Symmetry. An Introduction to Group Theory and Its Applications

254 Pages·1963·7.936 MB·English

by  R. McWeeny and H. Jones (Auth.)

THE INTERNATIONAL ENCYCLOPEDIA
OF PHYSICAL CHEMISTRY AND CHEMICAL PHYSICS
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University)
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SYMMETRY
AN INTRODUCTION TO GROUP THEORY
AND ITS APPLICATIONS
BY
R. MCWEENY, B.SC, D.PHIL.
READER IN QUANTUM THEORY
UNIVERSITY OP EEELE, ENGLAND
PERGAMON PRESS
OXFORD . LONDON . NEW YORK . PARIS
1963
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Library of Congress Catalogue Card Number 63-10017
Set in Modern 7-11 on 13 pt.
and printed in Great Britain by The Whitefiriars Press Ltd.
THE INTERNATIONAL ENCYCLOPEDIA
OF PHYSICAL CHEMISTRY AND CHEMICAL PHYSICS
Editors-in-chief
E. A. GUGGENHEIM J. E. MAYER
READING LA JOLLA
F. C. TOMPKINS
LONDON
Chairman of the Editorial Advisory Group
ROBERT MAXWELL
PUBLISHER AT PERGAMON PRESS
List of Topics and Editors
Mathematical Techniques H. JONES, London
Classical and Quantum Mechanics R. MCWEENY, Keele
Electronic Structure of Atoms C. A. HUTCHISON, JR., Chicago
Molecular Binding J. W. LINNETT, Oxford
Molecular Properties
(a) Electronic J. W. LINNETT, Oxford
(6) Non-electronic N. SHEPPARD, Cambridge
6. Kinetic Theory of Gases E. A. GUGGENHEIM, Reading
7. Classical Thermodynamics D. H. EVERETT, Bristol
8. Statistical Mechanics J. E. MAYER, La Jolla
9. Transport Phenomena J. C. MCCOUBREY, Birmingham
10. The Fluid State J. S. ROWLINSON, London
11. The Ideal Crystalline State M. BLACKMAN, London
12. Imperfections in Solids Editor to be appointed
13. Mixtures, Solutions, Chemical
and Phase Equilibria M. L. MCGLASHAN, Reading
14. Properties of Interfaces D. H. EVERETT, Bristol
15. Equilibrium Properties of R. A. ROBINSON, Washington, D.C.
Electrolyte Solutions
16. Transport Properties of R. H. STOKES, Annidale
Electrolytes
17. Macromolecules C. E. H. BAWN, Liverpool
18. Dielectric and Magnetic Properties J. W. STOUT, Chicago
19. Gas Kinetics A. TROTMAN-DICKENSON, Aberystwyth
20. Solution Kinetics R. M. NOYES, Eiujene
21. Solid and Surface Kinetics F. C. TOMPKINS, London
22. Radiation Chemistry R. S. LIVINGSTON, Minneapolis
INTRODUCTION
THE International Encyclopedia of Physical Chemistry and Chemical
Physics is a comprehensive and modern account of all aspects of the
domain of science between chemistry and physics, and is written
primarily for the graduate and research worker. The Editors-in-Chief,
Professor E. A. GUGGENHEIM, Professor J. E. MAYER and Professor
F. C. TOMPKINS, have grouped the subject matter in some twenty
groups (General Topics), each having its own editor. The complete
work consists of about one hundred volumes, each volume being
restricted to around two hundred pages and having a large measure of
independence. Particular importance has been given to the exposition
of the fundamental bases of each topic and to the development of the
theoretical aspects; experimental details of an essentially practical
nature are not emphasized although the theoretical background of
techniques and procedures is fully developed.
The Encyclopedia is written throughout in English and the recom­
mendations of the International Union of Pure and Applied Chemistry
on notation and cognate matters in physical chemistry are adopted.
Abbreviations for names of journals are in accordance with The World
List of Scientific Periodicals.
VÜ
PREFACE
THE OPEBATIONAL principles underlying the construction of sym­
metrical patterns were certainly known to the Egyptians. They were
formulated symbolically during the nineteenth century and played
their part in the development of the theory of groups. But only during
the past thirty-five years has the immense importance of symmetry in
physics and chemistry been fully recognized.
The value of group theoretical methods is now generally accepted.
In chemical physics alone, the symmetries of atoms, molecules and
crystals are sufficient to determine the basic selection and intensity
rules of atomic spectra, and of the electronic, infra-red and Raman
spectra of molecules and crystals. The vector model for coupling
angular momenta, the properties of spin, the Zeeman and Stark
effects, the splitting of energy levels by a crystalline environment, and
even the nature of the periodic table may all be traced back to
symmetry of one kind or another.
Many excellent textbooks on group theory and its applications are
already available. They range from rigorous but formal presentations
to highly condensed accounts which deal with particular applications
and give only a superficial treatment of underlying concepts. The
first kind of approach is unpalatable, except to the professional
mathematician ; the second provides a useful vocabulary and a set of
working rules—but no real understanding of group theory. The
present book is intended primarily for physics and chemistry graduates
who possess a fair amount of mathematical skill but lack the formal
equipment demanded by the standard texts (by Wigner, Weyl and
others). Accordingly, the elementary ideas of both group theory and
representation theory (which, incidentally, provides the basic mathe­
matical tools of quantum mechanics) are developed in a leisurely but
reasonably thorough way, to a point at which the reader should be
able to proceed easily to more elaborate applications. For this purpose,
emphasis is placed upon the finite groups which describe the symmetry
of regular polyhedra and of repeating patterns. By restricting the scope
in this way, it is possible to include geometrical illustrations of all
the main processes. In fact, over a hundred fully worked examples
have been incorporated into the text.
xiii
XIV PREFACE
A discussion of the permutation group and of the (continuous)
rotation groups would have carried this volume far beyond its pre­
scribed length. However, there is much to be said for dealing with
special groups in the context in which they occur. A study of the
rotation group, for example, is essential to any full discussion of the
quantum mechanical central field problem. And such applications
present no real difficulty once the basic principles have been grasped.
The book is constructed so that it may be read at various levels and
with emphasis on any one of the main fields of application. Chapter 1
is concerned mainly with elementary concepts and definitions; Chapter
2 with the necessary theory of vector spaces (though sections 2.8-10
may be omitted in a first reading). Chapters 3 and 4 are complementary
to 1 and 2: they provide the reader with an opportunity of actually
working with groups and representations, respectively, until the ideas
already introduced are fully assimilated. The more formal theory of
irreducible representations is confined to Chapter 5. In a first reading
it would be sufficient to grasp the main ideas behind the orthogonality
relations (5.3.1), passing then to section 5.4 for the properties of group
characters, and finally to the construction of irreducible basic vectors
according to (5.7.8) and the examples which follow. The rest of the
book deals with applications of the theory. Chapter 6 is concerned
largely with quadratic forms, illustrated by applications to crystal
properties and to molecular vibrations. Chapter 7 deals with the
symmetry properties of functions, with special emphasis on the eigen­
value equation in quantum mechanics. Chapter 8 covers more advanced
applications, including the detailed analysis of tensor properties and
tensor operators.
Much of the material in this book has been presented to final year
imdergraduates and to graduate students in the Departments of
Mathematics, Physics and Chemistry at the University of Keele, and
the exposition has often been guided by their response. It is a pleasure
to acknowledge this enjoyable and valuable contact, and also the many
conversations with colleagues both here and at the University of
Uppsala. Finally, anyone writing a book in this field must be deeply
conscious of his debt to those who pioneered the subject, over thirty
years ago; I should like to acknowledge, in particular, that the books
by Wigner and Weyl have been a constant influence.
R. MCWEENY
CHAPTER 1
GROUPS
1.1· Symbols and the Group Property
A large part of mathematics involves thç translation of everyday
experience into symbols which are then combined and manipulated,
according to determinable rules, in order to yield useful conclusions.
In counting, the symbols we use stand for numbers and we make such
statements as 2 + 3 = 5 without giving much thought to the meaning
of either the symbols themselves or the signs + (which indicates some
kind of combination) and = (which indicates some kind of equivalence).
In group theory we use symbols in a much wider sense. They may, for
instance, stand for geometrical operations such as rotations of a rigid
body ; and the notions of combination and equivalence must then be
defined operationally before we can start translating our observations
into symbols. We do arithmetic without much thought only because
we are so familiar with the operational definitions, which are far from
trivial, which we learnt as children. But it is worth reminding ourselves
how we began to use symbols.
How did we learn to count? Perhaps we took sets of beads, as in
Fig. 1.1, giving each set a name 1, 2, 3, . . . (the " whole numbers "). A
set of cows, for example, can then be given the name 3, or said to
• · · · · · ·
• ·· ·· ·· ·· ··
I 2 3 4 5 6
FIG. 1.1. Sets of objects representing the whole numbers.
contain 3 cows, if its members can be put in " one-to-one correspon­
dence " with the beads of the set named 3 (a bead for each cow, no
cows or beads left over). The same number is associated with different
sets if, and only if, their members can be put in one-to-one corre­
spondence : in this case the numbers of objects in the different sets are
equal. If x objects in one set can be related in this way to y objects in
another set we write x = y. If the numbers of fingers on my two hands
are x and y, I can say x = y because I can put them into one-to-one
correspondence : and I can say x = y = 5 because I can put the
1
2 SYMMETRY
members of either set in one-to-one correspondence with those in the
set named 5 in Fig. 1.1. This provides an operational definition of the
symbol =. We observe that the sets in Fig. 1.1 have been given
distinct names because none can be put in one-to-one correspondence
with any other : l ^ f 2 ^ £ 3 ^ 4 . . .. Numbers may be combined under
addition (or " added "), for which we usually use the symbol + , by
putting together different sets to make a new set. If we put together a
set of 4 objects and a set of 1 object the resultant set is said to contain
(4 + 1) objects : but there is another name for the number of objects
in this set because it can be put in one-to-one correspondence with the
set of 5 objects. Hence the different collections contain equal numbers
of objects and we write 4+1 = 5. The whole numbers are con­
veniently arranged in the ordered sequence (Fig. 1.1) such that
1 + 1 = 2, 2+1 = 3, 3+1 = 4, etc., so that sets associated with
successive numbers are related by the addition of 1 object. Generally,
we say that if the members of sets containing x and y objects, when
put together to form a new set, can be put into one-to-one corre­
spondence with those of a set of z objects, then x + y = z. The opera­
tional meaning of the law of combination (indicated by the + ) and of
the equivalence ( = ) is now absolutely clear. But, of course, the
terminology is quite arbitrary : instead of 2 + 3 = 5 we could just
as well write
2 combined with 3 gives 5 or 2 ! 3 : 5
What matters is that we agree upon (i) what the symbols stand for,
(ii) what we shall understand by saying two of them are equal, or
equivalent, and (iii) what we shall understand by combining them.
In group theory we deal with collections of symbols, A, ß, . . . ,
R, . . . , which do not necessarily stand for numbers and which are
accordingly set in distinctive type (gill sans—instead of the usual italic
letters). We refer to the members of the collection as " elements "
and often denote the whole collection by showing one or more typical
elements in braces, {R} or {A, ß, . . . , R, . . . }. The elements may,
for example, represent geometrical operations such as rotations of a
rigid body, and the law of combination is then non-arithmetic. Never­
theless rotations can be compounded in the sense that one (R) followed
by a second (S) gives the same final result as a third (7) : the italic
phrases describe, respectively, the law of combination (sequential
performance) and the nature of the equivalence. In order not to
restrict the law of combination we shall normally leave the question
completely open, simply writing side by side the two symbols to be
GROUPS 3
combined and, purely formally, calling the result of combination their
" product ". We adopt the = sign to denote equivalence in whatever
sense may be appropriate. In dealing with any particular collection of
elements, the meaning of the term " product " and the significance of
the = sign will be agreed upon at the outset.
When the elements of a collection are the natural numbers and the
law of combination is addition one important property is at once
evident. Three numbers may be combined, by applying the law of
combination twice, in either of two ways : with the general terminology
these are
(i) (abc) = a(bc) or (ii) (abc) = {ab)c
1 2
and the results are always identical. To give an example
(i) (2 3 4^ = 2(34) = 27 = 9 (ii) (234) = (23)4 = 54 = 9
2
The order in which the terms of a " product " are dealt with is therefore
irrelevant and there is no need to bracket together the pairs to show
how a triple product must be interpreted. The law of combination is
said to be associative. In this case there is no ambiguity and an ordered
product of any number of elements is uniquely equivalent to a certain
single element. This is a fundamental property of all the collections
we shall study and is assumed in defining the group property :
Any collection of elements {A, ß, . . . , R, . . . } has the
group property if an associative law of combination is
defined such that for any ordered pair R, S there is a
unique product, written RS, which (in some agreed sense) * * ' '
is equivalent to some single element T which is also in
the collection.
The condition that the combination of two elements should yield an
element of the same kind (i.e. also in the collection) is itself non-trivial :
the collection is then closed and we refer to the closure property. It
should be noticed that the law of combination refers to an ordered
pair, so that the possibility RS ^ SR is admitted from the start. It
must also be stressed that a collection with the group property is not
a group unless further conditions are fulfilled. Before introducing
these we give examples of various collections with the group property :
EXAMPLE 1.1. We first consider the natural numbers under addition.
This collection has already been mentioned in defining the group
property—which it evidently possesses. In dealing with such collec-
B