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Hyperspherical Harmonics Reidel Texts in the Mathematical Sciences A Graduate-Level Book Series Hyperspherical Harmonics Applications in Quantum Theory by John Avery Department of Physical Chemistry, H. C. 0rsted Institute, University of Copenhagen, Denmark KLUWER ACADEMIC PUBLISHERS DORDRECHT / BOSTON / LONDON Library of Congress Cataloging in Publication Data Avery, John, 1934- Hyperspherlcal har~onlcs ; applicatlons ln quantum theory / by John Avery. p. cm. -- (Reldel texts in the ~athematical sclences) Bibllography; p. Includes lndex. ISBN-13: 978-94-010-7544-2 1. Schrodlnger equatlon. 2. Spherical harmonlcs. 3. Quantum theory. 4. QuantuN chenlstry. I. Tltle. II. Serles. QC174.2S.W28A94 1989 530. 1 '24--dc19 89-31039 CIP ISBN-I3: 978-94-010-7544-2 e-ISBN-13: 978-94-009-2323-2 DOl: 10.1007/978-94-009-2323 -2 Published by Kluwer Academic Publishers, P.O. Box 17, 3300 AA Dordrecht, The Netherlands. Kluwer Academic Publishers incorporates the publishing programmes of D. Reidel, Martinus Nijhoff, Dr W. Junk and MTP Press. Sold and distributed in the U.S.A. and Canada by Kluwer Academic Publishers, 101 Philip Drive, Norwell, MA 02061, U.S.A. In all other countries, sold and distributed by Kluwer Academic Publishers Group, P.O. Box 322,3300 AH Dordrecht, The Netherlands. printed Oil acidfi'ee paper All Rights Reserved © 1989 by Kluwer Academic Publishers Softcover reprint of the hardcover 1s t edition 1989 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner. TO Professor Sir Geoffrey Wilkinson, F.R.S. and to Lady Lise Wilkinson with thanks for their kindness over many years TABLE OF CONTENTS Introduction. • • • . • • • • • . • • • • • • • • • • • . • • • • . • . • • • • • • . • • • • • • ix Harmonic polynomials.""",,"""""""""""""""""""""""""""""" 1 Generalized angular momentum ••••••.•••••••••••••••••••• 11 Gegenbauer polynomials................................. 25 Fourier transforms in d dimensions •••.••••••••••••••••• 47 Fock's treatment of hydrogenlike atoms and its generalization." "" "" """" """" """"""""""""""""""""""""""" 59 Many-dimensional hydrogenlike wave functions in direct space"""""""""""""""""",,.,,"""""""""""""""""""""" 7 7 Solutions to the reciprocal-space Schrodinger equation for the many-center Coulomb problem ••••.••..•• 93 Matrix representations of many-particle Hamiltonians in hyper spherical coordinates •••••••••.•••••••••...•..• 105 Iteration of integral forms of the Schrodinger equation.""" "" "",,"""" """"""""""""" """ """""""""" """""""" 127 Symmetry-adapted hyperspherical harmonics ••••••••••••.• 141 The adiabatic approximation............................ 175 Appendix A: Angular integrals in a 6-dimensional space ...• "" ...•.•.•....•.••..••••...• " ••.••...•.•.....• " 189 Appendix B: Matrix elements of the total orbital angular momentum opera tor. • • • • • • • • • • • • • • • • • • • • • • • • • • . •. 199 Appendix C: Evaluation of the transformation matrix U,,""""""""""""""""""""""""""""""""""""""""""""" 205 Appendix D: Expansion of a function about another c en ter" " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " "" 2 0 9 Appendix E: The set of many-dimensional hydrogenlike wave functions of constant ko •••••••••••••••••••••••••• 213 References" " " " " " " " " " " " " " " " " " " " " " . " " " " " " " " " " " " " " " " " " " " "" 219 Index. " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " "" 247 A molecular orbital in reciprocal space (see Chapter 7). I NTRODUCTI ON P.A.M. Dirac once remarked that the Schrodinger equation solved "all of chemistry and most of physics". It is certainly true that if one could find sufficiently accurate solutions to the Schrodinger equation (or Dirac's relativistic improvement of it), most of the properties of physical and chemical systems could be calculated from first principles. However, the Schrodinger equation for many-particle systems has proved to be difficult to solve without the help of simplifying approximations. In quantum chemistry, the most common approach has been to separate the motion of the nuclei from the electronic wave equation by means of the Born-Oppenheimer approximation. The many electron Schrodinger equation is then reduced to a one electron equation by means of the Hartree-Fock approximation. Finally, the deficiencies of the Hartree Fock approximation are usually corrected by a configuration interaction calculation. In order to adequately describe electron correlation, it is usually necessary to include a very large number of configurations, so that calculations in quantum chemistry can strain the capacity of even the largest currently available electronic computers. Recently, in nuclear and atomic physics and in quantum chemistry, there has been a fresh approach to the problem of solving the many-particle Schrodinger equation. In this new approach, one tries to solve the Schrodinger equation of an N-particle system directly in a space of dimension d = 3N, without the use of simplifying approximations such as the Born-oppenheimer approximation or the Hartree-Fock approximation. To do this, one has to become accustomed to IX x HYPERSPHERICAL HARMONICS working in a space of high dimension. Every physicist and chemist is familiar with the beauty and utility of spherical coordinates and spherical harmonics in a 3-dimensional space. Since we now wish to solve the wave equation in a space of high dimensionality, it is natural to turn for help to hyperspherical coordinates and hyper spherical harmonics, which are the d-dimensional generalizations of the familiar 3-dimensional spherical coordinates and spherical harmonics. Interestingly, it turns out that each of the familiar theorems for 3-dimensional spherical harmonics has a d-dimensional generalization. Thus, for example, we know that l/I~-~' I, (the Green's function of the Laplacian operator) can be expanded in terms of Legendre polynomials: 1 00 l: r> .Q,=O Similarly, the function l/I~-~' Id-; (which is the Green's function of the generalized Laplacian operator in a d-dimensional space), can be expanded in terms of Gegenbauer polynomials. (equation (3-5)): Just as the Legendre polynomials are eigenfunctions of the angular momentum operator L2, so the Gegenbauer polynomials are eigenfunctions of the generalized angular momentum operator A2 INTRODUCTION xi where d 3 3)2 L ( x - - x- i>j i3xj j3xi Thus the Gegenbauer polynomials play a role in the theory of hyper spherical harmonics which is analogous to the role played by Legendre polynomials in the familiar theory of 3-dimensional spherical harmonics; and when d = 3, the Gegenbauer polynomials reduce to Legendre polynomials. The familiar sum rule, in 'lrlhich a sum of spherical harmonics is expressed as a Legendre polynomial, also has a d-dimensional generalization, in which a sum of hyper spherical harmonics is expressed as a Gegenbauer polynomial (equation (3-27»: The hyper spherical harmonics which appear in this sum rule are eigenfunctions of the generalized angular monentum operator A2, chosen in such a way as to fulfil the orthonormality relation: VIe are all familiar with the fact that a plane wave can be expanded in terms of spherical Bessel functions and either Legendre polynomials or spherical harmonics in a 3-dimensional space. Similarly, one finds that a d-dimensional plane wave can be expanded in terms of