Abram I. Fet Group Theory of Chemical Elements De Gruyter Studies in Mathematical Physics Edited by Michael Efroimsky, Bethesda, Maryland, USA Leonard Gamberg, Reading, Pennsylvania, USA Dmitry Gitman, São Paulo, Brazil Alexander Lazarian, Madison, Wisconsin, USA Boris Smirnov, Moscow, Russia Volume 34 Abram I. Fet Group Theory of Chemical Elements Structure and Properties of Elements and Compounds An electronic version of this book is freely available, thanks to the support of libra- ries working with Knowledge Unlatched. KU is a collaborative initiative designed to make high quality books Open Access. More information about the initiative can be found at www.knowledgeunlatched.org Physics and astronomy classification scheme 2010 02.20.-a, 03.65.Fd, 31.10.+z, 31.15.xh, 82.20.-w Author Abram I. Fet † Translator Vladimir Slepkov Tsvetnoj proezd 17 Kvartira 60 Novosibirsk 630090 Russia [email protected] This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs 4.0 License, as of February 23, 2017. For details go to http://creativecommons.org/licenses/by-nc-nd/4.0/. ISBN 978-3-11-047518-0 e-ISBN (PDF) 978-3-11-047623-1 e-ISBN (EPUB) 978-3-11-047520-3 Set-ISBN 978-3-11-047624-8 ISSN 2194–3532 Library of Congress Cataloging-in-Publication Data A CIP catalog record for this book has been applied for at the Library of Congress. Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2016 Walter de Gruyter GmbH, Berlin/Boston Typesetting: Konvertus, Haarlem Printing and binding: CPI books GmbH, Leck ♾ Printed on acid-free paper Printed in Germany www.degruyter.com Contents Preface to the English edition VII Introduction 1 1 Symmetries of a quantum system 5 1.1. Rotation group in quantum mechanics 5 1.2. Electron in the Coulomb field 14 1.3. Broken symmetry 19 2 Observables of a quantum system 28 2.1. Deriving observables from a symmetry group 28 2.2. Quantum numbers of symmetry groups and their physical meaning 35 3 Lie groups and Lie algebras 39 3.1. Lie groups 39 3.2. Lie algebras 46 3.3. Representations of Lie groups 59 3.4. Universal enveloping algebras and Casimir operators 71 3.5. Tensor product of representations 76 4 The principles of particle classification 81 4.1. The concept of spin 81 4.2. Isotopic spin 85 4.3. SU(3) group 90 4.4. Baryon octet and decuplet 95 4.5. Mass formula in the SU(3) symmetry 101 4.6. SU(6) group 104 4.7. Classification principles in a quantum theory 108 5 The symmetry group of chemical elements 114 5.1. Description of the system of elements 114 5.2. The conformal group 119 5.3. A special representation of the conformal group 126 5.4. The symmetry group of the system of elements 131 5.5. Mass formula for atomic weights 141 VI Contents 6 Classification and chemical properties of elements 145 6.1. Small multiplets and chemical properties 145 6.2. Operators of chemical affinity 150 Appendix A. Fock’s energy spectrum of the hydrogen atom 158 A1. Schrödinger equation in the momentum representation 158 A2. Fock transformation 160 A3. Hydrogen spectrum 165 Appendix B. Representations of some groups 169 B1. Local representations of SO(3) and SO(4) groups 169 B2. Spherical functions and reduction of representations 172 B3. Some representations of SU(3) group 179 References 182 Preface to the English edition It is well known that multi-electron Schrödinger equation can give only approximate solutions. Empirical and semi-empirical methods used to find them are not accu- rate and hence they describe the properties and behavior of chemical elements only approximately. The author of this book applied group-theoretical approach, which proved to be effective in building the system of hadrons, to describe qualitatively the properties of chemical compounds which cannot be derived from the equations. This is the main result of his work. This book is useful in yet another respect. It makes the theory accessible to the reading audience with a general chemical background. The author starts with common, familiar subjects and introduces further the ideas of the physics of symme- try as naturally as possible, so that its origin and meaning can be understood without profound background in mathematical apparatus of quantum physics. This book is in no way purely theoretical. Rather, it has strong connections with experimentally measured properties of chemical elements and their compounds. To compare the theory with experiments, the author considered the properties of ele- ments that are associated with their chemical behavior: ionization potentials, atomic volumes, enthalpies of formation of the elements, polarizability of the atoms, their boiling and melting points, enthalpy of vaporization, electron affinity, energies of ionic lattices, and bonding energies of diatomic molecules and some lanthanide com- pounds. The obtained confirmations of the theory suggested that it would be useful to understand the properties of chemical compounds. These hopes largely justified. This book can be regarded a generalization of the ideas presented in the existing literature on this subject. I express my deep gratitude to my colleagues from different countries whose judgements strengthened my intention to publish the book in English. Rem Khlebopros Introduction This book presents a group description of chemical elements considered as states of a quantum system. Atoms of different elements are viewed as elements of a vector space associated with an irreducible unitary representation of some symmetry group. The elements of the group are carried into each other by operators defined on this group. This approach fundamentally differs from the traditional application of group theory to atomic physics, where a symmetry group is used to carry into each other dif- ferent excited states of the same atom. Besides, we describe heavy particles (hadrons) as representations of unitary groups. This approach follows the model proposed inde- pendently by Gell-Mann and Néeman and developed later in the works of Okubo, Gürsey and Radicati, Pais and other authors. The book “Unitary symmetry theory” by Yu. Rumer and A. Fet was published in 1970, in which these ideas were discussed systematically. While working on the book, Rumer had an idea to apply the same approach to the system of chemical elements. After getting over some difficulties, we laid down a group systematization of elements in our joint work (Rumer and Fet, 1971). Already in this article, we emphasized that the approach implies considering symmetry as related to the atom as a whole rather than to its electron shells only. As a group symmetry, we chose a two-dimensional covering group of SO(4) group. In 1972, B. Konopel’chenko extended the symmetry group up to the conformal group (Rumer and Fet, 1971). Barut (1972) and Barrondo and Novaro (1972) independently described the symmetry from the viewpoint of electron shells. Apparently, this clear differentiation between the symmetry of the elements and the traditional shell model helped us to achieve a clearer understanding of the obtained classification. In particular, I suggested a group description of chemical affinity in Fet (1974, 1975), and a mass formula for atoms in Fet (1979a, 1981). While refining the classification, we noticed, apparently for the first time, that the proper- ties of elements change with some regularity. In particular, each p-, d-, and f-family is distinctly divided into two subfamilies where the properties of the elements change by different laws. The results were compared with experimental data by Byakov et al. (1976) and later, with additional material, by Sorokin. Discussions of the method in scientific articles usually imply a sufficient knowl- edge of Lee groups. However, physicists and chemists usually meet in practice only representations of the three-dimensional rotation group, as it was elaborated by Wigner and Weil in the late 1920s. That is why I wanted to write a book that will be useful for this broad audience rather for a narrow circle of theorists. The work met serious difficulties since the available literature on Lee groups did not provide easy understanding of the ideas of symmetry which have acquired much importance in the Physics these last decades. What we have in this area are either systematic treatises for expert mathematicians or manuals for physicists which imitate these treatises or offer a number of separate applications to special cases. Therefore, I tried to state the main concepts of group theory and Lee algebras briefly, and only as far as it is 2 Introduction necessary to understand the symmetry of particles assuming that the reader has physical or chemical education and some basic knowledge of quantum mechanics. Group theory is explained from the very beginning, starting from the definition of a group. However, it would be desirable for the reader to have some preliminary idea of using rotation group for classification of atomic and molecular energy spectra. Group methods are introduced as necessary and are always illustrated by physical examples. Especially, the physical background of spin and unitary spin is discussed in detail. Therefore, the presentation of the material is “genetic” and inductive rather than deductive, that is, the main attention is paid to the origin and meaning of the concepts rather to the formal exactness of the constructions. In particular, mathemat- ical rigor, though followed when possible, is not an end in itself. We cite only those theorems which are constantly used further, their meaning is discussed in detail but the proofs are usually omitted as well as cumbersome calculations which are cited, when necessary, in the Appendices. First chapter of this book is an introduction to quantum chemical group methods and starts with the traditional applications of the rotation group to the hydrogen atom.