z ..D:,(R)
or D;,(Rj(ZII- Z ,,)
=0
i= 1,2,... , n j = 1.2•... , n.
(A.7-1.2)
Consider one specific diagonal element of Z. say th~ ~t Zu: if it is different from all the others. then eqn (A.7-1.2) shows that if. = 1 and that ifj = 1
D:.,(R) = 0
j = 2, 3•..•
11
D;,(R) = 0
i = 2,3 •...
11
so that the first row and column except the diagonal element are zero and hence the D'(R) are in block form and all the D(R) have been ~duced. by the transformation X-1D(R)X. But we stated that the D(R) are llTedUCIble and therefore Zu cannot be different from the other diagonal eleme~ts. By extension we find. therefore, that all the diagonal elements of the diagonal
141
Irreducibla Represantations and Charactar Tabla.
matrix Z are the same:
).
Z=
0
0
o ).
0
Irreducible Raprell8ntations and Cbaractar Tablas
OalJe (a). If n.. = n.. then A and AA t are square matrices and the determinant of AA t is = ).E
det(AAt) = det().E)
00).
det(A)det(At) = )..... {det(A)}"
where ). = constant. and hence
(2) Theorem II This theorem states that if for some group there are two differem irreducible representations r" and r· with matrices D"(R) and D'(R) of dimension n.. and n. respectively and if a rectangular matrix A exists such that AD'(R) = D"(R)A for all R, then Case (a) if nIl = n •• either det(A) =1= 0 and the two representations are equivalent or else A = 0 (the null matrix), Case (b) if n" =1= n., A = O. Without lOBS of generality we can assume that the D"(R) and D·(R) are unitary and n. < nIl' Then: for all R
(A.7-1.3)
and taking adjoints (see eqn (4-3.15»
and since D"(R) and DY(R) are unitary
(A.7-1.4) Since the inverses R-l are all. by definition. operations of the group. eqn (A.7-1.3) implies and multiplying by At we get:
ADY(R-1)At = Y(R-1)AAt.
(A.7-1.5)
Comparing eqns (A.7-1.4) and (A.7.1.5) we obtain
= U(R-1)AAt.
AAt =).E
and the two representations are equivalent. If ). = O. then AA t = 0 and
LAilA:~ = 0 f
i = 1,2,... nIl
k= 1.2•... n"
if i = k. this becomes
which is only possible if all Au = 0, hence A = O. Oalle (b). If n y < n... then A has n y columns and n.. rows and we can fill A out to a n"xn.. square matrix B by adding (1O.. -n.) columns of zeros. Clearly, BBt = AAt and since det(BBt) = {det(B)}" = 0
= O. but by eqn (A.7-1.6)
hence). = 0 and. recalling Case (a). A = O.
and multiplying both sides of this equation by A, we get
where ). is a constant.
(all R)
det(AAt) = det(I.E) = )..... (see eqn. (6-5.2»
AAtD"(R-1)
D·(R) = A-1D"(R)A
so det(AAt)
D·(R)t At = AtU(R)t
Thus, by Theorem I.
(see Appendix A.4-6)
= I......
If ). =1= 0, then det(A) =1= 0 and A has an invel'tle and we can write
A = X(I.E)X- 1 = ).E.
AD'(R) = Y(R)A
141
(A.7-1.6)
(3) Proof of the Great Orlhogonality Theorem This theorem states that
where D"(R) and D·(R) are the matrices for two non-equivalent irreducible representations of dimension nIl and n. for the group '§ of order g. There are two parts to this proof. (i) Take the irreducible representation with matrices D(S) of dimensions n x n. where S runs over the operations of the group '§ and constnJCt the matrix (A.7-1.7) where X is any arbitrary matrix and the summation is over all the operations of the group. We can show that
AD(R)
= D(R)A
(all R)
142
In8ducibla Rapr_ntetlall8 and Charact8r Taili.
Irr8ducibla R.p~lIIlt8tion8 and Charact8r Taill..
and hence A = AE (see Theorem I). We do this through the following steps
=
D(R)A
and eqn (A.7.1.8) beoomes ~ Dilt(S)Dos,(S-l) = (g/n)
D(R) ~ D(S)XD(S-l) S
or for the pth irreducible representation
1
= ~ D(R)D(S)XD(S-1)D(R- )D(R)t
=
= 0 unless p = k and q = m and X lf .. = I.
D"(R)A = ~ D"(R)D"(S)XD'(S-l) = ~ D"(R)D"(S)XD·(S-1)D·(R-1)D·(R) = [ ; D"(RS)XD.{(RS)-l}]D'(R) =
= AD'(R)
i = 1,2•... n..
(A.7-I.8)
k
=
p
••
(A.7-1.10)
ft.,
j = 1.2•... n •.
i
D.if (S)D m ,(S-l) = nAif..
~ (~,D ...(S-l)D
ifm •
2 D ..lt (S-lS)
A.7-2.
=2s a.. = g amlt
(g = number of elements in ~)
=
(g/n)
: Note that D(R-l)D(R) = D(R-'R)
(A.7-1.11)
When p = ". or r" = r' (the same representation) we have eqn (A.7.1.9) and when p =1= v we have eqn (A.7-I.IO). Eqn (A.7-I.ll) represents the Great Orthogonality Theorem.
s
Aif ..
Combining eqns (A.7-1.9) and (A.7.1.1O) and replacing the symbol S by the symbol R, we have:
:j: ~(R)D:..,(R-l) = (g/n,,) a". a., a
if
and hence
= 1.2•... n"
m = 1, 2
S i-I
nAifO' =
[~D"(T)XD'(r-l)] D.(R)
[these steps are similar to the ones we carried out in Part (il]. As D"(R) and D'(R) are chosen to be non-equivalent. Theorem II requires that A = 0, hence, choosing the matrix X as before. we get
Setting i = j and summing over i, we obtain
or.
(A.7·1.9)
where r" and r' are two non-equivalent irreducible representations of the group '[§ with dimensions n .. and n.. and X is any arbitrary matri:l:. Matrix A satisfies Theorem II since
The penultimate step is true since as S runs through the operations of the group 80. by the Rearrangement Theorem (Appendix A.3.1). does RS(= T). Replacing T by S gives the final step. Therefore A = AE. the value of A depending on the choice of X. Let us choose X to ha.ve a.ll zero elements except in the kth row and mth column and let this exception be X ltm = 1. The value of A under thuSe circumstances, we will symbolize as Alf... From eqn (A.7·1.7) and using the rule for matrix multiplication. we can obtain i = 1,2 n j = 1.2 n·
~
a_ a.,.
A = ~ D"(S)XD'(S-l)
[~D(T)XD(T-l)]D(R)
i = 1, 2,~~. on j = 1,2•... n·
r":
s (il) Construct the matrix
= AD(R).
Or, since X""
altos a.,.
~ DJ'lt(S)D:.,(S-l) = (g/n..)
= [~D(RS)XD{(RSrl}]D(R)
143
=
a"", = D(E)
(g/n)
= E.
altO'
PrDDfthat~n: = 9
"
The proof that the sum of the squares of the dimensions of the irreducible representations is equal to the order oithe group has three parts: (I) introduction of the regular representation, (2) the Celebrated Theorem, (3) the final steps.
144
Irreducible RlIJIr..entationa and Character Tables
Irreducible Representations and Character Tabln
(1) The regular repruentation The regular representation is a reducible representation composed of matrices constructed as follows: first write down the group multiplication table in such a way that the order of the rows corresponds to the inverses of the operations heading the columns; in this way E will appear only along the diagonal of the table. For example, from Table 3-4.2 we would have
E-1(= A-1(= 8-1 (= C-1(= D-l(= F-l(=
E) A) B) C) F) D)
E
ABC
D
F
E A B C F D
A E F D B C
B D E
D B C A E F
C A B D E
C
F D E A B
F C A
F
UO"(A) =
1
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
1
0
Q
0
0
1
0
0
001
•
;-.
i
n:"(R..R.) =
{~
JY:;"(R.) =
{~
if if
R;;1R; = R. R;;1 R; ~ R.
R"i1R , = R. R"i1RI ~ R •.
{I0
if R;;IR, = R.R. if R;;l R I ~ R.R.
which when compared with eqn (A.7.2.3) shows that eqn (A.7-2.2) is true and hence the matrices do form a representation of the group.
(2) The Oelebrated Theorem This theorem states that the number of times each irreducible representation r" occurs in the regular representation r- is equal to the dimension ofr" (fl.,,). This is easily proved by using eqn (7-4.2). We find, since X"'''(E) = fI and if R ~ E, zro"(R) = 0, that a = g-1 "
(A.7.2.2)
= (3) Proof that
L I'I~ =
g
L X"'''(R)x''(R)* JI
=
g-1X'''(E)X''(E)* g-1gn"
=n,..
"
By its construction the dimension of r"" is equal to the order of the group g but it must also be equal to the sum of the dimensions of all of the irreducible representations to which it can be reduced, that is
La"n" =
but since for
r"" a" =
(A.7-2.3)
g
" "", we obtain k~
From the wa.y we have set up the matrices, we know that
R;;1R, = R.R. R;;1R, #- R.R.
if if
By the Rearrangement Theorem (Appendix A.3.I) there will only be one value ofjfor which Ri1RI = R. and only one value ofj forwhichRl 1R, = R •. The sum over j in eqn (A.7.2.2) will vanish unless for some single j value boeh DW'(R.) and ~e(R.) are simultaneously non-zero and this will occur, if at all, if k::'lR )(R71R ) _ R R (~; ;1•• or R;;1R, = R.R.
000
~(R.R.) = L~(R.)~(R.).
0
J' ( .
re"(R )D"'''(R ) = I-l~; • ;1 •
It is clear that x-tEl = fI and that if R ~ E then r""(R) = 0, since only D1'Oll(E) has non-zero elements on the diagonaJ and it has unity g times (g = order of the group). We must now confirm that the matrices formed in the above way do indeed form a. representation of the group. We have to prove that if the operations of the group are R 1, R •• . ,. R., then a = 1,2, g (A.7.2.1) b = 1,2, g or, in terms of the matrix elements,
if if
u,... R ) = {I
in which case the sum will equal unity. So we have
Next, the matrix of the regular representation of the operation R is formed from the resulting table by replacing R by unity and aU the other operations by zero. :Iror example, in the above case we have 0
a.nd
146
n,."- g.
A. 1-3. Proof that the nu..lI., of irreducible repmentations I equals the number" cla_ k This proof is quite complex and we will first summarize the definitions (some of which have appeared before, see § 3.5) which are to be used. 0, = any operation in the ith ClaBB C 1 = E, class I is the identity operation
Ir.....ucibl. Repres.....tions and Charact.r Tables
Ineducible Repr. . .tlltions and Charact. Tabl..
14&
c•. =
any operation in the collection of inverses of the operations in the ith class Rf = jth operation of the mth class R = any operation of the group, irrespective of its class K. = the sum of the operations of the ith cl&88
KI=E K,. = the sum of the inverses of the operations of the ith cl&88 g, = the number of elements in the ith class gl = l.
The proof of k = r has two pa.rts of which the first is to prove that K,K1 is solely composed of complele elasses. k
(1) Proof that K,KJ = ,,_1 ~ cw.K" To understand what it is we wish to prove, consider the symmetric tripod point group ~Iv for which there are three classes and define
Kl=E Ka=
cr~
+ 0';+0';
K.= C.+ C:.
Then
since, for example,
KIKI = KIK. = KIK. = KaKa = K.K. = K.K. =
Kl K. K. 3K1 +3K. 2K. 2Kl +K.
K.K. = (cr~+cr;+o';)(a~+cr;+cr;')
As there is no opportunity for duplieation, they are therefore the same the ith claBS, hence
=E+~+~+~+E+~+~+~+E = 3E+3(C.+C~)
= 3Kl +3K. and we see that the product of any two classes produces a linear combination of complete classes. Consider the transformation R-IC,R for all operations of the ith class with some operation R of the group; the operations produced by this trans· formation are (a) equal in number to the number in the ith class; (b) all different (from the uniqueneBS of group multiplication which is itself a result of the Rearrangement Theorem); (c) members of the ith c1aBS (by definition).
&8
R-IK,R = K,.
Now, we know that K,K1 must, at least, be some linear combination of the operations of the group, 80:
K,K 1 = ... +aRr'+bR;'+ ... (A.7-3.I) where Rl." and Rf are two operations of the mth ClaBS, conjugate to each other by definition R-IRr'R = R;', (for at least one R ofthe group) (A.7-3.2) and a and b are positive integers. If we use the same R &8 in eqn (A.7.3.2) to transform the left.hand side of eqn (A,7-3.I), we get R-IK,K1R = R-IK,RR-IKJR
= K,K 1 = ... +aRr'+bRi+... (since R-IK,R = K" etc.) and transforming the right.hand side of eqn (A.7·3.I), we get Therefore
...+aR-IRr'R+bR-lRiR
=
... +aRi+ .. ··
... aRr'+bR:'+ ... = ... +aR;'+ ... and hence a = b. By generalizing this re~lUlt we see that every operation in the mth class occurs equally often in K,K1, henee, K,K1 consists only of whole claMes It
1:cH "K"
(A.7-3.3)
•• DY(R:") 1:
(A.7-3.4)
K.K 1 = where c's" are positive integers.
(2) Proof ,hat k = r We will define the matrix
~;
= (J;o;+a~O':+(J~O':'+o;cr~+G;O';+~~
+ o';a~ + cr;'cr;+ cr;'a;'
147
,.-1
by:
~; =
..- I
that is, ~, is the sum of all the matrices representing the operations of the ith olass in the r y irreducible representation. Then commutes with DY(R) for all R, since
A,
•• A; =!
DV(R:")
.....1
=
!•• !••
DY(R-IR:"R)
..-I
=
DY(R-l)DY(R:,.)DY(R)
...-1
= DY(R- I ){
~
DY(R:")}DY(R)
...-1
= D¥(R)-l A;DY(R)
(see Part (1)
148
Irreducible Representations and Character Tabl..
Irreducible Representations and Character Tabl..
[note that D'(R-I)
=
and
D'(R)-I, see eqn (6-5.2)] and therefore
DY(R) ai = aiDY(R). Because of this commutative property, we can use Theorem I (Appendix A.7-1(1» and write: ai = AiE (A.7-S.5)
where A:E is a constant matrix. Eqn (A.7-3.5) implies that the characters satisfy g,x'(Ci ) =
XNfr(R)
= {; ~: =
k
Y
= Aix'(E)
v-l
From eqn (A.7-3.3) (see Part (1» we get, by using the r' representation, (Ii 111 9' II' Ai as = ~ D'(R~) ~ D'(R~) = ~ ~ DY(R~Rn 111=11=1
1-1
m-l
k
D-l
and
~X'(Ci)XY(Cj) =
y_l
q_1
k
or
cOIgIU c{J1U
Y
h
(gIUI) b w
and for a unitary representation, since X'(R-I) = X·(R)· (soo the line below eqn (7-3.4»,
LCiif'a: ..-1
=
= =
and using eqn (A.7-3.S),
= ~ Cop~ D'(R:) ,,_1
R =1= E R = E
gpi ~X'(C,)X'(Ci) = ~ Cil .,fJ"gb pl
so:
(A 7 3 6) . - .
if if
{O g
eqn (A.7-3.9) now becomes
A:n,
A." - g. X·(C.) XY(E) .
R =1= E R=E
we have
iXY(E)XY(R) ._1
149
r
~XY(C,)X·(C/)· = (glgj) bw ,_I
This final equation shows that Ie r-dimensional orthogonal vectors can be formed by using the r oharacters X·(C.), 11 = 1,2.... r, as components and hence k <; r. But we have already shown (eqn (7-3.7» that r <; k, 80 therefore k = r as well as
and from eqn (A.7·S.6) ~
giX'(Ci ) giX'(C i ) x'(E)
x'(E)
=
gpX'(Cp)
"""e'i"
XY(E)
_1
k
gS,x'(C,)xY(C,) = xY(E) ~ cij.,fJpx·(Cp ). p_I
k
(A.7-3.7)
~X'(C,)X'(Cj)"
_1
=
(glgl)bii .
(A.7-3.1O)
In eqn (A.7·3.S) E appears in K,K j only if j = i', (that is the inverses of the opera.tions in K, all appear in one class and we denote the sum of the inverses by K,.) and then E appears U. times. Therefore, Ci:11 --
o {gi
j =1= i'
J =
t.
7.1. Given the characters X of a. reducible representation r of the indicated point group ~ for the various 01_ of '5 in the order in which the.se classes
to =
I to .. = r (r = the number of
appear in the character table, find the number of times each irreducible repre.sentation ooours in r.
If we Bum eqn (A.7-3.7) over all ", from irreducible representations) we get k
y
y
fliUj ~ X·(C.)X'(Cj ) = ~ Cii.,fJP~XY(E)X·(Cp). 11'_1
Since
y
X,e"{R) = ~a,xY(R)
,-I y
= ~n.xY(R)
._1 y
= ~X'(E)xY(R) y_I
PROBLEMS
(A.7-3.S)
.",
p=l
(rr'" =
.._1
regular representation)
(A.7-3.9)
(a) (b) (c) (d)
'll'zv ::c = 4, -2,0, -2, 'll'Sh ::c = 4, I, 1,2, -I, -1. !i'd X ~ 6. 0, -2, 0, -2, O. O. /lIh
::c
=
15, 0, -1,1,1, -3.0,5. -1,3.
7.2. Consider the four functions of Problem 5.2 which form a. basis for a reducible representation r of !i•. Using projection operators find the orthononnal basis functions which reduce r. Assume 1/;'//) = {jij' 7.3. Show that the characters of 'i'... obey the orthogonality rules of eqns (7.3.5) and (A.7-3.IO).
150
Irreducibl. RlIpr••lItlItiana .nd Ch....ct.r rabl.
7.4. How mWlY timQ8 does each irreducible representation of the 'itIT point group occur in the nine·dimensional representation found in Problem 6.31 7.6. Consider the group whoee group table is
E
A
B
c
E
E
A
A B C
A C
B E
E
C A
C B A
B C
B
E
write out the matrices Wld characters for the regular representation of this group. 7.6. Determine the irreducible representations to which the following real
orbitals belong for the indicated point group: (a) Pl' PI' P. in !p. Wld !PI.' (b) ~. do. cia, d •• d. in ~h'
(e) d l
•
(d) d l •
do, cia, d., d. in !p... do. cia. d •• d. in ~ d'
8. Representations and quantum mechanics B-1. InUaductian IN this chapter we introduce the SchrOdinger equation; this equation is fundamental to all applications of quantum mechanics to chemical problems. For molecules of chemical interest it is an equation which is exceedingly difficult to solve and any possible simplifications due to the symmetry of the system concerned are very welcome. We are able to introduce symmetry, and thereby the results of the previous chapters, by proving one single but immensely valuable fact: the transformation operators 0. commute with the Hamiltonian operator, JI'. It is by this subtle thread that we can then deduce some of the properties of the solutions of the Schriidinger equation without even solving it. Further, we will find in this chapter that wavefunctions (nuclear or electronic) must be functions which form bases for the irreducible representations of the point group to which the molecule belongs. With this knowledge we are able to determine which integrals over molecular wavefunctions are necessarily zero and this in turn (next chapter) leads to well known spectroscopic selection rules. B-2. Th. invarianca of Hamiltonian
OPIII'BtOrs
und.r 0.
Aside from relativistic and quantum electrodynamic effects. a single molecule in free space is completely described by the Schrodinger equation (8-2.1) where .71', the Hamiltonian, is an operator defined by certain quantum mechanical rules and which can be determined solely from a knowledge of the number of electrons and nuclei and the charges and masses of the nuclei. 'Y, the total wavefunction, is a function of the coordinates of the electrons and nuclei; it defines through further quantum mechanical rules all the properties of the molecule. E is a constant and is the total energy of the molecule. Eqn (8-2.1) is similar to eqn (4-4.1) and E and 'Yare ca.lled the eigenvalues and eigenfunctions of the SchrOdinger equation. The equation is solved by finding functions '¥ such that when they are acted upon by .71' the functions are simply regenerated multiplied by a constant. For molecules containing more tha.n three particles, solution of Schriidinger equation is no easy matter (because of
RepresentatiDna and auantum Mechania
1&2
R.p......tiona and auantum Mechanics
ma.thematical difficulties) and one hB.8 to resort to approximate methods. The reader might note that if one is interested in systems which are changing with time, then eqn (8-2.1) takes a slightly different form: the time-dependent Schrodinger equation and the eigenfunctions of this equation are functions of time. One way of simplifying eqn (8-2.1) is to use the Born-Oppenheimer approximation; we will not go into the details of this approximation but baldly state the results which come from its application; these are: '¥
=
where explicitly 11'•• is a function of the coordinates of the electrons alone and '/'"..e is a function of the coordinates of the nuclei alone. IT we consider a molecule with 11. electrons and N nuclei, we can let X.I symbolize the collection of 3n electronic coordinates :z:ill, :I:~l). :I:~lJ, :1::"1, III~'). Z~"I, •.. z1"), z~"l. z~,,) and X nue symbolize the collection of 3N nuclear coordinates (or displacements of the nuclei from certain equilibrium positions) E~II, '~'). ,~", ~~." E~Nl, ~iN); so that
,:Il,
,i"l ... ,:Nl,
'/'.1 = tpo.(X•• , X "ue)
and
lpuuc
=
'/'"..e(X"ue)
and we refer to X •• as the electronic configuration and to X nue B.8 the nuclear configuration. In the Born-Oppenheimer approximation '/'•• and lI'"ue are determined by two separate equations, an electronic equation H.I'P.I = E ol lp.1 and a nuclear equation
H...ue'Pnue
=
E,/,uuc
(8-2.2) (8-2.3)
where H.I = T.1 +Vol Tel = electronic kinetic-energy operator -h' " 8n-Om '_I
= - - ,}; V:
=
oz~".
+
Oz~·'·
+
0'" iJzi'"
-h' N i ,
= 8.".' -}; -V, ._, M.
(8-2.7)
(8-2.8)
Vu"e = E.I(X"ue) +nuclear repulsion terms =
V: =
mass of the ith nucleus Laplacian operator for the ith nucleus
iJI
=
o~10 I
iJI
0"
+ iJ,~'1 I + o,i'l I .
(8-2.9)
Essentially Chapter 9 is concerned with the solution of the nuclear equation (eqn (8-2.3», which involves the subject of molecular vibrations, and Chapter 10 deals with examples of the solution of the electronic equation (eqn (8-2.2». The reader will have observed that the eigenvalues of the electronic equation E OI which occur in V"ue are normally required before the nuclear equation can be solved, the latter equation providing the final total molecular energy E. From our point of view the most significant thing about the Hamiltonian operators H.I and H"ue is that they both commute with the operators OR' we say that H•• and H nue are invariant under all symmetry transformation operators of the point group of the molecular framework (8-2.10) ORH•./(X••) = HolO.!(X••) and (8-2.11) O.Hnuc!(X"ue) = H"ueO.!(Xnuc ),
(8-2.5)
ORT•./(Xo.) = T.IO.!(X. I ) O.Tnuc!(Xnuc) = T...ueO.!(X"ue)
(8-2.12) (8-2.13)
O.V•.J(X.I ) = V.IO.!(X.,)
(8-2.14)
O.V"ue!(Xu..e) = VnuoO.!(Xnue)
(8-2.15)
Laplacian operator for the ith electron 3"
T"ue+ Vuuo T uue = nuclee.r kinetic-energy operator =
where !(X••) and !(Xnue ) are functions of the electronic and nuclear coordinates respectively. We can prove the truth of eqns (8-2.10) and (8-2.11) by showing that
and electron-nuclear terms) m = mass of the electron
as
H""e
(8-2.4)
V•• = potential-energy operator (contains electr<>n-electron
V: =
and
M,
tp.''/'"ue
153
(8-2.6)
and this is done in Appendix A.8-1. Now consider the situation when there are degenera.te solutions to the equation Hlp
=
Etp
(8-2.16)
154
R.p.......t.tio... and Ouantum Mllch.nics
Representations .nd au.ntuM MICh.nia
(this equation stands for either eqn (8-2.2) or (8-2.3) and what follows is good for both electronic and nuclear C&8Ils). We have
H¥'r H¥';
=
E y ¥':
= Ey
¥'; etc.
and since H is a linear operator (see § 2-2) any linear combination of etc. will also be a solution with the same eigenvalue E y, i.e.
¥';, ¥';.
Therefore, all the solutions with E = E y form a function space associated with the energy E y (see § 5-5) and if n y of them are linearly independent, the space will be ny-dimensional. y If we can show that the functions (all and j = 1. 2•...• ny) also belong to the function space for energy E •• then we can write
¥';. ,,;..... ¥':
0.,,;
0.,,;
=
~ D;i(R),,:. • _1
R
j = 1,2..... n•
(8-2.17)
and the ,,; will form a basis for a representation (with matrices DY(R)) of the point group whose operations are R. It is easy to prove this in the following way: H(O.'P;)
=
O.(H'I'i)
=
O,.Ey"';
(H and O. commute)
= E.(O.,,;)
and therefore 0.",; is a solution of eqn (8-2.16) with an eigenvalue E y and it is consequently a member of the function space; it can therefore be written in the form of eqn (8-2.17). We sa.y that the linearly independent degenerate wavefunctions for energy level E. form a basis for the representation r Y • We as_me that all the degenerate wavefunctions associated with E. can be obtained by all the O. acting on a given wavefunction; this is known as normal degeneracy. Any degenerate wavefunctions which cannot be obtained in this way we consider to be accidentaUy degenerate, i.e. accidental degeneracy has no obvious origin in the symmetry of the system. Barring such an accidental degeneracy. the representations produced by eqn (8-2.17) will be irreducible since no smaller matrioes could express the most general transformation. We can think of aocidental degeneracy as the numerical coincidence of a number of different energy levels. each with its own irreduoible representation.
155
Or. put another way, if the matrices in eqn (8-2.17) are not irreducible then the degeneracy is deemed to be accidental.t Since we know a.ll the irreducible representations of a point group. we can tell a lot about the possible solutions to eqn (8-2.16), without actually solving it. For example. for ammonia which belongs to the ~sv point group. we know from Table 7-7.1 that its electronic and nuclear wavefunctions must fall into the following three categories: those which are non-degenerate and totally symmetric (unchanged by all the transformation operators 0.), they will belong to the rAJ representation; those which are non-degenerate. unchanged by 0c. but which change sign under 0" • they will belong to rAt; pairs which are doubly degenerate and under both OCt and 0a. each wavefunction of a pair produces a linear combination of itself and its partner, they will belong to rEo
.-3.
Direct product representations within II group It is always possihle to form a new, and in general reducible, representation r of a given point group from any two given representations r" and r Y of the group. This is done by forming a new function space for which the basis functions are all possible products of the basis functions of r" and r Y• Let the basis functions of r" and r' be
If.J:, .. ·!:'.. and respectively, where n. and n. are the dimensions of r" and r v • Then Y the basis functions for r will be JrJi (i = 1,2•...• n,,; j = I, 2, ... n,). We will put these functions in what is called dictionary order gl = g...+l •
t Examples
..
•
= •
Jm,
g. =
1m. g...
+1
•
•
•
•
•
•
fm. = J:I;,
..
•
•
•
g..v g.... •
• Yn1J'rIl'
JrJ;, = 1:1;.
=
= f~,.f~",
of accidental degeneracy are rare except in problems in which the Hamiltonian involves & continuously variable parameter, such as the strength of an electric or magnetic field. In 8uoh oaeee accidental degeneracy can occur for certain speoifio values of the parameter in queetion at which & pair of energy levels crOM. Suoh degeneraoy is exoeptiona.l t unprediotable, and easily recognised when it does occur. The reader might note that some authors olBSllify the degeneracy of hydrogenic wavefunctions of the same 110 value &iii aooidental.. However, this in fact is not aooidenta.l degeneracy since Fool< (Z.it8clInjt fur Phyfti<. 98. HIS (1936») has shown that the degeneracy aan be considered to arise from a four·dimensional rotational symmetry of the Hamiltonian in momentum "p""'. A complete di.scusaion of group theory and the hydrogen atom h .... been given by Bander and Itzykson (Roview8 oj modern PhY8k8. 98. 330 (1968».
158
R......ntatiollS Inti Qulntum MlIChlnics
ReprBS8ntatiollS and Quantum MlIl:hanics
then gives the following table:
so that with this convention our new basiB functioDB are
The new functioDB will form a basis for a n"n.-dimensional representation which we will symbolize by (8-3.1) f"'®r'is called the direct product (or Kronecker product) of the representations r" and r'. The sign ® does not mean multiplication, it is simply a signal that the direct product of two representations has been formed in the manner given above. That the g functions do indeed form a basiB for a representation of the point group can be verified by considering the effect of the transformation operators OR (8-3.2)
or, if we define
(8-3.3)
i
j
r
1 1
1
2
1 2
1 2 2
n,
71.
1 2
71,+1 n,+2
2
n,
2710.
nIl
n.
n.n.,
The same table holds true with i, j, and r replaced by p, q, and 8 respectively.· . A d As an example, consider the direct product of a 2 X 2 matrIx an a 3 x3 matrix B, then eqn (8-3.3) gives:
r
ll
nJl"V
01ffJ. = ~ D~'(R)go
0_'
(8-3.4)
where, by using dictionary order, the sum over p and q becomes a sum over 8. The n"n, product-functions g. are thus transformed into linear combinations of one another by the transformation operators 0lf of the group; they therefore form a basis for the r (= r" ® r') n"n.dimensional representation of the point group. The matrix D"0'(R) that corresponds to the symmetry operation R is a square matrix of order n"n. and its elements are the (n"n.)· possible products of each of the elements of D"(R) with each of the elements of D·(R). The matrix U""'(R) iB clilled the direct product of
n:
71;
U(R) and D'(R):
D"®'(R) = D"(R) ®D·(R).
(8-3.5)
The direct product of two matrices is quite different from the ordinary matrix product. First, let us consider how the indices of the va.rious matrix elements are related. By comparing eqns (8-3.2) and (8-3.4) we have ~"'J' _ Ji i - g . and
157
f':[:, = g••
Hence the subscript r in ~'(R) is determined by the subscripts i and j, while 8 is determined by p and q. The dictionary-order convention
B u B.. B 13 B n B•• B 18 B a, B.. Baa
AII\l ®
A..
An
A u B l1 AuBa AllBn A liB•• AuBn A liB•• AIlBn A"B18 AnBn AnB.. AIlB•• AnB u
A ll B 13 AllB.. AuB"" A 81B 18 AnB•• AIlBn
AIIBll AIIBn A l I B n A II B 8I A l IB n A liB.. AuB u A ••B II A••B n A••B•• A ••B n A..B..
AliB,. AaB.. A liB•• A ••B•• A••B•• A..B••
(8-3.5)
which can be partitioned into four sub-matrices
AuB I A 13B A®B
=
\
~nB-I·-~~~~
and is very different from ordinary matrix multiplication. The characters of the direct product representation will be n~n.,
"~ ft.
~_1
t_11_1
X"""'(R) = ~ D:r(R) = ~ ~D:;(R)D;i(R) = X"(R);(R)
(8-3.6)
158
Representations entl Quentum Mechanics Representations and Quantum Mechenics
(the easiest way of confirming the subsoripting used here is to look at eqn (8-3.5». We oan, of course, take the direct product of more than two representations, for example r = r~®rp®r" (8-3.7) will, by extension, also be a representation of the point group. We simply apply the direct produot rule twioe. The oharaoters for this representation will be Z~@P®V(R) =
where 'P·(X) and 'P'(X) are electronic or nuolear wavefunctions whioh belong to energy levels E~ and E p respeotively and form a basis for the irreducible representations r· and r p • X is the usual eleotronio or nuclear configuration and Fl(X) is a given funotion of X belonging to a function space whioh we will assume generates the r A irreducible representation. The integration is carried out over all of the electronic or nuclear coordinates. It is apparent that the integrand, 'P~(X)·Fl(X)'PP(X),
Z~(R)z"(R)ZV(R).
In general, the direct product representations are reducible and using the formulae of § 7-4 we have, if r' are irreducible representations
rp®r v
=
• .Ia.r·, '_1
is one of the p basis functions for the direct product representation I"'*®rl®r . (Note that 'P·(X)* belongs to r"~)t If we carry out the reduction of this representation
(8-3.9)
(k = number of classes = number of irreduoible representations)
•
I"'*®rl®rp
=
aoro+a,r'+...
~hen the original bam functions, 'P~(X)*FA(X)'f'P(X), can be expressed
(8-3.10)
terms of the basis functions which generate the irreducible representations ro, r' etc. i.e.
i=l
V'~(X)* Fl(X)'PP(X} =
(8-3.1I) This teohnique of decomposing a direct product representation will be of great use in the next section. The reader is oautioned that the term direct product has a second meaning in group theory. If the group '§ has the elements gl' g., ... g.. and the group.Jt" has the elements hi' h., ... , 11.", and if g.h/< = h1og, for all i and k, then the group whose elements are = =
1,2, 1,2,
(8-4.1)
10
ZP@'(R) = .Ia.X'(R),
i k
1&9
n m
is said to be the direct produot of group ~ and .Jt", e.g. ~'d = ~.®~''t. ~." = 'j'.®'j'•. t We have already come aoross this situation in § 7-8. 8-4. Venishing integrels
In this section we derive certain rules whioh will determine whether or not an integral over given eleotronic or nuclear wavefunctions vanishes; from such rules we can deduce spectroscopic selection rules. Consider the integral
f 'P~(X)* Fl(X)'f'P(X) d.,.
t The reader may check that the operations of 'if, (E and i) commute with all the operations of tj. and that the operations of 'if. (E and 0h) commute with all the operatio"," of 'I,.
cJ" +c,J' +...
(8-4.2)
If we apply to the integrand the projection operator PP (see § 7-6) where PP =
and it happens that
.I XP(R}·O" "
PP<'f'~(X)·FA{X}!pP(X}} = 0
then I'" does not appear in eqn (8-4.1) andr does not appear in eqn p (8-4.2). If r-*®rA®r does not contain r l , the totally symmetric representation (for which ZI(R) = 1 for all R and pI =.I 0) we will have Il '"
o=
PI{'f'''(X)* Fl{X)'f'P(X)} =
.I 0,,<'P"(X)* Fl(X}'f'P(X}}.
(8-4.3)
"
But the Oil are such (see § 5-7) that for any function Q(X)
f Q(X} d.,. f O"Q(X') d.,.' = f O"Q(X} d.,., =
therefore, summing over all R, 9
f Q(X) dT = f ~ O"Q(X) d.,.
where 9 is the order of the group, and consequently
f 'P"(X)· FA(X)'f'P(X} d.,.
= g-l
f -t O,,{'f'~(X)· F'(X)'PP(X)} d.,..
t ro- is the irreducible representation whose matrioe. D"(R)* are the oonjugate Complex of the matrioe. of roo If D"(R) are real then ro. = ro.
168
1&1
H . -••• i.....nd QuantuM MlICIIDnics
Hence from eqn (8-4.3) we have the important result that the integral
and if the coordinate system has mutually perpendicular axes, the matrU D(R) will be orthogonal. so that
(this is a. sufficient but not necessary condition). This same oondition may be expressed in an a.lternative way. If %,,(C;) is the character for any element in the ith class in the 1''' irreducible representation, then
i = 1,2,3 = 1,2,3. Now, by definition, if J is some function of the coordinates J(X1>
f ,,~(X)· F(X) "p(X) dT will be zero if 1'1 does not appear in 1'''. ~ 1'A~ 1' p
x~·@"(C,) = X"'(C,)·X"{C,)
(note that X~·(C;) = conjugate oomplex of X~(C,) = X"(C,).) and the number of times the totally symmetric irreducible representation 1'1 occurs in the reduction of r"'. ~ 1''' is
a,
= g-I
L" g,X"'·@"(C,)XI(C,)·
i_I
= g-'
L~ g,X"'{C,)·X"(C,)
i_I
and recalling eqn (7-3.5) and the fact that 1''' and 1''' are irreducible representations, we have (8-4.4)
Hence 1'1 appears once in r"'·~1'1' if fl = a and not at all if fl ¥ a. p Now consider the direct product representation 1'''.Q'il1'AQ'ilr . !fin the reduction of rt~rp the representation r" does not occur, then by eqn (8-4.4), a, = 0 and r"'.~rA~rp does not contain r l and
Jtp"(X)· FA(X)!pP(X) d1'
.i D,~(R)D/~(R)
j
(A.8-1.2) x •• x.)
O. V"J(x1• XI' x;)
V"J(xl , x.' %.) %2' %;). The right-hand side of this equation has the form V"g'. where =
= V"Ollf(x~,
(A.S-l.3)
g' = 0.I(X1, X;. x;) and V· refers to differentiation with respect to Xl. x.' X •• Now og' ~' oxi ()g' ()XI ()g' ox.
=OX1 - -ox~+ox. - ox~ - +ox;-ox~ ox~
and since, by eqn (A.8-l.I), ()X~
- ' = D,,,(R)
we have
()x" ()g'. og' = L D,~(R)-, . '_lOX,
Differentiating once more with respect to
x~
gives
(8-4.5)
p
unless r"' = r , i.e. unless the two wavefunctions belong to the same irreducible representation. (see eqn (A.8-l.2»
Appendix
A.I-1. Proofohqns (1-2.12)'. (1-2.15) The proof of these equations follows that given by Schonland. To prove eqn (8-2.12), consider first a single point with coordinates x1> x •• and X •• Under the operation R this point moves to x~. where, by eqn (5-2.17), i = 1,2,3 (A.8-U) xl = LDH(R)xi ,
x;,
•
i-I
t See page 218.
x;
then
O.J(xi,. xl, X;) = J(~, x.' x.) and if we form a new function V·J, where V· is the Laplacian operator, then:
ox"
is zero. So that reduction of rA~1'p and checking whether it oontains r" or not is all that is required to see if the integral vanishes. Also, if FA(X) is replaced by an operator H which belongs to the totally symmetric irreducible representation P[X'(R) = 1, all R]r then
= 6'1
k-I
182
R.,resentlltiollS end Ouentum MlIClleniCII
Likewise V"!(x~,
x;, xa) =
V'"!(x~,
[o.(V'"f)](xi,
and since ~,
183
x;, x;) and eqn (A.8.l.3) becomes
X8' x~)
V,I[(O./) (x~, x8' x;)]
=
XI, Z8 occur throughout this equation. we conclude that 0. V"! =
V"OIlJ.
(A.8-l.4)
Taking an equation like eqn (A.8.l.4) for each electron and multiplying by -h"/8-rr"m and adding we obtain eqn (8.2.12). To prove eqn (8-2.13), let us suppose that R, when it is applied to the nuclear fl'&mework, changes any general nuclear oonfiguration from X U1l0 to X~uo' then if the base vectors are transferred as in § 5.4(2) (see also Fig. 5-4.3), we have, in terms of coordinates I'&ther than base vectors,
~.)' =
f D;/(R)~~P),
i = 1,2,3
(A.8.l.5)
I-I
where & displacement from the equilibrium position of nucleus q has been transferred to where p was before the operation was carried out [in § 54(2) we combined the N equations (one for each nucleus) like eqn (A.8-l.5) for the base vectors together to obtain a 3N.dimensional matrix]. A slight change in the d6rivation of eqn (A.8.l.4) then leads to
case is the specifio nuolear oonfiguration llJIed to define the molecule's symmetry. If a symmetry opel'&tion R is first applied to the whole moleoule. &11 partioles (electrons and nuclei), then the relative positions of the particles are unohanged and so is V. l V.I(X. I• X uuo ) = V.I(X~I' X~uo)' If we now apply Jrl to the nuclei aZo- then, since this only interchanges like nuolei and by definition leaves the nuclear framework physically unchanged, V. l still remains the same V. I (X.1> X uue ) = Vel(X~I' X~ue) =
the proof of eqn (8.2.15).
PROBLEMS 8.1. To what irreducible representations oan the following direct product
representations be reduced for the specified point group? (a)
0. V:f(Xuuo ) = V:O.!(Xuuo )'
Because of the nature of R, p and q must be physicaJly identical and therefore have the same mass, so that
V.I(X~I' X nuo )'
So that for the fixed nuclear configuration which defines the molecule'a symmetry, the change of electronio configuration caused by R, X. I -.. X;I' leaves V. I unchanged. The rest of the proof of eqn (8.2.14) is the same as
(b) (e)
rA,®r.d" rA,®r.d., r.d.®rE• rE®rE for r E' ®rE', r.d; ®r.d;. r.d;®rE- for !jab rE,®rE" rE,®rE., rE.®rE• for ~'T'
~aT
8.2. To what irreducible representation must.,. belong if the integral
f
OR-!.. V:! (Xnuo ) = -!.. V:Oll!(Xnllo )
Mp M. and eqn (8.2.13) follows by addition. Let us now consider eqn (8.2.15). Vnuo is solely a function of the relative positions of the nuclei, Le. Vnuo = Vuuo(Xuuo)' Any symmetry operation must leave theae relative poeitions, and hence Vnuo' unaltered, i.e. if under R any general nuclear configuration X nuo becomea X~uo then Vnuo(Xnue) = Vnue(X~no)'
(A.8-1.6)
From the definition of O. we have
or and
O.Vnu.cX~uo) =
Vnuo(Xnuo)
=
V nuo(X~uo)
O.Vnuo = Vuuo O.Vnuo!(Xnuo) = °IlVnuoOll!(Xnuo)
= VnuoOll!(XnDe) which is eqn (8-2.15). Last we must prove eqn (8.2.14). is a function of the relative positions of the electrons and nuclei, that is V. I = V_I(X_ I , X nuo ) where X nuo in this
V_,
tp'(X) " pA(X)tpP(X) d,.
is to be non· zero in the following casee?
r A= rEo r p = r.d,. r.d., r B" r B• r A = r E ,.; r p = rEo. r d r A= 1'2"'; r p = r.do. r E, r T" 1'2"'.
(a) ~.v
(b) !jfJl (e)
Mol_lar Vibfltiona
Or we can use the so-called mass-weighted displacement coordinates
9. Molecular vibrations
with velocities: 9-1. Introduction IN this chapter we apply the results of the previous chapters to the problem of molecular vibrations. Before doing 80, however, it is necessary to have some knowledge ofthe quantum-mechanical equations which govern the way in which a molecule vibrates. We find that the solution of these equations is greatly simplified by changing the coordinates of the nuclei from Cartesian coordinates to a new type. defined in a special way. called the normal coordinates. This change is no more mysterious than changing. say, from Cartesian coordinates to polar coordinates when solving the Schrodinger equation for the hydrogen atom; the basic principle is the same, namely the mathematics is made easier. So we start this chapter with a discussion of normal coordinates. We then discover an extremely important fact; each normal coordinate belongs to one of the irreducible representations of the point group of the molecule concerned and is a part of a basis which can be used to produce that representation. Because of their relationship with the normal coordinates, the vibrational wavefunctions associated with the fundamental vibrational energy levels also behave in the same way. We are therefore able to classify both the normal coordinates and fundamental vibrational wavefunctions according to their symmetry species and to predict from the character tables the degeneracies and symmetry types which can, in principle. exist. Furthermore, knowledge of the irreducible representations to which the vibrational wavefunctions belong coupled with the vanishing integral rule tells us a good deal about the infra-red and Raman spectra of the molecule under consideration.
9·2. Normal coordinates If we consider a molecule with N nuclei, then the displacements of the nuclei from their equilibrium positions in Cartesian coordinates can be written as ~(1) ~Ul .(1) .ell ~(NI ~1
J
li"2
'''-8
'''-1 , .... "1
E~l). E~l), E~ll, E~.)
q~l), q~ll. q~ll, ... q~NI
(9-2.1)
q~ll, q~ll. g~ll •... 4~N)
(9-2.2)
where
q~O = Mt~}')
and M, is the mass of the ith nucleus. In actual fact it will be more convenient to let the subscript on the g's and q's run over all the coordinates and velocities, Le. from 1 to 3N, so that we have:
q1' q., g., ... q.N and
4.. g•• g••... '.iON
in place of eqns (9-2.1) and (9-2.2). In classical terms, if we use the mass-weighted Oartesian displacement coordinates, the kinetic energy of the moving nuclei ist (9-2.3) (these terms are of the familiar !mil" type) and the potential energy. relative to its value when the nuclei are in their equilibrium positions, is V, which can be expanded in a Taylor series as:
V
=
IN
(oV)
ON aN (
O"V )
~ g,+i~ ~ - - gOj+'" '-1 oq, 0 '-1 1-1 og,oqj 0
(9-2.4)
where the subscript 0 denotes that the derivative is evaluated when the nuclei are in their equilibrium positions. Since. by definition, V is minimal for the equilibrium configuration, we know that
_ ( oV) og, 0 -
0
i = 1,2•... 3N
(9-2.5)
and if we replace the second derivatives (which are called the harmonic force constants and are intrinsic properties of the molecule under consideration) by o·V ). i = 1. 2, ... 3N (9-2.6) B'j = (-0 og,ogj :J. = 1.2,... 3N and stop the expansion after the quadratic terms (the harmonic oscillator approximation), we have (9-2.7)
and the corresponding velocities as where
115
,...
~~N)
t T is the olasaical analogue of the quantum mechanical opezator T a •• defined in eqn (8-2.7). :: If the potential energy of the nuolei in their equilibrium positions is W .. q , then V+W•• = Va••, where V n • is defined ineqn (8-2.7).
117
1••
For thie set of 3N simultaneoue equations to have non-trivial solutions for the C_. the following equation must hold true (eee Appendix
The classical equation of motion for the moving nuclei is d de
(aT) av aq_ +ag( =
0
~
= 1.2•... 3N
(9-2.8)
d"ql ~ ~(f d'l
aN
•
IN
+1_1 ~ B,RI =
~
0
=
1. 2 ... 3N.
det(B-AE) = 0
(9-2.15)
where B is the matrix formed from the elements B_, and E ie the unit matrix. There will be 3N roots (values of A) of eqn (9-2.15) which, when found. can be used in turn to solve eqns (9-2.14) for the C_ (one ad-
and using eqns (9-2.3) and (9-2.7) this becomes 1_1
A.4-3(a)):
(9-2.9)
IN
ditional equation, a normalization equation. ~ ~ = 1 is required to i_I
Now let us choose a set of 3N coefficients C 1• C I •••• and CIN euch that when each of the eqns (9-2.9) is multiplied by the appropriate C i and the 3N equations are added, we obtain (9-2.10)
determine all of the 3N C'e). Since there are 3N A valuee, there are 3N eete of C( which will produce eqn (9-2.10). For convenience, we will add a subecript to A and Q to distinguish the different solutions and an additional subscript to the C's to ehow with which A value they are aIISOciated, i.e. IN
Al : Cu,
where
__1
Cu.···
C IN1 :
Q1 = ~
Cn
C INI
Q. = ~
(9-2.11)
(i.e. Q is a linear combination of the mase-weighted displacement coordinates) and A is a constant. There will be. in fact. 3N ways of ma.king the choice of the 3N coefficients. We can see the reason for this by looking at the equalities which muet exist between eqns (9-2.9) and (9-2.10). that ie we must have aN
IN
~ C_ 1_1 ~ ~(f
__1
CaN ) ""1 ~ hRI
dlg, ""1 lU
d
=
lU
j
and
IN
-1
=
1, 2, ... 3N
(9-2.12)
aN
IN
aN
AI: C n
,
•···
:
__1
C_IQ_
IN
AaN
: C UN • C UN , .. · C aNIN : QO N =
l
or
C i1Qi
~ Cn~_· '_1
The Ql' Q., ... QsN are called normal coordinate8 and what we have done is to transform the coordinates q_ to another set Q, auch that eqn (9-2.10) ia true. We can form the matrix C by using the coefficienta for each .( value as colum1J8:
~ C_ 1-4 ~ B_nl = A1_1 ~ hnl
__1
or
IN
__1
~
C_B_,
=
j = 1.2.... 3N.
lk ,
From eqn (9-2.12) we get
(9·2.13)
C_ = AJ
CaN 1
IN
Q
and hence
and by combining eqns (9-2.12) and (9.2.13) we have IN
~ (B_'-.(~_/)O.
=
0
j = 1.2•... 3N.
CaNIH
0INI
and since B ia eymmetric, thie matrix will be orthogonal (see Appendix AA-3(c». As well as satisfying
= ~On, 1-1
__1
C=
(9-2.14)
dlQ, dt l
+ A,Q_
= 0
~ =
1.2•... 3N
(9-2.16)
..
,
Mol_lar Vlbrlltloll8
we have .t, > 0 (Q~ is always positive). Now the only way in which
the normal coordinates also satisfy:
aN
aN
T=iLQ" ._1 i and
(9-2.17)
aN
V
=
1'_1 L A,Q:
(9-2.18)
(these equations are proved in Appendix A.9-1). The solutions of the equations of motion (eqn (9-2.16)) are easily found to be: Q. = A, cos (;Jt+e.) i = 1,2.... 3N (9-2.19) where A, and e, are constants and t is the time. Since aN
Q,
=
L
1_1
0 1&1
i = I. 2, ... 3N
(9-2.20)
!<~ A.Q~ can be zero for at least six of the Q,'s having non-zero values, if A, > 0, is for there to be at least six A, which are zero. In fact, there must be exactly six A, which are zero because if there were more one would be able to carry out a normal mode which is not necessariIy a combination of translations and rotations and still have V = 0, this cannot be done because if the bond lengths or the angles change then
V> O. We therefore aasociateQI' QI•..• QaN-o and AI' AI •... AIN_O (all positive) with vibrations and QaN-i. QaN-4•... Q IN and AaN_I = AaN-4 = ... AIN with translations and rotations and write
V
aN
L O#Q. ,_1
j
=
1.2'00' SN
1: Ol/A, cos (Att+e.)
and if all the normal coordinates are zero except one. say QI< (that is A, = 0 except for i = k), then
=
OI~1< cos (Att +eA)
(9-2.21)
and each m&8ll weighted Cartesian displacement coordinate is varying sinusoidally with time with a frequency of "1< where 2"'''A = At. Such a motion is called a fiONIWol11Wde and there are clearly 3N such modes, each one associated with one of the 3N normal ooordinates. Some nuclear displacements will be such that the bond lengths and angles in the molecule are unchanged from their equilibrium values and V is consequently unchanged; such will be the case for translation and rotation of the molecule as a whole. For a non-linear molecule, a rigid movement of the molecule as a whole may be expressed as a combination of tranalations along, and of rotations about, three chosen axes. To describe any general translation-rotation movement we will necessarily require six coordinates and therefore at least six normal coordinates; hence V is zero for at least six of the Q, having non-zero values. Since V is measured relative to the minimum potential energy (the value for the equilibrium nuclear framework) V > 0 and since aN
V
=i <-1 1: A.Q:.
=t '_I L
A,Q:.
aN-I
<_1
ql
0
For a linear molecule there are only two independent rotations and an argument similar to the one above leads to:
aN
=
=
aN....
and 0 is orthogonal, we have ql =
,.1
V
=1 '_I L
.t,Q:.
The point of changing from Cartesian displacement ooordinates to normal coordinates is that it brings about a great simplliication of the vibrational equation. Furthermore, we will see that the normal coordinates provide a basis for a representation of the point group to which molecule belongs. 8-3. The vibrational equation
The nuclear equation (eqn (8-2.3)) written out in full is -hi
- 'IT 8 I
N
1
,L M _1
-I
V:V'nuc(Xnuo)
+ VnuoV'nuo(X
DD . )
= EV'nuo(Xnuo )
(9-3.1)
where the symbols have been defined in § 8-2. It is possible to approximate V'nUO as the product of three functions, one a function of the coordinates describing translation V'u, another a function of the coordinates describing rotation V'ro& and the third a function of the normal coordinates describing vibration V'Vlb. that is:
""'10
= 1lrtp"'~"vlb.
Eqn (9-3.1) can then be separated into three eigenvalue equations for the three types of motion. The three eigenvalues will be W" (translational energyl, W"" (rotational energy) and W"lb (vibrational energy)
Moleculor Vibroti_
178
Mol_lor Vibrations
and the total molecular energy E is given by
E = Wi< +
wrot +
W'Ylb +
where: N, is a normalizing constant and is chosen such that
weq
where WO<1 is the energy of the molecule when the nuclei are kept fued in their equilibrium positioIl8 (V in eqn (9-2.4) is relative to WO<1). Of the three eigenvalue equatioIl8, the one of interest to us is the vibrational equation. It has a particularly simple form when normal coordinates are employed because the classical kinetic and potential energies then have no cross terms (see eqns (9-2.17) and (9-2.18» and this fact leads to a simple form for their quantum mechanical analogues (the kinetic energy and potential energy operators). The vibrational equation is thus aN.....
o.
:aN.....)
8 .(
L aQ.f ,,"Ib(Q., ... Q.N.....) + ,....: ._1
1 "_1 L J..R:
wvlb -
,,'Vlb(Q., ... Q.N-.)
=
0
(9-3.2)
and if we replace WVib by a sum of terms IN-fl W"'b =
L
i_I
W(
+«>
f ,,:(Q() dQ,
= 1,
-00
lX,
_ 2.". '. __-",_' 4."..". -hA,
At
", = ~ (fundamental frequency),
2 'IT n, = 0, 1,2, ... (vibrational quantum number), Hn,(:r:) = a Hermite orthogonal polynomial, e.g. Ho(:r:) = I, H 1 (:r:) = 2:r:, H.(x) = 4x·-2, H.(x) = Sx·-12x, etc. The fundamental frequencies '" (i = 1,2, ... 3N -6) are related to A, and since J.., are the roots of det(B-lE) = 0, ", are related to the matrix B and to the molecular force constantsB'I' Hence the vibrational energy levels for a non-linear polyatomic molecule in the harmonic oscillator approximation are given by .N.....
and "v1b(Qt> .•• Q.N-
W"lb =
L (n.+1)1111, '_1
II ,-. ",,(Q()
"vib
and divide by "vlb(Ql"oo Q.N-I), then we have aN-fl{
1
~1 ",(Q,)
d.tp,(Q,)
dQ:
8.".. • } +];1 (W,- iJ..,Q,)
=
Each term of this sum is a function of just one normal coordinate and is independent of all the other terms. Since the sum is zero, each term must be zero _1_d·tp,(Q,) 8r(W _11Q") = 0 tp,(Q,) dQ: +"," 'I ( , or d'~~Q,) +8;: (W,-i1,Q:)tp,(Q,) = 0 (9-3.3)
Eqn (9-3.3) is the same as the well known one-dimensional harmonic oscillator equation and has as its solutioIl8 Y',(Q,) = N, exp( -ll1,Q~)Hn.(<
and
W,
=
(n,+i)h",
= " ..
,n.......,._. =
N exp
.N..... ).N-
i_I'
(9-3.7)
and both W"'b and ",VIb are characterized by the values of the 3N-6 vibrational quantum numbers 11.., 11.., .•• naN_I' We see from eqn (9-3.7) that the lowest energy state, the ground state, occurs when n. = n. = ... naN_I = 0 and the energy is then
o.
i = 1,2•..• 3N -6.
(9-3.6)
and the corresponding vibrational wavefunctions are given by
IN-fl
"v1b(Q., ... Q.N-I) =
171
(9-3.4)
aN-
wrb = '_I ~ ih",.
(9-3.8)
~Ib is called the zero-point energy. The ground state wavefunction is ".Ib =
N exp (_iaNL.....ct,Q:)
'_I
(9-3.9)
where N is a normalization constant. The vibrational energy levels where all the quantum numbers are zero except for one which is unity are called the funda1Mntal leve18. There will, in principle. be 3N - 6 such levels and the energy of the pth one (11.. = 0, n. = 0'00' n" = 1'00' n.N _ I = 0) will be IN.....
~b
=
L ih",+(1 +1)11,,,,, = wrb+lall". ,,.,,
(9-3.10)
172
Moleculer VibretioDS
Moleculer Vibretlolll
Therefore the frequency of the radiation absorbed or emitted in a transition betwoon the ground state and a fundament&1 level is (w;tb - w:;")/h = "... the pth fundamental frequency. The corresponding wavefunction will be V'~b
=
V'00- ..1 ..•• oc V';"'Hl(ot~Q ..) oc V''ff'Q...
We can change the basis of the above representation by switching to normal coordinates (or normal coordinate vectom) and obtain a representation 1'" which ~ equivalent to ro (soo § 6-2). The change of basis is defined by (see eqn (9-2.20))
(9-3.11)
The infra-red and Raman spectra of molecules are dominated by transitions between the ground state and the fundamental levels but. in practice. the number of fundamental frequencies observed does not reach 3N -6 since (a) some of the A, are identical (leading to degenerate fundamental levels) and (b) selection rules forbid certain transitions. Both (a) and (b) are determined by the symmetry of the molecule. 9-4. Th' ro (Dr J:'3Nj repr_ntlliion The mass weighted displacementa of the nuclei of a molecule from their equilibrium positions q, can be used to generate a representation of the point group to which the molecule belongs. If under the symmetry operation R, the mass weighted nuclear displacements ql' g••... qlN
become
or
= 1,2•... 3N
= i_I 20u ql'
i
=
1.2•... 3N
and the matrices D"(R) of the representation ro, are found from
r". which is equivalent to
.N
Q~ = r_l 2D;'(R)Qr
or
(9-4.3)
IN
RQ_ = Q~ =
2 D;'.,(R)Qr'
(9-4.4)
r-l
Since the representations are equivalent. the D"(R) matrices can be found from the DO(R) matrices by a similarity transformation and in Appendix A.9-2 we show that the matrix which does the tra.DsfornUng is the matrix 0 (formed from the coefficienta Ou) so that D"(R) =O-IDO{R)O
(all R).
(9-4.S)
aN
= 2D:/(R)QI' I-I
i
=
1.2, ... aN
(9-4.1)
and the matrices DO(R) will form a representation r· of the point group. This is just a generalization of what we did for a single point in § 5-2. In some books the notation DIN{R) and r lN is used for this representation. Alternatively, one can set up mass weighted base vectom with their origins at the equilibrium nuclear positions, and transform these vectom with the symmetry operation R. then
Since both 0 and DO(R) are orthogonal so is D"(R) (see Appendix A.4-4(g». As an example. let us consider three sets of base vectors associated with three identicaJ nuclei which. in their equilibrium positions. are at the corners of an equilateral triangle (such a system belongs to the ~Ih point group); cf. Fig. 5-4.3. For the symmetry operation C. we have from eqn (5-4.3)
q" q., ... q.N
(if we have three base vectom for each of the N nuclei) will become
qi. and
i aN
Q.
then we can write
q;
113
1_1
0
0
0
0
-1/2
-../3/2
0
0
0
0
0
0
0 --../3/2
-1/2
0
0
0
0
0
0
0
0
0
1
-../3/2 0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
v'3/2 0
0
0
0
0
0
0
0
1
0
0
0
-1/2
q~, ... q~
DO{C.)
aN
Rq, = q; = 2DMR)ql'
0
i
=
1.2, ... aN
(9-4.2)
(see § 5-4) and we will obtain a representation r o identical to the one before. The matrices DO(R) will be unitary and r o will be a unitary representation (as the matrix elementa are real. the DO(R) will. in actual fact. be orthogonal).
=
-v'3/2 -1/2 0
0
1
0
0
0
0
-1/2
0
0
0
0
0
0
-v'3j2 -1/2 0
0
0
MDleculsr Vibrations
MDlecular Vibrations
174
If we make some reasonable &88umptions about the force oonst&nts BIJ for three nuclei in such a framework and solve the equations: det(B-.tlC) aN
~(BfI-A~"'1)0'~ = ._1
0,
0,
=
j = 1,2, k = 1, 2,
k
=
aN 3N
1,2, ... 3N
we obtain:
and since 0 is orthogonal (see Appendix A.4-3(c» 0- 1 = O. We can now evaluate D"(C.) from: D"(C.) = 0-l])O(C.)0 = ODO(C.)O o o 0 1 0 0 0 o o 0
-1/2
-V3/2
o
0
V3/2
-1/2
0
0
0
0
o o o o o
Ql = (-2ql+q.+y'3q.+q,-V3q8)/2V3
(AI = !1)
0
0
Q.
=
(2q.-v3q.-q.+y'3q7-q8)/2V3
(A.
0
0
Q.
=
(-2Ql+Q.-V3Q.+Q,+v3q.)/2V3
(A.
=!.) =!.)
Q.
=
(2Q.+2Q.+2q.)/2V3
(A.
=
o o
o o
0)
Q. = (-2Ql- 2Q.-2Q7)/2y'3
(A. = 0)
Q.= (2q.+v3q.-Q&-V3Q7-Q.)/2V3
(A. = 0)
Q7 = (q.+q.+q.)/V3
(A, = 0)
Q.
(A.
=
(As
= 0)
=
(2q.-q.-Q.)/V6
Q. = (q.-q.)/V2
0)
where the numerical values Of!l and!. depend on the particular choice of force constants, and where A. to A. correspond to translations and rotations. Hence
0= -2
0
-2
0
-2
0
0
0
0
2
0
2
0
2
0
0
0
0
0
0
0
0
0
2
2V2
0
1
-VB
1
0
-2
V3
0
0
0
Va
-1
-VB
2
0
-1
0
0
0
0
0
0
0
0
0
2
-V2
V6
1
V3
1
0
-2
-y'3
0
0
0
-V3
-1
va
2
0
-1
0
0
0
1
2v3
0
0
0
0
0
0
0
2
-V2
-V6
175
o
o o
0
o
o
o o
0
-1/2
V3/2
0
0
-V3/2
-1/2
0
0
o o o o
o
1
o
o o o
o o o
1
o o o o o
o o o o o o
0
-1/2
-V3/2
0
V3/2
-1/2 (9-4.6)
It is apparent that D"(C.) is in block form and since the same block form appears for all the other symmetry operations of the point group, the r o representation has been reduced by the change to the normal coordinate basis. That such a reduction will always occur, is a point taken up in the next section. Needless to say r o and r" have identical characters i.e. X'(R) = X"(R), for all R. The reader might note that once the matrix C has been determined it is possible, with the aid of eqn (9-2.21) to give a 'picture' of a normal mode. This is done by depicting the displacement of each nucleus, at some instant in the course of the normal vibration, by an arrow whose length is proportional to the displacement. In Fig. 9-4.1 pictures are given for the first three normal modes of the previous example.
1-5. TIl, rlduetion of I'll The previous example was not an exception: it is generally true that r o can be reduced by a change of coordinates (or base vectors) from
(,\,=1.) FlO. 9-4.1. The first three normal modeo for the system described in § 9-4.
177
Molecular Vibrations
176
q to Q (or q to Q). As has been mentioned before. the same A value may oocur more than once in the solution of eqn (9-2.15) (in fact. we know for sure that A = 0 will occur six times), so let us consider that there are M distinct A vsJues AI' .ill.... and AM and group together those normal coordinates which are associated with the same .il vs.lue by subscripting them in the following way:
V(Q) = V(Q'). (see also eqn (A.8-1.6». Now VCQ')
M~
=
i ...-,1 ~ __1 2. ApQ~"!"'1
QIm, QUI)"" QU", I: Al Q.llJ' QIII).··· QI(..I:
and
AI
hence, comparing terms, (9-U)
Since D"CR) is orthogonal M....
~
I
__1 .-,1
D:I""~lkICR)D:liI~lk)(R) =
M
tt.,.
~ ~ D:("')~!kl(R){D"(R)};;tk)"W
.-1 1'=1
= dp,.d"",
where n, is the number of norms.l coordinates associated with.it or the number of times A, occurs in the solution of eqn (9-2.15). v We can then replace eqn (9-2.18) by
and multiplying eqn (9-5.4) by D;(i)~(kl(R) and summing over a and k. we obtain: M~
Mna
~ 2. ApD;I"')VIJ)(R)6p,,6 m ; = ~ ~ AoD:(f)~(k)CR)d'~"fk
1'-1
(9-5.1) and eqn (9-4.3) by (9-5.2) and we can prove that if p
",_1 k_l
t.lt-]
or
*
A"D;W'lIl(R)
So that if p #
If
=
A.D:wv(J)(R).
and consequently A"
*A
o•
then
D:wolll(R} = 0 If,
(for all R).
(9-5.3)
The implication of this last equation is that the D"(R) are in block form with each n, x n y block corresponding to n y identical A vs.Iues, i.e. each block corresponds to a set of degenerate norms.l coordinates. We can prove eqn (9-5.3) as follows. Let Q stand for a set of normal coordinates which reflect the displacements of the nuclei from their equilibrium positions in some generaJ nuclear configuration X nac and similarly let Q' define these displacements after they have been transferred by R to other (but identical) nuclei. Then the relative positions of the nuclei are unchanged by R and since V is a function solely of these relative positions (see the footnote to eqn (9-2.7». we must have
and we have shown the truth of eqn (9-5.3). Thus DI(R) [0] O-'D"(R)O = D"(R) =
[OJ
[0]
DI(R)
[OJ
[0]
[0]
DI(R)
with the same block form for all R, where D'(R) is a n. xny matrix y associated with Ay which represents R in the representation r • [0) is a rectangular array with s.ll the elements zero. It turns out (see the next section) that the DYCR} can be assumed to be irreducible representations for the point group concerned.
178
Moleculer Vibretions
Moleeurer Vibreti.ns
Our previous example of three nuclei at the comers of an equilateral triangle (their equilibrium positions) belongs to the ~:u. point group and eqn (9-4.6) can now be written as: D"(C.)
=
~I'(C.)
[0]
[0] [0] [0] [0]
DB'(C.)
[0]
[0]
[0] [0] [0]
[0] [0] Dlf'(C.)
[0] [0] [0]
[0] [0] [0]
[0] [0] [0] [0]
[0] [0]
[0]
zytl·(C.)
[0]
[0]
[0]
D8"(C.)
~"(C.)
[0]
[0]
There are similar equations for the other operations of the point group.t Therefore, Q1 forms a basis for the ~I' representation and is a vibrational normal coordinate, Q. and Q. together form a basis for the rll'representation and they are also vibrational normal coordinates, the rest correspond to A = 0 and are tranelational or rotational normal coordinates, Q. and Q, belong to ['8', Q. to pAl'. Q. to ~I' and Q. and Q. to rE'. 9-6. The classification of nomer coordinata We are now in the position of being able to determine the irreducible representations to which the different normal coordinates belong. From this knowledge it will be possible to find out which of the fundamental frequencies are infra-red or Raman active. The reduction of 1'" (which is equivalent to 1''') gives
1'''(''''' I'll) = 1'1 E9 1'.... ED
r M - 1 ED r t ••
= f'T ED pt.•
where I't.. corresponds to the translational and rotational normal coordinates (A = 0) and I'v to the vibrational normal coordinates. Of course, in the above sequence some of the I"s may be the same even though the normal coordinates correspond to different ;"s; there is no reason why more than one set of normal coordinates should not belong to the same representation. As was mentioned before, we will assume that 1'1, 1'..... f'M-1 are irreducible representations, that is we will ignore the possible occurrence of accidental degeneracies where the ;". are equal not because of symmetry but because a fortuitous set of
t
To """ign to the individual blooks the correct symmetry epeoiee, one simply finds tho character of each block for each R and compare~ the result with the £it.b ch&r&Cter table.
179
force constants B,/ exists. On the other hand r be reducible. From the characters X"(R), zt"(R) we can find ZV(R) by nsing tor will
XV(R) = z"(R)-zt"(R)
and from the standard decomposition formula (eqn (7-4.2» we can then find the number of distinct fundamental levels and their degeneracies. The irreducible representation r' which is generated by (low (i = 1,2, no) is called the symmetry 8pecitJ8 of the coordinate(s) (low (i = 1,2, no) and it determines the vibrational selection rules. There are simple formulae for finding the character of ro, which can be written as IN XO(R)
>=
1: D~,(R).
'_1
In the first place, only the nuclei which are unmoved by R can contribute to the diagonal of DO(R) and, referring to § 5-2, for these there is a contribution per nucleus of: R X(R)
I
E 3
C(9)
C1
S(8)
1+20086
1
-1+2cos8
i -3
Since E = C(2..), C1 = S(2..) and i = S(..), we can abbreviate things even further and put XO(R) = n.x(R)
where
(9-6.1)
X(R) = 1 +2008 8 for R = C(8) x(R) = -1+2 C08 8 for R = S(8)
n.
= number of unmoved nuclei.
It is also quite straight forward to determine the character of r 4,r. For a non-linear molecule there are three tra.nalational normal coordinates and three rotational normal coordinates and we write rt.r = I'tEDr". The translational motion corresponds to a displacement of the molecule in some arbitrary direction and it can be depicted by a single vector showing the displacement of the centre of mass. Let this vector be :e1 +ge. +ze. (:I:. g, and z are Cartesian coordinates). We have shown before that a symmetry operation will transform this vector to :z;'81 +y'e.+z'e.
where :z;', g', and z' are certain linear combinations of:z;, g. and z. Thus the three translational coordinates will generate exactly the same representation that is generated by position vectors in physical space
180
MolllCUlar Vibrations
Molecular Vibrations
and we conclude that the set of coordinates x, y, and z forms a basis for a three-dimensional representation of the point group; we call this representation pt and l(R) will be identical to the X(R) given below eqn (9-6.1). The irreducible representation(s) to which x, y, and z belong are usually given in the character tables.t But note that 'x; y' in a character table has a different meaning from '(x, y)'; the absence of parentheses indicates that x forms a basis for a one-dimensional irreducible representation and y forms a basis for the same irreducible representation as x, while the presence of parentheses indicates that x and y together form a basis for a two-dimehsional irreducible representation. e
181
like R~ are caJled axial vectors or pseudo-vectors. The symmetry operations transform each of three axial vectors representing molecular rotation about the x, y. and z axes into linear combinations of one another, thereby generating the representation pr, which corresponds to the molecular rotational modes. The characters of pr are related to those of pt. Consider first the effect on, say R~, of a C(O) rotation about some axis. not necessarily the x axis. This rotation will move the rotation displacement vectors in such a manner as to transform R~ into a vector R:, where R: is the vector obtained by applying C(O) directly to R... Thus, for proper rotations. the matrices describing how R.., R v, and R. transform are exactly those matrices that describe how ordinary position vectors along the x, y, and z axes transform; the matrix in the representation rr corresponding to any C(O) is therefore the same as the matrix in pt that corresponds to C(O) and the characters for proper rotations are the sa.me for rr as for r'. Now consider the effect on R", of a S(O) operation. We have S(O) = aC(O).
(a)
FIG. 9-6.1. Effeot of B reflection on (a) a position vector; (b) an axial V'&otor.
We must also find the symmetry species of the rotation and we follow the treatment given by Levine. An arbitrary rigid rotation of the molecule can be resolved into rotations by various angles about the three translational axes x, y. and z. We can represent the rotation about the x axis by a single vecllor R. pointing along the x axis. The length of R~ is proportional to the angle of rotation and its direction is given by the direction the thumb points when the fingers of the right hand curl in the direction of rotation. The effect of any symmetry operation on the nuclear displacement vectors of a rotation defines its effect on RIO' Thus, the C. rotation about the z axis sends the displacement vectors for rotation about the x axis into their negatives, thereby reversing the direction of this rotation; hence R., whose direction is defined relative to the direction of rotation is reversed in direction. Reflection in the yz plane a v< has no effect on the displacement vectors for rotation about the x axis; hence R~ is unchanged by this symmetry operation. Thus the effect of au on R~ differs from its effect on a vector pointing along the x axis and representing translational motion; the latter vector is changed to its negative by a v< (see Fig. 9-6.1). Vectors t Proper orientation of 61' e 2 and e. will ensure that r t is in reduced form.
The C(O) part has the same effect on R. as on a translational vector along the x axis. However the effect of a reflection on R~ is opposite to its effect on a translationa.l vector along the x axis. Thus the matrices describing the effect of S(O) on the rotation vectors are the negatives of the matrices describing the effect of S(O) on the translational vectors. The character of any S(O) for pr is the negative of its character in Summarizing, we have:
re.
X'{C(O)} = Xt{C(O)} Xr{S(O)} = -Xt{S(O)}
zt·r(R) = 2(1 +2 oos 0) Z'·r(R) = 0
if R = C(O) if R = S(8).
If we return to our equilateral triangle example and apply the above rules, we obtain the numbers in Table 9-6.1. Using this table in conjunction with the ~Ih character table (Table 9-7.2) and the decomposition rule:
we obtain
pt =
rr and
pB' $
rA,'
= pA,' $ pr
p ... = p-d,' $ pH'.
112
Molecular VibrlltloM
Molacular Vibrations 9-6.1
TABLE
TABLE
r o repruentaticm oj the !?JII> point
Character, jor the 9.
E
X'(C,) %,(C,)
ZY(C,)
-1 -1 -1
0 0 0
9 3 3
x"(C,)
2S.
8cr.
1 -1 3
A' A' E'
,•
1 1 -1
0 -I 1
3
0
3
9,..
a.
SC,
2C,
9-7.2
Character table jor
group
B. (z.
A" A"
,•
0
z (BI
E'
E
yl
SC,
all
2S,
3<>v
1
I
1 1 -1 -1 -1
-1 0
2
1 1 -1 1 1 -1
12
0
2
1 1 •
B.)
XO(C;)-CO:-
~3h
2C,
1 1
113
1
-1
1
0
2
1 -1
-1 -1
0
-2
I
0
-2
4
-2
2
-1 1
8-7. Further au..pl.. of nor..al coordina.. cla..ifil:lltion
In this section three more examples of normal coordinate classifica.tion are given. We consider (1) H,O. (2) CO:- and (S) CH•. (1) The character table for the point group of H,O 'if,. is given in Table 9-7.1 and below it we show the characters for the r o representation (found from eqn (9-6.1». From these characters we obtain
(2) The character table for the point group of CO:-(!~Oh) is given in Table 9-7.2 and below it we show the characters for the r o representation. From these characters we obtain
r o = r..t·' EEl r..t,' EEl srB'
I" =
and
rr
=
rA' Ell r • ED r , rA' Ell r B , Ell r B ,. B
Hence, by subtracting these from
rv
=
and Hence
and there are two distinct non-degenerate vibrational modes of species r..t>, and r..t,' and two distinct doubly degenerate modes with the same symmetry species J'B'. (3) The character table for the point group of CH. (9""d) is given in Table 9-7.S and below it we show the characters for the r o representation. From these characters we obtain
2rAt Ell r B ,.
So that there are three distinct (having different l's or ,,'s) non-degenerate vibrational normal modes with symmetry species rAt (two) and rB' (one).
ro
ay
1
I I -1 -I
1 -1
z; B. y;R"
1 1 1
1 -I -1 I
x"(C,)-R,O
9
-1
1 -1
= GI • (perpendicular to the plane); G~ = G I • (molecu.... plane).
S
molecular
r T,
=
TABLE
,
Gy
t Gy
Ell r E Ell r T , EEl 3rT , rt
C,
•B.
rAt
9-7,1
E
At A, B, B,
=
and, by inspection of the table,
Character table jor 'if...f i',y
=~'
B
r o we have
TABLE
EEl r..t,r-.. = r..t,' EEl rr. r' = r..t,' Ell 2rB' EEl r..t," rt
rr are also given
2r..t,' EEl r H "
and, by inspection of the table,
r o = srA • EEl rA' EEl 2rB· EEl srB,. As is usual practice, the representatioD8 for r' and in the character table and we directly obtain
$
9-7.3
Character table for ord
or.
E
SC,
SC,
6S,
6cr.
I
1 -1
-1
T1(R., B." R~) T , (.,. Y. z)
1 1 -1
1
E
1 1 2
2
0
3 S
0 0
-I -1
1 -1
0 -I 1
15
0
-1
-1
3
A,
A,
;c'(C,>-CH,
I
1M and Hence
r·
=
rr = r T ,. rAJ e rE e
".'acula, Vibratlo.. which, written in the style of § 9-5, becomes
2rr· where
and the nine vibrational normal coordinates (or modes) can be elaasified as follows: one gives rise to ~', two are doubly degenerate and give rise to rH, three are triply degenerate and give rise to r T ., and the last three give rise to the same representation r T • but with a different 1 (or v) value from the previous one.
'PJ lb = N exp( - i
L «R~) ...1
AI-I
..p
,,-
_1
L,ct." L ~(m)
(9-9.1)
u:d'"
Q~l'"
=
l.n:..
(R)Q"cm)
(9-9.2)
which is the 'reduced ., f . . vel'Slon 0 eqn (9-4.3) and where lY'(R)' th matnx representmg R in the th' d . 18 e int . P Jrre uClble representation of the po gro~p .which the molecule belongs. Since the Dft(R) are ortho _ onaI, the mdlvldual blocks e g U(R) t I b g hence • . . . mns a so e orthogonal and
t:o
"p
Qplml =
~I D:..(R)Q~(.,).
(9-9.3)
If we substitute eqn (9-9.3) in eqn (9-9.1), we find 'Prb(Q) = N
eXP(-i1'~p I 1'-1
I I DPm(R)D" (R)Q'
11I-11_11:_1
i
1:".
p(f)
Q'
)
p(~)
where the Q . v1b(Q' . (9-9.4) m 'Po ) mdicates that the wavefunction take th al o f the normal c rdin tea b {. e v ues Since JY(R)' oorth a e ore the symmetry operation R is applied IS 0 ogonal .
lI"" Dfm (R)D:",(R) =
~/.
and we can replace eqn (9-9.4) by
ii\"1-1":1 ! ~.,I.Q'"en Q' ) N exp(-tr ct." ! Q~~.,) = 1J'J 1b(Q').
'PKib(Q) = N exp ( =
aN....
exp( - t
Under the symmetry operation R the normal coordinate Q - t formed to Q' hi h linear combination of Q "U'I IS rans• pC., w C 18 a Q d smce ~hese coordinates form a basis for the pth 'b"ll"", an sentatlOn, we have UCl e repre-
.-B. CI_ification of .... vibrationall••ls Classifying the vibrational energy levels means finding out to which irreducible representation of the molecular point group the vibrational wavefunction(s) associated with a given level belong. First let us oonsider the vibrational ground state. The co~sponding wavefunction is (see eqn (9-3.9»
=
211' 4 .." ct." =-1' h" =h v".
B-1. Nor...lcoordin.... for lin••r..olacal.
Linear moleculee belong to the ~.... point group if they are unsymmetrical and to the ~
N
vlb
'Po
116
~a;~:efine
_1
,,_1
pC.,
,t-l
4
the symmetry transformation operator 0. in the usual O.'P~ib(Q') = 'Po1b(Q),
we have from eqn (9-9.5) or
(9-95)
O.'P~ib(Q') = 'PKlb(Q') lb ib O.'PO = 'PK (all R) .
18&
Molecular Vibrations
Molecular Vibrations
Consequently ",Tib generates the identical representation 1'1 of the point group and we say that the ground state vibrational wavefunction is totally symmetric. The fundamental vibrational energy levels lie at an energy 11."" above the ground state. Each level has associated with it a certain fundamental frequency"" and therefore a certain 1 value 1" and each 1" has associated with it nil normal coordinates Q"C"'1 (m = 1.2•... nil)' Coupled with each of these normal coordinates there is a vibrational wavefunction "'::. which is proportional to ",;lbQ"(,,,) (see eqn (9-3.1I». Thus the fundamental levels have an nil-fold degeneracy; that is there are nIl wavefunctions having the same energy. If we write these degenerate wavefunctions as ~(Q) = N",X1b(Q)Q"C"'1
m = 1.2•... n"
where N is a normalization constant, and substitute eqns (9-9.3) and (9-9.5) we get
Consider a set of mutually perpendicular axes (specified by the vectors e 1 , e., and e.) with their origins at that point in the molecule which is unmoved by all the symmetry operations of the point group and let us refer the position of any nucleus to a set of axes (parallel to e 1 , e., and e.) whose origin is at the equilibrium position of the nucleus. Then the electric dipole moment of the molecule when the nuclei and electrons are in positions described by BOme configuration X, is the vector where
N
Pt(X)
= ,,_1 2 a::e"
z:
and is the coordinate of the pth nucleus in the direction of e" and e" is the effective charge on the pth nucleus. Quantum mechanics tells us that the probability of an electric dipole induced transition occurring between the states described by "'.. and "'''' is proportional to
•
Hence
~JJ "':Pt"'m dT)
and OJt~ =
"" D:",(RM· 2
...1
Therefore the wavefunctions '1'::' (m = 1,2, ... nil) form a basis for the irreducible representation 1''', the same representation to which the normal coordinates Q,,(l)"" Q"l"p) associated with the fundamental frequency"" belong. This result is tremendously useful. it not only leads to selection rules for vibrational spectroscopy but also, as was the case with eleotronic wavefunctions (see § 8-2), allows us to predict from inspection of the character table the degeneracies and symmetries which are allowed for the fundamental vibrational wavefunctions of any particular molecule. 9-1 D. loha-red spllCtla
Ifincident radiation with a frequency equal to one of the fundamental frequencies falls on a molecule, it may make a transition from the ground state to the appropriate fundamental level. These normal frequencies usually occur in the infra-red spectral region. The probability of such a transition occurring, however, depends on the relationship between the molecule's electric dipole moment (as a function of the nuclear coordinates) and the wavefunctions of the ground state and of the fundamental level.
187
•
where the integration is over the whole range of the nuclear coordinates. Consequently a transition from the vibrational ground state to the pth fundamental level is forbidden (has zero probability) if
J",X
i1 )·
Pt~ d.,.
=
0
for all m values (1, 2, ... '1',,) and all k values (1, 2, and 3). The vectors e 1 • e., and e. generate the same representation r t as the translational normal coordinates and therefore
Rei =
•
2 DMR)el 1_1
and it can easily be shown that
O.lSt =
•
2 D~,,(R)p; 1_1
k
=
1,2,3.
Therefore the p,,(X) form a basis for r~. There will be a particular choice of axes e 1 , e., and e. which will reduce r t to the irreducible representations which it contains and this choice is the same as the choice of translational axes which reduces r t • Therefore PI belongs to the same irreducible representation as z, p. belongs to the same one as 11 and P. belongs to the same one as z, and these can be found by consulting the appropriate character table.
188
Molec:ulor Vibrotions
Molec:ulor Vibrotians
Since 'i'Y:" belongs to P and 'i':' belongs to § 8-4 dictate that 'f~lb. Pavf". dor = 0
f
r p•
the conolusions of
unless r p coincides with the irreducible representation to which Pa belongs. We can therefore obtain the selection rule which governs the transitions from the ground state to the fundamental levels: ", is only infra-red active if its symmetry species r' is the same &8 one of the irreducible representations contained in r t • An alternative but not so general selection rule (it is restricted to the harmonic oscillator approximation) is that J'i'-:?b*Pa'i'':.. dT is zero if oPaloQ':.. (evaluated for the equilibrium nuclear configuration) is zero, i.e. if there is no linear dependence of the dipole moment on the normal coordinate Q::'. This selection rule may be found by making a Taylor series expansion of f'a(X) in the normal coordinates Ql> Q••... QSN-I (we revert to our initial notation) : pa
=
p~+
sN-4(O L /Qa)"Ql +.... ~_l
k = 1,2.3
x f'i'l(O, Q.)*Q''/'I(O, Ql) dQ.f'i''''(O. Q..)*,/,,,,(n,,,, Q",) dQ.. + +
(:~:r
enf
'i',(0, Q,)*,/,.(O. Q,) dQ,}f ,/,..(0, Q",)*Q..,/,,,,(n,,,. Q",) dQ.. +
+higher terms. and if n", =1= f
'i'~Ib·Pa'i'", dT
II
'_1
'i',(n" Q,)
where n, is the ith vibrational quantum number, then the ground state wavefunction is
and the wavefunctions for excited states in which only one vibrational quantum number is non-zero are 'f'm = Since
{rrl' ,,. ..f"(o. Q'»)'f'..(n... Q...).
f ,/,,(j. Q.)*'i',(k. Q,) dQ, = ~ia
(the ha.rmonic oscillator functions are orthonorma.l) we have
(:~:rf 'i'",(0, Q",)*Q..'i'",(nm• Qm) dQ",
=
k = 1,2,3
and hence. if
(:~:)" =
0. the probability of the transition from the
ground state to the excited state occurring is zero, i.e. for the transition to take place the dipole moment must change linearly with the normal coordinate Q... Since ,/,..(0, Q..)Q", is proportional to 'i'.. (I, Q",) and the functions ,/,,,,(n,,,, Qm) are orthogonal. it is clear that
f ,/,..(0, Qm)*Q..'f'm(n.., Q..) dQ... ° =
where the superscripts indicate evaluation at the equilibrium configuraN.....
(9-10.1)
° the first two terms and the higher terms are zero and
t
ation. If the vibrational wavefunctions are written as
189
unless n", = 1. Hence only transitions to the fundamental levels are allowed from the ground state. This is strictly true only within the approximations we have been making and. in reality, transitions to overtone levels (n.. =1= 1) do take place (see § 9-13). Furthermore. transitions from the ground state to excited states for which more than one vibrational quantum number is non-zero are forbidden, e.g. if n 1 =1= and n. =1= 0 in the excited state, then either or both of the integrals
°
f'i'I(O. Ql)*'i'1(111 • Q.) dQ. and f 'f'.(0. Q.)*'f'.(n•• Q.) dQ. will appear in each term of eqn (9-10.1) and hence the probability of the transition is zero. Again this is only strictly true within the approximations we have been making and transitions to so-called combination levels do in actual practice occur (see § 9-13). 9-11. Roman speetro
When an incident beam of radiation of frequency" falls on a molecule, some radiation is scattered and in this scattered radiation we get. as well &8". frequencies "±", where 'lip is a fundamental frequency. This is called the Raman effect and when a fundamental frequency appears in the Raman spectrum it is said to be Raman active.
190
Mol_lar Vibrations
Molllt:Ular Vibrations
An incident radiation field with an electric vector E induces a dipole moment M in a molecule. The components of Mare: k = 1,2,3
where E
=
and
E.e. +E,e,+E.e. k j
= =
1,2,3 1,2,3
and exn' ex.., ex••, exU> ex••, and ex.. define the polarizability of the molecule. The latter are transformed by the symmetry operations R in the same way as z:, z:, z:, z.z,' x.z., and ZtZ. where z.' Z2' and z. are the coordinates of a point in physical space (see Appendix A.9-3). The six polarizability functions generate a reducible representation which we will call I'" and the character tables give the irreducible representations to which z: (or x'), z: (or y'), x: (or Z2), x.x, (or zy), x,x. (or xz), x.z. (or yz), or the necessary combinations, belong and therefore give the decomposition of I"'. For example, for CH. (9"a) x'+y'+Z2 belongs to r.4" 2z'-x 2-y' and x"_g' to r E and xy, xz, and yz to r T " hence
r"
=
r-A' EEl rE EEl r T ,.
Quantum mechanics tells us that the probability that Raman scattering involves the fundamental frequency v, depends on the integrals i = 1,2,3
finding the number of active fundamentals and their symmetries for the two molecules. CH. belongs to the 9"d point group (the symmetry elements are shown in Fig. 3-6.2) and the reduction of r o was carried out in § 9-7(3) with the result that ro = pd, EEl r E EEl rT' EEl 3rT ,. Furthermore, inspection of the character table shows that
and henoo
r'" = r.4, EEl r
E $
2r T ,.
Also from the character table in Appendix I, we have
r-
= r.4, $
rE
Ei)
r
T ,.
So that the non-degenerate fundamental level which belongs to ~, will only be Raman active (r.4, is contained in r" but not in r t ), the doubly-degenerate fundamental level which belongs to r E will also only be Raman active (rE is contained in I'" but not in r t ) and the two triply-degenerate fundamental levels which belong to r T , will be both infra-red and Raman active (r T , is contained in both r' and r"). CH.D belongs to the 'if. v point group and the C-D axis is the G. axis. The characters for the r o representation may be found by using eqn (9-6.1) and they are
~
m = 1,2, ... n,
'_E
~
2C_.__3_0_",_
15
0
3
These characters together with those of the irreducible representations of the point group may be fed into eqn (7-4.2) and the reduction of r. carried out, when this is done we obtain
and since by inspection of the character table r
and 9-12. Th. ioha-rad and Raman spactra of CH. and CH.D In this section we will determine the differences in the infra-red and Raman spectra of methane CR. and monodeuteromethane CH.D by
r t = r T, rr = r T ,
and
j = 1,2,3
and therefore v, is only Raman active if r' coincides with one of the irreducible representations contained in I'" (remember that 'I''tb belongs to Pl. This rule is equivalent to saying that ", is only Raman active if the polarizability changes during the pth normal vibration. In a molecule with a centre of inversion, the irreducible representations in rt are of u-type and those in I'" are of g-type and sinoo r' cannot ooincide with both a u- and a g-type irreducible representation, no fundamental frequency for this type of molecule can be both infrared and Raman active.
191
we have Furthermore,
t
= r.4,
rr =
$
r
E
rE
pdt $
rv = 3r.4, $ arE.
r"
=
2r A ,
$
2r E •
112
Molecular VibratiollS
113
Therefore the three non-degenerate fundamental levels belonging to ~1 are both infra-red and Raman active and the three doubly degenerate fundamental levels belonging to r E are also both infra-red and Raman active. So for CHaD the number of fundamental frequencies which appear in the infra-red spectrum and the Raman spectrum are the same, whereas for CH. this is not so. This is sufficient information to distinguish between the two molecules. 1-13. Combination and overtone levels end Fermi resonance If it is possible to excite two normal modes simultaneously then a transition can occur to what is called a combination level. Such a level will be characterized by a set of vibrational quantum numbers which are all zero except for two which are unity (see eqn (9-3.6)); it will lie at an energy of h(vp+v,,) above the ground state where vI' and v" are the relevant fundamental frequencies. The corresponding vibrational wavefunctions will be of the form
i = 1,2, ... n p j
1,2•... n.
=
and as there are 7/,,,7/,,, products of Qp(j) and Q,,(l) the combination level, in the harmonic oscillator approximation, will have a degeneracy equal to npn". The npn. wavefunctions for a given combination level, taken together, will form a basis for the direct product representation r" 0 r". This representation will. in general, be reducible. There will therefore be combinations of the functions N'P~lb(Q)Q"(j)Q,,w which will form bases for the irreducible representations contained in r p 0 r". In the harmonic oscillator approximation these combinations are all degenerate; however. if anharmonio terms such as aN aN aN (
t
2: 2: 2: f
I
k
aa V
J.
aq., oqI aq
0
An OTertone level is charaoterized by a set of vibrational quantum numbers which are all zero except one which has a value greater than unity, say m. If the quantum number which is non-zero corresponds to a fundamental frequency vI" then from eqn (9-3.6) we see that the overtone level will lie mv" above the ground state. It can be shown that if the pth fundamental level is non-degenerate then for m even the overtone level belongs to the totally symmetric representation and for m odd it belongs to the same representation as the pth fundamental level. If v I' is degenerate, then the symmetry species of the overtone is difficult to determine and once again anharmonic effects destroy the degeneracy predioted by the harmonio oscillator approximation and new levels are oreated which belong to some definite symmetry species. Though the probability of a transition from the ground state to an overtone level is zero in the harmonic oscillator approximation, the probability can be quite high when anharmonic terms are taken into account. If an active fundamental level happens to lie close to an overtone or combination level with the aame symmetry species, then anharmonic terms in the potential V (see eqn (9-2.4)) will have the effect of mixing the two levels. Two new levels will be produced whose wavefunctions consist of approximately equal amounts of the wavefunctions belonging to the fundamental and the overtone or combination level. In such circumstances rather than having one strong (fundamental) and one weak (overtone or combination) transition there will be two strong transitions lying close together in enllrgy. This phenomena is cal1~d Fermi reaonance.
Appendices A.9-'. Proof of eqns (9-2.17) and (1-2.18) We ha.ve:
Mjq"
are inoluded in eqn (9-2.7), then this degeneracy is lifted and in place of a single combination level there will be a group oflevels with energies approximately h(vp+vo ) greater than that ofthe ground state and there will be one such level for each irreducible representation contained in r p 0r" (see § 8-3). If any of the irreducible representations coincide with those found in r t or r« then a frequency approximately equal to v" +v" will occur in the infra-red or Raman speotrum. Since these transitions are forbidden in the harmonic oscillator approximation they will be weaker than the fundamental ones.
But since 0 is orthogona.l and
and hence we obtain eqn (9.2.17).
194
Molecula' Vibrations
Mol_la, Vibrations
195
A.I-3. Sr-1Htry propertiw of polarizabllity functions Let R transform the nuclear--electronic configuration X to X' and the electrio veotor E to E'. then
From eqn (9-2.13) we can get
a
and hence
E =
IB,e, ,_I
•
E' = IB~e, i-I
and
8
E;" = 1:D..;(R)E;
1.2,3.
m =
;-1
Also
1:• cx...(X')E;"
M.(X') =
.._1
a
8
=
which proves eqn (9-2.18).
1: cx...(X') 1_1 1: D",;(R)E; ,"_1
and
A.9-2. ProDfthlt D (R) = O-lD"(R)O We have. by definition,
I
i
D'J,(R)q;
= 1,2•...• 3N
(A.9-2.J)
aN
;-1
or
aN
q;
1: (0_,
=
1
) ..
1: D",,(R)",-1 1: 1lt,..(X')1: D",;(R)E;. 1·-1 ;_1
8
8
= IIlt,,;(X)EJ ;-1
so
8
a
i-I
m-l
llt",s(X) =
;0...
8
a
M.,(X)
1: OJlq;
Q. =
M.,(X) =
But
;-1
and
,-1
therefore.
aN
Rq, =
a
1: D.,,(R)M.(X')
M.,(X) =
and since, by definition of
1: 1: D",(R)D mf(R)Ilt,..(X') 0.,
(A.9-2.2) (0 is orthogonal). Substituting eqn (A.9-2.2) twice into eqn (A.9-2.1), we have aN
aN
we have O~"iX') =
aN
I O~RO", = 1=1 1: D~,(R) 1: 0;",0", .1:_1
a
8
1: 1: D""(R)D..;(R)cx....(X'). i_I tn_I
Now consider the functions
t'JIa=l
aN
aN
1: 0,,,, J:-1
aN
aN
1: D;'",(R)O. = 1: D~,(R) I
r-l
J-1
0;... 0 ...
".-1
and by equating the coefficients of the Q'S: aN
1: O,,,,D:.'rc(R) =
I
and hence and
D'J,(R)O;,
1_1
aN k-l
I
aN
D:.'rc(R)(o-l)1H =
X a• X a) =
1: (o-l),;D'J,(R)
XrcX;
k = 1.2,3 j = 1,2.3
it is easily shown that •
O.I.,;(x~. x~. x~) =
ON
k-l
''''f(X "
a
1: 1: D,,,(R)Dmf(R)/,.. (x~. x~, ~). ,:-1,,"=1
Therefore the effect of OR on a polarizability function Ilt"f is the same as its effect on xrcXs'
1_1
D"(R)o-l = o-lD"(R) D"(R) = o-lD"(R)O.
PROBLEMS 9.1. For ethylene: (a) detennine the point group;
196
Molecular Vibrations (b) determine the nmnber and symmetry of ths vibrational normal co(c)
ordinates; determlne the spectroscopic activity of each fundamentalleve!.
9.2. Show on the basis of infra-red and Raman spectra that it is possible to distinguish between the two crown forms of octachlorocyclooctane, one in which the hydrogen atoms are all equatorial (~....) and the other in which the hydrogen atoms are alternating between axial and equatorial positions (~.v). 9.3. Discuss how the cis and trans isomers of K.F. can be distinguished by infrared and Raman measurements. 9.4. What will be the infra-red and Raman activity of the four fundlWlental levels of COt-7 9.5. Determine XO and carry out the reduction of (a) NB. (~3V)' (b) XeOF. (~.v), (c) PtCl:- (9.h ), (d) trans-glyoxal ('€.,,).
ro for the following molecules:
10. Molecular orbital theory 10-1. Inttoduetion IN this chapter we will consider how to apply a knowledge of symmetry and its ramifications to the determination of electronic wavefunctions. We will do so by looking at a particular kind of approximate electronic wavefunction for conjugated molecules. The SchrOdinger equation for the electrons of a molecule, our starting point, is just as hard to solve as the SchrOdinger equation for the nuclei. In the latter case we were able to find approximate solutions by replacing the true potential energy V with a sum of quadratic terms in the nuclear coordinates (l~ ~ B./NIJ)' In dealing with the electronic case we must, •
J
for example, for a molecule like benzene, make a whole series of fairly drastic approximations. First we consider the electronic wavefunction to be made up of molecular orbitals (approximation I), then we restrict the form of the molecular orbitals (MOs) to linear combinations of atomic orbitals (approximation 2), then we separate out the part of the wavefunction ooncerned with a-electrons and deal only with the or-electron part (approximation 3), finally we sol ve the appropriate equations by making assumptions about certain integrals over the 1T-electronic MOs (approximation 4). The final step brings us to the Huckel molecular orbital method, which is familiar to all chemists. Symmetry enters the approximate solution of the electronic SchrOdinger equation in two ways. In the first place, the exact MOs are eigenfunctions of an operator which commutes with all O. of the point group concerned, they therefore generate irreducible representations of that point group (see Chapter 8) and can be classified accordingly. The same is true for the approximate MOs and consequently one constructs them from combinations of atomic orbitals (symmetry orbitals) which generate irreducible representations. In the second place, the Hamiltonian operators which occur and commute with all O. belong to the totally symmetric irreducible representation r 1 (see Appendix A.I0-3) and integrals over them fV'··Hv/' dT vanish unless r' = r" {see eqn (8-4.5». Thus, in carrying out an approximate solution of the electronic SchrOdinger equation, changing to a set of basis functions which belong to the irreducible representations will allow us, by inspection, to put many of the integrals which occur equal to zero. There will also, because of this, be an
Mol.cul.r Orbit.' ThBOrY
198
Mol_l.r Orbit.1 Thnry
immediate factorization of the equations. Two examples. benzene and the trivinylmethyl radical, will be considered in detail. 18-2. lb. H.rtrBHOck .pproxim.tion The starting point for any molecular electronic problem is the electronic SchrOdinger equation; H'Y(I, 2•... n) = E'F(I, 2•... n). In this equation H. the Hamiltonian. is defined by precise quantum mechanical rules and can be written in atomic units (AppendixA.IO-I) as
H
..
= ~hp+ ~ ~ ..._1
where
.. ..
II-I
N
I/r,..
(10-2.1)
ZJr,...
(10-2.2)
.,>"
h,. = -IV:- ._1 ~
. .
In eqns (10-2.1) and (10-2.2) n is the number of electrons in the molecule rIO' is the distance between electron p and eleotron v, ~ ~ l/rp • is the
,._1,,>,. potential-energy operator due to interactions between the electrons, is a Laplacian operator involving the coordinates of electron p, -! is the kinetic-energy operator (in atomic units) for electron p, r ... is the distance between electron f' and nucleus ex, Z. is the charge
V;
V;
N
on nucleus ex and -
~
0_1
ZJr,.. is the potential-energy operator
arising
from the interactions of electron f' with all the N nuclei. 'Y(I. 2, ... n) is the electronic wavefunction and explicitly is a function of the coordinates of all n electrons; in this notation the coordinates of a given electron are symbolized by a single number. E is the total electronic energy of the molecule. The Hartre&-Fock (HF) method. or self consistent field (SCF) method as it is sometimes called, approximates 'Y(I. 2,... n) by expressing it solely in terms of functions each of which contains the coordinates of just one electron; these functions are called moluular orbita18 (MOs). This is an approximation because in reality the position of one electron is always correlated with the positions of the others. 110 that the function which describes a given electron cannot be independent of the functions describing the other electrons. It is for this reason that the error in the electronic energy in the HF approximation is called the correlation energy. The actual way in which the MOs are put together to form 'Y(1,2.... on) is restricted by two fundamental laws. One is that an electronic
199
wavefunction must not distinguish. by treating in a different manner, one electron from another; this is the law of indistinguishability of identical particles. The other law is that the electronic wavefunction must change its sign when two electrons are interchanged; trus is the antisymmetry law and in fact it leads to the Pauli exclusion principle. Both these laws are satisfied by expreBBing '1"(1, 2, ... n) in terms of the MOs with a Slater determinant;
..(I)t <1>..(2)
'Y(l, 2•... n) = I/vn!
. (10-2.3)
,(n)
.(n)
..(n)
In this determinant «(j) symbolizes the ith MO as a well-defined function of the coordinates of electron j; we say that electron j is occupying the MO <1>(. If the determinant is multiplied out there will be n I terms and each term corresponds to one of the n! permutations of the n electrons amongst the n MOs; since all permutations are included. every electron is treated equally and the indistinguishability law is satisfied. If two electrons are interchanged then, for a Slater determinant. this is equivalent to interchanging two rows and if two rows of a determinant are exchanged it changes sign, hence the antisymmetry law is also satisfied. e.g. if we exchange electrons I and 2, we get <1>.. (2) <1>..(1)
'Y(l!,t, ... on)
=
(l/vn!)
-l/vn! l(n) -'Y(I,2, ... on).
t This equation
is often abbreviated
.(n)
SIll
'1'"(1.2•... n} ~ 1~,(I}~.(2}~.(3) ... ~.(n}l.
,,(n)
Malacular Orbital Theory
200
Molacular Orbital Thaory
That '1'(1,2•... n) necessarily satisfies the Pauli exolusion principle is evident from the fact that if two MOs are absolutely identical (including their spin components), then so are two columns in the Slater determinant and a determinant with two identical coluDlll8 is zero. The I/vn! factor preceding the determinant takes account of normalization, since if the MOs are orthonormal. this factor ensures that 11)'1'(1, 2•... 11) dT. dT•... dT.. = l.
J... ff'Y·('.2 ....
The way in which the Hartree-Fock MOs are determined is by using the variational method, that is the form of each MO is varied until the integral J... JJ'Y·(l,2•... n)H'¥(I, 2•... 11.) dTI dT•... dT" is as low as possible: 6
{f..,ff '1'·(1,2•... n)H'I"(I. 2 •... n) dT. dT•... dT,,)
=
o. (10-2.4)
We will call the MOs that satisfy eqn (10-2.4) with respect to wmpkte variation in their form the exact MOs. By introducing eqn (10-2.3) into eqn (10-2.4) it is possible to arrive at a simple set of 11 one-electron eigenvalue equations. called the Hartree-Fock equations: HOrr(i)fb,(i) = 8,fb,(i)
i = 1.2•... 11
(10-2.5)
where Hou(i) (explicitly defined in the next section) is an effective Hamiltonian operator related to H and involving the coordinates of electron i. The eigenvalues 8, are constants called orbital energies. Once the MOs have been found. the total electronio energy is obtained from the equation:
E
=
201
methane a MO might be a non-degenerate 'a l ' orbital or a triply degenerate 't.' orbital etc. 10-3. lb, LeAD MO approximation Rather than find the most perfect MOs which satisfy egn (10-2.4) (or eqns (10-2.5». it is common practice to replace them by particular mathematical functions of a restricted nature. These functions will generally contain certain parameters which can then be optimized in accordance with eqn (10-2.4). Since these MOs are not completely flexible, we will have introduced a further approximation. the severity of which is determined by the degree of inflexibility in the form of our chosen functions. Typical of this kind of approximation is the one which expresses the apCIU part of the MOs as various linear combinations of atomic orbits.la centred on the same or different nuclei in the molecule. We write the space part of each of the approximate MOs as
...
([>j(i) =
L
O,j4>.(i)
(10-3.1)
.1-=1
where the "'. are atomic orbitals and the an are linear coefficients. These fbJ are called Linear Combination of Atomic Orbitals Molecular Orbitals (LCAO MOs) and if they are introduced into the Hartree-Fock equations (eqns (10-2.5». a simple set of equations (the Hartree-FookRoothaan equations) is obtained which can be used to determine the optimum coefficients 00/' For those systems where the space part of each MO is doubly occupied. i.e. there are two electrons in each fb J with spin a: and spin {J respectively 80 that the complete MOs including spin are different. the total wavefunction is
f. .. ff '1'·(1. 2•... n)H'I"(I. 2•... 11.) dTI dT•... dT".
Since it can be shown that HOtl(i), like the original Hamiltonian H. commutes with the transformation operators 011 for all operations R of the point group to which the molecule belongs, the MOs associated with a given orbital energy will form a function space whose basis generates a definite irreducible representation of the point group. This is exactly parallel to the situation for the exact total electronic wavefunotions. The exact MOs may therefore be classified according to the irreducible representation to which they belong and usually the symbol for the irreducible representation (in lower case type) is used to label the MO with which it is associated. For example, for the point group:T4 of
and we have the following equations:
...
~ (H~f!-e,SJ.JO~, = 0
~_l
j = 1,2'00' m
i = 1.2•... m
( 10-3.2)
where (10-3.3)
(10-3.4)
202
Molel:1ller Orbital Theory
Molecular Orbital Theory
J.(~) and K,(~) are the Coulomb and exchange operator respectively. and are defined by the equations:
J,( 1 )4>~( 1)
and
f
= ( l1lt(2)l1l,(2)riil d'-lj4>~( 1)
K.(l)4>~(I) =
(f 11l:(2)4>~(2)"1:d'-ljell,(I).
Non-trivial solutions of eqns (10-3.2) can be obtained provided that the eigenvalues E" the LOAO MO orbital energies and approximations to the E, of eqn (10-2.5). satisfy the equation det(H;~f-e,Ss~) = O.
(10-3.5)
With this proviso. eqns (10-3.2) coupled with a normalization equation can be solved to produce m sets of coefficients (each set corresponding to a particular MO and orbital energy) from which we can choose '11,/2 which correspond to the lowest orbital energies and to those Mas which are occupied in the electronic ground state. The total electronic energy is then E = =
where and
f. ..ff '1"*(1.2•... n)H'I"(I. 2, ... n) d'-I d'-I ... d,-" 2
n/2
,./11 fit/a
i-I
i-I i-I
2. E,- 2. 2. (2J,s-K;j)
J,s = K" =
203
a priori that the approximate MOs do behave in this way, the calculations are greatly simplified because the vanishing integral rule comes into play. The way in which one makes sure that the approximate MOs form bases for the irreducible representations is by first forming linear combinations of atomic orbitals which do. These combinations are called. appropriately. symmetry orbita18 and the coefficients of the atomic orbitals of which they are composed are totally determined by symmetry arguments. We will write a symmetry orbital as 4>; =
2. c••4>. •
(10-3.7)
where 4>. is an atomic orbital. The Mas are then formed from the symmetry orbitals by (10-3.8) and the coefficients 0;1 and total electronic energy are determined in the same fashion as before but with O;s replacing 0 Ii and 4>: replacing 4>•. The simplification which results by doing this will become clear in § 10-6 and § 10-7. 10-4. Tho ",-electron approximation
(10-3.6)
ff eIlt(l)eIl;(2)r1f eIl,(I)l1ls(2) d'-I d'-I ff 11l,*(I)l1lf(2)r1l ell,(2)eIl (l) d'-I d'-I' l
j
An alternative notation for the preceding equations is given in Appendix A.I0-2. The reader should note that eqns (10-3.2) have to be solved iteratively since the coefficients 0# appear in the operators J,(f.4) and K,(~) and hence in He"(~) and H;;f. What is done, therefore. is to gue88 sets of coefficients and with them calculate H:~. then solve eqns (10-3.2) for a new set of coefficients. These new coefficients can then be used as input to H;~f and the proce88 repeated until the input and output coefficients are consistent. In the above equations. integration over the spin parts of the Mas has been carried out a.nd the l1l j refer only to the space part of the Mas. In the previous section we stated that the exact MOs belonging to a given orbital energy must form a basis for one of the irreducible representations of the point group to which the molecule belongs and the same is true for the approximate Mas. Furthermore. if one ensures
We now consider conjugated systems and approximate things even further by focusing attention .upon only the or-electrons of such systems. The valence electrons of conjugated systems fall into two classes: O'-electrons and or-electrons. The O'-electrons are assumed to be fairly strongly localized in individual bonds and described by orbitals of O'-type symmetry (using the notation of linear molecules); they normally do not participate in those chemical reactions which do not involve bond breaking and they are regarded as relatively unreactive. The 1T-electrons, on the other hand, are highly delocalized over the carbon framework and play an important role in all reactions; they are often referred to as mobile electrons. In organic chemistry many of the properties of conjugated molecules can empirically be ascribed to the or-electrons alone and this indicates that it is not unreasonable in quantum mechanics to treat the or-electrons in an explicit fashion and to simply regard the O'-electrons as providing some kind of background potential field for them. The quantum mechanical separation of the electrons of a molecule into 0'- and or-electrons is known as O'-or separability. We therefore start the quantum mechanical treatment of conjugated systems by expre88ing the total electronic wavefunction in terms of a wavefunction for the O'-electrons and a wavefunction for the 1T-electrons: 'I"{I. 2•... n) = A(O'.
or)'I"~{I,
2 .....n ..)'I".(I. 2 ..... n.)
MolllClllar Orbital Theory
Molecular Orbital ThllOry
where n. and n. are the number of a- and ..-electrons respectively. A(a, ..) is an antisymmetrizer, which is an operator which 'exchanges' electrons between '1"" and '1"•. It works in the same way as the Slater determinant did in § 10-2. In fact we could have written '1"(1.2 ••.. n)= A{<1>I(I)11l.(2) ... <1>,,(n)}
H = II:+H. where
II:
=
~
~
H.=
" ..
~
,._1
Y>"
N
-12. V:- 2. ~ Z.jr p .+ ~ 2. 1jr pv ~_l
p=1 «=-1
II: + }; ..-1 ~ Ijrpv ~1
and ~ = -
t
'IF
ftr
N
"r ~
,._1
(1:-1
111=-1 Y>I'
~ and l ( refer respectively to a and .".-electrons exclusively and the Hamiltonian H y can also be written as ~
~
n.
H. = 2. h~re+ ~ 2. I/r p • ,._1
h",i're
=
(10-4.1 )
,M=-lv>p.
where
-tV:-
...V
".,.
111-1
",_1
~ Z./r p.+ ~ l/r pv.
(10-4.2)
....
term, ~ I/r"., not present in hI' which comes from the interactions
._1
between the ,uth ..-electron and the n.. a-electrons. The total electronic energy E is given by the sum of two terms
E and
E. = E. =
=
E .. +E
y
f '1":Ir.'I"" d'T. f 'I";H.'I". d'T•.
= A{<1>Hl)<1>;(2) ...
<1>~..(n.)}
a.J4>.(k)
<1>:(k) = ~
•
(10-4.4)
4>.(k) = .".-atomic orbital
(or, for symmetry orbitals, l1lj(k) =
!
a~4>;(k)).
10-5. Hiick.1 molucul.r orbital mBthod Our approximations so far (the orbital approximation. LCAO MO approximation, .".-electron approximation) have led us to a .".-electronio wavefunction composed of LCAO MOs which, in turn, are composed of ..-electron atomic orbitals. 'Ve still, however, have to solve the HartreeFock-Roothaan equations in order to find the orbital energies and coefficients in the MOs and this requires the calculation of integrals like (cf. eqns (10-3.3»: Ii:~r.• = ,p1(I)W rr ' Y (I),pk(l) d'T 1 and
8 Jk
=
f 4>:(l)4>k(l) d'Tl'
In these integrals the additional superscript .". indicates that we are now within the framework of the ..-electron approximation and that essentially H has been replaced by H y (see eqn (10-4.1)) and consequently Wff(!l) (see eqn (10-3.4» by Herr'.(!l) = h~ore+ ~ (2J;(,u)-K,(,u)}.
These last two equations have the same form as eqns (10-2.1) and (10-2.2) except that the core Hamiltonian h,;,re includes an additional
where
'l"y
where
f
~ V:- ~ ~ Z.Jrpa+ ~ 2.1/rpv.
_-1
equations (eqns (10-3.2) to (10-3.4» are applied to the ..-electrons by replacing H by H. and 'I" by '1"•• where '1".. is written as
and
in place of eqn (10-2.3). The .".-electron approximation is then defined as that approximation in which the electronic wavefunctions for some set of molecular states are separable with the same '1".. for all of them. The total Hamiltonian H is then separated into two parts:
20&
,
These integrals are difficult to evaluate exactly and the Huckel molecular orbital method centres on approximations to them. It is assumed that each carbon atom contributes one ..-electron and one 2p. atomic orbital to the system, so that <1>Hk) =
5: a ._1
d
4>.(k)
where n. equals the number of carbon atoms and 4>. is a 2p. orbital located at the 8 carbon atom. The theory then makes the following important approximations: (10-4.3)
Within the framework of the ..-electron approximation E .. is assumed to be simply a constant and the expression for E. is used to find the optimum .".-electron LCAO MOs; that is. the Hartree-Fock-Roothaan
H:;r,r = ex (if rand
8
signify nearest neighbour carbon atoms)
(otherwise)
208
MoleculDr Orhital Theory
MolDculDr Orhital Theory
a. and p are called the Coulombic and resonance integrals respectively and they are strictly empirical quantities which are determined by comparing the results of the theory with experimental data. With these approximations the equation which corresponds to eqn (10-3.5) d et(Heff... rs) - 0 ik - 8 ik -
then the equation which determines the .".-electron orbital energies det(H;lf'" -e"Sik ) = 0
k
given by E ..
ft~11 ej -
=2I
i_I
at,. The
total .".-electron energy is then
... /1 ft. II
I I
"ef2
(Ui/-K,/)
=2I
&'_1 1_1
8j -G.
(10-5.1)
a_I
Since G is 8.BBumed to be constant for all electronic states of a given molecule, the important part of E r is the sum of the .".-electron orbital energies. If symmetry orbitals are used in place of atomic orbitals, then ~:.r and Sik will become integrals over these orbitals and they will have to be broken down to integrals over the atomic orbitals before the Hiickel approximations are made.
10-8. Hiickll ..ollculDr orbitDI ..lthod for be_I We will consider the application of the Hiickel molecular orbital method to the benzene molecule and we will first see what happens when we do not make use of symmetry. The benzene molecule has a framework of six carbon atoms at the corners of a hexagon and each carbon atom contributes one .".-electron. The .".·electron MOs will be constructed from six 2p. atomic orbitals, each located at one of the carbon atoms, thus, • II>j =
tf>.
.I G.d>., ._1
= (2P.).·
If we use the following Hiickel approximations
H'f!·" =
{Po
(r and
8
nearest neighbours)
(otherwise)
(10-6.1)
becomes eqn (10-6.2). This equation can be simplified by dividing each
is solved. The roots of this equation correspond to the .".-electron orbital-energies e~ and they will be functions of a. and {J. Finally, the equations ~ (Heff... "S)G IN -- 0 .4 ik - e, ik J. = 12 . , ... n. are solved for the coefficients
207
element by
0
0
p
0
0
0
{J
0
0
fJ
CIt-e"
fJ
()
0
0
{J
(X-B"
{J
0
0
0
fJ
lX-e-
/X-err
{J
p
at-e"
{J
0
fJ
cx_e
0
0
0
fJ
0
Tr
fJ and letting
=0
(10-6.2)
(a.-er)!fJ
z =
to give eqn (10-6.3) which can be solved and the six roots z •• z ..... , z. x
1
0
0
0
1
1
x
1
0
0
0
0
1
x
1
0
0
0
0
1
x
1
0
0
0
0
1
x
1
1
0
0
0
1
x
=0
(10-6.3)
(and hence the six ...·electron orbital energies) determined. The solution of eqn (10-6.3) requires multiplying out the determinant, obtaining a sixth order polynomial equation in z and then finding the six roots. This can be quite time consuming. Now let us see what happens if we apply symmetry rules to the problem. Essentially what we will do is to write
11>1
=
•
I O;I; ._1
where the ; (symmetry orbitals) are symmetry-adapted combinations of 2p. atomic orbitals which generate irreducible representations of the point group to which the molecule belongs. This is the same thing as s~ying that we will change the basis functions used for .lI>r from tf>. to 4>•. Though benzene belongs to the ~llh( = ~. ® ~I) pomt group, we can, in fact, get all the information we require from the simpler point group ~., to which benzene also belongs.
Mol_lor Orbitel Th8Dry
ZOI
Mol.euler Orbitel Tb8Dry
The six 2p••"omio orbitals (4)., 4>_, ... 4>.) form a baais for a reducible representation r AO of ~., since by applying the usual techniques (§ 5-7) we find that the transformation operators 011 transform 4>, either into itself or the negative of itself or into one of the other five atomic orbitals or the negative of one of the others: OR4>, = ±4>, or
201
(the ~ indicates. for example, the three operators associated with the three 0; symmetry elements) pB.
=
0E-(OC. +Oc.-.) +(Oc.+Oc.-')-Oc.- ~ 0C.'+ ~ Oc.-
pEl = 20E+(Oc.+ O c.-.)-(Oc.+ O c.-.)-20 c • pEl = 20E-(Oc.+ O c.-.)-(Oc.+ O c.-.)+20 c•.
Since there is only one baais function for the one-dimensional irreducible representations r.d· and r B•• we need only apply p.d. and pB. to one of the starting functions 4>.:
0R4>. = ±4>.
so that
pd.. = .+(4)_+4>.)+(4>.+4>.)+4>.
The diagonal elements of the matrices nA°(R) will only be non-zero if an orbital is transformed into the positive or negative of itself. hence we obtain the following characters for r AO ~
E
•
6
o
c.
3C~
3C;
o
-2
o
o
0;
0;
pE'4>1 = 2.+4>.-4>.-24>.-4>.+4>., pE'4>. = . +2,+4>.-4>.-2.-4>., pE'4>. = 24>.-4>.-4>.+24>.-4>.-4>., pE.4>. = -4>. +2.-.-4>.+24>6-"'"
0;
and
Since in Hiickel molecular orbital theory it is assumed that
"'I'
and therefore it must be possible to find combinations of 4>., ... "'. which will serve as bases for the irreducible representations pd., pE., rEI, rEt of ~•. [The same combinations will also necessarily generate irreducible representations of ~.h and since each 4>. changes sign under 0 .... the corresponding irreducible representations from g.h must be such that Z(O'h) is negative.t From the character table for ~8h it is clear that we must have
J4>r4>J dT == ~<J' the symmetry orbitals are readily normalized. For example. for the first symmetry orbital the normalization constant N is given by the equation I
=
JN*2(4)1 +"'.+4>.+4>. +4>.+4>.)*N2(4)1 +"'. +4>. +. +4>.+4>.) dT
= 4N-
To find the partioular combinations of .p, which form a basis for r B ., rEI and rE" we make use of the projection operator technique and define the following operators (see eqn (7-6.6)):
r.d·,
pI' =
:! Z"(R)*OR It
pd,
=
0E+(OC.+Oc.-')+(Oc.+Oc.-')+Oc.-:! OC.'-:! 0c..
tSee footnote on page 216.
=
and these two linear combinations will form a basis for rd. and r B•. respectively. For rEI and rEi one must apply pEl and pEl to at least two 4>. In order to produce two linearly-independent basis functions for each of these two-dimensional irreducible representations. Hence
(the 0r-0.-G. axis is perpendicular to the molecule and through its centre, the three and three axes are in the molecular plane with the axes running through opposite carbon atoms and the axes bisecting opposite bonds). Using these characters and the~. character table (see Appendix I), the standard reduction formula (eqn (7-4.2» leads to: rAO = r.d.®rBtEfjre:'(!lrE,
0;
=
pB'4>.
-( -4>1-4>_-4>.) -( -4>.-4>. -.) 2(.+4>.+4>.+4>.+4>.+4>.) 2(4).-4>_+4>.-4>.+4>.-4>.)
• •
• •
~ ~f 4>r4>JdT = 4N':! ~d"
l-lJ_I
'_I
1_'
=
24N-
and therefore N = (24)-t and the normalized symmetry orbital is (4)1 +"'.+4>. +"'. +4>. +4>.)/-..16 . It is convenient if the symmetry orbitals belonging to a degenerate irreducible representation are made orthogonal to each other and this is achieVed in the present case by taking combinations which are the sum
210
Molacular Orbital TIIaory
Molecular Orllital lb_"
and difference of the original combinations. This works since if F and G are two real normalized functions. then
f(F+G)(F-G)
d'T =
=
then we obtain eqn (10-6.4) which is in block form. Any determinant in
fF' d'T+ fGF dT- fFG d'T-IG' d'T 1-1
=0. The orthonormal symmetry orbita.1s are therefore:
(.pI +4>.+.p.+.p.+4>.+4>.)/v'6, 4>~ = (4)I-4>.+.p.-4>.+4>5-4>.)/v'6, ~ = (4)1-4>.-24>.-4>.+4>5+ 24>.)/211'3, .p~ = (4)1 + 4>.-.p. - 4>.)/2, 4>~ = (.pI +.p.-24>.+4>.+4>,-24>.)/2"';3. .p~ = (.p1-4>.+4>.-4>.)/2.
4>i
=
(pdS) (r
BS
)
(r&'l) (r,el) (rES) (rEs)
These six orthonormal functions are an equivalent orthonormal basis to that of 4>, (they describe the same function space) and if we use them in place of 4>, by writing
fIl1
=
•
~ O~/4>;
=t f (4)1 +.p. +4>. +4>. + 4>,+4>.)HeCC'V(4)1 +4>.+4>.+4>. +4>.++.) d'T =«~+p+p+p+«+p+p+«+p+p+«+p+p+«+p+p+p+~
«-2p H;" = «+P H~ = «+P H!.. = «-p H~ = «-p
and,mostimportanUy. 0
for i =l=j
S;I = ~'I'
If we divide each element in the determinant by
z
0
0
0
o o o
x-2
0
0
0
0
0
z+1
0
0
0
0
0
x+l
0
0
o
0
0
0
x-I
0
o
0
0
0
0
x-I
=0
(10-6.4)
Al
[0]
[0]
[0]
A.
[0]
[0]
[0]
A,
then IAII = 0 or IA.I = 0 or IA.I from eqn (10-6.4) we obtain:
=
=0
0 etc. (see eqn (0-9.8». So that
= (a._ar)/p,
or or
z-2 = 0 x+l = 0
(twice)
x-I = 0
(twice).
It is clear that eqn (10-6.4) is much easier to solve than eqn (10-6.3). though the results, of course, are identical. In general, by using symmetry orbitals .p: which are a basis for one of the irreducible representations of the point group, the matrix whose elements are
(H'.-evS'.)
H~ =
H~I =
0
z+2 =0
f 4>i· Hefc.v4>~ d'T
and
0
block form can be factorized. for example if
or
where H;. and 8~. have the same form as H:f:" and SI. except that in the integrands each .p, is now replaced by 4>;. Thus:
=«+2P
z+2
a-J.
then eqn (10-6.1) becomes det(H;.-ewS'.) = 0
H~, =
211
p and put
will be in block form with each block corresponding to the symmetry orbitals which belong to a given irreducible representation. This occurs since the Hamiltonian H eff•• commutes with all O. and' this means that it belongs to the totally symmetric representation r 1 (see Appendix A.I0-3). The vanishing integral rule (§ 8-4) then predicts that Jt/>;.Jrff.r4>~ d'T is zero if 4>; and"';' belong to different irreducible representations. Similar arguments also hold for S;•. The 'IT-electron orbital energies for benzene are therefore, in order of increasing energy: «+2P, «+P, «+P. «-P. «-P, and «-2p (P is negative) and these energy levels are labelled by using the lower case
212
Mol_lar Orbital Thaory
Molecular Orbital Thaory
notation of the irreducible representations of ~Ih which are associated with them: ,,-2fJ ?I-fJ %+f3 " +2f3
TABLE 10-7.1 .f>. undet' O. Jor the trivinylmethyl radical
TrafUlJormatiO'fl oj
( b o.)
-
-
CfJ:
=
(e I.) (ao w )
4>:
. .. . . . "'. "'"
Os
.;, .;. .;. .;.
(e •• )
The crosses indicate that two electrons are fed into the non-degenerate a... level (one with spin cx:, the other with spin fJ) and that four electrons are fed into the doubly degenerate e, • level; the whole making up the ground state 7T-electron configuration (a tu )·(e, .)·. Ignoring G (see eqn (10-5.1)), the 1T-electron energy for the ground state will be 6cx:+8,8 which, when compared with the 7T-electron energy of three ethylene molecules (6cx:+6,8). shows that the delocalization energy of benzene is 2,8. In this problcm wc are fortunate that the factorization of the determinant is complete and as a consequence the 1T-electron MOs are. in fact. identical with the symmetry orbitals:
';1
.pI .p.
.;, .; .; .;. .;. ';1 .;
oc. Dc: .;. .;. .;, .; .;.
4>•
.;
.;,
.; .;. .p.
.p.
split into two one-dimensiona.l representations (rE , and rE·), where the characters of one are merely the conjugate complexes of the characters of the other. Use of the reduction formula (eqn (7-4.2)) leads to
rA.O
=
aJU6:l2rE'EEl2rE.
that the seven atomic orbitals can be combined into seven symmetry orbitals: three forming a basis for r..t. two for r E , and two for r E•. These symmetry adapted combinations are found by using the projection operators: p" = ~ %,,(R)·O. 80
R
on the .pi' The following linearly independent combinations are then obtained:
= .f>, +4>. +4>., p..t4>. = 4>.+.f>.+.f>., p..t4>. = .f>.+.f>.+.f>•• pE,.f>. = .f>, + e*~ + ecf>3, pE'.f>. = .f>. + e*cPf> + ecP6' pE.",. = .f>, +.e~+e*cf>3, pE.",. = .f>. +;£<1>5 + e*6' pA""
10-7. Huckel molecular orbital method for the trivinylmllthyl radical The trivinylmethyl radical ·C(CH=CH.la has seven carbon atoms and seven 1T-electrons and belongs to the .@'h point group. We will however use the lower symmetry point group 'if. to which the molecule also belongs. The labeling of the carbon atoms is shown in Fig. 10-7.1. In Table 10-7.1 we show how the seven 2p. atomic orbitals (.p1> .p..... .p.) transform under the operators O. and from these results we obtain the characters of r AO ; they are given together with the 'ifa character table in Table 10-7.2. It will be noticed that the r E repres~ntation haa been
TA.BLE 10-7.2
OharMter table 'ira and ZAO Jor the trivinylmethyl radicalt E
A &
Fla. 10-7.1. The trivinyhnethyl radioa.l.
213
J &, IE.
1 1 1
1
"e "
"I
"
t" = exp(2m/3) = ih/3i-l) ~. ~ exp( -2..i/3) = -ttl + v'3i) ~+e· - - I
214
Mol_I., Drbi..1Tb..ry
Mol_l.r Orbiul Th80ry
where s = exp(21ri!3) and all other combinations, e.g. pAt{>., will either be one of these combinations or BOme combination of these combinations. When normalized, the symmetry orbitals are:
(t{>.+t{>.+t{>.)!Y3, (t{>,+t{>.+t{>.)/y3, t{>., t{>~ = (. +s*t{>. + st/>.)/Y3, t{>~ = (t{>.+s*t/>.+s,p.)fy3, ,p~ = (t/>. +s,p.+s*,p.)!y3, 4>:. = (t/>,+s,p.+s*,p.)/y3. t{>;. = ~ = ~ =
+ + + + + + + -t + +
+
+
+ or
ell:
for
s;
for s; =
e:
I: :1
x (10-7.2)
=0
(10-7.3)
= 0
= 0
and
f(~'+B~+B."'.)rr"'.(~'+B.~'+B~.)d ...
a~: = 1,
~
rz.,
=
a~
= - y3!2,
= 0,
= -(y3/2)t/>~+0 x ,p~+l,p;
rz.-2p, ail = ell;
=
y2/4,
a~.
a~. = 1/2
= <+.++.+t/>.-4>,)/2
= -I!y2,
a;. =
y6/4
(t/>t+t/>.+,p.-2t/>.-2t/>.-2t/>.+3t/>,)!2Y6.
rz.+2p a~a
r". and r". type oym-
f ~~·rr"··~;d...
=
and
= v'2/..,
11>; =
= 1/y2, a;s = y6/4, (t/>1+,p.+t/>.+2t/>,+2t/>.+2t/>.+3tf>,)/2Y6. a~.
Multiplying out eqn (10-7.2), we get zO-1 = 0 or z. = I and -lor = rz.-P and s: = rz.+P, thus
z. =
=Cl
-fJ
y3
Clearly it is much easier to solve one 3 x 3 and two 2 x 2 determinantal equations than the 7 x 7 determinantal equation which occurs when no use is made of symmetry. Multiplying out eqn (10-7.1), we get zS-4
O~.
f~~'~'''~~dT
=.
o
(10-7.1)
y3 =0
1: :1
we obtain for
+
- I f (~. +B~+.·~.)H"'··(~.+.·~+ ...) dT =
x
L
t Care baa to be taken with the ooDjugate oomp1e%.... in the metry orbit&ls, e.g.
H;,
1
0
and using the normalization equation
where the first block corresponds to the three symmetry orbitals of p4 type, the second block to the two symmetry orbitals of r B, type and the third block to the two symmetry orbitals of rB· type. If we evaluate the elements of this determinant in accord with the Huckel approximations, divide each element by p, replace (rz.-s·)/P by x and factorize the determinant into its three parts, we obtaint eqn
a.nd.
I
~
+ +
=
x
L (H;~-s~S;~)O~,
+
H;.
(10-7.1) for the p4 symmetry orbitals, eqn (10-7.2) for the rE, symmetry orbitals, and eqn (10-7.3) for the pEl symmetry orbitals.
(rA) (rA) (rA) (pE') (rE .) (pE,) (rB,)
Since ft/>:t/>I dT = 6;/ (Huckel approximation), it is easily confirmed that these t/>; satisfy fr/>;*4>; dT = 1. Because the symmetry orbitals belong to the irreducible representations of W., the det(H;~- s· S;~) will be in the following block form :
215
and
s:
11>~
= (t/>.+e*,p.+et/>.-t/>.-&*,+.-s+.)!Y6
ell:
=
(,p.+s*.+e4>.+4>,+s*t/>.+st/>.>ly6. e; = rz.-P and s; = rz.+P, and (t/>.+er/>.+s*,p.-r/>,-et/>.-e*r/>.)!Y6
From eqn (10-7.3) we get
11>; =
Mol_lar Drbitol Theory
Mol.lI.r Drbltol Theary
218
&Dd
rB·.
(t+ e"'.+e*"'.+"'.+e.+a*.)/v'6.
~; =
The ....-electron MOs ~: and ~: are degenerate, in that they correspond to the same ....-eleotron orbital energy (a. - p). As is the case with any degenerate wavefunctions, any combina.tion of ~: and ~: will be equally valid. It is convenient to combine ~: and ~: in such a way as to obtain real MOs, this can be done by: ~:(real) = (11l:+~:)/v'2 = ~:(real) = (~:-~:)/v'2i =
(21-4>0-.-2"'.+"'.+"'.)/v'12 (<1>.-4>.-4>.+<1>.)/2
(note that e+s* = -1). Similarly for ~: and ~: (both corresponding to an orbital energy of ot+P), we can find the following two real MOs: ~:(real) = (11l:+~;)/v'2 =
I1lHreal)
=
(11l:-~;)/v'2i
(21-.-4>.+2"'.-"'.-"'.)"/12 = (.-4>.+4>.-"'.)/2.
The ....-electron orbital energy level diagram will be: a -2fJ a-fJ
(II)
- --
'"
"'+~
'" + 2fJ
(e)
-
The electronic configuration of the ground state will therefore be
(la;)0(le")4(2a;)'. Appandica
A.10-t. Atomic units The atomic unit of length is the radius of the first Bohr orbit in the hydrogen atom when the reduced mass of the electron is replaced by the rest mass me' Thus the atomic unit of length is 4 ....·m..e• The atomic unit of energy is
A.10-2. An altamativa notation for tha LeAD MD method
(a)
Define the MOB by
= -~~ i = 1,2,:.. 7 Ill:. and ~; change sign: i
=
4>p
.
F~. = H~.
1,2,3
cJ); are unchanged: 11l~ i = 1, 2, 3.
p
2p. orbitala ...... perpend.i<>Ular to the moIeeular plane a" a.nd
0",,4>,
=
-4>,.
p•
=
I
(fl11 Au) =
This information is sufficient to classify the MOs with respect to the irreducible representations of 9&h and using the character table, we see that ~;, IP;. and ~; belong to JUI" and ~:, ~:, cJ):, and Ill; belong to
t The
=
h~ =
and that under 0"., ~:, ~:, a.nd O",~~ =
p
are atotnic orbitals, then the coefficients Cp , are determined by ..l (F".-8.S~.)O.1 = 0
O",,~~
-~~
'" .l C~,4>~
Ill, =
where
Oc.~~ =
0·052918 nm.
(a) (e)
E r = 7a.+8p-G. If we consider the 9 1h point group, we find that under the transformation operator O"h all the molecular orbitals change sign:t ~:,
=
e"la. = 27·210 eVt = 1 a.u. (of energy). This is just twice the ionization potential of the hydrogen atom if the reduced mass of the electron is replaced by the rest mass. One atomic unit of energy is equivalent to twice the Rydberg constant for infinite mass. When atomic units are used, one sets e = h/2.... = m.. = 1 in quantum mechanical equations. For example -h'V"/87T"m e becomes -1'\7". The advantage of atomic units is that if all the calculations are directly expressed in such units, the results do not vary with the subsequent revision of the numerical values of the fundamental constants.
where
"
h'
a. =
and if we distinguish the MOs of the same symmetry by preceding the irreducible representation notation by a. number which ascends with increasing orbital energy, the 17-electron configuration in the ground state will be (la)·(le)·(2a)1 and the total ...-electronic energy will be
and that under 0c
217
S~. = t
1 eV = 1·60219 x 10-1 • J.
H".+
fl = 1,2,..., m i = I, 2,e .. , m
.
~ ~.. P ...{{fl11 I Au)-i{flu I A11)}
f4>:(I)h <1>.(I) dT1 -tV;-._1 .lZ.lrJl« 1
N
"t.
2.l C:fJ"
(n
=
number of electrons)
,-1 :{I)4>.(Wi;:{2)4>,(2) dT1 dT.
ff f :{I)4>.(1) dT1'
218
Molecular Orbital Theory
The total electronio energy is given by: E =
i; i; p ..fi...+! i;i; i i; p ...P).~{(IlV I Aa) IA
"
P
000
=
.,..
'"
..
}.
I
11. Hybrid orbitals
i(!Ja AV)}
IF
m. m
2.~ ,,_1 e,-! ~.!.!.! ,. .., .t _ p ...P).~{(IlV I Aa)-lCua J A")}.
The reader may confirm that the content of these equations is the same as that of eqns (10·3.1) to (10.3.6). A.10-3. Proof that the matrixelemants of an oparator Hwhich commutes with all OR of a group vanish between functions belonging to different
irreducibla representatione Let 'I'~ be a set of functions belonging to the irreducible representation r" and H an operator which commutes with all the transformation operators OR, then and 01l(H~) = HORv!: =
H(~ D7,(R)tp'i)
11-1. Introduction IN this chapter we explore how symmetry considerations can be applied to one of the most pervasive concepts in all of chemistry: bonding between atoms by the sharing of pairs of electrons. Though the idea of an electron-pair bond was first introduced in 1916 by G. N. Lewis, it was only after the advent of quantum mechanics that it could be given a proper theoretical basis. This came about through the development of two theories: valence bond (VB) theory and localized MO theory; both of which describe the electron pair in terms of orbitals of the component atoms of the bond. In VB theory the pair of bonding electrons in the bond A~B of some polyatomic molecule is described by the wavefunction
= ~ D7.(R)(R'I'7)· J
Consequently, the functions Htp't also form a basis for invoking the vanishing integral rule, the integral
J'1';* vanishes unless
r
y
=
H'I': dT =
r"
and hence by
J
'P;*(H'I',)" dT
r".
'1"(1)'1"(2)
r 0r"
. If we consider H'P': in the direct product representation H then smce HY": ~long to P', rH~p. = r" and therefore r H = r 1. Hence, any operator which commutes WIth all OR of a point group can be said to
belong to the totally symmetric irreducible representation
10.1.
r1.
PROBLEM For the following molecules, determine the point group and the synunetry of the MOs for the ".·electrons, and, using Hiickel theory obtain the MOs and orbital energies: ' (a) trans-l,3-butadiene. (b) ethylene, (e) cyclobutadiene, (d) cyclopentadienyl radical C,H•• (e) naphthalene, (1) phenanthrene.
where 'P A. is an orbital centred on nucleus A and 'PB is an orbital centred on nucleus B and the 1 and 2 indicate the two electrons (we ignore electron-spin considerations). In localized MO theory the electron pair is described by the wavefunction
where 'I" is a localized MO extending over both nucleus A and nucleus B and which can be synthesized from an orbital centred on A( 'I'.a.) and an orbital centred on B ('PB), i.e. (11-1.1)
where C 1 and c. are numerical coefficients. Both of these bond descriptions are approximations and at first sight appear to be quite different, but, if we carry the approximations a stage further, the methods converge and become completely equivalent. For this reason we will only consider one of them and choose for our purposes the localized MO method. When considering a polyatomic molecule, the general MO method (see Chapter 10) would describe the n electrons of the molecule by the
220
Hybrid O....i••I.
Hybrid Orlli••I.
wavefunotion (see eqn (10-2.3»
'1'(1,2, ... 71) =
I/Vn1 <1>1(71)
1Il.(n)
<1>A(n)
where the «11, are MOs which extend over the entire molecule, not just a single bond as in localized MO theory, and can be approximated by linear combinations of atomic orbitals centred on all the nuclei. Indeed, most quantum mechanical calculations done to-day use suoh wavefunctions. Clearly, the localized MO method, where the electrons in a polyatomic molecule are divided up into bond pairs, each described by MOs of the form of eqn (11-1.1), is an approximation to this more general treatment. The question arises therefore: why do we bother with itt The answer is two-fold. In the first place, chemical intuition and experience tells us that many properties of molecules are properties of the bonds and that these properties are often constant from one molecule to another, e.g. the existence of a characteristic infra-red absorption band near 3 pm due to a C-H valence stretching mode is used to detect the presence of C-H bonds in an unknown molecule. Such constancy would seem to imply localized distributions of charge which are transferable and which could be adequately described by localized MOs. In the second place, localized MOs are easier to imagine and handle and they preserve the conventional idea of a bona which is typified by the symbol A-B. Symmetry plays an important role in localized MO theory since the orbitals UBed in the construction of the MOs "P A and 'PB of eqn (ll-1.I), must be symmetric about the bond axis (for the present we will limit our discussion to a-bonding). The most natural, though not mandatory, building blocks to use for "PA and 'l'B are the atomic orbitals (AOs) of the component atoms (A and B).Jn some cases there is available a single AO on A and a. single AD on B, both of which are symmetric about the bond axis and therefore meet our requirements. But more often, and particularly when A has to form several bonds, there are not the required number of atomic orbitals with the appropriate symmetry and it is necessary to synthesize 'PA (or 'PB) from several ADs of A (or B). For example methane CH. is a tetrahedral molecule with four equivalent C-H bonds pointing to the corners of a tetrahedron and each localized MD is made up of an orbital from the
221
carbon atom and an orbital from the appropriate hydrogen atom. The oontribution from eaoh hydrogen atom oan be taken as a Is hydrogen AO and these will be symmetric about the appropriate bond axis; however, amongst the AOs of the carbon atom Is, 2s, 2P.., 2pw' and 2p. there are not four which are equivalent and symmetric about the four bond axes. We are therefore forced into taking combinations of these primitive orbitals if we wish to have four equivalent and symmetrio orbitals; this procedure is called hybridization and the combinations are called hybrid orbitals. If we restrict ourselves to the broad class of molecules which have a unique central atom A surrounded by a set of other atoms which are bonded to A but not to each other (e.g. mononuclear co-ordination complexes, NO;, SO:-, BF., PF., CH., CHCI., etc.), then the symmetry of the molecule will determine which AOs on atom A should be combined (§ 11-3) and in what proportions (§ 11-5). If there is more than one combination of AOs on atom A having the correct symmetry, and this will usually be the case, then arguments of a more chemical nature will have to be invoked in order to decide which is the most appropriate combination. A necessary prelude to determining the combinations of AOs which give a hybrid orbital of correct symmetry is the classification of the AOs of the central atom A in terms of the irreducible representations of the point group to which the molecule belongs. This is discussed in § 11-2. In § 11-4 we consider 'IT-bonding systems and in the final section we discuss the relationship between localized and non-localized MO theory. The reader who is not familiar with the background of this chapter, and it has only been summarized in the preceding paragraphs, is recommended to read C. A. Coulson's excellent book: Valence. 11-2. Tr.nsform••ion prop....i.. of ••omic orbitals
In constructing a localized MO for the bond A-B it is necessary to specify an orbital centred on A ('PA) and an orbital centred on B ('I'B)' In principle, provided symmetry about the bond axis is preserved (we are still considering only a-bonded systems), our choice of 'I'A and 'PB is not restricted and we could use any well-defined mathematical function or combination of functions. Common sense, however, dictates that the most sensible functions to use for this purpose are the AOs of the free atoms A and B. There are three reasons why this is a sensible choice: one mathematical, one chemical, and one practical. The mathematical reason is that the ADs of a given free atom form what is known as a complete set, that is any function can be produced
222
Hyllrid OrllitaJa
Hybrid Orbitala
by taking a combination of them; 80 we know that it is mathematically possible to replace Y'A' whatever its form, by a combination of A's AOs. The chemical re&80n is that the bond A-B is chemically formed by combining atom A with atom B and we expect the electronic distribution, at le&8t close to the nuclei, to be similar in the bond to what it is in the free atoms. The third reason is that we know from atomic calculations the energy order of the AOs and we expect that the lowest energy MOs will be those formed from the lowest energy AOs. This fact can often help us decide which AOs to choose for the construction of an MO when symmetry arguments leave the matter ambiguous. Having decided to use AOs (or combinations of them) for Y'A and 'i'D' we will now look at the form these take. They are approximate solutions to the Schrodinger equation for the atom in question. The Schrodinger equation for many-electron atoma is usually solved approximately by writing the total electronic wavefunction as the product of one-electron functions rpi (these are the AOs). Each AO . is a function of the polar coordinates r, 0, and rp (see Fig. 11-2.1) of a single electron and can be written &8
.p.
=
R,(r)Y.(O,
.p).
The radial functions R.(r) will be different for different atom8. Only for the hydrogen atom is the exact analytical form of the Ri(r)'s known. For other atoms the R.(r)'s will be approximate and their form will depend on the method used to find them. They might be analytical functions (e.g. Slater orbitals) or tabulated sets of numbers (e.g. numerical Hartree-Fock orbitals).
223
TABLlIl 11-2.1
Angular fUndiona (un-1Wrmalizea) for s, p, d, ana f
orbitala Symbol
no &ngUlar dependenOB sin OOOB .p 8in 0 sin .p cos 8 3008'0-1 .in·O OOB 2.p oin'O sin 2.p .in 0 OOB 0 00• .p .in 0 OOB 0 Bin .p .in OOOB .p(5 Bin'O 00B·.p-3) .in O.in .p(6 .in·O sin',p - 3) /j 008'0-3 OOB 0 .in 0 OOB ,p(oOB'O-oin'O sin',p) .in O.in "'(oOB'O-sin'O OOB',p) sin'O OOB 0 OOB 2'" .in·O OOB 0 .in 2
8
P.
P.
P.
d.,.I_' or d.' d,,>.....
<4.
£',,,'-8,.::1 1 or (.'I
f.c ••I_ a,..) or f. a (••,.I_I,.', or f. a f.c.·~)
c,.,.......,
f.,,,-.I. f ...
The angular functions Y,(O, , then changing to the Cartesian coordinates:z:, 1/. and z where: :z: = r sin 0 cos and z = r cos 0 gives us: sin"O sin 2
=
2 sin"O sin cos cos
= 2(sin 0
= 2(:z:/r)(y/r) = (2{rl ):z:y
=/(r):z:y
,
" "
" ,
' ....
y FIG. 11-2.1. The ooordinate system for atomio orbita1s.
and any orbital with this angular dependence is given the subscript :Z:1/. IT we restrict ourselves &8 before to those molecules which have a unique central atom A surrounded by a set of other atoms which are bonded to A, then in order to &8certam which AOs of A can be used to produce a Y'A which is symmetric about a particular bond axis, it is necessary to know to which irreducible representations of the molecule's point group the AOs of A belong, i.e. for which irreducible representations they form a baais. That they must form the baais of 80me representation of the molecular point group follows from the fact that
225
Hybrid Orbital.
H"rld 0 rbil.l. TABLE
The symmetry species oj the s, p, and d
11-2.2 orbitala for different point group8
•
A"
P.
E'
A..
A ..
..4'"
T,
T,.
P. B
A
B
B,
B.
B,
111
111 B
A
B.
B 111 d 1f 1'
.A.
111
IiI
E
IiI,
IiI
B
since atom A is unique, it must lie on all the planes and axes of symmetry which the molecule possesses. We have considered this question for p- and d-orbitals using the ~u point group in § 7-9 (see also § 5-9) and the teohnique used there can be applied to all orbitals and all point groups. The results are given in Table 11-2.2; they are also incorporated in the oharacter tables in Appendix I at the end of the book. .As an example of using Table 11-2.2, consider the phosphorus atom in PF•. This moleoule belongs to the ~'h point group and we immediately see that the phosphorus atomic orbitals belong to the following irreducible representations:
rA":
r B ': rA t -: rr:
s, d.t (P., P.), (d.,-.", d...)
P.
(d.•• d ••)
where the parentheses indicate that the two orbitals inside them togethu form a basis for the given two-dimensional·irreduoible representation. The reader should note that no transformation operator O. can alter the radial function R,(r) of an orbitalt and consequently the symmetry properties of the AOs are completely defined by the angular functions. Y,(8,
t The only exoeptioD to thic rule ic the hydrogen atom, _
the footnote OD page 156.
etc.) will belong to the symmetry species designated in Table 11-2.2. Similarly the symmetry species of an orbital will be independent of the atom involved and of the principle quantum number of the shell to which it belongs. 11-3. Hybrid arbitsls for cr-bandinllsystems A noaal plane or surface is the locus of all points at which a wavefunction has zero amplitude as a result of its changing sign on passing from one side of the surface to the other; the probability of finding an eleotron on such a surface is zero. a-Bonds and a-orbitals are defined as those having no nodal surface which contains the bond axis; such bonds and orbitals will be symmetrio about the bond axis. In this section we oonsider which AOs of a central atom A (whioh is bonded to a set of other atoms) can be combined to form a hybrid orbital which is symmetric about the bond axis and therefore capable of a-bonding. We will explain how to do this by taking the specifio example of methane. Methane has a central carbon atom which is a-bonded to four hydrogen atoms with each a-bond pointing to one of the corners of a tetrahedron. We therefore require four hybrid orbitals on the carbon atom which similarly point to the corners of a tetrahedron. Since the four bonds are indistinguishable. the four hybrids must be equivalent, that is to say they must be identical in all respects except for their orientation. For the reasons given in § 11-2, they will be taken to be linear combinations of the atomic orbitals of carbon, which are
22&
Hybrid Orbitllis
Hybrid Orbi••ls
themselves bases for certain irreduoible representations of the point group of the molecule ~d' The hybrid orbitals. &8 we will see in a moment. form the basis for 0. reducible representation of the~d point group and we will give this representation the symbol ~b. Using the equations in § 7-4 we can reduce ~b and find the irreducible representations which it contains. Let us suppose we find
r h7b
=
r1
227
It is clear that only if v, is unshifted by R will Dl'r'(R) be non-zero and then it will be unity. Consequently i'Yb(R) is equal to the number of vectors which are unshifted by R (this line of argument is analogous to that used in considering the representation ro in § 9-6). Proceeding in this way. we find the character below for r h7b for the ~d point E
8CI
4
1
(j;) r~
where r 1 and r" are two irreducible representations of ~d' This implies (see Chapter 6) that there are combinations of hybrid orbitals which
XhTb(C,)
0
0
2
group. Using this information together with the character table for and eqn (7-4.2), we find
~d
f'hTb
= r.d, EEl r T ,.
Recalling our earlier discussion, this equation means that the four AOs which are combined to make the set of four hybrid orbitals must be 1 chosen 80 &8 to include one orbital which belongs to the repreT sentation and a set of three orbitals which belong to the r , representation. Reference to Table 11-2.2 shows that the AOs fall into the categories below for the ~d point group. So we can combine an s-orbital
r..c
FIG. 11-3.1. A Bet of veators v" v•• v.. and v. representing the four a.hybrid orbitals used by oarbon to bond the four hydrogono in methane.
r..c,
will serve as a basis for the £,1 and r~ irreducible representations of Now we can rever8e the argument and say that there are combinations of those functions which belong to 1'1 and r l which will serve as a basis for r hyb and which can be taken as the hybrid functions with the symmetry properties which we desire. We conclude. therefore. that we should take linear combinations of those atomic orbitals which belong to f'l and 1'0. First then. for methane, we must obtain ph7 b • To do this let us associate with each carbon hybrid orbital a vector pointing in the appropriate direction and let us label these vectors VI' va. Va. V. (see Fig. 11-3.1). All of the symmetry properties of the four hybrid orbitals will be identical to those of the four vectors. Thfl reducible representation r hyb using these vectors (or hybrids) as a basis can be obtained from Rv. = L DN'b(R)v, i = 1,2.... 4
s
~d'
•
1_1
(compare with eqn (9-4.2», but since we only wish to carry out the reduotion of rhYb, we only need the character of ~7b, which is given by t'Yb(R)
= L• ~r(R). '_1
r T, (Pc' P.' P.) (d.., dc., d n
rE (d." d..-..') )
with the three p-orbitals in four different ways to produce four equivalent hybrids, called Spl hybrids, or we can combine an s-orbital with the three d-orbitals (d... de.. d ••) in four different ways to form four equivalent sd a hybrids. Both sets of hybrids will have the correct symmetry properties and will point in tetrahedral directions. This is &8 far as symmetry arguments alone will take us and we can only conclude that the most general solution to the problem would be a set of hybrid orbitals which are linear combinations of both possibilities. namely
'Po = a(spl) +b(sdl)
(plus additional terms if f. g •... orbitals are taken into account). The numerical coefficients a and b might be determined by some qua.ntum mechanical technique. However. it is at this stage that our chemical intuition can be used. or rather our knowledge concerning the relative energies of the AOs in a free carbon atom, and we can predict that b is quite small compared with a. If we assume that the Is orbital cannot be used on the grounds that it is a1rea.dy occupied by two core (or non-bonding) eleotrons then the
228
Hybrid Orbl..,.
spS hybrids can be constructed from the 2s- and 2p-orbitals of carbon. The lowest energy, and therefore most stable, d-orbitals available to carbon are the 3d-orbitals, so that the most stable sd" hybrids would be constructed from the 2s, 3d"", 3d",., and 3d". orbitals. However, in the carbon atom the 3d orbitals lie about 230 kcalfmole higher than the 2p-orbitals. Therefore, in order for the bonds formed by using sd" hybrids to be more stable than a set using sp" hybrids, each sd" bond would have to be about 3x 230/4 ,..., 170 kcal/mole stronger than each spa bond. This is higWy unlikely and even a limited usage of sd' hybrids will
(d••, d".)
(I) (2) (3) (4) (6) (6)
v, B Ii
v,
H FLO. 11-3.2. A set of vectors Vi. v., ... , and V6 representing the five a-hybrid orbitals used by A to bond the five B atoms which surround A in B trigonal bipyramid structure.
be of very minor importance. We conclude that for CH. the hybrids are of Bpi character. However, for a species like MnO~ the sd" hybrids will predominate beoause in this case the lowest available d-orbitals are 3d and the lowest available p-orbitals are 4p and the 3d-orbitals are of lower energy than the 4p-orbitals. As a second example, let us consider a molecule with the formula AB. having the symmetry of a trigonal bipyramid .@ah' The vector system is shown in Fig. 11-3.2. The set of five hybrid orbitals (or vectors) on A form a basis for a reducible representation of the ~Il> point group, with the following character: . .@"h
E
2C,
3C.
0.
28.
30.
XhYb(C,)
6
2
I
3
0
3
From this we deduce that: ['byb =
two orbitals which belong to p"',', one orbital which belongs to ~. and a set of two orbitals which together belong to rr. The atomio orbitals of A fall into the following categories for the ~Ih point group (see Table 11-2.2):
Therefore, any of the following combinations of AOs will produoe appropriate (from a symmetry point of view) hybrid orbitals:
B
H
229
2P-!l' (j;) ['..f.' (j;)
pw
and that the set of five atomic orbitals on A which are combined to produce the set of five hybrid orbitals must be chosen 80 as to include
(n+l)s, P.' P.' and P., (n + I)s, P.' d••, and d,.'-"o, 00 0°, (n+l)doo, P., P., and P., nd.o (n+l)d.., P., d,.., and d.o.....o, s, d .., P., P., and P.' s, d.o, P., d..., and d.o....... nil,
1'18,
In molecules whioh are known to have a trigonal bipyramid structure, energy criteria make it unlikely that the first four combinations are important. In PF. combination (5) is the most probable and the hybrids are labelled dsp'. In gaseous MoCl. it is likely that a combination of (6) aIUi (6) are used, since the 4d and 6p AOs of Mo are close in energy. The hybrid formed from the orbitals in (6) is labelled dasp and the most general form for ¥'vo in MoCl. is "110 = a(dsp')+b(dlsp) with a !::! b. In Table 11-3.1 hybrid orbitals for a selection of geometries are tabulated.
11-4. Hybrid nital. for ...-lIonding.,......
In contrast to u-orbitals and u-bonds, a ...·orbital or ...-bond is defined as one whioh has OM nodal surfaoe or plane oontaining the bond axis. (The reader might note that, though we will not deal with them, a-orbitals and a-bonds have ttDO nodal surfaoee which intersect on the bond axis.) If a ...-bond is to be formed in MO fashion by combining two orbitals, one on each of the two bonded atoms, it is obviously necessary that each orbital have ..-charaoter with respect to the bond axis and that their two nodal planes coincide. The formation of a ..-type bonding KO from two ",-type AOs is shown in Fig. 11.4.1, where the plus and minus signa refer to the sign of the wa.vefunction.
230
Hybrid Orbitala
Hybrid Ortlital. TAllLE 11-3.1 Different hybrid Qrbitals
planes with the 2n 1T-AOs of the B's. Each bond will then consist of two shared pairs of electrons (a double bond). In this section we are concerned with the choice of the AOs of A which, when formed into linear combinations, provide the 2n hybrid orbitals of the appropriate symmetry. As an example of an AB.. molecule, we will discuss the planar symmetrical molecule BCl a which belongs to the ~ah point group. First we assign to each chlorine atom a pair of mutua.lly perpendicular
Numb.reJ equivalent orbiWl4 2
3
4
6
6
Deoignalion
Geometry
sp dp sp' dp' do' d' p' d'p sp' sd' dopa dip. d' clap' d'sp dtsp" d's dlip. d'p d:lp' d' d'sp' dfosp d'p d·p·
lin...... lin...... trigon&! pl&ne trigonal plane trigonal plane trigonal plane trigonal pyramid trigon&! pyramid tetrahedral tetrahedral tetragonal plane tetragonal plane tetragonal pyramid trigonal bipyramid trigonal bipyramid tetragonal pyramid tetragonal pyramid tetragonal pyramid tetragonal pyramid pentagonal plane pentagonal pyramid octahedron trigonal prism. trigonal prism. trigonal an tiprism
----".
CI
-------- ----------- "
. . . . . . 1::: -----
...... ........
('J
,/
",'//
B
......
...... '
,
",,,,,,
; ''
(') FlO. 11·4.2. Vectors repreeenting ,,-orbitals on the Cl ..toms in BCI•.
Let us consider a molecule AB n in which each atom B is bonded to atom A. We will assume that each atom B has available two .,.-AOs which are orthogonal to each other, i.e. their nodal planes are mutually perpendicular (they might, for example, be a 2p.. and a 2pw AO). If we wish to form 2n .,.-bonds, requiring 2n .,.-MOs, then we must furnish 2n .,.-AOs or hybrids on A which match up with respect to their nodal
+
.. +
atomic orbitals
231
molecular orbital
Fro. 11-4.1. The formation of a TT-type bonding molecular orbital.
vectors to represent the 1T-AOs of" the chlorine atoms. Each vector points towards the positive lobe of the orbital and, in a pair, one vector is perpendicular to the molecular plane and the other is in the molecular plane and perpendicular to the relevant bond axis (see Fig. 11-4.2). This particular arrangement is chosen simply for convenience and other arrangements are equally valid provided that the two vectors on a given chlorine atom are mutually perpendicular and in a plane perpendicular to the B-Cl axis. The six necessary hybrid orbitals on the boron atom can also be assigned vectors. If 1T-bonds are to be formed, these vectors must have the same orientation as the six vectors on the chlorine atoms. If we followed in the footsteps of § 11-3, we would now construct the reducible representation rJ'Tb from a consideration of how the six vectors on the boron atom change under the symmetry operations of the ~3h point group. However, it is clear that since the six vectors on the chlorine atoms match the six on the boron atom, exactly the same representation p'"b can be found by using these vectors instead. Since it is less confusing to have three pairs of vectors separated in space than six originating from one point, we will take this latter approach.
232
Hybrid Orlllta'.
Hyllrld Orll ital.
(We could have done exactly the same thing for methane in § 11-3, it would simply have meant reversing the vectors in Fig. 11-3.1.) Except for this change, we find t"'b(R) in the same way as before. We note, however, that this time the direction of a vector may be reversed as the result of a symmetry operation and in such a case there will be a contribution of -1 to the character of that operation. Furthermore, we immediately see that in carrying out the different symmetry operations, no vllctor perpendicular to the molecular plane is ever interchanged with one in the molecular plane and vice versa. This implies two things: the representation rJ':rb is at once in a partially reduced form (the matrices are already in block form, each consisting of two blocks); and the vectors perpendicular to the molecular plane on their own form a basis for a reducible representation of ~Ib (which we will call r:~:,,) and the vectors in the molecular plane on their own also form a basis for a reducible representation of !'J.h (which we will call rJ':rb ). nece68arily p1&De '
rb:rb
=
r:~:"$r::~
Z33
d-orbital fulfills this requirement, though there are two pairs of atomic orbitals, (P., Pw) and (d,.I-..I, dew), which belong to pr, the other necessary component. Therefore we cannot form a set of three equivalent w-bonds in the molecular plane. However, this does not mean no w-bonds in the plane can be formed or that only two of the chlorine atoms can be w-bonded in the plane. It simply means that there can only be two w-bonds in the plane 8hared equally amongst the three
•.
The following results are obtained for BCl.: ~.b
E
2C.
3CI
Gb
2S.
3Gv
Xbyb(C.)
6
0
-2
0
0
0
X:~~p(C,)
3
0
-1
-3
0
1
X;r..':..(C.)
3
0
-1
3
0
-1
These characters, using the standard decomposition formula, lead to r:~.t;. = r~I'
and
®rB "
FlO. 11·4.3. Veotors representing .".·orbitals on the B atoms in an octahedral AB .. moleoule.
r::~. = r~··
Therefore, in order for boron to form a w-bond perpendicular to the molecular plane to each of the chlorine atoms, it must use three hybrid orbitals constructed from one AD which belongs to ~I' and a pair of ADs which belong to rr. Table 11-2.2 shows that only the following orbitals meet these requirements: r~"
rr
p.
(d,.., dr.)
and a set of three equivalent hybrids (pdl type) using these orbitals is the only one possible for the 'perpendicular' w-bonds. For the hybrids which are in the molecular plane, we require an atomic orbital which belongs to r d ,' and inspection of Table 11-2.2 shows that no S-, po. or
chlorine atoms; these two w-bonds using the rE' orbitals. This type of situation arises quite often. Now let us consider the important case of an octahedral AB. molecule. If we associate two mutually perpendicular vectors with each atom B as in Fig. 11-4.3, we obtain the following character for rh:rb: ~
E
8~
3~
6~
6~
-4
o
o
i
o
8~
3~
o
o
6~
6~
o
o
For this class of molecule the vectors do not fall into two categories and we obtain a single reducible representation:
234
Hybrid Orbitals
Hybrid Orllitals
For the irreducible representations in this symbolic equation, inspection of Table 11-2.2 shows that we have the following s-, p-, and d-orbitals:
(p~,
none
P., P.)
none
Since the p-orbitals on A have most likely been used up in a-bonding, we have only the three T a• d-orbitals for '17-bonding. We therefore conclude that there can be three '17-bonds shared equally amongst the six A-B pairs.
11-6. The m8t1Jem8ticai form of hybrid orbital. So far we have only considered whick AOs are required for the construction of hybrid orbitals of the appropriate symmetry. We now will show how we can obtain explicit mathematical expressions for the hybrid orbitals which will allow us to see exactly how much each AO contributes. Though hybrid orbitals are most frequently used in qualitative discussions of bonding, they do have their quantitative use when one carries out an exact MO calculation and when one deals with coordination compounds, where it is often necessary to use hybrid orbitals for evaluating overlap integrals which are often related to bond strengths; in these situations the explicit expressions are required. As an example, let us consider a symmetric planar ABa molecule belonging to the 9}ah point group. Using the techniques of § 11-3, we find that the three hybrid orbitals of A 'P.. 1fa, and 1fa which form a-bonds with the three B atoms, are composed of one AO which belongs to ['A." and a pair which belong to ['E'. By use of the projection operator technique (see § 7-6) we can project out of 1f.. 1fa, and 1fa. functions which belong to the irreducible representations p",.' and rE and, by equating these functions with the AOs which are being used, we can obtain equations mathematically linking the hybrids with the AOs. The projection operator corresponding to the ",th irreducible representation is: pI' = !x"(R)*O.
pA"1fl pB' 'PI
4('Pl +'P.+'Pa) 2(2'Pl -1fa - 'Pa) pB''P. = 2(2'P.-1fl-'Pa).
11-5.1
TABLE
Tran8Jormation oj the ~ah hybridB under O.Jor all R oj 9}Sh
'1'. '1''1'1
E
C.
c:
c. o
Clb
c••
\'.
'P.
'1'1 '1'1
'P1
'1'. '1', '1'.
\01 '1'-
'1'.
\01 \0. '1'1
\01 '1'. '1'1
'P.
<Jh
'1'. '1',
SI
S"I
cr.o
cr.b
cr.o
\01
'1'. '1'. '1'1
'1'. '1', '1'.
'1'1
'P_ 'P. 'PI
'PI 'P.
'P,
'PI 'P.
Applying pA.' to 1fa and 1fa is not necessary since the same combination as pA,' 1fl will be produced. The two combinations obtained from pB' are by inspection, linearly independent and a third combination which can be found by applying pB" to 1fa, will be a combination of these two. If the combinations are normalized with the assumption that
I 'P,'PI d .. =
6",
then we have: (1fl + 1fa + 'Pa)1 v 3 and
(2'Pl-'Pa-'Pa)/v 6} (2'P. -1fl -1fa)1 V 6
(belonging to
['B').
Under the 9}ah point group, an s-orbital belongs to ['A.' and the pair of p-orbitals p~ and p. belongs to rE'. If these orbitals have been used to construct the three hybrid orbitals (sp' type), then we can
y
"
.......... / I
•
(see eqn (7-6.6)) and in Table 11-5.1 the results of applying OR to the three hybrid orbitals are given (the directions of the hybrids are shown in Fig. llo5.I). From this information we deduce that
23&
/
(l'2"
I ,,
," .........
(c·ont.ains u vb )
...
l#l .............. B
O'
C:!a
(contains 0'",.)
=
=
FlO. I1·5.1. Directions of the hybrids and '" ami y axe. for an ABI molecule belonging to !$,b'
238
Hyllrid Orbi..l.
Hybrid Orbl....
immediately identify, since it is the only one of pA,' symmetry, (11'1 +11'. + 11'.)/V3 (11-1>.1)
We can also identify the other two combinations (2'P,-'P.~11'.)/V6 and (211't-11't-11'.)/V6 with two normalized combinations of pz and P. orbitals, since both pairs form a basis for rE', i.e. (211'1 -VJ. - 'P.)! V6
=
±M +bn-i(Cltpz +blP.)
(211't -'PI -tpa)!V6 = ±(a:+h:)-i(aapz+h.p.). The :e and '!I axes are shown in Fig. 11-5.1. To establish the values of the coefficients a" a., h t , and b. it is necessary to investigate the detailed effect of one or more of the transformation operators O. of 9Jah on the pair of combinations. The operator 0 ... (see Fig. 11-5.1 for the definition of the tTy.. plane) leaves (2'Pl-tp.-tp.) unchanged, therefore it must leave (a 1 P.+b1 P.) unchanged: 0 •.,.(a1 Pz+ b (1152) IP.) = atp.+ htP.· - . But O.....P..
=
(2", -11'. - 'Pa)! V6
=
±P.
and since, by inspection of Fig. 11-5.1, 11'.. -VJa, and - 'Pa have positive z components, we take the positive sign, i.e. (11-5.4) Now consider the operator O.Yb' This leaves (2'P.-'P,-'Pa), and consequently (a.p.+b.p.), unchanged. But O.YbP .. = (-p.. +V3P.)/2
and
P.
O"'bP • = (V3P..+p.)!2
and hence
(a.p.+b.p.) = {( -a. + V3b.)p.. +
(vaa. +b.)p.}!2.
Since p. and P. are linearly independent, we can equate the coefficients of P.. and likewise those of P. and obtain a. = bat V3. We conclude that (2'1'. -tpl-tp.)!V6 =
±(P. + V3P.)!2.
-(Pz+V3P.)/2.
=
(11'1-11'a)!V 2.
(11-5.5)
Eqns (11-5.1), (11-5.4), and (11-5.5) may be brought together in the single matrix equation s
I/V3
P.
I!V3
I!V3
'Pt
2/V6
-I!V6
-I!V6
'P.
0
-I!V2
I!V2
tp.
P. which on inversion leads to
-1
s
VJl
I!V3
1/V3
1!V3
'P.
2!V6
-I/V6
-I!V6
'PI
0
-I/V2
I!V2
1/V3
2!V6
0
s
I/V3
-1!V6
-1/V2
1!V3
-I!";6
I/V2
P" p.
(11-5.3)
and by comparing eqns (11-5.2) and (11-5.3) we see that bt = O.
=
Combining this equation with eqn (11-5.4), we get
P" and O.vaP. = -P., hence: O...(a,p..+blP.) = a t p.. -b1P.
Therefore
Since tp. and -'Pa have negative '!I components (11'1 has none), we take the negative sign, i.e. (2'Pa-1pl-tpa)/V6
with the s-orbital, i.e.
237
P.. P.
(Note that since the 3 x 3 matrix is orthogonal, its inverse is simply its transpose.) So we finally achieve the following mathematical expressions for 'PI' 'PI' and 'P.:
11't
=
(V2s+2p,,)/V6,
'P. 'Pa
=
(V2s-p.. -V3P.)!V6, (V2s-p.. +V3P.)/V6.
=
In this particular example we could have avoided some of the labour involved in finding the combinations of hybrid orbitals which are equal to P.. and P., by using the 't'. point group (to which the molecule also belongs), For this point group, the two-dimensional representation, the cause of all the trouble, can be expressed as two complex one-dimensionl representations. The orbitals P.. and P" are then just as easy to obtain as the s-orbital. Any complex numbers which result are eliminated at the end of the treatment by addition and subtraction of the orbitals formed. This is the technique which was used in § 10-7 to find the 1T-molecular orbitals of the trivinylmethyl radical. It is, however, of no avail when dealing with point groups which have three-dimensional irreducible representations as in our next example, CR•.
231
Hybrid Ortlitals
Hybrid 0 rbitBls (-'I', +3'1'. -'1'. -Y'.){y'12 = ±(a:+b:+c:)-l(a.p.. +b.p. +c.P.)
C'l
\\
\
.",........ '
jZ
(-'I', -Y'.+3Y'a -Y',)!y'12 = ±(a: +b:+c:)-l(aap.. +bap. +c.P.)
" , )~H!.._ ----/-~71 I
" ... ""."
I
I
:
Y'I
(11-5.9) Under the operator 0e.. (see Fig. 11-5.2 for the Ca. axis). the left hand side of eqn (11-5.7) is unchanged and since
:
!
I
0e...P.. = P..
I :
I
('~.p.--+11---- 11
1---- ---L-~ H I I
,
(11-5.8)
,,C""
. .------1~. --~.0'.--1 HI
,;
.... '
,; H------------1,-''''
Since, by inspection of Fig. 11-5.2, '1', and -'1'. have positive :c com-
"
p.one~ta and the :c components of Y'a andY', cancel, we take the positive
C"" FIG. 11-6_2. Directiorul of the hybrids s.nd "'. Y. and. axes for CR•.
For methane we have seen that there are four hybrid orbitals '1'1' '1'.' '1'., andY', (see Fig. 11-5.2), each composed of an s-orbital belonging to pd. and three p-orbitals P.., P., and Po> belonging to r T .; for the choice of :c, 1/, and z axes see Fig. 11-5.2. Using the relevant projection operators, we obta.in the following combinations:
Y'I-Y'.-Y'.-Y',)!y'12 = (P.. +P.+P.)!V3.
..
·.b
(-VJ, +SY'. -Y'.-Y',){-V 12 ='±(P.. +P. -P.)!y'S. Inspection of Fig. 11-5.2 justifies the negative sign and we have
(-1Pt-Y'.+3tp.-Y',)!y'12 = (P.. -P.-P.){-V3.
P.. = (Y',-Y'o+Y'a-Y'.){2 P. = (Y'I-Y'o-Y'.+Y',){2 P. = (Y', +Y'a-Y'a-Y'.)/2
r T .).
Cs.c
Z'J
s
t
(11-5.6)
P..
t
(3Y', -Y'I-Y'a-Y'.)/y'12 = ±(a~+b~+c~)-·(alP.. +blP. +c1P.) (11-5.7)
P.
t I
We can therefore establish that
v'
(11-5.12)
(11-5.13)
and these equations together with eqn (11-5.6) can be written as
( -Y'I-tpa+ 3 Y'a-Y',){y'12
(Y'I +tpl+tp.+tp,)/2 = s
°
(11-5.11) on P P
Eqns (11-5.10) to (11-5.12) can be solved to yield p". P., and P.:
) (belonging to
(11-5.10)
Under the operator 0e'b' the left hand side of eqn (11-5.8) is unchanged and since 0e ~P.. = -p, 0e P v = p OCab. p = -p11" we «' find that (a.p.. +b.p.+coP.) becomes (-a.p.+b.p.. -c.P.). Therefore, a. = bo' b. = -c.' c. = -a. and
Consideration of eqn (11-5.9) and the effect of operator an d P. leads to
The last three combinations are linearly independent a.nd the normalized combinations are: (belonging to r Ll ,) (Y'1 +Y'I +Y'a +VJ,)!2 (3Y',-VJI-VJa-Y',)/y'12
(3
(-'I" +31p.-Y'I-Y',)!y'12 = -(P.. +P.-p.){y'3.
6('1'1 + '1'. + Y'. + '1',) 2(3tpl-tpl-tpa-Y',) 2(-tp,+3Y'.-Y'.-Y',) 2( -VJ,-VJ.+3Y'.-Y',).
(-Y'1 +3Y'I-Y'a-Y',){-V 12
0e.aP. = P.,
(3Y"-Y'.-Y'a-Y',){y'12 = ±(P.. +P.+p.){y'3. 4
SIgn, I.e.
= = = =
0e..P. = P..
(a,p.. +b,p.+c,P.) becomes (alP. +b,P.. +c,P.). Therefore a, = c b = a c, = bl and '" ,.
'
~IY'I pT'Y'I pT'Y'I pT· VJ •
239
P.
i -t -t
t t -t -t 1
1
'PI
-t t
'Po
-,
'Pa
'P,
240
Hybrid Orbitals
Hprlll Orlll.. ls
or
'PI
I
I
i
'PI
1 -I
I
I -t
'l'a
I I
-i
-I -I
1 -I
'P.
t
i
-1
tit
and
S
d..
p.
°c..
p. p.
s
t -t - l i p . t i
-i
t -I
i
-I -1
P. P.
(note that the 4 x 4 matrix is orthogonal, so that its inverse equals its transpose). Hence we obtain the final relationships: '1'1 = (s+P.+P.+p.)/2, '1'1 = (s-P.-P.+p.)/2,
(11-5.14)
'l'a = (s+P,,-P.-p.)/2,
'1'. = (s -P. +P. -p.)/2.
We saw in § 11-3 that a set of equivalent tetrahedral hybrid orbitals could also be constructed from a set of s. d"". d.., and orbitals. Mathematical expressions for these hybrids (sd") can be obtained from eqns (11-5.14) by changing P. to P.. to d.., and P. to d..,. That these are the correct changes can be deduced from the fact that the operators Oc. .,Oc ' and OcIe acting on the column matrix .Ib
a..
a.•.
P. P.. P. produce, in each case, a matrix which is identical with the one obtained when the same operator acts on the column matri~
d,. d.. For exam pIe,
241
0
0
-1
d,.
d..
1
0
0
<1...
d""
0
-1
0
d..
11-8. Reletionship bear_localized end non-localized ...I_ler orbi..1theory Loca.lized MO theory uaing hybrid orbitals is essentially a special case of non-localized MO theory. In the latter theory, if we construct MOs from AOs scattered over the entire molecule, we do not need to make any initial assumptions about which AOs are being used to bond particular atoms together, nor for that matter need we consider the symmetry properties of the AOs which are used. All these things will be taken care of by the quantum mechanical method used to determine the coefficients of the AOs in the linear combinations: if an AO is of inappropriate symmetry, its coefficient will turn out to be zero and if an AO plays a small role in bonding, its coefficient win turn out to be small. However, it is clear that the major contribution to a non-localized MO will be similar to the localized MO formed from appropriate hybrid orbitals, since both, in the long run, have to describe the same thing: the electronic distribution.in the bonding regions. We can therefore utilize our knowledge of hybrid orbitals when we construct non-localized MOs by ignoring those AOs which on the basis of the simpler theory would not contribute significantly. Such an approach not only confortns with our intuitive understanding of bonding, but also, by cutting out that which is irrelevant, saves time in carrying out the calculations. We will return to the more general MO theory again in the next chapter. The key virtue of this theory is that, unlike the hybridization approach, it affords a mechanism for making direct energy calculations. As a final note, it should be emphasized that like the phenomenon of resonance, hybridization is not a real physical process (atoms don't hybridize any more than molecules resonate). It is a man-made process for describing an already existing situation, the molecular bond, when the simple model using single AOs fails to work.
d""
PROBLEMS
P.
0
0
-1
p"
Oc•• P..
1
0
0
P.
P.
0
-1
0
P.
11.1. Determine the irreducible repreIIeJltations of r d to which f-orbita1a belong. 11.2. Show that for a molecule of octahedral symmetry the a-bonding hybrid orbitals on the central atom are composed of six atomic orbita1a: s, p". P., P.:' d ••• and d..--.•.
242
Hybrid Orbitals
11.3. Determine what type of ".·bonding hybrid orbitals C&Il be formed for the square plEmM AB. molooule which belongs to the 9 •• point group. 11.4. Show that for the square planar AB. molecule .. possible set of four ahybrid orbitals on A is composed of the atomic orbitals: s, d,,---ol, PlO. and Po. Find explicit expressions for the four hybrid orbitals.
12. Transition-metal chemistry 12-1. Introduction SINCE the 1950's there has been a dramatio revival of interest in inorganio ohemistry. This has largely been due to the synthesis of many transition-metal compounds and oomplexes and the development of theories to explain their properties. Nearly aU of these compounds have a high degree of symmetry and in this ohapter we see how the group theoretical rules whioh we have previously developed can reduce the effort involved in calculating their properties. In some cases the reduction in labour is quite startling. Although strictly speaking the transition metals are defined as those which, as element8, have partly filled d- or f-shells, it is more common also to include in the definition elements which have partly filled d- or f-shells in any of their commonly occurring oxidation states. The atoms or molecules which are bonded to the metal in a transition metal compound are caUed Uganda. In broad terms, there are two very different theories which have been used to interpret the properties of transition-metal compounds; MO theory and crystal field theory.' MO theory is the more general of the two and, in the long run, the one most capable of giving the best results. However, in its application, the calculations which are entailed are very lengthy and only now, with the introduction of high-speed computers, are reliable results being produced. In § 12-2 and § 12-3 we will see how valuable symmetry principles can be in a MO treatment of octahedral MX. and tetrahedral MX. molecules. We choose these two types of molecule because they occur often in transition-metal chemistry and also because they are prototypes to which many molecules of lower symmetry can be referred. In § 12-4 we construct MOs for ferrocene, which is an example of a sandwich compound. Here the ligands are carbocyclic rings and so we are able to build upon the techniques which were developed in Chapter 10 for such rings. Crystal field theory has its origins in Hans Bethe's famous 1929 paper Splitting oj terms in Crystal8. In that paper Bethe demonstrated what happens to the various states of an ion when it is placed in a crystalline environment of definite symmetry. Later, John Van Vleck showed that the results of that investigation would apply equally well to a transition-metal compound if it could be approximated as a metal ion surrounded by ligands which only interact electr08taticaUy with the
244
Transition-Metal Chemistry
ion. One of the key elements in the application of crystal field theory to transition-metal compounds is the construction of so-called correlation diagrams. These diagrams relate the splitting of the electronic states of the transition-metal ion under the perturbation of a set of ligands to a parameter which reflects the strength of the electrostatic interaction between thc ion and the ligands. They resemble the'well known correlation diagrams for the formation of a united atom from the separated atoms of a diatomic molecule and like them depend on a non-crossing rule for states of the same symmetry and multiplicity. In § 12-5 and § 12-6 we develop some necessary preliminary steps before constructing, in § 12-7, a correlation diagram for a metal ion which has two d-electrons in the valence shell and is surrounded by an octahedral or a tetrahedral set of ligands_ In § 12-8 and § 12-9, the way in which correlation diagrams can predict the spectral and magnetic properties of transition-metal compounds and complexes is described. It is clcar from the start that the approximations involved in applying crystal field theory to transition-metal compounds are extreme and the results of MO theory confirm this. In very few cases can the ligands be considered simply as point charges or point dipoles which interact only electrostatically with the metal. In general the electrons of the ligands will be distributed throughout the molecule and there will be covalent (electron sharing) as well as electrostatic interactions. However, crystal field theory can, to some extent, be adapted to handle this, the real situation, by reinterpreting the parameters which occur. We let these parameters become strictly semi-empirical quantities and give them values which can no longer be ascribed solely to electrostatic interactions. This adaptation is called ligand jield theoryt and it is briefly considered at the end of the chapter. Two excellent books which offer useful background material to this chapter are: An introduction to ligand jields by B. N. Figgia (published by Interscience Publishers) and Atomic and molecular orbital theory by P. O'D. Offenhartz (published by the McGraw-Hill Book Co,). 12-2. LCAD MDs for octahedral compounds Let us consider the construction of molecular orbitals from linear combinations of atomic orbitals (LCAO MOs) for an octahedral molecule t Somo authors have used this tCI'In to describe all aspects of the manner in which an ion or atom is influenced by ligands. With this definition, crystal field theory is & special case of ligand field theory and the latter is indistinguishable from MO theory.
Trensition-Metel Chemistry
246
or ion MX. where M is a transition metal and the ligands X are bonded solely to M. We begin by selecting the AOs to be used. The transition metal in ita ground state will have an inner core of electrons with a noble-gas electronic configuration, which we will assume does not participate in bonding, and a number of valence electrons which occupy 00- and (71 + l)s-AOs. Since, for the transition metals, the (71 + I)p-orbitals are close in energy to both the 00- and (71+ I)s-orbitals, we will aBeume that the metal M contributes five do, one s-, and three
1
x.
"'. y
FIG. 12-2.1. p·type ligand orbitals for an octahedral MX. compound.
p-orbitals to the MOs. Of course. the selection of AOs of M is made easier by knowing which orbitals participate in the localized molecular orbital bonding scheme (see the previous chapter). We will assume that each ligand X contributes one s-orbital and three p-orbitals. We will distinguish the ligand orbitals from the metal orbitals by a subscript (1,2, ... 6) which indicates the ligand with which they are associated. The three p-orbitals on each X may be given the symbols a• .... and ...' and associated with a set of three mutually perpendicular vectors &8 shown in Fj~. 12-2.1. The a-type p-orbitals are chosen to have their positive lobes directed towards M. We therefore have a grand total of 33 atomic orbitals: s, P.., Pw' P., d ••, d.,.-.., d.w, d.,., d w.; 8,. a" "'j' ...; (i = 1, 2, ... 6). Application of the LCAO MO method without applying the principles of symmetry would therefore lead to an equation (see eqn (10-3.5)) det(H~~t-ESi') = 0,
(12-2.1)
248
TrllIlSition-Metel CII...istry
Trensition-Mml Ch...istry
whioh involves a 33 X 33 determinant. Fortunately, by using the techniques we have previously disoU886d, we can avoid the Herculean task of solving such an equation. What we do is to create linear combinations of the 33 ADs in such a way that they form bases for the irreducible representations of the /!Ih point group to which the moleoule belongs. Using these symmetry adapted combinations in place of the original 33 ADs immediately leads to the factorization of eqn (12-2.1). (cf. the benzene problem in § 10-6). To find the symmetry adapted combinations we first oonsider the application of the transformation operators O. to the 3S atomic orbitals. We see at once that for all symmetry operations R of the /!Ih point group, the central atom M is left unchanged and consequently any O. will only transform metal orbitals into metal orbitals (or combinations of metal orbitals) and ligand orbitals into ligand orbitals (or combinations of ligand orbitals). ThUJl. we can immediately reduce pAO (the reducible representation using all 33 atomic orbitals) to the form: rAO = J'M e r x where I'M: is a reducible representation using the nine atomic orbitals of M 88 basis functions and r X is a reducible representation using the 24 atomic orbitals of the six ligands 88 b88is functions. The further reduction of I'M: is an identical problem to that discUJlBed in § 11-2. In that section we cl88sified the atomic orbitals according to the irreducible representations of various point groups. For /!Ih we found
s
belongs to pd".
(P., PO', P.) (d••• d".-w') (cl.•• d••• d ••)
belongs to belongs to belongs to
r T ••, rE., r T •••
Hence we can write I'M = pd.. ffi 1'7'••
e rEo e rT••.
The further reduction of r is more complicateet. In the first place no transformation operator O. can change a ligand s-orbital into a ligand p-orbital (or combination of ligand p-orbitals), therefore the s-orbitals themselves form a representation I'" of /!Ih' Furthermore, for the particular choice of orientation of the p-orbitals (or vectors) in Fig. 12-2.1, no O. can change a a-type p-orbital into a .".-type one (or combination of .".-type p-orbitals), so we have two further representations I'" and r' and overall we can write X
247
ro, I"', and r' we must apply O. for all R to (S" so, ... s.). 0'.) and (""1' .".~, ""0' .".;•... .".., .".~) respeotively and thereby find z'(R), X'(R), X'(R) for all R. Using the equations: To reduce (0'1'
a., ...
O..
"or UI
=
and
L
1_1 • orll
X"(R) =
L
i_I
DMR)
(or its vector analogue)
~.(R).
this is a straight forward procedure and we obtain /!Ih X"(C.) X·(C.) X·(C.)
E 6 6 12
SCa SCo 6C.
6C~
i
SS.
3erh
6S.
6erd
2 2 -4
0 0 0
0 0 0
0 0 0
4 4 0
0 0 0
2 2 0
0 0 0
2 2 0
The decomposition rule (eqn (7-4.2» in conjunction with these results shows that T
r' = pd•• ffi rEo ffi r ." r' = pA,. ffi rEo ffi r T .., r' = r T •• $ rr.. ffi 1'7'••
ffi
r T •••
What do these symbolic equations mean1 Consider for example the first one. This shows that it is possible to find a new basis for r' in which the basis funotions are linear combinations of SI' s., ... s. such that the matrices of 1'"• .D"(R), have a completely reduced form. There will be one linear combination which is a basis for 1'..4", two linear combinations which form a basis for rEo and three linear combinations which form a basis for r T ... The other two symbolic equations can be interpreted in the same fashion. The actual linear combinations whi.ch reduce 1'", I""', and 1''' could be found by the projection operator technique of § 7-6 but because of the large number of symmetry operations contained in /!Ih and because of the problems connected with the multi-dimensional representations (similar to those encountered in § 11-5), to do so would require much time and even more patience. Fortunately, we can find the correct combinations by inspection. Any symmetry operation of /!Ih simply permutes the ADs s" s..... s. amongst themselves, so that if O. is applied to (Sl +s.+s. +s. +8.+8.) only the order of the orbitals within the bracket is changed and O.(SI +s.+s.+s. +s. +s.)
=
(SI +80 +S. +s. +s. +s.)
TransitlDD-IIetal Chemistry
248
for all R. Consequently the normalized basis function (8B8uming J,p,.p, d'T = 6"• .p, = any ligand orbital) for the identical representation rA'· is ~ •• = (s,+sl+sa+ s.+ s .+ s .)/v'6. To find a pair of linear combinations of s" Sl.... s. that form a basis for rEs (and there are many), we can look for a pair which mirror the rB'. AOs of the central atom, i.e. d •• and do........ The d ••-orbital has the symmetry of the expression 3z l -r" = 2zl -zl _ y l and it is clear that the s-orbital combination which reflects this ist
'P"E.Ul
= (2s.+2s.-s,-sl-sa- s .)/v'12.
The d........ orbital has positive lobes along the positive and negative z-axes and negative lobes along the positive and negative g-axes, therefore this will be mirrored by the combination:
Likewise, comparing the r •• combinations with d.w, d ••, and find (like d.w) 'Pi-.. Ul = (1l'; -1l';+ 1l'i-1l';)/2 (like d",.) 'PT•• CI) = (1l',-1l'I+1l'.- )/2 (like d••). 'Ph,CI) = (1l"-"'0+1l'~- ;)/2 T
rx·,
'Ph.w = (1l'a - .... -.".; + "'~)/2 'P~ ••1Il = (..., -"'1 -1l'6 + .".,)/2 V'~hCS) = (.".~ -1l'; -.".; + ""~)/2
.
rx
'P~•• Cll =
(s,-sl)/v'2 (8a -s.)/ v'2 'Ph.la) = (s.-s.)/v'2 'f'~•• (I)
=
(like Pc) (like P.) (like P.).
That these combinations do indeed behave like their oentral atom oounterparts may be verified by applying the operators 0 •. One can also confirm that, like the AOs of M, the combinations are orthogonal to each other. It is clear that the required combinations of 0'" 0'1"" 0'. can be obtained direotly from the previous combinations by replacing s by 0'. For the representations based on the ,"- and ,"'-orbitals, we have to find combinations for r T ••, r T .. and r T ... The r Tlo combinations must mirror the P., p". and P. AOs of M. The ligand orbitals '"~' '"~' '"., and '". all point in the z direction, ...:, ...~, ...~, and- ...~ all point in the g direotion and ...,. '"I' "'a. and 1l'0 in the z direction. Hence suitable normalized combinations are
rx·.,
'P~••11I = (1l'i+"';+,".+1l'.)/2
"~••III = ("'~+1l';+1l'~+1l';)/2 "P~"ca) = (1l', +1l'1 +"'1 +1l'0)/2
t The set
(like P.) (like p,,) (like P.)·
of n funotiOIlll which logetAor form a basis for the n-dimensional irreduOl"bI" representation will be labelled ",-(1). 'I"u•••... '\0"'.)0 TbiJ!I notation does not imply
r-
that there is an irredueible representation
rll-I' J.
we
For the r T •• and combinations there are no central atom counterparts and consequently, as they therefore do not 'mix' with the central a.tom orbitals, they do not take part in the bonding. Nonetheless, appropriate r T ., combinations can be found. The axial veotors R .. R". and R. (see § 9-6) form a basis for r T1• and using this piece of information, we find the following r T •• combinations which mirror R., R", and R.:
'f''lr II' ... (s, +sa-sa-s.)/2.
The three combinations of s" Sl' ..• s. whioh form a basis for r T .. can be ohosen to behave like the p"" P., P. AOs of M (since these also lo ), hence, by inspection, form a basis for
d..••
(like R",) (like R.) (like R.).
T
The three r .. combinations must be orthogonal to the previous nine combinations. Such a set is "P}.,I')
F
(1l'~+1l';-
-."..)/2 ; - ...;)/2 (1l', + 1l'1 -1l's - "'0)/2.
1p~•• (I) = (1l'~ +1l'~-
"Pi-.,la,
=
What we can now say ia that if we replace the initial basis of 33 atomic orbitals by the nine ADs of M and the 24 combinations of ligand ADs ('P~. ' 'P~ I<J l = 1, 2, 'P;', w i = 1. 2. 3, 'P~. ' "8 W l = 1,2, " " , " "PT.,c'I' 'PTlyl<)' 'PT•• c", 'PT••w, ,,70••1<) l = 1, 2, 3), the determinant of eqn (12-2.1) will be in block form and the equation can be factorized. The reason this happens is the same as in the benzene example (see § 10-6): a combination of the vanishing integral rule (§ 8-4) and the fact that the effective Hamiltonian belongs to r' (the totally symmetric representation of any point group). Thus we will have: (1) A 3 x3 block corresponding to rA'· which will produoe three non-degenerate energy levels (a,.-type) with MOs formed from a metal s-orbital, a combination of liga.nd a-orbitals (~•.) and a combination of ligand a-type p-orbitals ('P~•.)' (2) Three equivalent 4 x4 blocks corresponding to r T ••• One block will produce four energy levels with MOs formed from a metal p.-orbital. a combination of ligand s-orbitals ("PT••Ill)' a combination of ligand a-type p-orbitals ('PT.,l1I) and a combination of
.
250
TransitiDn-MlIt8l Chemistry
ligand 1T-type p-orbitals ('1';',.(11). The other two blocks will produce identical energy levels and values for the coefficients of the symmetry adapted basis functions (which are p ,,,,Ip , .. r ~ ..T ]."1(1) TJ'T ..(I)' V'TbU) In one case and P., 1p;'lIlU)' tpT11l (3), 1p;tu UI)' in the other). Consequently we need only solve eqn (12-2.1) for one of the blocks. Overall, we will have four triply-degenerate energy levels (t1 .. -type). (3) Two equivalent 3 X 3 blocks corresponding to r E ., one will involve d ••, 'I'~.(l), and fJ'E.(lI and the other d,..-••, fJ'~ (0)' and fJ'E.Ul· They will both produce identical energy lev~ls and identical values for the coefficients of the symmetry-adapted basis functions in the final MOs. Again. it is neoessary to solve eqn (12-2.1) for only one of the blocks. Overall we will have three doubly-degenerate energy levels (e.-type). (4) Three equivalent 2 X 2 blocks corresponding to r T ••, one will involv..e d.. and 'l'T.,ll)' another de. and "';'.,(1)' and a third <1". and fJ'p"m· These three blocks will lead to identical energy levels and coefficients. Putting the results together we will have two triply-degenerate energy levels (to.-type). (5) Six 1 X 1 blocks, three corresponding to rT'. and three correT sponding to r ••• These will provide one triply-degenerate level of each type (t l • and t•.> and the final MOs will only involve ligand 1T-type p-orbitals and will simply be the symmetry adapted combinations themselves ("';"0«) and fJ'T..W' i = I. 2, 3). . The upshot of all this is quite astounding: a problem which initially Involved a 33 x 33 determinant has been reduced to one involving one 4 x 4, two 3 X 3 and one 2 X 2 ·determinants. This is as far as we can go with symmetry arguments alone. the next step involves calculating. or estimating in a semi-empirical fashion, the required matrix elements Hf"r and 8 st . This is beyond the scope of the present book, however. once it is done eqn (12-2.1) can he solved and the energy levels determined. Radical approximations are often made in finding H~f: and Sit, e.g. the orr-diagonal elements H~r,c (j =1= k) are frequently made simply proportional to the corresponding overlap integrals Sit' and therefore the reader is advised to treat such LCAO MO results critically. Only recently have ab initio LCAO MO calculations been attempted for transition-metal compounds·t The energy-level diagram shown in Fig. 12-2.2 is a summary of the
t
S.... for example, the calculation on NiF:- by J. W. Moskowitz. C. Hollist&r, C. J Hornback, and H. Basch (Journal of chemical phyllic6. 53. 2570 (1970)).
Tr8nsitiDn-MlIt8l Chemistry
251
p
p(,,-type)t
•
p(a-type)t
d
•
Central atom "nergy level.
Molecular orbital energy I"vel.
Ligand energy level.
Flo_ 12.2.2. The energy level diagram for an ootahedrallllX. moleoul.. or ion.
conclusions given above. Though this diagram is only schematic, the order of the energy levels is that accepted by most inorganic chemists. 12-3. LCAO MOs fDr tlltrahedral compounds We will not consider the LCAO MO treatment of the tetrahedral molecule MX. in detail, since the steps involved are essentially the same as those for the MX. case. The central atom M can use S-, p- and d-orbitals and for the :Td point group these are classified as follows:
belong to pA."
s (d••, d,.cv')
belong to ['E,
(p", P•• p.)
belong to
belong to
(d.... d,..,
r T ., r T ••
If we restrict ourselves to only u-bonding, the four X ligands will each use a s-Qrhital and a a-type p-orbital (one which points towa.rds t Theee two l&Vel. are only ll&p&l'&ted for viBuaI eonvenienoe.
252
M). These ligand orbitals form a basis for the reducible representation
r
of .:rd' which can be broken down as: r X = I'" EB rr where I'" = pd. Ell r T , X
1'"'
=
253
Transition-Metel Chemistry
Transition-Metal Chemistry
pd. EB
r T ,.
The symmetry adapted linear combinations of the eight ligand orbitals which form a basis for these irreducible representations are found in the same way as in the octahedral MX. case. They are: ~. = (Sl +s.+so+s.)/2. 'P;',w = (S, -s. +so -s.)/2 'P;'.<., = (Sl -80 -so +s.)/2 'P~,CIJ = (Sl +s. -so -s.)/2
(like PlO)' (like P.). (like Pal.
and analogous combinations of 0'1' ct., ct., and ct. for 'P~.' 'Pr.m' 'PT,II)' and 't'~,(.).t Thus, neglecting the use of any 1T-type p-orbitaIa on the ligands, we have: (1) Three non-degenerate a 1 -type energy leveIa with MOs formed from the s. ~. and 'P~. orbitals. (2) One doubly-degenerate e-type energy level, where the molecular orbitals are simply the do. and d,..-., AOs of M. (3) Four triply-degenerate t.-type energy levels, where for each level the three degenerate MOs are composed of combinations of (a) P.., 't';',(1) and 'PT,U)' (b) P., d." tlT.(1) and'PT.I.I' (c) P., d,.". tp;',CO) and "T,lll· There have been several all inito LCAO MO SCF calculations on tetrahedra.l transition-metal complexes: for example, see the work of Hillier and Saunders (Molecula,. phy8ics 22, 1025 (1970», and Demuynok and Veillard (Theoretica chi mica acta 28, 241 (1973)) on nickel carbonyl Ni(CO)•.
dibenzenechromium [(C.H.>'Cr].t As an example of the construotion of Mas for this type of compound we sha.1l consider ferrocene. Ferrocene. if we assume the two rings to be staggered.::: belongs to the point group ~'d (see Fig. 3-6.4) and we begin by forming linear combinations of the ten 1T-type p-orbitals of the two carbocyclic rings which belong to the irreducible representations of ~.d. We then combine these 1T-MOs with the metal AOs. matching the symmetries as we do so, in order to create MOs for the entire system. We exclude from our treatment any possible bonding between the iron atom and the rings through use of the a-electrons of the latter. The ten 1T-type p-orbitals form the basis for a reducible representation rr of the ~.d point group with the following character: ~6d
E
10
2C. 2C: 5C;
o
o
o
i
o
2S~0
2S10
o
o
5a d 2
and hence, using the standard reduction procedure:
rr
= pd•• EEl pd.. (£)
r E•• (£) r E•• Ell rE,. EB rE ,-.
The exact form of the linear combinations which belong to these six irreducible representations can be found by using the techniques which were applied to benzene in § 10-6. If the 1T-type p-orbitals are labelled as "'~, "'~. t/>~, ... t/>~ (see Fig. 12-4.1) and have their positive
"'1' "'.,...
a.,.
12-4. LCAD MDs far sandwich compounds An important class of organo-metallic compounds 'are the so-ca.1led metal-sandwich compounds. These compounds have the formula (CnH.),M and consist of a transition metal atom M sandwiched symmetrically between two parallel carbocyclic ring systems, e.g. ferrocene. or. to give it its proper name. dicyclopentadienyliron [(C.H.>.Fe]. and t The X atoms ..... numbered in the same sense as the hydrogens in Fig. 11·6.2. The reader should be able to see for himseIrwby the set of equations Cor "~l' ";,111' ";a(I). _d .,;'.", itt identioal inform with the set of equations given byeqn (1l-6.S) and eqn (11-6.13).
2 y
View {rom above
91'
FIG. 12.4.1. Directions and labels of the "._type p.orbital. of ferrocene and a set of fro
Y. and
Z 8X88.
t Dibenzeneohromium was discovered by Hein in 1919 but Dot reoognized &8
So sand.. wicb compound until the 19.50's. t Experimentally it is not clear- whether the staggered or eoliplied fonn is the mON Bt...able; they are very- olose in energy. Most derivatives or ferrooene show the eclipsed conforInstion in the solid state, but there is evidenoe that farrooane itself is staggered.
TrBllllition-Metal Ch8mistry
254
Transition-Metal Chemistry
lobes directed towards the opposite ring, then the norma.lized or-MOs are: 'P.A.•• = (.;. +';1 +';1 +.;c +';.+ef>1 +.;~ +~+.;a.p~)/v'lO, 'P.A. I• = {(.;, +.pl+.pl+.p. +.p.) -(';i +.p~+~+.p~+.;m/v'IO, 'PE.,(1) = 'P, +'P;,
'PE.,111 = 'PI +'P~, 'PE•• ll) = 'P,-",i, 1I'E••III = 'PI-V'~' 'PE I ,(') = 'PI +'P~' 'PE.. ll) = 'P4 +'P~' 'PE.. (1) = 'PI-'P~' 'PEI.lBl = 'Pc -'P~' where 'P. = {"', + (COB W)¢>.+(COS 2w)¢>.+(COS 2w)';.+(cos w)¢>.}/v'O
'PI = {(sin W).p. + (sin 2w)t/>1 -(sin 2w)4>c-(sin w)¢>.}/v'O lJI. = {"'. + (COS 2w).p.+(COS w)¢>,,+(cos W)';4 + (COB 2w)"'.}/v'5 'P. = {(sin 2w)¢>. -(sin W)¢>. + (sin w).pc -(sin 2w).p.}/v'5 and w = 271"/5. 'P~ is obtained from 'P" by replacing .p~ by';;, i = I, 2, ... 5.
For the metal atom, iron, the valence orbitals are the five 3dorbitals, the 4s-orbital and the three 4p-orbitals. They belong (see Table 11-2.2) to the following irreducible representations of ~6d 4s, 3d,. (3d.,., 3d••) (d"".3d••_ w') 4p. (4P.,4P.)
belong to ~I', belong to pE", belong to r E .. , belongs to r..4,., belong to rE,•.
The original set of 19 orbitals would have led to a 19 X 19 determinant in eqn (12-2.1), but now instead, by using the equally valid set of symmetry adapted orbitals, we have: (I) A 3 x 3 determinant corresponding to r.A.', which will produce three non-degenerate energy levels (a••-type) and a corresponding set of MOs formed from combinations of the 4s, 3d••, andV', ~" orbitals. (2) Two equivalent 2 X 2 determinants corresponding to rE',. One determinant will produce two energy levels and two MOs formed
255
from the 3d.,. and 'PElom orbitals (that this is the correct MO of the r E " pair to match up with 3d,.. is verified by inspection of Fig. 12-4.2, i.e. 'PE (1) is positive (negative) where 3d.,. is positive (negative)). The other determinant will produce an identical set of energy levels with molecular orbitals formed from the 3d". and 'PE,,(I) orbitals (the values of the coefficients of these component functions will be identical with those obtained from the first determinant). Together, there will be two doubly-degenerate energy levels of the e.. -type. (3) Two equivalent 2 x2 determinants corresponding to rEI" one 'mixing' the 3d.,•....,. and 'P E ()) orbitals and the other 'mixing' the 3d,." and 'PEI,lal orbitals '(see Fig. 12-4.2 for the matching). These will provide two doubly-degenerate energy levels of ea.type. (4) One 2 x2 determinant corresponding to r.A.••. The two MOs formed will be mixtures of the 4p. and 'PA.. orbitals and the two energy levels will be non-degenerate and of the al..-type. (D) Two equivalent 2 X 2 determinants corresponding to rE'·, one 'mixing' the 4p., and 'PEh(l) orbitals (see Fig. 12-4.2) and the other 'mixing' the 4p. and 'PE (II orbitals. These will lead to two doubly-degenerate energy l;vels of the e,..-type. (6) Two equivalent I X I determinants corresponding to r E •• which will produce a doubly-degenerate energy level of the el,,-type. The MOs will be the pure ligand MOs 'PH (1) and 'PE (2) (there are no metal orbitals of r E •• symmetry) "~nd oonsequently they do not participate in the bonding of the iron atom to the rings. If certain a.ssumptions are made about the matrix elements H~;: and 8 1" in these determinants, then the energy levels for the valence electrons in ferrocene can be calculated. An energy level diagram, based on the results of such a calculation, is shown in Fig. 12-4.3. This diagram implies that the electronic configuration for the 18 bonding electrons of ferrocene is la:. la~.. le~ .. 1~. 2a:. Ie:., each individual MO accommodating two electrons of opposite spin. The reader is warned that there is much disagreement about the exact order of the MO energy levels in ferrocene since they depend rather critically on the assumptions made about H;1,f and 8 1", In 1972, however, Veillard and co-workerst carried out a strictly ab initio calculation and made no such assumptions. Their results are likely to be more reliable than the previous ones.
t
M.- M. Coutiere. J. Demuynok and A. Veillard, TMor-.tica <"'mica acta 27.281 (1972).
2&1
TransltiOll-M...1C.....
,.try
Transition-Matal Chemistry
•
r _-=2a::2:.:.
2&1
-'~.::::::::::~=== p
h~-r----S
l'
~.,,.(II
•
Ring:
:'Ilolecular orbit...ls
molCf'ular
orlJituls Flo.
<$ +
-
FIG. 12-4.2. POIIitive and negative regiona of the ",·moleoul&r orbitaIa for the two rings in Cerrooene.
12-4.3~
An energy level diagram for
Iron atomic orbitals ferroeene~
12-5. Crystal field splitting The central concern of crystal field theory is what happens to the electronic states of an ion when it is placed in some perturbing symmetric environment. If we assume that a set of ligands placed symmetrically about a trarulition-metal ion interact only electrostatically with the electrons of that ion, then the answer to this question is relevant to our study of transition-metal compounds.
258
Transition-Meta' Chemistry
'We begin by considering an ion which, outside of a totally symmetric closed shell (noble gas) electronic configuration, has a single electron in a d-orbitaI. If the ion is in its free state, then the d-orbital could equally well be anyone of the five degenerate ones d." d.,.-••, d.,., d.,., and d.•. If, however, the ion is placed in sayan octahedral environment, then the five d-orbitals are no longer equivalent since, as we have already seen, they belong to different irreducible representations of the (Ph point group i.e. the first two belong to rE· and the other three to r T ••• They will therefore interact with their octahedral environment differently and where there was once a single five-fold degenerate energy level for the electron, there will now be two possible energy levels, one doubly degenerate and the other triply degenerate. We say that the energy level has been split and we represent the process ,--
2, 3, or 4 etc. Just as there are, for example, for a single electron with an orbital angular-momentum quantum number of l= 2, five degenerate d-orbitals, so there are five degenerate total electronic wavefunctions for a many-electron atom or ion in a D (L = 2) state.t Furthermore, it can be shown that the symmetry properties of the five total-electronic TABLE 12-5.1 Splitting oj electronic states by different symmetrical environmentst Splil!iI,_ .tat.. :
Fru
l!I h
ion slalu
e,
-<
S P D
EfI' Tall
l'
Atil"
G
H
d - - - -_ _
.'Y.
!iI •
A. T. E. T,
AStI',E.
A,. T t • P a
Alii!"
T .. A1",E., T ttr • T ••
A.,E, T.
A ....(2), A •• ,
E., T u (2).
E, T.(2), T,
All Til T t .,
All
Alp' B 1v' B ..,E rr
Ttl
At., A. II • E •. T. II •
t9h environment
diagrammatically as shown. The new energy levels are labelled, in lower case letters, with the irreducible representation with which they are associated. Likewise, in a tetrahedral environment the five d-orbitals will be split into a doubly-degenerate e-pair (d••, del-v') and a triply degenerate t.-set (d..., de.. d ••). In § 11-2 we showed how all AOs can be classified according to the irreducible representations of the different molecular point groups. Therefore, if we consult Table 11-2.2, we can determine the splitting of the energy level of a single electron for any particular perturbing environment. If we have more than one d-electron outside a closed shell electronic configuration, then in general there will be several electronic states for the free ion. These states will have different electronic energies and will be characterized by their term symbols; oB+'TJ (see Appendix A.12-1 for a brief review of atomic-spectroscopic notation). In a given term-symbol, T will be S, P, D, F, or G etc. depending on whether the total electronic orbital angular-momentum quantum number L is 0, I,
A" A,(2), E(2) A.(2). A •• E(3)
B 1 • AI. B,. E(2) A,(2), A" B1' B a, E(2) At. A a (2}, Bl' B,. E(3)
Au. A 1JII (2). B I ,. Raj'_ E.(3) A 1 .. (2), A a•• 8 ..(2).8..(2). E.(3)
T .. (2)
t
A. A"E AI' B l •
B 1 •• B. II , E.(2)
AIIAi,E, T., T.(2)
!iI••
A. A.,E A,.E(2)
B t ,. B zl1 ' E ..(2)
T ..
'---------/., fn-e ion
259
Transition-Meta. Chemistry
Ba,E
A,. A.(2), E(4) A ,(3). A ,(2), E(4)
A,(2), A,. B,(2), B.(2), E(3)
The states are derived from d" configurations only and honce for 0b. and
!i'tb
tho
split states all have g character. :t The numbers in parentheses refer to the number of states of the given symmetry, e.g. the 13 degenerate total waVefUD
r'
~
3r
A
,
EB 2rA , EB 4r".
wavefunctions belonging to a D state are the same as those of the five d-orbitals. Necessarily then, a D state will split under an (T)h perturbing environment into two states. Tho new states will again be labelled by the releva.nt irreducible representations, but this time upper case letters will be used. In Table 12·5.1 the states into which the states of a. free ion are split under a variety of environments are listed. Since a chemical environment does not normally interact directly with electron spins, the spin multiplicity of a state is unaffected by the splitting and the split states will have the same multiplicity as the parent free ion state. The quantum number J also remains unaltered. For this reason, multiplicities and J values are left out in Table 12-5.1. tEach wavefunction will correspond to a different value of the quantum number M z. (the equivalent to m, for .. single electron) and will be .. linear combination of Slater determinanta, composed of atomio orbita1s~ which have the same value of M c.0
260
Transition-Metal Chemistry
Transition-Metal Chamistry
12-6. Order af orbital energy levels in I:I'Ystllll field theory Our object in discussing crystal field theory is to construct energy level diagrams (correlation diagrams) which show how the energies of the various electronic states into which the free-ion terms split depend on the strength of the interaction of the ion with its environment. Before we can do this, however, it is nccessary to have some idea about the relative order of the energy levels of the d-orbitals in tetrahedral and octahedral surroundings (our discussion of correlation diagrams will be restricted to ions with a d" configuration and environments with a d or (!Ih symmetry). Such relative order concerns quantum mcchanics rather than symmetry or group theory and a rigorous treatment is beyond the scope of this book. Nonetheless. if we are willing to forego rigour, a simple model using crystal field principles will establish all we need to know for the purposes of constructing correlation diagrams. For an octahedral or tetrahedral environment we have already seen that five, originally degenerate. d-orbitals are split into two sets which have different energies: one set is d._ and <1,,'-.' (called e. or e orbitals, depending on the environment) and the other is d"., d"., and d., (called t•• or t. orbitals). To investigate the relative order of the two new energy levels, it is only necessary to look at the effect of the environment on one orbital of each set, say d"'_1f_ and d"._ In crystal field theory each ligand is approximated as a point charge or a point dipole and each metal-ligand interaction is taken to bc purely electrostatic. So our problem is reduced to one of investigating the effect of point charges (or point dipoles) arranged tetrahedrally or octahedrally about an electron in a d"o_._ or a d",. orbital. Since the electron has a negative charge and since we will assume thc ligands to be anions or dipolar molecules with their negative ends pointed towards the central ion, an octahedral arrangement of ligands will clearly result in greater electrostatic repulsive forces for an electron in a d""--1f.(e.) orbital than for an electron in a
e·
261
~ ~ ~ p x--8
I
I I
:r
e
F fa. 12-6.1. d:r,2_,,2 and d:1:~ orbitals Burrounded by an ootahedral arrangement of hgandB. The ligands which are directly a.bove and below the origin of the axes are not shown.
Free ion FlO.
]2-6.2. The relative energies o-f d-orbitals in octahedral environment.s.
•
!I
•
0
:r
.eh
a.nd tetrahedral ffd
Y
0
X
•
0
•
d.l'!f
12-6.3. d..t,a_.f1 and du,.. orbitals surrounded by a tetrahedral arrangoment of ligands. The black circles indica.te ligands below the x:y plane and tho shaded ciroles indicate ligands above it. See Fig. 11-5.2 for the relations between the axes and the ligands.
FlO.
262
Tr.nsitian- Metel Chemistry
263
Transition-Metal Chemistry
d,.,~, orbital will be more stable than one in a d... orbital. The energy difference is given the symbol ~t or 10Dq (see Fig. 12-6.2).
12-7. Carrel.tion diagrams
We are now in a position to construct diagrams which show how the energies of the various states into which free ion states are split by surrounding ligands depend on the strength of the interaction between the ion and the ligands. To start with these diagrams are qualitative but they become more quantitative when we introduce a particular value for the separation, ~t or ~o' of the two sets of d-orbitals as a measure of the strength of the interaction (~t for a tetrahedral environment and ~o for an octahedral environment). We will restrict our discussion to ions having a d n electronic configuration and surroundings having a d or (!)h symmetry. The first thing we have to do is to establish the anchor points of the correlation diagram, that is to say the relative positions of the spectroscopic states under the extreme conditions of an infinitely weak interaction and of an infinitely strong interaction. Consider an ion with a d" electronic configuration, the states of the free ion are, in order of •• lncreasmg energy, SF, lD, sP, lG, and lS. This order and the precise energies are found from atomic spectroscopic measurements.t However the order of the statea can be partially established by the application of Hund's rule: (a) the state with maximal multiplicity lies lowest, (b) if there are several states with maximal multiplicity that with the largest value of L lies lowest. If we apply a weak octahedral field to these states they will be split into new statea in accordance with the results which have been given in Table 12-5.1. Since the perturbation to the ion is small, the relative energies of the new states may be calculated by the quantum-mechanical method known as perturbation theory. It is found that the new states are ordered, with respect to energy, as shown below. (The'S and sp states are not split.) The actual magnitude of the gaps between the states is a function of ~o. So much for the left-hand (weak-field) side of the correlation diagram for a d' ion in an octahedral environment. Now we must consider an infinitely-strong interaction. Here the situation is directly contrary to
.:r
t The standard compilation of experimental atomic energy levels is C. E. Moore's Atomic !CflMgy Le",,~. N.B.S. Circular 467. The .tate. which emanate from a given
el.ctrOD1~ co~tIO~ are
determined by the familiar vector mcrlel rule coupled with the Paull exoluslon prmolple. The reader might note that the vector model rule i.......lf can be derived from group theoretical principles. However. the derivation involves the full rotation group whieh is beyond the soope of this book.
the earlier one. We now assume that the interelectronic repulsions in the ion are small compared with the infinitely-strong crystal field. In other words we have a splitting of the free-ion states which is very large compared to the separation between them. The wavefunctions which are constructed for the free ion and are based on relatively strong interelectronic repulsions (see the footnote on page 259) are quite unsuitable for this new situation and instead infinitely-strong-field configurations become the natural choice. An infinitely-strong-field configuration)s obtained by assigning each d-electron to either at•• or an e. orbital. For a d' ion in an octahedral environment, both the electrons go into the t •• orbital to form the lowcst lying configuration: t:•. The next highest configuration will be t~.e~ and after that Thc energy separation between these configurations will be ~o or 10Dq (see Fig. 12-6.2). Now let us consider what happens as we weaken this infinitely-strong interaction with the environment to just a strong interaction. The interelectronic repulsions, in a relative scnse, start to increase as the electrons start to influence each other. As they begin to couple, a set of states of the entire configuration will be produced i.e. there is a splitting into the strong-field states. The new states and their energies can be found by treating the interelectronic repulsions as perturbations on the infinitely-strong-field configurations. The qualitative picture, however, can be obtained by the use of group theory in the following way. If the states which arise from a given configuration are known, then the statea which arise from the configuration with one additional electron must have the symmetry labels of the irreducible representations which are contained in the direct product (see § 8-3) between the symmetry labels (i.e. irreducible representations) of the initial states and the added electron. With only one d-electron, there are no interelectronic repulsions, so the symmetry labels of the states of d 1 must "be the same as those of the electron. Thus the ts. and e. electrons of d l
e;.
(!)h give rise to 'T,. and ·E. statea, respectively. For a d' configuration, to obtain the states of it is necessary to take the direct produet between T •• (the symmetry label of the state of t~.) and T •• (the symmetry label of the added electron). Using the
in
t:.
264
Transition-M"al Ch...lstry
Transition-Meta' Chamlstry
techniques established in § 8-3, we find
rT,"
I8i
rT··
rT,"
=
El>
rT," El> rEI
El>
t:.
that the infinitely-strong-field configuration gives rise to states having symmetries T I ., T •• , E. and AI.' Similarly, the first excited configuration, t~.e:, leads to T I • and T •• states since
80
rT·.
I8i
rEI
=
r T ..
and the second excited configuration, states since -nE r E I I8i .1-nE - . = .1 - I $
tit)
rT,"
e:, leads to E" r .4
II
El>
t:.
t=,.
and if we require that the degeneracy for the strong-field case remain at 15, then 3a+3b+2c+d = 15, with a, b, c, and d each equal to either 1 or 3. This equation has three possible solutions: abc d . 1
1
~)
1 1
3 1
1 3
1 3.
(3)
ilJITJ
3.
5.
.As well as the symmetry labels of these strong-field states, we also require the multiplicities. The completely general method of determining these is beyond the scope of this book, so we will confine ourselves to consideration of a. case which can be resolved on the basis of some simple arguments. Consider first the configuration and let the three t •• orbitals be represented by three boxes. In Fig. 12-7.1 it is shown that, if an electron with spin quantum number m. = I is represented by an arrow pointing upwards and one with spin quantum number m. = -I by an arrow pointing downwards, then the number of ways of arranging the arrows in the boxes is 15. This corresponds to the number of distinct wavefunctions for .As the field strength is decreased this total degeneracy must remain at 15. We now recall that a T-type state is of three-fold degeneracy, an E-type state of two-fold degeneracy, an A-type state is non-degenerate and also that only triplet or singlet multiplicities can arise from two electrons. Therefore, if the required multiplicities a, b, c, and d are attached to the states in the following way:
1
2.
t.
.......
3
mr::IJ
AI.' and A_.
.1 - Or •
(1)
I.
r..4··
That solution (2) is the correct one, we will discover only when we finally set up the correlation diagram. For the configuration t~.e~ it is possible to write 24 wavefunctions (for each of the six ways of putting an electron in the t •• set, there are
6.
7.
[]IIJ [ITI[] [[0[]
ITIITJ CDITI
8.
[IT][]
9.
GTIIJ
10.
II. 12. 13. 14. 15.
28&
DITD DIID DITO
DIJD DIJD IT1IIl
FIG. 12-7.1. Symbolic wavefunations for the
t:.. configuration.
four ways of putting one in the e. set). Hence, if a and b are the multiplicities of the states T h and T •• respectively, we have
3a+3b = 24 which is satisfied, for example, by a = 4 and b = 4. But since we have already stated that the multiplicities are restricted to 1 and 3, this result is unacceptable. We can extract ourselves from this dilemma by assuming that we have in fact four states IT,., "T•• , IT I,> and 'T_•.
2&1
TrllIIsitian-Mlt.1 Chemistry
Transitian ·Met.1 Chemistry
[Choosing eight singlet states is ruled out on the grounds that we would then have more states in the strong field than in the weak field (this will become apparent when the final diagram is set up).] For the configuration it is possible to construot six wavefunotions and if a, b, and c are the multiplioities of E., A,., and A •• respeotively, then 2a+b+c = 6
e:
Freeion terms
Weak crystal field
Intermediate crystal field
Strongfield
217
Strong·field configurations
terms
'S
for which there are two solutions:
a
b
c
(1)
1
1
3
(2)
1
3
1
Again the correlation diagram itself will dictate that solution (1) is the oorreot one. The order of the states derived from a given infinitely-strong-field configuration is given by a modified Hund's rule: (1) states with the highest multiplicity lie lowest, (2) for states with equal multiplicity, the ones with highest orbital degeneracy (T > E > A) tend to lie lower. Any ambiguities which remain after the application of this rule, can only be resolved by recourse to detailed quantum mechanioal calculations . Once the two sides of a correlation diagram have been established, the states of the same symmetry and multiplioity are conneoted by straight lines in such a way as to observe the non-orossing rule: identical states cannot cross as the strength of the interaction is changed. When this is done we have completed the correlation diagram. The assignment of multiplicities can now be settled. For a d' ion in an octahedral environment there are no states in the weak crystal field and thus solution (3) for the t~. configuration is ruled out since it includes suoh a state. Also the highest of the sT,. states in the weak crystal-field must connect with the highest 8T.. state in the strong field, namely the one arising from the t~.e~ oonfiguration, this leaves the other weak crystal field sT,. state with oruy the possibility of connecting with the T t • state from t=" thus this state must be a triplet and solution (2) is the correct one. Finally, the fact that the only state in the weak crystal field is a triplet requires that we accept solution (1) for the configuration. A correlation diagram for a d' ion (e.g. VS+) in an octahedral environment is shown in Fig. 12-7.2. What this diagram does is to demonstrate how the energy levels of the free ion behave as a function of the strength (~o) of the ion's interaction with a set of octahedrally disposed ligands.
-A,.
A,.
e:
FIG. 12-7.2. Correlation diagram (not to scale) for a d l ion in &n octahedral environment. Adapted from B. N. Figgia Imrodvction to ligand fieldB.
If ~o is known for a particular ion and set of ligands, then a correlation diagram will immediately predict the order of the ion's energy levels. For a d 8 ion in a tetrahedral environment, exaotly the same procedure oan be carried out. The free ion states will be the same as in the octahedral case. The type of states produced from a particular free-ion
288
state bJ the weak crystal-field will be the same as before except for the dropping of the subscript g (see Table 11-2.2). The order of the states from a particular parent state. however, will be reversed (we come back to this point in a moment). The infinitely-strong-field configurations will be reversed in accordance with Fig. 12-6.2. The strong field states derived from a particular infinitely-strong field configuration will be in the same order as before. The complete diagram is given in Fig. 12-7.3. One immediate deduction which can be made from these two correlation diagrams is that the ground state in both cases remains a triplet no matter what the strength of the interaction (aT,. in one case and "A. in the other). We therefore expect. for example, tetrahedral and octahedral complexes of V3+. in the crystal field approximation. to have two unpaired electrons. Indeed, this is known to be the case for the octahedral complexes e.g. (NH.)V(SO.)•. 12H.O. A useful relationship for constructing correlation diagrams for other d"-type ions is the hole formalism, according to which the d 1o - n electronic configuration will behave in exactly the same way as the d n configuration except that the energies of interaction with the environment will have the opposite sign. Essentially, we treat the n holes in the d shell as n 'positrons'. The change of sign of the interaction will have the effect of reversing the order of the infinitely-strong-field configurations (the stability of the e. and t•• levels is reversed). However, since the 'interpositronic' repulsions are the same as the interelectronic repulsions. the perturbations these cause when relaxing the infinitelystrong-field are the same and the order of the states in the strong field for a particular parent configuration is the same for both d" and d 'o-" ions. The free-ion states will be the same in both cases but the weakfield environmental perturbations will be of opposite sign. so that for any given parent state the order of the weak-field states is reversed. These relationships are summarized in Table 12-7.l.t All that has just been stated for changing a d'" correlation diagram to a d'O->I one with the Bame environment, applies equally well to changing a d" diagram for an octahedral environment to one for a tetrahedral environment. We have already seen that infinitely-strongfield configurations are reversed by such an environmental change (Fig. 12-6.2) and. if we assume therefore that the environmental perturbation in going from the free-ion to the weak-field case is also reversed, then we can conclude that the order of states emanating
t A precise and formal diso\l8.8ion of the hole form.alism is given by J. S. Griffith: TM eheory oftranrition-metal8 ion", C&mbridge University Pre.... 1961.
281
Trensitian-Mm' Che..istry
Trensitian-Mml Chemistry Freeion t.errns
IS
Weak .crystal field
Intermediate ery8tal field
Strongfield
Strong-field configura.t.ions
terms
1.1,
It'
.3.. iU(:I't'using-----... FlO. 12-7.3. Correlation diagram (not to scale) for a d l ion in a tetrahedr&l environment. Adapted from B. N. Figgi. Introduction to ligand f"'ld8.
from a particular parent free-ion state will. in the weak field. be the opposite in a Y d environment to that in an (!)h environment. The free ion states and the interelectronic-repulsion perturbation are the same in both cases. Hence, Table 12-7.1 applies also to the (!)h - Yd change. It should now be clear that if we change both the configuration, d"_ d 10-" (i.e. change the ion), and the environmental symmetry,
270
Transitlon-Metlll Ch.misby
Trensltlon·MIt8I Ch.mistry
271
.rd ...... /!Ih'
~ I::: gf
. 0
~
.b
." ."... ."t~ $
0
C
......
-0
1::.;;
then the correlation diagram is unaltered (except for the obvious and minor change of adding or dropping the subscript 9 on various symbols). We can express this result by d"(oct) ... d1H·(tetr) and d"(tetr) ~ d10-"(oct).
These relationships show that far fewer individual correlation diagrams need be constructed from scratch than might have first been anticipated.
~:;
CQ. .~
I::
.,.-
~~'"
"
C
""2 =~ o '" .~ §.
..,., ~= 1:: ,-
~5
~
.,:,
.",
...-.;
i!'
E
en
"I
H
~
e-.
."
...-.;
..
-'"
=:'"
]c ~
0
e--=
c '"
.-2-t: " >1"' C -
~~
.=c ~
;.;
..
Oil
:ii
..c
~
12-8. Spletrlll proplrtles
One of the most important applications of correlation diagrams concerns the interpretation of the spectral properties of transitionmetal complexes. The visible and near ultra-violet spectra of transitionmetal comple;'tes can generally be assigned to transitions from the ground state to the excited states of the metal ion (mainly d-d transitions). There are two selection rules for these transitions: the spin selection rule and the Laporte rule. The spin selection rule states that no transition can occur between states of different multiplicity i.e. M = O. Transitions which violate this rule are generally so weak that they can usually be ignored. The Laporte rule states that transitions between states of the same parity, u or g. are forbidden i.e. u ..... 9 and 9 ..... u but 9 +> 9 and 'U +> 'U. This rule follows from the symmetry of the environment and the invoking of the Born-Oppenheimer approximation. But since, due to vibrations, the environment will not always be strictly symmetrical, these forbidden transitions will in fact occur, though rather weakly (oscillator strengths of the order of 10-0 ). All the states of a transitionmetal ion in an octahedral environment are 9 states, so that it will be these weak symmetry forbidden transitions (called d-d transitions) that will be of most interest to us when we study the spectra of octahedral complexes. We will exclude from our discussion the so-called charge-transfer bands. These relate to the transfer of electrons from the surrounding ligands to the metal ion or vice versa. They may be fully allowed and hence have greater intensities than the d-d transitions. They usually, though not always, occur at high enough energies and with such high intensities that they are not too easily confused with the d-d bands. A third type of transition, transitions occurring within the ligands, will also be ignored. By consulting the appropriate correlation diagram, it is possible to see what kind of d-d spectrum a transition-metal ion in a given environment should have. For qualitative predictions we can Ulle diagrams of
272
Transition-Mete' Chamistry
Transition-Mete' Chemistry
the kind which were developed in the last seotion. However, for quantitative predictions it is necessary to use the so-called Tanabe-Sugano correlation diagrams (J. phys. soc. Japan 9, 753 (1954». These diagrams are based on proper quantum-mechanical caloulations of the energy levels of a d ft system in the presence of both interelectronic repulsions and crystal fields of medium strength. Such calculations are very difficult to carry out and we will simply discuss the form of the results. It turns out that the energy of each state depends not only on the field strength (as measured by ~o or ~t) but also on two electronicrepulsion parameters Band 0 called Racah parameters. (B and 0 are related to the Slater-Condon parameters F. and F C by the equations: B = F.-SF c, 0 = 35Fc.) In Tanabe--Sugano diagrams it is assumed that 0 is directly proportional to B with a proportionality constant which has a fixed value for each diagram (the diagrams are apparently not too sensitive to the value of this proportionality constant). Furthermore, the diagrams are made independent of B by plotting EfB against ~ofB (or ~fB) rather than E, the energy, against ~o (or -\). Consequently, to obtain from a given diagram the relative energies of the states of a metal ion-ligand system, it is necessary to specify both B and ~o (or ~t)' This is usually done by using two pieces of experimental data, e.g. by fitting two d-d transitions to the appropriate Tanabe--Sugano diagram. Now let us consider some particular cases. [V(H.O).P+ is a d' ion in an octahedral environment and the pertinent qualitative correlation diagram, Fig. 12-7.2, shows that there should be three spin-allowed transitions: from the 3T, .(F) ground state to the excited states sT••(F), 3T 1 .(P) and ·A•• (F); the symbol in brackets, in each case, denotes the parent state of the free ion. Experimentally, aqueous solutions of trivalent vanadium salts show two absorption bands, one at 17200 cm- l and the other at 25 600 em-I; these give rise to the green colour of such solutions. If we specify the complex (Le. determine ~o and B) by fitting the transitions ·T•• (F) +-- 'Ttg(F) and ~Tlo(P) +-- sTI.(F) to 17200 and 25600 cm- l respectively, then ~o is found to be 18600 cm- I and B to be 665 em-I. With these values the transition sA ••(F) +-- T,.(It') is predicted to lie in the region of 36 000 em -1. Unfortunately this cannot be verified as there is a very strong charge-transfer band in the same region. However, in the solid state, particularly for V3+ in AI.O., where charge transfer occurs at a higher energy, a weak band at about the right position has been found. Since the oxygen ligand atoms in the AI.O. structure are known to produce about the same value of ~o as water molecules, this can be considered as partial experimental confirmation of the assignments for [V(HaO).)" +-. An aqueous solution of
a
273
V8+ salt also shows some very weak bands (/ "'" 10- in the 2000030 000 cm-1 region; these a.re thought to be due to spin-forbidden transitions to excited singlet states. A very well studied group of complexes are those with d' configurations in an octahedral environment. We have not shown the correlation diagram for this case, but the important features of such a diagram are a cA •• (F) ground state and three other excited quartet states which, in order of increasing energy, are cTI.(F). CT lo (F), and cTtg(P); furthermore, none of these states cross each other as the strength of the interaction changes. As an example, we take the case of [Cr(H.O).l8+. The aqueous solutions of salts of trivalent chromium are green in colour as a result of absorption bands at 17000, 24000, and 37 000 cm-I.(there are also two very weak spin forbidden bands at 15000 and 22000 em-I). If the complex is specified by fitting the transitions cT•• (F) +-- cA••(F) and cT1 .(F) +-- cA •• (F) to 17000 and 24 000 cm- l respectively, then 11. 0 has to be 17 000 cm-1 and B has to be 695 em-I. The transition cTI.
The same correlation diagram can be used for the tetrahedral complexes of COH (d 7 ) and, as an example, we consider [CoCl.ls-. The spectrum of [CoCI.J"- in RCl solutions shows bands at 5800 and 15000 cm- I and some weak absorption in the 17000-23 000 cm- I region. If the 'TI(F) __ cA.(F) transitiont is assigned to the 5800 cm- I band, then it is found to be impossible to fit the remaining bands. However, the assignment: 'T,(F)
+--
fA.(F)
5800 cm- I
CT1 (P)
+--
'A.(F)
15 000 cm- I
can be fitted to the correlation diagram and this leads to a value of 3200 cm- I for ~ and 730 cm- I for B. With these values, the transition CTI(F) +-- 'AI(F) is predicted to lie at 3300 em-I. Studies in the infrared region show a band at 3500 cm- I which can probably be assigned to this transition. There are no other spin-allowed transitions, so the bands in the 17 000-23 000 cm-1 region must be assigned to spinforbidden transitions. 12-11. Magnstic properties
The effective magnetic moment of a transition metal ion !-len is defined by the equation !-Ierr = 2'84(X T )i
t
Note that the
f}
Bohr magnetons
subscript il9 not applica.ble in the tetrahedral Ol\se.
214
Transition-Mda' Chelllistry
Tranaltlon-Mda' Chemistry
where % is the molar magnetio susceptibility and T the temperature. Different formulae oan be derived whioh relate 140ft to the ion's angular momentum quantum numbers L, 8, and J (definitions of L, S, and J are given in Appendix A.12-I). The particular formula to be used depends on how far the exoited states of the ion lie above the ground state in oomparison with leT. If the excited states are separated from the ground state by an amount much larger than leT, then flerr = g{J(J + I)}.
No.o/d .lecWon6 I
2 3
4
where g, the Lande splitting faotor, is given by
g
=
1 +{(8(8+1)-L(L+ I)+J(J+l)}/2J(J+ 1).
Ion
or:+
y>+ Cr'+ Mo'+ Mn'+ Re'+ Cr'+ Mn" Ru"+ Os"
I'>
MD"
Hence, for a system in whioh the ion has no orbital angular momentum: L = 0, J = 8, g = 2 and
Fe +
flerr = 2{8(8 +1)}•.
Rul + 0.'+
Also, under these ciroumstances, if n is the number of electrons in the ion with unpaired spin, then 8 = n/2 and
l
6
*'
tSee P.J. Stiles, Mol. Phy•. , ].5, 405 (1968).
Fe l + Co'+ Rh'+
lr'+
pelf = {n(n +2)}•.
This is known as the 8pin-only formula. Many ions which in the free state have orbital angular momentum (L 0), lose it, completely or partially, when incorporated into a complex. This phenomenon is called orbital angular momentum quenching. It can be shown that for A states the quenching is complete and that for T and E states it is incomplete. t Because of this, the spin-only formula applies to more situations than might hlLve been expected. The number of unpaired spins n for an ion in its ground state is obviously determined by the multiplicity, 28+1, of that state and vice versa. The multiplicity is written in the upper left hand corner of the symbol for the state. Thus the ground stlLte symbol can be read off from the appropriate correlation diagram and, if spin-only conditions apply, the effective mlLgnetic moment can be immediately determined. For octahedral d 1 , d', dO, dO, and d· cases, the ground state is derived from the lowest term of the free ion for all values of the crystal field strength (defined by 6.0 ), Henoe the multiplioity, the number of unpaired spins and, if the spin-only formula applies, the effective magnetio moment must all be the same as that of the free ion, no matter how strong the interaction between the ion and the ligands. For ootahedral d~, d 5 , d", and d 7 oases the ground state is derived from
27&
TA.BLE 12-9.1 Magnetic propertie8 of 80me tranBition-metal ion8
7
Pt·+ Co'·
8
Ni'+
9
Cu'·
Compound CaTi(SO.>•.12H.O (NH,)V(SO,)•. 12H.O KCr(SO.l••I2H.O K.MoCl. BaMnF.
CesReC1.
Cr(SO,).6H.O Mn(aoaol. K,Mn(CNl. KoRuCl. K.OeC1. KoMn(SO.1o.6H.O K.Mn(CNl•.3H.O KFe(SO.l•. 12H.O KoFe(CNl. Ru(NH.)•.Cl. O.(NH.l•.Cl. K.Fe(CNl. (NH.l.Fe(SO.l••6H,O Co (NH,l•.Cl, Rh(NH,) •.Cl, KolrCl•.SHoO K.PtCl. K,BaCo(NO,). (NH.).Co(SO.J..6H.O (NH.).Ni(SO.),.6H,O [(C.H.l,Nl,NiCl, K.Cu(SO.),.6Ro°
.faI.
Ground n
'T..
I
• 'Ph 41A., ·A u 41A•• "A".
2 3 3 3 3 4 4 2 2 2
'E, 'E,
IP." 'p•• IT•• 41,Ah
I)
'T•• ·A .. liP.,
1 5
IP••
tP•• LA ••
"p•• JA ••
IA., LA •• IA I •
• E, ·'1'1. "AIIIl'
'p. 'E,
I I I
0 4 0 0 0 0 I
3 2 2 I
{..(n
+ 2)}t
1·73 2·83 3·87 3·87 3·87 3·87 4·90 4·90 2·83 2·83 2·83 11.92 1·73 0·92 1·73 1·73 1·73 0'0 4·90 0·0 0·0 0·0 0·0 1·73 3·87 2·83 2·83 1·73
_
(exptl 300K 1·84 2·80 3·84 3·79 3·80 3·lll. 4·82 4·86 3'110 2·96 1·110 5.92 2·18 /j·89 2·25 IH3 1·81 0'35 11'47 0'46 0'311 0'0 0·0 1'81 5'10 3'23 3'89 1'91
the lowest free-ion state only out to a certain critical ~o value, beyond which a state of lower multiplioity, originating in a higher free-ion state, drops below it and henoe becomes the ground stste.t So in these cases the multiplicity, the number of unpaired spina and the effective magnetic moment will depend on 6.0 and therefore on the nature of the ligands. For strong interactions between the ion and its environment (~o large) there will be fewer unpaired spins than for weak interactions (.6." sma.l1). Similar predictions can be made for ions iii. tetrahedral environments. In Table 12-9.1 calculated (spin-only formula) and experimental effective magnetic moments are listed for a number of ions, they are in accord with the previous disoussion.
t Studi~ of the Ol'088-over point have been made by E. Konig; Theor. CIKm. ACUI 26, 311 (1972).
1Jee,
for example.
27&
TraMition-Metal Chemistry
Transition-Matal Chamistry
12-10 Ligand field thaory In the introduction to this chapter we stated that the approximations made in applying crystal field theory to most transition-metal complexes and compounds are extreme. The question which arises is: can we modify the theory so as to take account of its known defectsl The answer is a qualified 'yes'. Essentially, what we must do is to drop the assumption that the metal ion's partially-filled shell consists solely of its d- or f-orbitals and allow for the overlap between the orbitals of the ion and those of the ligands (MO calculations show that there invariably is such an overlap). Doing this has two consequences. We can no longer consider the crystal field parameters .:l.. or A.t (and, if TanabeSugano diagrams are used, B) within the framework of simple electrostatics and they lose their initial significance and become quite arbitrary parameters to be adjusted in any way neoossary.t In other words, the corrections due to the approximations are assimilated in these parameters. Further, in the construction of the correlation diagrams, the separations of the energies of the free-ion states become adjustable and are not taken as the observed values given by atomic spectroscopy. With the exception of these changes, the practical development of ligand field theory and crystal field theory are the same. App8l1dill A.12-1. Spactroscopic statas and term symbols for many-alactron atoms or ions So far in this book we have only discussed non-relativistic Hamiltonian operators but when atomic or molecular spectra are considered it is necessary to account for relativistic effects. These lea.d to additional terms in the Hamiltonian operator which can be related to the following phenomena: (1) The coupling of spin and orbital angular moments among the electrons. (2) The coupling of spin angular moments among the electrons. (3) Interactions among the orbital magnetic moments of the electrons. (4) The coupling of spin angular momenta among the nuclei. (5) The coupling of spin angular moments of the electrons with spin angular momenta of the nuclei. (6) The coupling of nuclear-spin angular momenta with electron-orbitsl angular moments. (7) Nuclear electric-quadrupole-moment interactions. As well as these additional terms there will also be changes to the Hamil· tonian operator due to the relativistic change of electron ma.ss with velocity. In ordinary optical spectroscopy the first two phenomena, (I) and (2), are the most important, leading to changes to the non-relativistic energy levels which are observable (effects (4) and (5) are important in n.m.r. and e.s.r. spectroscopy). t For example. the parameters oan be adjusted 80 as to reproduce the experimental d-d spectral transitions. This, in faot~
WBo!J
what was done in § 12-8.
277
For these reasons the electronic energies, and therefore the electronic states, of a many-electron atom or ion will depend upon the electronic spins and how the Spin angular momenta a.re couplod with the orbital angular ~oments. The coupling scheme which is most appropriate for our purposes IS ~own as L-S (or Russell-Saunders) coupling. It first couples the electronic spm angular momenta together, then the electronic orbital angular moments together a~d .finally coupl~ these totsl momenta together. Like all coupling schemes, It IS an apprOXImation. Associated with the spin and orbital a~lar momenta of a single electron are quantum numbers land 8, respectIvely, and for a n-electron system there are equivalent quantum numbers Land S. The quantum number L defines the total orbital angular momentum and its allowed values are
L = ll+1.+ ... lft' 11 +1.+ ... lft-I, ... , -(ll+1.+ ... lft) where I, is the orbitsl quantum number of the ith electron. Capital letter symbols are assigned to ststes having different L values as follows: L=O symbol = S
I 2 3 4 PDF G
5 H
6
I.
The quantum number S defines the total electronic spin angular momentum and its allowed values are S = nJ2, (nJ2)-I,
, 1/2
S = nJ2, (n/2)-I,
,0
(ifn is odd) (if n is even).
In L-S coupling, the total electronic a.ngular momentum (spin and orbital) is defined by the quantum number J whose allowed values are L+S, L+S~I,... , IL-SI. TABLE
A.12-l.I
Spectroscopic terms ari8ing from equivalent ekctronic configurations in L-S coupling oonftguraticm s' p or p. p' or p'" p' d or dd' or d l
d a or d T d& or d S
d'
L-S termat
'S
'p IS, lD, ap
'P, ID,.8
'D IS ID IG Ip IF ID(2),'·P:"F,'IG. 'H, "PI 4}1'" 'S(2), 'D(2), 'F. IG(2). '1. 'P(2), 'D, 'F(2), 'G, 'H. 'D 'S, 'p. 'D(3), 'F(2), 'G(2), 'H, Ir. 4P~ "D, "F• .foG, 'S
t The number in parentheBeB is the number of distinct terms with the 8&me Land S quantuIn numbers. For oaoh distinct term thero will be different states correaponding to the different possible J valu81!1.
Tr.nsilion- Mml Chlmistry
278
'8
{
Is' 2s" 2p' 3s" 3p'
'D 'p
'p Ip"
{
:ap: levels
A.l2-!.!. Levels for the silioon atom.
J therefore can have 28+1 values if L > 8 and 2L+l values if L < 8. The number 28+1 is called the multiplicity. As the electronio energy of an atom or ion will depend on the quantum numbers L, S, and J, we designate the various energy states which may arise from a given electronio configuration by what is known as a speotrosoopic term symbol: IS+IT J
where T = S, P, D, ... as L = 0, I, 2, .... When &ll the electrons have different prinoipal quantum numbers there are no restrictions on the oombi. nations of Land S, but, if this is not so, BOrne combinations will be excluded by virtue of the Pauli Principle. In Table A.12-1.1 the spectroscopic states of common configurations of electrons with the same prinoipal quantum number are shown. The reader should note that we are only concerned with that part of an electronic configuration which is outside of a.ny closed shells (noble-gas structures). The latter are spherically symmetrical a.nd do not play any role in the effects which are of interest to us in this ohapter. In Fig. A.12-1.1, as an example of the above notation, the hierarohy of levels for the ground state configuration of silicon is shown;
PROBLEMS 12.1. Determine the qualitative fonn of the molecular orbitals for the squareplanar complex Ni(CN)/-. (Ass11Ille that each CN ligand provides one
a-type and two ...-type orbitals to the system.) 12.2. Detennine the qualitative fonn of the molecular orbitals for the tetra· hedral molecule MnO•. [Ass11Ille that each oxygen atom provides just three
p-orbitala (set these up 80 that one points towards the Mn and the other two are perpendicular to each other and to the Mn-O axis) and that the Mn atom provides 4s and 3d orbitals.] You will be on the right track if you find that pr _ r.d, E9 rTl
rr _ r E
(9
r T,
(9
r T ••
12.3. Determine the qualitative fonn of the molecular orbitals for the eclipsed conformation of ferrooene. 12.4. For an octahedral environment the d-orbitals are split into two sets
(d
"
and d.I I); how would they be split for a square-planar environment!
12.5. Set up a qualitative oorrelation diagram for the d 3 configuration in an
octahedral environment.
Appendix I:
Character Tables
'D.
L· S terms
configuration FIG.
'8"
The x, y, z axes referred to in these tables are a set of three mutually perpendicular axes chosen as follows: (I) ~.: the z axis is perpendicular to the reflection plane. (2) Groups with one main axis of symmetry: the z axis points along the main axis of symmetry and, whore applicable, the x axis lies in one of the
280
Appendix 1 : Cheracter Tables
Appendix 1 : Character Tables
I: Groups "ff'., ttl" and'tf'n (11. = 2,3,4,5,6) E ttl. G'h
A' AN
1 1
1 -1
ttll
E
i
A. AM
1
1
1
-1
ttl.
E
C,
A B
1 1
1 -1
"ff'.
E
A
1
C. 1
e
x; y; R.
R;r;R,,; R.
1
A
1 1
1 -1
1 1 -1
e:}
(x, y); (R., R.)
C', z; R~
1 -1
1
1
1
A
1
e
G
e' e'· e· e'· e' e' e· e e'.} e'· e e* e"
E
C.
Ei
{~
E.
G
x'+Y'; 2' x'-y'; xy
C:
E
1 -1
(x'-y', xy); (X2, Y2)
c·•
"ff'.
1 1
e -e· e· -e -e· -e -e -e·
c,
exp(21Ti/5)
C:
1
1
1
1
-1
n
-e -e· -e· -e} -e -e·
z; R.
x'+y'; z'
(x, y); (R", R.)
(xz, Y2) (x'-y", xy)
1 -1
1/2
A.
1 1
E
2
1 1 -1
1 -1 0
12.
E
2C.. C,
Ai
1
A,
1
1 1
Bi B,
1
-1
1
-1
1 1 1 1
E
2
0
-2
12.
E
2C.
2C~
5C~
Al
1
1
A. Ei E.
1
1
1 1
~1
2 2
2 cos IX 2 cos 2<:t.
2 COB 2<:t. 2 cos a.
!ii.
E
2C.
2C.
C.
3C~
3C;
1 1 1 1
1
1
1
1
1 1
1 -1
1 -1
-1
1 1 -1 -1
-1
1 -1
-1
-1
-2 2
0 0
Bi B, E, E,
exp(27Ti/6)
x; R z
x2;
z; RI.
Ai
(xz, yz)
e=
y; R.
y2; xy X2
1
-1 -1 1
3C~
(x, y); (R., R.)
•
1 -1
2C.
Al
C·
1 1 -1 -1
E
x"+y'; z'
(""-y", xy)
1 1 1 1
12.
z; R~
-1 -1 -1 1
1
(X2, Y2)
e=
;*)
C.
exp(21Ti/3)
x'+y'; z.
C:
-1
1 1
xz; yz
(x, y); (R", R.)
G -i
B
x 2 ; y2; Z2; xy
-:}
E
A
xy; xz; yz
2; R.
C,
ttl.
Bi B,
e=
C,
E,
Z2;
c:
E
(~
Xl; yZ;
z; R. x; y; R 2 ; R 1I
ttl,
E1
A
x;y;z
G e·
C.
y"; Z2; xy "'2; y2
B.
E
B
Z2;
z; R:c; R"
II: Groups12n (11. = 2, 3,4,5,6) = "f/ E C,(2) C.(y) Calx)
~,
A,
2 2
-1
1
-1
Z2
x 2 +y'!.;
ZZ
z; R z
(x, y); (R z ' R.) 2C~
2C;
1 --1
1 -1 -1 1
1 -1 0
0
(x'-y', xy); (X2, Y2)
x'+y'; 2' 2; R z
X2 _
(x, y); (R., R.)
y2
xy (X2, Y2) <:t. = 72 0
1
x"+y'; 2' z; R z
0 0
(x, y); (R., R.)
(X2, Y2) (x'-y', xy)
""+y'; 2' z; R"
1 0 (x, y); (R", R.) 0
(X2, yz) (x"-y', xy)
281
Appendix 1 : Character Tebl..
282
III: Group8 ~.." (n = 2,3.4,5.6) E C. O',,(xz) O',,(yz) 1 1
1 1
1 1
~1
1 -1 1
-1
-1
~3"
E
2C.
30'.
A, A. E
1 1
1
1
1 -1
2
-1
0
'iffy
E
2C.
C.
A, A, B, BI E
1 1
1 1
1 1
1 1
2
z
xl!; yt; zt
-1
R.
zy
-1 1
x; Rfl
1
R. (x, y); (R.., R.)
1
1 -1 -1
-1
1
-1 0
1
-1
1
2
0
0
~."
2C:
A,
1 1
1
1 1
AI
2 2
1 2 cos a: 2 CQS 2a:
~3h
E
C. C:
z
x'+y';
Zl
R. x 2 _y" zy
(x, y); (R.., R.)
(xz, yz) a:
50'"
z
1
-1
2 cos 2a: 2 CQS a:
=
E"
g • .0e
'If.,.
E
72° Zl
(=,yz) (x"_y', xy)
E
2C.
2C.
C.
30'"
30'<1
A,
1 1
1
1
1
1 1
-1 -1 1
1 -1 1 -1
1 -1 -1
0 0
0 0
AI
B, B. E, E.
2 2
1
-1
1 1 1 -1 1 -1 -1 -2 -1 2
z
X·+y"; z.
R, (x, y); (R... R.)
(xz. yz) (x"_y', xy)
:O} -1 _.O}
. -.
-1
-
C·
•
j
1
1
1
1
-1
1 -1 -1 1 1 -1 -1
-1
1 1
-i i
1 1
1 -1
-1 -1
-i
-1
1 1
Au
B.. Eu E
'if••
c.
C: C: c:
I
(x. y)
I I I I I -I -I -I -I
0"
0°
I
I
I
I
B"
B'
B'
B
B"
1 1 -1 -1 -1 -1 1 1
1
-1 -i
-1 1 -i
1 -1
-~}
sl
-I
-e -e'
-e'
-B'
-BO -8
-06"
I
I: .0 -.0 P-.0 -.0 8
-0 -I!
I -. I I
1 1
I: .•0 -.0 1 -I
1 -e
-E -Bo
I -I
-1
-I -I
-B
-eo
1 -I;.
.0
-e'·
-e
-.0
I -. I I 1 -1 I -I - I -I E* -1 -1 -e -1 -e· -1 1 -£* - 8 -I I - . -e· -1
•
1
-,e. I -.
- I -6
-1 -B-
-.0
-I
1
-1
-E
.0 •
.' • •
s*
e~
.xp(~i'G)
;rl-tyt; zl
R. (",1/)
(zt_yt;~
xy)
l'
-B
1 -1
-e* -8
s:
:'j • e'.) -I -I -e'· _BOj -.0 -B"I s. s.
-e
-1
-e·
(x, y)
• (R•• H.)
(z£~
1/1'.)
-B'
-I
1
-u s: -B'
1 0
(=, zy)
z
-8
-1
eO
e
(R•• R.)
-1 1
eO
CJh
I I I -I
X"+y";z" xl_y'; zy
0'° 0'
0'
-I
R,
1
o·
0
-I
(xz. yz)
S.
0'1>
.' .'0 .' .' .' .'0 .'• .' .' S: s: Co C, C, C: C:
-. E,. P-.0
E ,•
S:
s.
a.
x"+y';z' (xl_y'. xy)
z
-1
0 0'
'if•• E
B.
i -i
0 0°
E; E;
E .. A.
-1
g ... .' • g ." •.0 ." g • .' ."e' •.0 g ." • .0 I
E; E;
A. B. E lO
i -i 1
exp(21Ti/3)
R.
(R.., R.)
-EO
1
g g
•=
1
B.
A'
1
1
.0•
1 -1 -1 -1
CI
S:
A.
A'
'if••
S.
1 1
EO
C.
z x;y
1
1
1
Xi; y3; ZB; xy xz;yz
R. R..; R.
0'1>
g .0• •.0 1 1 1
E.
x"+y';
R, (x, y); (R.., R.)
0 0
1
A"
20'd
-1 1
2C.
(X"_y", xy); (xz, yz)
1 -1 -1
Bu
E'
Zl
1
1 -1 1 -1
A' x'+y';
2, 3. 4. 5. 6) O'h
1 -1 -1
1 1 1 1
Au
yz
z
E
E, E.
A. B.
=
y; R..
20'v
=
IV: Group8 ~"h (n j 'if. h E C.
~I.
A, A. B, B.
Zl3
Appendix 1 : Charecter Tabl..
1 -8·
:°1
-1
e
R. (R••
H.l
-.0I
.
_.oj :.}
= exp(2'1Ti/6)
:;rl-+yl; 2'
(.... 11·) (.'-11·, xy)
-I!
1 -. -1 -1 - I 1 -1 1 1 EO 1 eO -1
•
B
(x,
111
V: GrOUP6 !i:I..h (110 = 2,3,4,5,6) r h E C,(z) CIty) C,(x)
A.
1 1 -1 -1 1 1 -1 -1
1 1 1 1 1 1 1 1
B to B a•
A" BI"
B,.. B."
1 -1 1 -1 1 -1 1 -1
E
2C.
ac,
A'I 1 A' 1 E'• 2 A"I 1 A" 1 E'• 2
1 1 -1 1 1 -1
1 -1
1
0
2 -1 -1 -2
~.h
!i'41b
c.
1
,
Sl/Ob E
A;
A~
0
2a. 1
I I
-I
1 1 1
-1
I
-I -I I
-I
1
-2
0
0
I I I I -2
I -1 1 -1
I -I -I I
0
0
20:
2C,
2
1
A•• 1 B.. 1 B•• I E•• 2 E •• l! AlY 1 A •• 1 B ,• I
B•• 1 E•• 2 E•• l!
-1
0 -I -1 1
-I
1
-1 -1
-,
2 cos ex 2C08 2a I oae 2(1 2
1 1 -1 -1 1
-I 1 1 -1 -I 1 -1
1 1 -1
0 0
1 -1 0 0
0
a.
60. I -1
Sl/•• E 20. 2C. C. A ••
,
0 0
-1
-1
1
-2008241
-2 ooeoc
0
1
-I
-2
0 0
1 0 0
-1 -2 -2
1 1 1 1
2
1 1 -1 -1 1
I 1
1 1 1
-1
1 -I
-1 -2
-1 -1 -1 1 1 -1
-1 -I
1
I
-1
-1 -I I
,
-1 -1 1 1
2 -2
A." E"
1 1 2 1 1 2
1 1 -1 1 1 -1
~Od
E
25.
1 1 1 1 2 2 2
1 1 -1 -1 -yl2 0 -y2
zl_yl
BI
B,
'"II (zz, y.)
E1 E. E. ex = 72°
3a4
Jav
1
1 -1 -1
-1 1 -1
21
1
+y'; .'
R.
•
CR•• R..)
("" y.)
%'+11';
R.
Ii'
0 0
0 0
1
-I I
-1 1
I -I
0 0
0 0
(.,. II)
1
I I 2 coact 2 oos Sex 1 1
E •• 2 Sl/.. E
2S1 ..
B,
B.
E. E, E.
1 1 1
, 1
2 2 2 2
0
1
1 -1 -1
V·1 0
-I -V3
0 -2 0
2e: 1
1 2 coe 2(1 2 008 (X; 1 1 , COlI 2at 2:
-1
0
1 -1
x'-y'
x'+y'; z' R. (R., R.)
0
4ad
1 1 -1 -1 -y2 0 y2
1 1 1 1 -2 2 -2
1 -1 1 -1
1 -1 -1 1
0 0 0
0 0 0
0 0
I -I
-1
0 0
-2
2: 008 ex 2 oos 2ex.
-I -I -2:ooeoc -2: 008 Sa.
-2
2S:.
1
1
I I
1 -1 -1
C. 1 1
1 1
-1
1
0
-1
-V3 -2
-2
-1
1
-2 -1
0 2 0
2
0 -1
-1
-1
V3
2 -2
,
-,
z (x, y) (xI-y", xy)
60;
:r;s+y'i
c06lX
-1 -1 2«-Io"'cr;
R, (R•• Ho)
0 0
2«
008
2
a.- 724
Gad I -I
.2
-I 1
-2008
0 0
•
('" y)
Gad
1
1
-1 1 -1
-1
0 0 0 0 0
(=, yz)
(R.. R.)
1 1
I I
-I
,2;2+y'; z' R,
15••
2S: D 1 1 2 2
(xI-y'. xy); (=, yz)
z (x, y)
0
4C~
I -I
(=, yz)
(x, y); (R., R.)
C,
6C;
xy
z
-1 1
1 1 -1
1
xI+y'; z' R.
2S:
008 IX
1 1
0
1 1 -1 -1 -1 1
2C. 2S, 2C. 1 1 1
1 -1 -1 1 3ad
2C.
1 1 1 1
1 -1 1 -1
i 25. 1 1 2 -1 -1 -2
1 -1 0 1 -1
'C.
2ooeIX. 2 coo 2<x
E, ("..I/O) (:1:--1/', q)
,
A•• I E ,• 2
1f. (R.. R.)
I
I
Eu E •• 2
A. A.
1
-1
A •• A ••
A,.
(%t_y'.~)
-I
a.
A h
A.
(". r)
0
2 -1 -1 -1
,
COI«
-.2:6082cr;
!S. 2S,
2C. 3C~
Sl/", E
-2 008 a
1 -1 -1
-1
t
-1 -1
0 0
1
:2" OOIJ 2«
2<x
E
A to E.
1 1 1 1 -2
0
!i:lad
zl+y3 j c'
Ga.
2S:
2008 OC , COB
X
(",y)
1
3C; 3C;
-I
0
-1
I -I
I
0
1
0 0
,
1 -I
1
2
z Y
1
1
2 -1 -1 -2 -2
1
-I
(R., R.)
1
1 1 -1 -1
-1 I I
-1
,
R,
I
-2
I -I
-1
-1
-1 I -1 I
B,
Al
1 0
BI
E
(=, yz)
1 -1 -1
0
A.
= yz
1 1 -1 -1
1 1 1 1 2
Al
xy
(xI-y', xy)
1
1 -1 -1
1
-I
xl; y"; z'
R. R. R.
.....
15.
1 -1 1 -1
1 1
-1 1 -1
-, -I
1 -1 -1 1 -1 1 1 -1
AI"
z (R." R.)
1 1 1
1
1 1
0
a.
I
1 -1 1 -1 -1 1 -1 1
xI+y'; z'
1
1
'I; , E; , A; A; E; , E;
-1 -1
-1 1 0
1 1 -1 -1 -1 -1 1 1
R. (x, y)
2S,
1
A"
1 -1
1
-I B.. 1 -I
A•• 1 B lY 1 B... 1 E.
1 1 -1 -1 -1 1
1
I
11:.
3a..
2C;
1
0 1 1
25.
1
1 -1 0
1 1 1 1 -1 -1 -1 -1
1 -1 1
20,
1 1
,
1 -1 -1 1 1 -1 -1 1
2C;
E
Au A.. Bu I
ah
VI: GrO'Up6 !i:l nd (110 = 2,3,4,5,6) = r d E 25. C, 2C~ 2crd
~'d
i a(xy) a(xz) a(yz)
!i:l1h =
B I•
21ti
Appendix 1 : Cherel:tlr rebl.
Appendix 1 : Chereeter rebl.
ZI4
-I 1 0 0 0 0 0
%'+1/ 1 ;
.II
R.
•
(",II)
(R•• R ..)
(zl_W', sy) (zz, yz)
.II
(=. p)
(Zl_.'.q)
288
Appendix 1 : Character Telll..
= 4.6,8)
VII: Group" 9".. (n E CI S.
B
E
9".
A. E.
A" E.. 9".
A
1 1
{~
1 -1 i -i
E
C, C:
1
1 e
n
e*
e*
1
1
{~
e*
E
1 1 -1 -1
1
e 1
e* e
e
S.
C.
B
1 1
E,
,~
e e*
i -i -1 -1 -i
EI
-i -eo -e
-:}
i
Xl_YO; xy
(x, y); (Re • R.)
(xz, yz)
S.
•
1 -1 -eo -e
-i i B
eO
e
1
R,
e e*
:*}
(Re , R.)
-1
z
-e
(x, y)
-eO}
C. 1 1
-1 -1
S'
•
1 -1 -e _eo
1
1 -1 -1
-i
e* e
C:
= exp(27Ti/3)
(~_y", xy);
s'•
(xz, yz)
e = exp(27Ti/8)
1 1 R, x"+y"; z· 1 -1 z -i (x, y); :*} (R., R,,) i -1 (x'-yO, xy) -1 i (xz, yz) -i -eo
-u -B}
a.ntI (}b 3CI 6S.
E Tt, Ft To, F o
1 1 -1
3 3
0 0
1 1 2 -1 -1
()
E
8CI
A, AI E T I • F,
1 1 2 3
1 1 -1
T I , F,
3
0
-1
-1 1
3CI
6C.
6C~
1 -1
1 -1
0
1 1 2 -1
0
-1
E
8C,
1
1
1
1
A ..
1
1
-1
E.
2 3 3
1 -1
2
E.
1
TIM' Fl. 3
aG, eC,
0
0
1
-1
-1
1
60; 1 -1
•
(2zl -xl-y', xI_y') (R., R., R,) (x, y, z)
(2z2 -xl-y', XI_yO) (Re , R., R,); (x, y, z) (xy,
es,
30b
6S,
1 1
I I
1 -1
-I
-1
1 1
1 1
1 -1
2
0
0
-1 -2
-1 -1 1
0 0
-1 -1
1 -1
-1 1
-3 -3
0 0
-I 1 1 -1
3 3 -1
(xy, xz, yz)
~+yO+ZI
0 1 -1
-I
0
I I
~+yO+zl
0 0
• -I
TiM" ~'t. 3
1 -1
1
-1 1
~
I
6cJd
0
All T t •• F., T.l'. F I , A IO A IO
Zl7
()
1 1 2
At
A,
XO+y2; Zl
1
-1 -e -eo
s:
x"+y'; Zl
R. z
S'
1 1 1 -1 -1 -1
1 -1
{~ {~
1 -1
j
1 1
EI
VIII: Group Y d, Yd E 8CI
s:
9".
A
Appendix 1 : Cllerecur Tebl..
0 0
•
lla.
0
1 -1 -1 1
-I
-2
0
0
I
-1 1
I -I
-1 1
zt+Y'+a l
I -1
0
-I -I -I
=, yz)
(2a: I -Z'-1I 1• zl_yl) (Ra.R., R..)
1 -1 1
(ZY..... lI")
(~.
y,:)
288
AppllndiJl' : Character Talll..
IX: Groups'lJ"" .. aM~"'h E 2C(4)) 'lJ""" Al = L+ A, = L-
E 1 =TI E. = /l E. = l!I
1 1 2 2 2
.91oo~
E
1::1:;
1 I
Do,
I 2 2..".2<,6
II,
:E;:-
SC(~I
1 1 2..".~
1
:E;;
1
II.
2 200.'" 2 ..... 2<,6
A.
1
Bibliography
ooa.. 1 -1
1 1 2 cos 4> 2 COB 24> 2 cos 34>
0 0 0
... COD'v
i
1 ... -1
1 1
0 0
I I
1 ... -1
-1 -I
0 0
-I -2
z
x'+y';
R. (x, y); (R.., R~)
IS(~}
1 1 -2 cos 4-
2oo.~
-I -I
2 ••• .,. -2co.""
z·
(=, yz)
(x'-y', xy)
... roC; 1 ... - I 0
.xs+ys;: (R•• R.)
Q
... - I 1 0 0
:1-
R.
(%.11)
(n, yz) (zl_yl.%!f)
Bethe: SpliUifl{J of terma in crystals. Annalen der Physik 3, 133 (1929). Cotton: Chemical applicalitm8 of group theory (John Wiley). Ferraro and Ziomek: Introductory group tlteory (Plenum PrestI). Hall: Group theory aM aymmetry in chemi&try (McGraw-Hill). Hamermesh: Group theory (Addison-WeBley). Heine: Group tlteory in quantum mechanic8 (Pel'iamon PreBB). HochstraBBer: Molecular tupectB of aymmetry (Benjamin). Hollingsworth: Vectors, matricea, aM group theory (McGraw-Hill). Jaffo! and Orchin: Symmetry in chemi8try (John Wiley). Levine: Quantum chemi8try, Vola. I & II, (Allyn and Bacon). McWeeny: Symmetry-an introduction to group theory (Pergamon Press). Schonland: Molecular symmetry (Van Nostrand). Tinkha.m: Group theory aM quantum mechania (McGraw-Hill). Weyl: Thevry of groups aM quantum mechania (Dover Publications). Wigner: Group theory (Academic Preaa).
Answers to Selected Problems
[Answers to -selecte~ problems]
(c)
Chapter 2 2.2 2_3
(f)
CH 4 and SF6
(c)
Chapter 3 (a)
~.
(b)
1!13v
(c)
~h
(d)
~
(e)
~h
(f)
~
(g)
~.
(a)
~d
(b)
~2v
(c)
~
4.12
0
b- I
0
a-I
0
0
5
-3
1
-3
2
-I
0
0
(d)
~h
(e)
~
(c)
AI = I;
c, = O. c2 = O. C3 = 1
~=cos9+isina;
c,= Im,C2= im,C3=O
A3=cos9-isina;
c, = 1m. ~ = -i IYl. c3 = 0
A,=-15;
c 1 = (4 - i)/..{306. C2 = -17/-./306
~=3;
cl
o
D(E)
o
o
o
o
o
o
o
o
II
il/..{is, ~ = 11m
o
(a-ib)/D
-(c + id) /D
-roc + id) /D
(a+ ib)/D
o
o o o
o o
o
Chapter 4
o
o
~
(a)
= (4 -
o
o
D(C.>
3.6
col
Chapter 5 5.2
3.3
0
trans-Co(enhCNH3 )l+ 4.10
3.1
0
o o o
291
Answers to Selected Problems
Answers to Selected Problems
292
5.4
(a)
All matrices are diagonal. the three diagonal elements are: R=E; R = C 2(Z);
I, I. I -I, -1,1
R =Ciy); R = C 2(,,);
-1,1.-1
I, -1,-1 -1, -1.-1
R=i; R = <J("z); R = <J(yz);
Chapter 7 7.2
D(C 2)
II
1. 1. -I I, -I. I -I, I, 1
R = <J("y);
0
D(C;\)
III D(C,J
pod = 1"2 Ell rill Ell
rE
f A2 = (fl + f 2 + £3 + f 4 )12
0
0 0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
-I
0
0
-1
0
0
-l
0
0
-1
0
0
0
0
-I
0
0
-l
0
0
0
0
0
0
-l
0
0
-I
0
-l
0
0
0
0
II
II I;
II
fBI = (fl - £2 + f 3 • f 4 )12
f~ = (f l - f 3)/..fi
D(C2t>l
f~ = (f2 - f~/..fi 7.4 7.6
pod = 3 1"1 Ell 1"2 Ell 2 rill Ell n Jl2 Ca)
For~4'
P3 belongs to r'2 PI and P2 to r E
For%.
PI belongs to rJl3.
D(C2'.)
0
0
-I
0
0
-1
0
0
-1
0
0
0
0
-I
0
0
-I
0
0
-I
0
0
0
II
0
0
0
-1
,I
P2 belongs to rJl2. P3 belongs 10
rIl lu
Chapter 8 8.1
Cc)
D(Ci:t>l rBl ~ rB l = 1"1 Ell 1"2 Ell rBl ~
rEo
rEo ~ rEo 8.2
(b)
= rEI Ell = 1"1 Ell
<J = B I&, BOa or E I &
rEo
I
rEo r'2 Ell
rEt
Ii
II
293
Answers to Selecled Problems
294
Chapter 11
Chapter 9 9.2
For
Answers 10 Selected Problems
2)..,. the five B z and eight E l modes are infra-red active; the six AI' nine Ez
and eight
~
11.1
modes are Raman active.
f'(5,2. ,,2) fy(5rl· ,,2) fz(5z 2 _,,2)
For~2v, the
three Al and two B 2 modes are infra-ted active; the three AI' one A 2 and twO B 2 modes are Raman acti ve.
fz(x2. y2) fxyz;
10.1
Ca)
}r
l
~t
Chapter 12 '1\(...) = 0.526(<1>, + .)/.J2 + 0.851(<1>2 + ",,)/.,[2
al.:
mixturesofs. dzzand '1''::1& (three non-degenerate MOs)
+ .)/.J2 - 0.526(<1>2 + <1>3)/.J2
"-2.:
~2s (one non-degenerate ligand MO)
'I',(bs) = 0.851(<1>1 - .J/.J2 + 0.526(z - <1>3)/.J2
blS:
mixtures of d,2_'" and'l'gls (two non-degenerate MO.)
'1'2("') = 0.851(<1>,
'I'.(bs ) = 0.526(<1>1 - .J/.J2 - 0.851(<1>2 - J)/.J2 EI = a + 1.61813
Ez = a - 0.61813 a + 0.61813
12.1
b 2.:
mixtures of d,y and ~2. (two non-degenerate MOs)
es :
mixtures of d,z and ~. (I) (two non-degenerate MOs) and degenerate with the", two mixtures of d,z and
E, =
E. = a - 1.61813 Cd)
z
f,(zZ _y2) fy(zZ _x2)
Chapter 10
}r
'1'1 Cav = -IIi5 (I + z
+ <1>3 + <1>. + <1>5)
'1'2 (e'i) = --fiE (I + Ct<1>2 + C:zJ + cz. + CI4>5) '1', (e'i) = --fiE (CI1
+ <1>2 + CI<1>3 + cz. + C24>,,)
'1'4 (ev = --fiE (4)1 + cz<1>2 + CI<1>3 + CI. + cz4>,,) '1'" (ev = --fiE (cz1 + <1>2 + Cz<1>3 + cI4 + Ct,,) CI = cos(2lf,15) and c2 = cos(4l
a + 213 Ez = a + 0.61813 E4 = a - 1.61813
EI =
'PEg (2)
"-2u:
mixtures ofpz and ~2u (two non-degenerate MO.)
b 2u :
~2u (one non-degenerate ligand MO)
mixtures of p,. 'I'~ (I) and
'l'R, (I) (three non-degenerate MOs) and
degenerate with the", three mixtures of Py. 'I'~ (2) and
'PE. (2).
295
Index al> milia oaloula.tioWl. 260. 262. 266.
Abelian group, 31, 32. acoidental degeneracy. 154. 156. adjoint of a matrix, 55, 59. algebra of symmetry operations. 16. alternating axio of symmetry, 13. asaooiative law, 10, 17, 2.5. atomio orbitala, olaesifiea.tion of, 221, 224, 246; tr~ormation properties, 221. atomio unit.a, 217. axial veotors, 181, 249. axis of synunetry. 2. basis for a representation, 84, 86, 90; d-orbital example. 9l!. basis funotion generating machine, 126. basis funotions, 103.
benzene. ground state oonfiguration of, 212. Bothe, 6. US. Biot.20. blockstruoture, 96, Ill, 171l, 177. l!1I. 214. Born-OppenheiJner approximation, 11l2.
C., 16. Cauoby.5. Cayley, 5. Celebrated Theorem, 143. 145. oentre ofoymmetry. 16. charaoter tables, 279; construotion of, 128, ISO. oharaoteristie equation, 56. oharaoters., 120; oomputer determination of, 131; of a representation, 120; orthogonality of, 122, 129. oharge.tranafer bands, 271.
Coulson, 221. crystal field splitting, 257. orystal field theory. 243, 260. decomposition rule, 124, 158, 160, 181, 191, 208, 213, 226. 227, 232. 247, 263. degeneracy, 88, 154; of hydrogeo.ic wave.. functioD8, 166. delooalization energy, 212. determina.ntal equation, 66, 68, 167, 202, 206.207, 210, 211l, 246, 260, 21l4. determ.i.nanta, lSO, 97; as representations, 97. determination of x"(R), 179. determination or iJTeduoible represen· tationa, 134. diagonal matrioea, 68. diagona.lization, 68. 139. diotionary order, llil5. dipole moments, 19. direot product, oharacters, 1.67" of two matrices, 166; reduotion of, 168, 169; representation of, 165,218, 264. distributive law. 10. d.orbitaJ.s .... buie of repl"88eD.t&tion. 92. E.16. effective magnetio moment, 273. eigenvalues, 88. eigenveotors, theorema, 63. electric> dipole moment, 187. electronic equation, 162. enan tiomers, 20. equivalent atoms. 4. equivalent representations. lOS, 106, 1IIl, 124. exohange operator, 202.
x"(R), 179.
oJasees, 82,67,121; oomplete, 146. ola8&i1l.cation of atomio orbitals. 221. 224. olassification of vibrational levels, 184. onfactorll,60. oombination leve18, 189, 192. oommutation, 10, 16,63,58, 161, 197. oomplete oIas-. 146. complete ...to, l!21. oonjugate complex of a matrix, 69. oonjugation, 31. oore Hamiltonian, 204. oorrela.tion diagramB. 260, 262. oorrelation energy. 198. Coulomb operator, 202. Coulombie integrai8, ~06.
Fermi resonance, 192. F iggis, 244. free ion states, 262. Frobenius, 6. function spaoe. 72, 86. fundamental frequenoi.... 172. fundamental levels, 171. Galois, Il. I'" repre"""tation, 172; reduc>tion of, 176. generating :machine for baaia function_, 126. gwad•• 132. Grvat Orthogonality Thea.....". 1I8, 138; proof. 141.
2U
Ind811
Indu
ground _tate oonfiguration, Cor benzene, 212; for trivinylmethyl radioal. 216. group, definition of, 2"'. 25; order, 31; properties of, 31. group tahle. 27; for "'.0.79; for .ymmetrio tripod,29. H&miltonian opers.tors, 88. 161, 1113, 197; oore, 204; oommutation with 011, 200. 218; invaris.n.oe of, 1151. harmonic force oonsta.nts, 16~. harmonic 08ci Uator, approximation, 1605; equa.tion, 170. Hartree-Fock, approximation, 198; equatioD.8, 200; orbitals, 222. Ha.rtre&-Fook-Roothaan equations. 201, 204. 2011. Hermite polynomialo. 171. Hermitian matorices. 66, 69, 64:. 66~ hole formaliom, 268. homomorphism, SO. 48, 97, 100. Huckel moleoule.r orbital method, 205; for benzene, 206; for trivinyhnethyl radical. 212. hybrid orbitals. 219, 221; for "..bonding systems, 2215; for v-bonding tJYBtems, 229; geometry of, 230; mathematioal form of. 234. hybridization. 221.
t, 16. identity element, 11. 25. identity matrix. 58. identity operation, II, 77. indistinguiohability of identioal partiol.... 199. infinite point groupo, 133. infra·red activity, 178. infrs.·red opectra, 186; of CH, and CH,D. 190. inverse element, 2-6. inverse of a matrix, 54, 56. 237. inverse opers.tion, 15, 17, 77. invereea of operations, 146. :irreduoible representations. 103.. 111, 118; determination of, 134; for
LCAO MO approximation, 201; alternative notation, 21 7; for octahedral oompounda, 244:. for II&I1dwicb compounds, 252; for tetrahedral compounds, 261. Lewis. 219.
LiB,6. ligand field theory, 244, 276. ligands, 243. linear equations, 63. lin independence, 86, 1114. lin opere.toro, 9, 100, lli4. looalized moleoular orbital tboory, 219, 220; relotionship with non.localized moleoulor orbital theory, 241. L-S ooupling, 227. magnetio moment, effective, 273. magnetio propert;ies of transition-metal oomplexes, 273. mas.·weighted diopl.aoement ooordin..tee, 1611. :IIl&trioes, 48; addition, 61; adjoint, 56, 69; algebra of, 61; assooiative law, 55; oommutation, 66, 138, 140; oonjugate oomplex, 69; determination of inverse, 61; diagono.l, 58; diagonalization, 58, 66, 67; direct produot of, 1116; distrib. utive la.w, 66; division, 54; eigenvalue equation, 55; eigenvalues, til); eigenvectors, 66; equality, l51; Hermitian, 68, 59, 64, U; identity, 58; inve1"88 ofJ 64,. 115.237; multiplication, 52; non-oingulo.r. 54; order of, 50; orthogonal, 61i real, 69; aquare J 49; square root oft Ill); 9ubtractioDt 61; symmetric, 69; trace, 66; tl'&D8POoe of, 115. 59; unitary, 117, 611. matrix representation, 72; for ~Ih, 78; for ~p , 78, 108; :£rom base veotorB J 82; from poaition vectors, 'IS. mirror image, 20. molooular orbitals, 197, 198; degeneracy, 216; for benzene, 206; for octah..dral oompound.8, 244; for B&D.dwioh oompounds, 252; for tetrahedral oom· pounds, 251; for trivinylmethyl radioal, 212; theory. 197. molecular vibratioIUl, 164. Mozart, 3. Mulliken'. notation, 131.
non.ero-mg rule, 244. normaJ. coordinates. 1M, 167; ol&88ification, 178, 182; for linear molecules, 184. norm..1 modea, 168. 175. norrnaJization. 56, 209. nuclear equation, 162. null matrix, 60.
Offenhs.rtz, 2«. opers.tor algebra, 8. oparator, kinetic-.energy. 162; potentialenergy, 1112. optioal activity, 20. orbital energies, 200. 202. order of a group, 31. orthogonal eigenvec-tors, 57, 60. orthogonal matrice., 61. orthogonal vectors, 119, 122, 149. orthogonality theorem, 118, 138, 141. orthonormal basis funotions. 87. orthonormal funotiona. 87. overtone levels, 189, 192:. Puteur, 20. Pauli exclusion prinoiple, 199. ",-eleotron a.ppro:rima.tion, 203. ",-electrons, 2mL plane of oymmetry, 2, 13. point groups, 24, 26; classification of, 35; detertnination of, 38, 46: infinite, 133. polarizability, 190, 195. position veotor8:1' 73. principal axis, 11. produot law, 9. projeotion operators, 126, 208, 213, 234. quenohing J orbita.l angular :momentum,
274.
Raman activity, 178, 190. Raman opectr.., 189; of CH" CH,D, 1110. real matrices, 69. Rearrangement Theorem, 29,40.116,142, 145. reducible representations, 103, 110. Ill, 123, 208, 226. reduction of a representation, 123, 125, 208. reflection operation, 13, 76. regular representation. 143. 144, 148. representation theory, 6. representatioIW. basis for, 84, 85 90; determ.in&nts 88, 97; di.reot produot, 165. 218, 264; ro, 172. reoonanoe integral, 206. rotational energy. 169.. rotational norm.a.l coordinates1 168, 178. rotation operation, 11, 74. rotation-reflection operation, 13, 77.. J
S •• I6. ooalor produot, 86, 90, 113. Schmidt orthogono.lization, 109, 113. Schoenflies notation, 35, 36.
ZII
SohrOdinger equa.tion, 161. 197. 222. Sohur'a leIIlID&8. 138. aeleotion rules. 151. 158, 188. self ooIlBilltent field method, 198. a~h 16.. ..,., 16. av,16. a-bonda. 225. a -electrons, 203. a-orbitals, 22L'L (1-" oeparability, 203. similarity transfonna.tions, ~7, 6li. 106. 109. 116. 120, 121, 173. Slater determinante, 199. Slater orbitalo, 222. .pace group., 26. opeotral properti"" of tr......ition·metal complexes, 271spectroscopio states, 276. .pherical harmonioo, 223. spin, 201, 202; multiplioity. 264: BElleotion rul.... 271. .s.pin-only fonnula, 274. square matrix, 49. square root of a matrix, 115. sum law, 9. Sylow. 6. .symbol list, xv. symmetric. matrix. 59. symmetry eleIDent.. 7. symmetry operations, 7, 10; algebra of. 15; for synu:netrio tripod. 27.. sym...metry operator, 10. .ymmetry orbitalo. 203. 206, 207, 210, 212,246. term oytnbolo, 258, 276, 278. tote1ly .ymmetrio repr"""ntation, 1119. trace of a matrix. 66. 120. trBnBfor:m.ation operators, 88, 89, 105; for the 'C, • point group, 92. trans!ormation properties of atomic orbital., 221. tranoform., 31. transition-metaJ complexes, 243. tranalational energy, 169. traIl8lational norm.al coordinates, 168, 178 .. transpose of &. ma.trix, liS, 59. trivinylmethyl radioal, ground "tate oon· figuration for, 216. ungM'ade, 132. unit matrix, 68.
unitary matrio.... 57, 60, 65, 67, 90. unitary operators, 9, 90.. unitary representatioD.8, 103, 108, 113, 1111, 118. unitary transformations, 57, 67. unpaired .pino, 274.
31G
Ind.
v8.l.enoe bond theory, 219. vanishing integral rule. IllS. 160, 211. V"", Vl&Ok. 243. vibrational energy, 169. vib.....tional oneJ'llY levela. 171; olaoBifi· cation of, 184. vibrational equation, HJ9, 170.
vibrational normaJ coordinate•• 178. vibratjons, 164. wavefunotions, 88. Weyl.6. Wigner, 4, 6.
