Group Theory in Quantum Mechanics: An Introduction to Its Present Usage 0486458784, 9780486458786

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nHuntQr

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Vo

t--Ie e II

GROUP THEORY IN QUANTUM MECHANICS An Introduction to

its Present Usage by

VOLKER HEINE University of Cambridge

PERGA~fON

PRESS

LONDON · OXFORD · NEW YORK · PARIS

1960

PERGAMON PRESS LTD. 4 ct 6 Fitzroy Square, London, W.I.

Headington HiU Hall, Oxford

PERGAMON PRESS INC. 122 EaBlo5tn, Stred, New ~or1c 22, N.Y. 1404 New York Avenue N.W., WOBhington 6, D.O. P.O. Ben; 47716, Lo8 ..4.ngdu, OaUJomta

PERGAMON PRESS S.A.R.L. 24 Rus au :Bcoles, Pan. V,

PERGAMON PRESS G.m.h.H. Koi8e1-BtrCJ88e 76, Fran1c!'Url-am.Main

Copyright

© 1960 VOWR HEmE

Library of Congress Card Number 59-10523

Set by Santype Ltd., 45-55 Brown Street, Salisbury Printed by J. W. Arrowsmith Ltd., Winterstoke Road, Bristol 3

CONTENTS FAGB

PREFACE

vii

ix

NOTATION

I. SYMMETRY TRANSFORMATIONS 1. 2. 3. 4. 5. 6.

The uses of symmetry properties Expressing symmetry operations mathematically Symmetry transformations of the Hamiltonian Groups of symmetry transformations Group representations Applications to quantum mechanics

. II. 7. 8. 9. 10. 11. 12. 13.

14. 15. 16. 17.

THE QUANTU:M THEORY OF A FREE ATOM

Some simple groups and·representations The irreducible representations of the full rotation group Reduction of tho product representation D(/) X DU/) Quantum mechanics of a free atom; orbital degeneracy Quantum mechanics of a free atom including spin The effect of the exclusion principle Calculating matrix elements and selection rules

III.

1 3 6 12 24 41

48 52 67

73 78 89 99

THE REPRESENTATIONS OF FINITE GROUPS

Group oharacters Product groups Point-groups The relationship between group theory a.nd tho Dirac method

113 125 128 143 .

IV. FURTHER ASPECTS OF THE THEORY OF FREE ATOMS AND IONS 18. 19. 20. 21.

Paramagnetic ions in crystalline fields Time-reversal and Kmmers' theorem Wigner and Racah coefficionts Hyperflne structure

v.

148 164

176 189

THE STRUCTURE AND VIBRATIONS OF MOLECULES

22. 23.

Valence bond orbitals and molecular orbitals Molecular vibrations 24. Infra-red and Ram~ spectra

206

229

245

·VI. SOLID STATE PHYSICS 25. 26. 27.

Brillouin zone theory of simple structures Further aspects of Brillouin zone theory Tensor properties of crystals

v

265 284304

vi

CONTE~~S

V1I. 28. 29. 30.

The isotopic spin fornudism Nuclear forces Reactions VIII.

31. 32. 33. 34.

NUCLEAR PHYSICS

PAGE

313 321 334

RELA'!"IVISTIC QU"ANTUM MECHANICS

The representations of the Lorentz groLi'" The Dirac equation Beta decay Positronium

351 363 384

397

APPENDICES Matrix algebra Hornomorphism and isomorphisln Theorems on vector spac0s and group representations Schur'slemrna Irreducible representations of Abelian groups Momenta and infinitesimal transformations The simple harmcnic oscillator The irreducible representations of the complete Lorentz group Table of Wigner coefficients (jj' mtn'IJjl) J. Notation for the thirty-two crystal point-groups K. Charaoter ta.bles for the crysta.l point.groups L. Character tables for the axial rotation group and derived groups

A. B. C. D. E. F. G. H. I.

LIST

O~.,

GENERAL

REF}1~RENCES,

WITH REVIEWS

404 410 412 418 420 422 424 428 432 446 448 455 457

BIBLIOGRAPHY

459

SUBJECT INDEX

464

PREFACE The object of this book is to introduee the t,hret:: main uses of

group theory in quantum mechanic3, which are: firstly, to label energy levels and the corresponding e.igenEt:,atk~;:t; secondly, to discuss qualitatively the splitting of energy levels at; one starts from Q·n approximate Hamiltonian and adds correction terlns; and thirdly, t.o aid in the evaluation of matrix elements of all ki.nds, and in particular to provide general selection rule",~ for: the non-zero ones. The theIne is to Sh01V how all this ig (1.chie'v'8d by oonsidering t.he symmetry properties of the Hamilwninn and the wa.y in which these 8'yrrlillptries are reflected in the "rav." funntiolls. In Chapter I the necessary mathenlatical concepts n.r(~ introoueed in as elementary and illustrative a manner as possible. with the llroofs of some of the fundamental theorems being relegid:ed to an appenctix. The three U8(lf~ of group theory above are Hl~6trated. in detail in Chapter II by a. fairly quick run through the theory of atomic energy levels and tran8itions. This topic is particula;rly suitable for illustrative purposes, fy:(~an8P most of the resu1fr are familiar from the usual vector Inodel of the atoln but are o. d.ved here in a rigorous and precise \,~ay. Also most of it, e.g. the ~f:tr.oduction of spin functions and the exclusion principle, is fundanh~,rttal to all the later more advanced topics. Chapter III is a rep0f..:1t,ory for the theory of group characters, the crystallographic pOllit .. gl'OUpS and nlinor pOUlts required in some of the later aI)plications. Thus, after selected readjngs from chapter III according to his field of interest., the reader is ready to jump immediately to any of the applications of the theory covered in later chapters, nau}~ly· further tDpics in the theory of atomic energy levels (Oh",pter I'T), t.he electronic structure and vibrations of molecules ((Jhapt:er V \. Kolid state phYRics (Chap~ ter VI), nuclear physics (Chapter \111,l, and relativistic quantum mechanics (Chapter VIII). The level of t.he text is that of a course for re"earch students in physics and chemistry, such as is now offered in many Universities. A previous course in quantum theory, ba~ed on a text such as Schiff Quantunt Mechanics, is assumed, but the !natrix algebra required is included as an appendix. In selt'cting the nlaterial for the applications in various branches of physics and chemistry in Chapters IV to 'VIII, I have restricted mysf'lf as far as possible to topics satisfying three criteria: (i) the topics should be simple vii

VBl

PREFACE

(:Lpp1ioa:t/i(~n.~

ths/t illu.strate basic principles, rather than compIica,ted ~~Jxf1:!J1ples df.~jigne,d. to overa\ve the rea,deL' with the power of group vhe.nry; (ii) the material should be intrinsically interesting and of title sort, that is suitable for inclnsion in a general course of advanced ,rluan~uJn mechanics; and (iii) topics nlust not involve too much p.pecialized background kno\vledge of particular branches of physics. !:?he .view adopted throughout is t.ha,t group theory is not just a specialized 1;'001 for solving a few of the more difficult and intricate problems in quantuDl tbe,)fY. In advanced quantum mechanics practiclA,H.y all general statenlents that can be made about a com .. plica/ted systeln depend o.n its synlmetry properties, and the use of group representations i:~ just a systematic, unified way of thinklllg about and exploiting Ji..;hese s~(anJnetries. For this reason I have not heE-dtaJted to include gj e~.p::.t:· results for vvhich one could easily produce ad hoc l}roofs froI1l fiest, pri.nciples: indeed. it must always remain true that the use of group theory could be circumvented by detailed aJ.gebraic considerations nn ahnost all occasions. However, the au.tJio~~ js convinced that. tIle essential ideas of group theory are 8ldlicieutly simple to make the time spent on acquiring this way of thio1.i.ng ,veIl worfJJ ~Yhile. A series of ex.D,mplei~ is appended to each section. Some of these are 8.in.l~ple drill in the concepts introduced in the section; others, pal'tieul~~.rl,y in later ohapters, indicate extensions of the theory and fw.,ther applic3,tions. rrhose marked with an asterisk are more difficult or requh'~ rtdditiortal reading, and are often suitable as topics for revie,\' eSSt~.ys (a,has t~rm papers). \iVith th.e th.ree; criteria for selection mentioned above, it has of conr~e been quite h~lPDB8ible to do real justice to any of the a,pplieationf!. to variou:1 branches of physics and chemistry that are t •.:H~~~bed on in Chap{:Pfs IV to VIII. This appears to me wlavoidable becauBe of the amount of background knowledge required for many applications. It nlerely highlights the fact that in each of these specialized subjects there is a need for a monograph ",-hieh uses group theory from the b~ginning as naturally and as freely as the Schrodinger equation itself. In this field the chenlists have already led the way,t and the author hopes that t~e present book may hasten the day "then the sanle applies in physics by providing a convenient basic reference text. It is a pleasure to acknowledge my indebtedness to Professor B. L. Van der Waerden \vhose elegant book first inspired nly interest in this subject. Also I am very ,grateful to Dr. S. F. Boys, l

t See Eyring, WaJl,er and Kimball (1944) Quanturn Ghemi8try; and Wilson, Deoius and Cross (1955) Jf olecular Vibrations.

ix

PREFACE

Dr. G. Chew, Dr. R. Karplus, Dr. M. A. Ruderman, Dr. M. TInkham and Mr. D. Twose, who have either patientl),- helped me to understand aspects of their special subject, or have read parts of ,the marIUscript and made helpful commentso I alll indebted to Mrs. M. Rogers and Mrs. M. l\'Iiller for undertaking the typing of the Dlanuscript, and to M".r. J. G. Collins SIld .l\fr. D. A. Goodings who have generously helped with the correction of proofs. Dr. E. R. (~ob.en h3-5 kindly allowed the reproduction of his tables of \Vigner coefficient.s, and D. Van Nostrand Co. similarly a figure . Cambridge,

v.

~Jngland,

HEINE

NOTATION Note: e is taken as the charge on the proton: all angulf1r moment·um operators such as 1.., == (Lx) L y , L z} hnve the dilnensions of angular Inomentum and thus contain a factor n, (except in § 18), whereas" the quantuIIl nunlbers L, jlfL, etc., are of course pure numbers .

Chapter 1

SYMME'fRY TRANSFORMATIONS IN QUANTUM MECHANICS 1. The lJ"ses of Synlmetry Pl'operties Although this book has been titled ('Introduction to the Present ·Use of Group Theory in Quantum Mechanics" in acoordti,nce with customary uRage, a rath.er more descrlI)tive title 'vo~~ld h.!lve been "The Consequen(~cF\ of Symmetry in QU8utnnl JYlecIJani·'",ij1 rl--le fact that these sytllllietry properties D)nn \vhat Ir:H~t..h~rnat..i,::jancl ha.ve terrned "groups" lS re~nji incidental from a J?hy~i~;dst~& pO'int of view, th.ough it is vita.l to t)>1e mathemati.cal iGl'rn of ~,l ~"theory, Tt.h~ ir. fact the sy7nnu:;trie:.~ of quantum rne(~hani(~r-.d SYBn:;tOLt thftt \ve shaH he lllterested in, . J '11 ijstrate IJl . a pre1·Immar~r · BU.np1e eXfiJmp,es L 1. fh) to11li\V1J.1g~nrec way ~"hat is mt~aTtt by symmetry p1'operti~& >J.nd \lvhat their ID.ain consequences are. (i) It can be shown that the ,\Vl1ve function~ ¥/(rv T 2 ) (without spin) of a helitnn. atoin a.re of tV'!O type8~ ~)~~ mraetri~ and anti· symmetric, according to whether Y

•

"0

'"

I. "

•

or vlhere r! and r 2 are the position vectors of the t\VO electrons (Schiff 1955, p. 234). rrhe corresponding states of the atofll are aJAo referred

to as symnletric and 8nti .. symmetrie. Thus the eige!lfuncti{)l1[; turn out to have ,vell defined symmetry properties v-rhieh ean there" fore bp, used in elassifying and distinguishing ;;r.H the different eigenstates. (U) l"here are three 2[1 \vave functioni~ for a hydrugr-n atnln~ t/J(2px) =-= xJ(r), ,,,,here f(r) is a particular fUHction of r = l rl only (Schiff 195!-"

p. 85). No\v in a free atom there 3ire no special directions and 'Vb can choose and lahel the x·) y- and z-axes as we please, so that the three functions (1.1) must all correspond ·to the same energy level. If, however, W'6 apply a magnetic fi(,·id in some particular direction. the argument no long0r holrl.s, so t.hat \ve may expect the energy level to be ~,pHt into several different levels, up to tbree in number, 1

2

GROUP THEORY IN QUANTUM MECILA.NICS

In this kind of way the symmetry properties of the eigenfunctions ~an determine the degeneraey of an energy level, and how such 8, degenerate level may split a·s a result of some additional perturbation. (ill) The probability t·hE~t, the outer electroll of a sodium atom jumps from the state .pi to t.he state t#2 with the ernission of radiation polarized in the x-direction is proportional to the square of ".'0

M

00

GO

J J J"1* "2 dx dy dz

=

X

(1.2)

-OCi -00 -00

(Schiff 1955, p. 253). If the two states are the 48 and 3s ones, if;l and .p2 are functions of r only. To calculate M in this case, we make the change of variable x' = -x in (1.2) and obtain j f = -M, i.e. M(4s, 38) = O. This trallsition probability is therefore determined purely by symmetty. The situation is rather different when the transition proba,bilit~l is not zero. Suppose ifJl and ¢12 are the 4p:t and 38 wave functions X/l(lt) and f2(r). 'Ithen (1.2) beconles co

M(4pxJ 3s)

=

ro

00

J J JA*(r) x f2(r) dx dy dz. 2

.By ulaking the change of variable x' ClrtA

be replaced by

y2

or similarly by ;i! Z ~n the j"unciion 1

6

GROUP THEORY IN QUANTUM MECHANICS

and eh'U8 expreas f in terms of X, Y, Z. This results in a function of X, Y, Z which in general diIJplays a different functional form !rorn f(x, y, z). For instance applying R(tX, z) to t4e function (x - y)2, 'w'e obtain . (x - y)t

= [(X cos a; = [X(cos tX -

Y sin tX) - (X sin C( + Y cos sin tX) - Y(cos tX + sin tX)]2,

~)]2

(2.3)

which is a different function. of X. Y, Z. Similarly we can apply a transformation to each side of an equation. For instance the equation

!

(2.4)

(x - y)2 = 2(x - y)

beoomest (cos

IX

a~ -

sin

IX

=

O~T)[X(COS 2[X(cos

tX -

IX-

sin

sin .:t) --- Y(cos

ttl ,---

JT(COS ex

IX

+ sin

1%)]2

+ sin «)],

(2.5)

which is still a correct equation as ca,n easily be verified. PROBLEMs

2.1 Apply the transfonnation R(rx, z) (equation (2.2») to each

+

of the following functions: (a) exp x; (by (x iy)2; (c) x 2 -t- y2 + Z2; (d) xl(r), yf(r), zJ (-r). 2.2 Write down the linear transformation that corresponds t,O a rotation of tX about the y-axis, and apply it to each of the functions of problem 2.1. 2.3 The Schrodinger equation for a simple harmonie oseillator of frequency w is

(- :~ :;: + tmw2X2)~(X) =

E",(x).

where t/J(x) is an eigenfunction belonging to the enerffjT value E. By operating on the equation -W~HJl the tra.nsformation l' ..-= --X, show that {I( --- x) is aJso an eigenfunction belonging to t.he sa[':~e energy level and so are ¥-,(x) + if;{ -x) and if;(x) - ¢;( ··_-x).

3. Symlnctry Transformations of the Hamiltonian \V(~

the

shn]l

'fJ.O\Y c~pply

tirne.,~ndijponder,t

linear transforrnations like R(a, z}' (2.2) to Sehrodinge:r t~quation (3.1 ) ax is

gho~"n

in

,~tny

elementary calculus

te}~t,.

7

SYMMETRY TRA.NSFORMATIONS

where .Tt' is the Hamiltonian operator and E the energy value belonging to the eigenfun\.t~ion tfo. It is convenient to consider first the effect of a transformation on the Hamiltonian :Tf. The Hamiltonian for an atom with 11, electrons, considering the nucleus as fixed and omitting Spitl dependent terms, is (Schiff 1955, p. 284)

(3.2) where m is the mass of an electron, e the charge on a proton, and 2

82 ()2 4.-_+2

0 V,,2 _ _ Ii

-

OXl

I

Oy,2

(3.3)

&,2 1

(a.4a)

If we apply the transformation R(rx, z) (2.2) to the co-ordinates (XI, Yt, z,) of each of the n electrons, we have r,2

= Xt 2 + yl2 + Z~2 (X, cos C( = XI! + 1",2

:=

Y, sin «)1

+ (Xc sin. + C(

Y, cos cx)2

+

Z,2

+ Zl~.

Similarly r'1 2 = (Xi - XJ)2

+ (Y( --

Yj)2 -t- (Z, - Zj)2,

(3.4b)

and it can easily be shown thatt 02

62

02

02

-;-a + -!l..c/Yi..•..2 + v~ ~.2 = ;X 2 fiX, u (

(32

82

+ 7~y2 + '~r;~-;;. (, I. v.u,'"

(3J)

Thus 8ubstitut,ing these relations into (3*2), we see that the Jlarnil~ toman has precisely the Sftme form ~dlcn ex~n'essed in t.erms of t.he (X1' l?t, Zi) co-ordinates as in term8 of t.he (l"i, YI" Zt) ocj .. ord~tes, i.e. (3.6)

hy gaJyjn~ that the transformation R(a., z) leaves ,;!P(3.2) unchu,llged, cr I\((I., z) leaves ~ invariant, or.?R is invariant 'U/i'!..der R(a, zj, or li(r.x . z) is a symm.etry transformation of 7t'. A sy'mtneJry tran·~for7natio/n (if a Jlamiltonian is defined as a linear

This is

e:x.pr(;~~~d

t The t1"an.sforrnstion of difreeentiu.1 operators is discussed in any elementary ceJculns text.

8

GROUP THEORY IN QUANTUM MECHANICS

traM/ormation of co-ordinates which leaves that Ilamiltonian int'ar'iunt in the 8ense of equation (3.6). The reason for applying linear transformations like R(a., z) (2.2) to & Hamiltonian noW' becomes a little clearer. We have seen that R(a., z) leaves the Hamiltonian (3.2) invariant. Howeyer, R(a., z) applied to the eigenfunctions of the Hamiltonian docs not in general leave them invariant.. r.JOnsider for insta.noo the 21) 'vav,:.': functions for a hydrogen atom (example (ii) of §l). R(a., z) applied to x/(r) gives (X cos (X - Y sin r.t) j(R) l\'hich haa a different functional form. In particular for IX = 90 0 we obtain - .. Yf(R) so that ll!(fX, ~;) has changed one eigenfunction into another. More generally con.. sider a Schrodinger equation Jf7(Xt, ?/I, Zt) .pl(Xi! Yi) z,) = EtPl(Xf, !fl., Z,),

(3.7)

Applying nny symmetry trallSformation 1.1 l\1C obtain

,J'f(Xt, Y!, Zt)

tf;~(X~~

Jr{, Zd

=

E.p2(X f , J7'"

Z1)~

\vhere YJ 2 in general he.s a different functional form from

(3~8) VJ1'

'rhus

~~(XiJ

Y t , Zt) is an eigenfwlction of £~(X(, Y,: Z,), but since £;(Xt, }"t, Zl) and Jt'(Xj, y,~ Zt) have tJle same (orIll, we can also 8ay from (3.8) that ¢2(Xl, '!If-t Zl) is an eigenfUftlctian.. ()f eYC'{Xil YI, z~) a.rut belongs to the sante eigenvalue E as An ~,}t.ernative Inet.hod of vlording this argurnent is tlJ say that sinoe (3.8) is a differentia} eq1l8,~ tion in terms of thE: varialJles Xt~ Yt't Z" "ie can replace X(.~ X"'i, Z; hy XI, y" Zl or any other Ret of f~ymbols throughout without upsetting the validity of the equation. "rhus (3.8) becomes

+1·

~(Xf'

y", z,) tP2

(Xt~ !It ,Zi) :.:~> EtP2(Xt,

y" Zj),

(3.9)

which is just our previous conclusion expressed in symbols. Thus we. see that tke aymrneiry tra~~lo-rmation,s of a Harniltoni.fllz, can be

used to relaie the different eigen;fu/lwtions of one energy le?,el to one aTloth~r and hence to label thern. and to discuss the degree of degeneraey of the energy level. Before \fe cain pursue this further (§ 6), we InU&t dir;cuss in grentei~ ,3etail the SYlTIJ11etry t,ransformations of Ifauliltoruans {§§ :J and 4/~ ,'}nd their ptTect on '\~tfiVe fllnctior:.s (§ 5). l'he Ha1uiltonia n (it2) hH,~, two ot.her type8 of sylnnletl'Y transforp.J.c.ttion besides the rota,tion R. The transforrnation

r

(Xl' Yl' Zl) =

(X"

I (X2' Y2' Z2; ~-::: (Xl~

Y g•

Z-;;-----

J'l' Zl)

l

I;--.____ (Xi, Yt, Zt) = (X i, L, Z;), 'i=~ 3, ± .. . ?t. JI . ____ . _. _____ ___________ ~__.

(3,10)

9

SYMMETRY TRANSFORMATIONS

is called the interchange or permutation of the co-ordinates 1 and 2, a.nd is a symmetry transformation of (3.2,) as is obvious by inspection. Similarly any permutation of the co-ordinates Xi, y.", Zt, i = 1 to n, is a symmetry transformation. The other synlmetry transformation is the inversion trruudormation

n

en:

Xi

=

--X..

Yt

= -Y"

Zj

= -Zt

for all

....--.

i.]

(3.11)

p;-

This can be cQrrlbined with the rotations. 'An ordinary rotation such as (2.2) is called a proper l~otation, and the combination of a proper rotation with the inversion n is called an i1r~proper rotation.. As a particular example of an improper rotation, we have IIR(180°, x) which is just tIle reflectio:n mx in the mirror plane x = 0, i.e.

~:

(X" y"

Zj)

= (-Xt,

Y.~ Z,)

for aU i.

I

(3.12)

It can easily be verified that all inlproper rotations, as well as proper ones, leave the Hamiltonian (3.2) invariant. However, there are many simple and important transformations that are not symmetry transformations of (3.2), for lllstance the transformation to cylindrical polar co-ordinates (3.13)

This transformation is in an:r case not a linear one because it illvolv~ products of R, with trigonometric functions of Also V,I becomes

e,.

1.

a (' t a)

B, aBc R oBt !

1

8

2

8

2

+ R,'!. 08,2 + OZ.I'

(3.14)

which is not identical in fornl with (3.3)$ so that (3.13) is Mt a 8ymmdry transformation. Of course we may wish to express the Hamiltonian (3.2) in terms of cylindrical polar co-ordinates for some problem, but in the future we shall refer to such a transformation as a change to polar co-ordinates, so as to avoid confusion with symmetry transformations which we will be considering 80 much that it will be convenient to refer to the latter simply as transformations. We must now indicate briefly what the symmetry transformations are for the Hamiltonians of physical systems besides free atoms and ions which we have been considering so far. An atom has complete spherical symmetry, i.e. it is invariant to any rotation about any axis (cf. problem 3.7), so that it has a higher degree of symmetry than molecules and crystal lattices which are usually only invariant

10

GROUP THEORY IN QUANTUM MECHANIOS

to certain rotations about certain axes (cf. problems 3.4 and 3.5). Thus the latter have some of the symmetry transformations of the atom, but not any radically new ones except for the translational symmetry of a crystal lattice. We ha.ve therefore already mentioned in connection with (3.2) almost all the types of symmetry transformation which we shall discuss. To sum up, the form of a Hamiltonian remains unchanged by certain linear transformations which are caJIed symmetry transformations of the Hamiltonian. Symmetry transformations in general change the eigenfunctions of one energy level into one another. PBoBLEMS

3.1 Show that the following co-ordinate changes are not symmetry transformations of the Hamiltonian (3.2).

= (2X" 2YI, 2ZI), i = 1 to n. (b) (Xt, Yl' %1) = (--Xl' _}7l' -. Zl)' (Xt, Yi, z,) = (Xi, Y" Z,), i = 2 to n. (c) Xi = exp X(, y, = exp Y" z, = exp Zl, (a) (Xl, '!iI, z,)

i = 1 to n. (d) Xl' YI' Zt given in terms of Xl' Y I , Zl by equation (2.2), (Xi, Yt, Zi) = (X(, }"'l, Zt), i = 2 to n. (e) XI = R, sin 8, cos Jse a Hanultonian ~ is invariant under eaell of two sylnIn.etry t~-~:nlsformations F and J..'1. We shall first sho\v that the combinetl transformation SF (first F, then S second) is also a Elvm,n~et:iY transformation. Let the co-ordinates Xl' Yl~ Zl' 2.'i·_~ ?!~ •• Zn of ihe Hamiltonian be written for convenience Ql' Q2' . . , q~ln., and let. .F be the transformation

the

.,

'\.

'

;., ~

...... ' I

T., \ ...15 - I\

when -;n~itten in terms of the summation convention (a.'ppe!ldix t),ud S be the transformation

:Now the transformation SF means to substitute first for-

~(i),

tr. ~

l'

SYMMETRY TRANSFORMATIONS

19

in terms of the Q, using (4.15) and then to substitute for the Qt further in terms of some new variabl~ VI. where

Q, =

8,jvjo

(4.17)

Since F and S are both symmetry transformations, 3t'(qt)

== JIC(Qi) = Jt'( VI),

(4.18)

that the composite transformation SF from the q, direct to the Vi is also a symmetry tral1~formation. Thus the symmetry transformations satisfy the group requirements (a) and (b) above. We can indeed write down the transfol·mation SF explicitly by eliminate ing the Q, from (4.15) and (4.17), i.e. SF is 80

(4.19) Further we always have the identity tra'iUljorrnation qf == Q"

i = 1 to 3n,

(4.20)

having the property (4.8) of the unit element E, which verifies (0). As regards (d), if we substitute for the q, in the initial. Hamiltonian in terms of the Q, using F (4.15) and obtain jf'(Q,), then we can get back to ;tf'(q,) by solvillg (4.15) for the Q, and substituting into ~(Qd. But this is just applying the transformation F-l (4.21) which undoes the effect of F, and this is therefore also a symmetry transformation. In (4.21) the Q, are now the initial variables and the q1 the new ones, and F-l is the inverse of the matrix F'1- To make the argument quite rigorous, we 110te that all the transformations in which we are interested are unitary (cf. appendix A, problem A.9), whence IFI =1= 0 and (4.15) can actually be inverted to give (4.21). This verifies (d), and (e) can easily be verified by writing out the transformation TSF

q"

= F"kSldTzjQj

in full without using the summation convention and noting thaLJ d~ not matter where the brackets of (4.10) are inserted. This proves the theorem. It is now possible to give a precise meaning to the expression "the s:ymmetry properties of a Hamiltonian'·' which has been used in a descriptive way up till now. The symmRJry propertie& of a HamiUonian consist of the group of all 81Jm m.etry transformations oj the HamiUonian. We shall now investigate the group of symmetry transformations of the Hamiltonian (4.12) in greater detail. Out of the six elements

20

GROUP THEORY IN QUANTUM MECHANICS

(4.13), the elements E', A' and B' form a group in themRelves a.s can easily be seen from the first three rows and columns of Table 1. These elements chosen from the Ligger group (4.13) are said to foml a 8 uhgroup of the larger group. Another subgroup of (4.13) is the group (E', K'), another one (E', L') etc. Similarly, (4.13) does not include all the possible symnletry transformations of the Hamiltonian, but is a subgroup of the group of all its sylnmetry transformations. ~"or instance, a synlmetry traruiformation not included in (4.13) is the reflection 1

(x, y, z) == (-X, Y, Z)

in the plane x = 0, i.e. in the plane kOz (Fig. 3). Further sylnmetry trllnsformatioIl2 are the reflections in the planes lOz, mOz and z == 0" and combinations of these ,vith the rotations (4.13). 'fhese all together forra ,the group of 12 elements called the point-group 6tn2 of tran.sforD1.~tiollB, and they would appear to be all the sym .. metry transforIll:-t.tiol18 that the IIamiltonian (4.12) has. Incidentally, there is no certaul method of ensuring that one has found all the symmetry transformations of a given Hamiltonian instead of just a subgroup: one can only try all the transformations one can think of. Some of the symmetry transformations may not be at all obvious, one case being the symmetry property.of the pure Coulomb l/r field iIl a hydrogen atom which gives rise to the degeneracy of all levels with the same principal quantum number n· irrespective of their angular momentum quantum number l, unlike the more general situation in an alkali atom (Schiff 1955, p. 86; Fock 1935). Thus some of the more subtle symmetry transformations of certain systems have only been discovered relatively recently (Jauch and Rohrlich 1955, p. 143; Baker 1956; problem 24.11). Hwe now consider n electrons in the field of the three protons or of three identical charges of any magnitude similarly arranged, the Hamiltonian for this svstem would have the transformations 6m2 applied to each set (Xj, y" Zt), i = 1 to 1l, as symmetry transformations. It also hn,s the n! pernlutatiol1 transformations of the n variables XI etc., and all combinatioIl8 between the perlnutations and tile point-group 6m2 transformations. Thus the group of all s:ymmetry transfornlations would have a large nunlber (12n!) of elements, but it would be a simple cOlllbination of the groups 6m2 and Pn. ,,I

lsomorphum It was shown above that the elements of the group (4.1), say the group ~, and t.hose of the group (4.13), 05' say, both multiply in

SY~Il\rETRY

TRANS:FORl\IATIONS

21

the sarne way according to Table 1. 'J'his relationship bet","een Q; and (B' is called an isornorphi&rn and can be described by saying that the elements E, A, B, ~ .. of ® can be paired off ,vith the elements E', A', B', . " . of (5' ~uch that the relationships between E, ",4, B, ... a,s regards multiplication are in ever};'" way the sarne as the relationships between E') ...4.', B' ~ ~ . . . Actually it requires more care to define isomorphism precisely and to distinguish it from the related concept of homomorphism (of. appendix B), but the above description is sufficient for the present considerations. rrha.t 2' · • • cf>n. We caD. constrnct drfi orthogonal set ' J' • . • 4>' ft as follows. Put CP'l = 4>1' then '", is orthogonal to 4) 1: Then put 4>'3 == 4'3 'l + b~lP''& with bl and ba such that r;'s is orthogonal to rp'1 a.nd ¢'f;3 etc~ '.rhe fUl'H:tions 4>'1' 4>'2' " .• ""n can then also be nornl~lUzed~ Problems 5.8 to 5.13 are based 011 the above concepts. t

TrtJR8forrruJtio""ii in ff, t,'Jector space We shall be interoiSted 8Jmost exclusively in vector spa~~cE: j,~ {t, = ej1>b for ~l'4> = Cj T tfol

= CjDt1 (T)cp1

and thus Tcp belongs to !t. Thus ~ny traI,;~'~" ;TQfi-tio:n of ("55 turns every vector of R into another vector also tH;~,,)Lig:ing to R, and "\'Ve say that the space !t is invariant under the grov;p cf, of transformations. If n is a one dimensional space (4)1)' the vect{1r 4>1 is said to be an invariant vector . This implies that TCPl = f.(~C·H{Sl but the constant. c(T) is not necessarily unity. There is another way of looking at transJ~)r1nations of functions ITl the vector space n. which we sholl mention briefly but have no occa.siOIl to use except in §§ 20 and 32. (:~;:~r~fdder a!ly function rP ==- Cj'j given by' (5,7). t~ro\V the cP'j C8~n be used as new base vectors in the spa,ce~ ii·nft ~he untransformed ... ,• J.. b e express~..u . ...1 as a. 1·Inear eOllh.Hna 1··f 1 functIon 'fJ can tl0l~. ()./ rJle:rH \"f1)~'~! ooeffi Clent.S C j, I.e.
as shown explicitly by (5.1 f)L and conversely the transformation properties uniquely determine the (':oofiicients 1->'ij as can easii,Y be proved in the rnanner of probl~nl D.2. \\Te therefore have symbolically: tranf~fornlation

I

propertieR of the ~jt).)

J

~-~

102

GOOUP '.rREOB,Y IN qT.JAl'."'TUM DlBOlIANIOS

Similarly if t.he t/;'It; ~Jein.g ltstYl do n(~t· tt?~n8form in the standard WQ,y, then this can be tk.ken iut.o 8&~Count in the same manner. This gives the first extt{}nBion of the funtie.,.mental theorem: IIA. If the tPj{A) and ~t(S4") ltave not been, f;,lw..c:en to tra'n8forrn in tke st.(J/ftdard way bu;t i'lt .,01M, other m(J/n~ner, thR:n I A 8till holda ~ lIB. A180 the M1,j(p.r) are rorn,pletily determine-tl as regard8 their de.pende·nce on i and j by th.e transformation prope7iie.s oj tl"e !Pj (A) and tP,(J.Lf') , except Jor a oo'nlilant factor aAr • A further ext-enaion ia obtained by" taklllg llilear combinations

41,

= 22 p(jl-' 4>~(~) i IJ.

of 4>'8 from f3cveral irreducible representations. 'rhese axpanded in terms oJ-the Vis;

4>1

== 2122 M,l' p r) l/J, ~ f ~,* r«~~dTJ 8

(13.9)

104

GROUF THEORY IN QUANTUM MEOHANIOS

1"1 ==;

.-

~,,/!

2 (Xn

-I- iYn),

r_ 1

=

v! L (Xn -- iYn), n

11.

ro --

~z n,

(13.10)

~

n

and ,pi an.d ifJi nrc the final and initial states'. rfhe summatiOIL'i are over all electrons in the atom. Also (X = 0 applie..'J to radiation linea,rly polarized along tIle z-direction, and tJ., = 1, -1 give the IJroba .. bilities for rad.iation circularly polarized about the z~axis. In the Zeelnan effect (problem 13.6) the linearly and circularly polarized radiaJ,ion, a,re 1mown as the 7T and u components respectivel:r. It is no'w siulple to write down selection rules stating ,v'hen coeffioients ( CT1MJll rex

fJ2M J2 >oc (lJad.fJIIJl.ZJ1Jl).

(13.15)

as follows from (9.S) by the same argument.

Although (13.15) determines the relative values of the matrix elements complet,ely, we shall now calculat,e the ones for AJ = 0 transitions (13.12) in an alternative way purely to illustrate the use of the fundalnantal theorem. The .pi, »Pi are now two sets of functions ~Ml and « = .'EC(IXifi, be n mutually orthogonal linear combinations. Further let Mi'h M'tlfJ be matrix elements of any operator with respect to the tPl,: CPa;- Then in matrix notation M' = 0* MO. If .pt, CPtl are normalized Jt/J, *~, d T = JCPt!* about it, and [Rot(S)]-l moves it back to its original position ;2' Thus all rotations by one angle cP about diflerem axe8 belong to the same class of the full 'rotation group. By tonsidering Rot{cfo, z)um = etmq,um, we obtain for the characters of the irreducible representation DU) of the full rotation group

= mI; j

X(t/J)

+ t)rfo

sin (j exp(tm4) = sin i .

t/J'

(14.4)

The spherical harmonics YZ m transform according to D(U under the full rotation group, and therefore they also form the basis of a representation D of some finite point-group. Now the matrices of D are just the appropriate matrices taken from D(l), so that the charaoters of D are given by (14.4) with j =~. The representation can no~ be reduced by (14.2). Consider for instance the spherical hannonics with l = 2 transforming according to a representation D under the group 32. For A and _K, ~ is -1200 and 1800 respectively and X(E) = 5, X(A) = -1, X(K) = 1, whence froIn (14.2) and Table 8,

D

= r +r +.1.

Thus if we have the character table of some group-and this can usually be looked up, for instance in appendix K or L-we can find the ilTeducible components from (14.2) of any given representation. We have therefore achieved the main purpose of this section. The remainder of the section is devoted to deducing some of the other important properties of characters which are required for other applications .

OrthogoDOJity relationa Consider the matrix P

= 2: D(A)(T-1) 0 D(lJ.)(T), T

(14.5)

118

GROUP THEORY IN QUANTUM: MEOHANlOS

where D(AJ, D(IA) are two particular irreducible representations of a group of h, elements. 0 is any matrix, and the summation is over all group elements 'J'~ We have

P D(P)(S)

= D(A)(S) t

D(A)(S-lT-l) f!D(/I.)('l'S).

Here S is any group element, and S-lT-l = (TB)-l (problem A.5). Now the elemen. TS are just the group elements realTanged" for there are k of them and we cannot have TS = RS unless T = R. Hence the summation over T can be rep1&oed by a summation over TS and we ha.V'e P D(P)(S) = D(}.)(l9)P.

P therefore satisfies the oollt.) dT = O. If .D(A} is the identity representation, then the i and j suffices become superfluous and (14.12) becomes 11, fcfo(.\) dT == h

J

cp(A)

dT

and the integral in general has some arbitrary non-zero value. If we now ta,ke cp t.o be some function in a vectc;r space !\ transforming according to a reducible representation D, then we can express 4> in terms of the irreducible base vectors of ~, and hence Je/> dT = 0 if D does not contain the identity representation. If D contains the identity representation n times, f cP d T is (1 linear combination of 1~ undetermined integrals. We have therefore proved the theorem of § 13 on the evaluation of matrix elements in the particular form of result (13.8c). Let us consider the reduction of D = D(A)* X D(J.L), where D{~) and DCu.) are irreducible representations. The characters of D are given by (14.3). Suppose D contains the ideIltity representation c times. The characters of the identity representation are all unity, so that from (14.10) c is given by c =

~

L1·

X(A)*(T)if.ll(T)

T

= Si\", from

(14.9).

(14.13)

121

REPRESENTATIONS OF FINITE GROUPS

In words, D * X D(p,) contains the identity representation once, if and only if D()..) == D(p,). This is just the result of problem 13.13. With this and (13.8e) already proved, (13.8b) and part III of the theorem of § 13 follo"\\T immediately, and parts I and II then as special cases. The regular. representation Drf:g of a group is defined as follows. First write out the group multiplication table, rearranging the rows so that all E's occur 'On the diagonal, as shown in Table 9 for the 9 Construction of Regular Represent,ation TABLE

E

A

E

A

A-l= B

B

E

K

L

= Tt applied second

]\.1

A

E

A

E A

A E A

E

-- T-l 1

applied first

point· group 32 (4.1). The columns are labelled by the elements Tt and the rows by Tj-l. Then the matrix Dreg(A) is obtained by putting all A's in the table equal to one and all other elements zero. Thus D~jeg(A) = 1 when T,Tj - 1 = A, i.e. when ATj

=

T t == Df;g(A)T~.

Thus the regular representation is a representation among the group elements considered as base vectors. N ow by construction, Xreg(E) = hand Xreg(T) = 0 for T =1= E. We may reduce Dreg, and applying (14.10) we find c).. = n"o I.e. every irreducible repre8entation D(A> is contained in the regular representation n).. times, where n).. is the dimension of D(M. Counting up the total number of base vectors, ,ve obtain as a by-product the relation (14.14)

122

GBOUP THEORY IN QUANTUM MECHANIOS

In the reduction, let Y,(M), ;, == 1, 2, . ' . denote the

rth

set of base

vectors transform.lng according to DOt}. We shall now construct ~ll tra.nsformations P that commute with every elemept of the grou.p, and that a.re linear 9Qmbinations of the group. By the latter condition P tr~orms every vector space y"Ar), i == 1, 2, " int.() itaelf. Hence P is represented with respect to the Y,(Ar) by a matrix in reduced form. Since we also require it tAl commute wi1Jl all group elementa, it must, by Schur}s lemma (appendix D), have the particular form diag {aA1E, aA2E, ... ~ ap1E, a~, .. .} where the E's are littJe unit matrices along the diagona.I. Further it ~\lSt retain thia form if we choose nfiW base Y,{M), vtrhence aJtr = aM = a~ say. Th:ue W'~ can vectors Yf(Ar"' write P in the form Ii

•

±

(14.158)

where th~ &A are opera.'tors such that E y1 Y,(Ar) = Y"v.r), E"Y,(Pf') := OJ and where the Q.\ are arbitrary constants. E It is the unit operator inside all the subspaoos transforming a.coording to D(A). Now we can also construct P in a, different way. Since P commutes with any element S of the group, it must contain the elements T and BPS,....l an equal number of times. Hence P can be expressed in the form

P

== !btMk,

(14.15b)

k

where M k is t.he sum of all elements in the kin cl~~~. 'rhus P can be considered as a vector in a space spanned by tbf: E).. or the M k • Either way the dimension of this spa,oo has to be the same, wllence comparing (14.153) and (14.15b) we have

I:~11.~~

of i~~ucible repr~ion$

= nttmb~ oJ cla&Ye8:l (14.16)

Let the number of elements in the can be written

kth

class be h't. Then (14.9)

(14.17)

If we define tIle matrix [] such that

E,EPRESENTATIONS OF :r.tNITE GROUPS

123

then (14.16) shows that U is square. Also (144117) becomes O·U = E. Hen,ce U is unitary and we also have U*O = l!i, which when written out becomes (14.18)

Thf'.Jl6 are the ortlwgoPA.lity relations of the 8econd 1ci1Ul. The unitary property of [T also ensures that the ~imultaneous equations (14.2) always have a unique- solution [namely (14.10)]. Thus we ha.ve an alternative proof of 'I'heorem 3, appendb.. a, namely that the reduction of a ~presentation is unique, &palt from equivalence. As all application of the a,bove formulae, we shall set up from first principles the character tahle for the group 32" As already deduced, there are three olasses, (E), (Ll, B), (K, .L, X). From (14.16) there 8;Fe three irreducible representations, aud (14.13) becomes nll + na' + na 2 = 6. The only solution is 'Itt = nl = 1, "8 = 2. We always have the identity :representation wit4' 0haracters 1, 1, 1. Since 1(2 ::::::. E~ .A 3 = E, the only possibilities for the characters of the second representation. are 1; j ~ (J) or w 2 ; ±l; where cu 8 == 1. Relation (14.17) with A = 1, I'~ = 2 gives l!t 1, -1. as the only possibility. The third repI-esenta;t~iQ.a has X(E) = 2. Then (14.18) 'with j = 1, k = 2 and j == 1, it: = 3 determines the other two chaI'acte:rs, namely -1 and o. 'l'b:is gives just the oharacter table as shown in Table 8. For more complicated groups the result of problem 14. 9 can be v~ry helpful. "Referenees More detailed accounts are givell of the properties of characters iIl SpeiBer (1937, pp. 166-175), of the rn.e1Jhod of characters applied to continuous groups in Wigner (1931), a.11d of characters in general in Littlewood (1940). A systema'tic method of setting up the oha;ra:./i:~x table~ is 8iven by Bhagavarltam and Venk&tarayudu (1951, p. 2(4).

Summary Group charact,ers have been defined, and used to label represents,... t.iQn.s, partiC1l1arly the irfeducible represel1tations. Any representa.. tiun may be reduced using (14.2) and (14 . 10). ]UnctiOD8 with given transformation pro~rtie& rnay be determiIled using the projec~on operBft'()rS ( 14.11). The tv."o types of orthogoD.d.lity relations and other results m~y be used to derive the chara,o'OOllJ of all the irre.d.ucible representations of a, group.

124

GROUP THEORY IN QUANTUl\-I

MECH.A~ICS

PROBLE!tfS

14.1 Tile spherical harmonics of order 3 transform according to D under the point-group 32. Reduce D. 14.2 A rare-earth ion in the free state is in a state with J = 1 with energy Eo. In a crystal it is in an electric potential with symmetry 32 and mean zero. Sho,\v that the level is split into two levels E 1 , E 2 , and determine the ratio (El - E o)/(E 2 - Eo) (cf. § 6, also problem 13.12). 14.3 Prove equation (9~2) using group characters. 14.4 u+, u_ transform according to D(1/2) of the full rotation group and according to D under the point-group 32. Try to reduce D. What has gone wrong? 14.5 Use (14.11) to find base vectors in .the space (X2, y2, Z2, yz, zx, xy) transforming under the group 32 according to the irreduc .. ible component representations (cf. equation (5.20)). 14.6 Shew that in an Abelian group each elenlent forms a class by itself. Hence deduce that all the irreducible representations are one-dimensional. 14.7 From (14.10) show that Ilx(T) 12 = h 2: CJ\2. Hence show T

A

that a necessary and sufficient condition for a representation to be irreducible is that Ilx(T) 12 == h. 14.8 Show (14.14) is a particular case of (14.18). Also deduce that ~ riN{(A)(T) = h if T = E, and is zero other\vise.

" 14.9

Let 0, be the ith class of a group. The product CtCj denotes all elements obtained by multiplying those of Ot by those of OJ_ Show that these ca.n be grouped into classes. Let the class Ok appear Cijk times. This is written Gi(}j == 2 CijkGk. 'Vith this notation, show that k hihjXi(A)X/A>

== X(A)(E)

2: CijkhkXk(A). k

Hint: show M k == TJkE (14.15b) as regards its operating on irreducible vector spaces; TJiTJj == 2: CtjkTJk; the character of M k is 7]kX(A)(E) = hiXt(A) (Speiser 1937). 14.10 Set up the character table for the point-group 422. This is the group of proper rotations that leaves a square invariallt. Use any of the relations of this chapter and of problems 14.7, 14.8, 14.9, and trial and error to set up the character table for this group. Verify the result of problem 14.9 in a few cases. Also reduce all the product representations D('A) X D(p,). 14.11 cp,(A) and ifs/IJ.) transform irreducibly according to D(A) and DC/-L). ShowthatMtj = fcpt*¢'jdT == f(Tcpt}*(Tifsj)dT. Hence using the

REPRESENTATIONS OF FINrrE GROUPS

125

orthogonality relations show that .1.1fij = 0 when DCA) and D(Il-) are inequivalent. Deduce also the result IB of the fundamental theorem of § 13. 14.12 In the fundalnental theorenl of § 13, in (13.8) and problem .l3.1~~, in (14.12), (14.13) and prohlern 14.11, \ve have a series of closely related results with at least t1VO independent possible lines of proof. Summarize how these are all logically inter-related. 14.13 Suppose you a.re given thp characters of an irreducible representation. How would you constl.'uet in a systeInatic wayan actual set of matrices to constitute th(j representation 1 14.14* Discuss the application. of gtl.)UP eharacter techniques to continuous groups, in particular to the rota.tion group (Wigner 1931, p. 97). 14.15 A set of base vectors CPi transform aecordulg to the irreducible representation Dt/I\}(T) of SCIne group of transformations ']1. Show from the orthogonality :relation (14.8) that

cPi

:2 Dtj(A)*( 1') T? is given byt (16.8)

where y is the angle between the

;1 and ;2 axes.

Consider the case

t The reader can prove this formula as follows. Let T 1 and T 2 be the dyadios representing the first and second rotations (see problem 8.4). The dyadio T.· T 1 th~n represents the resultant rotation, and its scalar (sum of diagonal elements) is therefore 1 + 2 cos CPs; i.e. T 2 " iTl = 1 + 2 cos tPs- This equation reduces to (16.8) after some manipulation. For a more geometrical proof, see Buerger (1956, p. 38).

134

GROUP THE&.Y III QUANTUM MEOHANICS

when ~1 and ;1 are bo~1i\{.6-fold axes. Putting 0 ~1 = 60°, tPl = 120 we oht&in respectively

cfol == .pI == 60° and

i - t cos y, cos i4/ ~ = 1,\/3 - lv'S cos i'~ cos a

(iii

cos i& = 0,

(iii)

FrO:rl1

~16.9a),

(i) cos y

1,

(iJ

=

:=:

COS ~/8

= 0; Iv/3 ;

cos

1'$'8 =

cos

W>'a == -iV'3 .

these solutions give re3pective]y

1, Y = 0°;

(ii) cos r = ---I, y

= 180°;

(iii) oos y

=3

II

Case (iii) is meaningle6s~ and in either of tbe other ea.ses th(~ '<es 1;1' ;~ OOiufJide. Thus f~ere can be at Inoat one 6..fold axis in a :point-" group.

10

TABI.lE

8,

8a

0,

8.

(i) (il) (ill) (iv)

0 1 0 0

0

(v) (vi)

()

0 0 0 1 0

0 0 0 0 1 1 0 0

(vii) (viii) (bt) (x) (xi)

,6 4 6 3 3 3

0

1 0 0 0 0

0

1 3 0 0 0

4:

1. 4 D .-~

.

~

.............

~

~

..

-

__ •

~

:.III

~

•

~

_ _ _ _ _. . _ _

(

~

-

_ _ _

"

Number of Axe~ in Proper Poin.t-group8 ....

........_~J,4'

Pf'int.grO\1~?,

1 I S ;£

6 622 42a

,)

4S2 32

0 0

28 222

..

WOe can now tabulate all the possible combinations of axes. If there is only one axis w'e have the point. . groups (i) to (v) (Table J.O)~ We next consider the case when there is 81 6-fold axis; there oan only be one as we have just pr4Jved. F~rom this it follows that there

REFBESENTATIONS OF FINITE GROUPS

135

oan be no separate 3-fold axis since this would require at least three 6-fold axes to give a 3-fold symmetry. Likewise there can be no four-fold axes. Then (16.7) gives only case (vi) (Table 10) with 8 6 > 0, and in enumerating the other possibilities we shall assume without further mention that 8, = o. Consi.der the case of a single 4-fold axis. There can be no 3-fold ax~ 1)~3ause these would lead to more than one 4 ..fold axis, but from (16.7) there &Te four 2-fold axes, giving carse (vii). If there are tw·o or more 4·f()ld axes, it

can

easil~y

be shown from (16.8) that they intersect at 90°. Now from (16.8) or problem 8.41 R(90'\ x) R(90°, y) is a 1200 rotationr Hence we have a 3 ..fold axis and 8 third 4-fold axis in the z-direction. There can be no more 4.fold axes because there are no further direotions a,t right angles -to each of the x .. , 1/" and z-axes. Furt.her thel~e must be f01Ir 3..fold axes to preserve the four-fold symmetry, which gives case (viii). This exhausts the cases with 4-fold axes. If there is one 3 .. fold~ an'S, any 2.,fold axes must be perpendicular to it and from (16.7) there are three of them, giving oase (ix). If there are more than one 3·fold axes, (16.8) gives cos i' = ±i. Further rpl = cPt = 120°, COB 'Y = 1 giv-es cfos = 180°, i.e. a 2-fold axis. Then from (16~7) there must be three 2 . .fold axes, which in turn generate four 3-fold aXt18. The latter are oriented witII respect to one another as -the bod~y diagonals of a cube, and. the (~OB 'Y = ±l leaves no room for more. Hence we have only case (x) . }--rom (16 . 7) there remains only the possibility of three 2..fold axes, case (xi). Having derived all the possible numbers of axes, it can now be shown that for each case of Table 10 there is essentiaJly only one way of arranging the axes in spa.ca. This follows frolu elementary geometrical reasoning. For instance in case (x), the restriction that the three-fold axes intersect at eos-1(±1) allows only one arrangement of them. The two..fold axes must lie eitJl£-J" perpendicular to each three-fold ~xis or bisecting the I)Jngie between two of them. }"or there to be only two-fold axes it can 4?asily be seen that the latter alternative must hold, whio.h fixes the.ir direotions uniquely. This completes the derivation of all proper point.. groups, and the ones with itnptoper axes may be derived from them as

tm-ee

already mentioned (Zachariasen 1945, p. 40; problem 16.4).

Character tGbles anil other point-groups Besides the crystallographic point-groups) there are ot,her finite subgroups of the full rotation group. Using the same Ilotation as before, an n-fold rotation axis Til is a cyclic point-group for any value of .,z,. We"Oan, add 2 .. fold axes perpeIlflicular to it to obtain the diWral group n2. Similarly by adding mirror planes (inlproper 10

136

GROUP THEORY

I~

QUANTUl\I MECHANICS

2-fold axes) we obtain the groups n, n2, n/m, nm, n/mm, ii, fi, 2m, etc. These are not all different, but which ones are the same depends on whether we have n = 2q + 1, 4q, or 4q + 2 where q is arl integer (cf. Fig. 9). Apart from these there is only one other proper point-group, and that corresponds to the S3'!ll.nletry of a regula,r icosahedron (Murnaghan 1938, p. 336). In addition to the above fillite poillt-groups, there are several subgroups of the full rotation group that are associated with the axial rotation group. The latter has the symbol ro in the present notation, and the other (lerived groups are oo/m, 002, rom and oo/mm. These are very important for describing diatomic molecules and we shall now derive their character tables. Firstly 00 is a cyclic group and its irreducible representations have been derived in § 7. Each rotation forms a class of its own (cf. problem 14.6), and the cha.racter table is as shown in Table 11 and appendix L. 11 Character Tables of 00 and oom TABLE

The group

Irred. rep.

00

X(E)

X[R(cP, z)]

1 1 I

1 exp(ikq,) exp( -ik 1)

The group oom

Irred. rep.

x(E)

x[R(q" z)]

x(m z )

A+

1 I 2

1 1 2 cos kq,

1 -1 0

A_ E1c (k

~

1)

Note: The notation for the irreducible representations is not quite the standard notation of a,ppendices K and L, but is I.oore suited to the present discussion.

Since there are an infinite number of classes we expect an infinite number of irreducible representations. Consider next the group oom, operating on some invariant vector space. This space can first be reduced according to the axial rotation group, and let ifJl be a vector in it which transforms according to the representation

REPRESENTATIONS OF FINITE GR01JPS

137

exp( ike/» of Table 11. In spherical polar co-ordinates, the reflection fiz perpendicular to the x-axis indU\1es t,he tra,nsfornlation

e=

fJ,

R( , Z)~2

=

fl) __ ... ___

'J,J

---~..

r/)

't .

If we put then we have R( cp, Z )In,t'Y}l = mxR( -rf>, ~·)ifJl = rnx exp( ---'ilcli~)111 -- exp( ;k~l\.f. \f' j(.i'1' --

- - CI

(16.10)

lIence ~J2 belongs to the representation eXr\ -'ik4» of 00, so tha,t if k =F 0 it is linearly independent of ~'-iJ.. However, froni (16.10) tP2 belongs to the saIne irreducible vector s.~:)aA~0 as tPl' an.rl we have a t,,\vodiJnensional irreducible repreSclltation .Er> 'the cha.racter of .R(cp, z) is exp(ik~6)

-f- exp( -ik¢;)

-:.~:2

cos k4>,

and t.he character Df rc.:c, is 0 because it just interchanges ~1 and 2f2" For k == 0, ~l arld ~J2 need not be linearly independent. Wecan al"\\Tays forIn the syn~nletric t'lJld antisymmetric linear combinations tf] :.± CP2' which give t,vo one-dimensional representations. In this way \:ve obtain the cOll1plete \Jharacter table shoVirn in Table 11. 'The theory for the other groups derived from co is similar. The group 002 is isomorphic \vith oonl and hence has the salne character ta.ble ..Also oo/m is the direct product 00 X (E, 11lZ ) , and oo/nlnl is the direct product of com with the inversion group I =--= (E, II). Hence their character tables can be "Titten down froln the prescription of § 15 or appendix K. Similar considerations apply to thp finite groups n, n2, n/m, nm, n/mm, etc. IIo¥/ever there is the followillg difference. For n even, k in (16.1 0) rnay have the value in, so t.hat exp( -ik1» = ±1 for all possible values 27rr/l~ of cl~. Hence tPl and ¢;2 in (16.10) need not be linearly indepf:.'ndent and ,ve obtain two oI!.e-dimensional representations instead of a two-dimensiona.l onen In this \vay the irreducible represelltatJions of all the point-groups except the cubic ones (and the icosahedral ones) can easily be obtained. The character tables for these remaining ones can be determined from the relations of § 14.

Double-valued (or spin) representations Let u+, u_ be the spin functions transforming according to D(1/2) under the full rotation group. Now this representation is doublevalued, with the b3Jse vectors changing sign under a rotation of

360(;' a.bout i~ny a,xk (§ 8)~ so that 'u+, u_ must also transform aC'Jord· it1g to ~h double ..vsJ.ued representation under any '?Oint.g:oupe ~rh(~) doubla . . valued irredueible representations of all the cry,:,ta} !loint..groupa ha:Vt3 been tabulated by Koster (1957). We 3hall now show how the double-valued representations may be fOUtld~ following the lnethod of Opecho,,~ski (1940) based on the ~rlier work of Bathe (1929). 1'hese representations can be fOWld for the ojiclio ,groups 'n~ tIle dihedral groups 'n2 and asaociat~d groups bj'" 'n~ing hiJf integer values of k In (16410) ttnd p:fooeeding as bafore~ liowev'er the following device is m.ore gc:neral and OOZlvement" (1onsider a point-group 9 of proper rotations, and thl3 oorresponding group Q) of matrices taken. :from the representation DillS} (8~24) of the fuH T01~ation groupe Ole.at-ly these matrioes satisfy all the g-rvup requu:e:m.t3nt'3) because (8.24) forms 8, representatioZ:~1 l1:owev(,~r because of t·he -1- or - sign i.1'1 (8.24), i.e. the "double .. valtle{llleBS of DC1/';.'), (f, contains twice ~ "many elements 8,-8 9 does. C,)rlllequently anj" Bingle-valued represent;l.tion of Q5 automatically flJrJahes B, dOtl.ble. . valued representation of 9. l'he single-valued l'Ot;prr?J!entations of CO may of course be fOUIld by the methods of § 14 in It st!~aightforward ·way. The group of matrioes :; in the notation of Table 12.

COl18tru~lt; i,he eha;ract,.eTs r:'),fJ~~s:ent~))t';ou~ ofMl{' group ":t2~t

16,."1

fl,,,

of the extra d.ouble-v.luad

REPRESENTATIONS OF FINITE GROlJPS

16.8 Rand 8 are any two elements of a, group in the same oIaas. BIlow that there exists an element T such that P-1S'I' :c B~ 16.9 Show that Ul any represe'utation X(.R)* =: X(R-l). Wb.a.t are the oonsequences of this for th.e oharaeters of 1800 ro'W,;tions in single-valued and double-valued irreducible represeJltations (Opechowshi 1940) 1 16.10 Show how to construct polynomials of degree n Wllioh transform according to a, given representation of & point~group (Olson and Rodrigues, 1957)"

17. The Relationship Between Group Theory and the Dirac Method In this book we have developed the group theoretical methoo. tor sorting out and labelling a complete set of fUD.t.ttio~ usually the eigenfunctions of a Hamiltonian. rrhis oontrat-lts a·t first sight wIth the more usual procedure, developed by Dirac (1958» in whioh one uses as a complete set of' functions the 8imul~'\noous eigenfunotions of a set· of commuting operators (se. forex8,mple Schiff 1951$, p.143). The purpose of the prest}nt section :is to rel::.\.te these t·wo 6:pproaohel, t and show that they are oomplet(~iy equivalent, In brief, Dirac proooed.~ as lollowB. Let 4s be an eigemunotJon of two operators A and B belongiIlg to eigenvalues a, b; It follows that

A,Bt/s = Ab~ =. bat/J = rJ,B~ = lJfI4t (.4B - BA)+ :::;; o.

= BAt/I,

This suggests that simultaneous elgen£unc'bions like ~ are moat likely t.o exist if AB - BA = 0, i . e. if th.e two operat()tS cominute. In. fact it can be shown (Dirac 1058, p. 4£;) that the SimultaI100US eigenfunctions of two commuting operators form a complete set of functions. By using 8e'V'eral commuting ollemtors, we can arrange it that the ieta of eigenvalues of two different function.s are always different. This then giTeS 8 definite way of achieving a sorted and labelled complete set of functions. One of the operators is usually !Chosen to be th~ Hamiltonian, so that the eigenfunctions are sorted out according to their energies . In thIs case the other opera..tors are

t The results of this section are not mwd else'where in the book.. It haa been included for the benefit of those readers whose original introduction to qntmtuIn luechanies was through Dit'aC'a book, or wb;,) for ot,her roosoDS like to think in t i'IT:1S ':if comp'ete sets of oomm.uting op~ato~c>

144

GROUP TIIEORY IN QUANTUM MECHANICS

const.ants of the motion in the quantum me cba,ni cal sense. In detail, for any operator A not depending explicitly on tiIne, we have (Schiff 1955, p. 134) d I dt (A) =-= in (3,

-t) =

F 4l (r) sin 3 () ei3f!>u_

t Throughout this section we shall drop the factor" from all angular momenta for the sake of custom and convenienoe. Thus w6useJ.,ls, i for I, tot' I z , orb, t' etc.

THEORY OF

is the

=

.)id.er time-reversal symI[tetry explicitly 101' Aystems having a low spatial symmetry and '~ring spin {lepende.nt wave functions~ In other cases the degerlel',ney' prodn):":d by tiD.le,.:reversal is usually produced. also by SOllle ~J\atial syrnl~letI'Y. "fhese pointrS will become clearer as we proceed -o/Pith the detailed developnlertt. Tl.e time-reversal o]'Jerator The tiu\o· deperlden.t. Sc;hrodinger equation is

(ft -- iii !)Y'(t) = o. }1.ecapitulating briefly t-,he initial ar€,'llmellt, the operator in this . equDition is clear13r not nrvariant under the simple time ..inlve.fsion rrnbst,).t.u.t.ion t -~ -to 1'hus if there is to be a time-reversal transfornl;~tioll~ it Joust h;9,ve a more complicated form, and we define the time .. ret'ersal operat.or T by the rela,tion

~!p(r"

ad.

t)

~ Y'*(r,. ~d' :~ ~J

(19.5a)

Here 1ve have ahlJreviated all the electron co-ordinates rh (fzli1 r2,

= (£'r - ift'i)T4>,

as follows as an extension of (19.8) or alternatively by writing 4> == 4>r + i!foio Thus we have

T:I/'T-l =

~*,

i.e. we obtain the time-reversed operator T~T-l by replacing every operat~r in Ye by its complex conjugate. For instance a potential V{r) is real and remains unchanged. The lllomentum px -=---::. -ilia/ox is pure imaginary and changes sign, p -7 -po Also the angular momentuln opera.tors change sign. For the 'orbital angular momen.. tum this follows from. the definition 1 = r /\ p, but for the spin angular roo:cuentum it is necessary to go back to ita definition (11.6) as foUows. A -rotation R is a real operator,t and it follows from (8,1) that il; is real, and that all angular momenta nIF. are pure lmaginal')"r operators. Since:Yt' is a real function of r, p, S, we have (19.15) Kra~ers' theore~

This theorem states: in the presence of any elect"lo potential but in the absence of an external magnetic field, every energy level of a system with an odd number of electrons is n-fold degenerate, where n is an even nunlber (not necessarily the saIne for each level). We first note that in the absence of an external magIletic field, the H~Jnilt()nian conttl,ins only even powers of the momenta. This is

t This notation differs from tho one used for instance by Dirao (1958) who uses "real" to describe an operator with all real eigenvalues. ! This is obvious as regards its action on an ordinary function J (x, y, z) because it is a, real transformation of the real co-ordinates :t!, y, z. It is also true with respeot to its (;tract on bpin functions, for consider it operating on the real function (u,+ U+ *): R(u+ -t. u+ *) -= (au+ + a*u+ *) + (00_ + b*u_ *) == real. Incidentally, note that in deriving this result we have used (19.10), which explains why we use the definiti,)11 (19.11) of complex conjugation derived from (19.1(\), rather than using (19.9)~

+

170

GROUP THEORY IN QtrANTUM MECHANIOS

so for the kinetio energy p2/2m and for all the spin-orbit and spin.. bpin coupling terms of (11.8), (11.9) like 1,· Sf, rt A Pj· S" 8}, etc, Thus (19.16)

8,·

or from (19.15)

T.7eT-l =:1f.

(19.17)

In the presenoe of an external magnetic field H, the Hamiltonian (l,ontains the term (e/2mc)H· (L + 28) (Schiff 1955, p . 292) which is linear in the angular momenta, so that (19.16) does not hold. This shows why the theorem must be restricted to systems not in an ext.ernal magnetic field. Note, however, that interactions such as the spin-orbit coupling whicl~ depend on the internal magnetic iieids of the system generated by the moving electrons are invariaIlt uuder time-reversal, since these internal fields change sign if the luomenta of all the partioles are reversed. This is all completely analogous to the situation in classical mechanics (problem i9.4). It follows from (19.17) that if eYe", = E.p, thell £lT~1 = ET~, HO that tP and Tt/J are eigenfunctions belonging to the same energy h~vel. For this to give a degeneracy we have to show that they are Jinearly independent. Suppose (19.18) where

(X

is some constant.

T2cp

Then

= TtXt/J =

f.X*T~ =:-

:if.*oof;.

For a system with an odd number of electrons, this gives a contradiction to T2+ = -ifs (19.13b) since «*tX is positive and cannot possibly equal -1. Thus (19.18) is false, and .jJ and T.p are linearly independent. Since T2ifJ = -t/s, the degeneracy of every energy level is even, which proves the theorem.

Energy level degeneracies 14}t us consider an atom in a J = 3/2 state placed in an electric field with point-group symmetry 32 due to its neighbo~ in a crystal or molecule, and let us determine the possible splitting of the level in the manner of §§ 6 and 14. The J = 3/2 states transform according to a double-valued representation r under rotations, and their oha-racters for the double-group of 32 are (§ 16 and equation (14.4))

X(E) 4

X(E)

-4:

X(B) -1

X(A) 1

X(K)

x(K)

o

o

171

THEORY OF FREE .ATOMS AND TONS

FrOln 'liable 12 we ha.ve (19.19)

r

so that we might expect the level to split irrt:f' tTf{O singlets (r4~ 5) and Olle doublet (rs ). Ilowever we have not considered the full symluetry group of th.e HalTI.iltonian. becau~e tJhi~ also includes time reversal, and K..ramers' theorelH ShO"\YH lihat the greatest splitting possible is iu.t-o doublets. Henc~ lJ) this example time-reversal lea.ds to ~1 degeneraoy, at least het~yeen 1'4 t'tn Icy)(edES Icy).

This i~ the Illore usua,l definition of t.he Ra,cah coefficients. 20.7 By cOlnparing problem 20.6 "\vith equation (20.32), show that W(abcdef) = (-1)e+f-b-C1V(afedcb). Similarly prove t.hat

W(abcdef) == W(badcef) === lV(cdabeJ) == JV(acbdJe)

= 20.8

(-l)e+f-a-dW(ebcJad) === (-l)e+f-b-CW(aejdbc).

Wiite down the value of the Ina.trix elenlent

*Dclassr/> dT = 0, I because 4>*4> has even parity w 2 = + 1, and Dclass has odd parity, i.e. changes sign under ri =->- -rio The expectation value of the classical orbital dipole moment Dclass is thus always zero. However, there still relnains the possibility that the nucleons have an intrinsic dipole moment, in the same way as electrons and nucleons have. intrinsic spin angular momenta and spin magnetic moments. Suppose this is represented by an operator D sPin . For us to interpret this quantity as an electric dipole moment) it must manifest itself through the occurrence in the Hamiltonian of an interaction term

-8 · (Dclass

+ D sPin)

with an external electric field 8. Now all kno,vn electromagnetic interactions are invariant under inversion, so that D sPln must behave in the same way as Dclass under inversion. Consequently the expectation va.Iue of Dsptn is also zero, and nuclei exhibit no permanent electric dipole moments. t In the same way it follows,that nuclei can have no electric multipole moments of order 2l where l is odd, and no magnetic multipole

t The same result applies of oourse to any system interacting with parityoonserving forces. In this strict sense the ground state of an asymmetrical molecule does not have a pennanent dipole moment, and it corresponds to 8, spherically symmetric wave function with the molecule pointing in no particular direction in spa,ce. However a molecule has a large number of rotational levels separated from the ground state by energies normally small compared with kT (§§ 23 and 24). This gives the system certain properties at ordinary temperatures which are analogous to those of a classical dipole moment.

193

THEORY OF FREE ATOMS AND IONS

moments of order 2l where l is even. The other Inoments are allowed, e.g. magnetic dipole, electric quadrupole r etc. (}'or definitiollS of higher order mult.ipole moments, see Stratton (1941) and. Sehwartz (1955).) A nuclear 2'-pole moment must therefore be an electric one if l is even and a magn.et.ic one if l iq odd. lIenee the "'~vord s electric" and "magnetic" are usually omitted without can sing any eonfusion. It should be I10ted how the proof that the nuclei hftYC no dipole moments depends on. the fact that the intelrnal \vave functions 4> have a definite parity ±l, which in turn follow(;.d frr-rn. the jnvariance under inversion of the nuclear interactions fl ij. Rea.l nuclear interactions nan be divided hl'to three classes; tbe ~t,r(Jng interantions of the order of 10 l\tJeV. which are responsih!e for nuclear binding and most other properties, the electromagnetic intera{;tJc.ns of the orde~ of 1 MeV., and the Inuch weaker interClA~tjonf:; of the order of 1 eVe ,vhich are responsible for beta decay (~29). Of these, the last is nof, invariant Ullder inv-ersion, so that nucl~i are expe~ted to have some electric dipole moment which, howevel'$ i;:t too small to detect experinlentally (Ramsey 1953, Lee and 'Yang 195(5). H

Limitation oj multipole mORlents by 1 Let Men be an electric or rnagnetic multipole Inoment of order 2~. '.rh.i8 (Jan be expressed as a linear eombination of 2l +- 1 irreducibl~ multipole moments Mm H ) which transform according to the irreducible representation DU) of the rotation group (Stratton 1941, p. 182). The products Mrr"ulCPtnt(I,M j ) transform aecording to \

LD(j) ,vhere (§ 9)

I + l. lIenee by the fundamental theorem of § 13, the ground state matrix elements = (const)(I, M j

IIJ1, M'l).

'l'he :proporti.onality const.ant is conventionally written as JLn/(1iI), t\nd since ,\ve a·re: only' concerned with the ground state of the internal 11amHtonian ;?lnt, -we can wlite effectively

I

(21.1 )

ttn = f£n 11,1Thifi

p:tOY8S that p'n and I are always parallel. Similarly Helect, rnag11etjc field at the nucleus due to the orbital motion and spin ID/),ignetie mom.ents of the electrons, is proportional to their angular 7nCl1.1.f~ntur.n J', provided we restrict ourselves to considering only one pfJ!rticul~r (say the lowest) electronic energy level:

th~:

Hel ect

=

(const)~J.

(21.7)

Now th.e magnetic interaction energy between the electrons and the nucleus is (21.8) Yf'M = -f.Ln • Helect, fUld t~it.er

substituting (21.1) a,nd (21.7) this

-----,

beconl',~~;

. . drM = aI ",J J-Ln (.1'1' Helect • J

LWher~a= -iii"

J2

I

I>

(2] .9)

·

J

(21.10)

"'t/Ve now turn to col1t4':)ider the kind of energy levels given by an lllt.craetion Ha,miltoni3~n of the form (21.9). We defer to the next subsection the problem of calculating Hel ect so that I1n can be det.ermhled from (21.10) once a has been measured experimentally. \~ve consider only the set of low lying states. 1p(I, .l.7J!I[; J, M J)

== 4>n~cl(I,

MI)ifielect(J, MJ)

in ,vhlch the nucleus and the electrons are both in their respective ground states ~nllcl(l) MI) and tPelect(J, MJ). The interaction

195

THRORY OF FREE ATOMS AND IONS

;;reM

(21.9) is invariant under combined rotationy of the electronic and nuclear co·ordinates together, so that the eigenstates of the whole atom carl be desigllated by combined angular momentum quantum numbers F, M F .

= 2: (IJMIMJiFMF)tp(I')4.~fl,J,:JIJ)J ,l.1f,[/,MJ II - JI ~ F ~ I + J. (21.11) The relative energiesE(F) of these (2F + I}-fold degencratelevela s.rt: P(I,J,F,MF)

obtained by applying the perturbation Jt'1t! (21 ~9) to the first order of approximation E(F) = (IJFMplaI· J!lJFMlj1>~ These Inatrix elements can be caJculated by' exactl)" the same prooedure as was used in § 13 to calculate the futo structure splitting due to the spin-orbit coupling {L· S. If F = 1.+ J is the total angular mom.entum of the atom, we have

21 ~ J = F2 - II - J2

(21.12)

and hence

I E(F) == iali2[F(~ + 1) -

1(1

+ 1) -

J(J

'~-I)O

(21.13)

Usually it is the difference between successive levels that is measured by lluclear resonance techniques. E(F) - E(F - 1)

= a1i,2F.

Thus the intervals should bear simple numerical raJios to one another, and an~y deviation from these simple ratios indicates some omission in the theory, e.g. the existence of a quadrupole moment and second order perturbation effects from £'Mo If the atolll is in all externally applied magnetie :fi.eld Hot the Hamiltonian ma.'1t contain also intera.ction terms --p.. Iio for both the electronic and the nuclear magnetic moments. For the same resson that I'n and I are parallel, the electronic magnetic mornent p.tJ is parallel to J. ]"rom problem 13.6, it is given by

Ile

=

J -gL/1 A

where gL is the Lande g.. factor

gL

=1+

J(J

-t- 1) + S(S + 1) 2J(J

+ 1)

L(L

+. 1)

196

GROLT TllEOm: IN QtrAl'rtUM MECHA'NICS

and f1 the electronic Bohr magneton. 'rhe IIamiltonian then becomes

== aI · .J + (gL/3/Ii)J • Ho -- ILn · Ho.

£

(21.14)

Iler~ (.l~1

is of the order of the nuclear Bohr magneton e1i/2mn c 1{/hjch l~ Hbont 2000 thnes smaller than the electronic Bohr magnetoll because \rf 111.0 larger nlaSR n~p, of a nucleon. \Ve sha.ll therefore neglect the last tcrrl1 in (21.14). \Ve first eonsirler the ease of a ycry "Teak external field «:: 100 gauss) so that. plIo WC1( ML. ][ s:=-= :r: '),I'H(J.}f 't' J

~

'\...

i--

.A

::.:::

0 ill ~ ,

...,

=:-:

+

_~_~

~,

,

,

abbreviated to ,,~rhfTC tht'~e

±

1,'( I)P is the antjs~~mlnetrizing operatol. v':""e (~an group into terms by fol1o,vlng Slater's procedure (~3pe Yl'.:tlJle 5);; arid

obtain the terlns 311, 31:+, llI, 11:+ as shovm in Table 20. Th.ese 'l"ABLE 20 Low~est

Terms for T\vo Separated Chlorine and Hydrogen

Atoms Atomio states Cl H (I (1 (1 (0

T) -)

+) +) (0 +) (0 -)

(0 (0 (0 (0 (0 (0

+) +) -) +) -) +)

t(a) 311; (b)

Total ML

Total _l"ls

'rermt

0 0

1 0 0 1 0

0

0

a a b c c d

1 1 1

111; (0)

31:+; (d) 1£+.

Th~ M L l'efers to t,he representation for rotations abo utI the linfJ joining tho atoms. All the ...11L = 0 functions are even under the reflection mx.

all have the same energy in the limit of infinite separation. One of these is a 1}]+ state, the same as the state of the molecule, and since we are dealing with the lo,vest energy states these join up in the sense of Fig. 21. That is, the molecular state goes over into a state of the separated atoms which leaves each atom in its lawest state. This therefore gives the limiting energy of E(a) as a -+ 00. It should be noted that if 1E+ had not occulTed among the terms found from the ground states of the atoms, then as a ~ 00 one or

226

OROUP THEORY IN QUANTUM MECHANICS

more of the atoms would have been left in an excited state. An example of the latter situation. is the methane molecule (problem 22.14)~

Clearly similar argunlents can be applied to the way the oneelectron orbitals change with interatomic distance, and we catl also investigate what happena in the limit a -;. 0 as the nuclei of the two atoms coa.lesce into a new single nucleus.

Spin.orbit coupling So far we have based our discussion on the Hamiltonian (22.1) without spin-orbit couplitig, and all symmetr~y· cop.siderations have referred to the orbital part of the Wave function only. Howev:er, if the spin-orbit coupling is important, the Hamiltonian is not invariant under orbital transformations alone, and we have to apply the symmetry transformationS to the orbital and spin variables simultaneously. As with the full rotation group, the spin funotions u+, u_ transfornl according to double-valued representatioIL~, and thus so do all spin-orbitals. It has already been sho,Vll in § 16 hO\\T to work out these double-valued representatiollS of the pointgroups, and we shall not consider the matter further . As mentioned there, they have been tabulated for all crystallographic point.. groups by Koster (1957). Further references The whole subject of the quantum mechanical basis of molecula.r structure is covered in rather greater detail than here by Eyring, 'Valter and Kimball (1944). Pauling (1939) and (k)ulson (1952) appl~y' the quantum mechanical ideas to give a description of the chemical properties of a wide range of compounds. Van der Waerden (1932) discusses the spectra and symmetries of diatomic moleoules in great detail. r.JOulson and Longuet.. Higgins (1947-48) treat the chemical properties of conjugated hydrocarbons in terms of molecular orbitals. Roothan (1951) reviews some aspects of molecular orbital theory. Boys, Cook, Reeves and Shavitt (1956) show how one can calculate the properties of simple molecules quantitatively from first principles by systematically approximating to the exact wave function by a linear combination of many determinants. Mott and Stevens (1907) discuss the use of localized versus extended orbitals in solids. SumllUlry

\Ve have discussed the different types of orbitals, atomic, directed valence, valence bond, molecular and Bloch orbitaJs, and in what

STRUC'l'iJRE AND VIBRATIONS OF MOLECULF!S

227

physical situations they are used to describe the electrun'} i.n Iuolecules and solids. Their symmetry properties have been used to de~cribe, classify, and enumerate them. We have also cortsidered the sym .. nletry of the whole mo]eculat' ~·aye function, and how this affects tho way the energy changes as the lnolecule is pulled apart iJlto single atoms. PROBLEMS

22.1 (~aleulate the kinet1c energies of the orbitals hand Jf, (22.4), and hence show that ~fb ha3 the lO"\l'er kinetic energy'. 22.2 Sho'H that a set of three dit'ccted -7aJe1l0e orbitals making 1200 an.gles with one another in the x, y-p}une (Fig. 17b), call bt: constructed out of ifJ(2s), ¥,(2px) , ifJ(2plI) atolr:.ie orbitals. 22~3 Show that the foHovliug are four di:-ected valence orbitals making 900 angles with one anoth.er in a plh~ne (Fig. 17a):

,pA = ific =

.p(28}

+ ¥J(2px),

if;(2lJ)

+ ~b{21~11)1

q;R

if1(2s) - tP large .ll. (Coulson and l;"ishel~ IH":i!~j.) l'r()U~: the linear cornbil)ation lJI AO,l -- 1J'~'AO,2 is required to give an {? == 0 function: the corresponding t~ =--= 1 function~; tefer to excited states of the molecule. 22.8 -rhe radi.i of the 4/ and 68 shells of the gadoliniunl atom are ~Lout 0.3 and 2 Augstrom. units. In gadolinilnu meta], the uistanee between near~st neighhour atoms is 3.6 .l\.U. What types of orbital would :rou expect to use to describe the Inagnetic 4f p:lectrou.s alld the condu.0tion 68, Hp electrons in the rnet.al res .. p€ct'ively~

22.9* ""hat types of v.;·ave funct.ion are comrnonly used for the ele, z)] = 6 cos ~ 3, X[IR(c/>, z)] = -2 cos cp - 1,

X(2:,;) = -1, X(mx) = 3,

whence fronl the character table for oojmm (appendix I.A) Dtot = A 1g

+ 2A 2U + Ell + 2EIU •

We subtract out the three translational components in accordance Wlt.u. t'able 23 and the character table

and t.he two rotational co-ordinates D rot

=

E 1g ,

242

GROUP THEORY IN QUAN1'Ul\I MEC;HANICS

and obtain (23.24)

"fhere are therefore two non-degetlerat,e DiOdes (A 19 alld .A!u) and a pair of degenerate ones (E1U )' Of these AIU is Raman but not infra..red active, and the others are infra-red but not Raman active. The normal modes are pictured in Fig. 25.

The ammon.ia inversion. line At the beginning of this section we adopted the model that the electrons in a molecule move very rapidly around the nuclei, whose vibrations in comparison are slow semi-static displacements. The wave function corresponding to this model has the form (23.25a)

and it can indeed be shown that this is in general a very good -&pproximation to the wave function. Here tPelcct is a wave function in the electronic co-ordinates with the nuclear co-ordinates RGt entering only as fixed parameters. The approximate separation (23~7) ot: the nuclear Hamiltonian further implies that tPnucI can be written approximately as

r"

~------------------------------.

(23.25b)

However, our argument so far implicitly assumes that the lluclei never penetrate far from a definite equilibrium configuration, as '\vitnessed for instance by the fact that our discussion lIas been euuched in terms of small displacements. If each llucleus departs ar~)itrarily from its position by something of the order of the internuclear distance, then it is difficult to regard such a displacement as the sum of a rotation of the whole molecule and an iIlt.ernal distortion, nor are the approximations (23.25) valid any more. For this reason, our model breaks down when a molecule oan take up more than one distinct equilibrium configuration. "t\.n ex tun pIe of this is the ammonia molecule (Fig. 26) in which the nitrogen nucleus can either be above or below the place of the three hydrogen nuclei (position Nt or N a). We would expect that the nitrogen nucleus has quite a large potential hamer to- overcome in passing from one position to the other', so that this transition will be slow compared with the vibrational frequencies. We can therefore write down two approximate wave function tPnuel,l and t/lnucl,. to describe the molecule vibrating about the t~ro equilibrium

24~{

STR1JCTURE AND VI.HRA1'IONS OF MOLECULES

H

........... "

~~17 I /

...... ,-....

'.......

- /"

H

,/

N2 FIG. 26* The two possible positions Nl and Ns of the nitrogen nucleus in an ammonia molecule.

positions respectively_ However, . the transition from N! to N'I. is a, symmetrY transformation for the system of nuclei (though it is not a symmetry of each equilibrium configuration individually), and the whole moler.nl~ ws.:ve function m'ust be either the symmetric or antisylnmetrlc COInhiJ~B,tion, (23.26)

Now the time t for ~ =T~n!}itii}ri Xi-)1!l Nl to Na is very long, and the energy difference ..:.1E ~ njt bet-ween. the two states (23.26) is rather small . This has two consequel1~~~~L Firstly', the transition Nl to Ns gives an extra spectral line, naL::.f~l:f t,il? ~u.. called inversion line of ammonia which is in the triicrowave iJ .2;) ~:in) region of the spectrum, compared with the vibration frequencie.s: "YhJ~h lie in the infra-red. This is the line used. in the ammonia m&.33r. Secondly, it meallS the rest of the vibrational spectrum is just thz:t of one configura. tion ~lone, with possibly some fine structure of lliagnitude .t.1E. Other examples of molecules with several equilibrium configura .. tiona occur among organic molecules with hindered rotation. For instance ir~ 0 2016 , the two C CIa groups are not free to rotate about the central C-C bond ~ecause the big chlorine atoms get in one aIlother's way. However, the two groups can take up three diff~rent relative positions, in which the two sets of chlorine atoms interlock. References For the separation (23.25a) bet,,~een the electronic and nuclear motion, the rea~er is referred to Eyring, Walter and Kimball (1944, p. 190) and to the origirlal paper by' Born and Oppenheimer 1927. The separation of the lluclear Halnilto;uian into the centre of mass, the rotational and vibrational mot.ions is discussed hy Wilson, Decius and Cross (1955, Chapter 11) and by Bodi and Curtis (1956). '.rhe energy' levels of the rotational HamiltoTila.l1

244

GROUP THEORY IN QUANTUM MECHANICS

are conveniently summarizlJd by 'VilSOll et al. (1955, appelldix 16), the resuJts for a linear molecule being derived by Eyring et al. (1944, p. 72). r:rhe whole subject of 11101ecular vibrations is very ful1~y and clearly discussed in t.he book by "Tilson, Decius and Cross (1955). "r,vo shorter reviews having a scope simila.r to the present section al e Rosenthal and Murphy (1936) and Sponer and TeJler (1941). g

Sum.rnary

The lnotion of the lluclei in a molecule can be represented approsinlately as the ~Uln of three parts; a sidewaYR translation of the \\rhole molecule, a rotation of the molecule as a whole, and interna,l vib.ra~ tions or distortions of the molecule. The latter are express'ed hl terrrlS of the vibrational normal modes which ha.ve differeIlt synllnetries. rfhis symmetry determin.eB \vhether or not a particular vibraiioIl gives B line in the infrtl-red spectrum and thE" Raman spectl'ulll of the ~ubstance. PROBLEMS

23.1

What are the symluetry groups for the followlllg molecules: C2H 2J H 2 , H 20, DHO (heavy water), NHa, CH 4 , benzene, cyclohexane? Note: H 20 is not a linear lnolecule because of the lone pair electrons on the oxygen; benzene is planar. 23.2 Enumerate the symmetries of the norlnal vibrations of the above molecules and illustrate some of thenl by figures. DetermiIle which ones contribute as fundamentals to the infra-red and Raman spectra. 23.3 Prove that every element of a real one-dimensional representation is ±l. Hint: use the unitary property of problem C.2. 23.4 By looking at the character of mz, sort out the normal modes of benzene into those that are vibrat~ons in the plane of the molecule and those perpendicular to it. 23.5* Tn ordet' to interpret the vibration spectra of a complh~ated molecule, one usually Inakes a theoretical calculation of the frequencies from a theoretical lllodel, in \vhich SODle force constant~ are assumed for t.he various inter-atomic bonds to represeut their resistance to stretching or bending. Give an account of how the normal modes and frequencies are calculated from the model, and iIl particular of how group theory can help in the calculation. Reference: Wilson, Decius and Cross (1955). 23.6* Outline how the energy levels of a rigidly rotating molecule are derived (rigid rotor), distinguishing between the linear, spherical, symmetric and asymmet.ric ca.ses. Obtain the degenera,cies of the

STB,UOTURE AND VIBRATIONS OF MOLECULES

245

levels, and select.ion rules for transitions between them< References: CrlIf'3imir 1931; Wilson et al. 1955, appendix 16; Herzberg 1945. 23,7* Show how to incorporat.e the nuclear spins (§ 21) into the \va,ye function (23.25) of a whole molecule, and discuss their effect on the spectra of dlatonlic molecules with particular reference to . d 0 216" The nuc1· ' - splIlS ~ "1 .ti ~j "'[]'"I) r:L , .-0 2) an el 0 f H " D 0 16 nave 0f t, 1, 0 respectiv~Jy~ and D stands for a deuterium atom (Eyring et ale 1944, p. 265). "If'';"

24.

Infra-red and Raman Spectra

In the last section the Wi\,ve function for a ';\Thole rnolecule was 1r''1'it.ten in the approximate form

F= ~Clect(rt~.J

"'tra.ns(Rcm ). "'rot(f)t) PVlb(Qp)·

(24.1)

Here the vibrational "\vave function

[~"'vlb(nlo n2. • • · 1taN-6} ]

(24,.2a)

corresponds t.o an excited stat.e of the molecule with nfJ quanta of energy in the fith normal mode of vibration, the erlergy of the state being

I

E(nfJ} =

t (np + i)!iW,B'\

(24.2b)

The existence of these vibrational excited states and the transitions between them give rise t,o various types of line spectra, of which we shall discuss the infra-red absorption spectrum and the Raman spectrum.

Infra-red absorption spectrum If two energy levels El and E2 of the molecule are separated by an amount (24.3)

then the nlolecule can absorb light of frequency wand make a, transition from the state El to E 2 • For light polarized in the Xdirection, the transition probability, and hence the absorption coefficient of the bulk liquid or gas, is proportional to the modul.us squared of the matrix elemerlt

(2I D xI 1) = flJ'2*DxP1 dv.

(24.4)

246

GROUP THEORY IN QUANTUM MEOHANIOS

.AnalogotL~ly

to the atomie case (13.7), Dx is the X the dipole moment operator D ~:: --e 2: r~

CODIIJOIlent

+ e L: ZaRtI'

of

(24~5a)

where the sllJnmation is over all t.he electtrons (i) and nuclei ({%) Z« being the atomic nurnber. 'fhe formula (24~4) can be reduced using the wave functions (24.1). In almost all cases, the electronic energy levels of a molecule are separated by about 1 electron volt (10' em-I) or more, corr~ .. sI»onding to visible and ultra-violet frequencies. Infra-red light has insufficient energy to excite the electrons to above the ground state "'eo, so that in' -an infra-red. transition "~e must ha'Va if;elect., 1

== ",elect, I = ¥Seo-

Twning now to the translational wave flmction, we have •

+trans = exp .(ik· Rem)

for a {reely moving molecule, so that

+ .d) =

+trans{R«

exp(ik· A)4ttr~n8(RG&).

Since D is invariant under 8r translation A of the molecule, we have from the fundamental theorem (13.8b) that "'tranS'll and tPtrans,'lt IDJ.lst belong to t,he 3}tillA t,ra;nsl.ation.a,l representa,t,iOTi.S for (24 ...1:) to he n.Ol1."ZerO, i~e. lye :;lluBt ha:ve kl :::: k 2• We consider finally the r\)u~t;loual and Yibratio.n;~l oo·ordinatee, 8;nd refer D iJO the x, y, z co-of-ditlate = =

I+:lb,1 "'ViM I (l/J ~lb,l)· ILJ

V ib,1l

d'lin

(24.8)

JL1 dVn>

is zero or non ..zero. From the matrix. element theorem (13.8b), (24.8) can be non-zero only if the representation of iLl is contained in the reduction of tht) representa,tion D(vJb,2} X D(Vlb,l)*" More.. over, a.~1 uanalin Stl.r.J~ Sj'lllrnetry >J~rgume:utg, v~le ~;al1 aS~!F~ITae t}Utt th(~ t;ornreroo is H:lf:!c 'tl. .(V:~; i"i;~ if the represeIltatio:n of fl-J is (~ont.a,ined in .Df'{·lbJ·~~ >:: J~)t"Vffj,li:'jl,,. theIl '\~te expect; (24~8) to be ni~Il"zero (apart fi-;)ID e~e('ld(jntal oa:~tc.:1) &fi Inng l1B we· h.a/l~ iJv}lu~~8d all ~)le symluetry IJ1"ope'r!;ies of tlhe By[!t(~In in ,the g~~'OUp (b to tty'hleb tho r~PJ.'ese:nt.ations l·efel>.. ]l'um (.24 . 5&)'1 the P1 traI12fornl like /i, "'9'{;ctJor in. tll?t sa.rn.t8 way ~ 'If ,." fT·:;cA, .,), h,',:, q~I':~J't j~,t1"""{11"'dl""1l('1 ""~"" ·!·'t"u 'V'Apl'}":I..f.~{""tlf.f!I+1i}~··1 ).-;r »l veer.} t'if':l"'t~ J! 4~ "" .. cWt.'" J ,'tf~'.l. '-"'1:':> ~J lI1.. '·' .... "'V~ ;;M'l........ 11]SO in (~i·.£) ijhere is ~ BUnlJJ.Ul'tion over j", ~1() iJh~_~,t the t,~~ansjtit[)n pro'ballility 1B flnit~; if a,,;~ Je.ast one of ,tho mltt..1:~,_~~ eiemel1tb ... ,'

,t'.

t.,. . .

•

r.;>«.;tI,,' ..

.r,.v':'-lf

"l~ itJ-~n i~iff:;;~re{£-'akorption U;;--;;;;~'1;;~-:iJ;;'--;;1 'he tran-s'ition

I tw?u.~, D(~i'f.HJt,) Of, ul~,or Jk~}' fLy,

l

1.P

flz ?,,8

),(11ib,Z}

1~"""UlI_.

"';1'#

,,'..

to

lif a·ruJ, on.lJi i}'! tluz

e ~f t~ di~.e ~rwn;.e~ wmprm~~8 J m:r CQnta·~n.ed ~n the re4uctwn (~f the. ,eprue1l.tattOn r#Vlb,l

.pvib,1

1"eJweo2.'1l1ta.. , r

V':vlb,li* •

_ _ nr_ _ _

~""11

Raman spectrum In t.he Raman effect, one shines light of a fixed frequency WI usually from the mercury spectrum on to a suhstance, and one observes light of a cE.fferent frequency W2 emerging scattered by the substance. The molecules of the substance meanwhile make a vibrational and/or rotational transitioll lJI1 to ljl?,.' the total energy of course being conserved

IliCtJ; + El

= liCtJ 2

+ E 2• ]

Thus it is the frequency shift (wI -- (02) that corresponds to the energy difference E2 - E10 The intensity' 1'}. of the scattered. light is given by , (24.10)

Here ct21 is a component of the polarizability tensor of the molecule when. the nuclei are in fixed positions Rcx- The suffices 1 and 2 refer to the directions X, Y or Z of polarization of the incident and emergent beams. As well as being derivable quantum mechanically as in problem 24.11, (24.10) is suggested by a sen1i-classical argument. The electric field fit of the incident beam induces a dipole moment in the nlolecule of strength f.L =

at!1 -

This oscillating dipole radiates energy at a rate proportional to 11'2, and Itfll2 is proportional to the indicent intensity. Thus

249

STRUCTURE AND VIBRATIONS OF :Y.IOI.iECUJ-lES

Since the molecule is itself vibloating \\71th frequencies combinatioll frequencies

W,h

the various (24.11)

occur among the radiated light. Just as in the direct optical ahsorption theory the classical JLX gets replaced thp, quantunl ll1echanical matrix eJement (24.6), so analogously C(?l gPtJ8 replaced hy7

= .r tP:Ot,21U lJit#rot,l dv

I¢':ib,2

IXij tPvib,l d'1:.

(i, j 6urumed)

As ill the case of infra-red absorption, the first integral deterlnines the purely rotational Ralnall effect and the rotational fine structure of each vibrational Raman line, though these effects are only rarely observable in practice. The last integral in (24.12) detel'tuines the selection rules for the possible vibrational transitions. The cti vely like X2, y2, Z2~ X!/, yz, zx. Analogously to (24.9)) we obtain the rule:

r i

a po.s8ib/r:. vibrational tmnsit1;on tPvib,l to tPVib,2 can give 'rise to a line in the Rarnan .spectrU'fn -if and only if the 'representation D,;: both absent from a spectrum. It also means that in a, Raman .Ipectrum the lines are symmetrically placed about the exciting

"'0

t 'l'his follows from the fact that is non-degenerate and therefore trans.. ,:rnlft 8,ooording to a one-dimensional representation, X(T) = exp (itx) say. For a ,'na-dhnensional representation we have X*(T) X(T) = X(T-~l) X(T) = X(E) =- 1.

frequency w, the abS011)tioll and emission of a gi~fell eIE.~'itgJ J,Hle!'~1 ence tEl -- E 2 , being citlher both p~sible or both. forhidden 'prOCeg3CB. We now mention two complieatio:n.8. Firstly the expansion \jt45~8) contains ot,ller terms besides aN and (a*)N, for insUl,nce tha tt:,rm aN-1a,* which by the same argunlerlt. is soor~ to give all (.1V -~- 2)quantum trallsition. If the corresponding frequency WJ~ ~ non· degenerate; it ean easily be 8h~)"\rvr~t that tt.1is (N - 2)-quantuRl transition wou:.1d alreflAiy have been found in analysing the term q:'-2. In th~~ case of degenerate frequencies however, it is possible to obtain new lines this way. ~rhe seeoIld OOlnplication arises ortly witb d.oublj" or triply degeIlera:te frp:querlci~. Consider the term Q{32qy. }""'rom (28.33) this gives the transitioI1S

E2 - El

=

(2Te86!ltation, then qtJ' transforms according to the identity representation. 'l'hus qfJN a.nd --2

if

transform in the same way, and are either both present or both absent from the exptID8ions (24.14) and (24.19).

256

GROUP 'l'HEORY IN QUANTUM: MECHANICS

where 0 is the distance t.hrough which the atoms vibrate. Also ~ ...VIJ

. r'J

~...

~

I

vqfJ

b

r~

C

t".

-d- -

oqj' cq{j

--,

I

llLj (ml/2R)N

~----

where R is the internuclf~a r dist.ance. Thus the intensit.v of the lines is proportional to

~"e

call estima,te S as f()llow~. Classically, when the atoIHS are at the end of their swing, all the energy is in the form of potential energy and E ~; (n !)Iku ==: i1nw 2 82 ,

-r

typica.l vaJue.s n ::=: f)!, Titui == 200 c1n- 1, n~ =-= 15 proton masses, R == ] O-scln~ we filld SIR ~ 0 -I, so that succeRsive sets of iV-quantum t.ra,nsitions differ in intensity by a factor of about {Sf R)2 ~ 1/100. v\7'e 8haH se:(~ below that the presence of anharluonicity increa.ses the intlensitie~ of Home lines rather, but it nevertheless remains true that tr£Hlbit.ions ~ith high values of N becolne increasingly \ve3.tker. Ifor eXfunple, in the observed spectrurrl of CO! as listed by Herzberg (1~}~-5, p. 274), the line with highest N is a 7-quantum transition. Lljl these intensities h[V'7e to he tHultiplied, of course, by the fraction 11 of rnolecules act.uaHy ill the state ¢JVibtl initially) and hence able to make the particular trHJJ.sitioIl. If -the sample is in equUibrium at an absolute temI)erature Tot then ~ra,king

(24.34)

Molecular vibrations ha''V'e 8,11 eneI'gy of about 2000 em-I, wherea.~, at room temperature k:F =-== 200 cm- I . ~rhus tran8itions fron,-, exci.ted states win tend to be very ,veale 'Iihis id seen very clearly .tn t,h..,c R aman e.~_lec·, .-r t \~/.~erf; . 1" t,!le lUles on tine fllp:~n. ~]'f~quency 81"de 0 f . >. I' .. , .. k ". ) are .~:·lUC11\ "\veak er t h an TiLlIRltl011 \:-HH,tA~t;G eE', nnes the InOlnen1, those on the IO~N ffequcncy ~idf; (Stoke~; lines)] vh,~~ intensity ratio of cOl'resporlciing line'.; ~}eing 'i

1

i'

c:

.r.

'::'i.,L

AlIharnwDicity

We shall now examine one of the simplifying assumptions that has been Inade throughout § 2H and the present seetioll so fa-r. In § 23, we started wit.tl all arhii;rn.ry potential V{~) governing the

S1.'RUCTURE AND VJBRA.'fIONS O"D~ MOLECULES

:~57

nuclear Inotions, and expanded this about the equilibriulll COlIfiguration as a p()~'er series (2!l.6) in the QfJ. Assuming that the amplitude 0 of a nuclear vibration is very small compared "\vith -the internuclear distance R, ,ve retain in £Jvilt (23.9) onl)' the quadratic terlns, a.nd t·hi~ then led to the shnple harlnonic oscillators Hanliltonian (23.1~) or (24.25). IIo\vever, the assulupt.ion is not strietly va.lid because ,~~c showed above th8 t 0/ R ~~ 0·1. The t.rue situatiun is that the vibrations are sufficientl.y small for (24.25) to be a· very good first approxiJnation to the IIftmiltonian, but that the real Haluiltonian contains anharmonic perturbing terms such as (24.35)

whic:-h' have SIna.U but observable ~ftects. rrllese effects are of t1VO kinds, (i) an inrrease in the intensity of some lines, Hind (ii) a splitting of some degenerate levels and lines. As regards intensities, the JJreSeIlce of perturbing anharmonic terlns in the Ilamiltonian means that the eigenstates ~~]] not have the sinlple harDl0mc oscillator form (24.26), and those argument~ ,,,hieh depend on this specific forIn have to be re~examined. }"'or instance the selection rule (24.17) no longer holds, and the terlu qe in. the expansions of f11 (24.14) and (Xtj (2t.t19) can give l~ise to various lilIes in addition to the fundanlental frequenc)~ (Of'. 1'hese other lines "rill have an intensity proportional to the amount of anharrnorncity in the pot,ential. r.fhis can be seen explicitly by considering the gJB.e 3 term in (24.35). "'e restrict ourselves to the case of a single frequency w and drop the suffix fie Froln ordinary perturbation theory and the selection rules (24.31) for the ID8)triX elementR of the perturbation, ,ve have that the perturbation rnixes wa.ve functions if;n wjth L1n:-== ±3 (Schiff 1949, p. 151). The perturbed wave functions are

+ cn+at#n+8 -t- other terms -t- d n + if;n+l + other terlns,

¢JpertCn)

:=-...:

Cn¢;/ft

+ .1:)

::.:::=

dn + 4(!·rt+:l

~/'pert(n

1

where en ~ d n +4 ~ 1, Cn+3 ~ d n +1 ~ g, and where the wave functions 011 the right .. hand side are the unperturbed ones (24.24). If the expansion of Itj cOlltains the terln Aq, then we have

f "';'rt(n + 4)/Li¢pert(n) dv ~ ACn+3d"H f ifi:Hqifin+3 dv + Acnd n+1 f ifi:+tqifin dv oc g, so that the Jinenr terlll Aq in the expansion of I1-J leads to a 4-quantum tra.nsition with intensit,y proportiona.l to g2(O/ R)2. In this kind of

258

GROUP THEORY IN QUANTlJM MEOHANIOS

way the allowed N.quantum trsJusitions with .high N usually have their intenSities mOl'eased consider~bly above that expected from the unperturbed simple barnlomc oscillator calculation . .Some of the energy levels (24.26a) of the simple harmonic Or."Scillator Hamiltonian are highly degenerate, and the a.nbarmonic pert1urbation (24.35) produces a splitting in it. Consider a pair of degenerate frequencies Wfj = (I))' = W 88,Y. The energy E(n)

= (nlJ + l)nwlJ -t. (ny + !)ntciy

=:;:

,nJl

+ ny + l)Aw (24.36a)

depends only on the total number of quanta f~ = nl3 + n.y. The level is therefore (n + I)-fold degenerate, t colTesponding to the pairs of values of nfl and ny, n, n --- 1. . . , n -.~,. 9.a.J, .... ) 1, O·, n ,.,a n" = 0, 1, 2J ~ . . , in -- l:t n. ~

(24.36b)

For the point-groups, the irrodl1cible representations are at most a-dimensional, so that for 11, greate'f than 2 or 3, the wave functions of the level must form the basis of a reducible representation. 'I1his will be split by the perturbation (24.H5) into irreducible components in accordance with the general theory of § 6. The splittings are usually up to about 50 em-I. For example in ozone the 1«- :.=: 2 level of the degenerate freqllencies W2 = Wa (24.16) is three-fold degenerate in the simple harmonic oscillator approximation, th~ . wave functions being (24.37j

These transform according to the representation A.' 1 + E' of the symmetry group 6m2 (see below), and the anharmonic perturbation therefore splits the level into a non-degenera.te one (L4'l) and a doubly degenerate one (E /). 'The spectral lines are split correspondingly.

Symmetry 01 eM wave fUBC'iona In order to study the splittings of the energy levels by anhar .. monicity in detail, it is necessary to know the transformation properties of the wave functions (24.26b). From (24.27), +0 remains invariant under the symmetry group of the molecule just like the

t Allihough this degeneracy is obvious by inspection, it is an int~resting exercise to derive it group theoretioally .. (See problema 24.9 and 24.10.)

259

STRUOTURE AND VIBRATIONS OF MOLECULES

potential energy I 2: wfJ2q,. does. We also havet PfJ = till (23.1.2), so that PfJ transforms in the same way as qn, and henoe so do af3 and a{J* (24.22). If we write the representation of qj3 and al3 as D({J), we have that the state zp(n~) (24.26b) with energy ~ (nf) !)/i,wfj transfomls according to the product representation

+

D

= D(njl)

(24.38)

X D(n,,) X ••• ,

where D( nIJ) = D(fJ) X D(IJ) X Di tranSfotnl according to D, the products CPirp; are said to transfornl according to the symnlet.rical product D >< D (sym). This i~ egual to the ordinary product D X D with 80rne of the irreducible eonlpOn~]\V3 dropped out, because CPi,pj and ¢i¢t are not} independent quan tities (see problenl 9.4·), The saIne 8.rpplies to repeated products _D >< D X D X ... (sJrn). ()leariy 'In i~he ease of degenerate frequenc.i{-~8 it is the symnletricaJ produet (24.3H) th.H.t we ha.ve to use in (2o'.t38) to ohtain . t'Inn. to'he .Lt;O t [1,11 representa. From (14}l) the characters of the ordIna.ry nth product are [X(1 ' )]n, where the X(,T) are the chftra;cters of DC13 ), but ,from \vhat has "been said above thjg CJlnUCl!, he 81pplied to the sYlrnnetric product D(n) )fnd \ve ha,ve to dev~elop a n('\,\" forlnula for the eharRcters of D(n} which ,ve denote hy X