Group Theory and Chemistry David M. Bishop OepartmentofChem~uy
University of Ottawa
Dover Publications, Inc. New York ...
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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
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. HilJtori0./
~
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)
-
, 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
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
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, 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- (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
'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
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)
! 0 and since aN
V
=i -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 ~ ~( 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{I(I)11l.(2) ... ,,(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{Hl);(2) ...
~..(n.)}
a.J4>.(k)
:(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 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>. +;£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
= .-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.-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.iUlar 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 .(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(71)
1Il.(n)
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,,>.....
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 +.;~ +~+.;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 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.•.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 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(, + .)/.J2 + 0.851(2 + ",,)/.,[2
al.:
mixturesofs. dzzand '1''::1& (three non-degenerate MOs)
+ .)/.J2 - 0.526(2 + 3)/.J2
"-2.:
~2s (one non-degenerate ligand MO)
'I',(bs) = 0.851(1 - .J/.J2 + 0.526(z - 3)/.J2
blS:
mixtures of d,2_'" and'l'gls (two non-degenerate MO.)
'1'2("') = 0.851(,
'I'.(bs ) = 0.526(1 - .J/.J2 - 0.851(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
+ 3 + . + 5)
'1'2 (e'i) = --fiE (I + Ct2 + C:zJ + cz. + CI4>5) '1', (e'i) = --fiE (CI1
+ 2 + CI3 + cz. + C24>,,)
'1'4 (ev = --fiE (4)1 + cz2 + CI3 + CI. + cz4>,,) '1'" (ev = --fiE (cz1 + 2 + Cz3 + cI4 + Ct,,) CI = cos(2lf,15) and c2 = cos(4l 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