THE INTERNATIONAL ENCYCLOPEDIA OF PHYSICAL CHEMISTRY AND CHEMICAL PHYSICS Members of the Honorary Editorial Advisory Board J. N. AGAR, Cambridge R. S. MULLTKEN, Chicago R. M. BARRER, London R. G. W. NORRISH, Cambridge C. E. H. BAWN, Liverpool R. S. NYHOLM, London N. S. BAYLISS, Western Australia J. T. G. OVERBEEK, Utrecht R. P. BELL, Oxford K. S. PITZER, Rice University, Houston C. F. BÖTTCHER, Leiden J. R. PLATT, Chicago F. P. BOWDEN, Cambridge G. PORTER, Sheffield G. M. BURNETT, Aberdeen I. PRIGOGINE, Brussels (Free J. A. V. BUTLER, London University) C. A. COULSON, Oxford J. S. COURTNEY-PRATT, New Jersey R. E. RICHARDS, Oxford D. P. CRAIG, London Sro ERIC RIDEAL, London F. S. DAINTON, Leeds J. MONTEATH ROBERTSON, Glasgow C. W. DA VIES, London E. G. ROCHOW, Harvard B. V. DERJAGUIN, MOSCOW G. SCATCHARD, Massachusetts Institute M. J. S. DEWAR, Chicago of Technology G. DUYCKAERTS, Liège GLENN T. SEABORG, California D. D. ELEY, Nottingham (Berkeley) H. EYRING, Utah N. SHEPPARD, Cambridge P. J. FLORY, Mellon Institute R. M. Fuoss, Yale R. SMOLUCHOWSKI, Princetown P. A. GIGUERE, Laval H. STAMMREICH, &5O POUZO W. GROTH, Bonn SIR HUGH TAYLOR, Princeton J. GUERON, Paris H. G. THODE, McMaster C. KEMBALL, Queen's, Belfast H. W. THOMPSON, Oxford J. A. A. KETELAAR, Amsterdam D. TURNBULL, (?.i£., Schenectady G. B. KISTIAKOWSKY, Harvard H. C. LONGUET-HIGGINS, Cambridge A. R. J. P. UBBELOHDE, London R. C. LORD, Massachusetts Institute of H. C. UREY. California (La Jolla) Technology E. J. W. VERWEY, Philips, Eindhoven M. MAGAT, Paris B. VODAR, Laboratoire de BeUevue, R. MECKE, Freiburg France SIR HARRY MELVILLE, D.S.I.R., London M. KENT WILSON, Tw/te S. MIZUSHIMA, Tokyo W. F. K. WYNNE-JONES, King's J. A. MORRISON, N.R.C., Ottawa College^ Newcastle-upon-Tyne 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 PERGAMON PRESS LTD. Headington Hill Hall, Oxford 4 and 5 Fitzroy Square, London, W.l PERGAMON PRESS INC. 122 East 55th Street, New York 22, N.Y. GAUTHIER-VILLARS ED. 55 Quai des Grands-Augustine, Paris, 6e PERGAMON PRESS G.m.b.H. Kaiserstrasse 75, Frankfurt am Main Distributed in the Western Hemisphere by THE MACMILLAN COMPANY-NEW YORK pursuant to a special arrangement with Pergamon Press Limited Copyright © 1963 PERGAMON PRESS LTD. 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