Table Of ContentPoint. Gr?up Symmetry
Apphcatl0ns
Methods and Tables
Point. Gr?up Symmetry
Apphcatl0ns
Methods and Tables
Philip H. Butler
University 0] Canterbury
Christchurch, New Zealand
PLENUM PRESS • NEW YORK AND LONDON
Library of Congress Cataloging in Publication Data
Butler, Philip H
Point group symmetry applications.
Bibliography: p.
IncIudes index.
1. Symmetry groups. 2. Point defects. 3. Crystallography, Mathematical. I.
Title.
QD9ll.B89 530.1 '2 80-17947
ISBN-13: 978-1-4613-3143-8 e-ISBN-13: 978-1-4613-3141-4
DOI: 10.1007/978-1-4613-3141-4
© 1981 Plenum Press, NewYork
Softcover reprint of the hardcover 1s t edition 1981
A Division of Plenum Publishing Corporation
233 Spring Street, New York, N. Y. 10013
All rights reserved
No part of this book may be reproduced, stored in a retrieval system, or transmitted,
in any form or by any means, electronic, mechanical, photocopying, microfilming,
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Preface
The mathematical apparatus of group theory is a means of exploring and
exploiting physical and algebraic structure in physical and chemical prob
lems. The existence of structure in the physical processes leads to structure
in the solutions. For group theory to be useful this structure need not be an
exact symmetry, although as examples of exact symmetries we have that
the identity of electrons leads to permutation symmetries in many-electron
wave functions, the spatial structure of crystals leads to the Bloch theory of
crystal eigenfunctions, and the rotational invariance of the hydrogenic
Hamiltonian leads to its factorization into angular and radial parts.
In the 1930's Wigner extended what is known to mathematicians as
the theory of group representations and the theory of group algebras to
study the coupling coefficients of angular momentum, relating various
properties of the coefficients to the properties of the abstract group of
rotations in 3-space. In 1949 Racah, in a paper on rare earth spectra,
showed that similar coefficients occur in other situations. Immediately a
number of studies of the coefficients were begun, notably by Jahn, with his
applications in nuclear physics. In the years since then a large number of
physicists and chemists have added to the development of a general theory
of the coefficients, or have produced specialized tables for a specific
application. Applications now range from high-energy physics to biology.
Two unfortunate consequences of the wide range of applications
coupled with the simultaneous development of the appropriate mathe
matics are that separate notations and terminologies have been introduced
in the different areas, and that the most powerful techniques have not been
available to everyone. The theory of coupling coefficients for compact
groups has now advanced to a stage that I have felt it timely to produce a
v
vi Preface
complete set of coupling coefficients for all the point groups-those groups
that are subgroups of the group of orthogonal transformations in 3-space,
03' I have taken the opportunity to inc1ude a modem introduction to the
relevant group theory, together with a summary of the properties of the
coupling coefficients.
Group theory produces quantitative results. It does not produce only
selection rules. The key theorem is the Wigner-Eckart theorem. Although
the proof is rather short, the Wigner-Eckart theorem is a very significant
part of the book, for it is usually the key to the applications. The theorem
provides numerical relations between matrix elements; it is the link be
tween group theory as a piece of mathematics and group theory as a
simple computational too1. Some atomic and molecular applications are
inc1uded, but, needless to say, no attempt has been made to make the book
complete in this respect.
A serious barrier to the fruitful application of the Wigner-Eckart
theorem has been the existence of conflicts in the notations and phase
choices of previous workers. My notation and my phase choices are c10sely
based on the one book known in all subject areas, The 3j and 6j Symbols,
by Rotenberg, Bivins, Metropolis, and W ooten. That book gives a large
tabulation for the group 803 in the 802 (JM) basis. This book comple
ments the above in that we give as complete a set of tables for all other
bases of 03' namely the bases formed by using any of the 32 crystallo
graphic point groups or the noncrystallographic point groups (the icosa
hedral group K, Doo' and Dn and Cn, n finite).
It is a pleasure to thank many colleagues for the advice and assistance
given to me during the preparation of this book. Brian Wyboume, Mike
Reid, Ric Haase, Clare Churcher, Paul Bickerstaff, Alec Ford, Geoff
Stedman, and Susan Piepho deserve special mention. I wish to thank also:
Janet Warburton, who carefully and cheerfully typed and retyped a
difficult manuscript; Alan Wilkinson and others of the University Com
puter Centre who gave their computer expertise; Cliff Strange, of Printset
Processes, who taught me the basic typesetting skills; Betty Bruhns, of
Plenum, who shared the problems of table layout; and my wife Pamela,
who has given years of patient encouragement.
Philip Butler
Contents
1. Introduction ............................................ .
1.1. A Guide to the Use of the Text . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 2
1.2. Notation and Phase Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4
2. Basic Concepts ........................................... 7
2.1. Irrep Spaces .......................................... 8
2.2. Group-Subgroup Bases .................... . . . . . . . . . . . . .. 18
2.3. Coupling Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 23
2.4. Character Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 28
2.5. Complex Conjugation ................................... 36
2.6. Spin Irreps and Labeling Irreps. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 38
3. Thejm Factors andj Symbols ................................. 43
3.1. Thejm Factors ........................................ 44
3.2. Thej Symbols ......................................... 48
3.3. Properties ofjm Factors andj Symbols ....................... 55
3.4. Computational Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 68
3.5. Phase and Multiplicity Choices ............................. 71
4. The Wigner-Eckart Theorem ................................. 83
4.1. Basis Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 84
4.2. The Wigner-Eckart Theorem .............................. 86
4.3. Coupled Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.4. The Standard Racah Tensors of S03 . . . . . . . . . . . . . . . . . . . . . . . .. 94
5. 03 and Its Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 99
5.1. Direct Product Groups ................................... 100
vii
viii Contents
5.2. Isomorphie Subgroups ................................... 102
5.3. Specification of x, y, z Axes and Bases ........................ 107
6. Properties ofthe Dibedral Groups .............................. 129
6.1. The Structure of the Dihedral Groups ........................ l30
6.2. The Chain S03 :::lDoo :J S02 ............................... l32
6.3. Finite Subgroups of Doo ••••••••••••••••••.•••••.•.••••.•• l36
7. Fractional Parentage Coefficients .............................. l39
7.1. The Parentage Concept ................................... 140
7.2. Continuous Matrix Groups ................................ 142
7.3. Complete Parentage Schemes .............................. 146
7.4. Strong-Field Parentage Schemes ............................ 149
8. Time Reversal ............................................ 153
8.1. Time Reversal of States .................................. 153
8.2. Time Reversal and Selection Rules .......................... 160
9. Applications ............................................. 171
9.1. The Free-Ion Hamiltonian ................................ 171
9.2. The Zeeman Interaction .................................. 176
9.3. Ligand Fields ......................................... 182
10. Programming Notes ........................................ 187
11. Group Information Tables .................................... 189
11.1. The Cyclic Groups ..................................... 191
11.2. The Dihedral Groups ................................... 195
11.3. The Tetrahedral Groups ................................. 200
11.4. The Octahedral Groups ................................. 202
11.5. The Icosahedral Groups ................................. 205
12. Branching Rule Tables ...................................... 207
Introduction ............................................. 207
Tables ................................................. 210
13. jm Factor Tables .......................................... 217
Introduction ............................................. 217
Tables ................................................. 220
Contents ix
14. 3j and 6j Symbol Tables ..................................... 429
Introduction ............................................. 429
Tables .................. . ............................. 431
15. 9j Symbols ............................................. .463
Introduction ............................................. 463
Tables ................................................ .464
16. Bases in Tenns of Spherical Harmonics .......................... 513
16. I. Rotation Matrices in the J M Basis .......................... 514
16.2. Spherical Harmonies in Rectangular Coordinates ............... 518
Tables ................................................. 522
REFERENCES ................................................ 553
SUßJECT INDEX .............................................. 557
TAßLE INDEX ................................................ 566
1
Introduction
The immediate purpose of trus book is to present tables of vector coupling
coefficients for angular momentum, where the basis vectors are chosen to
have any one of the many possible point group symmetries. The quantum
theory of angular momentum is a major aigebraic tool in Condon and
Shortley's 1935 book, The Theory 01 Atomic Spectra. Wigner and Racah
(see Biedenharn and van Dam 1965) simplified the algebra of angular
momentum by applying various theorems of group theory, especially
Schur's two lemmas, to the rotation group in three dimensions. These
developments contributed to many advances in the theory of atomic and
nuc1ear spectra in the 1950s and 1960s (see especially Judd 1963 and
Vanagas 1972). Generalizations to coupling coefficients of other groups
were developed as part of this mathematical development toward obtain
ing solutions to these many-body quantum mechanical problems. Mucb. of
this early development of Wigner's and Racah's work was in extending
Racah's group-theoretic treatment of the coefficients of fractional paren
tage and their treatment of tensor operators.
Griffith (1962) led the way with respect to the topic of trus book. He
calculated tables of coupling and recoupling coefficients for various point
groups, and showed how the familiar techniques of angular momentum
could be carried over to point group symmetries. In 1965 the Racah-Wigner
calculus, as these methods are sometimes called, was shown to hold (with
appropriate modifications) for any group which has finite-dimensional
unitary representations (Derome 1966, Derome and Sharp 1965). These
papers were rather incomplete. They said that if certain free phases are
appropriately chosen, then certain simplifications occur, hut they did not
say how to make such choices. Also, like Wigner's preprint (1940), these
1
2 Cbaprer 1
papers did not discuss the question of tensor operators, and tensor opera
tors are an essential part of the Wigner-Eckart theorem.
In recent years many small tables of coupling coefficients for the
point groups have been published, but unfortunately few have used
Derome and Sharp's work. Phases have not been chosen such that a
consistent definition of 3jm factors can be made. A 3jm factor is defined
here as a symmetrized coupling factor (isoscalar factor). A 3jm is defined
so as to have much the same symmetry properties as Wigner's 3j symbol.
Indeed, we shall see that Wigner's 3j symbol is a special case, being a 3jm
factor for the S03 ::1 S02 group chain.
The textual material of this book should be sufficiently detailed to
enable a graduate student in physics or chemistry to leam and to be able
to use the essential group theory behind coupling coefficients. At the same
time the book has been arranged so that an experimentalist experienced in
the quantum theory of angular momentum can use the tables to apply the
methods of tensor operators in any point group basis with a minimum of
reading of the mathematical background.
In Section 1.1 we discuss in detail the layout of the various topics
within the book. In the other section of this introductory chapter various
phase conventions and notational choices are discussed. These two topics
arise in various places within the book, but it is useful to give a self
contained description of these matters. The guiding principle behind our
notations and conventions has been to treat Rotenberg, Bivins, Metropolis,
and Wooten (1959) as the standard for S03 ::1S02 choices.
The diagram techniques of angular momentum theory (Jucys,
Levinson, and Vanagas 1960) have been extended to the general case by
Stedman (1975, 1976). He has shown that these techniques merge with the
techniques of Feynman diagrams and he has chosen his notation for his
diagrams accordingly. Readers who prefer diagrammatic expressions to
algebraic expressions should find his papers readable.
1.1 A Guide to the Use of the Text
The text of this book has a single object, to enable the reader to make
the best use of the tables. The text starts with the assumption that the
reader knows littIe or no group theory but is familiar with elementary
quantum mechanics, for example, he has seen the standard derivation of
the solution of the hydrogenic Schrödinger equation. At the other end of
Description:The mathematical apparatus of group theory is a means of exploring and exploiting physical and algebraic structure in physical and chemical prob lems. The existence of structure in the physical processes leads to structure in the solutions. For group theory to be useful this structure need not be
