Elementary Molecular Quantum Mechanics: Mathematical Methods and Applications [2 ed.] 0444626476, 9780444626479

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Elementary Molecular Quantum Mechanics: Mathematical Methods and Applications Second Edition

Valerio Magnasco University of Genoa, Genoa, Italy

AMSTERDAM • BOSTON • HEIDELBERG • LONDON NEW YORK • OXFORD • PARIS • SAN DIEGO SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO

Elsevier Radarweg 29, PO Box 211, 1000 AE Amsterdam, The Netherlands The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, UK 225 Wyman Street, Waltham, MA 02451, USA Second Edition Ó 2013, 2007 Elsevier B.V. All rights reserved No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means electronic, mechanical, photocopying, recording or otherwise without the prior written permission of the publisher Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone (+44) (0) 1865 843830; fax (+44) (0) 1865 853333; email: [email protected]. Alternatively you can submit your request online by visiting the Elsevier web site at http://elsevier.com/locate/ permissions, and selecting Obtaining permission to use Elsevier material Notice No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. Because of rapid advances in the medical sciences, in particular, independent verification of diagnoses and drug dosages should be made British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress For information on all Elsevier publications visit our web site at store.elsevier.com Printed and bound in USA 13 14 15 16 17 10 9 8 7 6 5 4 3 2 1 ISBN: 978-0-444-62647-9

To Charles Alfred Coulson To Paola, my wife To my former Quantum Chemistry students Giorgio Aliani, Mauro Amelio, Luca Arcesi, Paolo Armanino, Cristina Artini, Roberto Austi, Fiorenza Azzurri, Monica Beggiato, Walter Bellini, Angelo Benazzo, Maurizio Benzo, Anna Berti, Luca Berti, Federico Bianchi, Anja Bijedic Mitranic, Sergio Bisio, Claudio Boffito, Vito Boido, Gabriella Borzone, Carlo Bottero, Andrea Briatore, Stefano Brusco, Santino Bruzzone, Giuseppe Burgisi, Vincenzo Buscaglia, Fabio Canepa, Maria Rosaria Carcassi, Francesca Carosi, Silvio Carrettin, Angela Cartosio, Massimo Casanova, Danilo Cassaglia, Rosana Cavalli, Marino Cervetto, Roberta Cesari, Silvia Ciuffini, Fabrizio Colace, Bianca Cosma, Camilla Costa, Aldo Curti, Marco Delfino, Giovanna Dellepiane, Fabrizio De Melas, Serena De Negri, Maria Grazia Desirello, Silvia De Vito, Ilaria Devoti, Daniele Duce, Roberto Eggenhoffner, Leandro Esposito, Sandro Ferraro, Alessandra Ferretti, Dino Romano Ferro, Giuseppe Figari, Maria Luisa Fornasini, Giuseppe Francese, Ambra Frare, Silvano Fuso, Massimo Gambetta, Silvia Garbarino, Paolo Gatti, Chiara Ghiron, Michele Giardina, Claudia Grandi, Paola Guarnone, Marina Guenza, Roberto La Ferla, Marco Leoncini, Serena Leporatti, Alessandra Leprini, Franco Lichene, Nara Liessi, Andrea Lionello, Romeo Losso, Mattia Lucchini, Costanzo Luzzago, Filippo Maido, Alberto Martinelli, Roberta Martolini, Bruno Matteazzi, Franco Merlo, Valeria Merlo, Giovan Battista Minetto, Tomaso Munari, Riccardo Musenich, Gian Franco Musso, Riccardo Narizzano, Marco Nencioni, Sergio Novelli, Riccardo Olivieri Colombo, Massimo Ottonelli, Donatella Paci, Silvia Palacio Azcona, Andrea Palenzona, Christopoulos Panagiotis, Laura Pardo, Paolo Parodi, Lise Pascale, Stefano Passalacqua, Elisa Pastore, Marta Patrone, Gianfranco Patrucco, Enrica Perrotta, Roberto Peverati, Matteo Piccardo, Giulio Pieretti, Francesca Pin, Arnaldo Rapallo, Marcello Rasparini, Anna Ravina, Renzo Rizzo, Caterina Rocca, Giuseppe Roncallo, Massimiliano Rossi, Marina Rui, Paolo Russo, Paolo Sacco, Carlo Scapolla, Daniele Scavo, Giovanna Schmid, Andrea Sciutto, Andrea Siciliano, Lucia Somma, Giuseppe Tassano, Anton Thumiger, Giuseppe Tonello, Giulio Andrea Tozzi, Riccardo Tubino, Alessandro Valente, Vito Vece, Massimo Vizza, Gilberto Vlaich, Ugo Zanelli, Maria Teresa Zannetti

Preface The idea of a book presenting in a simple way most of the essential mathematical tools beyond elementary calculus, and possibly entitled Mathematical Methods in Molecular Quantum Mechanics, grew for some time in the author’s mind during the many years of teaching and research at the University of Genoa. In 2007, Elsevier published a teaching book by the author for graduate university students of chemistry and physics entitled Elementary Methods of Molecular Quantum Mechanics, the book being intended as a bridge between the classic elementary Coulson’s Valence and the more advanced McWeeny’s Methods of Molecular Quantum Mechanics. After 5 years, this book necessarily needed some revision and restyling to eliminate various misprints. After an intense correspondence with the Elsevier’s editorial board, to avoid the narrowness of a single mathematical text, it was decided to merge the two books into the present single book, entitled Elementary Molecular Quantum Mechanics: Mathematical Methods and Applications. The new book is mostly based on the content of the old one, of which it maintains the interesting elementary approach to quantum chemical applications, and contains a new introductory part more specifically devoted to mathematical methods. So, the new book can be considered as a deeply revised edition of the former Elsevier’s book with a sensibly enlarged mathematical part. It is hoped that, in this way, the new book can meet better the students’ needs, enlarging its usability even for university students of applied maths, at least in their first years. The first part of the book is not intended to cover all mathematical topics, rather it is intended to offer the reader in a plane way and more in detail many of the building stones of the advanced mathematics needed in working applications of Quantum Mechanics to Quantum Chemistry. For the same reason, some important topics are omitted, such as functional, numerical, and computational analysis, but these arguments are so wide that they would imply separate books in themselves. For them, the reader should be addressed to specific treatises or to internet resources. The required background for students of physics and chemistry is still that of a few semester courses in mathematical analysis and physics. The book might help as an auxiliary advanced mathematical and physical text of about 60 h for university students of chemistry, physics, and applied maths. The main differences with the former Elsevier’s book are the following. (1) The shifting to the new Chapter 1 of both the fundamental approximation methods of Quantum Mechanics, the variational method and the various perturbative approaches (formerly partly given in Chapters 5 and 11), (2) the introduction of three new mathematical chapters devoted to the solution of the partial differential equations of Quantum Mechanics (Chapter 3), to the theory of special functions, including Fourier and Laplace transforms and Green’s functions (Chapter 4), to the functions of a complex variable and the evaluation of integrals of a real variable by the method of residues (Chapter 5), (3) the subdivision of the former unusually large Chapter 7 into the two new Chapters 14 and 16, the first devoted to the theory of model Hamiltonians (Hartree–Fock, Hu¨ckel and approximate MO methods), the second to the correlation problem and to the Post-Hartree–Fock methods, (4) the accurate revision of the old Chapter 12 on atomic and molecular interactions, which was completely re-written and updated in the new Chapter 17, (5) the addition of the Gaussian GTO approach besides the prevalent STO approach in the evaluation of multicentre molecular integrals in the new Chapter 18, (6) the addition of Chapter 19,

xix

xx

Preface

which introduces the reader to the still actually growing field of relativistic molecular quantum mechanics, starting from the Dirac’s equation for the relativistic electron up to its major applications to atomic and molecular systems, and, lastly, (7) a short outlook to the problem of molecular vibrations (Chapter 20) with particular emphasis to the vibronic interactions, a topic which was completely omitted in the former book. The original feature of the former book which completed each chapter with problems and solved problems was maintained and enlarged in the new book. The contents of the new book are given in the introductory section Contents, where all items in each chapter are specified in detail, so that they will be omitted here. A wide list of alphabetically ordered references, mostly taken from original research papers, and author and subject indices complete the book. Thanks are due to my son Mario and his daughter Laura, who prepared the drawings at the computer, to Professor Giuseppe Figari, who helped much during the many years of research at the Department of Chemistry of the University of Genoa, to Dr Michele Battezzati for useful suggestions and discussions on specific points, to Dr Tomas Martisius of Vilnius (Lithuania), who provided the software for translating from Latex to Word, which made this work possible, and to Dr Camilla Costa, who helped in the translation of the former published text from Latex to Word. Particularly warm thanks are due to the Elsevier’ Senior Acquisition Editor, Mr Adrian Shell, PhD, for his patience in discussing the best structure of the new book, and to Dr Egbert van Wezenbeek for his constant encouragement and support. Finally, I acknowledge the Italian Ministry for Education University and Research (MIUR) and the University of Genoa for their financial support during the many years of scientific research of the Genoa group. My last thanks are for my wife Paola, who supported me strongly during the long time this work was in progress. Valerio Magnasco Genoa, 30th November 2012

CHAPTER

Mathematical foundations and approximation methods

1

CHAPTER OUTLINE 1.1 Mathematical Foundations .................................................................................................................4 1.1.1 Regular functions ..........................................................................................................4 1.1.2 Schmidt orthogonalization..............................................................................................5 1.1.3 Lo¨wdin orthogonalization................................................................................................8 1.1.4 Set of orthonormal functions and basis set ......................................................................8 1.1.5 Linear operators ............................................................................................................9 1.1.6 Hermitian operators .....................................................................................................10 1.1.7 Expansion theorem ......................................................................................................14 1.1.8 Basic principles of quantum mechanics ........................................................................17 1.2 The Variational Method................................................................................................................... 19 1.2.1 Non-linear parameters .................................................................................................21 1.2.2 Linear parameters: the Ritz method ..............................................................................21 1.3 Perturbative Methods for Stationary States ...................................................................................... 25 1.3.1 RS perturbation theory.................................................................................................25 1.3.2 Second-order approximation methods in RS perturbation theory ......................................32 1.3.2.1 The Unso¨ld approximation ...................................................................................... 32 1.3.2.2 The Hylleraas method............................................................................................. 33 1.3.2.3 The Kirkwood method............................................................................................. 34 1.3.2.4 The Ritz method for E~ 2: linear pseudostates............................................................ 35 1.3.3 BW perturbation theory ................................................................................................37 1.3.4 Perturbation methods without partitioning of the Hamiltonian .........................................40 1.3.4.1 LS perturbation theory ............................................................................................ 41 1.3.4.2 EN perturbation theory ........................................................................................... 42 1.3.5 Perturbation theories including exchange (SAPTs) ..........................................................45 1.3.6 The moment method....................................................................................................53 1.4 The Wentzel–Kramers–Brillouin Method.......................................................................................... 55 1.5 Problems 1 .................................................................................................................................... 59 1.6 Solved Problems ............................................................................................................................ 62

In this chapter, after a short review of the mathematical and physical foundations of quantum mechanics, we shall be mostly concerned with the approximation methods on which all quantum chemistry applications are based. The variational method due to Rayleigh will be examined first, next Elementary Molecular Quantum Mechanics 2E. http://dx.doi.org/10.1016/B978-0-444-62647-9.00001-4 Ó 2013, 2007 Elsevier B.V. All rights reserved

3

4

CHAPTER 1 Mathematical foundations and approximation methods

different perturbative methods for stationary states, starting from the Rayleigh–Schroedinger (RS) perturbation theory, including some efficient methods of treating second-order quantities, up to the Brillouin–Wigner (BW) and Epstein–Nesbet (EN) theories, the symmetry-adapted theories including exchange (SAPT), and the recently developed moment method. A glance at the Wentzel–Kramers– Brillouin (WKB) method concludes this introductory chapter.

1.1 MATHEMATICAL FOUNDATIONS We begin by giving a few mathematical definitions. We denote by x either a single independent variable or a collective for many variables. Vectors and arguments of functions of more than one variable will be written in bold type: thus, we use r to denote a three-dimensional vector, say r ¼ ix þ jy þ kz in Cartesian coordinates, or the three variables (x,y,z) of a function f(x,y,z) ¼ f(r). Generalized coordinates and transformation from Cartesian to other coordinate systems will be examined in Chapter 2.

1.1.1 Regular functions In what follows, we shall be concerned only with regular functions of the general variable x. A function j(x), defined in a given domain D, is said regular if it satisfies the three constraints of being there: (1) single valued, (2) piecewise continuous with its first derivatives, and (3) quadratically integrable, i.e. it must be zero at infinity. So, not all functions that can exist mathematically are physically acceptable. Such functions are usually obtained, at least in principle, as solutions of differential equations of the second order like those examined in Chapter 3. According to Dirac, such functions can be treated as vectors having infinite components in an abstract linear space. Function j(x) can be complex, its complex conjugate being denoted by j*(x). Complex numbers and functions of a complex variable will be treated to some extent in Chapter 5. We often use the Dirac1 notation:


Function Complex conjugate

jðxÞ ¼ jji 0 ket j ðxÞ ¼ hjj 0 bra

(1)

The Dirac jji vectors have all the well-known properties of a linear vector space. The scalar (or inner) product (analogy between regular functions and complex vectors of infinite dimensions) of j* by j can then be written in the bra-ket (‘bracket’) form: Z

1

dx j ðxÞ jðxÞ ¼ hjjji ¼ finite number > 0

(2)

Dirac Paul Adrien Maurice 1902–1984, English physicist, Professor of mathematics at the University of Cambridge (UK); 1933 Nobel Prize for Physics.

1.1 Mathematical foundations

5

If: hjjji ¼ Nj

(3)

we say that function j(x) (the ket jji) is normalized to Nj (the norm of j). The function j can then be normalized to 1 by multiplying it by the normalization factor N ¼ ðNjÞ1=2. The scalar product of two regular functions j and 4 satisfies the Schwarz inequality: hjj4i2  hjjjih4j4i where the equality sign holds if and only if j and 4 are proportional. If: Z hjj4i ¼ dx j ðxÞ4ðxÞ ¼ 0

(4)

(5)

we say that 4 is orthogonal ðtÞ to j. If: hj0 j40 i ¼ Sðs0Þ 40

(6)

and j0

are not orthogonal, but can be orthogonalized in either of two ways. The primed notation will be mostly used for non-orthogonal functions and the unprimed notation for the orthogonal set. To put in evidence that we have two functions, it is convenient to denote j0 by c01 and 40 by c02 ; j by c1 and 4 by c2.

1.1.2 Schmidt orthogonalization The first orthogonalization method is the so-called unsymmetrical orthogonalization or Gram–Schmidt orthogonalization, in short Schmidt orthogonalization. It consists in choosing the linear combination:   (7) c1 ¼ c01 ; c2 ¼ N c02  Sc01 ; hc1 jc2 i ¼ 0 where N ¼ ð1  S2 Þ1=2 is the normalization factor. In fact, it is easily seen that if c01 and c02 are normalized to 1:      (8) hc1 jc2 i ¼ N c01 c02  Sc01 ¼ N S  S ¼ 0 The general Schmidt orthogonalization process for two functions is constructed as follows. If:        0 0  0 0 (9) c1 c1 ¼ c2 c2 ¼ 1; c01 c02 ¼ c02 c01 ¼ Sðs0Þ we take the new unprimed set:

(

c1 ¼ c01 c2 ¼ Ac02 þ Bc01

(10)

where we impose on c2 the conditions of orthogonality and normalization (in short, the orthonormality conditions). Then: 8    < hc jc i ¼ Ac0 þ Bc0 c0 ¼ AS þ B ¼ 0 2 1 2 1 1 (11)    : hc2 jc2 i ¼ Ac0 þ Bc0 Ac0 þ Bc0 ¼ A2 þ B2 þ 2ABS ¼ 1 2

1

2

1

6

CHAPTER 1 Mathematical foundations and approximation methods

We obtain the system:




AS þ B ¼ 0 A2 þ B2 þ 2ABS ¼ 1

which is solved in terms of the ratio (B/A): 8 B > > ¼ S As0 >

  B B > 2 > ¼ A2 1  S2 ¼ 1 þ2S :A 1 þ A A so that:

1=2  ; A ¼ 1  S2

 1=2 B ¼ S 1  S2

(12)

(13)

(14)

and the Schmidt orthogonalized two-set will be: 8 0 > < c1 ¼ c1

c0  Sc0 > : c2 ¼ p2ffiffiffiffiffiffiffiffiffiffiffiffiffi1 1  S2

(15)

Using the matrix form of Chapter 6, we can write Schmidt orthogonalization as the transformation between the two sets of functions described by (1  2) row vectors: !    0   1 S 1  S2 1=2  0 (16) ¼ c01 c02 OS ð c1 c2 Þ ¼ c1 c2   1=2 0 1  S2 where: OS ¼

1 0

 1=2 ! S 1  S2  1=2 1  S2

(17)

is the unsymmetrical (2  2) matrix doing Schmidt’s orthonormalization. For a set of three normalized but non-orthogonal functions ð c01 c02 c03 Þ having non-orthogonalities S12, S13, S23, respectively, it is better to orthogonalize in a first step the second function to the first one, next the third function to the resulting orthogonalized ones. In terms of the original set, Schmidt’s orthogonalization is given in this case by the (3  3) transformation:   (18) ð c1 c2 c3 Þ ¼ c01 c02 c03 OS where:

0

1

B B OS ¼ B 0 @ 0

N2 S21

   Skl ¼ c0k c0l

N2 0

N3 ðS31  S32 S21 Þ

1

C N3 ðS32  S31 S21 Þ C C A   N3 1  S221

k; l ¼ 1; 2; 3

 1=2 N2 ¼ 1  S221 ;

N3 ¼



 1=2 1  S221 1  S232  S231  S221 þ 2S32 S31 S21

(19)

1.1 Mathematical foundations

7

The orthonormality of the three-set can be checked with little labour using Eqns (18) and (19), and this is done in detail in Problem 1.1 at the end of this chapter. It is apparent how the complexity of the OS matrix rapidly increases. The general n-term Schmidt orthogonalization is given by a triangular transformation constructed stepwise as follows. The recurrence relation expressing ck , the k-th function of the orthonormalized set, in terms of c0k and c0l ðl < kÞ is found in Morse and Fesbach (1953) and is given by: ck ¼

1

k1 X

!1=2 S0lk 2

c0k

l¼1

where:



k1 X

! cl S0lk

k ¼ 2; 3; /; n

(20)

l¼1

   S0lk ¼ cl c0k

(21)

It was shown elsewhere (Magnasco et al., 1992a) that the Schmidt’s orthogonalization of the set of (n þ 1) Slater-type functions (STOs) with orbital exponent c ¼ 1: c0k ¼ Nk0 r k Rlm ðrÞj0

k ¼ 0; 1; 2; .; n

(22)

where Rlm(r) is a spherical tensor in real form2 and j0 , the 1s STO with orbital exponent c ¼ 1 gives the set of associated Laguerre functions of order (2l þ 2) first introduced by Lo¨wdin and Shull (1956) and Shull and Lo¨wdin (1959) in their studies on the natural orbitals in the quantum theory of two-electron systems, and later by Hirschfelder and Lo¨wdin (1959) in treating the long-range interaction between two hydrogen atoms. At variance with the hydrogen-like eigenfunctions3 having in their radial part the associated Laguerre polynomials of order (2l þ 2), L2lþ1 nþl ðrÞ, the Schmidt’s orthogonalized set is a complete set of associated Laguerre functions of order (2l þ 2), a discrete set of functions that avoids the treatment of the continuous part of the eigenspectrum. The orbitals (22) are normalized but non-orthogonal. When they are Schmidt-orthogonalized according to: !1=2 ! k1    k1 X X   0 0 2 0   cl ðrÞ cl c c ðrÞ  cl c (23) ck ðrÞ ¼ 1  k

l¼0

k

k

l¼0

by induction the following is obtained: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X ð1Þl ð2k þ 2l þ 2  2lÞ! 0 c ðrÞ ck ðrÞ ¼ k!ðk þ 2l þ 2Þ! l!ðk  lÞ!ðk þ 2l þ 2  lÞ! kl l¼0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8k! ð2rÞl L2lþ2 ¼ ð1Þk kþ2lþ2 ð2rÞexpðrÞYlm ðq; 4Þ ðk þ 2l þ 2Þ!3

2

(24)

See Section 4.11.2 of Chapter 4. These hydrogen-like (HAOs) orbitals are obtained as exact solutions of the radial Schroedinger eigenvalue equation for the atomic one-electron system in Section 3.7.1 of Chapter 3. 3

8

CHAPTER 1 Mathematical foundations and approximation methods

where Lba ðxÞ is the associated Laguerre polynomial of degree (a b) and order b in x, defined as:4 


ab X db db da a ð1Þk a a b La ðxÞ ¼ b La ðxÞ ¼ b expðxÞ a ½x expðxÞ ¼ ð1Þ a! xk a  b bþk dx k! dx dx k¼0 (25) Within the phase factor (1) , after changing k into (n  l  1), the set (24) is equivalent to the Lo¨wdin–Shull orthogonal set of associated Laguerre functions of order (2l þ 2). k

1.1.3 Lo¨wdin orthogonalization In this orthogonalization technique, all basis functions are treated on the same foot. In the matrix form of Chapter 6, if: c0y c0 ¼ M0 ¼ 1 þ S

(26)

is the metric matrix of the normalized non-orthogonal basis c0 , Lowdin’ symmetrical orthogonalization ¨

(Lo¨wdin, 1950) is given by the transformation: c ¼ c0 M01=2 ;

OL ¼ M01=2

(27)

as can be readily verified, since:

  cy c ¼ M01=2 c0y c0 M01=2 ¼ M01=2 M0 M01=2 ¼ 1

(28)

The technique for obtaining the inverse of the square root of the metric matrix, M01=2 , is described in Section 6.7 of Chapter 6 in connection with the study of the pseudoeigenvalue problem of the Hermitian matrix A0 of non-orthogonal metric M0 , and the case of n ¼ 2 is treated there in detail. The explicit form of the transformed functions in this case is: 8 > 0 a þ b þ c0 a  b > a ¼ ð1 þ SÞ1=2 ; b ¼ ð1  SÞ1=2 < c1 ¼ c1 2 2 2 >    > : c 2 ¼ c 0 a þ b þ c 0 a  b S ¼ c0  c 0 1 1 2 2 2 2 and the reason for the name symmetrical orthogonalization is evident.

1.1.4 Set of orthonormal functions and basis set Let: fck ðxÞg ¼ ðc1 c2 /ck /cl /Þ

(29)

be a set of functions. If: hck jcl i ¼ dkl

k; l ¼ 1; 2; $ $ $

where dkl is the Kronecker delta (1 if l ¼ k, 0 if l s k), the set is said orthonormal. 4

See Section 4.3.2 of Chapter 4.

(30)

1.1 Mathematical foundations

A set of functions is said linearly independent if: X ck ðxÞCk ¼ 0

9

(31)

k

implies, necessarily, Ck ¼ 0 for any k. For a set to be linearly independent, it will be sufficient that the determinant of the metric matrix M be different from zero: det Mkl s0

Mkl ¼ hck jcl i

(32)

A set of orthonormal functions is therefore a linearly independent set. A set of linearly independent functions forms a basis in the function space, and we can expand any function of that space into a linear combination of these basis functions. The expansion is unique.

1.1.5 Linear operators ^ (e.g. its derivative). It An operator A^ is a rule transforming a given function j into another function Aj ^ must be stressed that Aj is a new function. A linear operator A^ satisfies to: ^ 1 ðxÞ þ c2 ½Aj ^ 2 ðxÞ ^ 1 j1 ðxÞ þ c2 j2 ðxÞ ¼ c1 ½Aj A½c

(33)

where c1 and c2 are complex constants. The first and second derivatives of a function are simple examples of linear operators. ^ defined through its kernel Kðx; x0 Þ and A more complicated operator is the integral operator AðxÞ whose effect on a function J(x) is: Z Z ^ (34) AðxÞjðxÞ ¼ dx0 Kðx; x0 ÞP^xx0 jðxÞ ¼ dx0 Kðx; x0 Þjðx0 Þ ¼ 4ðxÞ The algebraic sum of two or more operators enjoys commutative and associative properties: ^ ^ ^ ^ ðA^ þ BÞjðxÞ ¼ AjðxÞ þ BjðxÞ ¼ ðB^ þ AÞjðxÞ ^ ^ ^ ^ ^ ðA^ þ B^ þ CÞjðxÞ ¼ ðA^ þ BÞjðxÞ þ CjðxÞ ¼ AjðxÞ þ ðB^ þ CÞjðxÞ

(35)

In general, the product of two operators is not commutative: ^ BjðxÞ ^ ^ ^ AjðxÞ A½ s B½

(36)

A^B^ ¼ B^A^

(37)

^ B ^ ¼ A^B^  B^A^ ½A;

(38)

where the inner operator acts first. If:

the two operators commute. The quantity: ^ B. ^ is called the commutator of the operators A; The quantity: ^ B ^ ¼ A^B^ þ B^A^ ½A; þ ^ B. ^ is called the anti-commutator of the operators A;

(39)

10

CHAPTER 1 Mathematical foundations and approximation methods

The equation: ^ AjðxÞ ¼ AjðxÞ

(40)

^ When Eqn (40) is satisfied, constant A is is called the eigenvalue equation for the linear operator A. ^ Usually A^ is a differential called the eigenvalue and function j the eigenfunction of the operator A. operator, and there may be a whole spectrum of eigenvalues, each one with its corresponding eigenfunction. The spectrum of the eigenvalues can be either discrete or continuous. An eigenvalue is said to be n-fold degenerate when n different linearly independent eigenfunctions belong to it. We shall see later that the Schroedinger equation for the amplitude j(x) is a typical eigenvalue equation where A^ ¼ H^ ¼ T^ þ V is the total energy operator (the Hamiltonian5), T^ being the kinetic energy operator and V the potential energy characterizing the system (a scalar quantity).

1.1.6 Hermitian operators A Hermitian6 operator is a linear operator satisfying the so-called turn-over rule: 8 ^ ¼ hAjj4i ^ > < hjjA4i Z Z  > ^ ^ ¼ dxðAjðxÞÞ 4ðxÞ : dxj ðxÞðA4ðxÞÞ

(41)

In matrix form, we say that matrix A is Hermitian if: A ¼ Ay

(42)

where Ay is the adjoint matrix.7 In fact, in terms of matrix elements, using Dirac’s notation:   Aj4 ¼ A4j ^ ¼ h4jAji ^  ¼ hAjj4i ^ hjjA4i

(43)

which is the turn-over rule (41). The Hermitian operators have the following properties: 1. Real eigenvalues 2. Orthogonal (or anyway orthogonalizable) eigenfunctions 3. Their eigenfunctions form a complete set. The first two properties can be easily derived from the definition. ^ ¼ hA4j4i ^ 1. h4jA4i Def. with: ^ ¼ A4 A4

5

^  ¼ A 4 ðA4Þ

A ¼ eigenvalue

After Hamilton William Rowan (Sir) 1805–1865, Irish astronomer mathematician and physicist, Professor of Astronomy at the Trinity College of Dublin (Ireland). 6 After Hermite Charles 1822–1901, French mathematician, Professor at the Sorbonne (Paris), Member of the Acade´mie des Sciences. 7 See Section 6.4 of Chapter 6.

1.1 Mathematical foundations

11

^ ¼ Ah4j4ihA4j4i ^ ¼ A h4j4i h4jA4i and by subtracting the first from the second equation: 0 ¼ ðA  AÞh4j4ih4j4is0 0 A ¼ A ^ m ¼ Am 4m Al ; Am are two eigenvalues ^ l ¼ Al 4l A4 2. A4 We have two cases: • Al s Am not degenerate eigenvalues       ^ m ¼ A m 4l  4 m 4l A4          ^ l  4 m ¼ A  4l  4 m ¼ A l 4 l  4 m A4 l and by subtracting:

       0 ¼ Al  Am 4l 4m Al  Am s0 0 4l 4m ¼ 0

• Al ¼ Am ¼ A degenerate eigenvalue ^ l ¼ A4l A4

^ m ¼ A4m A4

We have now two linearly independent functions belonging to the same eigenvalue, so that the demonstration above is no longer valid. But we can always orthogonalize 4m to 4l (e.g. by the Schmidt method) without changing the eigenvalue.     4l 4m 40l 40m 0 non-orthogonal set orthonormal set 8 0 < 4l ¼ 4l       4m  S4l : 40m ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi S ¼ 4l 4m ¼ 4m 4l s0 1  S2 It can be verified immediately that: 0 0 A^ 4l ¼ A4l ^ m  SðA4 ^ lÞ A4 4  S4 4m  S4l ^ 0 ¼ A^ pmffiffiffiffiffiffiffiffiffiffiffiffiffil ¼ p ffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ A pffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ A40m A4 m 2 2 1S 1S 1  S2

so that the orthogonal set belongs to the same degenerate eigenvalue. Completeness includes also the eigenfunctions belonging to the continuous part of the eigenvalue spectrum (see Section 1.1.7 of this chapter for a precise definition of completeness). 2 v v2 2 ^ ¼  Z 72 ; H^ ¼ T^ þ V, with i being the ; 7 ; T Hermitian operators are i , iV; vx2 2m vx imaginary unit ði2 ¼ 1Þ; V ¼ i

8

v v v þ j þ k , the gradient vector operator, 72 , the Laplacian8 vx vy vz

After De Laplace Pierre Simon 1749–1827, French mathematician and astronomer, Member of the Acade´mie des Sciences (Paris).

12

CHAPTER 1 Mathematical foundations and approximation methods

^ the kinetic energy operator for a particle of mass m with operator (in Cartesian coordinates), T, h ^ the Hamiltonian operator. the reduced Planck constant, and H, Z¼ 2p v and V are instead anti-Hermitian operators, for which: vx 8       >   > > < jv4 ¼  vj4  vx vx  (44) > > > : hjjV4i ¼ hVjj4i Worked out proofs on different Hermitian and anti-Hermitian operators are given as Problems in Section 1.2 of this chapter. We have seen that in Cartesian coordinates the vector operator9 V (the gradient) is defined as (Rutherford, 1962): v v v V¼i þj þk (45) vx vy vz Now, let f(x,y,z) be a scalar function of the space point P(r). Then: Vf ¼ i

vf vf vf þj þk vx vy vz

(46)

is a vector, the gradient of f. If f is a vector of components fx, fy, fz, we then have for the scalar product: V$f ¼

vfx vfy vfz þ þ ¼ div f vx vy vz

(47)

a scalar quantity, the divergence of f. As a particular case: V $ V ¼ V2 ¼

v2 v2 v2 þ 2þ 2 2 vx vy vz

(48)

is the Laplacian operator. Gradient and Laplacian operators can be expressed in generalized coordinates, and the transformations from Cartesian to spherical, spheroidal, and parabolic coordinates are given in Chapter 2. From the vector product of the vector operator V by the vector f we obtain a new vector, the curl or rotation of f (written curl f or rot f):   i   v V  f ¼   vx f x 9

j v vy fy

A vector whose components are operators.

 k  v  ¼ curl f ¼ i curlx f þ j curly f þ k curlz f vz   fz

(49)

1.1 Mathematical foundations

a vector operator with components:

8 vfz vfy > >  curlx f ¼ > > vy vz > > > < vfx vfz  curly f ¼ > vz vx > > > > > > : curlz f ¼ vfy  vfx vx vy

13

(50)

In quantum mechanics, the vector product of the position vector r by the linear momentum vector ^ operator iZV gives the angular momentum vector operator L:    i j k     ^ ¼ iZr  V ¼ iZ x y z  ¼ iL^x þ jL^y þ kL^z L (51) v v v    vx vy vz  with components:

v v ^ Lx ¼ iZ y  z ; vz vy

v v ^ Ly ¼ iZ z  x ; vx vz

v v ^ Lz ¼ iZ x  y vy vx

(52)

In the theory of angular momentum, frequent use is made of the ladder (or shift) operators: L^þ ¼ L^x þ iL^y ðstep-upÞ;

L^ ¼ L^x  iL^y ðstep-downÞ

(53)

10

also called raising and lowering operators, respectively. Angular momentum operators have the commutation relations: 8

  > L^x ; L^y ¼ iL^z ; L^y ; L^z ¼ iL^x ; ½L^z ; L^x  ¼ iL^y > > < ½L^z ; L^þ  ¼ L^þ ; ½L^z ; L^  ¼ L^ > > > : ^2 ^ ½L ; Lk  ¼ ½L^2 ; L^  ¼ 0 k ¼ x; y; z

(54)

^ The same commutation relations hold for the spin vector operator S. 2 It can be shown that, in spherical coordinates, the Laplacian 7 separates into a radial Laplacian 72r and an angular part depending on the square of the angular momentum operator L^2 (Problem 1.5): 72 ¼ 72r 

2 L^ =Z2 r2



1 v v2 2 v 2v þ r ¼ vr 2 r vr r 2 vr vr  

1 v v 1 v2 2 2 ^ L ¼ Z sin q þ sin q vq vq sin2 q v42 72r ¼

10

Note that the ladder operators are non-Hermitian.

(55) (56) (57)

14

CHAPTER 1 Mathematical foundations and approximation methods

To get rid of all fundamental physical constants in quantum chemistry calculations, it is convenient to introduce the system of atomic units11 (au), by posing: e ¼ Z ¼ m ¼ 4pε0 ¼ 1

(58)

The basic atomic units of charge, length, energy, and time are expressed in terms of the SI values for the fundamental physical constants (Mohr et al., 2008) by: 8 Charge; e > > > > > > > > >Length; bohr > >
> Energy; hartree > > > > > > > > > : Time

Eh ¼ s¼

Z2 ¼ 5:291772087  1011 m me2

1 e2 ¼ 4:359743802  1018 J 4pε0 a0

(59)

Z ¼ 2:418884331  1017 s Eh

At the end of a calculation in atomic units, the actual SI values can be obtained by taking into account the SI equivalents (59).

1.1.7 Expansion theorem Any regular function f (x) can be expanded exactly into the complete set of the eigenfunctions of any ^ Hermitian operator A. If: ^ k ðxÞ ¼ Ak 4k ðxÞ; A4 then: f ðxÞ ¼

X

A^y ¼ A^

4k ðxÞCk

(60)

(61)

k

where the expansion coefficients are given by: Z Ck ¼ dx0 4k ðx0 Þf ðx0 Þ ¼ h4k jf i

(62)

It is evident that: f ðxÞ ¼

X

j4k ðxÞih4k ðx0 Þj f ðx0 Þi ¼ f ðxÞ if

k

X

j4k ðxÞih4k ðx0 Þj ¼ ^1

(63)

k

namely, Eqn (63) becomes an identity if the set is complete (more precisely, S is the Dirac’s d-function, the kernel of the integral operator ^ 1). 11

Atomic units were first introduced by Hartree (1928).

1.1 Mathematical foundations

15

Some authors insert into Eqn (61) an integral sign to emphasize that integration over the continuous part of the eigenvalue spectrum must be included in the expansion, or we can write: Z X 4k ðxÞCk þ dEf ðEÞ (64) f ðxÞ ¼ k

When the set of functions f4k ðxÞg is not complete, truncation errors occur. A more precise definition of completeness can be given following Courant and Hilbert (1989). Consider the positive integral: 2  Z n   X   Ck 4k   0 (65) dx f    k¼1 Expanding the square and integrating term by term gives, using Dirac’s notation: 0  hfjfi  2

n X

Ck hf j4k i þ

k¼1

n X

jCk j2 ¼ Nf  2

k¼1

n n X X jCk j2 þ jCk j2 k¼1

(66)

k¼1

so that: n X jCk j2  Nf

(67)

k¼1

Since the norm of f(x) is independent of n, it follows what is known as the Bessel’s inequality: N X jCk j2  Nf

(68)

k¼1

which proves that the sum of the squared moduli of the coefficients in expansion (61) always converges. Integral (65) occurs when we try to approximate a function f(x) by a finite linear combination of n functions f4k ðxÞg, with constant coefficients Bk, such that the mean square error M, defined as: 2  Z   n X   Bk 4k   0 (69) M ¼ dx f    k¼1 is as small as possible. It can be shown that M takes on its least value for Bk ¼ Ck. This approximation is known as approximation by the least squares or approximation in the mean. This allows for a more precise definition of completeness of a given basis set. If, for a given orthonormal set of functions f4k ðxÞg function f(x) can be approximated in the mean to any desired degree of accuracy by choosing n large enough, namely if n can be chosen such that the mean square error is less than an arbitrarily small positive number, then the system of functions f4k ðxÞg is said to be complete. For a complete orthonormal set of basis functions, Bessel’s inequality (68) becomes an equality for every function f(x): N X jCk j2 ¼ Nf k¼1

or

X k

j4k ðxÞih4k ðx0 Þj ¼ ^1

(70)

16

CHAPTER 1 Mathematical foundations and approximation methods

Either of relations (70) is known as completeness relation. A sufficient condition for the completeness of the set of functions f4k ðxÞg is that the completeness relations (70) be satisfied for all continuous functions f(x). Further details are given in Courant and Hilbert (1989). Furthermore, we want to outline that the completeness relation on the left of Eqn (70) is equivalent to the introduction of the Dirac’s delta function12 dðx  x0 Þ as kernel of the identity integral operator ^1 (right of the equation): Z Z 0 0 ^ 0 ^ (71) 1f ðxÞ ¼ dx dðx  x ÞPxx f ðxÞ ¼ dx0 dðx  x0 Þf ðx0 Þ ¼ f ðxÞ In fact: X     Z Z X X   0 2 y 0    Ck Ck ¼ f ðxÞ 4k ðxÞ h4k ðx Þ f ðx Þ ¼ dxf ðxÞ dx0 dðx  x0 ÞP^xx0 f ðxÞ jCk j ¼ k

k

Z ¼

dxf  ðxÞ

Z

k

dx0 dðx  x0 Þf ðx0 Þ ¼

Z

dxf  ðxÞf ðxÞ ¼ h f j f i ¼ Nf

(72)

which is the completeness relation on the left of Eqn (70). Using the expansion theorem, we can pass from operators (acting on functions) to matrices (acting on vectors, see Chapter 6). Consider a finite n-dimensional set of basis functions 4 ¼ f4k ðxÞg k ¼ 1; 2; $$$; n. Then, if A^ is a Hermitian operator: X X ^ l i ¼ ð4AÞ ^ l ðxÞ ¼ 4k ðxÞAkl ¼ (73) A4 j4k ih4k jA4 l k

k

where the expansion coefficients have now two indices and are the elements of the square Hermitian matrix A (order n): Z ^ l i ¼ dx0 4 ðx0 Þ½A4 ^ l ðx0 Þ Akl ¼ h4k jA4 (74) k 0

A11 B A21 fAkl g 0 A ¼ B @$$$ An1

A12 A22 $$$ An2

$ $ $ $

$ $ $ $

$ $ $ $

1 A1n A2n C ^ C ¼ 4y A4 $$$A Ann

(75)

which is called the matrix representative of the operator A^ in the basis f4k ðxÞg; and we use matrix multiplication rules (Chapter 6). In this way, the eigenvalue equations of quantum mechanics transform into eigenvalue equations for the corresponding representative matrices. We must recall, however, that a complete set implies, according to Eqns (61) and (70), matrices of infinite order. So, using matrices of order n is equivalent to work into a finite n-dimensional subspace of the whole Hilbert space. Under a unitary transformation U of the basis functions 4 we obtain a new basis 40 such that: 40 ¼ 4U 12

See Section 4.8 of Chapter 4.

(76)

1.1 Mathematical foundations

Then, the representative A of the operator A^ is changed into:   ^ 0 ¼ Uy 4y A4 ^ U ¼ Uy AU A0 ¼ 40y A4

17

(77)

So, in giving matrix representatives it is always necessary to specify the basis set to which the representative refers.

1.1.8 Basic principles of quantum mechanics We find it convenient to state concisely here the basic principles of quantum mechanics in the form of three postulates, which we assume in an axiomatic way. Their physical significance will be examined later in Chapter 11. 1. Correspondence between physical observables and Hermitian operators In coordinate space, we have the basic correspondences: ( r ¼ ix þ jy þ kz 0 b r¼r

(78) p ¼ iZV p ¼ ipx þ jpy þ kpz 0 b h is the reduced Planck13 constant. More where i is the imaginary unit (i2 ¼ 1) and Z ¼ 2p complex observables are treated by repeated applications of the correspondences (78) under the constraint that the resulting quantum mechanical operators must be Hermitian.14 We see in this way from the very beginning that we have to introduce the concept of operators, particularly linear and Hermitian operators. Kinetic energy and Hamiltonian (total energy operator) for a particle of mass m in the potential V are examples already seen. For the one-electron atomic system (an electron attracted by a nucleus of charge Z) the Hamiltonian in au is: 1 Z H^ ¼  72  ¼ h^ 2 r

(79)

For the two-electron atomic system (two electrons attracted by a nucleus of charge Z and repelling each other through the Coulombic repulsion 1/r12) the Hamiltonian in au is:



1 2 Z 1 2 Z 1 1 ^ H ¼  71  þ  72  þ ¼ h^1 þ h^2 þ (80) 2 r1 2 r2 r12 r12 For the hydrogen molecule, in the Born–Oppenheimer approximation (Chapter 20), when the two hydrogen atoms are a distance R apart, the molecular Hamiltonian in au is:



1 1 1 1 1 1 1 1 1 1 H^ ¼  721   þ  722   þ þ ¼ h^1 þ h^2 þ þ 2 r1 rB1 2 r2 rA2 r12 R r12 R ¼ h^1A þ h^2B þ V (81) 13 Planck Max 1858–1947, German physicist, Professor at the Universities of Kiel and Berlin (Germany); 1918 Nobel Prize for Physics. 14 The quantities observable in physical experiments must be real.

18

CHAPTER 1 Mathematical foundations and approximation methods

where: 1 1 1 1 þ þ V¼  rB1 rA2 r12 R

(82)

is the interatomic potential. 2. State function and average value of observables We assume there is a state function (or wavefunction, in general complex) J(x,t) that describes in a probabilistic way the dynamical state of a microscopic system. In coordinate space, J is a regular function of coordinate x and time t such that: Jðx; tÞJ ðx; tÞdx ¼ probability at time t of finding the system in state J at dx provided J is normalized to 1:

Z

dxJ ðx; tÞJðx; tÞ ¼ 1

(83)

(84)

where integration covers the whole space. The average value of any physical observable15 A described by the Hermitian operator A^ is obtained from: R Z  ^ dxJ ðx; tÞAJðx; tÞ ^R Jðx; tÞJ ðx; tÞ ¼ dx A (85) hAi ¼ R dxJ ðx; tÞJðx; tÞ dxJ ðx; tÞJðx; tÞ where integration is extended over all space, and A^ acts always on J and not on J : The last expression above shows that A^ is weighted with the (normalized) probability density JJ. 3. Time evolution of the state function The state function J is obtained by solving the time-dependent Schroedinger16 equation: vJðx; tÞ ^ HJðx; tÞ ¼ iZ vt

(86)

a partial differential equation, which is second order in the space coordinate x and first order in the ^ so that the total energy E is seen to time t. This equation involves the Hamiltonian of the system H, play a fundamental role among all physical observables. If the Hamiltonian H^ does not depend explicitly on t (the case of stationary states), following the usual mathematical techniques of Chapter 3, the variables in Eqn (86) can be separated by writing J as the product of a space function jðxÞ and a time function g(t): Jðx; tÞ ¼ jðxÞgðtÞ (

ˇ

giving upon substitution:

H jðxÞ ¼ EjðxÞ gðtÞ ¼ g0 expðiutÞ

15

(87)

(88)

Its expectation value that can be observed by experiment. Schroedinger Erwin 1887–1961, Austrian physicist, Professor at the Universities of Breslaw, Zu¨rich, Berlin, Oxford, and Dublin; 1933 Nobel Prize for Physics. 16

1.2 The variational method

19

where E is the separation constant, g0 an integration constant, and u ¼ E=Z. The first of Eqn (88) is the eigenvalue equation for the total energy operator (the Hamiltonian) of the system, and jðxÞ is called the amplitude function. This is the Schroedinger equation that we must solve or approximate for the physical description of our systems. The second equation gives the time dependence of the stationary state, while a more general time dependence is fundamental in spectroscopy. It is immediately evident that for the stationary state the probability JJ dx is independent of time: Jðx; tÞJ ðx; tÞdx ¼ jJðx; tÞj2 dx ¼ jjðxÞj2 jg0 j2 dx

(89)

1.2 THE VARIATIONAL METHOD The variational method is the most powerful technique for doing working approximations when the Schroedinger eigenvalue equation cannot be solved exactly. Applications involve optimization of variational parameters, either non-linear (orbital exponents) or linear (the Ritz method). Many examples will be worked out in detail later in Chapter 13. Let 4 be a normalizable regular trial (or variational) function. We define the Rayleigh17 ratio as the functional:18 R ^ ^ dx4 ðxÞH4ðxÞ h4jHj4i ε½4 ¼ (90a) ¼ R  dx4 ðxÞ4ðxÞ h4j4i where x are the electronic coordinates and H^ the Hamiltonian of the system. Then: ε½4  E0

(91)

is the Rayleigh variational principle for the ground state, E0 being the true ground state energy; ε½4  E1 provided hj0 j4i ¼ 0

(92)

is the Rayleigh variational principle for the first excited state, provided the trial function 4 is taken orthogonal to the true ground state function j0 . The proofs of these statements are easily done by expanding 4 in the orthonormal set of the ^ eigenfunctions fjk g of H. We have for the ground state: X X 4¼ (93) jk Ck ¼ jk hjk j4i k

k

Then we have from Eqn (91): XX XX Ck Cl hjk jH^  E0 jjl i ¼ Ck Cl ðEl  E0 Þdkl ε  E0 ¼ h4jH^  E0 j4i ¼ ¼

X

k

2

l

jCk j ðEk  E0 Þ  0

k

l

(94)

k

and inequality (91) is proved. 17

Rayleigh John Strutt (Lord) 1842–1919, English physicist, Cavendish Professor of Physics at the University of Cambridge (UK); 1904 Nobel Prize for Physics. 18 A function of function 4ðxÞ. A domain of a functional is a set of admissible functions rather than a region of a coordinate space.

20

CHAPTER 1 Mathematical foundations and approximation methods

Next, to prove (92), we observe that expansion (93) can be written: X X X 4¼ jk hjk j4i ¼ j0 hj0 j4i þ jk C k ¼ jk Ck k

ks0

(95)

ks0

because of the orthogonality constraint, so that: ε  E1 ¼ h4jH^  E1 j4i ¼ ¼

X

XX

Ck Cl hjk jH^  E1 jjl i ¼

ks0 ls0

XX

Ck Cl ðEl  E1 Þdkl

ks0 ls0

2

jCk j ðEk  E1 Þ  0

(96)

ks0

all eigenvalues Ek being larger than or at most equal to E1, which proves inequality (92). Evaluation of the integrals in the Rayleigh ratio (90a) under the constraints of normalization and orthogonality gives therefore upper bounds to the energy of ground and excited state. This is of fundamental importance in applications, since the variational energy must always lie above the true energy. As general comments on the variational principle (91), we can say that: • The equality sign holds for the exact functions. • If the variational function is affected by a first-order error, the error in the variational energy is second order. Therefore, energy is always determined better than wavefunction. The same is true for the second-order energies of Section 1.3.2.2. In fact, let the variational function 4 differ from the true j0 by a small first-order function d: 4 ¼ j0 þ d

(97)

  ε  E0 ¼ h4jH^  E0 j4i ¼ hj0 þ djH^  E0 jj0 þ di ¼ hdjH^  E0 jdi ¼ O d2  0

(98)

Then, we have for the energy:

and the error in ε is second order in d. • The variational method privileges the regions of space near the nucleus, so that variationally determined wavefunctions may not be appropriate for dealing with the expectation values of operators that take large values far from the nucleus, like electric moments or polarizabilities. Variational approximations to energy and wavefunction can then be worked out simply by introducing variational parameters fcg in the trial function, then evaluating the integrals in functional (90a), giving in this way an ordinary function of the variational parameters fcg which can be minimized against the parameters. For N parameters ðc1 ; c2 ; /; cN Þ: R ^ dx4 ðx; c1 ; c2 ; /; cN ÞH4ðx; c1 ; c2 ; /; cN Þ (99) ¼ εðc1 ; c2 ; /; cN Þ ε½4 ¼ R  dx4 ðx; c1 ; c2 ; /; cN Þ4ðx; c1 ; c2 ; /; cN Þ vε vε vε ¼ ¼/¼ ¼0 vc1 vc2 vcN

(100)

1.2 The variational method

21

being the necessary conditions for the stationarity of the functional ε½4. Solution of the N Eqn (100) gives the set of N optimized parameters:   cðbestÞ ¼ c01 ; c02 ; /; c0N (101) and, substituting in Eqn (99), we obtain the best variational energy as:   εðbestÞ ¼ ε c01 ; c02 ; /; c0N

(102)

In this way, we obtain the best approximation compatible with the form assumed for the approximate trial function. Increasing the number of flexible parameters increases the accuracy of the variational result.19 For working approximations, use is made of some basis of regular functions (such as the exponentially decreasing Slater-type orbitals, STOs, or the Gaussian decreasing-type orbitals, GTOs, of Chapter 12), introducing either (1) non-linear (orbital exponents) or (2) linear variational parameters. Only in the latter case, the stationarity Eqn (100) can be solved in a standard way connected with the diagonalization of the matrix representative of the Hamiltonian H^ over the basis functions c.

1.2.1 Non-linear parameters In this case, we cannot obtain any standard equations for the optimization, which must usually be done by numerical methods (e.g. the simple Ransil (1960) method, useful for functions having a parabolic behaviour near the minimum, see Problem 13.5 of Chapter 13). The method is rather powerful in itself but, when many parameters are involved, there may be some troubles with the difficulty of avoiding spurious secondary minima in the energy hypersurface, which would spoil the numerical results. Examples for the ground and excited states of the particle in the box, the harmonic oscillator, the atomic one- and two-electron system, and the molecular one-electron system will be given in detail in Chapter 13.

1.2.2 Linear parameters: the Ritz method The method of linear combinations is due to the young Swiss mathematician Ritz (1909; see also Pauling and Wilson, 1935), and is usually referred to as Ritz’s method. Flexibility in the trial function is introduced through the coefficients of the linear combination of a given set of regular functions. Usually, the basis functions are fixed, but they can be successively optimized even with respect to the non-linear parameters present in their functional form. We shall see that the Ritz method is intimately connected with the problem of matrix diagonalization of Chapter 6. We consider a finite basis set of N orthonormal functions c, the problem being best treated in matrix form. The Rayleigh ratio (90a) can be written as: ε ¼ HM 1 19

(90b)

Using appropriate numerical methods, it is feasible today to optimize variational wavefunctions containing millions of terms (Roos, 1972).

22

CHAPTER 1 Mathematical foundations and approximation methods

where: ^ H ¼ h4jHj4i;

M ¼ h4j4i

(103)

If we introduce the set of N orthonormal functions as the (1  N) row matrix : c ¼ ðc1 c2 /cN Þ and the corresponding set of variational coefficients as the (N1) column matrix: 0 1 c1 B c2 C C c¼B @/A cN

(104)

(105)

then H and M in Eqn (103) can be written in terms of the (N  N) Hermitian matrices H and M: ^ ¼ cy Hc; H ¼ cy cy Hcc

M ¼ cy cy cc ¼ cy Mc ¼ cy 1c

(106)

where M ¼ 1 is the metric matrix of the basis functions c, and H the matrix representative of the Hamiltonian H^ over the basis functions. The matrix elements of matrices H and M are:      ^ n i; Mmn ¼ cm cn ¼ 1mn ¼ dmn (107) Hmn ¼ cm Hjc An infinitesimal first variation in the linear coefficients will induce an infinitesimal change in the energy functional (90b): dε ¼ dH $ M 1  H $ M 2 dM ¼ M 1 ðdH  εdMÞ

(108)

where to first order in dc (a column of infinitesimal variation of coefficients): dH ¼ dcy Hc þ cy Hdc;

dM ¼ dcy 1c þ cy 1dc

(109)

The necessary condition for ε being stationary against arbitrary variations in the coefficients yields the equation: dε ¼ 0 0 dH  εdM ¼ 0

(110)

dcy ðH  ε1Þc þ cy ðH  ε1Þdc ¼ 0

(111)

and in matrix form:

Because matrix H is Hermitian, the second term in Eqn (111) is the complex conjugate of the first, so that, since dcy is arbitrary, condition (111) takes the matrix form: ðH  ε1Þc ¼ 0 0 Hc ¼ εc

(112)

which is the eigenvalue equation for matrix H, Eqn (56) of Chapter 6. The variational determination of the linear coefficients under the constraint of orthonormality of the basis functions in the Ritz method is therefore equivalent to the problem of diagonalizing matrix H. Following what was said there, the homogeneous system (112) has non-trivial solutions if and only if: jH  ε1j ¼ 0

(113)

1.2 The variational method

23

The solution of the (N  N) secular Eqn (113) yields as best values for the variational energy (90b) N real roots, the eigenvalues εm ðm ¼ 1; 2; /; NÞ of the Hermitian matrix H, usually ordered in ascending order, the corresponding set of N eigenvectors cm ðm ¼ 1; 2; /; NÞ, one for each eigenvalue, and, finally, the N functions 4m ðm ¼ 1; 2; /; NÞ, which are the best variational approximation to the eigenfunctions obtained from the linear combination of the basis functions: ε1  ε 2  /  εN

(114)

c1 ; c2 ; /; cN

(115)

41 ;42 ; /; 4N

(116)

The Ritz method not only gives the best variational approximation to the ground state energy (the first eigenvalue ε1 ), but also approximations to the energy of the excited states. A theorem due to MacDonald (1933) states further that each of the ordered roots (114) gives an upper bound to the true energy of the respective excited state: ε1  E1 ; ε2  E2 ; /;

ε N  EN

(117)

MacDonald’s theorem does not include the case of degenerate eigenvalues. Davies (1960) has discussed the problem of the separation of degenerate eigenvalues in variational calculations. He showed that the occurrence of degenerate eigenvalues does not prevent the determination of upper bounds to exact eigenvalues in physical problems. The complete set of eigenvalues and eigenvectors of matrix H is given by the full eigenvalue equation: HC ¼ Cε

(118)

where C is the square matrix of the eigenvectors (the row matrix of the single column eigenvectors, which are denoted in bold type as column indices in the full matrix): 1 0 c11 c12 / c1N B c21 c22 / c2N C C (119) C ¼ ðc1 jc2 j/jcN Þ ¼ B @ $ $ $ $ A cN1 cN2 / cNN and ε is the diagonal matrix collecting the N eigenvalues: 0 1 ε1 0 / 0 B 0 ε2 / 0 C C ε¼B @$ $ / $ A 0 0 / εN

(120)

Since matrix C is unitary (Chapter 6): CCy ¼ Cy C ¼ 1 0 C1 ¼ Cy

(121)

the Hermitian matrix H can be brought to diagonal form through the unitary transformation with the complete matrix of its eigenvectors: Cy HC ¼ ε

(122)

24

CHAPTER 1 Mathematical foundations and approximation methods

The inverse transformation: H [ CεCy

(123)

allows to express matrix H in terms of its eigenvalues and eigenvectors. We now examine the second variation of the functional (90b). The second variations of H and M are: d2 H ¼ d2 cy Hc þ cy Hd2 c þ 2dcy Hdc 2 y

y

(124)

y

d M ¼ d c Mc þ c Md c þ 2dc Mdc 2

2

(125)

where d2 c is a column of infinitesimal second variations of coefficients, so that we obtain for the functional ε:     d2 ε ¼ M 1 d2 H  εd2 M  dε $ dM  M 2 dMðdH  εdMÞ ¼ M 1 d2 H  εd2 M (126) since: dε ¼ 0;

dH  εdM ¼ 0

(127)

at the stationary point. In matrix form, using Eqns (124) and (125): 

d2 ε ¼ M 1 d2 cy ðH  ε1Þc þ cy ðH  ε1Þd2 c þ 2dcy ðH  ε1Þdc ¼ 2M 1 dcy ðH  ε1Þdc2 ¼ M 1 dcy Cðε  ε1ÞCy dc where use was made of Eqn (112) and its dagger and, finally, of Eqn (123). If we pose: Cy dc ¼ dc0 ; dcy C ¼ dc0y where

dc0

(128)

(129)

is a new column of variations, the second variation of ε can be finally written as: d2 ε ¼ 2M 1 dc0y ðε  ε1Þdc0

(130)

In components: X X X  2       dc0m y εmn  εdmn dc0v ¼ 2M 1 dc0m y εm  ε dmn dc0v ¼ 2M 1 d2 ε ¼ 2M 1 dc0m  εm  ε m; n

m; n

m

(131) We then see that for the lowest eigenvalue ε1 of H the second variation of ε is positive, so that dε ¼ 0 corresponds here to a true minimum of the ground state energy:  N  X   2   0 2  (132) d ε ε¼ε1 ¼ 2M 1 dcm  εm  ε1 > 0 m¼1

We can do the same for all remaining eigenvalues of H, so that we conclude that all ordered roots of the secular Eqn (113) give true minima for the variational energies ordered in an ascending way. Many examples on the application of the Ritz method to atomic and molecular systems will be given in Chapter 13.

1.3 Perturbative methods for stationary states

25

1.3 PERTURBATIVE METHODS FOR STATIONARY STATES The Rayleigh variational method examined so far in the previous section is undoubtedly the most important approximation method, but the perturbation method first introduced by Schroedinger (1926a) and known as the RS perturbation theory is very important too, particularly when treating molecular properties and small interactions. In RS theory, the actual problem is related to that of a simpler one which has already been solved, and a solution is seeked by expanding energy and wavefunction in powers of a perturbation parameter l. The unperturbed problem must not be too different from the actual one, so that the difference between the two Hamiltonians can be treated as a small perturbation. Powers of l give the orders of the perturbation expansion and, at least in principle, corrections can be pushed to higher and higher orders so improving in a systematic way the accuracy of the results. Schroedinger (1926a) used his method for the calculation of the first-order Stark effect in the hydrogen atom.20 Standard RS perturbation theory for stationary states up to the third order with a short outline of higher orders is introduced in Section 1.3.1, recent advances in the calculation of second-order terms being presented in Section 1.3.2.4. Section 1.3.3 introduces a modification of the theory due to Lennard-Jones–Brillouin–Wigner, which is valid even for large perturbations. Next, Section 1.3.4 deals with the perturbation theories without partitioning of the Hamiltonian and Section 1.3.5 with what are known as the symmetry-adapted perturbation theories (SAPTs). Finally, Section 1.3.6 presents the moment method, a recent development of the theory, which is particularly useful for high orders of perturbation.

1.3.1 RS perturbation theory We want to solve the Schroedinger eigenvalue equation: ðH^  EÞj ¼ 0

(133)

for the Hermitian decomposition of the Hamiltonian H^ into: H^ ¼ H^0 þ lH^1

(134)

where (1) l is a parameter giving the orders of the perturbation theory, (2) H^0 is the unperturbed Hamiltonian, namely the Hamiltonian of the problem (either physical or model) already solved, and (3) H^1 is the small first-order difference between H^ and H^0 , called the perturbation. We expand both the eigenvalue E and the eigenfunction j into powers of l: 21

E ¼ E0 þ lE1 þ l2 E2 þ l3 E3 þ /

(135)

j ¼ j0 þ lj1 þ /

(136)

where the coefficients of the different powers of l are, respectively, the corrections of the various orders to energy and wavefunction (for instance, E2 is the second-order energy correction, j1 the 20 21

See Section 3.10 of Chapter 3. Orders are here carried by the suffixes or their sum.

26

CHAPTER 1 Mathematical foundations and approximation methods

first-order correction to the wavefunction, and so on). It is sometimes useful to define corrections up to a given order, which we write, for instance: Eð3Þ ¼ E0 þ E1 þ E2 þ E3 meaning that we add corrections up to the third order. By substituting the expansions into the Schroedinger Eqn (133):  

ðH^0  E0 Þ þ lðH^1  E1 Þ  l2 E2  l3 E3  / j0 þ lj1 þ l2 j2 þ / ¼ 0

(137)

(138)

separating the orders we obtain: 8 0 l ðH^0  E0 Þj0 ¼ 0 > > > > > > > lðH^0  E0 Þj1 þ ðH^1  E1 Þj0 ¼ 0 > < l2 ðH^0  E0 Þj2 þ ðH^1  E1 Þj1  E2 j0 ¼ 0 > > > 3 ^ > > l ðH 0  E0 Þj3 þ ðH^1  E1 Þj2  E2 j1  E3 j0 ¼ 0 > > > : /

(139)

which are known as RS perturbation equations of the various orders specified by the power of l. Because of the Hermitian property of H^0 , bracketing Eqn (139) on the left by hj0 j all the first terms in the RS equations are zero, and we are left with: 8 0 l hj0 jH^0  E0 jj0 i ¼ 0 > > > > > > > lhj0 jH^1  E1 jj0 i ¼ 0 > < l2 hj0 jH^1  E1 jj1 i  E2 hj0 jj0 i ¼ 0 > > > > > l3 hj0 jH^1  E1 jj2 i  E2 hj0 jj1 i  E3 hj0 jj0 i ¼ 0 > > > : $$$$ Taking j0 normalized to 1, we obtain the RS energy corrections of the various orders as: 8 0 l E0 ¼ hj0 jH^0 jj0 i > > > > > > > l E1 ¼ hj0 jH^1 jj0 i > < l2 E2 ¼ hj0 jH^1  E1 jj1 i ¼ hj1 jH^0  E0 jj1 i > > > > l3 E ¼ hj jH^  E jj i  E hj jj i ¼ hj jH^  E jj i > 3 1 2 2 0 1 1 1 > 0 1 1 1 > > : $$$$

(140)

(141)

E0 and E1 are the diagonal terms giving the average value of H^0 and H^1 , respectively, over the unperturbed function j0 , while E2 is given as a non-diagonal term, often referred to as transition integral, connecting j0 to j1 through the operator H^1 ; the expression in the third row of Eqn (141) showing that always E2 < 0 for the ground state. The equations above show that knowledge of j1 (the

1.3 Perturbative methods for stationary states

27

solution of the first-order RS differential equation) determines the energy corrections up to the third order.22 In expanding j to the various orders of perturbation theory: j ¼ j0 þ lj1 þ l2 j2 þ $ $ $

(142)

the normalization condition on the wavefunction j gives:    hjjji ¼ j0 þ lj1 þ l2 j2 þ /j0 þ lj1 þ l2 j2 þ / ¼ hj0 jj0 i þ l½hj0 jj1 i þ hj1 jj0 i þ l2 ½hj0 jj2 i þ hj2 jj0 i þ hj1 jj1 i þ / ¼ 1 meaning:

(143)

8 0 l hj0 jj0 i ¼ 1 > > > > > > lhj0 jj1 i þ hj1 jj0 i ¼ 0 > >
> > > 3 > > l hj0 jj3 i þ hj3 jj0 i þ hj1 jj2 i þ hj2 jj1 i ¼ 0 > > : /

(144)

Roughly speaking, the orthogonality conditions of the various orders mean that the corrections to the unperturbed wavefunction add to it only new features not containing anything of the original function. In general, orthogonality to order n can be written as: n X

hjk jjnk i ¼ 0 n > 0

(145)

k¼0

Orthogonality to first order gives: hj0 jj1 i þ hj1 jj0 i ¼ 0

(146)

2hj0 jj1 i ¼ 0 0 hj0 jj1 i ¼ 0

(147)

which, for real functions, means:

giving the familiar orthogonality condition on the first-order function. In general, if j1 is complex: j1 ¼ A þ iB;

A ¼ Reðj1 Þ;

B ¼ Imðj1 Þ

(148)

where A, B are real functions. Then, orthogonality to first order gives: hj0 jj1 i þ hj1 jj0 i ¼ ½hj0 jAi þ hAjj0 i þ i½hj0 jBi  hBjj0 i ¼ 0

(149)

where the last term is zero irrespective of the value of function B. The condition of strong orthogonality to first order takes then hj0 jBi ¼ 0 individually, which is simply satisfied by taking zero the imaginary part of j1 . We recall that all jn ðns0Þ corrections are not normalized (even if they are normalizable). 22

In general, jn determines E up to E2nþ1.

28

CHAPTER 1 Mathematical foundations and approximation methods

Before proceeding any further, we must explain the symmetric forms resulting for E2 and E3 in Eqn (141). Using the RS perturbation equations and taking into account the fact that the operators are Hermitian, it is possible to shift the order from the operator to the wavefunction, and vice versa, what is known as Dalgarno interchange theorem. In fact, we can write for the dagger of the first-order equation (the bra): hðH^0  E0 Þj1 j ¼ hðH^1  E1 Þj0 j

(150)

so that: E2 ¼ hj0 jH^1  E1 jj1 i ¼ hðH^1  E1 Þj0 jj1 i ¼ hðH^0  E0 Þj1 jj1 i ¼ hj1 jH^0  E0 jj1 i

(151)

Next, the third-order energy correction resulting from the third-order RS equation has the form: E3 ¼ hj0 jH^1  E1 jj2 i  E2 hj0 jj1 i

(152)

Making a repeated use of the Dalgarno’s interchange theorem, the following is obtained: E3 ¼ hðH^1  E1 Þj0 jj2 i  E2 hj0 jj1 i ¼ hðH^0  E0 Þj1 jj2 i  E2 hj0 jj1 i ¼ hj1 jH^0  E0 jj2 i  E2 hj0 jj1 i ¼ hj1 jH^1  E1 jj1 i  E2 ½hj0 jj1 i þ hj1 jj0 i ¼ hj1 jH^1  E1 jj1 i

(153)

where the last term in square parenthesis is zero because of the orthogonality condition in first order. Two important considerations must be done at this point. First, we must emphasize that the leading term of the RS perturbation Eqn (139), the zeroth-order equation ðH^0  E0 Þj0 ¼ 0, must be satisfied exactly, otherwise uncontrollable errors will affect the whole chain of equations. Second, it must be observed that only energy in first order gives an upper bound to the true energy of the ground state, so that the energy in second order, E(2), may lie below the true value.23 For a glance to the higher orders in the theory, it may help to rewrite RS theory in a formally more compact, even if less transparent, form. Let: G¼HE

(154)

be a Hermitian operator where H is the Hamiltonian and E the energy eigenvalue of the actual system ^ in state j, and where we omit for short the caret denoting the operator signs on H^ and G. The Schroedinger eigenvalue equation then takes the form: Gj ¼ 0

(155)

Expanding H,j,E,G in powers of the perturbation parameter l, Schroedinger Eqn (133) becomes: N X n¼0

ln

n X

Gk jnk ¼ 0

(156)

k¼0

To simplify further the writing, we introduce Dirac’s notation omitting the now useless symbol j. 23

This is particularly true when the correct value of E2 is determined.

1.3 Perturbative methods for stationary states

29

Equating to zero the coefficient of each power n of l, we obtain the Schroedinger’s perturbation equations of order n in the compact form: n X

Gk jn  ki ¼ 0

n ¼ 0; 1; 2; 3; /

(157)

k¼0

which must be solved under the orthonormality constraints: n X hkjn  ki ¼ dn0

(158)

k¼0

n ¼ 0 gives the normalization condition of the unperturbed function j0 ¼ j0i, while n > 0 gives the orthogonality conditions to order n: n X (159) hkjn  ki ¼ 0 k¼0

The corresponding energy corrections are obtained by bracketing with h0j on the left the RS Eqn (157): n n n1 X X X (160) h0jGk jn  ki ¼ h0jGk jn  ki ¼ h0jGn j0i þ h0jGk jn  ki ¼ 0 k¼0

k¼1

k¼1

where the term with k ¼ 0 is zero because j0 does satisfy the zeroth-order equation and G0 is Hermitian. Therefore, from the definition of G it follows that the n-th RS energy correction will be: n1 X (161) En ¼ h0jHn j0i þ h0jGk jn  ki k¼1

The first RS differential equations of the various orders, corresponding to the previous Eqn (139), are then: 8 0 l G0 j0i ¼ 0 > > > > 1 > > > < l G0 j1i þ G1 j0i ¼ 0 (162) l2 G0 j2i þ G1 j1i þ G2 j0i ¼ 0 > > > > l3 G0 j3i þ G1 j2i þ G2 j1i þ G3 j0i ¼ 0 > > > : / with the corresponding RS energy corrections (see previous Eqn (141)): 8 0 l E0 ¼ h0jH0 j0i > > > > > > l1 E1 ¼ h0jH1 j0i > > > > 2 > > > < l E2 ¼ h0jH2 j0i þ h0jG1 j1i ¼ h0jH2 j0i þ h1jG0 j1i > > > > > > > > > > > > > :

l3

E3 ¼ h0jH3 j0i þ h0jG1 j2i þ h0jG2 j1i ¼ h0jH3 j0i  h1jG0 j2i þ h0jG2 j1i

¼ h0jH3 j0i þ h1jG1 j1i þ h1jG2 j0i þ h0jG2 j1i $$$

(163)

30

CHAPTER 1 Mathematical foundations and approximation methods

where use was made of the perturbative equations of the first and second order. In the important case of intermolecular interactions, Hk ¼ 0 for k > 1, so that the terms involving H2 and H3 are absent and the last two terms of E3 are zero for the orthogonality to first order. The second- and third-order RS energy corrections then take the form: E2 ¼ h1jG0 j1i;

E3 ¼ h1jG1 j1i

(164)

Using repeatedly the perturbative equations of lower order and the orthogonality relations, simple algebraic manipulations give for the case Hk ¼ 0 for k > 1 the general formulae: E2n ¼ hn  1jG1 jni 

n X j¼2

E2nþ1 ¼ hnjG1 jni 

n X j¼2

Ej

Ej

n1 X

hkj2n  j  ki n  2

(165)

k¼nj n X

hkj2n þ 1  j  ki n  2

(166)

k¼nþ1j

The second terms in these equations are known as renormalization terms. These formulae express the well known fact that the perturbative correction of order n in the wavefunction determines the energy corrections up to order (2n þ 1). We now look in a little more detail at the first two orders of RS perturbation theory and at their applications. 1. First-order theory First-order RS theory is useful, for instance, in explaining Zeeman effect and the splitting of the multiplet structure in atoms, or in giving the Coulombic component of the interaction energy between atoms and molecules. An interesting case arises when the zeroth-order energy level E0 is N-fold degenerate, namely there is a set of N orthonormal (linearly independent) functions all belonging to the same eigenvalue E0: D  E  j0a j0b ¼ dab a; b ¼ 1; 2; /; N (167) In this case, we replace j0 by a linear combination of the N functions belonging to the degenerate eigenvalue, and the first-order RS perturbation equation can be written as: X ðH^0  E0 Þj1 þ ðH^1  E1 Þ j0b Cb ¼ 0 (168) b

Bracketing on the left by hj0a j, the first term is zero and we obtain: * + X 0 ^ 0 j a j H 1  E1 j jb Cb ¼ 0

(169)

b

giving the homogeneous system: i Xh ðH^1 Þab E1 dab Cb ¼ 0 b

(170)

1.3 Perturbative methods for stationary states

31

which, in the matrix form of Chapter 6, is written as: ðH1  E1 1ÞC ¼ 0

(171)

In this matrix equation, H1 is the (N  N) matrix representative of the perturbation H^1 over the N functions belonging to the degenerate eigenvalue E0. The corresponding secular equation: jH1  E1 1j ¼ 0 has N real roots:

(172)

a ¼ 1; 2; $ $ $ ; N

E1a

(173)

When all roots (173) are different, degeneracy in first order will be completely removed. A striking example of this first-order degenerate perturbation theory is offered by the Hu¨ckel’s theory of chain hydrocarbons (Chapter 14). In this case, the determinantal Eqn (172) is nothing but the Hu¨ckel determinantal equation arising from the Hu¨ckel determinant DN :    x 1 0 /    1 x 1 /  ¼0 DN ¼  (174)  / / / / / 0 1 x  where: x ¼

aε ; b

ε ¼ a þ xb

(175)

In these equations, a and b are the two parameters of Hu¨ckel’s theory and x is the p bond energy in units of b. Secular equations like Eqn (174) for the linear and closed polyene chain were first solved by Lennard-Jones (1937) and re-derived by Coulson (1938) and McWeeny (1979). They will be studied in detail in Sections 14.8.2 and 14.8.3 of Chapter 14 of this book. 2. Second-order theory RS theory in second order is of great importance when dealing with interatomic and intermolecular interactions and second-order electric properties of molecules (see Chapter 17). Here, we limit ourselves to present the second-order RS energy expression in terms of the formal expansion in eigenstates of H^0 . We expand the first-order function j1 in the complete set of the orthonormal eigenfunctions of H^0 : X  j00 Ck0 k0 > 0 j1 ¼ (176) k k0

Substituting in the first-order RS equation, we obtain: X   Ck0 Ek00  E0 j0k0 þ ðH^1  E1 Þj0 ¼ 0

(177)

Bracketing on the left by hj0k j Eqn (177) becomes: X     Ck0 Ek00  E0 dkk0 þ j0k jH^1  E1 jj0 ¼ 0

(178)

k0

k0

giving coefficient Ck as:



j0 jH^1  E1 jj0 Ck ¼  k 0 E k  E0

 k>0

(179)

32

CHAPTER 1 Mathematical foundations and approximation methods

First-order RS function and second-order RS energy then become:   X   j0k jH^1  E1 jj0 0  jk j1 ¼  Ek0  E0 k>0 E2 ¼ 

  X  j0k jH^1  E1 jj0 2 k>0

Ek0  E0

0 * + ! all    X 1 0 0  ^ ^ j0 jðH 1  E1 Þ ¼ jk jk ðH 1  E1 Þjj0 DE k E 1 1 D 1ðH^1  E1 Þjj0 i ¼  ¼  hj0 jðH^1  E1 Þ^ j0 jðH^1  E1 Þ2 jj0 DE DE (184)

1.3 Perturbative methods for stationary states

33

where ^ 1 is the identity operator.24 In this way, we are left with the evaluation of the expectation value of the squared perturbation over the unperturbed ground state function. No more than a rough estimate could be obtained by this formula due to the uncertainty in the value to be given to the average excitation energy DE. Unso¨ld approximation was put onto a firm quantitative basis by Kirkwood (1932) using the Hylleraas variational method which we explain in the next section.

1.3.2.2 The Hylleraas method Hylleraas (1930) introduced the second-order energy functional as: h i D E D E D E ~1 þ j ~1 ~1 ¼ j ~ 1 jH^0  E0 jj ~ 1 jH^1  E1 jj0 þ j0 jH^1  E1 jj E~2 j

(185)

~ 1 is a first-order variational approximation to j1 and E~2 a variational approximation to E2. where j If j1 is the exact first-order wavefunction, then functional E~2 coincides with the exact second-order correction to the energy, since: E~2 ½j1  ¼ hj1 jH^0  E0 jj1 i þ hj1 jH^1  E1 jj0 i þ hj0 jH^1  E1 jj1 i ¼ hj1 jðH^0  E0 Þj1 þ ðH^1  E1 Þj0 i þ hj0 jH^1  E1 jj1 i ¼ hj0 jH^1  E1 jj1 i ¼ E2

(186)

where the ket in the first term of the second row is zero because the first-order RS Eqn (139) is exactly satisfied. ~ 1 ¼ j1 þ dj1 is a well-behaved variational approximation differing from the Otherwise, if j exact first-order function j1 by the infinitesimal first-order function dj1 , it is readily shown that E~2 gives an upper bound to the true E2 and that the error in E~2 is second order in the error function dj1 . In fact: E~2 ½j1  ¼ hj1 þ dj1 jH^0  E0 jj1 þ dj1 i þ hj1 þ dj1 jH^1  E1 jj0 i þ hj0 jH^1  E1 jj1 þ dj1 i ¼ hdj1 jH^0  E0 jdj1 i þ hj0 jH^1  E1 jj1 i þ hj1 jðH^0  E0 Þj1 þ ðH^1  E1 Þj0 i þ hdj1 jðH^0  E0 Þj1 þ ðH^1  E1 Þj0 i þ hðH^0  E0 Þj1 þ ðH^1  E1 Þj0 jdj1 i ¼ hdj1 jH^0  E0 jdj1 i þ E2

(187)

where the last term of the second row and those of the third row are zero as explained before. Therefore:   E~2 ½j1   E2 ¼ hdj1 jH^0  E0 jdj1 i ¼ O d2  0 (188) and the theorem is proved. As already seen for the Rayleigh variational principle for the total energy, ~ 1 gives a second-order error in E~2 . So, even in second-order RS perturbation a first-order error in j theory, energy is determined better than wavefunction. 24

See Eqns (63) and (70)–(72) of Section 1.1.7.

34

CHAPTER 1 Mathematical foundations and approximation methods

~ 1 , we obtain for the change in E~2 to first order If we look for an arbitrary infinitesimal change in j ~1: in dj D E D E D E ~1 ~ 1 jH^0  E0 jj ~ 1 jH^1  E1 jj0 þ j ~ 1 jH^0  E0 jdj ~ 1 þ dj dE~2 ¼ dj D E ~1 ¼ 0 þ j0 jH^1  E1 jdj (189) ~ 1 i the giving as necessary condition for the stationarity of E~2 because of the arbitrariness of jdj Euler–Lagrange equation: ~ 1 þ ðH^1  E1 Þj0 ¼ 0 ðH^0  E0 Þj

(190)

~ 1 we find: ~ 1. To second order in dj which is nothing but the first-order equation for j   D E D E ~ 1 þ ðH^1  E1 Þj0 þ ðH^0  E0 Þj ~ 1 þ ðH^1  E1 Þj0 d2 j ~ 1 ðH^0  E0 Þj ~1 d2 E~2 ¼ d2 j E D E D ~ 1 ¼ 2 dj ~1  0 ~ 1 jH^0  E0 jdj ~ 1 jH^0  E0 jdj þ 2 dj

(191)

~ 1 and its complex conjugate. So, the second variation of because of the Euler–Lagrange equation for j E~2 is positive, and the stationary value for E~2 corresponds to a true minimum for this functional. The same result can be found from Eqns (189) and (191) in terms of functional derivatives. In fact we obtain from Eqn (189) as necessary condition for the stationarity of the functional (185) against ~ 1 j: arbitrary changes in the variational first-order function hdj dE~ ~ 1 þ ðH1  E1 Þj0 ¼ 0 D 2  ¼ ðH^0  E0 Þj ~ 1  dj

(192)

d2 E~2   ¼ ðH^0  E0 Þ  0 ED  ~ ~ 1  dj1 dj

(193)

and from Eqn (191):

since ðH^0  E0 Þ (the excitation energy operator) is positive definite. ^ second-order In the context of double perturbation theory, involving static perturbations V^ and W, bivariational functionals were introduced for studying magnetic properties of molecules and atomic polarizabilities (see, among others, Kolker and Karplus, 1964; Kolker and Michels, 1965).

1.3.2.3 The Kirkwood method

~ 1 in the form: Using the Unso¨ld approximation in the first-order variational function, we assume j   X   j0k jH^1  E1 jj0 1 0 j ~1 ¼  x j k DE εk k>0

all    X  j 0 j0  k

k

k

! ðH^1  E1 Þ j0 ¼ 

1 ^ ðH 1  E1 Þj0 DE (194)

1.3 Perturbative methods for stationary states

35

~ 1 the by the closure property. Therefore, Kirkwood (1932) suggested to use as a first approximation to j function: ~ 1 ¼ C H^1 j0 (195) j ~ 1 is not normalized. For real functions, the Hylleraas functional for where C is a linear coefficient and j Eqn (195) then becomes the ordinary function of C:   2  E~2 ðCÞ ¼ C2 hj0 jH^1 ðH^0  E0 ÞH^1 jj0 i þ 2C j0 H^1 j0 (196) which we minimize against C obtaining as necessary condition:   2  d E~2 ¼ 2Chj0 jH^1 ðH^0  E0 ÞH^1 jj0 i þ 2 j0 H^1 j0 ¼ 0 dC which gives the best C as:

   j0 H^21 j0 C¼ hj0 jH^1 ðH^0  E0 ÞH^1 jj0 i

(197)



Substituting in Eqn (196) we obtain as best E~2 :   2  2  j0 H^ j0  1 ~ E2 ðbestÞ ¼  hj0 jH^1 ðH^0  E0 ÞH^1 jj0 i

(198)

(199)

Comparison with Eqn (184) gives Kirkwood’s determination of Unso¨ld’s average excitation energy as: DEðUnsoldÞ ¼

hj0 jH^1 ðH^0  E0 ÞH^1 jj0 i   2  >0 j0 H^1 j0

(200)

However, Hirschfelder et al. (1964) observed that the denominator in Eqn (200) diverges when H^1 is the electron repulsion 1/r12, so that DE ¼ 0 and Kirkwood’s method cannot be used for this important perturbation.

1.3.2.4 The Ritz method for E~ 2: linear pseudostates

~ 1 as well. We expand the first-order The Ritz method of Section 1.2.2 of this chapter can be applied to j ~ trial function j1 into the set of N normalized basis functions which we write as the (1  N) row matrix c: c ¼ ðc1 c2 /cN Þ

(201)

where the functions are possibly orthogonal in themselves but must necessarily be orthogonal to j0 . We assume that the set (201) be orthonormal, namely: cy c ¼ 1;

cy j0 ¼ 0

(202)

If the cs are not orthogonal they must be preliminarly orthogonalized by the Schmidt method. ~ 1 in the finite set of the cs, we can write: By expanding j ~ 1 ¼ cC ¼ j

N X k¼1

ck Ck

(203)

36

CHAPTER 1 Mathematical foundations and approximation methods

Then, we construct the matrices: M ¼ cy ðH^0  E0 Þc

(204)

the (N  N) Hermitian matrix of the excitation energies, and: N ¼ cy ðH^1 j0 Þ

(205)

the (N  1) column vector of the transition elements of the perturbation. Then, the Ritz’s form of the Hylleraas functional (185) is E~2 ¼ Cy MC þ Cy N þ Ny C

(206)

dE~2 ¼ MC þ N ¼ 0 0 CðbestÞ ¼ M1 N dCy

(207)

which is minimum for:

giving as best variational approximation to E~2 : E~2 ðbestÞ ¼ Ny M1 N

(208)

It was recently shown by Magnasco and Battezzati (2007) that the ordinary form (208) of the Ritzoptimized second-order Hylleraas functional is a particular case of the general determinantal equation derived by Battezzati (1979) in his studies on bivariational functionals. The Hermitian matrix M can be reduced to diagonal form by a unitary transformation U among its basis functions c: j ¼ cU;

Uy MU ¼ ε;

Uy N ¼ Nj

where ε is here the (N  N) diagonal matrix of the (positive) excitation energies: 0 1 ε1 0 / 0 B 0 ε2 / 0 C C ε¼B @/ / / /A 0 0 / εN

(209)

(210)

The js are called pseudostates, and give E~2 in the form: y E~2 ðbestÞ ¼ Nj ε1 Nj ¼ 

N X jhjk jH^1 jj0 ij2 k¼1

εk

(211)

which is known as sum-over-pseudostates expression. Equation (211) has the same form as the analogous expression (181) arising from the complete discrete set of eigenstates of H^0 , but has definitely better convergence properties, reducing the infinite summation over eigenstates to a finite summation over pseudostates, and avoiding the need of considering the contribution from the continuous part of the spectrum. The examples of the dipole polarizability of the hydrogen atom and the C6 dispersion coefficient for the long-range interaction between two ground state hydrogen atoms will show the power of the pseudostate method, even at its simplest level, the two-term approximation, and will be given in detail in Section 13.7 of Chapter 13.

1.3 Perturbative methods for stationary states

37

1.3.3 BW perturbation theory We have already said that RS perturbation energy corrections converge rapidly when the perturbation is small. When this is not the case, it is possible to resort to a method, which was really first proposed by Lennard-Jones (1930) and later extended by Brillouin (1932, 1933) and Wigner (1935) so that it should appropriately be referred to as Lennard-Jones–Brillouin–Wigner perturbation theory. Nonetheless, following general use, we shall refer to it in short as the BW method. We follow here in an elementary way the historical development of the theory. The problem is to find a solution of the equation: ðH^0 þ V  EÞj ¼ 0

(212)

when we know the solution of the equation with V ¼ 0: ðH^0  EÞj ¼ 0

(213)

Following Lennard-Jones, we assume to know the whole set (possibly infinite) of eigenfunctions fji g and eigenvalues Ei of H^0 , where we omit for short the apex specifying that we are referring to unperturbed quantities. We further assume that there is no degeneracy in the eigenvalues so that there is only one eigenfunction for every eigenvalue Ei and we write V for H^1. We expand the wavefunction j of the actual system and Vji in the eigenfunctions of H^0 as: j¼

X

ji c i

(214)

i

Vji ¼

X

jj Vji

(215)

  Vji ¼ jj jVjji

(216)

j

where using Dirac’s notation:

Then: Vj ¼

X

ðVji Þci ¼

X

X

jj Vji

(217)

jj Vji ¼ 0

(218)

or interchanging i and j in the second term: X X X ci ðEi  EÞji þ cj ji Vij ¼ 0

(219)

i

ci

i

j

Substituting in Eqn (212) for j and Vj, it is found: X

ci ðEi  EÞji þ

i

X i

i

j

ci

X j

i

Equating coefficients of similar ji s the following is obtained: X ci ðE  Ei Þ ¼ cj Vij j ¼ 1; 2; / j

(220)

38

CHAPTER 1 Mathematical foundations and approximation methods

which is a system of equations determining the coefficients ci and the possible values of E. Written explicitly the homogeneous system is: 8 > > > > ½V11  ðE  E1 Þc1 þ V12 c2 þ V13 c3 þ / ¼ 0 < V21 c1 þ ½V22  ðE  E2 Þc2 þ V23 c3 þ / ¼ 0 > > V31 c1 þ V32 c2 þ ½V33  ðE  E3 Þc3 þ / ¼ 0 > > :/

(221)

The system (221) admits non-trivial solutions if and only if the determinant of the coefficients vanishes:   V11  ðE  E1 Þ V12   V  ðE  E2 Þ V 21 22   V V 31 32   /

V13 V23 V33  ðE  E3 Þ

 /  /  ¼0 /  

(222)

We thus have a determinant that may have an infinite number of rows and columns. The successive approximations of the Schroedinger method are then obtained as follows. First, to be definite, suppose that we want the perturbation of E1. If we neglect all Vij except V11 we get the first approximation: E ¼ E1 þ V11

(223)

which is Schroedinger’s result to first order. Next, neglect every Vij except those of the first row and column. We then obtain the bordered determinant (Chapter 6):   V11  ðE  E1 Þ   V21   V31  

V12 E2  E 0 /

V13 0 E3  E

 /  /  ¼0 /  

(224)

Denoting the determinant in Eqn (224) by D and its elements by Aij, the Cauchy expansion of this determinant according to the elements of the first row and column (Section 6.3 of Chapter 6) gives: D ¼ A11 ja11 j 

XX

      Ai1 A1j a11;ij  ¼ A11 ja11 j  A21 A12 a11;22   A31 A13 a11;33   /

i>1 j>1

¼ ðV11 þ E1  EÞ ðE2  EÞðE3  EÞ/  V21 V12 ðE3  EÞðE4  EÞ/  V31 V13 ðE2  EÞðE4  EÞ $ $ $ ¼ ðV11 þ E1  EÞ P  V21 V12

P P  V31 V13 / E2  E E3  E

(225)

1.3 Perturbative methods for stationary states

39

where ja11 j is the cofactor of A11 in D and all cofactors25 ja11;ij j are zero unless i ¼ j and we have put: P ¼ ðE2  EÞðE3  EÞðE4  EÞ/

(226)

Putting D ¼ 0, we obtain the expansion: ðV11 þ E1  EÞP 

X

V1j Vj1

j>1

P ¼0 Ej  E

(227)

Hence the equation to determine E is: E ¼ E1 þ V11 

X V1j Vj1 j>1

Ej  E

(228)

which is identical with the energy expression resulting from RS second-order theory except that E now appears in each denominator of the summations on the right-hand side. Equation (228) is the BW expression of the perturbed energy evaluated to the second order. As a first approximation, we put E ¼ E1 on the right-hand side, and, in a second approximation, we put E ¼ E1 þ V11 on the right getting an improved value for E on the left, and so on. The process can be continued until E is found to any required degree of accuracy. The greater the perturbation, the greater the number of elements that must be retained in the determinant, but, in principle, the method can be applied however large the perturbation. The theory for degenerate states can be developed precisely in the same way. When the energy values have been determined, the corresponding eigenfunctions are given by Eqn (214) where the coefficients are obtained in the usual way by substituting each energy value in the homogeneous system (221) or by a determinant of the type:    j1 j2 / ji /    V21 V22  ðE  E2 Þ / V2i /    (229) / J ¼     Vi1 V / V  ðE  E Þ / ii i i2     / Lennard-Jones (1930) applied this determinantal form of the theory to the perturbation of rotating polar molecules in an electric field, a problem arising in the theory of dielectrics (Debye, 1929). Lennard-Jones’ theory was extended by Brillouin (1933), who gave the energy expression up to fourth order in V:26 E ¼ E1 þ V11 

X V1r Vr1 r

Er  E

þ

XX r

s

XXX V1r Vrs Vs1 V1r Vrs Vst Vt1  þ/ ðEr  EÞðEs  EÞ ðE  EÞðE r s  EÞðEt  EÞ r s t (230)

25 26

The cofactors of Aij in ja11j. In Eqn (230), r,s,t>1.

40

CHAPTER 1 Mathematical foundations and approximation methods

and later, in a different way, by Wigner (1935), who gave the general BW formula for the energy as: E ¼ E1 þ V11 þ

2n X

Tjþ1 ðEÞ

(231)

V1m1 Vm1 m2 /Vmj1 mj     E  Em1 E  Em2 / E  Emj

(232)

j¼1

where: Tjþ1 ðEÞ ¼ ljþ1

X m1 /mj



Equations (230) and (231), like Eqn (228), can be solved by successive approximations. The series (232) converges even in cases in which the RS perturbation diverges. At least in principle, Eqns (230) and (231) could be obtained by the Lennard-Jones’ procedure simply by taking into account further rows and columns in the determinant of Eqn (222). Brillouin (1933) used this theory to treat the problem of free electrons in metals. An elegant derivation of the BW theory using matrix partitioning and density matrices techniques will be outlined in Section 1.3.4.2 of this chapter in the context of a configurational interaction leading to EN perturbation theory. The BW perturbation theory presented so far was initially believed to be superior to the RS perturbation theory because of its more rapid convergence, but, with the development of many-body theories in the second part of the 1950s by Brueckner (1955), Goldstone (1957), and Hubbard (1957), its shortcomings in the possibility of using it as an appropriate perturbation method in the many-body theory of electronic structure become readily apparent. This was because the presence of the exact energy in the denominators of the energy expression gives rise to unphysical terms that scale nonlinearly with the number of electrons as instead it should be for a correct many-body theory, where EfN. This size inconsistency of BW theory was the object of much attention by the researchers in the field, but we shall not dwell any further on this problem here, the interested reader being referred to the very interesting paper by Hubac and Wilson (2000) where use of BW perturbation theory for manybody systems was critically re-examined and where further references to the problem can be found. Corrections to the method of limited configuration interaction (CI) based on BW perturbation theory were derived by Hubac et al. (2000) and applied to the evaluation of correlation corrections to the ground state of the water molecule. In a recent paper by the Wilson’s group (Papp et al., 2007), where the many-body BW formalism is presented in terms of modern projector operator techniques (Lo¨wdin, 1962), second-order BW perturbation theory was suggested as a convenient working approach to the electron correlation problem for systems demanding use of multireference functions.

1.3.4 Perturbation methods without partitioning of the Hamiltonian In this section, we take into consideration two important perturbation methods where the partition of the Hamiltonian H^ into an unperturbed Hamiltonian H^0 and a perturbation H^1 ¼ V is unnecessary since it does not appear in the final perturbation formulae expressing either wavefunction or energy. This is of a particular importance in the perturbation theory of interatomic or intermolecular interactions, where the partition of the Hamiltonian into an unperturbed Hamiltonian referring to isolated

1.3 Perturbative methods for stationary states

41

not interacting atoms or molecules and a first-order perturbation representing the interatomic or intermolecular potential V causes serious symmetry problems, which will be examined to some extent in Section 1.3.5 of this chapter. We shall present, first, the method of Lebeda and Schrader (LS method) and, next, what is called the Epstein-Nesbet (EN) method.

1.3.4.1 LS perturbation theory Lebeda and Schrader (1968) suggested a method that does not require an explicit partitioning of the Hamiltonian and works whenever V is a scalar perturbation. Let: ^ 0i H00 ¼ hj0 jHjj

(233)

with: ðH^0  E0 Þj0 ¼ 0;

H^0 ¼ H^  V

(234)

where V is a scalar perturbation. Then, provided 4 is a real nodeless regular function such that: j1 ¼ 4j0 ; hj0 j4j0 i ¼ 0 we have:




ðV  E1 Þj0 ¼ ðH^  H00 Þj0 ^ 4j0 ðH^0  E0 Þj1 ¼ ½H;

(235)

(236)

^ 4 is the commutator of H^ and 4, giving the first three energy corrections of RS perturbation where ½H; theory in the form: 8 > > E1 ¼ hj0 jVjj0 i ¼ hj0 jH^  H^0 jj0 i ¼ H00  E0 > > > > > > ^ < E2 ¼ hj0 jV  E1 jj1 i ¼ hj0 jH^  H00 j4j0 i ¼ hj0 jHj4j 0i (237) > ^ ^ >  E ¼ H  H  H ¼  E E jV jj i h4j j j4j i h4j j j4j i hj 3 1 00 0 0 > 1 1 0 0 0 0 > > >  2  > > ^ ^ ^ : ¼ h4j0 jH  H00 j4j0 i  h4j0 j½H; 4jj0 i ¼ 4 j0 jH  H00 jj0 In this way, all RS energy corrections up to third order are reformulated in terms of matrix elements ^ involving j0 ; 4 and the whole unpartitioned Hamiltonian H. When j0 has one of the symmetries of the physical states described by the totally symmetric ^ the choice of a totally symmetric 4 will suffice to avoid any symmetry problem in the Hamiltonian H, perturbation treatment. Variational approximations to the second-order energy including exchange can then be obtained using the Hylleraas functional in the form: ^ 4j0 i þ h4j0 jH^  H00 jj0 i þ hj0 jH^  H00 j4j0 i E~2 ½4j0  ¼ h4j0 j½H;

(238)

42

CHAPTER 1 Mathematical foundations and approximation methods

Putting equal to zero the first functional derivative of E~2 with respect to h4j0 j (a bra), we obtain the first-order RS differential equation in the form: dE~2 ^ 4j0 þ ðH^  H00 Þj0 ¼ 0 ¼ ½H; dh4j0 j

(239)

which is the Euler–Lagrange equation in the search for the stationarity value of the functional (238). By noting that the first functional derivative (239) (a ket) can be written as: ^ 0 Hj dE~2 ^ þ ðH^  H00 Þj0 ¼ Hj4j 0 i  j4j0 i dh4j0 j j0

(240)

taking the further functional derivative of Eqn (240) (the second functional derivative) with respect to the ket j4j0 i we have: ^ 0 Hj d2 E~2 ðH^0 þ VÞj0 ¼ H^0 þ V  ¼ ðH^0  E0 Þ  0 ¼ H^  dj4j0 idh4j0 j j0 j0

(241)

which ensures the absolute minimum for the variational approximation to the second-order energy of the ground state since ðH^0  E0 Þ is a positive definite operator. Hence: E~2 ½4j0 stationary ¼ hj0 jH^  H00 j4j0 i ¼ minimum

(242)

The absolute minimum for E~2 gives therefore the best choice for the function 4 in the usual sense of variation theory. A convenient matrix formulation can be given to the theory when 4 is expanded into finite basis sets of appropriate functions (Siciliano, 1993). Either one-centre functions of the Lo¨wdin– Shull type or more elaborate two-centre functions in spheroidal coordinates of the Guillemin–Zener and of the James–Coolidge–Ko1os–Wolniewicz27 type were used in the applications to simple molecular systems (Magnasco et al., 1992b,c; 1993; 1994).

1.3.4.2 EN perturbation theory The EN perturbation theory is essentially a variant of RS perturbation theory in the study of configurational interaction corresponding to a new definition of the zeroth-order Hamiltonian yielding faster convergence than the usual Møller–Plesset partition of the exact Hamiltonian when applied to the correlation problem (Epstein, 1926; Nesbet, 1955a,b; Claverie et al., 1967). The method was first proposed by Epstein (1926) in a quantum mechanical study of the Stark effect in hydrogen according to the new Schroedinger’s theory, and later re-considered by Nesbet (1955a) in his studies of configurational interaction in orbital theories and applied (Nesbet, 1955b) to approximate SCF calculations for the excited electronic states of cis- and trans-1,3-butadiene. Nesbet applied the perturbation formula for the second-order energy to calculate the correlation effects in butadiene using denominators where the one-electron orbital energy differences were replaced by differences between expectation values of the whole Hamiltonian over excited manyelectron configurations. We shall present here a short derivation of the second-order theory in terms of the matrix partitioning technique (see Section 6.2 of Chapter 6) used by Lo¨wdin (1951, 1963) and McWeeny (1960). 27

In this case, the basis set for H2 included positive powers of the interelectron coordinate.

1.3 Perturbative methods for stationary states

43

We assume that the many-electron wavefunction j is built from some set of orthonormal N-electron functions, which we write in matrix form as the row vector F: F ¼ ðF0 F1 /Fk /Þ

(243)

Borrowing the terminology of orbital theories, we shall refer to F0 as the one configuration approximation while the refinement of admitting Fk ðk > 0Þ will be referred to as configuration interaction (CI). We shall refer to function j as to the multi-configuration wavefunction, which we write as: j ¼ FC

(244)

Best energy is obtained by the Ritz variational determination of the coefficients in Eqn (244) giving the matrix eigenvalue equation: HC ¼ EC

(245)

where C is the column vector of variational coefficients, the elements of matrix H being given in terms of the one- and two-electron transition density matrices rn ðklÞ (Chapter 14) as: ^ 1 Þr1 ðkljx1 ; x0 Þ þ 1 tr 1 r2 ðkljx1 ; x2 ; x0 ; x0 Þ ^ k i ¼ tr hðx Hlk ¼ hFl jHjF 1 1 2 2 r12

(246)

^ 1 Þis the one-electron bare nuclei Hamiltonian and 1/r12 the electron repulsion operator (a where hðx scalar quantity). Best energies and coefficients are then found by solving the standard homogeneous system: ðH  E1ÞC ¼ 0

(247)

jH  E1j ¼ 0

(248)

leading to the secular equation: It is convenient, however, to start from the one-configuration approximation for which C0 ¼ 1, Ck ¼ 0(k > 0) and obtain the accurate solution of Eqn (245) in the form of a series expansion. To this aim, the partitioning technique first proposed by Lo¨wdin (1951) is preferred to use of standard RS perturbation theory which frequently diverges. We first re-formulate Eqn (245) in terms of the system density matrix r. Multiplying both members of Eqn (245) by Cy gives the matrix equation: E¼

Cy HC ¼ trHr Cy C

(249)

where: r ¼ CCy =Cy C is the system density matrix. We now write Eqn (245) in the partitioned matrix form:





AA A A HAB H ¼ E B B HBA HBB

(250)

(251)

44

CHAPTER 1 Mathematical foundations and approximation methods

where A labels the starting one-configuration approximation and B the remainder multi-configuration functions whose effect is admitted as a perturbation. Assuming a non-degenerate one-configuration approximation, we write: HAA ¼ H00

(252)

that we call the unperturbed energy. Solving for E we obtain from Eqn (248):  1 E ¼ HAA  HAB HBB  E1 HBA ¼ f ðEÞ

(253)

Expanding the inverse matrix in Eqn (253) gives the BW series of Section 1.3.3. Taking C0 ¼ 1, the coefficients Ck(k > 0) are contained in the column matrix B which is given by:  1 B ¼  HBB  E1 HBA ¼ BðEÞ

(254)

Since E occurs in both sides of Eqns (253) and (254), it is natural to solve by iteration, what is best done by a second-order process (Lo¨wdin, 1963) giving the k-th energy iterate as: EðkÞ ¼ trHrðkÞ where the k-th iterate of the density matrix is given by: h i1 1 ðkÞ ðkÞy ðkÞ r ¼ 1þB B BðkÞ with B(k) defined by:

(255)

BðkÞy ðkÞ ðkÞy B B



   1 HBA BðkÞ ¼ B Eðk1Þ ¼ Eðk1Þ 1  HBB

(256)

(257)

The sequence: Eð0Þ ; Eð1Þ ; Eð2Þ ; / nearly always converges rapidly and its limit is an upper bound to an exact eigenvalue. Usually, taking: Eð0Þ ¼ H00

(258)

r ¼ rð1Þ

(259)

E ¼ Eð1Þ

(260)

the approximation:

is adequate and the first energy iterate:

gives an upper bound to the energy of the lowest state. Introducing a perturbation parameter x and posing Hkl ¼ xHkl ðx / 1Þ for ksl, it can be shown (McWeeny, 1960) that the first iterate in the density matrix corresponds to the inclusion of the exact

1.3 Perturbative methods for stationary states

45

first- and second-order terms of the usual theory plus part of all higher-order terms. The energy to Oðx2 Þ is then given by: X H0k Hk0   E ¼ H00  x2 þ O x3 ðx / 1Þ (261) H  H 00 k > 0 kk When basis (243) is non-orthogonal with non-orthogonality matrix S, Eqn (261) must be replaced by: E ¼ H00  x2

X jHk0  H00 Sk0 j2

Hkk  H00   ¼ E 0 þ E1 þ E2 þ O x 3

  þ O x3 ðx / 1Þ ¼ Eð0Þ þ Eð2Þ

k>0

(262)

This second-order EN energy expression is particularly useful in the perturbative calculation of interatomic and intermolecular interactions beyond the first order since (1) it avoids the need of choosing as unperturbed j0 an exact eigenfunction of the unperturbed Hamiltonian H^0 referring to isolated not interacting atoms or molecules, (2) its one-configuration energy (E(0) ¼ H00) gives, besides the energy of E0 the separate systems, the first-order interaction energy including all exchange effects,28 and (3) its multi-configurational part allows to evaluate through E2 the most important small second-order corrections including exchange. This theory was used by the author and coworkers (Magnasco and Figari, 1986) in reasonably accurate calculations29 of the inter-atomic interactions in the Van der Waals region for the simple model diatomics H2þ ð2 SÞ; H2 ð3 SÞ and He2 ð3 SÞ, in early intra-molecular calculations of rotational barriers of 19 single rotor molecules in the improved bond-orbital approximation (Musso and Magnasco, 1984) and in an investigation of the origin of the hydrogen bond in the water dimer (Magnasco et al., 1985).

1.3.5 Perturbation theories including exchange (SAPTs) Since the interaction energy between two atomic or molecular systems is a small quantity arising from the difference between two large quantities,30 perturbation methods are expected to be particularly appropriate for performing such calculations. Even if this was recognized long ago, just after the pioneering calculation by Heitler and London (1927) on the hydrogen molecule, by Eisenschitz and London (1930), who first considered the effect of electron exchange in RS perturbation theory, and in the later 1930s by Margenau (1939) in his classic paper on Van der Waals interactions, exchange perturbation theories (also called SAPTs) were extensively developed only since the last part of the 1960s following the work by Van der Avoird (1967), Murrell and Shaw (1967), Musher and Amos (1967), and Hirschfelder (1967a,b). The introduction of exchange into RS perturbation theory is however affected by the following paradox. If the whole Hamiltonian H^ is naturally decomposed into: H^ ¼ H^0 þ V 28

(263)

By far the largest contribution to the interaction energy. Pseudostates accounting for the first three non-expanded multipoles were included in the calculations. 30 The molecular energy of the interacting system as a whole and the energies pertaining to the separate atoms or molecules at infinity. 29

46

CHAPTER 1 Mathematical foundations and approximation methods

the unperturbed Hamiltonian H^0 describing separate not interacting atoms or molecules and the firstorder perturbation V which is the interatomic or intermolecular potential, each one of these components has a symmetry lower than the symmetry pertaining to the whole interacting system. As a consequence of this, the antisymmetrizer31 A^ commutes with the total Hamiltonian H^ but not with either of H^0 or V: ^ ¼0 ^ A ½H; (264) In fact, introducing the perturbation parameter l, we have: ^ ¼ ½H^0 ; A ^ þ l½V; A ^ ¼0 ½H^0 þ lV; A

(265)

^ ¼ l1 ½A; ^ V l0 ½H^0 ; A

(266)

meaning: ^ on the left should equal the first-order commutator l½A; ^ V so that the zeroth-order commutator ½H^0 ; A in the right-hand side of Eqn (266)! This means that in an exchange perturbation theory based on such premises orders are not uniquely defined and we cannot generate the perturbed solutions by continuously varying l from zero to unity (Hirschfelder, 1967a). Nonetheless, different working exchange perturbation theories were developed in the past, the choice among them being left to a balance between advantages and disadvantages of each theory. In this section, we present shortly three of the most popular SAPTs obtained from the form assumed by the arbitrary regular function F, defined below, which is different for each theory, limiting ourselves to the first few orders of each theory, more details being left to the PhD thesis of Andrea Siciliano (1993). Let: ðH^  EÞJ ¼ 0

(267)

be the Schroedinger equation to be solved. We assume that the exact wavefunction J can be obtained by acting with the projection operator A^ on a regular unsymmetrized primitive function j : ^ J ¼ Aj

(268)

where the symmetrizer (antisymmetrizer for electrons) projector has the properties: ^ A^2 ¼ A;

^ A^y ¼ A;

^ H ^ ¼0 ½A;

(269)

^ the following is Substituting in Eqn (267) and taking into account the commutation properties of A, obtained: ^ ¼ Að ^ H^  EÞj ¼ 0 ðH^  EÞAj

(270)

^ ðH^  EÞj ¼ ð1  AÞF

(271)

so that:

where F is an arbitrary regular function satisfying the boundary conditions of the problem. 31

A projection operator antisymmetrizing the individually antisymmetric unperturbed wavefunctions of A and B.

1.3 Perturbative methods for stationary states

47

Assuming the Hamiltonian decomposition (263), j0 is the unperturbed function satisfying the zeroth-order equation: (272) ðH^0  E0 Þj0 ¼ 0 with: E0 ¼ E0A þ E0B

(273)

the sum of the unperturbed energies pertaining to separate not interacting molecules, and V is the intermolecular potential. We start by assuming j in the product form: j ¼ j0 ¼ A0 B0

(274)

where A0 and B0 are the unperturbed wavefunctions of A and B, each individually antisymmetric in the respective electrons. We then expand E, j, F in powers of V: E ¼ E 0 þ E1 þ E2 þ /

(275)

DE ¼ E  E0 ¼ Eint

(276)

j ¼ j0 þ j 1 þ j 2 þ /

(277)

F ¼ F0 þ F1 þ F2 þ /

(278)

F0 ¼ 0

(279)

with: the interaction energy;

where we must have: in order to satisfy the zeroth-order equation. Substituting in Eqn (271) the following is obtained: ^ 1 þ F2 þ /Þ ½ðH^0 E0 Þ þ ðV  E1 Þ  E2  E3  /ðj0 þ j1 þ j2 þ /Þ ¼ ð1  AÞðF and, separating orders in V, we obtain the perturbative equations: 8 ðH^0  E0 Þj0 ¼ 0 > > > > > ^ > ^ > < ðH 0  E0 Þj1 þ ðV  E1 Þj0 ¼ ð1  AÞF1 $$$ > > > n X > > > ^ n ^ Ek jnk ¼ ð1  AÞF > : ðH 0  E0 Þjn þ ðV  E1 Þjn1 

(280)

(281) ðn  2Þ

k¼2

It is now evident that different exchange perturbation theories are obtained depending on the choice of function F. 1. Polarization (P) theory Taking Fn ¼ 0 for each order of the perturbation theory we obtain the so-called polarization approximation (Hirschfelder, 1967a,b), which turns into the ordinary RS perturbative theory

48

CHAPTER 1 Mathematical foundations and approximation methods

when H^1 ¼ V: The polarization approximation converges very slowly with the perturbation order (Jeziorski and Ko1os, 1982), but it is possible to show that it gives the exact interaction energy at infinite order (C-alasinski et al., 1977). 2. Murrell–Shaw–Musher–Amos (MS-MA) theory Even if its original derivation (Murrell and Shaw, 1967; Musher and Amos, 1967) is rather complicated, MS-MA theory is easily obtained by posing in Eqn (271): F ¼ ðH^  EÞj0

(282)

hj0 jji ¼ 1

(283)

hj0 jj0 i ¼ 1

(284)

hj0 jjn i ¼ 0

(285)

with the intermediate normalization:

whose meaning, order by order, is:

the normalization condition, and: the orthogonality condition (n  1). We notice that this orthogonality condition is stronger than that of ordinary RS perturbation theory. By expanding ðH^  EÞ the corrections of the various orders are found to be: 8 > > > F0 ¼ ðH^0  E0 Þj0 ¼ 0 < F1 ¼ ðV  E1 Þj0 > > > : Fn ¼ En j0 ðn  2Þ

(286)

giving the first two MS-MA perturbative equations as: ˇ

(

ðH^0  E0 Þj1 þ AðV  E1 Þj0 ¼ 0 ^ 0 ðH^0  E0 Þj2 þ ðV  E1 Þj1 ¼ E2 Aj

(287)

with the energy corrections:    8  > ^ ^ > E1 j0 jAj0 ¼ j0 jAVjj0 > > > > > >         < ^ 0 ¼ j0 jV  E1 jj1 ¼ j ^  E1 Þjj0 ¼ A^j ~ P jV  E1 jj0 ~ P jAðV E2 j0 jAj 1 1 > > > >          > > > P P  > ^ ^ ^ ~ ~ : E3 j0 jAj0 ¼ j0 jV  E1 jj2 ¼ Aj2 jV  E1 jj0  E2 Aj1 j0

(288)

1.3 Perturbative methods for stationary states

49

~ P satisfy the differential equation: where the functions j n ~ P þ ðV  E1 Þj ~P  ðH^0  E0 Þj n n1

n X

~P ¼ 0 Ek j nk

(289)

k¼2

a polarization-like equation with EnP ð¼ Encb Þ replaced by En ð¼ EnMSeMA Þ. We further notice that: ~ P ¼ jP j 1 1 the true first-order polarization function, which is solution of the differential equation:   ðH^0  E0 ÞjP1 þ V  E1cb j0 ¼ 0

(290)

(291)

It was shown by Battezzati and Magnasco (1977) that, at least in principle, the second-order MS–MA perturbative energy including exchange could be evaluated from the extrema (absolute minimum and absolute maximum) of a single bivariational functional. • Advantages of MS–MA perturbation theory are: – All energy corrections can be evaluated without solving equations involving the ^ The second-order energy including exchange depends only on jP and antisymmetrizer A: 1 can be calculated using the linear pseudostate technique of Section 1.3.2.4 of this chapter. The third-order energy requires knowledge of jP2 or, alternatively, knowledge of jP1 and j1 (Adams et al., 1984). – The exchange-overlap energy of any order vanishes exponentially for large R, so ensuring that the theory gives the correct asymptotic expansion in the limit of large R. – In the Van der Waals region MS–MA theory gives reasonable results already in second order, this being important in the calculations for complex systems. • Disadvantages of MS–MA perturbation theory are: – The MS–MA theory is not convergent from a strictly mathematical standpoint, even at large values of R (Jeziorski and Ko1os, 1982). A small fraction of the interaction energy cannot be obtained in any treatment of finite order. It can be said that the theory gives about 98% of the asymptotic value of E2exch-ov , while at the Van der Waals minimum this fraction is less than 0.01% of the interaction energy. – The MS–MA function truncated to a finite order has not the correct symmetry of the true wavefunction J (Weak Symmetry Forcing ¼ WSF). This is of importance in the calculation ^ 0 þ j1 of observables different from the energy, but not for calculating the energy with Aj in second order. – It is not easy to evaluate matrix elements involving exchange for many-electron wavefunctions, a difficulty common to all exchange perturbation theories based on a nonsymmetric H^0 . 3. Jeziorski and Ko1os (JK) theory This theory is also known as theory of the Intermediate Symmetry Forcing ¼ ISF. In this scheme, the first two energy perturbative orders coincide with MS–MA, whereas higher orders (n  2) are different. The JK scheme can be considered as an MS–MA theory modified so as to improve convergence.

50

CHAPTER 1 Mathematical foundations and approximation methods

The following is obtained by posing: ^ 0 þ j1 þ j2 þ /hj0 jjn i ¼ 0 j ¼ N0 Aj

ðn  1Þ

(292)

where: ^ 0 i1 N0 ¼ hj0 jAj

(293)

The first two perturbative equations are then: ( ^  E1 Þj0 ðH^0  E0 Þj1 ¼ N0 AðV ^ 0 ðH^0  E0 Þj2 þ ðV  E1 Þj1 ¼ N0 E2 Aj

(294)

The JK Eqn (294) differ from the corresponding MS–MA equations by the presence of the factor N0 in the last two terms. The first two JK energy corrections are: 8 ^ < E1 ¼ N0 hj0 jAVjj 0i  E D  (295) : E2 ¼ hj0 jV  E1 jj1 i ¼ N0 j0 ðH^0  E0 Þ1 AðV ^  E1 Þj0 The JK perturbative scheme converges faster than MS–MA, but many of the difficulties remain. 4. Eisenschitz–London–Hirschfelder–Van der Avoird (EL–HAV) theory This theory is obtained by posing in Eqn (271): Fn ¼ ðV  E1 Þjn1 

n X

Ek jnk

ðn  1Þ

(296)

k¼2

The first two EL–HAV perturbative equations are then: ( ^  E1 Þj0 ¼ 0 ðH^0  E0 Þj1 þ AðV ^ 0 ^  E1 Þj1 ¼ E2 Aj ðH^0  E0 Þj2 þ AðV

(297)

with the energy corrections: 8 ^ 0 i ¼ hj0 jAVjj ^ > E hj jAj 0i > < 1 0 ^ 0 i ¼ hAj ^ 1 jV  E1 jj0 i E2 hj0 jAj > > : ^ 0 i ¼ hAj ^ 1 jV  E1 jj1 i  E2 ½hj1 jAj ^ 0 i þ hj0 jAj ^ 1 i E3 hj0 jAj

(298)

While the first-order EL–HAV differential equation is the same as MS–MA, the second-order equation determining j2 is different as are the higher orders in the energy corrections. In EL–HAV theory, all energy corrections to the wavefunction are antisymmetrized (Strong Symmetry Forcing ¼ SSF), so that the wavefunction truncated to any finite order has the correct symmetry

1.3 Perturbative methods for stationary states

51

of the true J. Unfortunately, this excess of symmetry determines a wrong asymptotic behaviour of the second-order EL–HAV energy at large interatomic distances R. In fact, for ground state Hþ 2 , it was found (Chipman and Hirschfelder, 1973; Jeziorski, 1974): lim

R / large

E2ELeHAV ¼

41 P 3 P E z E 54 2 4 2

instead of the correct value E2P ¼ E2cb . So, at large distances, EL–HAV second-order theory gives about 75% of the correct second-order Coulombic (polarization) energy. Even if this behaviour may be corrected by including higher orders of perturbation theory, it remains an obstacle for calculations on many-electron systems. The general formalisms of the different SAPTs were compared theoretically by Chipman et al. (1973) and applied to analytical calculations on the Hþ 2 molecular ion by Chipman and Hirschfelder (1973). 5. Application to the many-electron systems In the perturbation theory of molecular interactions, the majority of the significant physical effects can be described in second order, provided the Coulombic energies (electrostatic, induction, dispersion), already accounting for the intrinsic antisymmetry of the wavefunctions of the interacting partners, are supplemented by terms arising from the antisymmetry of the total wavefunction and intermolecular electron exchange. First-order effects, which are dominant in short range, can be described by antisymmetrizing the product j0 of the individually antisymmetrized unperturbed wavefunctions of the separate molecules, say A0 and B0. The small second-order effects can then be calculated by seeking variational approximations to induction and dispersion energies in terms of linear pseudostates, as explained in Section 1.3.2.4 of this chapter and later applied to the calculation of long-range dispersion coefficients for atoms and molecules in Chapter 17 of this book. Second-order exchange effects could be included in a second time using, for instance, the MS–MA perturbation theory. As already seen, even if MS–MA is just one of the possible exchange perturbation theories, it owes its popularity to the fact that the energy corrections can be calculated without the need of solving perturbation ^ the operator that implies interchange of electrons equations involving the antisymmetrizer A, between different molecules. A more detailed account of the MS–MA perturbation theory for large molecular systems can be found elsewhere (Magnasco and McWeeny, 1991; Magnasco, 2007), where attention was focused on the physical meaning of the different components of the interaction energy. We want to outline here a few points that are particularly important in the application to large molecular systems. If A^ is the partial (idempotent) antisymmetrizer: ^ A^ ¼ Q1 ð1 þ PÞ;

Q1 ¼

NA !NB ! N!

(299)

and P^ is the operator interchanging electrons between different molecules: P^ ¼ 

ðAÞ X ðBÞ X i

j

P^ij þ

ðAÞ X ðBÞ X i > < H 0 ¼ 2 V  r > 2 2 >  > : H^1 ¼ B x2 þ y2 ¼ B r 2 sin2 q 8 8

(316)

where B is the strength of the uniform magnetic field directed along the z axis, the functions F are chosen in the form (Ferna´ndez and Morales, 1992): Fijnm ðr; q; 4Þ ¼ ðsin qÞi ðcos qÞj r n expðbrÞ expðim4Þ

(317)

where b is a positive scale factor to be determined, and i; j; n ¼ 0; 1; 2; / positive integers including zero. The q-parity of the functions F is: ( Fijnm ðr; q; 4Þ ¼ ð1Þi Fijnm ðr; q; 4Þ (318) Fijnm ðr; qþp; 4Þ ¼ ð1Þiþj Fijnm ðr; q; 4Þ The resulting moment recurrence relation can be simplified by choosing j to be 0 or 1 and b ¼ 1/N, where N ¼ 1; 2; 3; $$$ is the principal quantum number, so that j(j  1) ¼ 0 and b(n þ 1)  1 ¼ 0 when

1.4 The Wentzel–Kramers–Brillouin method

55

n ¼ N  1. Then, Ferna´ndez and Morales obtain for the energy shift the recurrence relation in terms of the moments (307): 1 DE ¼ E  E0 ¼ E þ 2 (319) 2N
1 nþ1N 1 DE ¼ Ai;n1 þ ½ði þ jÞði þ j þ 1Þ  nðn þ 1ÞAi;n2 Ai;n N 2 (320)   1 2 B2 2 Aiþ2;nþ2 þ m  i Ai2;n2 þ 8 2 The method allows the treatment of classes of states with common symmetry properties. The authors derive analytic expressions for higher-order energy shifts in terms of the principal quantum number N for different perturbation corrections to the energy of the unperturbed states, discovering some disagreement with the results of previous calculations. The moment method greatly facilitates the application of the perturbation theory to simple nonseparable quantum-mechanical systems, allowing the treatment of both degenerate and nondegenerate states and is suitable for numerical and analytic computations using computer algebra. By evaluating directly the expansion coefficients, the method efficiently bypasses the calculation of matrix elements and integrals required by other methods. Another important advantage of the moment method is that, when the perturbation series is divergent, as in the case of the Zeeman effect for the hydrogen atom in a uniform magnetic field, it is possible to introduce an adjustable parameter into the recurrence relation to obtain re-normalized series having improved convergence properties (Austin, 1984).

1.4 THE WENTZEL–KRAMERS–BRILLOUIN METHOD The Wentzel–Kramers–Brillouin (WKB) method is a way for finding an approximate solution to onedimensional Schroedinger’s equations when the wavefunction is expressed as an exponential function which is then expanded into powers of ðiZÞ, where Z is the reduced Planck constant and i the imaginary unit (Wentzel, 1926; Kramers, 1926; Brillouin, 1926). Even if it is rather complicated in its applications, requiring use of the techniques of integration in the complex plane of Chapter 5, it is of some interest for us in so far as it shows the connection existing between classical and quantum mechanics. It is also of importance in evaluating tunnelling effects across potential barriers. We give below a short historical account of the theory, mostly following the presentation of Wentzel’s paper (1926), which is the most appropriate for our purposes. More can be found in Section 28 of Schiff’s book (1955) on quantum mechanics. It can be easily verified that a solution jðxÞ of the one-dimensional Schroedinger equation: 1 d2 j 2m ¼  2 ðE  VÞ j dx2 Z

(321)

  Z i ydx Z

(322)

can be written in the form: jðxÞf exp

56

CHAPTER 1 Mathematical foundations and approximation methods

where:



1 dj dln j yðxÞ ¼ iZ ¼ iZ j dx dx

(323)

We remark that in Eqn (321):34 EV ¼T ¼

p2 2m

is the kinetic energy of the particle. Then:    



dy dj dj d2 j 1 1 dj 2 1 d2 j þ ¼ iZ  j2 þ j1 2 ¼ iZ 2  iZ dx dx dx dx j dx j dx2 Z  2   y 2m i ¼ iZ 2  2 ðE  VÞ ¼  y2  2mðE  VÞ Z Z Z

(324)

(325)

Multiplying both members of Eqn (325) by ðiZÞ we get the basic WKB equation: iZ where: p¼

dy ¼ p2  y 2 dx

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi 2mðE  VÞ ¼  2mT

(326)

(327)

is the classical momentum of the particle. From a mathematical standpoint, the basic WKB Eqn (326) is a Riccati first-order differential equation35 (Ince, 1963). We now expand y(x) in powers of ðiZÞ: yðxÞ ¼ y0 þ ðiZÞy1 þ ðiZÞ2 y2 þ /

(328)

dy dy0 dy1 dy2 þ ðiZÞ þ ðiZÞ2 þ/ ¼ dx dx dx dx

(329)

where we must remark that Eqn (328) is a semiconvergent or asymptotic expansion36 (Erde`lyi, 1956), accurate far from the ‘turning points’ where E ¼ V. Then: h i2 dy0 dy1 dy2 þ ðiZÞ2 þ ðiZÞ3 þ / ¼ p2  y0 þ ðiZÞy1 þ ðiZÞ2 y2 þ ðiZÞ3 y3 þ / ðiZÞ dx dx dx ¼ p2  y20  ðiZÞ2 y21  2ðiZÞy0 y1  2ðiZÞ2 y0 y2  2ðiZÞ3 y1 y2  2ðiZÞ3 y0 y3 þ / (330) 34

We omit for brevity the caret symbol on the impulse (linear momentum) operator p. The standard way of solution of a Riccati’s type differential equation requires knowledge of a particular solution to reduce it to a form amenable to integration by quadrature (Ince, 1963). Burkill (1962), instead, gives a solution in terms of infinite series by a method of successive approximations. 36 An asymptotic (or semiconvergent) expansion is a series which, though divergent, is such that the sum of a convenient number of terms gives a good approximation to the function which it represents (compare the Rn long-range expansion of the intermolecular potential in Chapter 17 of this book). 35

1.4 The Wentzel–Kramers–Brillouin method

57

and, collecting terms corresponding to same power of ðiZÞn : ðiZÞ0 p2  y20 ¼ 0 dy0 þ 2y0 y1 ¼ 0 dx

(332)

dy1 þ y21 þ 2y0 y2 ¼ 0 dx

(333)

dy2 þ 2y1 y2 þ 2y0 y3 ¼ 0 dx

(334)

ðiZÞ1 ðiZÞ2 ðiZÞ3

(331)

Therefore we obtain the equations determining the various coefficients of the expansion (328): pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi y0 ¼ p ¼  2mðE  VÞ ¼  2mT (335) 1 1 dy0 1 1 dp 1 dln p ¼ ¼ 2 y0 dx 2 p dx 2 dx

1 dy1 y21 þ y2 ¼  dx 2y0

y1 ¼ 

the yns being given in general by the recursion formula (Wentzel, 1926; Dunham, 1932a): n X yk ynk y0n1 ¼ 

(336)

(337)

(338)

k¼0

In this way, we obtain a hierarchy of equations giving a connection between classical mechanics and the different quantum theories of matter. Equation (335) is the classical result expressing the momentum of the particle, Eqn (336) gives the first quantum correction, Eqn (337) the second correction, and so on. In terms of the potential energy V (characterizing the system) and its first and second derivatives, V 0 and V 00 , we can write (see Problem 1.6): V0 (339) y1 ¼ 4ðE  VÞ h i 1 2 y2 ¼ ð2mÞ1=2 ðE  VÞ5=2 5ðV 0 Þ þ 4ðE  VÞV 00 (340) 32 Now: exp Z

Z y1 dx ¼



Z

dx

y0 dx ¼ exp

V0 1 ¼ 4ðE  VÞ 4

Z

Z

 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2mðE  VÞ dx

dðE  VÞ 1 ¼ EV 4

Z

dlnðE  VÞ ¼ lnðE  VÞ1=4

(341)

(342)

58

CHAPTER 1 Mathematical foundations and approximation methods

Z exp



h i y1 dx ¼ exp lnðE  VÞ1=4 ¼ ðE  VÞ1=4

giving as the first two-term approximation:  Z  Z  Z  Z pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi y1 dx ¼ ðE  VÞ1=4 exp 2mðE  VÞdx exp ydx y exp y0 dx $ exp    Z  Z pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i i 1=4 ydx y NðE  VÞ 2mðE  VÞdx exp jðxÞf exp Z Z

(343)

(344)

(345)

with the probability distribution: jjj2 ¼ N 2 ðE  VÞ1=2 z constant 

1 p

(346)

that coincides with the classical result. Quantization will result when we try to extend the wavefunction into the region where E < V (imaginary kinetic energy). The restriction imposed on E demands that: I ydx ¼ nh n ¼ 0; 1; 2; / (347) where the cyclic integral (the action integral) can be calculated only for conditionally periodic systems, and n is a positive integer. If we insert in Eqn (347) the first term of the series (328) for y, y ¼ y0 ¼ p, we obtain the condition of the old quantum theory: I pdx ¼ nh n ¼ 0; 1; 2; / (348) while the second term introduces the half-quantum numbers characteristic of the new quantum theory: I h (349) iZ y1 dx ¼  2 I I h ðy0  iZy1 Þdx ¼ pdx  ¼ nh (350) 2 so that:



I pdx ¼

nþ

1 h 2

(351)

to the second approximation. In deriving Eqn (349) (see Problem 1.7), the contour integration techniques were made use of in the complex plane of Chapter 5. For the effective evaluation of the action integrals, more must be known about the specific form of the potential energy V(x). Wentzel (1926) applied his method to the study of the radial equation

1.5 Problems 1

59

of the hydrogen atom and of the first- and second-order Stark effect in hydrogen. In the latter case, Wentzel’s results coincide with those obtained about at the same time by Waller (1926) from an expansion in powers of the strength F of the external electric field of the Schroedinger equation in parabolic coordinates. In later years, Duhnam (1932a) gave a somewhat critical treatment of the WKB method and applied his formulae to a study of the energy levels of a rotating vibrator (Dunham, 1926b).

1.5 PROBLEMS 1 1.1. Schmidt orthogonalize the three normalized non-orthogonal functions ðc01 c02 c03 Þ. Hint: Orthogonalize first function c02 to c01 ; next c03 to the resulting orthogonalized ones. 1.2. The Hermitian operators. • One-dimensional problems 1.2.1. Show that:

      d d  4 j ¼  4 j dx dx

so that d/dx is anti-Hermitian. 1.2.2. Show that:        d d  4i j ¼ i 4 j dx dx so that id/dx is Hermitian. 1.2.3. Show that:   2   2   d  d 4 2 j ¼ 4 j dx dx2  so that d2/dx2 is Hermitian. • Three-dimensional problems Let 4ðx; y; zÞ and jðx; y; zÞ be well-behaved functions of the three variables (x, y, z). 1.2.4. Show that:        v v   j ¼ i 4j 4 i vx vx  so that iv/vx is Hermitian. 1.2.5. Show that:   2   2   v  v 4 2 j ¼ 4j 2 vx vx so that v2/vx2 is Hermitian.

60

CHAPTER 1 Mathematical foundations and approximation methods v2 v2 v2 Proceeding in a similar way for the y,z-components, we can show that 72 ¼ 2 þ 2 þ 2 is vx vy vz Z2 Hermitian. Hence, T^ ¼  72 , the kinetic energy operator, is Hermitian. 2m Hint: Make use of the integration by parts. In fact: dðuvÞ ¼ udv þ vdu and by integration:

Z

Z udv ¼ uv 

vdu

where: u,v ¼ finite factors du, dv ¼ differential factors 1.3. Find the commutators of the components Lx, Ly, Lz of the angular momentum operator L with themselves and with L2 (here and in the following we omit for brevity the caret sign on the operators). Answer: 



Lx ; Ly ¼ iLz Ly ; Lz ¼ iLx ½Lz ; Lx  ¼ iLy that can be summarized as: L  L ¼ iL

2  2  2  L ; Lx ¼ L ; Ly ¼ L ; L z ¼ 0 We conclude that the three components of the angular momentum operator cannot be specified simultaneously, but we can exactly specify each individual component and the square of the angular momentum. Hint: Use the expressions of Lx, Ly, Lz in Cartesian coordinates. 1.4. Find the commutators of Lz and L2 with the ladder operators Lþ and L. Answer: ½Lz ; Lþ  ¼ Lþ ½Lz ; L  ¼ L

2   L ; Lþ ¼ L2 ; L ¼ 0 Hint: Use the commutators for the components of the angular momentum operator found in Problem 1.3. 1.5. Using Cartesian coordinates, find the relation connecting the Laplacian 72 to the square of the angular momentum operator L2. Answer: 72 ¼ 72r 

L2 r2

1.5 Problems 1

where 72r is the radial Laplacian: 72r ¼



1 v v2 2 v 2v r ¼ 2þ 2 vr r vr vr r vr

Hint: Start from the expression of L2 in Cartesian coordinates:  L2 ¼ L2x þ L2y þ L2z ¼ 

y





 v v 2 v v 2 v v 2 z þ z x þ x y vz vy vx vz vy vx

and calculate the derivatives taking into account that:



v vf vg ðfgÞ ¼ g þf vz vz vz

etc

where f,g are arbitrary functions of (x,y,z). Furthermore, notice that: r$V ¼ x

v v v v þy þz ¼r vx vy vz vr



v v v2 v r ¼ r2 2 þ r ðr $ VÞ ¼ r vr vr vr vr 2

1.6. Find the expressions of y1 and y2 in the series expansion of the action y(x). Answer: y1 ¼ y2 ¼ 

V0 4ðE  VÞ

h i 1 2 ð2mÞ1=2 ðE  VÞ5=2 5ðV 0 Þ þ 4ðE  VÞV 00 32

where: V0 ¼

dV ; dx

V 00 ¼

d2 V dx2

Hint: Integrate the corresponding differential equations. 1.7. Evaluate the phase integral. Answer: I h iZ y1 dx ¼  2 where h is the Planck constant. Hint: Use contour integration in the complex plane (Chapter 5 of this book).

61

62

CHAPTER 1 Mathematical foundations and approximation methods

1.6 SOLVED PROBLEMS 1.1. Schmidt orthogonalization of three functions. If ðc01 c02 c03 Þ is the set of three normalized but non-orthogonal functions having nonorthogonalities S12, S13, S23, respectively, the normalized Schmidt-orthogonalized set ðc1 c2 c3 Þ is given by: c1 ¼ c01

  c2 ¼ N2 c02  S12 c01   c3 ¼ N3 c03  Ac2  Bc1 where:  1=2 N2 ¼ 1  S212 "

1  S212 N3 ¼    1  S212 1  S213  ðS23  S12 S13 Þ2

#1=2

A ¼ N2 ðS23  S12 S13 Þ; B ¼ S13 The procedure is straightforward. The orthogonality and normalization of the transformed functions were checked explicitly in Problem 1.9 of Magnasco (2007). 1.2. Hermitian operators. 1.2.1.

   ZN ZN ZN  N d djðxÞ      4 j ¼ dx4 ðxÞ 4 ðx Þ dj ðxÞ ¼ 4 ðxÞ jðxÞ N  jðxÞ d 4 ðxÞ ¼ dx dx dv u u v v N

ZN ¼ N

N

d4 ðxÞ dx jðxÞ ¼  dx

N

ZN N

  

d4ðxÞ  d  dx jðxÞ ¼  4j dx dx 

so that d/dx is anti-Hermitian. Notice that, if A(x) and B(x) are real functions: 4ðxÞ ¼ AðxÞ þ iBðxÞ; 4 ðxÞ ¼ AðxÞ  iBðxÞ

d4ðxÞ dA dB d4ðxÞ  dA dB d4 ðxÞ dA dB ¼ þi i 0 ¼ i ¼ dx dx dx dx dx dx dx dx dx so that: d4 ðxÞ ¼ dx



d4ðxÞ  dx

du

1.6 Solved problems

1.2.2.



  ZN ZN  d djðxÞ j ¼ ¼ 4i dx4 ðxÞ i 4 ðx Þ d ðij ðx ÞÞ dx dx N

u

N

ZN

N ¼ 4 ðx Þ ðij ðx ÞÞ N  

v

u

dv

ZN



ðij ðx ÞÞ d 4 ðx Þ ¼  v

N

du

dx N

d4 ðxÞ ðijðxÞÞ dx

 

 d4ðxÞ  d  4j dx i jðxÞ ¼ i dx dx 

ZN ¼ N

so that id/dx is Hermitian. 1.2.3.

    2  ZN ZN

d d2 j x djðxÞ    4 2 j ¼ dx4 ðxÞ ¼ 4 ð Þ d x dx2 dx dx N

N

 ¼ 4 ðxÞ u

¼ N

dv

 ZN djðxÞ

djðxÞ N d4 ðx Þ dx dx N  v

ZN

u

N

d4 ðxÞ djðxÞ ¼ dx dx dx

du

v

ZN N

d4ðxÞ  djðxÞ dx dx dx

Now:  

 ZN 2 ZN d2  d 4ðxÞ d2 4 ðxÞ 4 j ¼ dx jðxÞ ¼ dx jðxÞ  dx2 dx2 dx2



N

ZN N

N



d4 ðxÞ d4 ðxÞ ¼ jðxÞ jðx Þ d dx dx u

u

dv

ZN ¼ N

so that d2/dx2 is Hermitian.

v

N ZN d4 ðxÞ djðxÞ     dx N

N

du

v

  2  d d4ðxÞ  djðxÞ ¼ 4 2 j dx dx dx dx

63

64

CHAPTER 1 Mathematical foundations and approximation methods

Notice that in all such examples, the term uvjN N vanishes for the regularity properties of functions 4ðxÞ; jðxÞ and their first derivatives. 1.2.4.

 Z Z ZN

   v vjðx; y; zÞ j ¼ dx dy dz 4 ðx; y; zÞ i 4i vx vx N

Z ZN ¼

Z Z ZN ¼ y;z

dy dz 4 d x ðij Þy;z u

N

dv

 x¼N  dy dz 4 ðx ; y ; z Þ ðij ðx ; y ; z Þ Þ  

u

N

x ¼ N

surfaceintegral

v

Z Z ZN 

dy dz ðij ðx ; y ; z Þ Þ N

v4

 ðx ; y ; z Þ

vx

v

¼ N

y;z

dx

du

volumeintegral

Z Z ZN

!

 

 v4ðx; y; zÞ  v  4j dx dy dz i jðx; y; zÞ ¼ i vx vx  y;z

since the surface integral vanishes at infinity. So, the operator iv=vx is Hermitian.

1.2.5.

  2  Z Z ZN

2 Z Z ZN v v jðx; y; zÞ  4 2 j ¼ dx dy dz 4 ðx; y; zÞ ¼ dy dz4 d x vx vx2 y;z u N

Z ZN ¼ N

N



 vjðx; y; zÞ x¼N dy dz 4 ðx ; y ; z Þ  vx y;z x¼N u

surface integral

 N

ðx ; y ; z Þ dy dz v j vx v

!

volumeintegral

Z Z ZN ¼ N

y;z

v4

 ðx ; y ; z Þ

vx

du



v4ðx; y; zÞ  vjðx; y; zÞ dx dy dz vx vx y;z y;z

!

dv



Z Z ZN

vj vx

! y;z

dx

y;z

1.6 Solved problems



  Z Z ZN

 2 Z Z ZN v2  v 4ðx; y; zÞ 4j ¼ dx dy dz jðx; y; zÞ dy dz j d x vx2  vx2 y;z N

Z ZN ¼ N

N



ðx; y; zÞ  dy dz jðx ; y ; z Þ v4 vx yz u

v

surface integral

Z Z ZN 

dy dz v 4 N

 ðx ; y ; z Þ

!

vx

y;z

¼ N

1.3.



v4ðx; y; zÞ dx dy dz vx

u

! y;z

dv

x¼N   

x¼N

ðx ; y ; z Þ v j vx

! y;z

dx

du

v

volume integral

Z Z ZN

v 4 vx

65

 y;z

vjðx; y; zÞ vx

y;z

  2  v ¼ 4 2 j vx









 v v v v v v v v Lx ; Ly ¼ Lx Ly  Ly Lx ¼  y  z z x þ z x y z vz vy vx vz vx vz vz vy 2 2 2 2

v v v v v  xy 2  z2 þ zx ¼  y þ yz vzvx vz vyvx vyvz vx

v2 v2 v2 v v2  z2  xy 2 þ x þ zx þ yz vxvz vxvy vz vzvy vy 

 v v v v ¼ iLz ¼x y ¼i i x y vy vx vy vx By cyclic permutation of (x,y,z), it follows:     

 Ly ; Lz ¼ iLx ½Lz ; Lx  ¼ iLy L2 ; Lx ¼ L2 Lx  Lx L2 ¼ L2x þ L2y þ L2z Lx  Lx L2x þ L2y þ L2z ¼ Ly Ly Lx þ Lz Lz Lx  Lx Ly Ly  Lx Lz Lz By adding and subtracting appropriate terms:

2  L ; Lx ¼ Ly Ly Lx  Ly Lx Ly þ Ly Lx Ly þ Lz Lz Lx  Lz Lx Lz þ Lz Lx Lz  Lx Ly Ly

 þ Ly Lx Ly  Ly Lx Ly  Lx Lz Lz þ Lz Lx Lz  Lz Lx Lz ¼ Ly Lx ; Ly þ Lz ½Lz ; Lx  

 Lx ; Ly Ly þ ½Lz ; Lx Lz ¼ iLy Lz þ iLz Ly  iLz Ly þ iLy Lz ¼ 0 so that L2 commutes with the Lx component. By cyclic permutation of the indices, it follows immediately:

2  2  2  L ; Lx ¼ L ; Ly ¼ L ; Lz ¼ 0

66

1.4.

CHAPTER 1 Mathematical foundations and approximation methods

     ½Lz ; Lþ  ¼ Lz Lþ  Lþ Lz ¼ Lz Lx þ iLy  Lx þ iLy Lz ¼ ½Lz ; Lx   i Ly ; Lz ¼ iLy  iðiLx Þ ¼ Lx þ iLy ¼ Lþ

     ½Lz ; L  ¼ Lz L  L Lz ¼ Lz Lx  iLy  Lx  iLy Lz ¼ ½Lz ; Lx  þ i Ly ; Lz ¼ iLy þ iðiLx Þ   ¼ Lx þ iLy ¼  Lx  iLy ¼ L so that:



2 L ; L ¼ 0

since L is a linear combination of Lx and Ly and L2 commutes with all components of L. 1.5.







v v v v v v v v z y z þ z x z x vz vy vz vy vx vz vx vz



 v v v v x y þ x y vy vx vx vx

 L2 ¼ L2x þ L2y þ L2z ¼ 

y





v v v v v2 v v2 v2 v v2  zy  z þ z2 2 y z y z ¼ y2 2  y  yz vz vzvy vyvz vy vz vy vz vy vy vz By cyclic permutation of (x,y,z), we immediately obtain:



v v v v v2 v v2 v2 v v2 z x  xz  x þ x2 2 z x ¼ z2 2  z  zx vx vxvz vzvx vz vx vz vx vz vz vx



v v v v v2 v v2 v2 v v2  yx  y þ y2 2 x y x y ¼ x2 2  x  xy vy vyvx vxvy vx vy vx vy vx vx vy By adding the three components altogether: 2

2

2

v v2 v2 v2 2 v 2 v þ y þ z þ/ þ þ þ L2 ¼ x2 vy2 vz2 vz2 vx2 vx2 vy2 v2 v2 v2 ¼ x2 V2 þ y2 V2 þ z2 V2  x2 2  y2 2  z2 2 þ / vx vy vz



2 v v v v v2 v2 v2 2 2 2v þ xz þ yx  x 2þx þ xy ¼r V  x þy þz vx vxvy vxvz vyvx vx vy vz vx

 2 2 2 2 2 v v v v v v v þ zx þ zy þ z2 2 þ z þ yz þ y2 2 þ y vy vyvz vzvx vzvy vz vy vz

2 v v v ¼ r 2 72  ðr $ VÞ  ðr $ VÞðr $ VÞ ¼ r2 72  r  r 2 2 þ r vr vr vr 2

v 2 v þ ¼ r 2 72  r 2 72r ¼ r 2 72  r 2 vr 2 r vr

1.6 Solved problems

Hence: 72 ¼ 7 r 

L2 r2

1.6. Coefficients y1 and y2 in the WKB series expansion of y(x). The equation for the momentum p gives (apart from the signs in front) ln p ¼

1 lnð2mE  2mVÞ 2

d ln p 1 2mV 0 V0 ¼ ¼ 2ðE  VÞ dx 2 2mE  2mV so that we obtain for y1: 1 d ln p V0 ¼ y1 ¼  4ðE  VÞ 2 dx which is the required Eqn (339). Then: ðy1 Þ2 ¼

1 0 2 ðV Þ ðE  VÞ2 16

h i dy1 1 00 1 1 2 ¼ V ðE  VÞ1  V 0 ðE  VÞ2 ðV 0 Þ ¼ ðE  VÞ2 ðV 0 Þ þ ðE  VÞV 00 dx 4 4 4 and from Eqn (337):   1 1 1 1 2 2 1 00 0 2 0 2 ðE  VÞ ðV Þ þ ðE  VÞ ðV Þ þ ðE  VÞ V y2 ¼  2p 16 4 4   h i 1 5 0 2 1 1 2 ðV Þ þ ðE  VÞV 00 ¼  ðE  VÞ2 5ðV 0 Þ þ 4ðE  VÞV 00 ¼  ðE  VÞ2 2p 16 4 32p which, upon substituting: p1 ¼ ½2mðE  VÞ1=2 ¼ ð2mÞ1=2 ðE  VÞ1=2 gives the required Eqn (340). 1.7. The cyclic integral. As far as integral (349) is concerned, we notice that from Eqn (339): I I I V0 1 dðE  VÞ y1 dx ¼ dx ¼ 4ðE  VÞ 4 EV Introduce now the complex plane of Figure 1.1, and pose the complex variable: E  V ¼ z ¼ r expði4Þ Then: dz ¼ ir expði4Þd4

67

68

CHAPTER 1 Mathematical foundations and approximation methods

FIGURE 1.1 Integration path for the complex variable z

giving:

I

so that:

Hence, we finally obtain:

dz ¼ z I

I

ir expði4Þd4 ¼i r expði4Þ

1 y1 dx ¼  $ 2 4

I

I d4 ¼ 2pi

dz ¼ ip z

I h iZ y1 dx ¼ iZðipÞ ¼  2

which is the required result (349).

CHAPTER

2

Coordinate systems

CHAPTER OUTLINE 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9

Introduction ................................................................................................................................... 69 Systems of Orthogonal Coordinates.................................................................................................. 71 Generalized Coordinates ................................................................................................................. 73 Cartesian Coordinates (x,y,z) .......................................................................................................... 74 Spherical Coordinates (r,q,4) ......................................................................................................... 74 Spheroidal Coordinates (m,n,4)....................................................................................................... 75 Parabolic Coordinates (x,h,4)......................................................................................................... 76 Problems 2 .................................................................................................................................... 78 Solved Problems ............................................................................................................................ 80

2.1 INTRODUCTION An excellent presentation of the subject is found in Chapter 5 of the classic book by Margenau and Murphy (1957). Here, we shall limit ourselves to introduce the operative definitions of the main coordinate systems most useful in quantum chemistry calculations. The choice of a convenient coordinate system depends on the symmetry of the potential entering the corresponding Schroedinger equation. For instance, the solution of the Schroedinger eigenvalue equation for the atomic one-electron system describing a single electron in the field of Z nuclear charges: ^ HjðrÞ ¼ EjðrÞ

(1)

where H^ is the Hamiltonian operator in atomic units: 1 Z H^ ¼  r2  2 r

(2)

is greatly simplified by the use of the spherical coordinates ðr; q; 4Þ, which reflect the symmetry of the central Coulomb potential. In these coordinates: r2 ¼ r2r 

2 L^ r2

Elementary Molecular Quantum Mechanics 2E. http://dx.doi.org/10.1016/B978-0-444-62647-9.00002-6
2013, 2007 Elsevier B.V. All rights reserved

(3)

69

70

CHAPTER 2 Coordinate systems

where: r2r is the radial Laplacian, and:


 1 v 2v ¼ 2 r r vr vr

(4)


 1 v v 1 v2 (5) sin q þ 2 sin q vq vq sin q v42 ^ in atomic units. All these quantities will be the square of the angular momentum vector operator L derived from the general formulae in Section 2.5. Using spherical coordinates, Eqn (1) separates into two different eigenvalue equations, one for the radial coordinate r and the other for the angular variables ðq; 4Þ involving the square of the angular 2 momentum L^ , as it will be shown in Section 3.3.3 of Chapter 3. As a second example, consider the Schroedinger Eqn (6) for the molecular one-electron system describing the motion of a single electron in the field of two nuclear charges ZA and ZB a distance R apart: 2 ^ L ^¼ L^ ¼ L$



^ Hjðr; RÞ ¼ Ejðr; RÞ where H^ is the electronic Hamiltonian operator: 1 ZA ZB H^ ¼  r2   rA rB 2

(6)

(7)

The potential of this system has now an ellipsoidal symmetry with the nuclei at the foci of the ellipse, so that Eqn (6) is separable in confocal spheroidal (elliptic) coordinates ðm; n; 4Þ. It will be shown later in Section 2.6 that in spheroidal coordinates with a ¼ R/2:   1 m 2  n2 v2 2 2 2 (8) rm þ rn þ 2 r ¼ 2 2 ðm  1Þð1  n2 Þ v42 a ðm  n2 Þ    v v  2 r2m ¼ m 1 (9) vm vm    v  2 2 v rv ¼ 1n (10) vn vn 1 1 ; ¼ rA aðm þ nÞ

1 1 ¼ rB aðm  nÞ

(11)

When r2 is expressed in confocal spheroidal coordinates, the eigenvalue Eqn (6) (a second-order partial differential equation in three variables, with R taken as a parameter) separates into three different one-dimensional eigenvalue equations, one for each variable, as it will be shown in Section 3.3.4 of Chapter 3 for the hydrogen molecular ion H2 þ (ZA ¼ ZB ¼ 1). As a last example, the Schroedinger equation for the atomic hydrogen in a uniform electric field of strength F directed along the z axis (Alexander, 1969): ^ Hjðr; FÞ ¼ Ejðr; FÞ

(12)

2.2 Systems of orthogonal coordinates

71

with: 1 1 H^ ¼  r2   Fz 2 r

(13)

where F, taken as a parameter, is separable in parabolic coordinates ðx; h; 4Þ giving the results of Section 3.3.5 of Chapter 3. In these coordinates, it will be seen later in Section 2.7 that: 


  4 v v v v 1 1 1 v2 x þ h þ þ (14) r2 ¼ x þ h vx vx vh vh 4 x h v42 and: 1 z ¼ ðx  hÞ 2

(15)

2.2 SYSTEMS OF ORTHOGONAL COORDINATES Orthogonal systems are those systems of coordinates where the surfaces, whose intersections determine the position of a given point P(r) in space, do intersect at right angles. As we have seen, most useful for our purposes are the Cartesian, spherical, confocal spheroidal (elliptic) and parabolic coordinates. We give below the definition and the interval of variation of each coordinate covering the whole space. • Cartesian coordinates : x; y; z x; y; z ˛ðN; NÞ

(16)

• Spherical coordinates : r; q; 4 rð0; NÞ; qð0; pÞ; 4ð0; 2pÞ

(17)

where (Figure 2.1): x ¼ r sin q cos 4;

y ¼ r sin q sin 4;

z ¼ r cos q

(18)

FIGURE 2.1 Spherical coordinate system. (Magnasco, V., 2009, Methods of Molecular Quantum Mechanics: An introduction to Electronic Molecular Structure, Wiley. Reprinted with permission from John Wiley and Sons).

72

CHAPTER 2 Coordinate systems

FIGURE 2.2 Spheroidal coordinate system. (Magnasco, V., 2009, Methods of Molecular Quantum Mechanics: An introduction to Electronic Molecular Structure, Wiley. Reprinted with permission from John Wiley and Sons).

with the inverse transformations: 1=2  ; r ¼ x2 þ y 2 þ z2 1

q ¼ cos1 ðz=rÞ;

4 ¼ tan1 ðy=xÞ

(19)

denoting the inverse trignometric function.1 • Spheroidal coordinates : m; n; 4 mð1; NÞ; nð1; 1Þ; 4ð0; 2pÞ

(20)

where (Figure 2.2): m¼

rA þ rB ; R

n¼

rA  rB ; R

4

with the inverse transformations: 8 < r ¼ R ðm þ nÞ; r ¼ R ðm  nÞ B A 2 2 : rA þ rB ¼ Rm; rA  rB ¼ Rn • Parabolic coordinates : x; h; 4 xð0; NÞ; hð0; NÞ; 4ð0; 2pÞ

(21)

(22)

(23)

where: x¼

pffiffiffiffiffi xh cos 4;

cos1 ðz=rÞ denotes the function arc cosðz=rÞ.

1

y¼

pffiffiffiffiffi xh sin 4;

1 z ¼ ðx  hÞ 2

(24)

2.3 Generalized coordinates

73

2.3 GENERALIZED COORDINATES Let: x ¼ xðq1 ; q2 ; q3 Þ;

y ¼ yðq1 ; q2 ; q3 Þ;

z ¼ zðq1 ; q2 ; q3 Þ

(25)

be the functional relations connecting the Cartesian coordinates (x,y,z) to the generalized coordinates (q1,q2,q3). Then:
2
2
2 vx vy vz 2 hi ¼ þ þ i ¼ 1; 2; 3 (26) vqi vqi vqi where:

vx vq 1 vx h1 h2 h3 ¼ jJj ¼ vq2 vx vq3

vy vq1 vy vq2 vy vq3

vz vq1 vz vq2 vz vq3

is the Jacobian2 determinant of the transformation. We notice that: vx vy vx vx vx vz vq vq vq vq vq vq 1 1 1 2 3 1 2 vx vy vy vy vy h1 vz J2 ¼ ¼ 0 vq2 vq2 vq2 vq1 vq2 vq3 0 vx vy vz vz vz vz vq3 vq3 vq3 vq1 vq2 vq3

(27)

0 h22 0

0 0 h23

(28)

In the second determinant we have interchanged rows and columns, the last result being a consequence of the fact that the coordinate systems are orthogonal so that all off-diagonal elements of the product are zero. In generalized coordinates, the infinitesimal volume element dr, the gradient ∇ and the Laplacian operator r2 are given by the symmetrical expressions: (29) dr ¼ h1 h2 h3 dq1 dq2 dq3

∇ = e1

∇2 = ∇ ⋅ ∇ =

2

1 h1h 2 h 3

1

h1 q1

+ e2

1

h2 q2

+ e3

1

h 3 q3

⎡ ⎛ h1h2 ⎞⎤ ⎛ h 2 h3 ⎞ ⎛ h 3 h1 ⎞ ⎢ ⎜ ⎟⎥ ⎜ ⎟+ ⎜ ⎟+ q2 ⎝ h 2 q2 ⎠ q3 ⎝ h 3 q3 ⎠ ⎥⎦ ⎢⎣ q1 ⎝ h1 q1 ⎠

After Jacobi Karl Gustav 1804–1851, German mathematician.

(30)

(31)

74

CHAPTER 2 Coordinate systems

Using formulae (29–31), we now calculate the infinitesimal volume element dr, the gradient r and the Laplacian operator r2 in Cartesian, spherical, spheroidal and parabolic coordinates.

2.4 CARTESIAN COORDINATES (x,y,z ) q1 ¼ x;

q2 ¼ y; q3 ¼ z 0 hx ¼ hy ¼ hz ¼ 1 8 dr ¼ dx dy dz > > > > > >

> > > v2 v2 v2 > > : r2 ¼ 2 þ 2 þ 2 vx vy vz

(32)

(33)

2.5 SPHERICAL COORDINATES (r,q,4) q1 ¼ r;

q2 ¼ q;

q3 ¼ 4

x ¼ r sin q cos 4; y ¼ r sin q sin 4; z ¼ r cos q 8 > vx vy vx > > ¼ sin q cos 4 ¼ sin q sin 4 ¼ cos q > > vr vr vr > > < vx vy vz ¼ r cos q cos 4 ¼ r cos q sin 4 ¼ r sin q > vq vq vq > > > vx vy vz > > > ¼0 : v4 ¼ r sin q sin 4 v4 ¼ r sin q cos 4 v4 Then:

(34) (35)

(36)

8 h2 ¼ sin2 qcos2 4 þ sin2 4 þ cos2 q ¼ 1 0 hr ¼ 1 < r   h2q ¼ r 2 cos2 q cos2 4 þ sin2 4 þ r 2 sin2 q ¼ r 2 0 hq ¼ r :   0 h4 ¼ r sin q h24 ¼ r 2 sin2 q sin2 4 þ cos2 4 ¼ r2 sin2 q

(37)

dr ¼ r2 dr sin q dq d4

(38)

so that:

∇ = er

r

+e

1 r

+e

1 r sin


2 

 1 v r sin q v v r sin q v v 1 v þ þ r 2 sin q vr 1 vr vq r vq v4 r sin q v4  

 2 L^ 1 v 1 1 v v 1 v2 2v 2 ¼ r r þ 2 sin q þ 2  ¼ 2 r r2 r vr vr r sin q vq vq sin q v42

(39)

r2 ¼

(40)

2.6 Spheroidal coordinates (m,n,4)

where: r2r ¼ is the radial Laplacian, and: 2 L^ ¼ 


 1 v v2 2 v 2v r ¼ 2þ 2 vr r vr vr r vr

75

(41)


 1 v v 1 v2 sin q þ 2 sin q vq vq sin q v42



(42)

the square of the angular momentum operator in atomic units. In this way, in spherical coordinates the radial Laplacian (which depends on the radial variable) separates from the angular momentum operator (which depends on the angular variables). The relation connecting the whole Laplacian r2 to the square of the angular momentum operator 2 ^ L using Cartesian coordinates was examined in detail in Problem 1.5 of the previous chapter of this book.

2.6 SPHEROIDAL COORDINATES (m,n,4) q1 ¼ m;

q2 ¼ n;

q3 ¼ 4

8 rA þ rB rA  rB > m¼ ; n¼ ; 4 > > < R R pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x ¼ a ðm2  1Þð1  n2 Þ cos 4; y ¼ a ðm2  1Þð1  n2 Þ sin 4; > > > : a ¼ R=2 8 > > > > > > > > > >
> > m2  1 vn > > > qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > vx > > 2 2 > : v4 ¼ a ðm  1Þð1  n Þ sin 4

(43)

z ¼ aðmn þ 1Þ

sffiffiffiffiffiffiffiffiffiffiffiffiffiffi vy 1  n2 sin 4 ¼am m2  1 vm sffiffiffiffiffiffiffiffiffiffiffiffiffiffi vy 1  n2 sin 4 ¼an m2  1 vn qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vy ¼ a ðm2  1Þð1  n2 Þ cos 4 v4

(44)

vz ¼an vm vz ¼am vn vz ¼0 v4

(45)

Then: 
   1  n2  2 m 2  n2 cos 4 þ sin2 4 þ n2 ¼ a2 2 h m 2 ¼ a2 m 2 2 m 1 m 1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m 2  n2 0 hm ¼ a m2  1

(46)

76

CHAPTER 2 Coordinate systems


2    m 1  2 m 2  n2 2 2 cos ¼ a2 hn 2 ¼ a2 n 2 4 þ sin 4 þ m 2 1  n2 1n rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m 2  n2 0 hn ¼ a 1  n2  

  h4 2 ¼ a2 m2  1 1  n2 sin2 4 þ cos2 4 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi    ¼ a2 m2  1 1  n2 0 h4 ¼ a ðm2  1Þð1  n2 Þ so that:

  dr ¼ a3 m2  n2 dm dn d4

1⎡ ∇ = ⎢e a⎢ ⎣

2 2

−1

− 2

1− 2

+e

2

− 2

+e

(47)

1 ( 2 − 1)(1 − 2 )

⎤ ⎥ ⎥ ⎦

 

  v  1 v  2 v  m2  n2 v2 2 v  1 m þ 1  n þ ðm2  1Þð1  n2 Þ v42 a2 ðm2  n2 Þ vm vm vn vn   1 m 2  n2 v2 2 2 r þ r þ ¼ 2 2 m n ðm2  1Þð1  n2 Þ v42 a ðm  n2 Þ

(48)

8 > > > r2 ¼ < > > > :

(49)

where: rm

2

rn


  v  v2  v  2 v ¼ þ 2m m 1 ¼ m2  1 2 vm vm vm vm

(50)


    v2 v  v 2 v ¼  2n 1n ¼ 1  n2 vn2 vn vn vn

(51)

2

2.7 PARABOLIC COORDINATES (x,h,4) Instead of the squared parabolic coordinates of Margenau and Murphy (1957, p. 185), we find it convenient to use here the system of parabolic coordinates introduced by Schroedinger (1926a) and Epstein (1926) and also used by Condon and Shortley (1963) and Alexander (1969): q1 ¼ x; x¼

pffiffiffiffiffi xh cos 4;

q2 ¼ h; y¼

q3 ¼ 4

pffiffiffiffiffi xh sin 4;

1 z ¼ ðx  hÞ 2

(52) (53)

2.7 Parabolic coordinates (x,h,4)

8 rffiffiffi > > vx 1 h > > ¼ cos 4 > > vx 2 x > > > s ffiffiffi > < vx 1 x ¼ cos 4 > vh 2 h > > > > > pffiffiffiffiffi > vx > > > : v4 ¼  xh sin 4 Then:3

rffiffiffi h vz 1 ¼ sin 4 x vx 2 sffiffiffi vy 1 x vz 1 ¼ sin 4 ¼ vh 2 h vh 2 vy pffiffiffiffiffi vz ¼ xh cos 4 ¼0 v4 v4

77

vy 1 ¼ vx 2

8 sffiffiffiffiffiffiffiffiffiffiffi > > 1 x þ h 1 xþh > > hx 2 ¼ 0 hx ¼ > > > 4 x 2 x > > < sffiffiffiffiffiffiffiffiffiffiffi 1 xþh 1 xþh > > 0 hh ¼ hh 2 ¼ > > 4 h 2 h > > > > > p ffiffiffiffiffi : 2 0 h4 ¼ xh h4 ¼ xh

(54)

(55)

so that: 1 dr ¼ ðx þ hÞdx dh d4 4 sffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffi 4x v 4h v 1 v þ eh þ e4 pffiffiffiffiffi V ¼ ex x þ h vx x þ h vh xh v4  


 4 v v v v 1 1 1 v2 x þ h þ þ r ¼ x þ h vx vx vh vh 4 x h v42 2

(56) (57)

(58)

The relations between wavefunctions expressed in spheroidal, spherical and parabolic coordinates were examined by Coulson and Robinson (1958) in their study on the hydrogen atom in spheroidal coordinates as resulting from the hydrogen molecular ion H2 þ at finite values of the internuclear distance R, by taking the limiting cases of R / 0 and R /N, respectively. Battezzati and Magnasco (2003) studied the behaviour of equations and solutions of the H2 þ problem at the limit of the united atom Heþ . It was shown there that, since when R / 0, m / 2r/R, n / cos q, the outer m-equation goes into the correct radial equation for the hydrogen-like system constant A / –l(l þ 1) and the parameter for the bound state ðE < 0Þ Heþ , with the separation pffiffiffi s þ 1 ¼ Zr=2p ¼ 2=2ZðEÞ1=2 / n, where n and l are the principal and orbital quantum numbers of the hydrogen-like system with nuclear charge Z, respectively. However, whereas the outer m-solution X(m) (Byers Brown and Steiner, 1966) goes into the correct radial function RðrÞfexpð2rÞ as R / 0, the inner n-solution has an infinitesimal behaviour different from the hydrogen-like Qðcos qÞfPm l ðcos qÞ, because the term (ZA  ZB)Rn, characteristic of the H-like system in 3

Alliluev and Malkin (1974) and Silverstone (1978), instead, use scaled parabolic coordinates differing from Eqn (53) by a constant factor.

78

CHAPTER 2 Coordinate systems

spheroidal coordinates, is missing in the H2 þ case (where ZA ¼ ZB ¼ 1). In other words, even if the differential equations become identical for R ¼ 0, the inner n-solutions for H2 þ and the Heþ hydrogenlike Qðcos qÞ have a different behaviour for infinitesimal R s 0.

2.8 PROBLEMS 2 2.1. Taking into account the relations between Cartesian and spherical coordinates, and their inverse relations, calculate the derivatives of ðr; q; 4Þ with respect to ðx; y; zÞ. Answer: vr x ¼ ¼ sin q cos 4 vx r vq cos q cos 4 ¼ vx r v4 sin 4 ¼ vx r sin q

vr y vr z ¼ ¼ sin q sin 4 ¼ ¼ cos q vy r vz r vq cos q sin 4 vq sin q ¼ ¼ vy r vz r v4 cos 4 v4 ¼ ¼0 vy r sin q vz

Hint: The relations between the two coordinate systems are: x ¼ r sin q cos 4; y ¼ r sin q sin 4; 1=2  2 ; q ¼ cos1 ðz=rÞ; r ¼ x þ y 2 þ z2

z ¼ r cos q 4 ¼ tan1 ðy=xÞ

We must remember the rules of derivation of the inverse trigonometric functions for u(x,y,z):  1=2 d tan1 u  1 d cos1 u ¼  1  u2 ¼ 1 þ u2 du du and that: v v vr v vq v v4 ¼ þ þ etc vx vr vx vq vx v4 vx In the following, we omit for brevity the caret sign on the angular momentum operators. 2.2. Using the results of Problem 2.1, find the expression in spherical coordinates of the three components Lx, L y, Lz of the angular momentum operator (here and in the following, we omit for brevity the caret sign on the operators). Answer:
 v v Lx ¼ i  sin 4  cot q cos 4 vq v4
 v v (59) Ly ¼ i cos 4  cot q sin 4 vq v4 Lz ¼ i

v v4

2.8 Problems 2

79

Hint: Start from the expression of the components Lx, Ly, Lz in Cartesian coordinates. 2.3. Using the results of Problem 2.2, find the expression of L2 (the square of the angular momentum operator) in spherical coordinates. Answer: 
2  
 1 v v 1 v2 v v 1 v2 ¼  (60) þ cot q sin q þ 2 þ L2 ¼  sin q vq vq vq sin2 q v42 sin q v42 vq2 Hint: Remember that: L2 ¼ L2x þ L2y þ L2z and that: L2x ð fgÞ ¼ Lx ½Lx ð fgÞ ¼ Lx ½gðLx f Þ þ f ðLx gÞ etc where f ; g are functions of angles q; 4. 2.4. Find the expression in spherical coordinates of the ladder operators Lþ ¼ Lx þ iLy and L ¼ Lx  iLy . Answer:
 v v þ i cot q Lþ ¼ expði4Þ vq v4
 (61) v v L ¼ expði4Þ  þ i cot q vq v4 Hint: Use the expression of Lx and L y in spherical coordinates as found in Problem 2.2. 2.5. Find the expression of L2 in spherical coordinates using the corresponding expressions of the ladder operators Lþ and L . Answer:   
2
 v v 1 v2 1 v v 1 v2 2 ¼ (62) þ cot q þ 2 sin q þ 2 L ¼ vq sin q v42 sin q vq vq sin q v42 vq2 Hint: Find first the relations between L2 and Lþ ; L : L2 ¼ L2x þ L2y þ L2z ¼ Lþ L  Lz þ L2z ¼ L Lþ þ Lz þ L2z Next use the expressions in spherical coordinates found in Problems 2.2 and 2.4. 2.6. Find the relation between ðx; y; zÞ and ðm; n; 4Þ. Answer: x¼

1=2  R  2 m  1 1  n2 cos 4; 2

y¼

1=2  R  2 m  1 1  n2 sin 4; 2

z¼

R ðmn þ 1Þ 2

80

CHAPTER 2 Coordinate systems

Hint: Let Pðx; y; zÞ be the point in the Cartesian coordinate system which corresponds to Pðm; n; 4Þ in spheroidal coordinates. Use the definitions of spheroidal and spherical coordinates of point P and the Carnot’s theorem for the angle qA ¼ r^A z. 2.7. Give a geometric derivation of the infinitesimal volume element dr in spherical coordinates. Hint: Consider the circular sectors having an infinitesimal basis.

2.9 SOLVED PROBLEMS 2.1. With reference to Figure 2.1, the inverse transformations are:  1=2

 1=2 r ¼ x 2 þ y2 þ z2 ; q ¼ cos1 z x2 þ y2 þ z2 ;

  4 ¼ tan1 yx1

1=2 vr 1 2 x vr y vr z ¼ x þ y2 þ z2 ¼ ; ¼ 2x ¼ ; vx 2 r vy r vz r   h     i1=2  vq 1  2 2 1=2 vu 2 2 2 2 1 2 2 3=2  z x þy þz ¼ 1u ¼ 1z x þy þz 2x vx vx 2 ¼ ¼

1=2
2  3=2 x þ y 2 þ z2  z2 zx x2 þ y2 þ z2 2 2 2 x þy þz zx ðx2 þ y2 Þ1=2



x2 þ y 2 þ z2

1

¼

r 2 cos q sin q cos 4 2 cos q cos 4 r ¼ r sin q r

  h     i1=2  vq 1  2 2 1=2 vu 2 2 2 2 1 2 2 3=2  z x þy þz ¼ 1u ¼ 1z x þy þz 2y vy vy 2 1=2
2  3=2 x þ y2 þ z2 ¼ yz x2 þ y2 þ z2 x2 þ y2 ¼

yz ðx2 þ

1=2 y2 Þ



x2 þ y2 þ z2

1

¼

r 2 cos q sin q sin 4 2 cos q sin 4 r ¼ r sin q r

 1=2 vu vq ¼  1  u2 vz vz   h  2 1 i1=2  2 1=2 1  2  2 2 3=2 2 2 2 2 2 x þy þz  z x þy þz 2z ¼ 1z x þy þz 2 1=2
2
2 2 2 2  2  x þ y2 þ z2 2 þ z2 1=2 x þ y þ z  z þ y x x2 þ y2 x 2 þ y 2 þ z2  2 1=2 x þ y2 r sin q sin q ¼ 2 ¼ 2 ¼ x þ y2 þ z2 r r ¼

2.9 Solved problems

81

 1 vu h 2 i1  2  v4  y x ¼ 1 þ u2 ¼ 1 þ yx1 vx vx
¼

1

y y r sin q sin 4 sin 4 ¼ ¼ 2 2  2 ¼ 2 x x þ y2 r sin q r sin q 1
2   1  v4  x þ y2 2 1 vu ¼ 1þu ¼ x x2 vy vy

x2 þ y2 x2

¼

x r sin q cos 4 cos 4 ¼ ¼ 2 2 x2 þ y2 r sin q r sin q 1 vu v4  ¼ 1 þ u2 ¼0 vz vz

2.2. 
v v Lx ¼ i y  z vz vy 

v vr v vq v v4 v vr v vq v ¼ i r sin q sin 4 þ þ  r cos q þ þ vr vz vq vz v4 vz vr vy vq vy v4 

v sin q v v cos q sin 4  r cos q sin q sin 4 þ ¼ i r sin q sin 4 cos q  vr r vq vr r 
v v v ¼ i sin2 q sin 4  cos2 q sin 4  cot q cos 4 vq vq v4 
v v ¼ i  sin 4  cot q cos 4 vq v4

v4 vy



v cos 4 v þ vq r sin q v4

where the results of Problem 2.1 were made use of.
 v v Ly ¼ i z  x vx vz 
 
v vr v vq v v4 v vr v vq v v4 þ þ  r sin q cos 4 þ þ ¼ i r cos q vr vx vq vx v4 vx vr vz vq vz v4 vz  
v cos q cos 4 v sin 4 v  ¼ i r cos q sin q cos 4 þ vr r vq r sin q v4 
v v ¼ i cos 4  cot q sin 4 vq v4



82

CHAPTER 2 Coordinate systems
 v v Lz ¼ i x  y vy vx 
 v vr v vq v v4 ¼ i r sin q cos 4 þ þ vr vy vq vy v4 vy
 v vr v vq v v4  r sin q sin 4 þ þ vr vx vq vx v4 vx 
 v cos q sin 4 v cos 4 v ¼ ir sin q cos 4 sin q sin 4 þ þ vr r vq r sin q v4
 v cos q cos 4 v sin 4 v   sin 4 sin q cos 4 þ vr r vq r sin q v4    v  2 v 2 ¼ i ¼ i cos 4 þ sin 4 v4 v4

2.3. L2x



v v v v sin 4 þ cot q cos 4 ¼  sin 4 þ cot q cos 4 vq v4 vq v4 v cot q ¼ 1  cot2 q vq

  v2 v  v 1  cot2 q þ cot q cos2 4 L2x ¼  sin2 4 2 þ sin 4 cos 4 v4 vq vq  v2 v þ cot2 q cos2 4 2  cot2 q cos 4 sin 4 v4 v4 L2y



 v v v v ¼  cos 4  cot q sin 4 cos 4  cot q sin 4 vq v4 vq v4   v2 v  v sin2 4 ¼  cos2 4 2  cos 4 sin 4 1  cot2 q þ cot q v4 vq vq  v2 v 2 2 2 þ cot q sin 4 2 þ cot q sin 4 cos 4 v4 v4
L2z

¼

v i v4


 v v2 i ¼ 2 v4 v4

2.9 Solved problems

83

2.4. Lþ ¼ Lx þ iLy 

v v v v þ cos 4  cot q sin 4 ¼ i sin 4 þ cot q cos 4 vq v4 vq v4 ¼ ðcos 4 þ i sin 4Þ

v v þ i cot qðcos 4 þ i sin 4Þ vq v4




v v þ i cot q ¼ expði4Þ vq v4



L ¼ Lx  iLy

 v v v v ¼ i sin 4 þ cot q cos 4  cos 4  cot q sin 4 vq v4 vq v4 ¼ ðcos 4  i sin 4Þ

v v þ i cot qðcos 4  i sin 4Þ vq v4


v v ¼ expði4Þ  þ i cot q vq v4 where Euler’s formulae for imaginary exponentials were used: expði4Þ ¼ cos 4  i sin 4 As a further exercise, derive in an elementary way Euler’s formulae (Hint: use series expansions for exponentials and trigonometric functions). 2.5.

   Lþ L  Lz þ L2z ¼ Lx þ iLy Lx  iLy  Lz þ L2z



 ¼ L2x þ L2y þ L2z  i Lx ; Ly  Lz ¼ L2x þ L2y þ L2z  iðiLz Þ  Lz ¼ L2x þ L2y þ L2z    L Lþ þ Lz þ L2z ¼ Lx  iLy Lx þ iLy þ Lz þ L2z



 ¼ L2x þ L2y þ L2z þ i Lx ; Ly þ Lz ¼ L2x þ L2y þ L2z þ iðiLz Þ þ Lz ¼ L2x þ L2y þ L2z First of all remember that: v cot q ¼ 1  cot2 q vq

84

CHAPTER 2 Coordinate systems

Then, using the results of Problems 2.2 and 2.4:



 v v v v þ i cot q expði4Þ  þ i cot q vq v4 vq v4 
 
v v v v  expði4Þ þ i cot q expði4Þ ¼ expði4Þ vq vq vq v4

 v v v v  expði4Þ þ i cot q i cot q expði4Þ þ i cot q v4 vq v4 v4   v2 v  ¼ expði4Þ  expði4Þ 2 þ i expði4Þ 1  cot2 q v4 vq 
v v2 v 2  expði4Þ cot q  cot q expði4Þ 2  iexpði4Þ v4 vq v4

Lþ L ¼ expði4Þ

¼

  v v2 v v2 v  cot q  cot2 q 2 þ i cot2 q  i 1 þ cot2 q 2 v4 v4 vq v4 vq v2 v v2 v v v2  cot q  cot2 q 2  i þ i  2 2 v4 vq v4 v4 v4 vq   2  v2 v v  ¼  2 þ cot q þ 1 þ cot2 q v42 vq vq

Lþ L  Lz þ L2z ¼ 


2  v v 1 v2 ¼ þ cot q þ 2 vq sin q v42 vq2  
 1 v v 1 v2 ¼ sin q þ 2 sin q vq vq sin q v42 2.6. With reference to Figure 2.2, the spheroidal coordinates are: rA þ rB ; R

m¼

n¼

rA  rB ; R

4

with the inverse transformations: R R ðm þ nÞ; rB ¼ ðm  nÞ 2 2 x ¼ rA sin qA cos 4; y ¼ rA sin qA sin 4

rA ¼ z ¼ rA cos qA ; The Carnot theorem then gives:

2 þ R2  2RrA cos qA rB2 ¼ rA

2.9 Solved problems

so that:

85


2 h i R ðm þ nÞ2  ðm  nÞ2 þ 4 2 2 2 r  rB þ R 2 ¼ cos qA ¼ A
2 2RrA R 4 ðm þ nÞ 2 m2 þ n2 þ 2mn  m2  n2 þ 2mn þ 4 mn þ 1 ¼ 4ðm þ nÞ mþn " #1=2  1=2 ðmn þ 1Þ2 sin qA ¼ 1  cos2 qA ¼ 1 ðm þ nÞ2 " #1=2   1=2 m 2  1 1  n2 m2 þ n2 þ 2mn  m2 n2  1  2mn ¼ ¼ mþn ðm þ nÞ2 ¼

Therefore we obtain: R mn þ 1 R ðm þ nÞ ¼ ðmn þ 1Þ 2 mþn 2 

 2 1=2 m  1 1  n2 R x ¼ rA sin qA cos 4 ¼ ðm þ nÞ cos 4 mþn 2  1=2 R  2 cos 4 ¼ m  1 1  n2 2  1=2 R  2 y ¼ rA sin qA sin 4 ¼ sin 4 m  1 1  n2 2 z ¼ rA cos qA ¼

2.7. This result corresponds to the infinitesimal volume of a solid with a curved basis ðr 2 dr sin qdqd4Þ having infinitesimal sides ðdrÞðr dqÞðr sin qd4Þ as Figure 2.3 shows. In fact,

FIGURE 2.3 From Cartesian to spherical elementary volume

86

CHAPTER 2 Coordinate systems

we have for the circular sectors with infinitesimal basis, the results of Figure 2.4. Since dq is infinitesimal, the series expansion for sin x (x small) gives sin xzx.

FIGURE 2.4 Circular sectors with infinitesimal bases

CHAPTER

Differential equations in quantum mechanics

3

CHAPTER OUTLINE 3.1 Introduction ................................................................................................................................... 88 3.2 Partial Differential Equations .......................................................................................................... 88 3.3 Separation of Variables .................................................................................................................. 89 3.3.1 The particle in a three-dimensional box .........................................................................89 3.3.2 The three-dimensional harmonic oscillator.....................................................................91 3.3.3 The atomic one-electron system....................................................................................91 3.3.4 The molecular one-electron system ...............................................................................93 3.3.5 The hydrogen atom in a uniform electric field ................................................................96 3.4 Solution by Series Expansion .......................................................................................................... 98 3.5 Solution Near Singular Points ....................................................................................................... 100 3.6 The One-dimensional Harmonic Oscillator...................................................................................... 101 3.7 The Atomic One-electron System ................................................................................................... 104 3.7.1 Solution of the radial equation....................................................................................105 3.7.2 Solution of the F-equation .........................................................................................109 3.7.3 Solution of the Q-equation .........................................................................................109 3.7.4 The hydrogen-like atomic orbitals ...............................................................................115 3.8 The Hydrogen Atom in an Electric Field ......................................................................................... 117 3.9 The Hydrogen Molecular Ion HD 2 ................................................................................................... 120 3.10 The Stark Effect in Atomic Hydrogen ........................................................................................... 124 3.10.1 Solution of the x-equation in the zero-field case .......................................................124 3.10.2 The first-order Stark effect......................................................................................128 3.11 Appendix: Checking the Solutions ............................................................................................... 131 3.11.1 The radial equation of the H-atom in spherical coordinates .......................................131 3.11.2 The Q-equation of the H-atom in spherical coordinates.............................................132 3.11.3 The x-equation of the H-atom in parabolic coordinates..............................................135 3.12 Problems 3 ................................................................................................................................ 135 3.13 Solved Problems ........................................................................................................................ 137

Elementary Molecular Quantum Mechanics 2E. http://dx.doi.org/10.1016/B978-0-444-62647-9.00003-8 Ó 2013, 2007 Elsevier B.V. All rights reserved

87

88

CHAPTER 3 Differential equations in quantum mechanics

3.1 INTRODUCTION Generally speaking, a function is a dependent variable, namely, a correspondence between numbers. All functions of quantum mechanics are regular functions, namely, mathematical functions physically restricted to be (1) single valued, (2) continuous with their first derivatives, and (3) quadratically integrable, i.e. vanishing at infinity. Algebraic equations have for solution numbers, whereas differential equations have for solution functions. A differential equation is an equation containing, besides the independent variable and the unknown function of this variable, also the derivatives of this function and its differentials (Smirnov, 1993b). The differential equations of quantum mechanics are linear equations of the second order, mostly of two kinds: (1) eigenvalue equations, like the Schroedinger equation for the hydrogen atom or the hydrogen molecular ion, which are second-order partial differential equations in three variables, and (2) non-homogeneous differential equations of the second order like those arising from the Rayleigh–Schroedinger (RS) perturbation theory of the hydrogen atom in an external electric or magnetic field.

3.2 PARTIAL DIFFERENTIAL EQUATIONS We speak of ordinary differential equation if the functions appearing in a differential equation depend just on a single independent variable. The order of a differential equation is the order of the highest derivative involved in it (Ince, 1963). If instead the differential equation contains the partial derivatives of functions of more variables, we speak of partial differential equation. A first example is given by the Schroedinger eigenvalue equation for the one-electron atomic system in internal coordinates1 describing a single electron at point r in space in the field of Z nuclear charges: ^ HjðrÞ ¼ EjðrÞ

(1a)

where H^ is the Hamiltonian operator in atomic units: 1 Z H^ ¼  V2  2 r

(2)

jðrÞ; the solution of Eqn (1a), is called the eigenfunction of the operator, and E is a constant called the eigenvalue of the operator. Equation (1a) is a very particular kind of differential equation, since when ^ we find the same function multiplied by the function is acted upon by the differential operator H, a constant. A second example is given by the Schroedinger equation for the molecular one-electron system describing the motion of a single electron in the field of two nuclear charges ZA and ZB a distance R apart: ^ Hjðr; RÞ ¼ E jðr; RÞ

(3)

with the electronic Hamiltonian operator given in atomic units by 1 ZA ZB H^ ¼  V2   rA rB 2 1

The separation of the motion of the centre of mass is described in Problem 3.2.

(4)

3.3 Separation of variables

89

For ZA ¼ ZB ¼ Z we have the corresponding homonuclear system, having a centre of symmetry, which for Z ¼ 1 reduces to the hydrogen molecular ion Hþ 2 . In the Born–Oppenheimer approximation for the separation of electronic from nuclear motion (Born and Oppenheimer, 1927)2 R is treated as a parameter specifying the relative position of the two nuclei.

3.3 SEPARATION OF VARIABLES Typical of any partial differential equation is the attempt to separate it into differential equations involving a lesser number of variables. When the partial differential equation has been fully reduced to a set of ordinary differential equations, we say we have accomplished the separation of the partial differential equation. Each separation involves a separation constant, while the integration constants depend on the boundary conditions imposed on the problem, taking in mind that the resultant solutions must be regular functions. To clearly illustrate the technique, we work out below the separation of variables in five cases, two in Cartesian coordinates, the particle in a three-dimensional box and the three-dimensional harmonic oscillator; one in spherical coordinates, the atomic oneelectron system; one in spheroidal coordinates, the molecular one-electron system; and the last one in parabolic coordinates, the hydrogen atom in a uniform electric field. We notice in passing that partial differential equations involving more than one electron (e.g. that of the He atom with N ¼ 2) are not separable.

3.3.1 The particle in a three-dimensional box The box is a system whose potential energy is zero for the particle inside a closed region and constant everywhere else. In Cartesian coordinates, consider a box of sides a, b, c along x, y, z, respectively. Then, the potential energy is assumed to be Vx ¼ 0 0 < x < a Vx elsewhere Vy ¼ 0 0 < y < b Vy elsewhere Vz ¼ 0 0 < z < c Vz elsewhere

(5)

where Vx, Vy , Vz are constant values, with V x þ Vy þ Vz ¼ V

(6)

The Schroedinger eigenvalue equation for the particle in three-dimensional space can then be written as V2 j þ 2ðE  VÞj ¼ 0

(7)

v2 v2 v2 þ þ vx2 vy2 vz2

(8)

where the Laplacian is V2 ¼ 2

For a variational derivation of the different Born–Oppenheimer approximations resulting from separation of nuclear from electronic motion, see Longuet-Higgins (1961) and Chapter 20 of this book.

90

CHAPTER 3 Differential equations in quantum mechanics

If we pose jðx; y; zÞ ¼ XðxÞYðyÞZðzÞ

(9)

in Cartesian coordinates, the three-dimensional Schroedinger eigenvalue Eqn (7) can be separated into three ordinary differential equations one in each single variable. Substituting Eqn (9) in Eqn (7) we obtain YZ


 v2 X v2 Y v2 Z þ ZX 2 þ XY 2 þ 2 E  Vx  Vy  Vz XYZ ¼ 0 2 vx vy vz

(10)

and, dividing through by XYZ
 1 v2 X 1 v2 Y 1 v2 Z þ þ 2 E  V  V þ 2Vz ¼ 2Ez ¼  x y X vx2 Y vy2 Z vz2

(11)

where 2Ez is a first separation constant. This is due to the fact that the equation in the left-hand member of Eqn (11) depends only on x, y, while the equation in the right-hand member depends only on z, so that both members, depending on independent variables, must be equal to a constant, which we called 2Ez. In this way, the differential equation in z is separated off, and we can write d2 Z þ 2ðEz  Vz ÞZ ¼ 0 dz2

(12)

where now the derivatives occurring in Eqn (12) are total derivatives. In a similar way, we obtain from Eqn (11) 1 v2 X 1 v2 Y þ 2ðE  E  V Þ ¼  þ 2Vy ¼ 2Ey z x X vx2 Y vy2

(13)

where 2Ey is a second separation constant. So we are left with
 1 d2 X þ 2 E  E z  Ey  Vx ¼ 0 2 X dx

(14)


 1 d2 Y þ 2 E  Ez  Ex  Vy ¼ 0 2 Y dy

(15)

d2 X þ 2ðEx  Vx ÞX ¼ 0 dx2

(16)


 d2 Y þ 2 Ey  Vy Y ¼ 0 2 dy

(17)

E  E z  Ey ¼ Ex

(18)

E  E z  Ex ¼ Ey

(19)

namely,

provided we pose

3.3 Separation of variables

91

namely, E ¼ E x þ Ey þ Ez

(20)

In this way, through the decomposition (9), we accomplished the separation of the three-dimensional partial differential Eqn (7) into three one-dimensional total differential equations in the independent variables, one in x, Eqn (16); one in y, Eqn (17); and one in z, Eqn (12). While Problem 3.3 studies the case of the free particle, further examples on the problem of the particle in one-dimensional boxes are given in Problems 3.4–3.6.

3.3.2 The three-dimensional harmonic oscillator This system is of importance in the theory of specific heats, crystalline solids and spectroscopy, and is best studied in Cartesian coordinates (Pauling and Wilson, 1935). It is characterized by the potential energy function: Vðx; y; zÞ ¼

1 1 1 k x x 2 þ k y y 2 þ k z z 2 ¼ V x þ Vy þ Vz 2 2 2

(21)

where kx, ky, kz are the harmonic force constants according to the three orthogonal directions. The potential V has the same decomposition as that given in Eqn (6) with a different meaning of the individual terms. The Schroedinger eigenvalue equation can be written as V2 jðx; y; zÞ þ 2ðE  VÞjðx; y; zÞ ¼ 0

(22)

and can be separated by posing, as before jðx; y; zÞ ¼ XðxÞ YðyÞ ZðzÞ

(23)

The calculation is entirely similar to that in Section 3.3.1, so that we omit all details giving only the final result: d2 X þ 2ðEx  Vx ÞX ¼ 0 dx2 
d2 Y þ 2 Ey  Vy Y ¼ 0 2 dy d2 Z þ 2ðEz  Vz ÞZ ¼ 0 dz2

(24) (25) (26)

and we have obtained the separation of the partial differential Eqn (22) for the three-dimensional harmonic oscillator into three one-dimensional differential equations, as we did before for the particle in the three-dimensional box.

3.3.3 The atomic one-electron system Separation of variables in Eqn (1a) (atomic units) is possible in terms of the spherical coordinates ðr; q; 4Þ which reflect the symmetry of the central Coulombic potential. In spherical coordinates, the

92

CHAPTER 3 Differential equations in quantum mechanics

Laplacian operator can be written as the sum of the radial Laplacian, V2r , and the square of the angular momentum operator, L^2 : 2 L^ (27) V2 ¼ V2r  2 r V2r

L^2 ¼ 

  1 v v2 2 v 2v ¼ 2 r ¼ 2þ vr r vr vr r vr

  2    1 v v 1 v2 v v 1 v2 ¼  þ cot q sin q þ 2 þ sin q vq vq vq sin2 q v42 sin q v42 vq2

(28)



(29)

The Schroedinger Eqn (1a) can hence be written as the partial differential equation in the three variables:    2 L^ 1 Z jðr; q; 4Þ ¼ 0 (30)  V2r þ 2  E þ 2r 2 r We now separate radial from angular variables by posing jðrÞ ¼ RðrÞ Yðq; 4Þ Substituting in Eqn (30) we obtain    1 Z R Y  V2r  E þ R ¼  2 L^2 Y 2 r 2r and, dividing through both members by RY/2r2    1 Z R 2r 2  V2r  E þ 2 L^ Y 2 r ¼ ¼ l R Y

(31)

(32)

(33)

Since the left-hand member of Eqn (33) depends only on the independent variable r and the righthand member on the independent variables q; 4, the equality being true for all values of the variables, both members must be equal to a constant which we call l. l is therefore a first separation constant of the partial differential Eqn (30). Hence we obtain the two separate differential equations:    1 l Z   V2r þ RðrÞ ¼ ERðrÞ (34) 2 2r 2 r L^2 Yðq; 4Þ ¼ lYðq; 4Þ

l0

(35)

Equation (34) is the one-dimensional Schroedinger eigenvalue equation for the electron in the spherical effective potential: l Z (36) Veff ðrÞ ¼ 2  2r r

3.3 Separation of variables

93

resulting from the balancing between the repulsive centrifugal potential l / 2r2 and the attractive Coulombic potential Z /r, while Eqn (35) is the eigenvalue equation for the square of the angular momentum operator L^2. If we write  2  v v 1 v2 þ þ cot q þ l Yðq; 4Þ ¼ 0 (37) vq sin2 q v42 vq2 Equation (37) further separates into two differential equations, one for each variable, by posing Yðq; 4Þ ¼ QðqÞFð4Þ In fact, we have

 F

 v2 v Q v2 F þ l Q ¼  þ cot q vq sin2 q v42 vq2

Multiplying both members by sin2 q=QF, we obtain   sin2 q v2 v 1 d2 F þ l Q ¼  þ cot q ¼ m2 Q vq F d42 vq2 where m2 is a second separation constant. Therefore, we finally obtain   d2 Q dQ m2 Q¼0 þ l 2 þ cot q dq sin q dq2 d2 F þ m2 F ¼ 0 d42

(38)

(39)

(40)

(41)

(42)

and the separation process is completed. In this way, the partial differential Eqn (1a) in three independent variables has been separated into three ordinary differential equations of the second order, one in each single variable.

3.3.4 The molecular one-electron system We shall limit ourselves to the case of the hydrogen molecular ion, H2þ , in which case ZA ¼ ZB ¼ Z ¼ 1. Equation (3) is separable in confocal spheroidal (elliptic) coordinates ðm; n; 4Þ (among others, Bates et al., 1953; Byers Brown and Steiner, 1966; Battezzati and Magnasco, 2003). In these coordinates rA ¼

R ðm þ nÞ; 2

rB ¼

R ðm  nÞ 2

(43)

the Laplacian V2 being given by V2 ¼

       v  4 v
2 v
m 2  n2 v2 2 v  1 m þ 1  n þ ðm2  1Þð1  n2 Þ v42 R2 ðm2  n2 Þ vm vm vn vn

(44)

94

CHAPTER 3 Differential equations in quantum mechanics

and the potential V by V¼

  2 1 1 4m þ ¼ R mþn mn Rðm2  n2 Þ

(45)

Equation (3) may be rewritten as V2 j þ 2ðE  VÞj ¼ 0

(46)

jðm; n; 4Þ ¼ MðmÞNðnÞFð4Þ

(47)

      vM  4 1 v
2 v
2 vN m NF þ 1  n MF  1 R2 m2  n2 vm vm vn vn     m 2  n2 v2 F 8m þ 2E MNF ¼ 0 MN þ þ ðm2  1Þð1  n2 Þ v42 Rðm2  n2 Þ

(48)

and is separated by posing

We obtain

R2 2 ðm  n2 Þ; we have 4        vN  vM v
2 v
m 1 þ MF 1  n2 NF vm vm vn vn   2   2 2 2  m n v F R E
2 2 þ 2Rm þ m  n MNF ¼ 0 þ MN ðm2  1Þð1  n2 Þ v42 2

Multiplying through by

Multiplying through by


(49)

ðm2  1Þð1  n2 Þ m 2  n2


      vM  m2  1 1  n2 v
2 v
2 vN NF  1 m þ MF 1  n vm vm vn vn m 2  n2   2 2  R E
2 v F þ 2Rm þ m  n2 MNF ¼ MN 2 2 v4

(50)

Dividing through by MNF



       vM  m2  1 1  n2 1 v
2 1 v
2 vN þ m 1 1n m 2  n2 M vm vm N vn vn   2 2  R E
2 1d F m  n2 ¼ þ 2Rm þ ¼ L2 2 F d42

(51)

3.3 Separation of variables

95

and the equation for Fð4Þ is separated off by taking L2 as a first separation constant. Then, we are left with d2 F þ L2 F ¼ 0 (52) d42        vN   vM 1 v
2 1 v
R2 E
2 m2  n2 ¼0 m 1 þ 1  n2 þ 2Rm þ m  n2  L 2 2 ðm  1Þð1  n2 Þ M vm vm N vn vn 2 (53) Since



m2  n2 ¼ m2  1 þ 1  n2

we can write for the last equation      vM  1 v
2 1 v
2 vN þ m 1 1n M vm vm N vn vn 

2   2 2


 
R E 2 2 2 m 1 þ 1n ¼0 m 1 þ 1n L þ 2Rm þ 2 ðm2  1Þð1  n2 Þ

(54)

(55)

      vM 1 v
2 1 v
2 vN m 1 þ 1n M vm vm N vn vn     R2 E
R2 E
2 L2 L2 þ 2Rm þ  m 1 þ 1  n2  2 ¼0 m  1 1  n2 2 2

(56)

    2  vM  1 v
2 R E
2 L2 þ m 1 m  1 þ 2Rm  2 m 1 M vm vm 2     2

  1 v vN R E L2 ¼ ¼C 1  n2  1  n2  1  n2 N vn vn 2

(57)

where C is a second separation constant. Then    2   dM  1 d
2 R E
2 L2 þ ¼0 m 1 m  1 þ 2Rm  C  2 M dm dm 2 m 1     2   1 d
R E
L2 2 dN 2 ¼0 1n þ 1n þC 1  n2 N dn dn 2

(58)

(59)

By putting R2 p2 ¼  E > 0 2 R 1þs¼ p

(60) (61)

96

CHAPTER 3 Differential equations in quantum mechanics

where E < 0 for bound states, we obtain the two separate second-order differential equations:      dMðmÞ 
d
2 L2 MðmÞ ¼ 0 m 1 þ  p2 m2  1 þ 2pð1 þ sÞm  C  2 m 1 dm dm      dNðnÞ
 d
L2 NðnÞ ¼ 0 1  n2 þ  p 2 1  n2 þ C  1  n2 dn dn

(62)

(63)

For Ls0, m ¼ 1 and jnj ¼ 1 are singular points of the differential equations.

3.3.5 The hydrogen atom in a uniform electric field This problem was studied by Schroedinger (1926a) in the third part of his series of papers on ‘Quantisierung als Eigenwertproblem’.3 We follow here, in part, Schroedinger presentation, putting l1 ¼ x and l2 ¼ h. See also Condon and Shortley (1963, pp. 398 ff) and Alexander (1969). The Schroedinger eigenvalue equation for the hydrogen atom in a uniform electric field of strength F directed along z is separable in the parabolic coordinates ðx; h; 4Þ introduced in Chapter 2. The purely perturbative approach will be followed instead in Section 3.8. We remind from Chapter 2 that pffiffiffiffiffi pffiffiffiffiffi 1 (64) x ¼ xh cos 4; y ¼ xh sin 4; z ¼ ðx  hÞ 2 so that x2 þ y2 ¼ xh r 2 ¼ x2 þ y 2 þ z2 ¼ x ¼ r þ z; and

1 ðx þ hÞ2 ; 4

h ¼ r  z;

(65) 1 r ¼ ðx þ hÞ 2 1 2 ¼ r xþh

       2 4 v v v v 1 1 1 v x þ h þ þ V ¼ x þ h vx vx vh vh 4 x h v42 2

(66)

(67)

(68)

The Schroedinger eigenvalue equation for the ground-state H-atom in the uniform electric field Fz directed along z is then written in atomic units as


2 (69) V þ 2 E þ r1  Fz j ¼ 0           4 v v v v 1 1 1 v2 2 xh þ  F j þ 2 E þ x þ h þ j¼0 x þ h vx vx vh vh 4 x h v42 xþh 2 3

II. Anwendung auf den Starkeffekt pp. 457–490.

(70)

3.3 Separation of variables

4 1 xþh           v v v v 1 1 1 v2 j E F
2 2 þ x þ h jþ þ ðx þ hÞ  x  h þ 1 j ¼ 0 vx vx vh vh 4 x h v42 2 4

97

and multiplying by

(71)

If we pose jðx; h; 4Þ ¼ MðxÞNðhÞFð4Þ substituting in Eqn (71), we have       v vM v vN MN 1 1 v2 F x þ MF h þ þ NF vx vx vh vh 4 x h v42    E F
þ ðx þ hÞ  x2  h2 þ 1 MNF ¼ 0 2 4

1 11 ðMNFÞ1, we obtain þ x h        1 1 1 1 v vM 1 v vN 4 þ x þ h x h M vx vx N vh vh  
 E F 1 d2 F ¼ m2 ðx þ hÞ  x2  h2 þ 1 ¼  þ 2 4 F d42

(72)

(73)

Multiplying through by 4

where m2 is a first separation constant. Then          m2 1 1 1 v vM 1 v vN E F
x þ h þ ðx þ hÞ  x2  h2  þ þ1 ¼0 M vx vx N vh vh 2 4 4 x h giving

where

    1 v vM F 2 E m2 1 x þ1 x þ  x þ x 4 M vx vx 4 2     1 v vN F 2 E m2 1 1þb h ¼ h  h þ h ¼ 4 N vh vh 4 2 2

(74)

(75)

(76)

1þb is a second separation constant, so finally giving the three separate differential equations 2 d2 Fð4Þ þ m2 Fð4Þ ¼ 0 (77) d42     d dMðxÞ F 2 E 1  b m2 1 MðxÞ ¼ 0 x x þ  x þ xþ  4 dx dx 4 2 2

(78)

    d dNðhÞ F 2 E 1 þ b m2 1 NðhÞ ¼ 0 h h þ h þ hþ  4 dh dh 4 2 2

(79)

98

CHAPTER 3 Differential equations in quantum mechanics

which are Schroedinger’s (1926a) Eqn (38) when expressed in atomic units, with n replaced by m, and 1b 1þb Eqns (1)–(3) in Alexander’s (1969) paper, provided we pose ¼ Z1 and ¼ Z2 . Equation 2 2 (77), concerning the cyclic variable 4 about the field axis, was already found for the one-electron atomic and molecular systems. In the zero-field case, F ¼ 0, and we obtain the Schroedinger eigenvalue equation for the free hydrogen atom in parabolic coordinates.

3.4 SOLUTION BY SERIES EXPANSION Using the usual primed notation for the derivatives, consider the ordinary differential equation of the second order for the unknown function y(x): pðxÞy00 ðxÞ þ qðxÞy0 ðxÞ þ rðxÞyðxÞ ¼ 0 (80) where x is the independent variable, p(x), q(x), r(x) are known coefficient functions, in the first instance assumed to be finite, one-valued and having derivatives of all orders in the interval (a, b) of definition of the variable x including the extrema. We try a solution of Eqn (80) in the form of a series expansion in powers of the variable x: N X ak x k (81) yðxÞ ¼ k¼0

Taking the first and second derivative of Eqn (81), we have N X k ak xk1 y0 ðxÞ ¼

(82)

k¼1

y00 ðxÞ ¼

N X

kðk  1Þak xk2

(83)

k¼2

and, substituting into Eqn (80) N N N X X X pðxÞ kðk  1Þak xk2 þ qðxÞ kak xk1 þ rðxÞ ak xk ¼ 0 k¼2

k¼1

(84)

k¼0

Since expansion (84) must be true for any value of the independent variable x, the coefficients of all powers of x must be identically zero. Putting equal to zero the coefficient of power xk, we have ðk þ 1Þðk þ 2ÞpðxÞakþ2 þ ðk þ 1ÞqðxÞakþ1 þ rðxÞak ¼ 0

(85)

and we obtain for the coefficients the three-term recurrence formula akþ2 ¼ 

1 qðxÞ 1 rðxÞ akþ1  ak k þ 2 pðxÞ ðk þ 1Þðk þ 2Þ pðxÞ

k ¼ 0; 1; 2; 3; /

(86)

namely, 1 qðxÞ 1 rðxÞ ak1  ak2 ak ¼  k pðxÞ ðk  1Þk pðxÞ

k ¼ 2; 3; 4; /

(87)

3.4 Solution by series expansion

99

which expresses the k-th coefficient in terms of the previous two, provided they are different from zero. Three-term recurrence formulae are difficult to manage with, but, fortunately, the regularity conditions which must be imposed on the functions because they become physically acceptable in most cases yield two-term recurrences associated to polynomial forms of the solutions, as we shall see later in this chapter. As an example, consider the second-order differential equation y00  a2 y ¼ 0

(88)

with the variable x ranging from 0 to N, and a a real constant, namely, Eqn (80) with pðxÞ ¼ 1;

qðxÞ ¼ 0;

rðxÞ ¼ a2

(89)

First, we change variable to u¼ax

(90)

d d du d ¼ ¼a dx du dx du

(91)

   2 2 2 d2 d d du du du d 2 d ¼ ¼ a ¼ dx2 du du dx dx du2 dx du2

(92)

obtaining

giving the transformed differential equation in the new variable u: y00 ðuÞ  yðuÞ ¼ 0

(93)

Using expansions (81)–(83), we have N N X X kðk  1Þak uk2  ak uk ¼ 0 k¼2

(94)

k¼0

and equating to zero the coefficient of the power uk akþ2 ¼

1 ak ðk þ 1Þðk þ 2Þ

k ¼ 0; 1; 2; 3; /

(95)

namely, ak ¼

1 ak2 kðk  1Þ

k ¼ 2; 3; 4; /

(96)

Putting a 0 ¼ a1 ¼ 1

(97)

we find a2 ¼

1 1 a0 ¼ ; 2$1 2!

a3 ¼

1 1 a1 ¼ ; 3$2 3!

a4 ¼

1 1 1 a2 ¼ a0 ¼ ; 4$3 4$3$2$1 4!

/

(98)

100

CHAPTER 3 Differential equations in quantum mechanics

and we obtain the particular solution y¼1þuþ

u2 u3 u4 þ þ þ / ¼ expðuÞ ¼ expðaxÞ 2! 3! 4!

(99)

which satisfies Eqn (93). A regular solution in the interval xð0; NÞ requires a < 0. In the following, we apply these considerations to a few cases derived from simple physical systems, where some difficulties will appear in the extrema of the interval of definition of the variable.

3.5 SOLUTION NEAR SINGULAR POINTS The problem of the singular points occurring in the solution of the differential equations of quantum mechanics is usually approached in terms of the indicial equations arising from the series expansion of the functions in terms of the independent variable (Margenau and Murphy, 1957; Ince, 1963; Rossetti, 1984). The form of the solution in the neighbourhood of a singularity is, in fact, different from that of a solution appropriate to an ordinary point, and, instead of initiating the expansion of the function y(x) with a constant term, as we did in Eqn (81), it is used in the expansion yðxÞ ¼

N X

ak xkþa

(100)

k¼0

with a a constant to be determined. To further clarify the point, we follow here the presentation given by Ince (1963). If (x  x0) is a factor of p(x) but not of q(x) and r(x), (x  x0) is said to be a singular point of the differential equation. If it is possible to reduce the differential Eqn (80) to the form ðx  x0 Þ2 p1 ðxÞy00 ðxÞ þ ðx  x0 Þq1 ðxÞy0 ðxÞ þ r1 ðxÞyðxÞ ¼ 0

(101)

p1 ðx0 Þs0; and p1 ðxÞ; q1 ðxÞ; r1 ðxÞ finite for x ¼ x0

(102)

with

then we can write yðxÞ ¼ ðx  x0 Þa f ðxÞ ¼ ðx  x0 Þa

N X

ak ðx  x0 Þk

(103)

k¼0

where (x  x0) is now a regular singular point, and f(x) can be expanded as a Taylor series in (x  x0). To determine a, we calculate the first and second derivatives of y(x), substitute into the differential equation, and equating to zero the coefficient of (x  x0)a (k ¼ 0), we are left with the so-called indicial equation: aða  1Þp1 ðxÞ þ aq1 ðxÞ þ r1 ðxÞ ¼ 0

(104)

which for x ¼ x0 gives the quadratic equation in a p1 ðx0 Þa2  ½ p1 ðx0 Þ  q1 ðx0 Þa þ r1 ðx0 Þ ¼ 0 whose roots give the two possible values for a.

(105)

3.6 The one-dimensional harmonic oscillator

101

This procedure is entirely general but is sometimes quite involved, as it is, for instance, in the case of the differential Eqn (41) in the q-variable arising in the quantum mechanical Schroedinger eigenvalue equation for the atomic one-electron system. In this case, a simpler approach is to study, first, the asymptotic form that the differential equations assume at the singularities, and, next, solve the differential equations in these asymptotic regions, the general regular solutions in the whole interval being then found by replacing integration constants by functions of the variables and finding the differential equations determining these functions. The resulting differential equations are now free from singularities and can be solved by the usual series expansion (81). We apply this technique, first, to the case of the one-dimensional harmonic oscillator, and, next, to the one-electron atomic and molecular systems (Magnasco, 2007, 2009, 2010b). In all cases where an analytic solution is possible, we check this solution by direct substitution into the original differential equation so as to verify that the equation is satisfied. These results are given in detail in Appendix 3.11.

3.6 THE ONE-DIMENSIONAL HARMONIC OSCILLATOR The Schroedinger eigenvalue equation for the one-dimensional harmonic oscillator is ^ HjðxÞ ¼ E jðxÞ

(106)

where 1 kx2 H^ ¼  V2 þ 2 2

NxN

is the Hamiltonian operator in atomic units and k is the force constant. Equation (106) can be written as
 j00 ðxÞ ¼ kx2  2E jðxÞ If we pose

pffiffiffi k ¼ b;

2E ¼ ð2n þ 1Þb

(107)

(108)

(109)

with n a positive integer to be determined, and change variable to z¼

pffiffiffi bx;

dz pffiffiffi ¼ b; dx

d2 z ¼0 dx2

(110)

so that d d dz pffiffiffi d ¼ ¼ b dx dz dx dz

(111)

      2   d2 d d dz dz dz dz d d d dz d2 ¼ ¼ b ¼ þ dx2 dz dz dx dx dz2 dx dx dz2 dz dz dx

(112)

102

CHAPTER 3 Differential equations in quantum mechanics

we obtain the transformed Schroedinger equation in the new variable z:


  j00 ðzÞ ¼ z2  1  2n jðzÞ jzj  N

(113)

For jzj/N we can neglect the last two terms in Eqn (113), so that the asymptotic differential equation is

which is satisfied by

j00 ðzÞ z z2 jðzÞ

(114)


jðzÞ z exp z2 =2

(115)

where only the regular solution has been retained. In fact j0 ¼ zj;


 j00 ¼ ðz j0  jÞ ¼ z2  1 j z z2 j

for jzj / N

So, the regular solution of the asymptotic Eqn (114) will be 
jðzÞ ¼ A exp z2 =2

(116)

(117)

where A is an integration constant. We now replace the integration constant by a function of the independent variable z, say H(z), so that the general solution will be
 jðzÞ ¼ exp z2 =2 HðzÞ (118) Calculating the first and second derivative of jðzÞ, we have
 j0 ðzÞ ¼ exp z2 =2 ðH 0  zHÞ



  j00 ðzÞ ¼ exp z2 =2 H 00  2zH 0 þ z2  1 H

(119) (120)

and, substituting in Eqn (113), we find the differential equation of the second order determining the unknown function H(z) H 00 ðzÞ  2zH 0 ðzÞ þ 2nHðzÞ ¼ 0

(121)

which, for n  0 integer, is the well-known Hermite differential equation (Sneddon, 1956), which we now solve by the usual series expansion HðzÞ ¼

N X

ak z k ;

k¼0

H 0 ðzÞ ¼

N X

k ak zk1 ;

H 00 ðzÞ ¼

k¼1

N X

kðk  1Þak zk2

(122)

k¼2

We find N X k¼2

kðk  1Þak zk2  2

N X k¼1

kak zk þ 2n

N X

kak zk ¼ 0

(123)

k¼0

giving as coefficient of zk ðk þ 2Þðk þ 1Þakþ2  2kak þ 2nak ¼ 0

(124)

3.6 The one-dimensional harmonic oscillator

103

So, we obtain the two-term recurrence formula for the coefficients: akþ2 ¼

2k  2n ak ðk þ 1Þðk þ 2Þ

k ¼ 0; 1; 2; 3; /

(125)

We now study the convergence of the series expansion for H(z) with increasing k. The ratio test gives for the coefficients

n 2k 1  akþ2 2k  2n 2 k  ¼ lim ¼ lim  (126) ¼ lim lim 1 2 k/N ak k/N ðk þ 1Þðk þ 2Þ k/N 2 k/N k 1þ k 1þ k k showing that it has the same limit as that of the function exp (2z), since expð2zÞ ¼ 1 þ 2z þ

ð2zÞ2 ð2zÞ3 ð2zÞk ð2zÞkþ1 þ þ/þ þ þ/ 2! 3! k! ðk þ 1Þ!

(127)

In fact, the limit of the ratio between two successive coefficients of this series gives lim

k/N

2kþ1 k! 2 2 ¼ lim ¼ lim ðk þ 1Þ! 2k k/N k þ 1 k/N k

(128)

The study of the convergence of the power series (122) shows therefore that it converges to the function exp (2z), a solution physically not acceptable for us since in this case H(z) would diverge at z ¼ N. So, the regularity conditions on the function H(z) require that expansion (122) should be truncated to a polynomial. This can be achieved if, for ak s0;

akþ2 ¼ akþ4 ¼ / ¼ 0

which, in turn, implies from Eqn (125) the necessary condition 2k  2n ¼ 0 0 kmax ¼ n

(129)

(130)

The physically acceptable solutions for the one-dimensional oscillator must hence include a polynomial of degree n at most. So, the physically acceptable solutions of the differential Eqn (121) will be given by even or odd polynomials of degree n: Hn ðzÞ ¼

n=2 X

a2k z2k þ

k¼0

ðn1Þ=2 X

a2kþ1 z2kþ1

(131)

k¼0

These polynomials differ by a constant, irrelevant from the standpoint of the differential Eqn (121), from the conventional Hermite polynomials given by Sneddon (1956). It is shown there that the proportionality constant is n! for n ¼ even (132) a0 ¼ ð1Þn=2 ðn=2Þ! and a1 ¼ ð1Þðn1Þ=2

2n! ½ðn  1Þ=2!

for n ¼ odd

(133)

104

CHAPTER 3 Differential equations in quantum mechanics

So, the final regular eigensolutions of the Schroedinger eigenvalue Eqn (113) for the onedimensional harmonic oscillator will be given by the Hermite functions:
 (134) jn ðzÞ ¼ exp z2 =2 Hn ðzÞ with the energy eigenvalues

  1 En ¼ n þ b 2

n ¼ 0; 1; 2; 3; /

(135)

The functions in Eqn (134) are orthogonal but not normalized, the normalization factor being found from the integral (Sneddon, 1956): ZN pffiffiffiffi pffiffiffiffi dzJn ðzÞJm ðzÞ ¼ 2n n! p dnm 0 Nn ¼ ð2n n! pÞ1=2 (136) N

As an example, we give the values of the first few coefficients derived from the recursion formula (125). 8 nðn  2Þ nðn  2Þðn  4Þ > k ¼ 2 a4 ¼ k ¼ 4 a6 ¼  a0 ; a0 < k ¼ 0 a2 ¼ n a0 ; 6 90 > 1n ð3  nÞð1  nÞ ð5  nÞð3  nÞð1  nÞ : a1 ; k ¼ 3 a 5 ¼ a 1 ; k ¼ 5 a7 ¼ a1 k ¼ 1 a3 ¼ 3 30 630 (137) Then we calculate two examples of even, H6, and odd, H7, polynomials, also giving the proportionality factors yielding correspondence with the Hermite polynomials. We have n n¼6 ¼ 3 k ¼ 0; 1; 2; 3 2   8 6 (138) H6 ðzÞ ¼ a0 þ a1 z þ a2 z2 þ a3 z3 ¼ 1  6z2 þ z4  z a0 15 ¼ 64z6  480z4 þ 720z2  120 n¼7

n1 ¼3 2

a0 ¼ 120

k ¼ 0; 1; 2; 3

H7 ðzÞ ¼ a1 z þ a3 z3 þ a5 z5 þ a7 z7 ¼

  4 8 7 z a1 z  2z3 þ z5  5 105

¼ 128z7  1344z5 þ 3360z3  1680z

(139)

a1 ¼ 1680

which coincide with the explicit expressions of the Hermite polynomials given in Sneddon (1956). More about Hermite polynomials and Hermite functions will be said in Chapter 4.

3.7 THE ATOMIC ONE-ELECTRON SYSTEM The Schroedinger eigenvalue equation for a single electron in the field of a nucleus of charge Z was introduced in Section 3.3.3, where it was shown how the three-dimensional partial differential

3.7 The atomic one-electron system

105

Eqn (30) in spherical coordinates was separated into the three ordinary differential equations, Eqn (34) for the radial variable r, Eqn (41) for the angular variable q, and Eqn (42) for the cyclic variable 4. We first consider the solution of the radial Eqn (34).

3.7.1 Solution of the radial equation The radial Eqn (34) is an ordinary differential equation of the second order which has different solutions according to the value assumed by the parameter E. We are interested here in the case E < 0, an electron bound to its nucleus of charge Z. Using the primed notation for the derivatives and assuming l ¼ l(l þ 1), with l a non-negative integer, we rewrite Eqn (34) as  


(140) R00 ðrÞ þ 2r 1 R0 ðrÞ þ 2 E þ Zr 1  lðl þ 1Þr 2 RðrÞ ¼ 0 We now introduce the new function P(r) through the relation RðrÞ ¼ PðrÞr1

(141)

with the derivatives R0 ¼ r 1 P0  r 2 P;

R00 ¼ r 1 P00  2r 2 P0 þ 2r 3 P

(142)

Substituting into Eqn (140), we obtain


  r 1 P00  2r 2 P0 þ 2r 3 P00 þ 2r 2 P0  2r 3 P00 þ r 1 2 E þ Zr 1  lðl þ 1Þr 2 P ¼ 0 namely,


  P00 ðrÞ þ 2 E þ Zr1  lðl þ 1Þr 2 PðrÞ ¼ 0

(143)

(144)

a differential equation where the first derivative is absent. In the scaled variable x¼

Z r n

(145)

if we pose Z2 E¼ 2 ¼ NN



Z2p

Z2p d4 expðim4Þexpðim4Þ ¼ N

d4 ¼ 2pN 2 ¼ 1

2

0

(177)

0

1 N ¼ pffiffiffiffiffiffi 2p

(178)

The regularity condition imposed on F(4) requires that6 Fð4 þ 2pÞ ¼ Fð4Þ

(179)

exp½imð4 þ 2pÞ ¼ expðim4Þ

(180)

since expði2pmÞ ¼ 1;

cos 2pm þ i sin 2pm ¼ 1

(181)

cos 2pm ¼ 1 0 m ¼ 0; 1; 2; / integer

(182)

by the Euler’s formula. Therefore

Hence, the normalized solution of Eqn (42) will be 1 Fm ð4Þ ¼ pffiffiffiffiffiffi expðim4Þ m ¼ 0; 1; 2; / 2p

(183)

F depends on the quantum number m which can take both positive and negative integer values, including zero.

3.7.3 Solution of the Q-equation If in Eqn (41) we change variable to x ¼ cos q 6

Remember that 4 is a cyclic variable.

1  x  1

(184)

110

CHAPTER 3 Differential equations in quantum mechanics

we have
1=2 d d d dx ¼ ¼  1  x2 dq dx dq dx      2 2  8 d2  d d dx dx dx d d d
dx 2 1=2 > ¼  ¼ 1  x > 2 < dq2 dx dx dq dq dq dx dx dx dq   > >

 d2
   2
d d : 2 1=2 1 2 1=2 2 d 1  x ¼ 1  x þ 1  x ð2xÞ x ¼ 1  x2 2 2 dx dx 2 dx dx

(185)

(186)

Hence, Eqn (41) becomes


   

 1=2 1=2 d m2  1  x2 Q¼0 QðxÞ þ l  2 1  x2 Q00 ðxÞ  xQ0 ðxÞ þ x 1  x2 dx sin q

finally giving the transformed differential equation in the new variable x  
 m2 QðxÞ ¼ 0 1  x2 Q00 ðxÞ  2x Q0 ðxÞ þ l  1  x2

(187)

(188)

We notice that x ¼ 1 are regular singular points of the differential equation. The approach leading to the indicial equation is useless if used directly, unless we do first a change of the independent variable x. This change is given, for instance, in Eyring et al. (1944), and consists in translating the variable x to the new variable u given by u ¼ x  1. The approach is tedious, however, and can be appropriately replaced by the study of the behaviour of the solution QðxÞ at the extrema of definition of the variable x, just as we did before for the radial equation. The behaviour of the function QðxÞ at jxj ¼ 1 can be obtained by considering the asymptotic form assumed by the differential Eqn (188), namely,


 m2 QðxÞ 1  x2 Q00 ðxÞ  2xQ0 ðxÞ z 1  x2

(189)

We easily see that the two solutions are ðm ¼ jmj > 0Þ: m=2
QðxÞ ¼ 1  x2 m=2
QðxÞ ¼ 1  x2

(190) (191)

Taking the first and second derivative of QðxÞ ¼ ð1  x2 Þm=2 :
m1 1 m
Q 1  x2 2 ð2xÞ ¼ mx 1  x2 2


1 2 1 0 Q00 ðxÞ ¼ m 1  x2 Q þ mx 1  x2 ð2xÞQ  mx 1  x2 Q


 1 2 2 ¼ m 1  x2 Q  2mx2 1  x2 Q þ m2 x2 1  x2 Q 

2 m2 x2  mx2  m Q ¼ 1  x2 Q0 ðxÞ ¼

(192)

(193)

3.7 The atomic one-electron system

111

so that



  1
2 2 1  x2 Q00 ðxÞ ¼ 1  x2 m x  mx2  m Q
1 Q 2xQ0 ðxÞ ¼ 2mx2 1  x2

(194) (195)

Substituting in Eqn (189) and multiplying both members by ð1  x2 ÞQ1, we find m2 x2 þ mx2  m z m2

(196)

which is true for jxj ¼ 1. Next, take the first and second derivatives of QðxÞ ¼ ð1  x2 Þm=2 : Q0 ðxÞ ¼ 


m  1 1 m
Q ð2xÞ ¼ mx 1  x2 1  x2 2 2




1 2 1 0 Q  mx 1  x2 ð2xÞQ þ mx 1  x2 Q Q00 ðxÞ ¼ m 1  x2 


2 1 2 2 2 2 2 2 2 ¼m 1x Q þ 2mx 1  x Qþm x 1x Q  

2 m2 x2 þ mx2 þ m Q ¼ 1  x2

(197)

(198)

so that


 1
2 2
 m x þ mx2 þ m Q 1  x2 Q00 ðxÞ ¼ 1  x2
1 Q 2xQ0 ðxÞ ¼ 2mx2 1  x2

(199) (200)

Substituting in Eqn (189) and multiplying both members by ð1  x2 ÞQ1, we find m2 x2  mx2 þ m z m2 which is true for jxj ¼ 1. So, the solution of Eqn (189) will be a linear combination of the two solutions:

m=2 m=2 QðxÞ ¼ A 1  x2 þ B 1  x2

(201)

(202)

where A, B are integration constants and B ¼ 0 for the regularity condition. Therefore, Eqn (190) is our asymptotic solution for jxj ¼ 1, and, as we did before for the radial solution, we look for a solution of the type
m=2 QðxÞ ¼ 1  x2 GðxÞ (203) where G(x) is a function to be determined under the regularity conditions. Taking the first and second derivative of Q(x), we have Q0 ðxÞ ¼


m  1 m=2 0
m=2 h 0 1 i
m
ð2xÞG þ 1  x2 1  x2 2 G ¼ 1  x2 G (204) G  mx 1  x2 2

112

CHAPTER 3 Differential equations in quantum mechanics

h m  1 1 i
m=2
m
ð2xÞ G0  mx 1  x2 1  x2 2 Q00 ðxÞ ¼ G þ 1  x2 2 h
i n 1 2 1 0 o

G  mx  1  x2 ð2xÞ G  mx 1  x2 G G00  m 1  x2 h

m=2 00 1 0
2 2 2 i
G þ m x þ mx2  m 1  x2 G ¼ 1  x2 G  2mx 1  x2

(205)

Substituting in Eqn (188) and dividing through by ð1  x2 Þm=2 we obtain




 1 1  x2 G00 ðxÞ  2mxG0 ðxÞ þ m2 x2 þ mx2  m 1  x2 GðxÞ from Q00
1 GðxÞ from Q0 2xG0 ðxÞ þ 2mx2 1  x2 h i
1 l  m2 1  x2 GðxÞ from Q

Adding all terms altogether, we finally obtain 
1  x2 G00 ðxÞ  2ðm þ 1ÞxG0 ðxÞ þ ½l  mðm þ 1ÞGðxÞ ¼ 0

(206) (207) (208)

(209)

which is the required differential equation determining the function G(x). Equation (209) is free from singularities, and can be solved by the usual series expansion in powers of the variable x of Section 3.4. Using the ordinary series expansion for the function G(x) GðxÞ ¼

N X

ak x k ;

G0 ðxÞ ¼

k¼0

N X

G00 ðxÞ ¼

k ak xk1 ;

k¼1

N X

kðk  1Þak xk2

(210)

k¼2

substituting into Eqn (209), we obtain N X k¼2

kðk  1Þak xk2 

N X

kðk  1Þak xk  2ðm þ 1Þ

k¼2

N X

kak xk þ ½l  mðþ1Þ

k¼1

N X

ak x k ¼ 0

(211)

k¼0

thereby giving for the coefficient of xk ðk þ 1Þðk þ 2Þakþ2  ðk  1Þkak  2ðm þ 1Þkak þ ½l  mðþ1Þak ¼ 0

(212)

In this way, we obtain the two-term recurrence formula for the coefficients: akþ2 ¼

ðk þ mÞðk þ m þ 1Þ  l ak ðk þ 1Þðk þ 2Þ

k ¼ 0; 1; 2; /

(213)

We see that we obtain an even (k ¼ 0, 2, 4,.) and an odd series (k ¼ 1, 3, 5,.), as we already obtained for the harmonic oscillator. We must now study the convergence of the series (210) when k/N. The ratio test shows that this series has the same asymptotic behaviour of the geometrical series of reason x2: N X xk ¼ lim Sn (214) S¼ k¼0

n/N

3.7 The atomic one-electron system

113

where Sn is the n-th reduced sum Sn ¼ 1 þ x þ x2 þ x3 þ / þ xn1 ¼

1  xn 1x

(215)

Now, lim Sn exists if jxj¼ The first-order Stark shift is hence given by

which is Schroedinger’s final result. In this equation, n is the principal quantum number, which is related to the parabolic quantum numbers by the relation (see Footnote 11) n ¼ k1 þ k2 þ m þ 1

(364)

Equation (363) shows that no Stark shift can occur if k2 ¼ k, and those states for which k1 > k2 have the lowest energy. Higher order terms in the Stark effect for hydrogen are discussed in detail in Condon and Shortley (1963). The theoretically calculated expression for the second-order Stark shifts for hydrogen was found to be (Wentzel, 1926; Waller, 1926)
 1 (365) E2 ¼  F 2 n4 17n2  3n2e  9m2 þ 19 16 where ne ¼ k1  k2 and for the third order (Doi, 1928; Alliluev and Malkin, 1974)12  3 3 7
F n ne 23n2  n2e þ 11m2 þ 39 E3 ¼ 32

(366)

(367)

Wentzel (1926) obtained expression (365) for the quadratic Stark effect using what became later known as the Wentzel–Kramers–Brillouin method,13 while Waller (1926) obtained the same result by a method where functions, energy eigenvalue and separation constants in parabolic coordinates were first expanded in powers of the field F, the equations obtained therefrom by equating to zero the coefficients of each power of F being then solved by successive approximations.14 Alliluev and 12

Condon and Shortley (1963) give a wrong value for the constant factor (71 instead of 39). See Section 1.4. 14 The first approximation gives the zero-field case; the second one the first-order Stark effect, linear in the field; the third one the second-order Stark effect, quadratic in the field. 13

3.11 Appendix: checking the solutions

131

Malkin (1974), using the dynamical symmetry group of the hydrogen atom, gave also explicitly the fourth-order term. Further higher order corrections (up to sixth order) were calculated by Nguyen et al. (1965) up to a paper by Silverstone (1978) who gave a perturbation theory of the Stark effect in hydrogen to arbitrarily high order. Silverstone derived a very compact formula for the N-th order energy correction in terms of an asymptotic expansion to the N-th order of separation constants and energy eigenvalues, which were shown to be polynomials of degree N þ 1 and N, respectively, in the parabolic quantum numbers k1, k2 and the magnetic quantum number m. These polynomial coefficients were tabulated up to the 17th order and reported explicitly in his paper up to the 10th order. Silverstone’s perturbative results for the energy of the hydrogen atom in the electric field were found to be in excellent agreement with the non-perturbative results by Alexander (1969).

3.11 APPENDIX: CHECKING THE SOLUTIONS In this appendix, we check in detail that the analytic forms of our polynomial solutions do satisfy their respective differential equations for the case of the H-atom in spherical and parabolic coordinates.

3.11.1 The radial equation of the H-atom in spherical coordinates We now check that our general solution (169) verifies the differential Eqn (147) in the scaled variable x. Calculating the first and second derivative of the function (169), we have nl1 X

 kak xk1 P0 ðxÞ ¼ 1 þ ðl þ 1Þx1 PðxÞ þ expðxÞxlþ1

(368)

k¼1



 P00 ðxÞ ¼ 1  ð2l þ 2Þx1 þ lðl þ 1Þx2 PðxÞ þ 2 þ ð2l þ 2Þx1  expðxÞxlþ1

nl1 X

kak xk1 þ expðxÞxlþ1

k¼1

nl1 X

kðk  1Þak xk2

(369)

k¼2



 x2 P00 ðxÞ ¼ x2  ð2l þ 2Þx þ lðl þ 1Þ PðxÞ þ 2x2 þ ð2l þ 2Þx  expðxÞxlþ1

nl1 X

kak xk1 þ x2 expðxÞxlþ1

k¼1

nl1 X

kðk  1Þak xk2

(370)

k¼2

Therefore, substituting into Eqn (147), dividing through by exp (x)xlþ1 and simplifying, we have x

nl1 X k¼2

kðk  1Þak xk2 þ ½ð2l þ 2Þ  2x

nl1 X k¼1

kak xk1 þ 2ðn  l  1Þ

nl1 X

ak x k ¼ 0

(371)

k¼0

which is Eqn (160) with the series for F(x) replaced by its polynomial of order (n  l  1) and its first and second derivatives.

132

CHAPTER 3 Differential equations in quantum mechanics

As a particular case, we now check that R51(x) does satisfy its differential equation. It will be sufficient to verify this condition for the function F51(x), which satisfies the differential equation 00 0 xF51 ðxÞ þ ð4  2xÞF51 ðxÞ þ 6F51 ðxÞ ¼ 0

with

3 3 1 3 x þ x2  x 2 5 15 3 6 1 F510 ðxÞ ¼  þ x  x2 2 5 5 6 2 F5100 ðxÞ ¼  x 5 5

F51 ðxÞ ¼ 1 

(372) (373) (374) (375)

Substituting into Eqn (372), we find       6 2 3 6 1 18 2 2 3 x  x2 þ ð4  2xÞ  þ x  x2 þ 6  9x þ x  x 5 5 2 5 5 5 5          6 24 2 4 12 18 2 2 3 2 3 2 þx þ 6  9x þ x  x ¼ 6 þ x þ þ 3 þ x    5 5 5 5 5 5 5 5     18 2 2 3 18 2 2 3 (376) x þ x þ 6  9x þ x  x ¼0 ¼ 6 þ 9x  5 5 5 5 and Eqn (372) is verified.

3.11.2 The Q-equation of the H-atom in spherical coordinates We check that our general solution (228) verifies the differential Eqn (209) in the variable x with l ¼ l (l þ 1). First, we do the calculation for the even polynomial. Calculating first and second derivatives of the function (228) for (l  m) ¼ even, we have X m1 m ðlmÞ=2
m
1  x2 2 ð2xÞQ þ 1  x2 2 ð2kÞa2k x2k1 Q0 ðxÞ ¼ 2 k¼1 (377) ðlmÞ=2

  P 1 m=2 ¼ mx 1  x2 Q þ 1  x2 ð2kÞa2k x2k1


Q00 ðxÞ ¼ m 1  x

 2 1

k¼1



2 1 0 Q þ mx 1  x2 ð2xÞQ  mx 1  x2 Q

ðlmÞ=2 X X
m m ðlmÞ=2 m
2 2 1 2k1 2 2 þ ð2xÞ ð2kÞa2k x þ 1x ð2kÞð2k  1Þa2k x2k2 1x 2 k¼1 k¼1 h
1
2
2 i
m=2 ¼ m 1  x2  2mx2 1  x2 þ m 2 x 2 1  x2 1  x2

X
1
m=2 ðlmÞ=2  2mx 1  x2 ð2kÞa2k x2k1 1  x2

1
þ 1  x2 1  x 2 1x

k¼1 ðlmÞ=2 X  2 m=2

ð2kÞð2k  1Þa2k x2k2

k¼1

(378)

3.11 Appendix: checking the solutions

133

namely,


00

Q ðxÞ ¼ 1  x


2 m=2

1x

"
 1 2




m x  mx  m 1  x 2 2

2

 2 1

ðlmÞ=2 X

a2k x2k  2mx

k¼0



ðlmÞ=2 X

X  ðlmÞ=2 ð2kÞa2k x2k1 þ 1  x2 ð2kÞð2k  1Þa2k x2k2


k¼1



00




1  x Q ðxÞ ¼ 1  x 2

(379)

k¼1

Hence


#

 2 m=2

"




X 
1 ðlmÞ=2 a2k x2k  2mx m2 x2  mx2  m 1  x2 k¼0



ðlmÞ=2 X

ð2kÞa2k x

2k1




þ 1x

k¼1

2



ðlmÞ=2 X

ð2kÞð2k  1Þa2k x

# 2k2

" # ðlmÞ=2 X X

 ðlmÞ=2 2 m=2 2 2 1 2k 2k1 a2k x  2x ð2kÞa2k x 2mx 1  x 2xQ ¼ 1  x 0

k¼0




1  x2

m=2 h

(380)

k¼1

(381)

k¼1

X
1 i ðlmÞ=2 a2k x2k lðl þ 1Þ  m2 1  x2

(382)

k¼0

Adding all terms altogether, we find h

1 i 1  x2 Q00 ðxÞ  2xQ0 ðxÞ þ lðl þ 1Þ  m2 1  x2 QðxÞ (  ðlmÞ=2 X 
m2 x2  mx2  m þ 2mx2  m2 2 m=2 þ lðl þ 1Þ a2k x2k ¼ 1x 1  x2 k¼0 ) ðlmÞ=2 X X
 ðlmÞ=2 2k1 2 2k2 ¼0 2ðm þ 1Þx ð2kÞa2k x þ 1x ð2kÞð2k  1Þa2k x k¼1

Noting that

(383)

k¼1


 m2 x2 þ mx2  m  m2 ¼ mðm þ 1Þ 1  x2

(384)

simplifying Eqn (383), we finally obtain


1  x2

X  ðlmÞ=2

ð2kÞð2k  1Þa2k x2k2  2ðm þ 1Þx

ðlmÞ=2 X

k¼1

þ½lðl þ 1Þ  mðm þ 1Þ

ð2kÞa2k x2k1

k¼1 ðlmÞ=2 X k¼0

a2k x2k ¼ 0

(385)

134

CHAPTER 3 Differential equations in quantum mechanics

which is Eqn (209) with the series G(x) replaced by its even polynomial of order ðl  mÞ and its first and second derivatives. We can proceed similarly for the odd polynomial, obtaining


1  x2

ðlm1Þ=2 X X  ðlm1Þ=2 ð2k þ 1Þð2kÞa2kþ1 x2k1  2ðm þ 1Þx ð2k þ 1Þa2kþ1 x2k k¼1

k¼1

þ½lðl þ 1Þ  mðm þ 1Þ

ðlm1Þ=2 X

a2kþ1 x2kþ1 ¼ 0

(386)

k¼0

which is Eqn (209) with the series G(x) replaced by its odd polynomial of order ðl  m Þ and its first and second derivatives. So, Eqn (228) is the complete regular solution of our differential Eqn (209). As two particular cases, we now check that Q53 ðxÞ and Q51 ðxÞ do satisfy their respective differential equations. First, take the function Q53 given by
3=2

3=2 G53 ðxÞ (387) Q53 ðxÞf 1  x2 1  9x2 ¼ 1  x2 For l¼5

m¼3

lm¼2

2ðm þ 1Þ ¼ 8 lðl þ 1Þ  mðm þ 1Þ ¼ 18

G53 must satisfy the differential equation 
1  x2 G0053 ðxÞ  8xG053 ðxÞ þ 18G53 ðxÞ ¼ 0

(388)

(389)

with G53 ðxÞf1  9x2

(390)

Calculating the first and second derivative of G53 ðxÞ, we have G053 ðxÞf  18x;

G0053 ðxÞf  18

(391)

so that
 

1  x2 G0053 ðxÞ  8xG053 ðxÞ þ 18G53 ðxÞ ¼ 18 1  x2  8xð18xÞ þ 18 1  9x2 ¼ 18 þ 18x2 þ 144x2 þ 18  162x2 ¼ 0 as it must be. As a further example, take the function Q51 given by 1=2

1=2
1  14x2 þ 21x4 ¼ 1  x2 G51 ðxÞ Q51 ðxÞf 1  x2

(392)

(393)

with G51 ðxÞf1  14x2 þ 21x4

(394)

For l¼5

m¼1

lm¼4

2ðm þ 2Þ ¼ 4 lðl þ 1Þ  mðm þ 1Þ ¼ 28

(395)

3.12 Problems 3

G51 must satisfy the differential equation
 1  x2 G0051 ðxÞ  4xG051 ðxÞ þ 28G51 ðxÞ ¼ 0

135

(396)

Taking the first and second derivative of G51(x), we have G051 f  28x þ 84x3 ; so that


G0051 f  28 þ 252x2

(397)

 1  x2 G0051 ðxÞ  4xG051 ðxÞ þ 28G51 ðxÞ



 ¼ 1  x2 28 þ 252x2  4x 28x þ 84x3 þ 28 1  14x2 þ 21x4


 ¼ 28 þ 252x2 þ 28x2  252x4 þ 112x2  336x4 þ 28  392x2 þ 588x4

 ¼ 28 þ 392x2  588x4 þ 28  392x2 þ 588x4 ¼ 0

(398)

and Eqn (396) is satisfied.

3.11.3 The x-equation of the H-atom in parabolic coordinates Lastly, we check that our general solution (331) verifies the differential Eqn (323). Taking the first and second derivative of (331), we have ( )   k1 k1 X 1 X 0 bþ1 k1 1 k kak x þ ðb þ 1Þx  ak x (399) U ðxÞ ¼ expðx=2Þx 2 k¼0 k¼1 ( 00

U ðxÞ ¼ expðx=2Þx

bþ1

k1 X

k1

X kðk  1Þak xk2 þ ð2b þ 2Þx1  1 kak xk1

k¼0

k¼1

)  k1 1 X 2 1 k þ bðb þ 1Þx  ðb þ 1Þx þ ak x 4 k¼0 

(400)

so that Eqn (323) becomes, after dividing through by exp (x/2)xb and simplifying k1 k1 k1 X X X kðk  1Þak xk2 þ ðm þ 1  xÞ kak xk1 þ k1 ak xk ¼ 0 x k¼2

k¼1

(401)

k¼0

which is Eqn (334) with the series for F(x) replaced by its polynomial Fk1 ðxÞ and its first and second derivatives.

3.12 PROBLEMS 3 3.1. Prove the trigonometrical identity tanða þ bÞ ¼

tan a þ tan b 1  tan a tan b

Hint: Use the elementary definition of the tangent and the addition formulae for the trigonometric functions.

136

CHAPTER 3 Differential equations in quantum mechanics

3.2. Separation of the motion of the centre of mass in the one-electron atomic system. Answer:   2 Z2 2 ^ ¼ WJ 0  Z V2X F ¼ EG F; V þ V j ¼ Ej HJ  2M 2m Hint: Change from the Cartesian coordinates of the two particles (nucleus þ electron) to a new system specifying the Cartesian coordinates of the centre of mass of the electron–nucleus system and the internal coordinates describing the motion of the electron with respect to the nucleus taken as origin of the new coordinate system, then substitute the product function JðX; xÞ ¼ FðX; Y; ZÞjðx; y; zÞ into the original Schroedinger equation. 3.3. Find the wavefunction for the free particle in one dimension. Answer: jðxÞ ¼ A expðiaxÞ þ B expðiaxÞ Hint: Solve the Schroedinger eigenvalue equation for a particle moving along a line x in a field-free space (V ¼ 0). 3.4. Find the wavefunction and energy of a particle in a one-dimensional box of side a with impenetrable walls. Answer:  1=2 p 2 jn ðxÞ ¼ C sin n x; C ¼ a a En ¼ n2

p2 2a2

n ¼ 1; 2; 3; /

Hint: Solve the Schroedinger eigenvalue equation using the boundary conditions for the problem. 3.5. Calculate the mean values and for a particle in a one-dimensional box of side a with impenetrable walls. Answer: a hxi ¼ hjjxjji ¼ 2   2  2   2  a 3 1 2 2 x ¼ jx j ¼ 3 2p n Hint: Use the wavefunction jn ¼ C sin ax with aa ¼ np (n ¼ non-zero integer) and the formulae for the corresponding definite integrals. 3.6. Find wavefunctions and energies for a particle in a one-dimensional box of finite height V0. Answer: We have different results according the relative values of E and V0. Hint: Solve the Schroedinger equation under the appropriate boundary conditions.

3.13 Solved problems

137

3.7. Evaluate the normalization integral for the H-atom in parabolic coordinates. Answer: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2k1 !k2 ! N¼ 4 n ½ðm þ k1 Þ!3 ½ðm þ k2 Þ!3 where n is the principal quantum number, k1, k2 the parabolic quantum numbers, and m the magnetic quantum number. Hint: Use the formulae given in Section 4.3.3. 3.8. Evaluate the z-integral for the H-atom in parabolic coordinates. Answer: hzi ¼ hjnkm jzjjnkm i ¼

3 nðk1  k2 Þ 2

Hint: Use the formulae given in Section 4.3.3.

3.13 SOLVED PROBLEMS 3.1. Using elementary trigonometric definitions and addition formulae we immediately find tanða þ bÞ ¼

sinða þ bÞ sin a cos b þ cos a sin b ¼ cosða þ bÞ cos a cos b  sin a sin b

sin a cos b þ cos a sin b tan a þ tan b cos a cos b ¼ ¼ cos a cos b  sin a sin b 1  tan a tan b cos a cos b which is the required trigonometric formula. 3.2. Separation of the motion of the centre of mass in the one-electron atomic system. The transformation for the Cartesian coordinates of the two-particle system is ðx1 ; y1 ; z1 ; x2 ; y2 ; z2 Þ / ðX; Y; Z; x; y; zÞ where the first set on the right-hand side (r.h.s.) denotes the Cartesian coordinates of the centre of mass, and the second gives the Cartesian coordinates of the electron with respect to the nucleus taken as origin of the second coordinate system (internal coordinates). Furthermore M ¼ m1 þ m2 ;

m¼

m1 m2 1 ¼ m2 z m2 m1 þ m2 1 þ m2 =m1

denote, respectively, the total mass (essentially that of the atomic nucleus) and the reduced mass (essentially the mass of the electron) of the system. For the x-component we have x ¼ x2  x1

MX ¼ m1 x1 þ m2 x2

138

CHAPTER 3 Differential equations in quantum mechanics

The inverse transformation is hence derived by solving the inhomogeneous system:     m1 m2  m1 x1 þ m2 x2 ¼ MX  ¼ m1 þ m2 ¼ M  D¼ 1 1  x1 þ x2 ¼ x where D is the determinant of the coefficients. Solutions are obtained by using Cramer’s rule:     1  MX m2  m2 1  m1 MX  m1 ¼X ¼Xþ x; x2 ¼  x x1 ¼    1 x M M M x M 1 Hence, the x-component of the kinetic energy in the new coordinate system will be T¼ ¼

m1 2 m2 2 m1 _ m2 2 m2 _ m1 2 X Xþ x_ þ x_ ¼ x_ þ x_ 2 1 2 2 2 M 2 M m1 þ m2 _ 2 m1 m2 ðm1 þ m2 Þ 2 M _ 2 m 2 x_ ¼ X þ X þ x_ 2 2M 2 2 2

the same being true for the remaining y- and z-components, so that the classical kinetic energy in the r.h.s. system of coordinates will be T¼

 M
_2 m
2 1 2 1 2 2 2 X þ Y_ þ Z_ þ x_ þ y_ 2 þ z_2 ¼ PX þ p 2 2 2M 2m

which is the kinetic energy of a particle of mass M placed in the centre of mass plus the kinetic energy of a particle of mass m in internal coordinates. Introducing the linear momentum operators

∇

∇

we obtain for the Hamiltonian operator of the two-particle nucleus þ electron system Z2 2 Z2 2 1 Ze2 H^ ¼  VX  V  2M 2m 4pε0 r where the last term is the Coulombic attraction of the electron by the nucleus in SI units. The corresponding Schroedinger equation ^ ¼ WJ HJ can then be separated by posing JðX; xÞ ¼ FðXÞjðxÞ In fact, we can write 

   Z2 2 Z2 2 V F þF  V þ V j ¼ WFj j  2M X 2m

3.13 Solved problems

139

Dividing both members through by Fj we obtain   Z2 2 Z2 2 V þV j  V F  2m 2M X ¼ W  ¼ EG j F Since the left-hand side of this equation depends only on (X, Y, Z), the r.h.s. only on (x, y, z), and the equality must be true for all values of the independent variables, the expression itself must be equal to a separation constant, say EG. Posing W ¼ EG þ E the Schroedinger equation separates into the two eigenvalue equations   Z2 2 Z2 2  VX F ¼ EG F; V þ V j ¼ Ej  2M 2m the first being the Schroedinger equation for a free particle of mass M describing the translational motion of the centre of mass, the second the Schroedinger equation for the motion of the electron with respect to the nucleus. In Sections 3.3.3 and 3.7 we solved the last eigenvalue equation, which, after introduction of the atomic units, becomes   1 2  V þ V j ¼ Ej 2 3.3. The free particle in one-dimension. Introducing the atomic units from the very beginning, the Schroedinger eigenvalue equation for a particle moving along a line x in a field-free space (V ¼ 0) is 

1 d2 j ¼ Ej 2 dx2

d2 j þ 2Ej ¼ 0 dx2 having the general integral in complex form jðxÞ ¼ A expðiaxÞ þ B expðiaxÞ where A, B are two integration constants. Evaluating the second derivative and substituting, we obtain the characteristic equation:15 pffiffiffiffiffiffi a2 þ 2E ¼ 0 0 a ¼ 2E Since there are no boundary conditions, except that the function j must be finite at x ¼ N, the quantity 2E must necessarily be real so that E must be positive.16 Changing the sign of a simply we theffiffiffiffiffiffiffiffiffiffi  signs in front of 2E in the equation. pffiffiffiffiffiffiffiffiffiffiffiffi pmeans ffiffiffiffiffiffiffiffiffi to interchange A with B. pSo, ffiffiffiffiffiffiffiffiffiffi ffi can omit p ffi For E < 0, a ¼ 2jEj ¼ i 2jEj, so that jðxÞ ¼ A expð 2jEjxÞ þ B expð 2jEjxÞ, the first term of which diverges at x ¼ N, the second at x ¼ N. The resulting wavefunction is hence oscillatory and, in real form, is a combination of sine and cosine functions. 15 16

140

CHAPTER 3 Differential equations in quantum mechanics

FIGURE 3.1 The three regions for the particle in the one-dimensional box with impenetrable walls

Since no further restrictions are imposed on E, we conclude that the energy has a continuous spectrum of eigenvalues, all positive values being allowed, and the energy is not quantized. Each component of j (the fundamental integrals) is separately eigenfunction of the linear d with eigenvalue a or a, respectively, the first describing momentum operator p^x ¼ i dx pffiffiffiffiffi pffiffiffiffiffiffi a particle moving along þx with a definite value a ¼ p2x ¼ 2E of the momentum and the second a particle moving along x with the same absolute value of the momentum. These quantities are the classical values of the momentum of a free particle having energy E ¼ p2x =2. Since the integral of jjj2 over all values of x between N and N is infinite, the function j cannot be normalized in the usual way. The problem of normalization of such a wavefunction is rather complicated and will not be further pursued here (Pauling and Wilson, 1935, and references therein; Kauzmann, 1957). Since the probability density of each component will be a constant independent of x, we have equal probabilities of finding the particle at any value dx. This means that the uncertainty in the position of the particle is infinite. This is in accord with Heisenberg’s principle since Dpx ¼ 0 so that DxDpx wh as it must be. 3.4. Particle in a one-dimensional box with impenetrable walls. With reference to Figure 3.1 we now consider the particle confined in the one-dimensional box of side a with impenetrable walls, which means that outside the box the potential is infinite, while in the box we assume V ¼ Vx ¼ 0. 1. Outside the box ðV ¼ NÞ In regions I and III, where V ¼ N, it must be jðxÞ ¼ 0 for all points, which means that the particle cannot be found outside the region 0  x  a. 2. Inside the box ðV ¼ 0Þ The Schroedinger equation is d2 j þ 2Ej ¼ 0 dx2

3.13 Solved problems

which has the solution

141

j ¼ A expðiaxÞ þ B expðiaxÞ

where A and B are integration constants. The value of the constant a is readily obtained since j0 ¼ ia½A expðiaxÞ  B expðiaxÞ; from which follows the characteristic equation a2 ¼ 2E 0 a ¼

j00 ¼ a2 j

pffiffiffiffiffiffi 2E

which is real since E > 0. We must now impose upon j the boundary conditions arising from the fact that we must join in a continuous way the solutions at the edge of the box, hence we must have jð0Þ ¼ jðaÞ ¼ 0. 1. First boundary condition jð0Þ ¼ 0 A þ B ¼ 0 0 B ¼ A jðxÞ ¼ A½expðiaxÞ  expðiaxÞ ¼ 2iA sin ax ¼ C sin ax where C is a normalization constant. 2. Second boundary condition jðaÞ ¼ 0 C sin aa ¼ 0

0 aa ¼ np n ¼ 1; 2; 3; /

so that the argument of the trigonometric function is quantized, giving  1=2 p 2 jn ðxÞ ¼ C sin n x; C ¼ a a The positive energy spectrum is now quantized according to En ¼

a2 p2 ¼ n2 2 2 2a

n ¼ 1; 2; 3; /

or, measuring the energy in units of ðp2 =2a2 Þ En ¼ n2 p2 =2a2 The energy levels and wavefunctions for the ground and the first two excited states of the particle in a box with impenetrable walls are given in Figure 3.2 for C ¼ 1. It is seen that the functions have (n  1) nodes, while only the functions having n ¼ even are zero at x ¼ a/2. All functions are normalized to unity and orthogonal to each other. In three dimensions, the corresponding results are jðx; y; zÞ ¼ Xnx ðxÞYny ðyÞZnz ðzÞ  1=2 8 p p p ¼ sin nx x$sin ny y$sin nz z abc a b c ! 2 p2 n2x ny n2z þ þ E ¼ E x þ Ey þ Ez ¼ 2 a2 b2 c2 These results are of great importance in the theory of the perfect gas.

142

CHAPTER 3 Differential equations in quantum mechanics

FIGURE 3.2 Energy levels (in units of p2 =2a2 ) and wavefunctions for the first three lowest states of the particle in a box with impenetrable walls (C ¼ 1)

3.5. Mean values hxi and hx2 i for a particle in a one-dimensional box of side a with impenetrable walls. The general indefinite integrals needed in this problem can be taken from Gradshteyn and Ryzhik’s (1980) tables: Z x2 x 1 dx sin2 x ¼  sin 2x  cos 2x 4 4 8 Z

  x3 x 1 2 1 x  sin 2x dx x sin x ¼  cos 2x  6 4 4 2 2

2

as can be easily verified by derivation of the integrand. We now make the change of variable: ax ¼ y;

y x¼ ; a

dy dx ¼ a

 x  y 0  0 a  aa

The definite integrals become  Zaa Za 1 1 y2 y 1 2 2  sin 2y  cos 2yjaa dxx sin ax ¼ 2 dyy sin y ¼ 2 0 a a 4 4 8 0

"

0

1 ðaaÞ2 ðaaÞ 1 1  sin 2aa  cos 2aa þ ¼ 2 4 a 4 8 8

#

3.13 Solved problems

143

and, for aa ¼ np Za

" # 1 ðaaÞ2 ðaaÞ 1 1  dx x sin ax ¼ 2 sin 2pn  cos 2pn þ 4 a 4 8 8 2

0

# " 1 ðaaÞ2 1 1 a2 ¼ 2  þ ¼ 4 4 a 8 8 Similarly Za

1 dx x sin ax ¼ 3 a 2

0

Zaa

2

dy y2 sin2 y 0

#   1 ðaaÞ3 ðaaÞ 1 2 1 ¼ 3  cos 2aa  ðaaÞ  sin 2aa 6 a 4 4 2 "

and, for aa ¼ np Za

" #   1 ðaaÞ3 aa a3 3  1 2 2 dx x sin ax ¼ 3 ¼ 6 6 a 4 2p n 2

0

2

Hence, we obtain Za < x >¼< jjxjj >¼ C

dx x sin2 ax ¼ C 2 $

2

a2 2 a2 a ¼ $ ¼ 4 a 4 2

0

Za < x2 >¼< jjx2 jj >¼ C 2

    a3 3 a2 3 1 2 2 ¼ 1 2 2 dx x2 sin2 ax ¼ C 2 $ 6 3 2p n 2p n

0

While the first result coincides with the classical average, the second differs from it by the second term in parenthesis, which becomes zero for high values of the quantum number n. The quantum corrections are 0.152, 0.038, 0.017,., 0.0015 for n ¼ 1, 2, 3,., 10, respectively. This is an example of the correspondence principle, according to which classical mechanics and quantum mechanics give the same result for systems in highly excited quantum states. 3.6. Wavefunctions and energies for a particle in a one-dimensional box of finite height V0. With reference to Figure 3.3, we now consider the case of a particle confined in a box of finite height V0, the potential being zero inside the box. The Schroedinger equation is d2 j ¼ 2ðV0  EÞj dx2

144

CHAPTER 3 Differential equations in quantum mechanics

FIGURE 3.3 The three regions for the particle in the one-dimensional box of finite height V0

with the solution jðxÞ ¼ A expðiaxÞ þ B expðiaxÞ with a2 ¼ 2ðE  V0 Þ;

a¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ðE  V0 Þ

where (E  V0) is the kinetic energy of the particle. Then h h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i jðxÞ ¼ A exp i 2ðE  V0 Þx þ B exp  i 2ðE  V0 Þx For E > V0, we have the free particle case discussed in detail by Kauzmann (1957). For E < V0, we have the bound particle case. In a classical description the particle does not have enough energy to escape from the box, but its quantum description is different. We have the following cases. 1. Outside the box (regions I and III) In this case (E < V0) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a ¼ i 2ðV0  EÞ ¼ ib; b ¼ 2ðV0  EÞ > 0 The boundary conditions give jI ¼ AI expðbxÞ þ BI expðbxÞ jI ¼ BI expðbxÞ jIII ¼ AIII expðbxÞ þ BIII expðbxÞ jIII ¼ AIII expðbxÞ

N  x < 0 since AI ¼ 0 since

BIII ¼ 0

There is a finite probability of finding the particle outside the box, in a region where the kinetic energy is negative (classically forbidden). As V0 /N, jI and jIII tend to be zero everywhere. 2. Inside the box (region II, where V ¼ 0) The particle is bound, being confined into the box. Then pffiffiffiffiffiffi a ¼ 2E real 0 E ¼ T > 0 jII ðxÞ ¼ A expðiaxÞ þ B expðiaxÞ

3.13 Solved problems

145

in complex form, or, in real form jII ðxÞ

¼ Aðcos ax þ i sin axÞ þ Bðcos ax  i sin axÞ ¼ ðA þ BÞ cos ax þ iðA  BÞ sin ax ¼ C sin ax þ D cos ax

or, introducing the amplitude AII and the phase ε jII ðxÞ ¼ AII sinðax þ εÞ since C ¼ AII cos ε;

D ¼ AII sin ε

Therefore (Figure 3.4) jII ð0Þ ¼ AII sin ε;

jII ðaÞ ¼ AII sinðaa þ εÞ

jII is not zero either at x ¼ 0 or x ¼ a, and the particle can be found in a region outside the box (quantum filtration). To make further progress we must now introduce the continuity conditions for the function and its first derivative at the boundaries. For x ¼ 0 jI ð0Þ ¼ jII ð0Þ     djI djII ¼ dx 0 dx 0 For x ¼ a jII ðaÞ ¼ jIII ðaÞ     djII djIII ¼ dx a dx a

FIGURE 3.4 Wavefunction for the ground state in the bound region II. The particle has now a not negligible probability of being found outside the box

146

CHAPTER 3 Differential equations in quantum mechanics

Let us calculate the first derivatives of the wavefunction: djI djII djIII ¼ bjI ; ¼ aAII cosðax þ εÞ; ¼ bjIII dx dx dx From the continuity conditions it follows then  BI ¼ AII sin ε AII sinðaa þ εÞ ¼ AIII expðbaÞ bBI ¼ aAII cos ε

aAII cosðaa þ εÞ ¼ bAII expðbaÞ

a tan ε ¼ ¼ b BI 1 ¼ sin ε ¼ ¼ AII cosec ε

rffiffiffiffiffiffiffiffiffiffiffiffiffiffi E V0  E

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v rffiffiffiffiffi u 1 a E u 1 ¼u ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 2 2 2 t 1 þ cot ε V0 b a2 þ b 1þ 2 a a tan ðaa þ εÞ ¼  b

Now, since (Problem 3.1) tan ðx þ yÞ ¼

tan x þ tan y 1  tan x tan y

a tan aa þ tan aa þ tan ε a b ¼ ¼ tan ðaa þ εÞ ¼ 1  tan aa$tan ε 1  a tan aa b b a formula which can be solved for tan aa giving  2 a a a tan aa  tan aa ¼  þ b b b 2a pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 EðV0  EÞ 2ab b tan aa ¼ ¼  2 ¼ 2 2E  V0 a  b2 a 1 b Therefore we obtain as quantization condition on the energy for the particle in a box of finite height pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi 2 EðV0  EÞ tan 2a2 E ¼ 2E  V0

3.13 Solved problems

147

Measuring E and V0 in units of 1/2a2, we get sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffi E E 1 2 V0 V0 pffiffiffiffi tan E ¼ E 2 1 V0 a trigonometric transcendental equation that can be solved numerically or graphically (Kauzmann, 1957) by seeking the intersection of the curves sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffi E E 2 1 V0 V0 pffiffiffiffi vs E tan E vs E and RðEÞ ¼ E 2 1 V0 The intersection points (for different values of V0) are the roots of the transcendental equation and, therefore, the permissible values for the eigenvalue E. We shall content ourselves here in seeing that for V0 very large ðV0 /NÞ pffiffiffiffi pffiffiffiffi tan E ¼ 0 E ¼ np 2 p n ¼ 1; 2; 3; / E ¼ n2 2 2a the result found previously in Problem 3.4. Schiff (1955) used translational symmetry arguments for working out an elegant graphical solution for the energy levels, while the wavefunctions are seen to fall into two classes, being even or odd with respect to the interchange of x into x. 3.7. The normalization integral for the H-atom in parabolic coordinates. The normalization integral for the wavefunction of the free hydrogen atom in parabolic coordinates is    I ¼ hjnkm jjnkm i ¼ Mnk1 m ðxÞNnk2 m ðhÞFm ð4ÞMnk1 m ðxÞNnk2 m ðhÞFm ð4Þ  m=2  2 h   

h

hi2 x x x h m=2 m dr Lmþk1 exp  exp  Lm mþk2 n 2n n n 2n n

Z ¼

N2

N2 ¼ 2m n

Z2p d4 0

¼

ZN

ZN dx

0

8 N

> P0 ¼ 1
> 4 2 5 3 > : P ¼ 35x
30x þ 3 P ¼ 63x
70x þ 15x 4 5 8 8 8   1=2 3
2


 1=2 1=2 > P11 ¼ 1
x2 5x
1 P12 ¼ 1
x2 3x P13 ¼ 1
x2 > < 2 > > :



 1=2 5
3 1=2 15
4 P14 ¼ 1
x2 P15 ¼ 1
x2 7x
3x 21x
14x2 þ 1 2 8 8   
2


 2 2 2 2 2 2 15 > > < P2 ¼ 1
x 3 P 3 ¼ 1
x 15x P4 ¼ 1
x 2 7x
1 > > :

 105
3
 P25 ¼ 1
x2 3x
x 2
  8 3
2 3=2 15 P 3 ¼ 1
x2 3=2 105x > 4 < P3 ¼ 1
x > :


 3=2 105
2 9x
1 P35 ¼ 1
x2 2 n  2

2 P 44 ¼ 1
x2 105 P 45 ¼ 1
x2 945x n
5=2 945 P55 ¼ 1
x2

(6)

154

CHAPTER 4 Special functions

In general l   l
k l! X ðx þ 1Þk l ðx
1Þ Pl ðxÞ ¼ l ðl
kÞ! k! 2 k¼0 k

Pm l ðxÞ where




¼ 1
x

 2 m=2

 l  l
k l! X ðx þ 1Þk
m l þ m ðx
1Þ k ðl
kÞ! ðk
mÞ! 2l k¼m

(7)

(8)

  n! n ¼ is the binomial coefficient (Abramowitz and Stegun, 1965) with n  k. k k!ðn
kÞ!

4.2.2 Recurrence relations for Legendre polynomials We give here without proof some recurrence relations occurring for the Legendre polynomials and their first derivatives in the primed notation (Sneddon, 1956): ðl þ 1ÞPlþ1 ðxÞ ¼ ð2l þ 1ÞxPl ðxÞ
lPl
1 ðxÞ

(9)

P0lþ1 ðxÞ
P0l
1 ðxÞ ¼ ð2l þ 1ÞPl ðxÞ

(10)

P0lþ1 ðxÞ
xP0l ðxÞ ¼ ðl þ 1ÞPl ðxÞ

(11)

4.2.3 Series of Legendre polynomials It may sometimes be useful to express a given function as a series of Legendre polynomials. Because of the linear independence of Pms, any polynomial P(x) of degree l in x can be expressed in terms of Legendre polynomials as l X PðxÞ ¼ Pk ðxÞck (12) k¼0

Now, if we want to expand any given function f(x), defined in the interval jxj  1, into a series of Legendre polynomials in the form N X Pk ðxÞck (13) f ðxÞ ¼ k¼0

assuming the series to be convergent in the given interval, we see that the expansion coefficients are given by Z1 Z1 N N X X 2 2 dlk ¼ cl (14) d x f ðxÞPl ðxÞ ¼ ck d x Pk ðxÞPl ðxÞ ¼ ck 2k þ 1 2l þ1 k¼0 k¼0
1


1

2l þ 1 cl ¼ 2

Z1 d x f ðxÞPl ðxÞ
1

(15)

4.2 Legendre functions

155

so that f(x) is given by the series N X

2k þ 1 Pk ðxÞ f ðxÞ ¼ 2 k¼0

Z1

dx0 f ðx0 ÞPk ðx0 Þ

(16)


1

4.2.4 Legendre functions of first and second kind Problems of mathematical physics often involves the Laplace’s equation: 72 VðrÞ ¼

v2 V v2 V v2 V þ 2 þ 2 ¼0 vx2 vy vz

(17)

which can be solved in spherical coordinates by putting Vðr; q; 4Þ ¼ RðrÞQðqÞFð4Þ. Factorization of function V implies the independence of the solutions with respect to the different variables, namely, the R(r) and F(4) terms can be treated as constants in the solution of the q-equation. If we put   1 d 1 d2 F 2 dR r ¼ nðn þ 1Þ; ¼
m2 R dr dr F d42 the solutions of these equations are (Hobson, 1956) RðrÞ ¼ c1 r n þ c2 r
n
1 ;

Fð4Þ ¼ c3 expðim4Þ

(18)

Two solutions of the Laplace’s Eqn (17) must hence have the form Vfrn Unm ðq; 4Þ;

Vfr
n
1 Unm ðq; 4Þ;

Unm ðq; 4Þ ¼ QðqÞexpðim4Þ

(19)

In this way, by multiplying both members by QðqÞ, Eqn (17) gives     1 d dQ m2 sin q þ nðn þ 1Þ
2 Q ¼ 0 sin q dq dq sin q which, by posing Q ¼ w;

cos q ¼ z; sin q ¼ ð1
z2 Þ1=2 , becomes      dw d
m2 w¼0 1
z2 þ nðn þ 1Þ
1
z2 dz dz

(20)

Equation (20) is known as associated Legendre’s differential equation (compare Eqn (188) of Chapter 3 with l ¼ n(n þ 1)). For m ¼ 0 we have the corresponding Legendre’s differential equation:    d
2 dw 1
z þ nðn þ 1Þw ¼ 0 (21) dz dz Even if variable z has been defined as the cosine of a real angle, being therefore real and restricted to the interval (
1,1), the solution of Eqn (21) will now be considered in a more general sense. Assuming the ascending power series expansion wðzÞ ¼ a0 þ a1 z þ a2 z2 þ /

(22)

156

CHAPTER 4 Special functions

we obtain in the usual way the two-term recurrence formula: akþ2 ¼

kðk þ 1Þ
nðn þ 1Þ ðn
kÞðn þ k þ 1Þ ak ¼
ak ðk þ 1Þðk þ 2Þ ðk þ 1Þðk þ 2Þ

k ¼ 0; 1; 2; /

giving the solution wðzÞ as   nðn þ 1Þ 2 nðn
2Þðn þ 1Þðn þ 3Þ 4 z þ z
/ wðzÞ ¼ a0 1
1$ 2 1$ 2$ 3$ 4   ðn
1Þðn þ 2Þ 2 ðn
1Þðn
3Þðn þ 2Þðn þ 4Þ 4 z þ z
/ þ a1 z 1
1$ 2$ 3 1$ 2$ 3$ 4$ 5 which, putting 2 F1 ¼ F, can be rewritten in terms of the hypergeometric functions of Section 4.5.1 as     n nþ1 1 2 n
1 nþ2 3 2 wðzÞ ¼ a0 F
; ; ; z þ a1 zF
; ; ;z 2 2 2 2 2 2 as can be easily seen. When n is a positive integer, one of the series terminates and, if n is even, the solution   n nþ1 1 2 ; ;z (23) w1 ðzÞ ¼ a0 F
; 2 2 2 becomes a polynomial of degree n in even powers of z2, while the remaining solution  n
1 nþ2 3 zF
; ; ; z2 is an infinite series which converges when jzj < 1. 2 2 2 For n ¼ odd, the finite solution is   n
1 nþ2 3 2 ; ; ;z (24) w2 ðzÞ ¼ a1 zF
2 2 2 which is a polynomial of degree n in odd powers of z, while the remaining solution  n nþ1 1 ; ; z2 is an infinite series which converges when jzj < 1. F
; 2 2 2 Now, let us try to obtain a solution of Legendre’s Eqn (21) in the form of a series of descending powers of z: wðzÞ ¼ b0 zq þ b2 zq
2 þ b4 zq
4 þ /

(25)

obtaining for the coefficients the two-term recurrence relation b2kþ2 ¼

ðq
2kÞðq
2k
1Þ b2k ðq
2k
2Þðq
2k
1Þ
nðn þ 1Þ

k ¼ 0; 1; 2; /

Since b
2 ¼ 0, we obtain from the resulting quadratic equation (Problem 4.1) ðq
nÞðq þ n þ 1Þ ¼ 0

4.2 Legendre functions

157

which is satisfied by the two values, q ¼ n and q ¼
n
1, giving rise to the two different solutions   n n
1 2n
1
2 (26) w3 ðzÞ ¼ bzn F
;
;
;z 2 2 2   n þ 1 n þ 2 2n þ 3
2
n
1 (27) w4 ðzÞ ¼ cz F ; ; ;z 2 2 2 By reversing the order of the terms in the series (Hobson, 1965), w3 is seen to be identical to w1 and w2 for n integer even and n integer odd, respectively, whereas w4 is an infinite series converging when jzj > 1. Putting both constants a0 and a1 of w1 and w2 equal to an ¼ ð
1Þ½n=2 n!=f2n ð½n=2!Þ2 g where [t] means ‘integer part of t’, constant b of w3 and constant c of w4 equal to (Sneddon, 1956) b¼

ð2nÞ!

; 2n ðn!Þ2

c¼

2n ðn!Þ2 ð2n þ 1Þ!

we can write the general solution of the Legendre’s differential Eqn (21) as wðzÞ ¼ APn ðzÞ þ BQn ðzÞ where A, B are arbitrary constants, and where   n n
1 2n
1
2 ; ;
;z Pn ðzÞ ¼ bzn F
;
2 2 2

Qn ðzÞ ¼ cz
n
1 F

(28)   n þ 1 n þ 2 2n þ 3
2 (29) ; ; ;z 2 2 2

are called Legendre’s functions of the first and second kind of degree n, respectively. For m s 0 the corresponding solutions of Eqn (20) are the associated Legendre’s functions of the first and second kind, respectively, which in un-normalized form are given as 8 m 
m ðzÞ ¼ 1
z2 m=2 d Pn ðzÞ z ˛ ð
1; 1Þ; > P > > n < dzm


m=2 dm Pn ðzÞ m Pbn ðzÞ ¼ z2
1 z ˛ð1; NÞ dzm

> m > > : Qm ðzÞ ¼
1
z2 m=2 d Qn ðzÞ z ˛ ð
1; 1Þ; n dzm

bm ðzÞ Q n




m=2 dm Pn ðzÞ ¼ z2
1 z ˛ð1; NÞ dzm

(30)

4.2.5 Neumann’s formula for the Legendre functions It is possible to show (Sneddon, 1956) that Qn(x) can be expressed by the integral Z1 1 Pn ðxÞ Qn ðxÞ ¼ dx jxj > 1 real 2 x
x

(31)


1

which is known as Neumann’s formula, and also that, for jxj > 1 Z1 1 xþ1 1 Pn ðxÞ
Pn ðxÞ
Qn ðxÞ ¼ Pn ðxÞln dx 2 x
1 2 x
x
1

(32)

158

CHAPTER 4 Special functions

An expression for Qn(x) with n integer useful for practical calculations is the following: Qn ðxÞ ¼

p 1 x þ 1 X 2n
4k
1 Pn ðxÞln
Pn
2k
1 ðxÞ 2 x
1 k¼0 ð2k þ 1Þðn
kÞ

(33)

where p ¼ ðn
1Þ=2 or p ¼ ðn
2Þ=2 according to n > 0 is odd or even. The first four Qns are then Q0 ðxÞ ¼

1 xþ1 ln ; 2 x
1

3 P1 ðxÞ 2 7 1 Q4 ðxÞ ¼ P4 ðxÞQ0 ðxÞ
P3 ðxÞ
P1 ðxÞ 4 3

Q1 ðxÞ ¼ P1 ðxÞQ0 ðxÞ
1;

5 1 Q3 ðxÞ ¼ P3 ðxÞQ0 ðxÞ
P2 ðxÞ
; 3 6

Q2 ðxÞ ¼ P2 ðxÞQ0 ðxÞ


(34)

4.3 LAGUERRE FUNCTIONS 4.3.1 Laguerre polynomials and Laguerre functions The Laguerre polynomials Ln(x) of degree n in x defined as (Sneddon, 1956) Ln ðxÞ ¼ expðxÞ

dn n ½x expð
xÞ dxn

(35)

are the solution of the Laguerre’s differential equation of the second order: x

d2 Ln ðxÞ dLn ðxÞ þ ð1
xÞ þ nLn ðxÞ ¼ 0 dx2 dx

(36)

where n is a positive integer. The explicit form for the first few Laguerre polynomials up to n ¼ 5 is L0 ðxÞ ¼ 1 L1 ðxÞ ¼ 1
x L2 ðxÞ ¼ 2
4x þ x2

(37)

L3 ðxÞ ¼ 6
18x þ 9x2
x3 L4 ðxÞ ¼ 24
96x þ 72x2
16x3 þ x4 L5 ðxÞ ¼ 120
600x þ 600x2
200x3 þ 25x4
x5 and, in general Ln ðxÞ ¼ n!

n X k¼0

ð
1Þk

  k n x k k!

(38)

We now verify in detail that L5(x) does verify the differential equation with n ¼ 5: xL005 ðxÞ þ ð1
xÞL05 ðxÞ þ 5L5 ðxÞ ¼ 0

(39)

4.3 Laguerre functions

159

In fact, using the primed notation for derivatives, we have from Eqn (37) L05 ðxÞ ¼
600 þ 1200x
600x2 þ 100x3
5x4 L005 ðxÞ

¼ 1200
1200x þ 300x
20x 2

3

(40) (41)

and substituting into Eqn (39) and adding all terms 1200x
1200x2 þ 300x3
20x4
600 þ 1200x
600x2 þ 100x3
5x4 þ 600x
1200x2 þ 600x3
100x4 þ 5x5

(42)

þ 600
3000x þ 3000x2
1000x3 þ 125x4
5x5 ¼ 0 as it must be. The Laguerre polynomials are not orthogonal in themselves, but the Laguerre functions 1 expð
x=2ÞLn ðxÞ n!

(43)

form an orthonormal set in the interval 0  x  N: ZN dx expð
xÞLn ðxÞLn0 ðxÞ ¼ ðn!Þ2 dnn0

(44)

0

as can be proved by integration by parts. Recurrence relations for the Laguerre polynomials are (Sneddon, 1956) Lnþ1 ðxÞ ¼ ð2n þ 1
xÞLn ðxÞ
n2 Ln
1 ðxÞ

(45)

L0n ðxÞ ¼ nL0n
1 ðxÞ
nLn
1 ðxÞ

 L00nþ1 ðxÞ ¼ ðn þ 1Þ xL00n ðxÞ
L0n ðxÞ

(46)

and for the derivatives

xL00n ðxÞ

¼ ðx


1ÞL0n ðxÞ
nLn ðxÞ

(47) (48)

the last equation being Eqn (36) defining Ln(x).

4.3.2 Associated Laguerre polynomials The m-th derivative (m  n) of the Laguerre polynomial Ln(x) of order n in x

dm Ln ðxÞ dm dn n m ¼ m expðxÞ n ½x expð
xÞ Ln ðxÞ ¼ dxm dx dx

(49)

160

CHAPTER 4 Special functions

is called the associated Laguerre polynomial Lm n ðxÞ of degree (n
m) and order m in x, and is the solution of the associated Laguerre’s differential equation of the second order: x

d2 Lm dLm ðxÞ n ðxÞ þ ðm þ 1
xÞ n þ ðn
mÞLm n ðxÞ ¼ 0 2 dx dx

(50)

where n, m (m  n) are positive integers. The explicit form of the first few associated Laguerre polynomials up to m ¼ 5 is L11 ðxÞ ¼
1 L12 ðxÞ ¼
4 þ 2x;

L22 ðxÞ ¼ 2

L13 ðxÞ ¼
18 þ 18x
3x2 ;

L23 ðxÞ ¼ 18
6x;

L14 ðxÞ ¼
96 þ 144x
48x2 þ 4x3 ;
 L34 ðxÞ ¼
96 þ 24x; L44 x ¼ 24

L33 ðxÞ ¼
6

L24 ðxÞ ¼ 144
96x þ 12x2 ;

L15 ðxÞ ¼
600 þ 1200x
600x2 þ 100x3
5x4 ; L35 ðxÞ ¼
1200 þ 600x
60x2 ;

L25 ðxÞ ¼ 1200
1200x þ 300x2
20x3 ;

L45 ðxÞ ¼ 600
120x;

and, in general m Lm n ðxÞ ¼ ð
1Þ n!

(51)

nX
m

ð
1Þk k! k¼0



L55 ðxÞ ¼
120  n xk mþk

(52)

Once the explicit form of a Laguerre polynomial is known, the corresponding associated Laguerre polynomial is readily evaluated through the appropriate derivative. We now verify in detail that L25 ðxÞ does verify the differential Eqn (50) with n ¼ 5, m ¼ 2, n
m ¼ 3: 00

0

xL25 ðxÞ þ ð3
xÞL25 ðxÞ þ 3L25 ðxÞ ¼ 0 Using the primed notation for the derivatives, we have from Eqn (51) 0

L25 ðxÞ ¼
1200 þ 600x
60x2 00 L25 ðxÞ

¼ 600
120x

and, substituting into the differential equation and adding all terms, we find

 600x
120x2 þ
3600 þ 1800x
180x2

 þ 1200x
600x2 þ 60x3 þ 3600
3600x þ 900x2
60x3 ¼ 0

(53) (54)

(55)

as it must be. In the previous chapter, we discussed two such associated Laguerre differential equations which we solved by the series expansion technique after solution of the asymptotic differential equations resulting at the singular points, (1) the radial equation for the atomic one-electron system and (2) the x-equation for the free hydrogen atom in parabolic coordinates.

4.3 Laguerre functions

161

Recurrence relations for the associated Laguerre polynomials are (Sneddon, 1956) m m
1 Lm ðxÞ
n2 Lm nþ1 ðxÞ ¼ ð2n þ 1
xÞLn ðxÞ
mLn n
1 ðxÞ

(56)

m m
1 Lm n ðxÞ ¼ nLn
1 ðxÞ
nLn
1 ðxÞ

(57)

4.3.3 Basic integrals over associated Laguerre functions The general integral over the product of two orthogonal Laguerre functions involving the associated Laguerre polynomials of degree k and k0 was originally given by Schroedinger (1926a) in terms of generalized binomial coefficients as ZN

0

m dx x p expð
xÞLm mþk ðxÞLm0 þk0 ðxÞ 0 0

0

¼ ð
1Þmþkþm þk p!ðm þ kÞ!ðm0 þ k0 Þ!

0 minðk;k X Þ

k¼0

 ð
1Þk

p
m k
k



p0
m0



k0
k


p
1 k

  i a generalized binomial coefficient defined by j 8   < 1 ðj ¼ 0Þ i ¼ iði
1Þ/ði
j þ 1Þ ðj > 0Þ j : j!



(58)

where p is a non-negative integer, and

(59)

with j a non-negative integer and i any integer. The generalized binomial coefficient becomes identical to an ordinary binomial coefficient whenever i  j. This formula was put into a more convenient form by Figari (2010) in terms of ordinary binomial coefficients as ZN

0

0

mþm m dx x p expð
xÞLm ðm þ kÞ!ðm0 þ k0 Þ! mþk ðxÞLm0 þk0 ðxÞ ¼ ð
1Þ

0



kþk X0 k¼0

ð
1Þk ðp þ kÞ!

0 minðk X;kÞ



mþk

1 ðk
sÞ!s! m þ k
s s¼maxð0;k
kÞ



m0 þ k 0 m0 þ s



(60)

For the calculation of the first-order Stark effect, we introduced in Chapter 3 the auxiliary integral Il(m,k) as ZN Il ðm; kÞ ¼ 0

2 dx x mþl expð
xÞ Lm mþk ðxÞ

(61)

162

CHAPTER 4 Special functions

From the formulae above, for m0 ¼ m; k0 ¼ k; p ¼ m þ l, we obtain ZN

2 dx xmþl expð
xÞ Lm Il ðm; kÞ ¼ mþk ðxÞ 0

¼ ðm þ lÞ!½ðm þ kÞ!

2

k X

" ð
1Þ

k
k

k¼0 k X

l

k¼maxð0;k
lÞ

k
k

¼ ðm þ lÞ!½ðm þ kÞ!2

¼ ðl!Þ2 ½ðm þ kÞ!2 since "

l k
k

"

# ¼

k X


m
l
1

!2

# ¼

#

k mþlþk

(62)

!

k

ðm þ l þ kÞ!

k¼maxð0;k
lÞ ½ðl


k þ kÞ!2 ½ðk
kÞ!2 k!

ðk ¼ kÞ

1

lðl
1Þ/½l
ðk
k
1Þ > : ðk
kÞ! 8  l! l > < ¼ k
k ðk
kÞ!ðl
k þ kÞ! ¼ > : 0


m
l
1 k

8 >
0 has no singularities for x / 0. Therefore, the function kn ðxÞ ¼ ð2=pÞ1=2 xn Kn ðxÞ

(169)

was called by Shavitt (1963) reduced Bessel function. For half-integer n k
1=2 ¼

kN
1 ðxÞ ¼ 2

expð
xÞ x

(170)

N expð
xÞ X ð2N
k
1Þ! k
N k 2 x x ðk
1Þ!ðN
kÞ! k¼1

(171)

The following recursion formula, practical for computational purposes, holds for all values of n (Magnus et al., 1966): x2 kn
1 ðxÞ ¼ knþ 3 ðxÞ
ð2n þ 1Þknþ 1 ðxÞ 2

2 3

2

2

Equation (164) is obtained in the solution of the Laplace’s equation in cylindrical coordinates. Incorrectly, Cha1asinski and Jeziorski call Kn(x) modified Bessel’s function of the third kind.

(172)

178

CHAPTER 4 Special functions

The reduced Bessel functions were extensively used by Steinborn and co-workers (Weniger and Steinborn, 1983; and references therein) in their studies of multicentre molecular integrals over Slatertype orbitals (STOs) and, more generally, exponential-type orbitals (ETOs) with their related anisotropic generalizations, the so-called B-functions. The B-functions are linear combinations of STOs, having a sensibly more complicated analytical structure than STOs but more appealing properties in multicentre problems, such as simpler Fourier transforms (FTs) and extremely compact convolution integrals (see Section 4.9).

4.7 FUNCTIONS DEFINED BY INTEGRALS 4.7.1 The gamma function The gamma function G is defined by the integral (Sneddon, 1956; Abramowitz and Stegun, 1965) ZN dx expð
xÞxn
1 (173) GðnÞ ¼ 0

where n > 0. A few properties of the gamma function are 1. Gð1Þ ¼ 1 2. Gðn þ 1Þ ¼ nGðnÞ the recurrence formula 3. Gðn þ 1Þ ¼ n! if n is positive integer   pffiffiffiffi 1 ¼ p 4. G 2     1 1 Gð2nÞ ¼ 22n
1 GðnÞG n þ 5. G 2 2 the duplication formula For n a positive integer the duplication formula becomes     1 1 ð2nÞ! ¼ 22n n!G n þ G 2 2 n!nx n/N ðx þ 1Þðx þ 2Þ/ðx þ nÞ the Euler’s formula

6. Gðx þ 1Þ ¼ lim

x>0

Some of these properties are easily derived in Problem 4.3 using the definition of the gamma function and the rule of integration by parts.

4.7.2 The incomplete gamma function The incomplete gamma function g(a,x) is defined by the integral (Abramowitz and Stegun, 1965) Zx gða; xÞ ¼ dt expð
tÞta
1 (174) 0

4.7 Functions defined by integrals

179

and is related to the Kummer’s confluent hypergeometric function of Section 4.5.2 by the relation gða; xÞ ¼ a
1 expð
xÞxa Mð1; a þ 1; xÞ # " a
1 k (175) X x ¼ ða
1Þ!½1
expð
xÞPa
1 ðxÞ ¼ ða
1Þ! 1
expð
xÞ k! k¼0 The difference between the gamma function GðaÞ and the incomplete gamma function g(a,x) defines the new function Gða; xÞ: ZN dt expð
tÞta
1 (176) Gða; xÞ ¼ GðaÞ
gða; xÞ ¼ x

which for a ¼
n negative integer can be expressed by # " n
1 X 1 n
1 k ðn
1
kÞ! k
n x ð
1Þ Gð
n; xÞ ¼ ð
1Þ Eið
xÞ þ ðn
1Þ!expð
xÞx n! ðn
1Þ! k¼0 "

¼

#

(177)

1 ð
1Þn
1 Eið
xÞ þ ðn
1Þ!expð
xÞx
n Pn
1 ðxÞ n!

where Ei(
x) is the exponential integral function defined by Eqn (185) of the next section.

4.7.3 From the gamma function to the exponential integral function From Eqn (6) of the previous section it is possible to derive an expression for Euler’s constant g, which is defined by the series   1 1 1 (178) g ¼ lim 1 þ þ þ / þ
ln n ¼ 0:577 215 665/ n/N 2 3 n In fact, taking the logarithm of (6) we have ln Gðx þ 1Þ ¼ lim ½ln n! þ x ln n
ln ðx þ 1Þ
ln ðx þ 2Þ
/
ln ðx þ nÞ n/N

and, taking its derivative   d 1 1 1 ln Gðx þ 1Þ ¼ lim ln n


/
n/N dx xþ1 xþ2 xþn Letting x/0 we obtain



d ln Gðx þ 1Þ g ¼
lim x/0 dx and, from the definition of Gðx þ 1Þ it follows ZN g ¼
dt expð
tÞln t 0

(179)

(180)

 (181)

(182)

180

CHAPTER 4 Special functions

Integrating by parts the function of x ZN ZN ZN N dt expð
tÞln t ¼
ln td½expð
tÞ ¼
ln t expð
tÞjx þ dt expð
tÞt
1 x

x

ZN

¼ ln x expð
xÞ þ

x

(183)

dt expð
tÞt
1

x

and, taking the limit for x/0, we find ZN ZN g ¼
dt expð
tÞ ln t ¼
lim dt expð
tÞln t x/0

0

2

¼
lim 4

3

ZN dt

x/0

x

expð
tÞ þ ln x5 ¼ lim ½Eið
xÞ
ln x x/0 t

(184)

x

where

ZN Eið
xÞ ¼


dt

expð
tÞ ¼
E1 ðxÞ t

(185)

x

is the exponential integral function (Sneddon, 1956; Abramowitz and Stegun, 1965). We notice that dEið
xÞ expð
xÞ ¼ dx x

(186)

4.7.4 The exponential integral function For E1(x) we have the series expansion E1 ðxÞ ¼
Eið
xÞ ¼
g
ln x
where g is Euler’s constant, or Eið
xÞ ¼ g þ ln x þ Hence

N X xn ð
1Þn n $ n! n¼1

N X xn ð
1Þn n $ n! n¼1

g ¼ lim ½Eið
xÞ
ln x x/0

(187)

(188)

(189)

This relation, which is nothing but Eqn (184) derived before, occurs in the calculation of the twocentre two-electron exchange integral ð1sA 1sB j1sA 1sB Þ between STO functions centred at nuclei A and B (Section 18.7.2 of Chapter 18 of this book) and its higher homologues. This integral was met by Heitler and London (1927) in their wave mechanical calculations on the ground state of the hydrogen molecule, and was first evaluated exactly by Sugiura (1927).

4.7 Functions defined by integrals

181

The complete form for this exchange integral, here given in the so-called charge density notation, is Z

Z ð1sA 1sB j1sA 1sB Þ ¼

Z

dr2

dr1

½1sA ðr2 Þ1sB ðr2 Þ ½1sA ðr1 Þ1sB ðr1 Þ r12

(190)

dr1 KAB ðr1 Þ½1sA ðr1 Þ1sB ðr1 Þ

¼

where KAB(r1) is the two-centre exchange potential at the space point r1 due to the two-centre charge distribution of electron 2 at r2: Z ½1sA ðr2 Þ1sB ðr2 Þ (191) KAB ðr1 Þ ¼ dr2 r12 The calculation of this difficult integral is based on the expansion of the inverse of the interelectron distance r12 in spheroidal coordinates and is explained in detail in Section 18.7.2 of Chapter 18 of this book. The series occurring for two-centre spherical 1s orbitals with equal orbital exponents breaks down after two terms, but an infinite number of terms is needed when the orbital exponents are different. This is the case of the four-centre two-electron integral evaluated by Musso and Magnasco (1971) in terms of a Gauss–Legendre four-dimensional numerical integration using appropriate recursion formulae for the auxiliary functions.

4.7.5 The generalized exponential integral function In the calculation of molecular multicentre exchange integrals over STOs (Magnasco et al., 1998; Magnasco et al., 1999; Magnasco and Rapallo, 2000) occurs the generalized exponential integral of order n En(r) (Abramowitz and Stegun, 1965) defined as ZN En ðrÞ ¼

dx expð
rxÞx
n

(192)

1

with n a non-negative integer and Re(r) > 0. The high accuracy needed in its numerical calculation can be achieved through multiple precision arithmetics using recurrence relations and accurate Gaussian quadrature techniques (Ralston, 1965; Demidovic and Maron, 1981; Bachvalov, 1981). It is of crucial importance to start the recursion with extremely accurate terms.

4.7.6 Further functions The basic indefinite integral occurring in atomic or molecular calculations involving STOs with exponential radial decay is defined by the primitive function Z n X n! xn
k ð
1Þk Fn ðxÞ ¼ dx expð
rxÞxn ¼ expð
rxÞ ðn
kÞ! ð
rÞkþ1 k¼0 # " n! ðrxÞn ðrxÞn
1 ðrxÞ2 þ þ/þ þ rx þ 1 ¼
nþ1 expð
rxÞ (193) n! ðn
1Þ! 2! r ¼


n! rnþ1

expð
rxÞ

n X ðrxÞk k¼0

k!

182

CHAPTER 4 Special functions

where n is a non-negative integer and r a real positive number, a result that can be obtained by repeated integration by parts. From these relations follow some definite integrals needed in one-centre atomic problems: " # Zu n k X n! ðruÞ dx expð
rxÞxn ¼ nþ1 1
expð
ruÞ (194) k! r k¼0 0

ZN dx expð
rxÞxn ¼ u

n! rnþ1

expð
ruÞ

n X ðruÞk k¼0

k!

(195)

giving by addition the well-known integral ZN dx expð
rxÞxn ¼ 0

n! rnþ1

(196)

and the following auxiliary functions needed in two-centre molecular problems (Roothaan, 1951b): ZN An ðrÞ ¼

dx expð
rxÞxn ¼ 1

n! rnþ1

expð
rÞ

n X rk k¼0

k! (197)

Z1 dx expð
rxÞxn ¼ ð
1Þnþ1 An ð
rÞ
An ðrÞ

Bn ðrÞ ¼
1

In some calculations are needed the integrals Zu n X n! ðruÞk dx expð
rxÞxn ¼ An ðrÞ
nþ1 expð
ruÞ k! r k¼0

(198)

1

ZN dx expð
rxÞxn ¼ u

n! rnþ1

expð
ruÞ

n X ðruÞk k¼0

k!

(199)

whose addition gives the An(r) function. Properties of the Bn(r) functions are Bn ð
rÞ ¼ ð
1Þn Bn ðrÞ;

Bn ð0Þ ¼

2 den nþ1

(200)

where e ¼ even. Recurrence relations between the auxiliary functions are sometimes needed in numerical calculations. For instance, it is easily shown from the definitions that 1 An ðrÞ ¼ ½nAn
1 ðrÞ þ rA0 ðrÞ r

(201)

4.7 Functions defined by integrals

183

In fact An ðrÞ ¼

n! rnþ1

n
1 k X r

expð
rÞ

k¼0

k!

"

¼

þ

n! rnþ1

n
1 k X r

expð
rÞ

rn n!

#

(202)

n ðn
1Þ! expð
rÞ 1 þ expð
rÞ ¼ ½nAn
1 ðrÞ þ rA0 ðrÞ k! r rn r r k¼0

The generalized auxiliary functions are defined as ZN m
dx expð
rxÞxj x2
1 Tjm ðrÞ ¼

(203)

1

Z1


m dx expð
rxÞxj 1
x2

(204)


m=2 m dx expð
rxÞxj 1
x2 Pl ðxÞ

(205)

Gjm ðrÞ ¼
1

Z1 Bljm ðrÞ

¼
1

ZN Hlmpq ðr1 ; r2 Þ ¼

ZN dx

1


m=2 m


Q^l ðx> ÞP^m dy expð
r1 xÞexpð
r2 yÞxp yq x2
1 y2
1 l ðx< Þ (206)

1

^m ^m j, l, m, p, q being non-negative integers, Re(r) > 0, Pm l ; Pl ; Ql normalized associated Legendre functions of first and second kind, x< and x> the lesser and the greater of x and y. Tjm(r) and Gjm(r) are generalizations of the elementary auxiliary functions Aj(r) and Bj(r) defined above, while Bljm ðrÞ and Hlmpq(r1,r2) were introduced by Ruedenberg (1951) as a generalization of the auxiliary integral ZN Hlpq ðrÞ ¼ Hlqp ðrÞ ¼

ZN dx

1

dy exp½
rðx þ yÞxp yq Q^l ðx> ÞP^l ðx< Þ

(207)

1

occurring in Rosen calculation (1931) of the exchange integrals over STOs in his quantum mechanical treatment of the ground state of the hydrogen molecule. It is seen that Tjm(r) and Gjm(r) can be written as finite sums of An and Bn functions:   m X mþk m Ajþ2k ðrÞ ð
1Þ (208) Tjm ðrÞ ¼ k k¼0   m X k m Bjþ2k ðrÞ ð
1Þ (209) Gjm ðrÞ ¼ k k¼0

While evaluation of Tjm(r) by Eqn (208) is straightforward through use of recurrence relations, for some values of indices j, m, Gjm(r) may be affected by strong numerical instabilities due to cancellation of terms of similar magnitude. Stability in the evaluation of Gjm(r) with very high numerical

184

CHAPTER 4 Special functions

accuracy (14–15 significant figures) can be obtained by expanding in series the exponential in Eqn (204). Similar problems are met in the evaluation of Bljm ðrÞ of Eqn (205), and again numerical stability to about 14–15 significant figures is achieved by the series expansion of the exponential. We may point out here that the accurate evaluation of the double integral (206) is the most timeconsuming step in the evaluation of two-centre molecular integrals in quantum chemistry calculations.

4.8 THE DIRAC d-FUNCTION The Dirac d-function is a distribution which may be visualized as an infinitely sharp Gaussian function (spike) selecting a given value of a function f(x), its main property being Z dx dðx
aÞf ðxÞ ¼ f ðaÞ (210) where the integration is over all possible values of definition of the variable x. The operation of multiplying f(x) by d(x
a) and integrating over all values of x is hence simply equivalent to replacing a for x in the argument of the original function. The Dirac d-function is met when expanding any regular function f(x) in the complete set f4k ðxÞg ^ say of the eigenfunctions of a Hermitian operator A, f ðxÞ ¼

X

4k ðxÞCk

(211)

k

the expansion coefficients being given by

Z

dx0 4k ðx0 Þf ðx0 Þ

Ck ¼

(212)

Using Dirac’s notation, Eqn (212) can be written as

and expansion Eqn (211) as

Ck ¼ h4k j f i

(213)

 X  jf ¼ j4k > h4k j f

(214)

k

where

X X 4k ðxÞ4k ðx0 Þ ¼ dðx
x0 Þ j4k >< 4k j ¼ k

(215)

k

if the set is complete (see Section 1.1.7 of Chapter 1). So, Eqn (215) expresses the completeness of the set f4k ðxÞg of regular functions in terms of the Dirac d-function which can hence be said to be the identity operator ^ 1. More precisely, dðx
x0 Þ is recognized as the kernel of the integral operator ^dðxÞ ¼ ^1: Z ^dðxÞf ðxÞ ¼ dx0 dðx
x0 Þf ðx0 Þ ¼ f ðxÞ (216)

4.9 The Fourier transform

185

Roughly speaking, as said before, we say that the Dirac d-function corresponds to the identity operator when working on continuous functions. It can be shown (Sneddon, 1956) that further properties of the Dirac d-function are dð
xÞ ¼ dðxÞ and, for a > 0 dðaxÞ ¼

1 dðxÞ; a


 1 d a2
x2 ¼ ½dðx
aÞ þ dðx þ aÞ 2a

(217)

(218)

It is also often said that the Dirac d-function is the derivative of the Heaviside unit function H(x), which is defined by the equations 1 if x > 0 HðxÞ ¼ (219) 0 if x < 0 a relation that can be precisely stated in terms of Stieltjes integration.4

4.9 THE FOURIER TRANSFORM The Fourier transform (FT) is of great interest in the calculation of multicentre molecular integrals, in the theory of intermolecular potentials and, in its discretized form (Griffiths, 1978), in recent applications to infrared spectroscopy. The FT was first suggested by Prosser and Blanchard (1962) to simplify the computation of molecular integrals over STOs and, more extensively, by Silverstone and co-workers (Todd et al., 1982) and by the Steinborn group (Filter and Steinborn, 1978; Weniger and Steinborn, 1983; Weniger et al., 1986) in the attempt to give compact analytical forms useful in the practical calculation of multicentre molecular integrals. With the help of the FT, some six-dimensional integrals in coordinate space with nonseparable integration variables can be transformed into three-dimensional integrals in momentum space where the integration variables are separated quite easily. The same is true for the application of the FT method to the study of intermolecular potentials, as we shall see more in detail in Section 17.8 of Chapter 17 of this book. The FT of the interparticle distance in the intermolecular potential allows for a generalized expansion converging for all intermolecular separations R and for separation of angle-dependent from R-dependent factors (Koide, 1976; Knowles and Meath, 1987; Magnasco and Figari, 1989). We now proceed to introducing the essential elements of the FT. The FT of a function f(x) of a real variable x is defined as ZN dx f ðxÞexpð
ipxÞ (220) FðpÞ ¼
N

where p is a parameter, whereas 1 f ðxÞ ¼ 2p

ZN dp Fð pÞexpðipxÞ
N

is called the inverse Fourier transform or Anti-Fourier transform. 4

For the definition of a Stieltjes integral, see Sneddon (1956) p. 162.

(221)

186

CHAPTER 4 Special functions

Equation (221) is obtained as the limit as L /N of the complex Fourier expansion of a function f(x) defined in the interval (
L, L): N  pk X ak exp i x (222) f ðxÞ ¼ L k¼
N the coefficients being given by 1 ak ¼ 2L

ZL

 pk x dx f ðxÞexp
i L


L

(223)

In fact, we can rewrite expansion (222) as N  pk ZL  pk 1 X f ðxÞ ¼ x x f ðxÞ exp i dx exp
i 2L k¼
N L L

(224)


L

and change variable to p¼

pk L

(225)

p increases in steps of unity by Dp ¼ pðk þ 1Þ
pðkÞ ¼

kþ1 k p 1 1 p
p¼ 0 ¼ Dp L L L 2L 2p

(226)

Then, using these relations, expansion (224) becomes ZL N 1 X f ðxÞ ¼ Dp expðipxÞ dx expð
ipxÞf ðxÞ 2p k¼
N

(227)


L

Now, taking the limit for L / N, Dp becomes infinitesimal, so that the sum over k becomes an integral: ZN N X lim Dp ¼ dp (228) lim Dp ¼ dp; L/N

L/N

k¼
N


N

and expansion (227) becomes 1 f ðxÞ ¼ 2p

ZN dp FðpÞexpðipxÞ

(229)


N

ZN dx f ðxÞexpð
ipxÞ

FðpÞ ¼
N

(230)

4.9 The Fourier transform

187

A more symmetrical definition of FT is 1 f ðxÞ ¼ pffiffiffiffiffiffi 2p

ZN

1 FðpÞ ¼ pffiffiffiffiffiffi 2p

dp FðpÞexpðipxÞ;
N

ZN dx f ðxÞexpð
ipxÞ

(231)


N

Using Eqn (231), we can write 1 f ðxÞ ¼ 2p

ZN dp expðipxÞ
N

ZN ¼
N

ZN

dx0 f ðx0 Þexpð
ipx0 Þ


N

dx0 f ðx0 Þ

1 2p

ZN

dp exp½iðx
x0 Þp ¼


N

ZN

(232) dx0 dðx
x0 Þf ðx0 Þ


N

where use was made of the integral representation of the Dirac d-function (Section 4.8) 1 dðx
x Þ ¼ 2p 0

ZN

dp exp½iðx
x0 Þp

These formulae can be extended to a three-dimensional space (r,k): Z 1 f ðrÞ ¼
pffiffiffiffiffiffi3 dk expðik $ rÞFðkÞ 2p 1 FðkÞ ¼
pffiffiffiffiffiffi3 2p

(233)


N

(234)

Z dr expð
ik $ rÞf ðrÞ

(235)

where (k $ r) is the scalar product of vectors k and r k $ r ¼ kx x þ ky y þ kz z

(236)

and dk ¼ dkx dky dkz ;

dr ¼ dx dy dz

(237)

From these relations, we have the generalization of the integral representation of the Dirac dfunction to three dimensions: 0

dðx
x Þ ¼

1 ð2pÞ3

ZN

dk expðiðx
x0 Þ $ kÞ

(238)


N

where the three-dimensional d-function has the properties Z Z dðxÞ ¼ 0 x s 0; dx dðxÞ ¼ 1; dx dðx
yÞf ðxÞ ¼ f ðyÞ

(239)

188

CHAPTER 4 Special functions

Of importance in the applications is the fact that the FT transforms convolutions into products and vice versa. The convolution (German / Faltung) of two functions f1(x) and f2(x) is denoted by f1(x) * f2(x) and is given by the integral relation (Rossetti, 1984) ZN dx0 f1 ðx
x0 Þf2 ðx0 Þ (240) f ðxÞ ¼ f1 ðxÞ  f2 ðxÞ ¼
N

The convolutory product has the properties f1 ðxÞ  f2 ðxÞ ¼ f2 ðxÞ  f1 ðxÞ

(241)

ðf1  f2 Þ  f3 ¼ f1  ðf2  f3 Þ

(242)

and Now F½ f1  f2  ¼

pffiffiffiffiffiffi 2p F½ f1  $ F½ f2 

(243)

and 1 F½f1 $ f2  ¼ pffiffiffiffiffiffi F½f1   F½f2  2p If f (x) is a real even function, its FT is real and is given by rffiffiffiffi ZN ZN 1 2 FðpÞ ¼ pffiffiffiffiffiffi dx cosðpxÞf ðxÞ ¼ dx cosðpxÞ f ðxÞ p 2p
N

(244)

(245)

0

If f (x) is a real odd function, its FT is purely imaginary and is given by rffiffiffiffi ZN ZN i 2 FðpÞ ¼
pffiffiffiffiffiffi dx sinðpxÞ f ðxÞ ¼
i dx sinðpxÞ f ðxÞ p 2p
N

(246)

0

A few examples of FTs are given in Problems 4.4–4.5 (Rossetti, 1984).

4.10 THE LAPLACE TRANSFORM The Laplace transform (LT) of a function F(t) of the real variable t is defined as ZN LðsÞ ¼

dt FðtÞexpð
stÞ

(247)

0

where s is a complex variable. The integral (247) converges whenever Re(s) > 0. F(t) is called the original function and L(s) the image function. Equation (247) can be rewritten as LðsÞ ¼ L^s FðsÞ

(248)

4.10 The Laplace transform

where L^s is the integral operator with kernel exp(
st) ZN ZN ^ ^ Ls FðsÞ ¼ dt expð
stÞPst FðsÞ ¼ dt expð
stÞFðtÞ ¼ LðsÞ 0

(249)

0

Fundamental properties of the LT are   d FðtÞ ¼ sLs ½FðtÞ
Fð0Þ Ls dt and, in general

189

(250)



 n
1 X dn F ðkÞ ð0Þsn
1
k Ls n FðtÞ ¼ sn Ls ½FðtÞ
dt k¼0

where the notation for the derivatives means F

ðnÞ



dn FðtÞ ð0Þ ¼ lim t/0 dtn

(251)

 (252)

The LT and its property (251) are of interest in the solution of the linear differential equations with constant coefficients such as those derived in chemical kinetics. More details about that can be found elsewhere (Rossetti, 1984). A few LTs, taken from Abramowitz and Stegun (1965), are given in Table 4.1. They are easily proved by integration as shown in Problem 4.6. Table 4.1 Table of Laplace Transforms L(s)

F(t)

1 s 1 s2 1 ðn ¼ 1; 2; 3; /Þ sn 1 aþs 1 ðn ¼ 1; 2; 3; /Þ ða þ sÞn

1 t t n
1 ðn
1Þ! exp(eat) t n
1 expð
atÞ ðn
1Þ!

1 ðbsaÞ ða þ sÞðb þ sÞ

expð
atÞ
expð
btÞ b
a

s
1=2

1 pffiffiffiffiffi pt


 1 s
nþ 2 1 a2 þ s2

1

2n t n
2 pffiffiffiffi 1$3$5/ð2n
1Þ p 1 sinðatÞ a

190

CHAPTER 4 Special functions

4.11 SPHERICAL TENSORS A n-rank tensor is a quantity having 3n components and can be regarded as a vector in a 3n-dimensional space whose components carry a representation of the rotation group (Section 8.15 of Chapter 8). Thus, scalars can be regarded as tensors of rank zero (n¼ 0), while a vector in three-space can be regarded as a first-rank tensor (n ¼ 1) having three components. An irreducible tensor operator T of rank k is an operator with (2 kþ 1) components Tkq transforming in a well-defined way under rotation of axes (Brink and Satchler, 1993). Spherical harmonics are examples of spherical tensors. The spherical tensors Rlm(r), also called regular solid harmonics (Brink and Satchler, 1993; Stone, 1996), can be given in complex or real form. Complex spherical tensors are eigenfunctions of the square of the angular momentum operator L^2 and of the z-component of the angular momentum operator L^z, while real spherical tensors are eigenfunctions only of the square of the angular momentum operator L^2. In atomic theory use is often made of a theorem connected to spherical tensors, the Wigner–Eckart theorem (Wigner, 1959; Eckart, 1930a), which states that    0 0 (253) lmTkq l m ¼ hlkTk kl0 ihl0 m0 kqjlmi where Tkq is the q-component of a spherical tensor T of rank k, hlkTk kl0i is a reduced matrix element independent of m and q, and hl0 m0 kqjlmi is the Clebsch–Gordan coefficient (Chapter 10) describing the coupling of the angular momentum eigenvector jl0 m0 i to jkqi to give a state with resultant angular momentum jlmi. Thus, all directional properties are contained in the Clebsch– Gordan coefficients, while the dynamical properties of the system appear in the scalar factor hlkTk kl0 i.

4.11.1 Spherical tensors in complex form The spherical tensors Rlm(r) in complex form are defined in terms of the (normalized) modified spherical harmonics (Brink and Satchler, 1993), eigenfunctions of the operators L^2 and L^z : 8 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > > ðl
mÞ! m > P ðxÞ½ð
1Þm expðim4Þ > > > ðl þ mÞ! l rffiffiffiffiffiffiffiffiffiffiffiffi > < 4p Ylm ðq; 4Þ ¼ r l Pl ðxÞ Rlm ðrÞ ¼ rl > 2l þ 1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > > ðl
jmjÞ! jmj > > > P ðxÞexpð
ijmj4Þ > : ðl þ jmjÞ! l

m>0 m¼0

(254)

m > ðl
mÞ! m > > 2 P ðxÞcos m4 > > ðl þ mÞ! l > > rffiffiffiffiffiffiffiffiffiffiffiffi > < 4p c;s Pl ðxÞ Ylm ðq; 4Þ ¼ r l Rlm ðrÞ ¼ rl > 2l þ 1 > s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > > > ðl þ mÞ! jmj > > > : 2 ðl
mÞ! Pl ðxÞsinjmj4

m>0 m¼0

(256)

m R20 ¼ 3z2
r 2 > > > 2 > > > < pffiffiffi pffiffiffi R21 ¼ 3z x; R21 ¼ 3yz > > > > pffiffiffi > > pffiffiffi > : R ¼ 3
x2
y2 ; R ¼ 3xy 22 22 2

(259)

8 5z2
3r 2 > > > R30 ¼ z > > 2 > > > > rffiffiffi rffiffiffi > > >   3
2 3
2 > 2 > > R31 ¼ 5z
r x; R31 ¼ 5z
r 2 y > < 8 8 pffiffiffiffiffi > > pffiffiffiffiffi  > 15
2 2 > > R x ¼
y ¼ 15xyz z; R 32 > 32 > 2 > > > > > rffiffiffi rffiffiffi > >   > 5
2 5
2 > 2 : R33 ¼ x
3y x; R33 ¼ 3x
y2 y 8 8

(260)

192

CHAPTER 4 Special functions

8 > > > > R40 > > > > > > > > > > R41 > > > > > > > < R42 > > > > > > > > > > R43 > > > > > > > > > > > : R44

8z4 þ 3x4 þ 3y4
24z2 x2
24y2 z2 þ 6x2 y2 8 pffiffiffi 2 pffiffiffi 2 5 4z
3x2
3y2 5 4z
3x2
3y2 zx; R41 ¼ pffiffiffi yz ¼ pffiffiffi 2 2 2 2 ¼

pffiffiffi 6z2
x2
y2 pffiffiffi 6z2
x2
y2
2  x
y2 ; R42 ¼ 5 xy 5 4 4 pffiffiffiffiffi 2 pffiffiffiffiffi 35 x
3y2 35 3x2
y2 z x; R43 ¼ pffiffiffi yz ¼ pffiffiffi 2 2 2 2 ¼

¼

pffiffiffiffiffi x4 þ y4
6x2 y2 ; 35 8

R44 ¼

(261)

pffiffiffiffiffi x2
y2 xy 35 2

The STOs of valence theory are real spherical tensors multiplied by a radial decay factor exp(
cr) with c a real non-zero positive number. Figures 4.1–4.5 give the angular form in space for the AOs with l ¼ 0, 1, 2, 3, 4 and m ¼ 0.5 The plots were obtained from the function ParametricPlot3D of Mathematica 6 (Wolfram Research, 2007) using a small software due to Ottonelli.

FIGURE 4.1 Three-dimensional representation in space of the s AO with l ¼ m ¼ 0 (For colour version of this figure, the reader is referred to the online version of this book.)

The form of complex and real spherical tensors is identical for m ¼ 0.

5

4.11 Spherical tensors

193

FIGURE 4.2 Three-dimensional representation in space of the p AO with l ¼ 1 m ¼ 0 (For colour version of this figure, the reader is referred to the online version of this book.)

FIGURE 4.3 Three-dimensional representation in space of the d AO with l ¼ 2 m ¼ 0 (For colour version of this figure, the reader is referred to the online version of this book.)

194

CHAPTER 4 Special functions

FIGURE 4.4 Three-dimensional representation in space of the f AO with l ¼ 3 m ¼ 0 (For colour version of this figure, the reader is referred to the online version of this book.)

FIGURE 4.5 Three-dimensional representation in space of the g AO with l ¼ 4 m ¼ 0 (For colour version of this figure, the reader is referred to the online version of this book.)

4.12 Orthogonal polynomials

195

4.11.3 Generalized spherical tensors The generalized spherical tensors in k-space (Magnasco and Figari, 1989) are spherical tensors whose rl radial part is replaced by (2lþ 1)!!jl(rk): rffiffiffiffiffiffiffiffiffiffiffiffi 4p Ylm ðq; 4Þ (262) Rlm ðr; kÞ ¼ ð2l þ 1Þ!!jl ðrkÞ 2l þ 1 where jl(rk) is a spherical Bessel function (Abramowitz and Stegun, 1965). They are related to the ordinary spherical tensors Rlm(r) by the relation   Rlm ðr; kÞ 1 vl ¼ R ðr; kÞ (263) Rlm ðrÞ ¼ lim lm k/0 kl l! vkl k¼0 The spherical Bessel functions were discussed in Section 4.6.3 and (2 l þ 1)!! is the double factorial function defined by Eqn (138). We then have rffiffiffiffiffiffiffiffiffiffiffiffi  r lþ2 klþ2 r lþ4 klþ4 4p l l þ
/ Ylm ðq; 4Þ (264) Rlm ðr; kÞ ¼ r k
2ð2l þ 3Þ 8ð2l þ 3Þð2l þ 5Þ 2l þ 1 an equation which is obtained from the representation (139) of jl(rk) as the infinite series jl ðrkÞ ¼

N X

ð
1Þp ðrkÞlþ2p ð2pÞ!!ð2p þ 2l þ 1Þ!! p¼0

(265)

4.12 ORTHOGONAL POLYNOMIALS We have seen in Section 4.2.3 that any regular function f(x), defined in the interval jxj < 1, can be expanded in series of Legendre polynomials. They form a complete set of orthogonal functions so that the expansion is unique. However, they are not the only possible set of orthogonal polynomials. Given a set of powers of the variable, say {xn}, it is always possible by Schmidt orthogonalization (Chapter 1) to obtain a complete set of polynomials {Pn(x)} of degree n, orthogonal and normalizable in the finite interval (a,b) and with respect to any weight function p(x), such that Zb ðPn ; Pm Þ ¼

dx Pn ðxÞPm ðxÞpðxÞ ¼ dnm

(266)

a

Any power xn can then be expressed in terms of the first (n þ 1) orthogonal polynomials, each polynomial being orthogonal to any polynomial of lower degree. It can be shown that the expansion is unique and the following important theorem exists (Rossetti, 1984): the zeros of a polynomial Pn(x) belonging to a system of polynomials {Pn(x)} orthogonal in L2 are all simple and lie inside the interval (a,b). The importance of orthogonal polynomials lies in the fact that the n-th reduced sum in the expansion of a continuous function f(x) allows for the interpolation of the function in at least (n þ 1) points within the interval (a,b).

196

CHAPTER 4 Special functions

In other words, if we have the expansion f ðxÞ ¼

N X

ck Pk ðxÞ

(267)

k¼0

where the polynomials are orthogonal according to Eqn (266), then the n-th reduced sum Sn ðxÞ ¼

n X

ck Pk ðxÞ

(268)

k¼0

coincides with the function f(x) in at least (n þ 1) points within the interval (a,b). This property is of great importance in practical applications, such as those involving numerical quadratures, e.g. the Gauss–Legendre quadrature. If X(x) is at most a quadratic polynomial with real roots, then it can be shown (Rossetti, 1984) that the Rodriguez formula holds: Pn ðxÞ ¼

1 1 dn ½ pðxÞX n ðxÞ Kn pðxÞ d xn

(269)

where p(x) is the weight function, which must be real and integrable in (a,b), and Kn is a constant factor. Possible forms of p(x) are pðxÞ ¼ ðb
xÞa ðx
aÞb

(270)

pðxÞ ¼ expð
xÞðx
aÞa

(271)


 pðxÞ ¼ exp
x2

(272)

with a and b real and >
1; with a >
1;

In Eqn (269), the constant factor Kn, called the standardization factor, defines in a unique way each different class of polynomials. The majority of orthogonal polynomials entering practical applications are given by the Rodriguez formula (269). They are called classical orthogonal polynomials, and correspond to three classes according to the form chosen from Eqn (270–272) for the weight function p(x). Most common is to choose the interval (a,b) as (
1,1),6 in which case XðxÞ ¼ 1
x2

(273)

and the weight function p(x) is pðxÞ ¼ ð1
xÞa ð1 þ xÞb

a; b >
1

(274)

Such polynomials are called Jacobi polynomials. In the interval (0,N), X(x) is given by XðxÞ ¼ x 6

It is always possible to reduce any interval (a,b) to (
1,1) by an appropriate linear transformation.

(275)

4.13 Pade´ approximants

197

with the weight factor pðxÞ ¼ expð
xÞxa

a >
1

(276)

Such polynomials are called Laguerre polynomials. They were treated in detail in Section 4.3. In the interval (
N,N), X(x) is given by with the weight factor

XðxÞ ¼ 1

(277)


 pðxÞ ¼ exp
x2

(278)

Such polynomials are called Hermite polynomials. They were treated in detail in Section 4.4. Specific values given to a and b originate particular polynomials. For instance, the choice a¼b¼l


1 2

l>


1 2

(279)

gives the Gegenbauer (or ultraspherical) polynomials, denoted by Cnl ðxÞ. Among these polynomials, particular importance have the Tchebichef polynomials (of first and second kind) which are obtained by choosing l ¼ 0, 1 and, perhaps the most important of all, the Legendre polynomials, obtained by choosing l ¼ 1/2. These polynomials were treated in detail in Section 4.2. So, we see that all polynomials found in the previous sections of this chapter, obtained as regular solutions of the appropriate differential equations of Chapter 3, can be reassigned to particular classes of the more general definition of orthogonal polynomials. Much more than this can be found in Chapter IV of Rossetti (1984).

4.13 PADE´ APPROXIMANTS Even if strictly pertinent to numerical analysis, a few elements of the Pade´ approximants technique (Baker and Gammel, 1975) may be of interest here. They are of importance, for instance, in assessing the form of the exchange-correlation functional (Vosko et al., 1980) in density functional theory, in Rayleigh–Schroedinger (RS) (Wilson et al., 1977) and Brillouin–Wigner (BW) (Bendazzoli et al., 1970) perturbation theories, and in the power expansion of time-dependent correlation functions (Paul, 1980). The Pade´ approximants technique deals with the possibility of reproducing in an efficient way a given function f(x) in terms of polynomials and it is an improvement over the Taylor approximants. We first recall a theorem due to Weierstrass which states that if a function f(x) is continuous at a point x0, interior to the domain of definition of the function, for any arbitrary positive ε it is possible to find a polynomial: n X ak ðx
x0 Þk (280) Pn ðx
x0 Þ ¼ k¼0

such that for any point x j f ðxÞ
Pn ðx
x0 Þj < ε

(281)

The degree of the polynomial and the values of its coefficients are flexible parameters that must be properly fixed so as to satisfy the required accuracy ε.

198

CHAPTER 4 Special functions

We examine first the Taylor approximant. The Taylor expansion of a function f(x) is the power series f ðxÞ ¼

N X

ak ðx
x0 Þk

(282)

k¼0

with ðkÞ

f ak ¼ 0 ; k!

ðkÞ f0

 ¼

dk f d xk

 (283) x¼x0

The degree of a Taylor polynomial can be unlimited as far as the function admits unlimited derivatives. Taylor’s expansion requires rather regular functions, having in x0 continuous derivatives up to the order corresponding to its polynomial approximant. Stopping the expansion at a finite degree n, we have a truncation error in the Taylor approximation, namely, we have a remainder Rn(x) which we can estimate, for instance, in the Lagrange form: Rn ðxÞ ¼ f ðxÞ


ðkÞ n X f0 f ðnþ1Þ ðxÞ ðx
x0 Þk ¼ ðx
x0 Þnþ1 k! ðn þ 1Þ! k¼0

(284)

for x ˛ ðx0 ; xÞ

(285)

The difference at the point x0 between the function and its polynomial approximant is therefore proportional to the value assumed, at a point x lying in-between x0 and x, by the derivative successive to the last derivative included in the polynomial. It is clear that if f (nþ1)(x) is limited in absolute value at the interior of the interval (x0,x), the error has an upper bound which can be evaluated exactly. We can see that f (nþ1)(x) is the factor to be included in the remainder if we write f ðxÞ


ðkÞ n X f0 ðx
x0 Þnþ1 ðx
x0 Þk
D ¼0 k! ðn þ 1Þ! k¼0

(286)

Then, using repeatedly Rolle’s theorem, it can be shown that f ðnþ1Þ ðxnþ1 Þ
D ¼ 0

(287)

(nþ1)

(xnþ1) where xnþ1 is a value (denoted by x in Eqn (284) for the so that D coincides with f remainder) lying in the interval (x0,x). The remainder is an indicator of the convergence properties of the series, which will be reproduced exactly if lim Rn ðxÞ ¼ 0

n/N

(288)

Let us now turn to the Pade´ approximants. In this case, the values of the function f(x) are estimated as the ratio [m,n] between two polynomials: f ðxÞ ¼

Qm ðx
x0 Þ ¼ ½m; n Tn ðx
x0 Þ

(289)

4.14 Green’s functions

where7 Qm ðx
x0 Þ ¼

m X

bk ðx
x0 Þk

199

(290)

k¼0

Tn ðx
x0 Þ ¼

n X

ck ðx
x0 Þk

ðc0 ¼ 1Þ

(291)

k¼0

The coefficients of the Pade´ expansion are obtained by the comparison with the corresponding Taylor expansion of the function ðkÞ N X f Qm ðx
x0 Þ ; ak ¼ 0 (292) ak ðx
x0 Þk ¼ k! Tn ðx
x0 Þ k¼0 N X

ak ðx
x0 Þk $

k¼0

n X

cl ðx
x0 Þl ¼

m X

bm ðx
x0 Þm

(293)

m¼0

l¼0

Since coefficients of the same power of (x
x0) must be identical, this gives m X bm ðm ¼ 0; 1; 2; /; mÞ am
l cl ¼ 0 ðm ¼ m þ 1; m þ 2; /; m þ nÞ

(294)

l¼0

cl ¼ 0

for l > n

(295)

In this way, we obtain an algebraic system where the m þ n þ 1 unknown parameters b0, b1,/, bm, c1, c2,/, cn are put in relation to the a0,a1,/,amþn parameters of the corresponding Taylor series. The connection between Pade´ and Taylor expansions is such that their difference depends on powers higher than m þ n: Xm XN Xn m N b ðx
x Þ X a c ðx
x0 Þk m 0 m¼0 k k¼mþnþ1 l¼0 k
l l Xn ak ðx
x0 Þ
Xn ¼ (296) c ðx
x0 Þl c ðx
x0 Þl k¼0 l¼0 l l¼0 l It is not always possible to associate a Pade´ approximant with given values of m and n to the Taylor expansion of a function, because sometimes system (294) does not admit a solution. However, if the Pade´ approximant exists, then it is a more flexible instrument than the Taylor series, which must be regarded as a particular case of Pade´ approximant having n ¼ 0 in the denominator of Eqn (289). Unfortunately, there does not exist any rule giving the best balance of the powers m and n of the two polynomials so as to give the most efficient approximation, but it is generally true that m ¼ n and m ¼ n  1 are appropriate choices.

4.14 GREEN’S FUNCTIONS Green’s functions originate in the study of differential equations and even if they occur in many problems of theoretical and mathematical physics (Morse and Fesbach, 1953; Courant and Hilbert, Choosing c0 ¼ 1 is not a constraint, since it corresponds to dividing each term of the fraction by c0 leaving the fraction unchanged. 7

200

CHAPTER 4 Special functions

1989) they are sensibly less popular in quantum chemistry, except perhaps for their use in propagator theory (Thouless, 1961; McWeeny, 1992). For a conceptually simple introduction to Green’s functions theory we follow here the presentation given by Mathews and Walker (1970). The theory is strictly connected with the solution of the inhomogeneous differential equations of mathematical physics arising when an eigenvalue problem is modified by some point field disturbance and rests on the fundamental statement that the series expansion of a regular function in the complete set of eigenfunctions of a Hermitian operator L^ converges absolutely and uniformly in the domain D of definition of the function. We assume that we know the whole set of orthogonal eigenfunctions {uk(x)} and eigenvalues lk of the operator L^ obtained as solutions of the eigenvalue equation ^ k ðxÞ ¼ lk uk ðxÞ; huk juk0 i ¼ dkk0 (297) Lu We want to solve for the unknown function u(x) the inhomogeneous differential equation: ^ LuðxÞ
luðxÞ ¼ f ðxÞ

(298)

where L^ is a linear Hermitian operator, l is a parameter, and u(x) and f(x) are regular functions defined over a domain D and subjected to the usual boundary conditions. We expand u(x) and the inhomogeneity f(x) in the eigenfunction (297) of the operator L^ as Z X (299) ak uk ðxÞ; ak ¼ d x0 uk ðx0 Þuðx0 Þ ¼ huk jui uðxÞ ¼ k

f ðxÞ ¼

X

Z bk uk ðxÞ;

bk ¼

k

so that Eqn (298) becomes

X

d x0 uk ðx0 Þf ðx0 Þ ¼ huk jf i

ak ðlk
lÞuk ðxÞ ¼

X

k

(301)

k

X

namely,

bk uk ðxÞ

(300)

ak ½ðlk
lÞ
bk uk ðxÞ ¼ 0

(302)

k

Since the set {uk(x)} is linearly independent, we obtain ak ½ðlk
lÞ
bk  ¼ 0 and, provided (lk
l) s 0 ak ¼

bk ¼ lk
l

Z

d x0

uk ðx0 Þf ðx0 Þ ¼ ðlk
lÞ
1 huk j f i lk
l

Therefore X

X

Z

uk ðx0 Þf ðx0 Þ lk
l k k Z Z X uk ðxÞ d x0 uk ðx0 Þf ðx0 Þ ¼ d x0 Gðx; x0 Þf ðx0 Þ ¼
l l k k

uðxÞ ¼

ak uk ðxÞ ¼

uk ðxÞ

(303)

(304)

d x0

D

(305)

4.14 Green’s functions

201

where Gðx; x0 Þ ¼

X uk ðxÞu ðx0 Þ k

k

lk
l

¼

X juk ðxÞ >< uk ðx0 Þj k

lk
l

(306)

is called the Green’s function of the operator L^. To emphasize the dependence of G on the value of the parameter l, the Green’s function is sometimes written as Gðx; x0 ; lÞ. It can be seen that the Green’s ^ function (306) is the kernel of the integral operator GðxÞ whose action on a function f(x) is Z ^ GðxÞf ðxÞ ¼ dx0 Gðx; x0 Þf ðx0 Þ ¼ uðxÞ (307) If f(x) is the Dirac’s d-function f ðxÞ ¼ dðx
x0 Þ then the solution of Eqn (298) can be written as Z uðxÞ ¼ dx0 Gðx; x0 Þdðx0
x0 Þ ¼ Gðx; x0 Þ

(308)

(309)

D

so that

Gðx; x0 Þ

is the solution of the differential equation ^ LGðx; x0 Þ
lGðx; x0 Þ ¼ dðx
x0 Þ

(310)

In other words, the physical meaning of the Green’s function is that of being the solution of problem (298) when the disturbance is the unit point source f ðxÞ ¼ dðx
x0 Þ. The problems arising when l equals one of the eigenvalues of L^ are avoided by imposing the orthogonality condition Z (311) dx uk ðxÞf ðxÞ ¼ 0 These considerations are readily extended to three-dimensional space simply by replacing x by r and dx by dr. In the RS perturbation theory of Chapter 1, the first-order equation ðH^0
E0 Þj1 þ ðV
E1 Þj0 ¼ 0

(312)

which must be solved under the orthogonality constraint hj0 jj1 i ¼ 0

(313)

j1 ¼
ðH^0
E0 Þ
1 ðV
E1 Þj0

(314)

has the formal solution Expanding j1 in the eigenstates fjk g of H^0 : Z X jk ðrÞ Z 0  0 0 j1 ¼
dr jk ðr ÞðV
E1 Þj0 ðr Þ ¼
dr0 Gðr; r0 ÞðV
E1 Þj0 ðr0 Þ
E E k 0 kðs 0Þ

(315)

202

CHAPTER 4 Special functions

the Green’s function Gðr; r0 Þ ¼

X jk ðrÞj ðr0 Þ k
E0 E k kðs 0Þ

(316)

is called the Rayleigh–Schroedinger resolvent of Eqn (312) (Courant and Hilbert, 1989; Hubac et al., 2000). Unfortunately, Eqns (314) and (316) are purely formal since, as already said in Chapter 1, the correct application of Eqn (315) requires consideration of the whole discrete spectrum belonging to the bounded eigenstates of H^0 as well as of its continuous spectrum belonging to the ionized state. The Green’s function in spherical coordinates for the ground state of the hydrogen atom was obtained by Hameka (1967) following earlier work by Meixner (1933), and resulted in extremely complicated sectorialized formulae involving products of confluent hypergeometric functions that cannot be reported here. Extension of the method to the ground state of helium (Hameka, 1968a) and lithium (Hameka, 1968b) atoms resulted in purely formal results not practical for numerical estimates. Molecular applications to the second-order induction energy of a hydrogen atom perturbed by a proton a distance R apart were done by Pan and Hameka (1968) and by Singh et al. (1970), who used a modified form of the Green’s functions obtained as solutions of differential equations rather than by direct summation of their spectral expansion (316). Cha1asinski and Jeziorski (1974) were the first who succeeded in calculating exactly the second-order induction interaction between the atoms in the hydrogen molecule using the Green’s functions of the hydrogen atom as explained in detail in Section 17.3.3 of Chapter 17 of this book. In the opinion of the author, the method of linear pseudostates presented in Section 1.3.2.4 of Chapter 1 of this book seems more valuable for obtaining practical approximations, even very accurate, to second-order quantities, as shown for the second-order energy in the hydrogen-like perturbation theory of the ground state of the helium atom (Byron and Joachain, 1967; Magnasco et al., 1992a), for the damping coefficients in the non-expanded second-order induction energy for Hþ 2 (Magnasco and Figari, 1987a), and from the early calculations of C6 dispersion coefficients for the long-range H–H interaction (Magnasco and Figari, 1987b) up to the recently very accurate results for the dispersion coefficients of simple atomic and molecular systems obtained by use of reduced pseudospectra (Magnasco and Figari, 2009).

4.15 PROBLEMS 4 4.1. Find the quadratic equation in q occurring in the Legendre’s equation expressed in the form of descending powers of the variable. Answer: qðq þ 1Þ
nðn þ 1Þ ¼ ðq
nÞðq þ n þ 1Þ ¼ 0 4.2. Solve the Bessel’s equation of integral order. Answer:   xn 1 2 Jn ðxÞ ¼ n 0 F1 n þ 1;
x 2 n! 4 Hint: Solve the differential Eqn (117) by the usual series expansion in the variable x.

4.16 Solved problems

203

4.3. Prove properties (1)–(4) of the gamma function. Hint: Use the definition of gamma function and the rule of integration by parts. 4.4. Find the FT of the exponential and the Gaussian function. Answer: 1. The exponential function  1=2 2 c FðpÞ ¼ p c 2 þ p2 2. The Gaussian function  2 2 c c p FðpÞ ¼ pffiffiffi exp
4 2 Hint: Evaluate the integrals defining the FT of these functions. 4.5. Find the Fourier and the anti-FT of the Coulombic potential. Answer: Z 1 expðik $ rÞ 1 dk ¼ FðrÞ ¼ 2 2p k2 r FðkÞ ¼

Z

1 ð2pÞ3=2

dr expð
ik $ rÞ

1 ¼ r

 1=2 2 1 p k2

Hint: Evaluate the integrals defining the FT and the anti-FT of 1/r. 4.6. Prove the LTs of Table 4.1. Hint: Use the definition of LT and ordinary integration rules.

4.16 SOLVED PROBLEMS 4.1. The quadratic equation occurring in Legendre’s equation. The recurrence relation gives for 2k ¼
2 b0 ¼

ðq þ 2Þðq þ 1Þ qðq þ 1Þ
nðn þ 1Þ b
2 0 b
2 ¼ b0 qðq þ 1Þ
nðn þ 1Þ ðq þ 2Þðq þ 1Þ

(317)

Since b0 s 0, the constraint b
2 ¼ 0 necessarily implies qðq þ 1Þ
nðn þ 1Þ ¼ 0

(318)

204

CHAPTER 4 Special functions

a quadratic equation in q whose solutions are q ¼ n and q ¼
n
1, and which can be written in terms of these solutions in the form: ðq
nÞðq þ n þ 1Þ ¼ 0

(319)

4.2. The Bessel’s functions of integral order. For n positive integer, the Bessel’s differential equation of the second order is
 x2 y00 ðxÞ þ xy0 ðxÞ þ x2
n2 yðxÞ ¼ 0

(320)

The solution near the singular point x ¼ 0 suggests the power expansion: N X yðxÞ ¼ ak xkþa

(321)

k¼0

where a is a constant to be determined by solution of the indicial equation. We have for the derivatives N N X X ðk þ aÞak xkþa
1 ; y00 ðxÞ ¼ ðk þ aÞðk þ a
1Þak xkþa
2 (322) y0 ðxÞ ¼ k¼0

k¼0

so that substituting in the differential equation N N N X X X ðk þ aÞðk þ a
1Þak xkþa þ ðk þ aÞak xkþa þ ak xkþaþ2
n2 ak xkþa ¼ 0

N X k¼0

k¼0

k¼0

(323)

k¼0

we obtain for the coefficient of xkþa the equation

h i ðk þ aÞðk þ a
1Þak þ ðk þ aÞak
n2 ak ¼ 0 0 ðk þ aÞ2
n2 ak ¼ 0

giving as solution of the indicial equation the two roots 
k ¼ 0 0 a2
n2 a0 ¼ 0 0 a ¼ n

(324)

(325)

since a0 s 0. The same result could have been obtained using Eqn (104) of Section 3.5 of the previous chapter by noting that p1 ðxÞ ¼ q1 ðxÞ ¼ 1;

r1 ðxÞ ¼
n2

for x0 ¼ 0

giving the same quadratic equation in a as before. Hence for a ¼ n the first solution of the Bessel’s equation can be written as 8 y ðxÞ ¼ xn FðxÞ > > < 1 
y01 ðxÞ ¼ xn F 0 þ nx
1 F > > : y00 ðxÞ ¼ xn F 00 þ 2nx
1 F 0 þ
n2
nF 

(326)

(327)

1

By substituting Eqn (327) into the Bessel’s Eqn (320) we obtain the differential equation determining the unknown function F(x): x2 F 00 þ ð2n þ 1ÞxF 0 þ x2 F ¼ 0

(328)

4.16 Solved problems

an equation which we solve by the usual power expansion as N N N X X X ak xk ; F 0 ðxÞ ¼ kak xk
1 ; F 00 ðxÞ ¼ kðk
1Þak xk
2 FðxÞ ¼ k¼0

k¼1

205

(329)

k¼2

therefore obtaining N X

kðk
1Þak xk þ ð2n þ 1Þ

k¼2

N X

kak xk þ

k¼1

N X

ak xkþ2 ¼ 0

(330)

k¼0

The coefficient of xk in expansion (330) is kðk
1Þak þ ð2n þ 1Þkak þ ak
2 ¼ 0

(331)

giving the recurrence relation ak ¼


ak
2 kð2n þ kÞ

(332)

Since series (329) starts with the term a0 s 0, we must take a1 ¼ 0 and, in order the equation above may be satisfied for all k  2, we must take 8 > < a2kþ1 ¼ 0 (333) k ¼ 1; 2; 3; / a2k
2 > : a2k ¼
2kð2n þ 2kÞ so that only even coefficients survive in the expansion. To relate the general coefficient a2k to the non-zero coefficient a0, we write down the first few coefficients which are given explicitly by a0 (334) k ¼ 1 a2 ¼
2 $ 1ð2n þ 2Þ k¼2 k¼3

a4 ¼


a2 ð
1Þ2 a0 ¼ 2 $ 2ð2n þ 4Þ 2 $ 2ð2n þ 4Þ $ 2 $ 1ð2n þ 2Þ

a4 ð
1Þ3 a0 ¼ a6 ¼
2 $ 3ð2n þ 6Þ 2 $ 3ð2n þ 6Þ $ 2 $ 2ð2n þ 4Þ $ 2 $ 1ð2n þ 2Þ

(335) (336)

Therefore the (2k)-th coefficient is given in terms of a0 as a2k ¼ ¼

ð
1Þk a0 2kð2n þ 2kÞ $ 2ðk
1Þð2n þ 2k
2Þ $ 2ðk
2Þð2n þ 2k
4Þ/2 $ 1ð2n þ 2Þ ð
1Þk a0 ½2kð2k
2Þð2k
4Þ/2 $ 1½ð2n þ 2kÞð2n þ 2k
2Þð2n þ 2k
4Þ/ð2n þ 2Þ

ð
1Þk a0 ¼ k ½2 kðk
1Þðk
2Þ/1½2k ðn þ kÞðn þ k
1Þðn þ k
2Þ/ðn þ 2Þðn þ 1Þ  k ð
1Þk a0 a0 1 ¼ ¼ 2k
4 2 k!½ðn þ 1Þðn þ 2Þ/ðn þ k
1Þðn þ kÞ k!ðn þ 1Þk where in the last expression we have introduced the Pochhammer’ symbol (96).

(337)

206

CHAPTER 4 Special functions

Taking a0 ¼¼ 1/(2nn!), we obtain for the general coefficient   k  1 1 1
k ¼ 0; 1; 2; / a2k ¼ n 2 n! k!ðn þ 1Þk 4

(338)

finally obtaining the solution for a ¼ n in the form of the infinite series in even powers of x  2 k N N X xn X 1 x y1 ðxÞ ¼ xn
a2k x2k ¼ n (339) 2 n! 4 k!ðn þ 1Þ k k¼0 k¼0 Writing explicitly the first few terms of the series N X a2k x2k ¼ a0 þ a2 x2 þ a4 x4 þ / k¼0

"

1 ð
1Þ2 x2 þ x4
/ ¼ a0 1
2ð2n þ 2Þ 2 $ 2ð2n þ 4Þ $ 2 $ 1ð2n þ 2Þ   2  2 2  1 x 1 x þ ¼ a0 1 þ


/ 4 4 nþ1 1 $ 2ðn þ 1Þðn þ 2Þ

# (340)

we see that the series in Eqn (340) can be expressed in terms of the hypergeometric function 0 F1 ðc; zÞ containing just the single parameter c: 1 1 (341) zþ z2 þ / 0 F1 ðc; zÞ ¼ 1 þ 1$c 1 $ 2cðc þ 1Þ which for c ¼ n þ 1 and z ¼
x2/4 becomes    2  2 2 x2 1 x 1 x n þ 1;
¼ 1 þ þ F þ/

1 0 4 4 4 1 $ ðn þ 1Þ 1 $ 2ðn þ 1Þðn þ 2Þ so that y1 ðxÞ ¼

  xn x2 ¼ Jn ðxÞ n þ 1;
F 1 2n n! 0 4

(342)

(343)

which is Eqn (118) of the main text. Noting that ðn þ kÞ! ¼ ðn þ kÞðn þ k
1Þðn þ k
2Þ/½n þ k
ðk
2Þ½n þ k
ðk
1Þðn þ k
kÞ! ¼ ðn þ kÞðn þ k
1Þðn þ k
2Þ/ðn þ 2Þðn þ 1Þn! ¼ ðn þ 1Þk n!

(344)

we can write for Jn(x) the equivalent expressions Jn ðxÞ ¼ ¼

 2 k  X  2 k N xn X 1 x x n N 1 x

¼ 2n n! k¼0 k!ðn þ 1Þk 4 4 2 k¼0 k!ðn þ kÞ! N X

ð
1Þk xnþ2k nþ2k k!ðn þ kÞ! 2 k¼0

(345)

4.16 Solved problems

207

Because of relation (125), for n integer, the two solutions Jn(x) and J
n(x) are not linearly independent. Relation (125) of the main text is most easily derived in terms of the coefficients (Bessel’s coefficients of order n) of tn and t
n in the symmetrical series expansion of the exponential, which gives (Sneddon, 1956)    N x  x X x 1 t
¼ exp t exp
t
1 ¼ Jn ðxÞtn (346) exp 2 t 2 2 n¼
N For n ¼ n not an integer, we simply replace factorials by gamma functions obtaining   xn x2 n þ 1;
Jn ðxÞ ¼ n F 1 2 Gðn þ 1Þ 0 4

(347)

The two solutions Jn(x) and J
n(x) are now linearly independent and we can write the general solution of Eqn (320) in the form yðxÞ ¼ AJn ðxÞ þ BJ
n ðxÞ

(348)

with A, B arbitrary constants. It can be shown that the series obtained in this way are convergent and differentiable for any value of x, so that our formal solutions are the solution of Bessel’s differential Eqn (117) of the main text. Generalizing what we have seen so far, we give below some current definitions of the particular solutions of the Bessel’s Eqn (320) with n ¼ n (Sneddon, 1956): Jn ðxÞ Bessel’s functions ðBFsÞ of the first kind of order v

(349)

Yn ðxÞ BFs of the second kind of order v ðor Weber’s BFsÞ

(350)

where Yn(x) is a rather complicated function containing a logarithmic part, with g the Euler’s constant of Section 4.7.3 8      n
1 > 2 1 1 X ðn
k
1Þ! 2 n
2k > > g þ ln x Jn ðxÞ
Yn ðxÞ ¼ > > < p 2 p k¼0 k! x (351) nþ2k > N k k  X X > 1 ð
1Þ 1 1 > > > x ½4ðn þ kÞ þ 4ðkÞ 0 4ðkÞ ¼ :
p k!ðn þ kÞ! 2 s s¼1 k¼0 8 < Hnð1Þ ðxÞ ¼ Jn ðxÞ þ iYn ðxÞ (352) BFs of the third kind of order v ðor Hankel’s BFsÞ : H ð2Þ ðxÞ ¼ J ðxÞ
iY ðxÞ n

n

n

ð1Þ

ð2Þ

where i is the imaginary unit. Jn(x) and Yn(x), as well as Hn ðxÞ and Hn ðxÞ, are functions linearly independent for any value of n. Expressions for the functions J0(x), Y0(x) of order zero are given in Sneddon (1956). Y0(x) is also called Neumann’s Bessel function of the second kind and zero order. Much the same can be said for the SBFs which are solutions of the differential equation

 (353) x2 j00n ðxÞ þ 2xj0n ðxÞ þ x2
nðn þ 1Þ jn ðxÞ ¼ 0 with n a non-negative integer.

208

CHAPTER 4 Special functions

Particular solutions are rffiffiffiffiffi p J 1 ðxÞ SBFs of the first kind of order n 2x nþ 2 rffiffiffiffiffi p Y 1 ðxÞ SBFs of the second kind of order n yn ðxÞ ¼ 2x nþ 2 8 rffiffiffiffiffi > p ð1Þ ð1Þ > > > < hn ðxÞ ¼ jn ðxÞ þ iyn ðxÞ ¼ 2x Hnþ 12 SBFs of the third kind of order n rffiffiffiffiffi > > p ð2Þ ð2Þ > > hn ðxÞ ¼ jn ðxÞ
iyn ðxÞ ¼ H 1 : 2x nþ 2 jn ðxÞ ¼

(354) (355)

(356)

4.3. Properties (1)–(4) of the gamma function. Properties (1)–(4) can be proved by integration. We have 1. n ¼ 1 ZN

ZN d x expð
xÞ ¼


Gð1Þ ¼ 0

0 d½expð
xÞ ¼
expð
xÞjN 0 ¼ expð
xÞjN ¼ 1

0

Reminding that dðuvÞ ¼ u dv þ v du 0 u dv ¼ dðuvÞ
v du where u is the finite factor and dv the differential factor, by integrating both members follows the rule of integration by parts: Z

Z u dv ¼ uv


v du

Then, integrating by parts 2. ZN Gðn þ 1Þ ¼

d x expð
xÞx

ðnþ1Þ
1

0

¼
xn expð
xÞjN 0 þn

ZN

ZN ¼

ZN d x expð
xÞx ¼
n

0

d x expð
xÞxn
1 ¼ nGðnÞ 0

xn d½expð
xÞ 0

4.16 Solved problems

209

By repeated integration by parts 3.

ZN Gðn þ 1Þ ¼

ZN d x expð
xÞx ¼


x d½expð
xÞ ¼
x

n

0

n

n

expð
xÞjN 0

ZN þn

0

ZN ¼
n

dx expð
xÞxn
1 0

xn
1 d½expð
xÞ ¼
nxn
1 expð
xÞjN 0 þn

0

ZN


expð
xÞd xn
1

0

ZN

ZN ¼ nðn
1Þ

n
2

dx expð
xÞx 0

2

¼ nðn
1Þðn
2Þ/4


ZN

¼ / ¼ nðn
1Þðn
2Þ/

dx expð
xÞxn
ðn
2Þ 0

3 x2 d½expð
xÞ5

0

8 9 ZN < N
2 = 2 ¼ nðn
1Þðn
2Þ/
x expð
xÞ0 þ expð
xÞd x : ; 0

ZN ¼ nðn
1Þðn
2Þ/2

2

d x expð
xÞx ¼ nðn
1Þðn
2Þ/24


0

2 ¼ nðn
1Þðn
2Þ/24
x expð
xÞjN 0 þ

ZN

3

ZN

3 xd½expð
xÞ5

0

d x expð
xÞ $ 15

0

2 ¼ nðn
1Þðn
2Þ/24


ZN

3 d½expð
xÞ5 ¼ nðn
1Þðn
2Þ/2 $ 1 ¼ n!

0

1 2   ZN ZN 1 1
1 2 d x expð
xÞx ¼ d x expð
xÞx
1=2 ¼ G 2

4. n ¼

0

0

Now, change variable to x ¼ u2 ;

d x ¼ 2u du;

u ¼ x1=2

the interval of definition of the new variable being the same as that of x. Then pffiffiffiffi   ZN ZN
2 
1
 1 p pffiffiffiffi 2u du exp
u u ¼ 2 du exp
u2 ¼ 2 ¼ G ¼ p 2 2 0

0

210

CHAPTER 4 Special functions

where use was made of the integral over Gaussian functions ZN


 ðn
1Þ!! du exp
cu2 un ¼ sðnÞ nþ1 ð2cÞ 2 0 rffiffiffiffi p for n ¼ even; sðnÞ ¼ 1 sðnÞ ¼ 2

for n ¼ odd

and (n
1)!! is the double factorial function defined by Eqn (138). Therefore, for n ¼ 0, c ¼ 1 rffiffiffiffi pffiffiffiffi ZN
2 p 1 1 p du exp
u ¼ pffiffiffi sð0Þ ¼ pffiffiffi ¼ 2 2 2 2 0   pffiffiffiffi 1 ¼ p follows. and G 2 By repeated integration by parts it is also possible to derive the duplication formula (5). The recurrence property (2) allows one to obtain the G functions whose argument is a negative fraction. For example, 7.

   1 1   G
þ1   G 2 2 1 1 G
¼   ¼   ¼
2G 1 1 2 2

2 2       3 1 1   G
þ1   G
G 3 4 1 2 2 2        G
¼ ¼ ¼ ¼ G 3 3 3 1 2 3 2



2 2 2 2 

This can be shown by use  of theprevious variable transformation followed by integration by 1 parts. We prove this for G
. In fact 2 1 2   ZN 1 d x expð
xÞx
3=2 ¼ G
2 n¼


0

x¼

u2 ;

d x ¼ 2u du;

x
3=2 ¼ u
3

4.16 Solved problems

211

Then, taking v ¼ exp(
u2) as finite factor and d(u
1) as differential factor   ZN ZN

 
 1 G
¼ 2 du exp
u2 u
2 ¼
2 exp
u2 d u
1 2 0

N
 ¼
2 exp
u2 u
1 0 þ 2

ZN

0


 u
1 d exp
u2

0

Now, the first term vanishes at infinity, and also at zero by the l’Hoˆpital’s rule, since
2  
1


2 
1  d exp
u du lim exp
u u ¼ lim u/0 u/0 du du


2  ¼ lim exp
u ð
2uÞ=1 ¼ 0 u/0

so that we obtain 

1 G
2



ZN ¼2

u


1




ZN



exp
u ð
2u duÞ ¼
4 2

0




du exp
u

2



  1 ¼
2G 2

0

as it must be. 4.4. Find the FT of the exponential and the Gaussian function. 1. The exponential function Let f ðxÞ ¼ expð
cjxjÞ

(357)

be the exponential function where c is a real positive parameter. Then its FT is 1 Fð pÞ ¼ pffiffiffiffiffiffi 2p

ZN d x expð
ipxÞexpð
cjxjÞ
N

2 0 3   Z ZN 1 4 1 1 1 þ d x exp½ðc
ipÞx þ d x exp½ðc þ ipÞx5 ¼ pffiffiffiffiffiffi ¼ pffiffiffiffiffiffi 2p 2p c
ip c þ ip
N

 1=2 2 c ¼ p c2 þ p 2

0

(358) 2. The Gaussian function Let


f ðxÞ ¼ exp
x2 =c2

(359)

212

CHAPTER 4 Special functions

be the Gaussian function where c is a real parameter. Then its FT is still a Gaussian function: 1 Fð pÞ ¼ pffiffiffiffiffiffi 2p

ZN


 d x expð
ipxÞexp
x2 =c2


N

 2 2  ZN  h   2 2 1 c p x x cp 2 i c c p p ffiffiffiffiffiffi p ffiffi ffi d ¼ exp
exp
¼ exp
þ i 4 4 c c 2 2p 2

(360)


N

4.5. The FT and the anti-FT of the Coulombic potential. With reference to Figure 4.6, choosing r as polar axis, we have in spherical coordinates k$r ¼ kr cos qk

(361)

dk ¼ k dk sin qk dqk d4k

(362)

2

Putting x ¼ cos qk ;

z ¼ kr;

dz ¼ r dk

(363)

we evaluate the integral extended to the whole space: 8 Z expðik $ rÞ > > I ¼ dk > > > k2 > > > > > Z1 Z2p ZN ZN > > > expðikrxÞ expðikrÞ
expð
ikrÞ < ¼ 2 dk k dx d4k ¼ 2p dk k2 ikr > 0
1 0 0 > > > > > ZN > > > 4p sin z 4p p 2p2 > > ¼ $ ¼ dz ¼ > > r r z r 2 : 0

FIGURE 4.6 The spherical coordinates ðk; qk ; 4k Þ of the wave vector k

(364)

4.16 Solved problems

Therefore, we obtain 1 1 ¼ 2 r 2p

Z dk

expðik $ rÞ k2

213

(365)

which is the form of the FT of r
1 (Koide, 1976). Even if it does not exist in the strict sense of ordinary analysis, the theory of generalized functions (Gel’fand and Shilov, 1964) shows that the divergent integral arising in the inverse FT of 1/r can be treated as well and is given by (Weniger and Steinborn, 1983):  1=2 Z 1 1 2 1 dr expð
ik $ rÞ (366) ¼ FðkÞ ¼ 2 3=2 r p k ð2pÞ 4.6. The LTs of Table 4.1. The first six integrals of Table 4.1 are immediately evaluated by recalling the well-known general formula ZN d x xn expð
axÞ ¼

n! anþ1

0

with n a non-negative integer and a a real positive number different from zero. Then, we obtain for the third integral in the table 1 LðsÞ ¼ ðn
1Þ!

ZN dt tn
1 expð
stÞ ¼

1 ðn
1Þ! 1 ¼ n $ ðn
1Þ! sn s

0

The fourth integral is evident, since ZN

ZN dt expð
atÞexpð
stÞ ¼

LðsÞ ¼ 0

dt exp½
ða þ sÞt ¼

1 aþs

0

For the sixth integral we have 1 LðsÞ ¼ b
a

ZN dt½expð
atÞ
expð
btÞexpð
stÞ 0

2 1 4 ¼ b
a

ZN dt exp½
ða þ sÞt
0

¼

ZN

3 dt exp½
ðb þ sÞt5 ¼

0

1 bþs
a
s 1 $ ¼ b
a ða þ sÞðb þ sÞ ða þ sÞðb þ sÞ

as it must be.

  1 1 1
b
a aþs bþs

214

CHAPTER 4 Special functions   1 The seventh integral is obtained much in the same way as we did before for the G function. 2 In fact 1 LðsÞ ¼ pffiffiffiffi p

ZN

dt t
1=2 expð
stÞ

0

Changing variable to st ¼ u2 ;

s dt ¼ 2u du;

u¼

pffiffiffiffi st;

t
1=2 ¼ s1=2 u
1

we have 1 LðsÞ ¼ pffiffiffiffi p

ZN 0


2u 1=2
1 2 s u exp
u2 ¼ pffiffiffiffiffiffi du s ps

ZN




du exp
u 0

2



pffiffiffiffi 2 1 p ¼ pffiffiffiffiffiffi$ ¼ pffiffi s ps 2

The last integral in the table is a little more complicated. It is, however, easily shown that the integrand is the primitive of the function f ðtÞ ¼


expð
stÞ½s sinðatÞ þ a cosðatÞ a2 þ s 2

as we can see by calculating the first derivative of f(t). Taking the definite integral, we then have  ZN expð
stÞ½s sinðatÞ þ a cosðatÞ dt sinðatÞexpð
stÞ ¼
 a2 þ s2 t¼N 0  expð
stÞ½s sinðatÞ þ a cosðatÞ a þ ¼ 2  a2 þ s2 a þ s2 t¼0 and the integral of the table follows.

CHAPTER

Functions of a complex variable

5

CHAPTER OUTLINE 5.1 Functions of a Complex Variable ................................................................................................... 215 5.1.1 Complex numbers......................................................................................................215 5.1.2 Functions of a complex variable..................................................................................217 5.1.3 Regular functions ......................................................................................................218 5.1.4 Elementary operations ...............................................................................................219 5.1.5 Power series of elementary functions...........................................................................219 5.1.6 Many-valued functions ...............................................................................................223 5.2 Complex Integral Calculus ............................................................................................................ 223 5.2.1 Line integrals ............................................................................................................223 5.2.2 Integrals in the complex plane and the Cauchy theorem................................................224 5.2.3 Integration over a not simply connected domain ...........................................................225 5.2.4 Cauchy’s integral representation .................................................................................227 5.2.5 Taylor’s expansion around a singularity........................................................................228 5.2.6 Laurent’s expansion...................................................................................................228 5.2.7 Zeros of a regular function..........................................................................................230 5.2.8 Analytic continuation .................................................................................................230 5.3 Calculus of Residues .................................................................................................................... 232 5.3.1 The residue theorem ..................................................................................................232 5.3.2 The Jordan lemma .....................................................................................................234 5.3.3 Sum of non-convergent series.....................................................................................235 5.3.4 Evaluation of integrals of functions of real variable.......................................................238 5.4 Problems 5 .................................................................................................................................. 241 5.5 Solved Problems .......................................................................................................................... 242

5.1 FUNCTIONS OF A COMPLEX VARIABLE 5.1.1 Complex numbers A number of the form z¼xþiy

(1)

where i is the imaginary unit ði2 ¼ 1Þ is called a complex number. x ¼ ReðzÞ Elementary Molecular Quantum Mechanics 2E. http://dx.doi.org/10.1016/B978-0-444-62647-9.00005-1
2013, 2007 Elsevier B.V. All rights reserved

(2)

215

216

CHAPTER 5 Functions of a complex variable

FIGURE 5.1 The two-dimensional space (Argand diagram) for the complex variable z

is a real number, called the real (Re) part of the complex number z, y ¼ ImðzÞ

(2) 1

is a real number, called the imaginary (Im) part of the complex number z. Hence, we might define a complex number as an ordered pair of real numbers, z ¼ ðx; yÞ. The complex number z ¼ x  i y is called the complex conjugate of z. The quantity pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jzj ¼ þ x2 þ y2 ¼ r

(3)

(4)

is called the modulus of the complex number z. Complex quantities are conveniently represented in the two-dimensional space of Figure 5.1 by the so-called Argand diagram, having x as abscissae (real) axis and y as ordinate (imaginary) axis. A complex number can also be represented in terms of radial and angular variables as z ¼ jzj expði4Þ

(5)

where jzj is the modulus of the complex number and 4, called the argument of z and denoted by arg z (period 2p), is the polar angle about the z axis, yielding the point z in the (x, y) plane through an anticlockwise (positive) rotation from x to y. For the complex numbers, the commutative and associative laws for addition and multiplication, as well as the distributive law, hold: z1 þ z 2 ¼ z 2 þ z 1 ;

z1 þ ðz2 þ z3 Þ ¼ ðz1 þ z2 Þ þ z3 ¼ z1 þ z2 þ z3

z1 z2 ¼ z2 z1 ; z1 ðz2 z3 Þ ¼ ðz1 z2 Þz3 ¼ z1 z2 z3 ðz1 þ z2 Þz3 ¼ z1 z3 þ z2 z3 1

(6)

Imaginary numbers were introduced for the first time by Raffaele Bombelli, Italian mathematician (Bologna, 1572), in the study of the algebraic equations of third and fourth degrees.

5.1 Functions of a complex variable

217

It is also possible to define the division of two complex numbers, z ¼ ðx; yÞ and z0 ¼ ðx0 ; y0 Þ, giving the complex number 2 ¼ ða; bÞ provided jz0 js0, in which case: a¼

xx0 þ yy0 ; x0 2 þ y0 2

b¼

x0 y  xy0 x0 2 þ y0 2

(7)

and 2 ¼ ða; bÞ is called the quotient z=z0 . In fact, we can write: 2¼

¼

z x þ iy x y ¼ 0 ¼ 0 þi 0 0 0 0 z x þ iy x þ iy x þ iy0 xðx0  iy0 Þ yðx0  iy0 Þ xx0 þ yy0 x0 y  xy0 þi 2 ¼ 2 þi 2 2 2 2 2 x0 þ y0 x0 þ y0 x0 þ y0 x0 þ y0 2

(8)

¼ a þ ib and Eqn (7) is proved. Complex numbers become necessary in the study of algebraic equations. Considering only real numbers, an equation like x2 þ 1 ¼ 0 would have no solution, while an equation like x3  1 ¼ 0 would have only one solution. The solutions of the first quadratic equation are evidently x ¼ i, those of the cubic equation are one real solution, x1 ¼ 1, and two complex conjugate solutions, pffiffiffi 1 x2;3 ¼ ð1  i 3Þ, as can be easily verified by direct substitution. 2

5.1.2 Functions of a complex variable If z ¼ x þ iy is a complex number, we say that f ðzÞ ¼ uðx; yÞ þ ivðx; yÞ

(9)

is a function of the complex variable z, if a correspondence exists between the values of z and f(z). In Eqn (9), u and v are real functions of the variables ðx; yÞ. In the complex plane we may have one-valued or many-valued functions, according if to a given value of z we may associate one or many values of f(z). In the following, we shall focus attention mostly on one-valued functions, even if it must be stressed that the important logarithm function is a many-valued function in the complex plane (Section 5.1.5). For functions defined in this way, the definition of continuity is exactly the same as that for functions of a real variable. So, we say that a function f(z), defined in a domain D, is continuous at the point z0 if lim f ðzÞ ¼ f ðz0 Þ

z/z0

(10)

It can be further proved that a function which is continuous in a closed domain is uniformly continuous there. This definition of continuity is equivalent to the statement that a continuous function of the complex variable z is a continuous function of the two variables x and y. Furthermore, we shall assume that the definition of the derivative of a single real variable is applicable to the functions of a complex variable: if f(z) is a one-valued function, defined in the domain

218

CHAPTER 5 Functions of a complex variable

D of the Argand diagram, then f(z) is differentiable at a point z0 of D if there exists a unique limit as z/z0 of the incremental ratio, and this limit is called the derivative of f(z) at z ¼ z0 :   f ðzÞ  f ðz0 Þ df ðzÞ ¼ ¼ f 0 ðz0 Þ (11) lim z/z0 z  z0 dz z¼z0

5.1.3 Regular functions Let f(z), defined by Eqn (9), be a complex function continuous with its first partial derivatives with respect to x and y in a domain D. A function of z which is one valued and differentiable at every point of domain D is said to be regular (or analytic or holomorphic) in domain D. A function may be differentiable in a domain except for a finite number of points, which are called singularities of f(z). The necessary conditions for a function f(z) to be a regular function are the so-called Cauchy–Riemann’s differential equations which are obtained in the following way. If f(z) is the function of the complex variable z ¼ x þ iy defined by Eqn (9), then the incremental ratio f ðz þ DzÞ  f ðzÞ Dz

(12)

where Dz ¼ Dx þ iDy must tend to a definite limit as Dz/0 in any way. If we take Dz to be wholly real, so that Dy ¼ 0, then uðx þ Dx; yÞ  uðx; yÞ vðx þ Dx; yÞ  vðx; yÞ þi Dx Dx

(13)

must tend to a definite limit as Dx / 0. It follows that the partial derivatives ðvu=vxÞ and ðvv=vxÞ vu vv must exist at the point ðx; yÞ and the limit is þ i . Similarly, if we take Dz to be wholly imaginary, vx vx so that Dx ¼ 0, then uðx; y þ DyÞ  uðx; yÞ vðx; y þ DyÞ  vðx; yÞ þi iDy iDy

(14)

must tend to a definite limit as Dy / 0. It follows that the partial derivatives ðvu=vyÞ and ðvv=vyÞ vv vu must exist at the point ðx; yÞ and the limit is  i . Since the two limits must be equal it follows vy vy that: vu vv ¼ ; vx vy

vu vv ¼ vy vx

(15)

and we obtain what are called Cauchy–Riemann’s differential equations, the regularity conditions for f(z). Cauchy–Riemann’s conditions allow for the uniqueness of the limit giving the first derivative of f(z), and Eqns (13) and (14) show that it can be obtained by the same procedure as that for functions of a real variable.

5.1 Functions of a complex variable

219

5.1.4 Elementary operations If f(z) is regular in the domain D and g(z) is regular in the domain D 0 , then f ðzÞ þ gðzÞ the sum of f plus g is a function regular in the domain

(16)

DXD 0 ;

f ðzÞ$gðzÞ

(17)

the product of f times g is a function regular in the domain DXD0 . For the sum and product of regular functions is therefore guaranteed the regularity of the functions in the intersection between the regularity domains of the original functions. It may be possible, however, that the regularity of the resulting function extends outside the region DXD0 . For instance, let f ðzÞ ¼ z gðzÞ ¼

regular everywhere

1 z

(18) regular for zs0

Then f ðzÞ þ gðzÞ ¼ z þ

1 z

(19)

is a function regular for zs0, whereas f ðzÞ$gðzÞ ¼ 1

(20)

is a function regular everywhere.

5.1.5 Power series of elementary functions Consider the series f ðzÞ ¼

N X

fk ðzÞ

(21)

k¼0

where the variable z may be complex. Then it is assumed that the ordinary rules for series expansion hold, especially for what convergence is concerned. We say that series (21) converges uniformly within a given domain of the definition of the complex variable z if for any positive arbitrary ε we can find an index n0 of the series such that for n > n0     n X   fk ðzÞ < ε (22)  f ðzÞ    k¼0 is true for any z belonging to the domain. A power series is obtained when fk ðzÞ ¼ ak zk . As in the theory of real functions, for any power series it is useful to define a radius of convergence R, such that for jzj < R the series will be absolutely and uniformly convergent, divergent if jzj > R, while if jzj ¼ R we cannot say anything.

220

CHAPTER 5 Functions of a complex variable

A useful convergence test for power series is the Cauchy–Hadamard test, which states that the convergence radius R can be obtained from i1 h (23) R ¼ lim ðjak jÞ1=k k/N

or the alternative ratio test, more useful in practice:

   ak   R ¼ lim  k/N akþ1 

We give here the radii of convergence of a few power series.   N X  1 k z : R ¼ lim   ¼ 1 k/N 1 k¼0   N k X ðk þ 1Þ! z   ¼ lim jk þ 1j ¼ N : R ¼ lim   k/N k/N k! k! k¼0 so that the series is convergent for any value of z,     N X  k!   1  k    ¼0 k!z : R ¼ lim  ¼ lim  k/N ðk þ 1Þ! k/N k þ 1 k¼0

(24)

(25)

(26)

(27)

so that the series converges at the origin. We now give without proof two important theorems (Phillips, 1954). P k If f ðzÞ ¼ N k¼0 ak z , then the function f(z) is a regular function at every point within the circle of convergence of the power series. Let f(z) be given by the series N X uk ðzÞ (28) f ðzÞ ¼ k¼0

Now, if each term uk ðzÞ is regular within a domain D, and if the series is convergent throughout every region D0 interior to D, then f(z) is regular within D and all its derivatives may be calculated by term-by-term differentiation. In the following, we shall briefly consider some elementary functions of a complex variable. 1. Rational functions P k A polynomial of degree m in z, Pm ðzÞ ¼ m k¼0 ak z , may be regarded as a power series which converges for all values of z. Rational functions of the type f ðzÞ ¼

Pm ðzÞ a0 þ a1 z þ . þ am zm ¼ b0 þ b1 z þ . þ b n z n Qn ðzÞ

(29)

are regular at all points of the plane at which the denominator does not vanish. If we choose a point z0 at which the denominator does not vanish, then f(z) may be expanded in a power series of the form N X ck ðz  z0 Þk (30) f ðzÞ ¼ k¼0

5.1 Functions of a complex variable

221

2. Exponential, circular and hyperbolic functions Defining exponential and circular (trigonometric) functions of the complex variable z as the sum functions of the corresponding series: N k X z (31) ez ¼ exp z ¼ k! k¼0 sin z ¼

N X

ð1Þk

k¼0

cos z ¼

N X

z2kþ1 ð2k þ 1Þ!

(32)

z2k ð2kÞ!

(33)

ð1Þk

k¼0

it can be shown (Phillips, 1954) that all properties typical of the same functions of real argument hold. So exp z$exp z0 ¼ expðz þ z0 Þ (34) az ¼ expðz ln aÞ

(35)

where ln a is the real natural logarithm of a, sinðzÞ ¼ sin z; the parity of circular functions,

cosðzÞ ¼ cos z

sin2 z þ cos2 z ¼ 1

(37)

sinðz  z0 Þ ¼ sin z cos z0  cos z sin z0 0

0

cosðz  z Þ ¼ cos z cos z Hsin z sin z

(36)

0

(38) (39)

the addition formulae for circular functions, cos z  i sin z ¼ expð izÞ 1 sin z ¼  i½expðizÞ  expðizÞ 2 1 cos z ¼ ½expðizÞ þ expðizÞ 2

(40) (41) (42)

Euler’s formulae, 1 sinh z ¼ ½expðzÞ  expðzÞ 2 1 cosh z ¼ ½expðzÞ þ expðzÞ 2 sinhðzÞ ¼ sinh z; coshðzÞ ¼ cosh z

(43) (44) (45)

the parity of hyperbolic functions, cosh2 z  sinh2 z ¼ 1

(46)

222

CHAPTER 5 Functions of a complex variable

The important relations (Problem 5.1) sin iz ¼ i sinh z;

cos iz ¼ cosh z

sinh iz ¼ i sin z;

cosh iz ¼ cos z

(47)

are useful for deducing the properties of circular functions of imaginary argument from those of hyperbolic functions and vice versa. Term-by-term differentiation of the power series shows that d d exp z ¼ exp z; expðazÞ ¼ a exp z dz dz d d sin z ¼ cos z; cos z ¼ sin z dz dz

(48) (49)

Furthermore, we notice the following. The function exp z has the period 2pi. If k is zero, or a positive or negative integer, we have exp z ¼ expðz þ 2piÞ ¼ expðz þ 2kpiÞ

(50)

In fact, if we increase z by 2pi, y increases by 2p and this leaves the values of cos y and sin y unchanged. Every value that exp z can assume is therefore taken in the interval p < y  p. Lastly, it can be seen that sin z  vanishes  if, and only if, z ¼ npðn ¼ 0; 1; 2; $$$Þ, and that 1 cos z vanishes if, and only if, z ¼ n þ p. 2 3. The logarithmic function When x is real and positive, the equation exp u ¼ x has one real solution u ¼ ln x. However, if z is complex but not zero, the equation exp w ¼ z has an infinite number of solutions, each of which can be called a logarithm of z. If w ¼ u þ iv

(51)

exp w ¼ exp u$expðivÞ ¼ exp uðcos v þ i sin vÞ ¼ z

(52)

we have

Hence it is seen that v is one of the values of arg z and exp u ¼ jzj. Hence u ¼ ln jzj. Every solution of Eqn (52) has therefore the form w ¼ lnjzj þ i arg z

(53)

Since arg z takes an infinite number of values, there is an infinite number of logarithms of the complex number z. We write Ln z ¼ lnjzj þ i arg z

(54)

so that Ln z is an infinitely many-valued function of z. Its principal value, which is obtained by giving arg z its principal value, is denoted by ln z and coincides with the ordinary logarithm when z is real and positive.

5.2 Complex integral calculus

223

5.1.6 Many-valued functions We have seen that a number of elementary functions, such as za (a not an integer), ln z, sin1 z are many-valued functions. It can be shown (Phillips, 1954) that the equation w2 ¼ z

(55)

has no continuous one-valued solutions defined over the entire complex plane, but it defines a twovalued function of the complex variable z. The two functions       pffiffi pffiffi pffiffi 1 1 1 i4 ; w2 ¼  r exp i 4þp ¼  rexp i4 (56) w1 ¼  r exp 2 2 2 are called the two branches of the two-valued function (55). Each of these branches is a one-valued function in the z-plane if we make a narrow slit, extending from the origin to infinity along the real positive x-axis, distinguishing the values of the function at points on the upper and lower edges of the cut. We observe that, if z describes a circle about any point a and the origin lies outside this circle, then arg z is not increased by 2p but returns to its initial value. Hence, the values of w1 and w2 are exchanged only when z turns about the origin. For this reason the point z ¼ 0 is called a branch point of the function pffiffi w ¼ z, and w1 and w2 are called its two branches. Since turning about z ¼ N means describing a large pffiffi circle about the origin, the point z ¼ N is also a conventional branch point for w ¼ z. For w ¼ ln z, since w ¼ ln r þ ið4 þ 2kpÞ;

(57)

where r and 4 are the radius and the angle in the Argand plane, every positive and negative integer k gives a branch, so ln z is an infinitely many-valued function of z. The points z ¼ 0 and z ¼ N are branch points.

5.2 COMPLEX INTEGRAL CALCULUS 5.2.1 Line integrals Any integral in the complex plane is a line integral in two variables along the path g. The integration path is specified in a parametric form (Figure 5.2):  xðtÞ g¼ (58) yðtÞ We shall focus attention on the case where x(t) and y(t) are continuous functions, with their first derivatives, of the variable t. A path having a definite tangent at any point is called a regular path. For the integrals in the complex plane are of interest only regular paths, such as a circle or a part of it, or a square of four regular sides. A line integral from point A to point B along the path g is given in general by ZB f ðx; yÞdgðx; yÞ AðgÞ

(59)

224

CHAPTER 5 Functions of a complex variable

FIGURE 5.2 An integration path g in the complex plane

The path is taken from A to B, while from B to A the integral changes its sign: ZA

ZB f ðx; yÞdgðx; yÞ ¼ 

BðgÞ

f ðx; yÞdgðx; yÞ

(60)

AðgÞ

If A ¼ B the path is closed, and in this case anticlockwise circulation along the path is chosen as positive circulation. We can rewrite Eqn (59) by explicitly specifying the parametric dependence on g: ZB

ZB f ðx; yÞdgðx; yÞ ¼ AðgÞ

 ZtB    vg vg vg dx vg dy dx þ dy ¼ þ dt f ðx; yÞ f ðx; yÞ vx vy vx dt vy dt

(61)

tA

AðgÞ

so that we see that the line integral (59) reduces to an integral in just the single variable t.

5.2.2 Integrals in the complex plane and the Cauchy theorem The integrals in the complex plane have a form similar to that given by Eqn (59) but include also the imaginary unit i: ZB

ZB dz f ðzÞ ¼ AðgÞ

ZB ðdx þ i dyÞðu þ ivÞ ¼

AðgÞ

½ðu dx  v dyÞ þ iðv dx þ u dyÞ AðgÞ

(62)

5.2 Complex integral calculus

An upper bound for the contour integral is given by the Darboux inequality:     ZB     dz f ðzÞ  Ml      AðgÞ

225

(63)

where M ¼ maxjf ðzÞj

z ˛g;

l ¼ g length

(64)

If f(z) is regular on a simply connected domain D and g is a closed path inside this domain, Cauchy’s theorem states that I dz f ðzÞ ¼ 0 (65) g

Cauchy’s theorem arises from the fact that a regular function does satisfy the Cauchy–Riemann conditions, Eqn (15), and exhibits a strict analogy with the exact differential of two variables in real space. An elementary proof is given in Phillips (1954). We have already said that the value of an integral of a regular function between any two points in the complex plane depends only on the points and not on the path connecting them. Given a point z0, the integral of the regular function f ðz0 Þ up to a second variable point z gives the function: Zz FðzÞ ¼

0

0

Zz

dz f ðz Þ ¼ z0

½ðu dx0  v dy0 Þ þ iðv dx0 þ u dy0 Þ ¼ Uðx; yÞ þ iVðx; yÞ

(66)

z0

The functions U and V arise from the separation of real from imaginary part in integral (66). Between U and V and their (exact) differentials appearing under the integral sign must hold the relations vU ¼ u; vx

vU ¼ v; vy

vV ¼ v; vx

vV ¼u vy

(67)

This means that functions U and V satisfy the conditions which make F(z) a regular function as f(z). Therefore, we conclude that dF vF vU vV ¼ ¼ þi ¼ u þ iv ¼ f ðzÞ dz vx vx vx

(68)

so that for the search of the primitive of a regular function of complex variable the same rules hold as for the functions in the real field.

5.2.3 Integration over a not simply connected domain If the closed integration path g is the border of a not simply connected domain containing a singularity, integral (65) is not zero in general, even if the function is regular along the whole path. It is still true, however, that the value of the integral is unique and common to all possible closed paths surrounding the “hole” responsible for the singularity.

226

CHAPTER 5 Functions of a complex variable

FIGURE 5.3 Disconnected paths g and g0 enclosing a singularity at point O

This can be seen by comparing any pair of arbitrary closed paths g and g0 as those in the drawing of Figure 5.3, where O is the point originating the disconnection of the domain (the singularity). In the drawing, the paths are left open but with an infinitesimal separation so that the integral from A to B practically coincides with that of the closed path g, the same being true for the integral from A0 to B0 with respect to the closed path g0 . The new path ABB0A0 A encloses a simply connected area over which we can therefore write Z Z Z Z Z ¼ þ þ þ ¼0 (69) ABB0 A0 A

BB0

AB

It is also true that

Z

B0 A 0

A0 A

Z þ

BB0

¼0

(70)

A0 A

since BB0 and A0 A practically correspond to the same path covered in opposite ways, and that Z Z Z Z ¼ ; ¼ (71) AB

g

B0 A 0

g0

where the sign is taken positive for the anticlockwise path, negative for the clockwise path. Then we conclude that Z Z ¼ g

(72)

g0

Since g and g0 are completely arbitrary, Eqn (72) states that the result of any cyclic integration of a regular function around a singularity (not simply connected area) lying at the interior of the integration domain is unique. As a simple example, let I dz (73) z C

5.2 Complex integral calculus

227

be the integral to be evaluated, where C denotes a path closed around the origin of axes, a point not simply connected in the domain of f (z), where the function is regular except for z ¼ 0. Since the integral we want must be zero, we choose a convenient closed integration path equivalent to C, for instance a circle of radius r around the origin. Introducing the polar coordinates r and 4: z ¼ r expði4Þ; dz ¼ ir expði4Þd4 ¼ iz d4 (74) we obtain for integral (73)

I

I dz ¼ i d4 ¼ 2pi z

(75)

C

5.2.4 Cauchy’s integral representation Let f(z) be a regular function in a simply connected domain D and g a closed path inside the domain. Then it is true that I 1 f ðz0 Þ (76) f ðzÞ ¼ dz0 0 2pi z z g

This is called Cauchy’s integral representation of the function f (z). This is true since the result of the integration is identical over any closed path containing point z inside it, so that we can choose as integration path a circle of infinitesimal radius centred at z. Over this path f ðz0 Þ is practically equal to f (z) everywhere. Therefore, we can write I I 1 f ðz0 Þ 1 dz0 ¼ (77) f ðzÞ dz0 0 z0  z 2pi z  z 2pi g

g

where the last integral is evaluated according to Eqn (75). The result is I 1 dz0 1 f ðzÞ ¼ f ðzÞ$2pi ¼ f ðzÞ 2pi z0  z 2pi

(78)

g

and Cauchy’s integral representation (76) is proved. It is also possible to show that the successive derivatives of f(z) can be obtained from Cauchy’s integral representation, using the formalism of the derivation under the integral sign: df 1 ¼ dz 2pi

I

dz0 f ðz0 Þ

d 0 1 ðz  zÞ1 ¼ dz 2pi

g 2

d f 1$2 ¼ dz2 2pi

g

I

dz0

g

dn f n! ¼ dzn 2pi

I g

I

dz0

f ðz0 Þ ðz0  zÞ3 f ðz0 Þ

ðz0  zÞnþ1

dz0

f ðz0 Þ ðz0

 zÞ2

(79)

(80)

(81)

228

CHAPTER 5 Functions of a complex variable

5.2.5 Taylor’s expansion around a singularity Let f (z) be a function regular in the simply connected domain D and limited by a circle C centred at the point z0. For each point inside C the integral Cauchy’s representation (76) holds. If jz0  z0 j is the radius of the integrand function, we can rewrite the denominator in Eqn (76) as  N  X 1 1 1 1 z  z0 k ¼ 0  ¼ ¼ (82) z  z0 z0  z ðz0  z0 Þ  ðz  z0 Þ z  z0 k¼0 z0  z0 ðz0  z0 Þ 1  0 z  z0 where for the second factor at denominator in Eqn (82) we used Taylor’s expansion: N X 1 ¼ tk 1  t k¼0

(83)

z  z   0 According to Eqn (25), expansion (82) has radius of convergence  0  < 1 since z0 lies on the z  z0 border of the circle whereas z is at its interior. Substituting expansion (82) into the Cauchy’s integral representation (76), it is obtained: I I N 0 1 1 X f ðz0 Þ k 0 f ðz Þ ¼ dz 0 ðz  z0 Þ dz0 (84) f ðzÞ ¼ 2pi z  z 2pi k¼0 ðz0  z0 Þkþ1 g

g

Recalling Eqn (81) for the nth derivative of f (z), we obtain the Taylor expansion in the well-known form N  k  X d f ðz  z0 Þk (85) f ðzÞ ¼ k k! dz z¼z0 k¼0 Examples of Taylor’s expansions in the complex plane, with radius of convergence R ¼ N, are expansions (31)–(33) of Section 5.1.5.

5.2.6 Laurent’s expansion Let f (z) be a function regular in the domain D except for a point z0 called the singularity. We take two circles C1 and C2 centred at z0 and construct the integration path of Figure 5.4 obtained by opening in an infinitesimal way C1 and C2 and taking an infinitesimal circle C3 around the point z. f ðz0 Þ In this way, we construct a circular closed path in a simply connected domain for the function 0 z z for which it must hold: I I I f ðz0 Þ f ðz0 Þ f ðz0 Þ  dz0 0  dz0 0 ¼0 (86) dz0 0 z z z z z z C2

C3

C1

Cauchy’s integral representation (76) hence allows one to write I f ðz0 Þ ¼ 2pi f ðzÞ dz0 0 z z C3

(87)

5.2 Complex integral calculus

229

FIGURE 5.4 The infinitesimal circle C3 around the point z

and therefore

2 f ðzÞ ¼

1 6 4 2pi

I C2

dz0

f ðz0 Þ z0  z

I 

3 dz0

f ðz0 Þ z0  z

7 5

(88)

C1

If for the denominators of the single integrals we use expansions similar to that used for the Taylor expansion, Eqn (82) gives z  z   0 1. Integral over C2, where  0  < 1: z  z0 1 1 ¼ ¼ z0  z ðz0  z0 Þ  ðz  z0 Þ

 N  1 1 X z  z0 k ¼ 0  z  z0 z  z0 k¼0 z0  z0 ðz0  z0 Þ 1  0 z  z0

(89)

 N  0 1 1 X z  z0 k   ¼ z0  z0 z  z0 k¼0 z  z0 ðz  z0 Þ 1  z  z0

(90)

z0  z   0 2. Integral over C1, where   < 1: z  z0 1 1 ¼ ¼ z0  z ðz0  z0 Þ  ðz  z0 Þ

Using these series expansions in Eqn (88), we obtain for f (z) the so-called Laurent’s expansion: 3 2 I I N N X 1 6X f ðz0 Þ 1 k7 (91) ðz  z0 Þk dz0 þ dz0 f ðz0 Þðz0  z0 Þ 5 f ðzÞ ¼ 4 kþ1 kþ1 2pi k¼0 Þ ðz0  z0 Þ ðz  z 0 k¼0 C2

C1

In expansion (91) positive and negative powers of the variable appear with coefficients that cannot be put in relation with the successive derivatives of the function. Laurent’ series is an extension to

230

CHAPTER 5 Functions of a complex variable

negative powers of Taylor’ series, to which it reduces if the singularity at z0 is absent, in which case the coefficients of the negative powers of ðz  z0 Þ are all zero.

5.2.7 Zeros of a regular function The point z ¼ z0 is a zero of order n for the regular function f (z) when f ðz0 Þ ¼ 0 with all its derivatives there up order ðn  1Þ. In the neighbourhood of a zero, which is a regular point, function f (z) can be represented by a Taylor series starting with the power ðz  z0 Þn, since all coefficients of the powers up to order ðn  1Þ are zero: I N 1 X f ðz0 Þ ðz  z0 Þk dz0 (92) f ðzÞ ¼ 2pi k¼n ðz0  z0 Þkþ1 C

It is possible to factor out the factor ðz  z0 Þn : f ðzÞ ¼

I N ðz  z0 Þn X f ðz0 Þ ðz  z0 Þk dz0 2pi k¼0 ðz0  z0 Þkþnþ1

(93)

C

5.2.8 Analytic continuation Let f1 ðzÞ and f2 ðzÞ be complex functions regular in the domains D1 and D2, respectively. Then, if a common intersection part exists where f1 ðzÞ ¼ f2 ðzÞ, we can regard the aggregate of values of f1 ðzÞ and f2 ðzÞ at points interior to D1 and D2 as a single regular function g(z). Thus, g(z) is regular in D ¼ D1 þ D2 , and gðzÞ ¼ f1 ðzÞ in D1 and gðzÞ ¼ f2 ðzÞ in D2. The function f2 ðzÞ may then be regarded as an extension of the domain of definition of f1 ðzÞ and is called an analytic continuation of f1 ðzÞ. In other words, we may say that f (z) and g(z) are the same regular function, and that f1 ðzÞ and f2 ðzÞ are the analytic continuation one of the other. Function f1 ðzÞ, initially assumed known only in D1, will now be known even in D1 XD2 , so that (94) f1 ðzÞ ¼ f2 ðzÞ for z˛D3D1 XD2 As an example, consider the power series (Rossetti, 1984) N X zk

(95)

k¼0

Function (95) (the geometric series) converges for jzj < 1 and represents there a well-defined regular function f (z). So, we can write   N X 1 zk ¼ for jzj < 1 (96) f ðzÞ ¼ 1z k¼0 Now, consider the integral ZN dt exp½ð1  zÞt 0

(97)

5.2 Complex integral calculus

231

which converges for any z such that Reð1  zÞ > 0 and, in the region ReðzÞ < 1, represents the welldetermined regular function g(z): ZN dt exp½ ð1  zÞt

gðzÞ ¼

 ¼

1 1z

 ReðzÞ < 1

(98)

0

Now notice that ZN

ZN dt exp½ð1  zÞt ¼

0

dt expðtÞ$expðztÞ 0

ZN ¼

dt expðtÞ 0

N k X z

k! k¼0

$tk ¼

N N k Z X z

k! k¼0

(99) dt tk expðtÞ

0

The step of interchanging summation by integral is possible only if the resulting series is convergent, namely, the last step is possible only in the region of intersection of the region of convergence of the integral ðReðzÞ < 1Þ with that of convergence of the series. To obtain explicitly the coefficients of the expansion we must calculate the integral: ZN dt tk expðtÞ ¼ k!

(100)

0

This can be done by repeated integrations by parts, or by noting that ZN

ZN dt t expðtÞ ¼ lim k

a/1

0

dk dt t expðatÞ ¼ lim ð1Þ k a/1 da

0

¼ lim ð1Þk a/1

ZN

k

k

dtexpðatÞ 0

(101)

dk ðaÞ1 ¼ lim ð1Þk ð1Þð 2Þ.ðkÞa1k ¼ k! a/1 dak

Using this result in Eqn (99), it is obtained: ZN dt exp½ð1  zÞt ¼ 0

N N k Z X z k¼0

k!

dt tk expðtÞ ¼

0

N X

zk

(102)

k¼0

this result being valid in the region: ðReðzÞ < 1ÞXðjzj < 1Þ

(103)

We then say that g(z) and f (z) are the same regular function since Eqns (96) and (98) coincide in the region jzj < 1. In other words, the integral representation (98) is the analytic continuation of function f (z), defined by the series (96) only in the region jzj < 1 to the whole region ReðzÞ < 1.

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CHAPTER 5 Functions of a complex variable

Lastly, we notice that the fact that f (z) and g(z) are the same function could be immediately evident, since the series (96) is summable so that 1 (104) f ðzÞ ¼ gðzÞ ¼ 1z The important point here is the fact that all we discussed above is completely independent of the explicit form of f (z) and g(z).

5.3 CALCULUS OF RESIDUES 5.3.1 The residue theorem We define pole of order n of the regular function f (z) a singular point z0 such that, in the Laurent expansion (91), all coefficients of the negative powers higher than n are zero. Then the series (91) becomes 2 3 I I N n1 0 X X 1 6 f ðz Þ 1 k7 ðz  z0 Þk dz0 þ dz0 f ðz0 Þðz0  z0 Þ 5 (105) f ðzÞ ¼ 4 kþ1 kþ1 0 2pi k¼0 ðz  z0 Þ k¼0 ðz  z0 Þ C2

C1

and factoring out the power ðz  z0 Þn : 2 3 I I N n 1 X 1 f ðz0 Þ k7 6X ðz  z0 Þk dz0 þ ðz  z0 Þnk1 dz0 f ðz0 Þðz0  z0 Þ 5 f ðzÞ ¼ n4 knþ1 2piðz  z0 Þ k¼n ðz0  z0 Þ k¼0 C2

C1

(106) The residue of a regular function f (z) at the isolated singularity z0 is the result of the integral of the function over a closed path g enclosing the singularity divided by 2pi: I 1 dz f ðzÞ (107) ½Res f ðzÞz¼z0 ¼ 2pi g

The residue is hence the coefficient of the power ðz  z0 Þ1 in Laurent’s expansion, and is zero if point z0 is a regular point. For calculating the residue in z0 (pole of order n), it will suffice to note that, from Eqn (106), the product ðz  z0 Þn f ðzÞ is a regular function, since it is given by a power series with positive exponents. Then, from Cauchy’s integral representation (76) we obtain I 1 ðz0  z0 Þn 0 f ðz Þ dz0 0 (108) ðz  z0 Þn f ðzÞ ¼ z z 2pi g

whose derivative can be calculated by means of Eqn (81), giving I n 0 dn1 ðn  1Þ! n 0 ðz  z0 Þ ½ðz  z Þ f ðzÞ ¼ dz f ðz0 Þ 0 dzn1 ðz0  zÞn 2pi g

(109)

5.3 Calculus of residues

233

FIGURE 5.5 The triangle g0 joining together the three small circles gk

Taking the limit for z/z0 it is then obtained: 1 dn1 1 lim n1 ½ðz  z0 Þn f ðzÞ ¼ z/z ðn  1Þ! 2pi 0 dz ¼

I

dz0

g

1 2pi

I

ðz0  z0 Þn f ðz0 Þ ðz0  z0 Þn

dz0 f ðz0 Þ ¼ ½Res f ðzÞz¼z0

(110)

g

If the pole is of order 1, Eqn (110) gives for the residue the simple result: ½Res f ðzÞz¼z0 ¼ lim ðz  z0 Þf ðzÞ

(111)

z/z0

Now, a theorem due to Cauchy, called the residue theorem, states that I dz f ðzÞ ¼ 2pi g

X

½Res f ðzÞz¼zk

(112)

k

namely, the integral of the function f (z) over a closed path g containing in it isolated singularities of f (z) is equal to 2pi times the sum of the residues at these points. The proof of this theorem is very simple. The path g can be deformed to give the combination between a number of small circles gk containing the singularities, as shown in Figure 5.5. Then I

I dz f ðzÞ ¼

g

dz f ðzÞ þ g0

3 I X k¼1 g k

dz f ðzÞ ¼ 0 þ 2pi

3 X k¼1

½Res f ðzÞz¼zk ¼ 2pi

3 X

½Res f ðzÞz¼zk

k¼1

(113) The paths g0 (triangle) and g1 ; g2 ; g3 (bumps) can be considered as closed paths because the small circles opened at z1 ; z2 ; z3 can be made infinitesimal.

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CHAPTER 5 Functions of a complex variable

5.3.2 The Jordan lemma Let gR be a semicircle of radius R (Figure 5.6) and f (z) a regular function such that lim jf ðzÞj$jzj ¼ 0

jzj/N

when arg z lies in the interval ð0; pÞ. Then, if we pose Z IR ¼ dz f ðzÞ

(114)

(115)

gR

we have the result known as Jordan lemma: lim IR ¼ 0

R/N

(116)

The Jordan lemma (116) can be easily proved in polar coordinates in the Argand plane, where z ¼ R expði4Þ;

dz ¼ iR expði4Þdz ¼ iz dz

(117)

so that we have Z IR ¼

Zp dz f ðzÞ ¼ iR

gR

d4 f ½R expði4Þexpði4Þ

(118)

0

We then have for the moduli the inequalities Zp jIR j  jiRj

d4jf ½R expði4Þj$jexpði4Þj

(119)

0

and considering that jiRj ¼ R;

jexpði4Þj ¼ 1;

jf ½R expði4Þj  MðRÞ

FIGURE 5.6 Semicircle in the Argand plane describing the closed path gR

(120)

5.3 Calculus of residues

235

where MðRÞ ¼ maxj f ½R expði4Þj

(121)

is the maximum value of the modulus of f when changing 4 at fixed R, we finally have Zp jIR j  R$MðRÞ

d4 ¼ R$MðRÞ$p

(122)

0

Since Eqn (114) has to be satisfied, we must also have lim R$MðRÞ ¼ 0

(123)

R/N

and therefore lim IR ¼ 0 as it must be. R/N

A like result is obtained for a semicircle in the imaginary half-plane lying below the x-axis, provided hypothesis (114) is satisfied for arg z lying in the interval ðp; 2pÞ.

5.3.3 Sum of non-convergent series Using the residue theorem, it is possible, in some cases, to give a precise meaning to infinite series which are not convergent in the ordinary sense. As an example, consider the following parametric integral: ZN SðhÞ ¼

ZN dz f ðzÞ ¼

0

dz 0

zh cosh pz

(124)

representing a well-defined function SðhÞ which is analytic in the region ReðhÞ > 1. The integral (124) is the integral of a many-valued function f(z), and it can be shown that (Rossetti, 1984) X p ½Res f ðzÞz¼zk (125) expð iphÞ SðhÞ ¼  sin ph k On the positive real x-axis for 0 < arg z < 2p the singularities of f (z), which are the zeros of cosh pz, occur at the points     1 1 (126) z ¼ zk ¼ k þ expðip=2Þ; z ¼ z0 k ¼ k þ expði3p=2Þ k ¼ 0; 1; 2; / 2 2 These points are all simple poles, so that the residue theorem for the first pole z ¼ zk gives 1 zh ¼ ðzk Þh lim z/z z/zk k p sinh pz cosh pz  h   1 expðiph=2Þ 1 h expðiph=2Þ k   ¼ kþ ¼ ð 1Þ k þ 1 2 2 ip p sinh ip k þ 2

½Res f ðzÞz¼zk ¼ lim ðz  zk Þ

(127)

236

CHAPTER 5 Functions of a complex variable

and, for z ¼ z0 k

 ½Res f ðzÞz¼z0 ¼ ð 1Þ

k

k

1 kþ 2

h

expði3ph=2Þ ip

(128)

Substituting into Eqn (125) we obtain   N pexpðiphÞ X 1 h1 ð1Þk k þ ½expðiph=2Þ  expði3ph=2Þ sin ph 2 ip k¼0    N  1 expðiph=2Þ  expðiph=2Þ X 1 h k 2 ð1Þ k þ ¼ sin ph 2i 2 k¼0   N 2sinðph=2Þ X 1 h ¼ ð1Þk k þ sin ph k¼0 2

SðhÞ ¼ 

namely SðhÞ ¼

  N X 1 1 h ð1Þk k þ cosðph=2Þ k¼0 2

(129)

(130)

The series occurring in the right-hand member of Eqn (130) is not convergent in an ordinary sense. However, since function SðhÞ is well behaved and analytic for ReðhÞ > 1, this means that   N X 1 h ð1Þk k þ ¼ cosðph=2ÞSðhÞ (131) 2 k¼0 the function on the right-hand side being perfectly defined for ReðhÞ > 1. What is not true in this case is the usual definition of the sum of the series as the infinite limit of the succession of its partial sums. Particular values of the parameter h give the “sums” of series (131). For instance, if h ¼ 0 Sð0Þ ¼

N X

ð1Þk

(132)

k¼0

For h ¼ 0, integral (124) can be evaluated in an elementary way. Thus, changing variable to 8 dy < expðpzÞ ¼ y; p dz ¼ y (133) : z ¼ 0 0 y ¼ 1; z ¼ N 0 y ¼ N we have ZN Sð0Þ ¼ 2

1

dz½expðpzÞ þ expð  pzÞ 0

2 ¼ p

ZN 1

   dy 1 1 2 1 N 2 p p 1 ¼ tan y ¼ yþ  ¼ y y p p 2 4 2 1 (134)

5.3 Calculus of residues

237

so that we finally obtain N X

1 2

(135)

  1 ð1Þk k þ ¼0 2 k¼0

(136)

Sð0Þ ¼

ð1Þk ¼

k¼0

h ¼ 1 gives instead

N X

so that N X k¼0

ð1Þk k þ

N N X 1X 1 ð1Þk ¼ 0 0 ð1Þk k ¼  2 k¼0 4 k¼0

(137)

These results were obtained for a particular case, but it can be shown that they could be obtained in general by a method due to Borel. The Borel summation consists in constructing the sum function of a series as the Laplace transform of a sum of a series which is summable in the ordinary sense. Thus, the power series N X

ak z k

(138)

k¼0

is an arbitrary function series which converges whenever

  an R ¼ lim  n/N a

jzj < R;

nþ1

   

(139)

where R is the radius of convergence. Defining the function 4ðtzÞ ¼

N X k¼0

ak

ðtzÞk k!

(140)

it can be shown (Rossetti, 1984) that, for jzj < R, the identity holds: N X

ZN ak z ¼ k

k¼0

dt expðtÞ4ðtzÞ

(141)

0

For every point z at which integral (141) exists, the Borel sum of series (138) is the value of the integral above. In practice, the Borel summation gives a method for the analytic continuation of the function f (z), given by the series (138) for jzj < R, to the whole region where integral (141) does exist. We now apply the Borel summation method to our previous series (131). Consider the series: N X ðztÞk k¼0

k!

¼ expðztÞ

(142)

238

CHAPTER 5 Functions of a complex variable

Multiplying both members by expðztÞ and integrating over dt for z < 1 it is obtained: ZN dt expðztÞ

N X ðztÞk

k!

k¼0

0

ZN ¼

dt exp½ðz  1Þt ¼

1 1z

(143)

0

If the resulting series is convergent, it is possible to interchange summation by integral and taking into account Eqn (100) we obtain N Z X

N

k¼0

0

N k X ðztÞk z expðtÞ ¼ dt k! k! k¼0

ZN dt tk expðtÞ ¼

N X

zk ¼

k¼0

0

1 1z

(144)

where use was made of the previous result if z < 1. For z ¼ 1, we immediately obtain N X

zk ¼

k¼0

N X

ð1Þk ¼

k¼0

1 1 ¼ 1  ð1Þ 2

(145)

which is the result of Eqn (135). As already said, this is the analytic continuation of the geometric series N X k¼0

zk ¼

1 1z

(146)

whose convergence radius is z < 1, and which has a simple pole for z ¼ 1. For z > 1 it is necessary to integrate from N to zero, and we obtain the analytic continuation of series (146) in that interval.

5.3.4 Evaluation of integrals of functions of real variable The residue theorem (112) is useful for the calculation of some integrals of functions of a real variable. Integrals such as Z2p d4 f ðcos 4; sin 4Þ

(147)

0

can be easily evaluated by substitution with a complex variable. Over the circle of radius 1 in the complex plane it is true that  z ¼ expði4Þ ¼ cos 4 þ i sin 4; z1 ¼ expði4Þ ¼ cos 4  i sin 4 (148) dz ¼ iexpði4Þd4 ¼ iz d4 so that cos 4 ¼

  1 1 zþ ; z z

sin 4 ¼

  1 1 z ; 2i z

d4 ¼

dz iz

(149)

A given integral transforms into an integral over the circle of radius 1 in the complex plane and can hence be studied by analysing the singularities of the function inside this circle.

5.3 Calculus of residues

239

Integrals over the real variable x like ZN dx f ðxÞ

(150)

N

can be seen as integrals over the complex variable z, if we pose I lim dz f ðzÞ R/N

(151)

gR

with gR a closed path like those of Figure 5.7, provided function f(z) satisfies the Jordan conditions in the half-plane corresponding to the chosen path. As R tends to infinity the contribution to the integration along the semicircle vanishes and the required value of the integral is the value remaining along the real x-axis. Even in this case, the evaluation of the integral over the real variable x reduces to the analysis of the isolated singularities enclosed in the interior of the integration path. As an example, consider the integral ZN I¼

dx ða2

þ

x2 Þðb2

þ x2 Þ

(152)

0

which occurs when the dispersion coefficients of two atoms are expressed in terms of their dynamic polarizabilities (Chapter 17). Integral (152) can be immediately related to integral (150) since ZN

dx 1 ¼ ða2 þ x2 Þðb2 þ x2 Þ 2

0

1 ¼ 2

ZN N

ZN N

dz ða2 þ z2 Þðb2 þ z2 Þ (153) dz ðz  iaÞðz þ iaÞðz  ibÞðz þ ibÞ

FIGURE 5.7 Closed integration paths for integrals over a real variable

240

CHAPTER 5 Functions of a complex variable

FIGURE 5.8 The two poles lying along the imaginary axis

and the conditions of the Jordan lemma are satisfied either in the positive complex half-plane (left in Figure 5.7) or in the negative one (right in Figure 5.7).2 Choosing the positive half-plane, we notice the presence on the imaginary axis of two poles corresponding to the points z ¼ ia and z ¼ ib of Figure 5.8. Therefore, we conclude that 1 I¼ 2

ZN N

dz ða2

þ

z2 Þðb2

þ

z2 Þ

 ¼ pi ½Res f ðzÞz¼ia þ ½Res f ðzÞz¼ib

(154)

where f ðzÞ ¼

1 ða2

þ

z2 Þðb2

þ

z2 Þ

¼

1 1 ðz  iaÞðz þ iaÞ ðz  ibÞðz þ ibÞ

(155)

The poles are both of order 1 and the corresponding residues are hence obtained through Eqn (111): 1

½Res f ðzÞz¼ia ¼ lim ðz  iaÞ f ðzÞ ¼

2iaðb2

½Res f ðzÞz¼ib ¼ lim ðz  ibÞ f ðzÞ ¼

2ibða2

z/ia

z/ib

 a2 Þ

(156)

 b2 Þ

(157)

1

Introducing the residues into Eqn (154) 1 I¼ 2

ZN N

  dz 1 1 p b þ a p ¼ pi  þ ¼ ¼ $ 2 2 2 2 2 2 2 2 2 2 ða þ z Þðb þ z Þ 2iaða  b Þ 2ibða  b Þ 2ab a  b 2abða þ bÞ (158)

The function to be integrated goes to zero as R4 .

2

5.4 Problems 5

241

finally giving the integral transform 1 2 ¼ aþb p

ZN dx

a b a2 þ x2 b2 þ x2

a; b ¼ Re > 0

(159)

0

It is worth noting that Eqn (159) could also be obtained in an elementary way without the use of the integration in the complex plane (Problem 17.2 of Chapter 17). Two similar integral transforms are given as Problems 5.2 and 5.3, respectively. In Problem 5.4 we evaluate the integral over three spherical Bessel functions in k-space needed in the theory of the generalized multipole expansion of the intermolecular potential of Chapter 17 (Magnasco and Figari, 1989).

5.4 PROBLEMS 5 5.1. Find the expression of the circular and hyperbolic functions of the imaginary argument. Answer: Expressions (47) of the main text. Hint: Use definitions (43) and (44) of the hyperbolic functions and the power series expansions (31)–(33) for exponential and circular functions. 5.2. Evaluate the integral ZN I ¼ dx

b ac  x2 p 1 ¼ b2 þ x2 ða2 þ x2 Þðc2 þ x2 Þ 2 ða þ bÞðb þ cÞ

0

5.3. Evaluate the integral ZN I ¼ dx

a2

a b c p aþbþc ¼ 2 2 2 2 2 þx b þx c þx 2 ða þ bÞðb þ cÞðc þ aÞ

0

5.4. Evaluate the k-integral over three spherical Bessel functions. Answer: ZN p ð2n2 þ 2n3 Þ!n2 !n3 ! r2 n2 r3 n3 dk jn1 ðr1 kÞjn2 ðr2 kÞjn3 ðr3 kÞ ¼ dn1 ;n2 þn3 2 ð2n2 þ 1Þ!ð2n3 þ 1Þ!ðn2 þ n3 Þ! r1 n2 þn3 þ1 0

Hint: Use contour integration in the complex plane and the residue theorem choosing the semispace of integration according to the boundary conditions of the problem.

242

CHAPTER 5 Functions of a complex variable

5.5 SOLVED PROBLEMS 5.1. The circular and hyperbolic functions of imaginary argument. Using definitions (43) and (44) of the hyperbolic functions and the power series expansions (31)–(33) for exponential and circular functions, we easily obtain for the corresponding functions of the imaginary argument sin iz ¼ i

N P

z2kþ1 ¼ i sinh z; k¼0 ð2k þ 1Þ!

sinh iz ¼ i

N P k¼0

ð1Þk

cos iz ¼

z2kþ1 ¼ i sin z; ð2k þ 1Þ!

N X z2k ¼ cosh z ð2kÞ! k¼0

cosh iz ¼

N X

ð1Þk

k¼0

z2k ¼ cos z ð2kÞ!

which are the results given in Eqn (47). 5.2. We must evaluate the integral ZN I¼

b ac  x2 1 ¼ dx 2 b þ x2 ða2 þ x2 Þðc2 þ x2 Þ 2

0

1 ¼ 2

ZN dz N

ZN dz N

b ac  z2 b2 þ z2 ða2 þ z2 Þðc2 þ z2 Þ

b ac  z2 ðz  ibÞðz þ ibÞ ðz  iaÞðz þ iaÞðz  icÞðz þ icÞ

where, on the imaginary axis in the positive complex half-plane of Figure 5.9, the integrand has the three poles of order 1, corresponding to the points z ¼ ia, z ¼¼ ib, and z ¼ ic, so that by the residue theorem Z X 1 1 dz f ðzÞ ¼ 2pi Res½ f ðzÞz¼zk I¼ 2 2 k C

 ¼ pi lim ðz  iaÞ z/ia

b ac  z2 ðz  ibÞðz þ ibÞ ðz  iaÞðz þ iaÞðz  icÞðz þ icÞ

b ac  z2 z/ib ðz  ibÞðz þ ibÞ ðz  iaÞðz þ iaÞðz  icÞðz þ icÞ  b ac  z2 þ lim ðz  icÞ z/ic ðz  ibÞðz þ ibÞ ðz  iaÞðz þ iaÞðz  icÞðz þ icÞ þ lim ðz  ibÞ

and, after little algebra   p b ac þ a2 ac þ b2 b ac þ c2  þ I¼ 2 a2  b2 aða2  c2 Þ ða2  b2 Þðb2  c2 Þ b2  c2 ða2  c2 Þc ¼

p ða  bÞðb  cÞ p 1 ¼ 2 ða2  b2 Þðb2  c2 Þ 2 ða þ bÞðb þ cÞ

5.5 Solved problems

243

FIGURE 5.9 The three poles lying along the imaginary axis

finally giving the integral transform: 1 2 ¼ ða þ bÞðb þ cÞ p

ZN dx

b2

b ac  x2 2 2 þ x ða þ x2 Þðc2 þ x2 Þ

0

5.3. Evaluate the symmetric integral ZN I¼

a b c 1 dx 2 ¼ a þ x2 b2 þ x2 c2 þ x2 2

0

¼ ¼

1 2 1 2

ZN dz N

dz N

a b c a2 þ z 2 b 2 þ z 2 c 2 þ z 2

a b c ðz  iaÞðz þ iaÞ ðz  ibÞðz þ ibÞ ðz  icÞðz þ icÞ

Z

dz f ðzÞ ¼ C

ZN

X 1 Res½ f ðzÞz¼zk 2pi 2 k

where, on the imaginary axis in the positive complex half-plane, the three poles of order 1 occur at the same points of Figure 5.9. Hence, by calculating the residues at these points, after some algebra we obtain I¼

p aþbþc 2 ða þ bÞðb þ cÞðc þ aÞ

giving the integral transform as aþbþc 2 ¼ ða þ bÞðb þ cÞðc þ aÞ p

ZN dx 0

a b c a2 þ x 2 b2 þ x 2 c 2 þ x 2

244

CHAPTER 5 Functions of a complex variable

This integral occurs in the three-body long-range dispersion interaction between three spherically symmetric atoms A, B, and C (Tang, 1969). 5.4. Evaluation of the k-integral over three spherical Bessel functions. We want to evaluate the k-integral over three spherical Bessel functions: ZN dk jn1 ðr1 kÞjn2 ðr2 kÞjn3 ðr3 kÞ 0

under the conditions

8 > < ðaÞ r1  r2 þ r3 ðbÞ n1 þ n2 þ n3 ¼ 2n > : ðcÞ n2 þ n3  n1  0

where n is a non-negative integer. Under these restrictions, the integral can be evaluated by contour integration in the complex plane. We recall, first, the finite representation of jn, Eqn (155), and, second, the parity of jn, Eqn (140) of Chapter 4. The parity of the jns under condition b allows one to extend the integration range over k to the whole real axis. Introducing the complex variable z ¼ x þ iy, we must perform the integration along the real x-axis, giving ZN

1 dk jn1 ðr1 kÞjn2 ðr2 kÞjn3 ðr3 kÞ ¼ 2

0

2 16 ¼ 4 lim 4 u/N

ZN dz jn1 ðr1 zÞjn2 ðr2 zÞjn3 ðr3 zÞ N

I

I dz Wn1 ðr1 zÞ expðir1 zÞjn2 ðr2 zÞjn3 ðr3 zÞ þ lim

v/N

C1

3 7 dz Wn1 ðr1 zÞ expðir1 zÞjn2 ðr2 zÞjn3 ðr3 zÞ5

C2

where C1 and C2 are closed paths in the positive and negative semispaces lying on the imaginary axis (Figure 5.10).

FIGURE 5.10 Closed paths for the integration in the complex plane. C2 encloses the singularity at the origin

5.5 Solved problems

245

Condition (a) ensures that r1 alone determines the algebraic signs of the arguments of the exponentials, so fixing the semispace appropriate to the integration. When r1 ¼ r2 þ r3 , both semispaces are acceptable. In fact, using the trigonometric expression for the complex variable z z ¼ jzjðcos 4 þ i sin 4Þ we see that expði r1 zÞ ¼ expði r1 jzjcos 4Þ$expði r1 jzjsin 4Þ the last term decreasing exponentially as jzj/N for 0 < 4 < p (upper plane). So, the spurious contribution of the semicircle in the upper plane vanishes when its radius u/N. Similar considerations hold for the semicircle in the lower plane, p < 4 < 2p. The residue theorem says then that the first integral above is zero whilst the second is simply 2p i times the residue of the integrand at z ¼ 0: ZN

h i 1 dk jn1 ðr1 kÞjn2 ðr2 kÞjn3 ðr3 kÞ ¼  2pi Res Wn1 ðr1 zÞexpði r1 zÞjn2 ðr2 zÞjn3 ðr3 zÞ z¼0 4

0

In this expression, the minus sign accounts for the clockwise circulation along the closed path. The residue is the coefficient of the z1 term in the Laurent’s expansion of the function around the singularity z ¼ 0. Expanding each function in the last expression in powers of its argument and retaining the lowest power in z, it is obtained: Wn1 ðr1 zÞexpðir1 zÞjn2 ðr2 zÞjn3 ðr3 zÞ ¼ in1 þ1 ¼i

ðiÞn1 ð2n1 Þ! ðr2 zÞn2 ðr3 zÞn3 ðr1 zÞn1 1 $1$ $ þ/ n ð2n2 þ 1Þ!! ð2n3 þ 1Þ!! 2 1 n1 !

r2n2 r3n3 n2 þn3 n1 1 ð2n1  1Þ!! z þ terms containing higher powers of z ð2n2 þ 1Þ!!ð2n3 þ 1Þ!! r1n1 þ1

A non-zero residue exists only if condition c is satisfied in the form n2 þ n3  n1 ¼ 0 so that h i Res Wn1 ðr1 zÞexpðir1 zÞjn2 ðr2 zÞjn3 ðr3 zÞ

z¼0

¼ dn1 ;n2 þn3 i

r2n2 r3n3 ð2n2 þ 2n3 Þ!n2 !n3 ! ð2n2 þ 1Þ!ð2n3 þ 1Þ!ðn2 þ n3 Þ! r1n2 þn3 þ1

giving the final result ZN dk jn1 ðr1 kÞjn2 ðr2 kÞjn3 ðr3 kÞ ¼ 0

r2n2 r3n3 p ð2n2 þ 2n3 Þ!n2 !n3 ! dn1 ;n2 þn3 2 ð2n2 þ 1Þ!ð2n3 þ 1Þ!ðn2 þ n3 Þ! r1n2 þn3 þ1

CHAPTER

Matrices

6

CHAPTER OUTLINE 6.1 6.2 6.3 6.4 6.5 6.6

6.7 6.8 6.9 6.10

6.11 6.12

Definitions and Elementary Properties ......................................................................................... 247 The Partitioning of Matrices ....................................................................................................... 248 Properties of Determinants ......................................................................................................... 250 Special Matrices........................................................................................................................ 255 The Matrix Eigenvalue Problem................................................................................................... 256 Functions of Hermitian Matrices ................................................................................................. 261 6.6.1 Analytic functions ...................................................................................................261 6.6.2 Projectors and canonical form ..................................................................................262 6.6.3 Examples................................................................................................................262 The Matrix Pseudoeigenvalue Problem ........................................................................................ 266 The Lagrange Interpolation Formula ............................................................................................ 268 The Cayley–Hamilton Theorem .................................................................................................... 270 The Eigenvalue Problem in Hu¨ckel’s Theory of the p Electrons of Benzene.................................... 270 6.10.1 General considerations...........................................................................................270 6.10.2 Unitary transformation diagonalizing the Hu¨ckel’s matrix ..........................................272 Problems 6 ................................................................................................................................ 273 Solved Problems ........................................................................................................................ 278

Matrices are the powerful algorithms connecting the differential equations of quantum mechanics to equations governed by the linear algebra of matrices and their transformations. After a short introduction on the elementary properties of matrices and determinants (Aitken, 1958; Margenau, 1961; Frazer et al., 1963; Hohn, 1964), we introduce special matrices and the matrix eigenvalue problem, as well as more advanced techniques and a chemical application to the p electrons of benzene.

6.1 DEFINITIONS AND ELEMENTARY PROPERTIES A matrix A of order ðm  nÞ is an array of numbers or functions ordered according to m rows and n columns: 0 1 A11 A12 / A1n B A21 A22 / A2n C C A¼B (1) @ / / / / A Am1 Am2 / Amn Elementary Molecular Quantum Mechanics 2E. http://dx.doi.org/10.1016/B978-0-444-62647-9.00006-3 Ó 2013, 2007 Elsevier B.V. All rights reserved

247

248

CHAPTER 6 Matrices

and can be denoted by its ij element ði ¼ 1; 2; $$$; m; j ¼ 1; 2; $$$; nÞ as:   A ¼ Aij

(2)

Matrix A is rectangular if nsm; square if n ¼ m: In square matrices, elements with j ¼ i are called diagonal. To any square matrix A we can associate two scalarP quantities, its determinant jAj ¼ det A (a number, if matrix elements are numbers) and its trace tr A ¼ i Aii , the sum over all diagonal elements. Two matrices A and B of the same order are equal if: B¼A

Bij ¼ Aij for all i; j

(3)

Matrices can be added (or subtracted) if they have the same order: AB¼C

Cij ¼ Aij  Bij

(4)

Addition and subtraction enjoy commutative and associative properties. Multiplying a matrix A by a complex number c implies multiplication of all elements of A by that number: cA ¼ B Bij ¼ cAij (5) The product rows by columns of two (or more) matrices A by B is possible if the matrices are conformable (the number of columns of A must equal the number of rows of B): Xn Xn Xp B ¼ C Cij ¼ A B ; A B C ¼ D Dij ¼ A B C A a¼1 ia aj a¼1 b¼1 ia ab bj mn np

mn np pq

mp

mq

(6) Matrix multiplication is usually not commutative, the quantity ½A; B ¼ AB  BA

(7)

being called the commutator of A and B. If ½A; B ¼ 0

(8)

matrices A and B commute. The product of more than two matrices enjoys the associative property: ABC ¼ ðABÞC ¼ AðBCÞ

(9)

The trace of a product of matrices is invariant under the cyclic permutation of its factors: tr ABC ¼ tr CAB ¼ tr BC A

(10)

6.2 THE PARTITIONING OF MATRICES It is sometimes useful to subdivide (or partition) matrices into rectangular blocks of elements, called submatrices. Submatrices hold all properties of matrices seen so far, in particular the need of the same order for algebraic addition and the conformability of the factor matrices for multiplication.

6.2 The partitioning of matrices

To take an example, let the ð3  3Þ matrix U 0 1 Uð3  3Þ [ @ 2 1 where the four submatrices are: Að2  2Þ ¼

1

3

2

1

Cð1  2Þ ¼ ð 1

1 Þ;

be partitioned into 1   3 0 A B 1 3A ¼ C D 1 1 ! ;

Bð2  1Þ [

0

249

(11) ! (12)

3

Dð1  1Þ ¼ ð1Þ

Following what was stated before, we shall say that two partitioned matrices are equal, if and only if their non-partitioned forms are equal. Addition of partitioned matrices:       A2 B2 A1 þ A2 B1 þ B2 A1 B1 þ ¼ (13) C1 D1 C2 D2 C1 þ C2 D1 þ D2 provided all submatrices have the same order. Multiplication of partitioned matrices is possible if the submatrices are conformable by treating the partitioned matrices if they were elements, so that, if ! 1 1 1 Að2  3Þ ¼ ¼ ð A1 A2 Þ 2 3 2 ! ! 1 1 1 ; A2 ð2  1Þ ¼ A1 ð2  2Þ ¼ 2 3 2 0 1 (14) 1 0 ! B1 B C C Bð3  2Þ [ B @0 1A ¼ B2 3 1 ! 1 0 B1 ð2  2Þ ¼ ; B2 ð1  2Þ ¼ ð 3 1 Þ 0 1 by block-matrix multiplication we obtain:    B1 1 ¼ A1 B1 þ A2 B2 ¼ ð A1 A2 Þ $ B2 2

1 3



the same result that would have been obtained by unpartitioned matrices A and B: 0   1 1 1 1 @ AB ¼ 0 2 3 2 3

 þ

3 6

1 2



 ¼

4 0 4 1

 ¼ AB

(15)

ordinary matrix multiplication between the 1  0 4 A 1 ¼ 4 1

0 1

 (16)

These results are true in general, provided the partitioning is so carried out that the submatrices to be multiplied are all conformable.

250

CHAPTER 6 Matrices

6.3 PROPERTIES OF DETERMINANTS Given a determinant jAj of order n (n rows and n columns), we call jAij j the minor of jAj, a determinant of order ðn  1Þ obtained from A by deleting row i and column j, and by jaij j ¼ ð1Þiþj jAij j the cofactor (signed minor) of jAj. The main properties of determinants will be shortly recalled here (Aitken, 1958). 1. Definition The determinant of a square matrix A   A11 A12   A21 A22 jAj ¼  / /  An1 An2

of order n can be written as:  / A1n   / A2n  X ¼  A1a A2b /Ann / /   / Ann

(17)

the first being a notation due to Cayley and used today as a standard notation, the second a notation due to earlier mathematicians (Laplace, Cauchy, Jacobi and others). In Eqn (17), the summation is over n! terms, being extended over all permutations ða b $$$ n Þ of the second (column) index of the elements Aij ; the sign being þ or  according to the parity of the permutation (þ if the permutation is even,  if the permutation is odd). 2. Expansion by rows or columns (ordinary expansion) A determinant can be expanded in an elementary way in terms of any of its rows or columns: jAj ¼

n X

Aij aij

i ¼ fixed

(18)

j¼1

is the expansion according to the ith row (we sum over all columns): jAj ¼

n X

Aij aij

j ¼ fixed

(19)

i¼1

is the expansion according to the jth column (we sum over all rows). 3. The expansion of a determinant of order n gives n! terms. This may be simply proved by induction using Eqn (17).        a1 b  ¼  a1 b1  ¼ a1 b2  b1 a2 ¼ ab  ba n ¼ 2 0 2! terms 2  a2 b2  the last expression being a shorthand notation1 which will be used in the following:   a b c     j a1 b2 c3 j ¼  a b c  ¼ aðbc  cbÞ  bðac  caÞ þ cðab  baÞ   a b c n ¼ 3 0 3  2! ¼ 3! terms 1

We observe that, when the elements of a determinant are numbers, a2 and b2 must be different from a1 and b1.

(20)

(21)

6.3 Properties of determinants

 a   a j a b c d j ¼  a  a    b c d a       ¼ a b c d   b  a    b c d a

 d   b c d   b c d  b c d   a c d     c d  þ c a   a c d b

251

c

b b b

  a b d     d   d a b   a b d

 c   c  c

(22)

n ¼ 4 0 4  3! ¼ 4! terms and so on. 4. Cauchy expansion The Cauchy expansion of a determinant jAj of order n according to the element h of a given row and the element k of a given column is given by: jAj ¼ Ahk jahk j 

XX i

  Aik Ahj ahk;ij  ish; jsk

(23)

j

where the cofactor jahk j of Ahk in jAj is a determinant of order ðn  1Þ, and jahk;ij j, the cofactor of Aij in jahk j, is a determinant of order (n  2), as shown in Figure 6.1.

FIGURE 6.1 Matrix elements in the Cauchy expansion of a determinant of order n

252

CHAPTER 6 Matrices

Example 1 Consider the Cauchy expansion of a determinant D3 of order 3 according to the elements of the first row and column: h¼k¼1    A11 A12 A13    3 P 3 P   D3 ¼  A21 A22 A23  ¼ A11 ja11 j  Ai1 A1j ja11;ij j   i¼2 j¼2 A  31 A32 A33    A22 A23              ¼ A11    A21 A12 a11;22  þ A13 a11;23   A31 A12 a11;32  þ A13 a11;33   A32 A33     A22 A23    ¼ A11    A21 ½A12 A33 þ A13 ðA32 Þ  A31 ½A12 ðA23 Þ þ A13 A22   A32 A33 

(24)

¼ A11 A22 A33  A11 A23 A32  A21 A12 A33 þ A21 A13 A32 þ A31 A12 A23  A31 A13 A22 and we obtain all terms of the ordinary expansion of D3, as can be easily checked. We notice that, while the cofactor ja11 j is a determinant of order 2, the cofactor ja11;ij j is a determinant of order 1, namely a number with the appropriate sign. To get ja11;22 j, we must delete the second row and the second column in D3 (where we have already deleted the first row and column) so that the only remaining element in the determinant of order 2 will be A33 which is of even place in D2 and must therefore be taken with the plus sign. In the same way, to get ja11;23 j, we must delete the second row and third column in D3 (where we have already deleted the first row and column) so that the only remaining element in the determinant of order 2 will be A32 which is of odd place in D2 and must therefore be taken with the minus sign. The same result holds true for any different choice of h and k. Example 2 Consider the Cauchy expansion of a determinant D3 of order 3 according to the elements of the first row and second column: h ¼ 1; k ¼ 2    A11 A12 A13    3     P P  D3 ¼  A21 A22 A23  ¼ A12 ja12 j  Ai2 A1j a12;ij  j¼1;3 i¼2    A31 A32 A33  !   A21 A23              ¼ A12     A22 A11 a12;21  þ A13 a12;23   A32 A11 a12;31  þ A13 a12;33   A31 A33  !   A21 A23    ¼ A12     A22 ½A11 ðA33 Þ þ A13 A31   A32 ½A11 A23 þ A13 ðA21 Þ  A31 A33  ¼ A12 A21 A33 þ A12 A23 A31 þ A22 A11 A33  A22 A13 A31  A32 A11 A23 þ A32 A13 A21 (25) and we obtain all terms of the ordinary expansion of D3 without omission or repetition.

6.3 Properties of determinants

253

To get ja12;21 j, we must delete the second row and the first column in D3 (where we have already deleted the first row and second column) so that the only remaining element in the determinant of order 2 will be A33 which is of even place in D2 and must therefore be taken with the minus sign because of the minus sign in front of the cofactor of order 2. In the same way, to get ja12;23 j, we must delete the second row and third column in D3 (where we have already deleted the first row and second column) so that the only remaining element in the determinant of order 2 will be A31 which is of odd place in D2 and must therefore be taken with the plus sign, because of the minus sign in front of the cofactor of order 2. This clearly illustrates the way of proceeding and the care that must be taken in determining the appropriate signs in the jahk;ij j. Example 3 Bordered determinants For a determinant jAj ¼ j A11 A22 A33 $$$ Ann j of order n consider the identity:      0 A12 A13 / A1n   A11 A12 A13 / A1n       A21 A22 A23 / A2n   A21 A22 A23 / A2n       A31 A32 A33 / A3n  ¼ A11 ja11 j þ  A31 A32 A33 / A3n          / /      An1 An2 An3 / Ann   An1 An2 An3 / Ann 

(26)

where the second determinant on the right is called a bordered determinant, if we consider it as formed by bordering j A22 A33 $$$ Ann j (a determinant of order n  1) by a given row, say the first row f 0 A12 A13 $$$ A1n g, and a given column, say the first column ½0 A21 A31 $$$ An1 . For easy writing, we have denoted here the row by f $$$ g and the column by ½ $$$ . For this determinant we obtain the Cauchy expansion according to the elements of the first row and column: XX   (27) Ai1 A1j a11;ij  is1; js1 jAj ¼ A11 ja11 j  i

If jaij j is the cofactor of (Hohn, 1964):  0   v1   v2     vn

j

Aij in jAj and n > 1; we have for the one-bordered determinant v1 A11 A21 An1

v2 A12 A22 / An2

/ / / /

 vn  A1n  n X   A2n  ¼  vi vj aij   i;j¼1  Ann 

(28)

In the same way, we have for the two-bordered determinant (Hohn, 1964):   A11   A21     An1   u1  v1

A12 A22 An2 u2 v2

/ / / / / /

A1n A2n

u1 u2

Ann un vn

un 0 0

 v1   v2   X    qij qkm aij;km  ¼ vn   1i > > > > ~ Bij ¼ Aji transpose

B ¼ Ay Bij ¼ Aji adjoint > > > > > : B ¼ A1 B ¼ ðdet AÞ1 a  inverse ij ji When performed twice, the operations ), w, y, Products have the properties:

1

(44)

1

C 0 C C ¼ l1 /C A l

/

(43)

(45)

(46)

restore the original matrix.

w ~ ðABÞy ¼ By Ay ~A ðABÞ ¼ B

ðABÞ1 ¼ B1 A1

(47)

Ay

(48)

6. Given a square matrix A of order n, if: A ¼ A

~ A

we say that A is real, symmetric, Hermitian (or self-adjoint), respectively. 7. If: ~ A1 ¼ A

Ay

(49)

we say that A is orthogonal or unitary, respectively.

6.5 THE MATRIX EIGENVALUE PROBLEM A system of linear inhomogeneous algebraic equations in the n unknowns ci ði ¼ 1; 2; .; nÞ can be written in matrix form as: Ac ¼ b

(50)

6.5 The matrix eigenvalue problem

if we introduce the matrices:

0

A11

B B A21 A¼B B/ @ An1 the square matrix of coefficients,

the column vector of the unknowns,

A12

/

A22

/

/

/

An2

/

A1n

257

1

C A2n C C /C A Ann

(51)

0

1 c1 B c2 C C c¼B @/A cn

(52)

1 b1 B b2 C C b¼B @/A bn

(53)

0

the column vector of the inhomogeneous terms, and adopt the matrix multiplication rule (6). Matrix Eqn (50) can be interpreted as a linear transformation on vector c, which is transformed into vector b under the action of matrix A. If A1 exists ðdet As0Þ, the solution of system (50) is given by: c ¼ A1 b

(54)

which is the well-known Cramer’s rule. When b is proportional to c through a number l: Ac ¼ lc

(55)

we obtain what is known as the eigenvalue equation for the square matrix A. By writing: ðA  l1Þc ¼ 0

(56)

we obtain a system of linear homogeneous algebraic equations in the unknowns c, which has nontrivial solutions if and only if the determinant of the coefficients vanishes:    A11  l A12 / A1n      A21 A22  l / A2n  ¼0 detðA  l1Þ ¼  (57) / / /   /    An1 An2 / Ann  l  Equation (57) is known as characteristic (or secular) equation of the square matrix A. Expanding the determinant we find a polynomial of degree n in l, called the characteristic polynomial of matrix A: detðA  l1Þ ¼ a0 þ a1 l þ a2 l2 þ / þ an1 ln1 þ an ln ¼ Pn ðlÞ

(58)

258

CHAPTER 6 Matrices

the equation: Pn ðlÞ ¼ 0

(59)

being an algebraic equation of degree n in l having n roots. Solution of Eqn (59) gives: l1 ; l2 ; .; ln n rootsðthe eigenvalues of AÞ c1 ; c2 ; .; cn n column coefficients ðthe eigenvectors of AÞ

(60)

If L is the diagonal matrix of the eigenvalues, Eqn (59) can be written in the alternative way: Pn ðlÞ ¼ ðl1  lÞðl2  lÞ/ðln  lÞ ¼ 0

(61)

and, on comparing the coefficients of the different powers of l: X a0 ¼ l1 l2 /ln ¼ det A; /; an1 ¼ ð1Þn1 li ; an ¼ ð1Þn

(62)

i

In general, the kth coefficient in Eqn (58) is given by: ak ¼ ð1Þk

n X

ðnkÞ

Mii

kn

(63)

i¼1 ðnkÞ

is the principal minor (namely, the minor along the diagonal) of order ðn  kÞ of where Mii matrix A. The whole set of the n eigenvalue equations for A: Ac1 ¼ l1 c1 ;

Ac2 ¼ l2 c2 ; /;

Acn ¼ ln cn

(64)

can be replaced by the full eigenvalue equation: AC ¼ CL

(65)

if we introduce the square matrices of order n: 0

l1

0

B B 0 l2 L¼B B/ / @ 0

0

/

0

1

/

C 0 C C; /C A

/

ln

/

0

c11

B Bc B 21 C ¼ ðc1 c2 /cn Þ ¼ B B B/ @ cn1

1

c12

/

c22

/

/

/

C c2n C C C C /C A

cn2

/

cnn

c1n

(66)

where L is the diagonal matrix of the n eigenvalues, and C is the row matrix of the n eigenvectors (a square matrix on the whole). If det Cs0;C1 exists, and the square matrix A can be brought to a diagonal form through the transformation C1 A C ¼ L a process which is called the diagonalization of matrix A.

(67)

6.5 The matrix eigenvalue problem

259

If A is Hermitian:3 A ¼ Ay

(68)

It is shown in Problems 6.1 and 6.2 of this chapter that the eigenvalues are real numbers and the eigenvectors are orthonormal (vectors normalized and orthogonal in pairs), namely: cym cn ¼ dmn

(69)

so that the complete matrix of the eigenvectors is a unitary matrix ðC1 ¼ Cy Þ: Cy C ¼ C Cy ¼ 1 0

C1 ¼ Cy

(70)

and Eqn (67) therefore becomes: Cy A C ¼ L

(71)

so that a Hermitian matrix A can be brought to diagonal form by a unitary transformation with the complete matrix of its eigenvectors. While further details and examples for the general ð2  2Þ Hermitian matrix are given as problems at the end of this chapter, we examine below the simple case of the ð2  2Þ Hermitian matrix A:   a b A¼ (72) b a where a,b are real positive numbers. The secular equation is:   a  l b   ¼0  b a  l

(73)

giving upon expansion the equation quadratic in l: ða  lÞ2 b2 ¼ ða  l þ bÞða  l  bÞ ¼ ðl1  lÞðl2  lÞ ¼ 0

(74)

with the roots (the eigenvalues): l1 ¼ a þ b;

l2 ¼ a  b

(75)

We now turn to the evaluation of the eigenvectors. 1. l1 ¼ a þ b (first eigenvalue)

(

ða  l1 Þ c1 þ bc2 ¼ 0 c21 þ c22 ¼ 1

(76)

We solve the linear homogeneous system Eqn (76) for the first eigenvalue with the additional constraint of coefficients normalization:4   c2 l1  a b ¼ ¼ 1 0 c2 ¼ c 1 ¼ (77) c1 1 b b 3 4

The name Hermitian arises from the French mathematician Charles Hermite (1822–1901). The solution of the homogeneous system is seen to give only the ratio c2/c1.

260

CHAPTER 6 Matrices

1 1 c1 ¼ pffiffiffi; c2 ¼ pffiffiffi 2 2 1 0 1 B pffiffi2ffi C C B c1 ¼ B C @ 1 A pffiffiffi 2

c21 þ c22 ¼ 2c21 ¼ 1 0

2. l2 ¼ a  b (second eigenvalue)

(78)

(79)

8 < ða  l2 Þ c1 þ bc2 ¼ 0

(80)

: c2 þ c2 ¼ 1 1 2

We now solve system Eqn (80) for the second eigenvalue with the additional constraint of coefficients normalization:   c2 l2  a b ¼  ¼ 1 0 c2 ¼ c1 ¼ (81) c1 2 b b 1 c21 þ c22 ¼ 2c21 ¼ 1 0 c1 ¼ c2 ¼ pffiffiffi 2 0 1 1 B pffiffi2ffi C B C c2 ¼ B C @ 1 A pffiffiffi 2 The orthonormal properties of the eigenvectors are easily checked: 1 0 1  B pffiffiffi C    1 1 1 1 1 1 B 2C y y p ffiffi ffi p ffiffi ffi  þ ¼ 1; c c ¼ ¼ c1 c1 ¼ pffiffiffi pffiffiffi B C 2 2 2 2 2 2 @ 1 A 2 2 pffiffiffi 2 1 1  B pffiffiffi C 1 1 1 2C pffiffiffi B C ¼  þ ¼ 0; B 2 2 2 @ 1 A pffiffiffi 2 0

cy1 c2 ¼



so that:

1 pffiffiffi 2

0

y



c2 c 1 ¼

1 B pffiffi2ffi B C ¼ ðc1 c2 Þ ¼ B @ 1 pffiffiffi 2

1 pffiffiffi 2

1 1 pffiffiffi C 2C C 1 A pffiffiffi 2

(82)

(83)

0

1 1 pffiffiffi B 2C 1 1 B C B C ¼ þ ¼1 @ 1 A 2 2 pffiffiffi 2 (84) 0 1 1  pffiffi2ffi C 1 B 1 1 C pffiffiffi B B C ¼ þ ¼0 2 2 2 @ 1 A pffiffiffi 2 (85)

(86)

6.6 Functions of Hermitian matrices

261

is a unitary matrix: Cy C ¼ CCy ¼ 1

(87)

6.6 FUNCTIONS OF HERMITIAN MATRICES 6.6.1 Analytic functions If A is a Hermitian matrix of order n, L the diagonal matrix of its n eigenvalues, and C the row matrix of its n eigenvectors, we have just seen from Eqn (71) that A can be brought to diagonal form by the unitary transformation with the complete matrix of its eigenvectors (in the last section of this chapter, we shall give the further example of the Hu¨ckel matrix H for the six p electrons of benzene). For A ¼ Ay, Eqn (71) gives for the inverse transformation:

Similarly:

A ¼ CLCy

(88)

  A2 ¼ CLCy CLCy ¼ CL2 Cy    An ¼ AA/A ¼ CLCy CLCy / CLCy ¼ CLn Cy

(89) (90)

In this way, we can define any analytic function (namely, a function expressible as a convergent power series) of the Hermitian matrix A in the form: FðAÞ ¼ CFðLÞCy

(91)

where F specifies the kind of function (inverse, square root, exponential, etc.). Examples are: A1 ¼ CL1 Cy

(92)

A1=2 ¼ CL1=2 Cy

(93)

provided A is positive definite (positive eigenvalues). In fact, it is easily proved that:  AA1 ¼ A CL1 Cy ¼ ðCLÞL1 Cy ¼ CCy ¼ 1   A1 A ¼ CL1 Cy A ¼ CL1 LCy ¼ CCy ¼ 1

(94)

Ay ¼ A;

(96)

(95)

where we have used: Ly ¼ L;

AC ¼ CL;

Cy A ¼ LCy

Similarly:

  A1=2 A1=2 ¼ CL1=2 Cy CL1=2 Cy ¼ CL1=2 L1=2 Cy ¼ CLCy ¼ A

(97)

262

CHAPTER 6 Matrices

6.6.2 Projectors and canonical form Let us introduce the square matrix of order n: Pm ¼ cm cym

(98)

as the projector corresponding to the eigenvalue lm : The projectors have the properties: Pm P m ¼ P m

(99)

Pm Pn ¼ 0ðnsmÞ

(100)

idempotency,

mutual exclusivity, and: n X

Pm ¼ 1

(101)

m¼1

completeness (or resolution of the identity). Then: A¼

n X

lm P m

(102)

m¼1

is called the canonical form of the Hermitian matrix A. In fact, from the nth eigenvalue equation for A, we obtain:

Acn ¼

X m

!

A cn ¼ ln cn

lm Pm cn ¼

X m

lm cm cym cn ¼

(103) X

lm cm dmn ¼ ln cn

(104)

m

For a function F of the Hermitian matrix A: FðAÞ ¼

n X  F lm P m

(105)

m¼1

is the canonical form of the analytic function F(A).

6.6.3 Examples We now apply these results to the simple case of the Hermitian matrix A of order 2 already given in Eqn (72):   a b A¼ (106) b a with a and b real positive numbers. The general Hermitian matrix A of order 2 will be examined in the problems at the end of this chapter.

6.6 Functions of Hermitian matrices

263

The secular equation: jA  l1j ¼ 0

(107)

has the real roots: l1 ¼ a þ b; and the orthonormal eigenvectors: 1 1 0 0 1 1 B pffiffi2ffi C Bpffiffi2ffi C C C B B c1 ¼ B ¼ ; c C C; B 2 @ 1 A @ 1 A pffiffiffi pffiffiffi 2 2

l2 ¼ a  b

(108)

1 1 1 B pffiffi2ffi pffiffi2ffi C C B c2 Þ ¼ B C @ 1 1 A pffiffiffi pffiffiffi 2 2 0

C ¼ ð c1

(109)

From eigenvectors Eqn (109) we now construct the two square symmetric matrices of order 2: 0 1 0 1 1 1 1 p ffiffi ffi   B 2C 1 B2 2C B C pffiffiffi p1ffiffiffi y C ¼B P 1 ¼ c1 c1 ¼ B (110) C @1 1A @ 1 A 2 2 pffiffiffi 2 2 2 0 1 0 1 1 1 1 p ffiffi ffi    B 2C B 2 2 C B C p1ffiffiffi p1ffiffiffi y C ¼B (111) P 2 ¼ c2 c2 ¼ B C @ 1 1 A @ 1 A 2 2 pffiffiffi  2 2 2 It is seen that the two matrices P1 and P2 do not admit inverse (the determinants of both are zero) and have the properties: 0 10 1 0 1 1 1 1 1 1 1 B 2 2 CB 2 2 C B 2 2 C CB C B C (112) P21 ¼ B @ 1 1 A @ 1 1 A ¼ @ 1 1 A ¼ P1 2 2 2 2 2 2 0 10 1 0 1 1 1 1 1 1 1    B2 B B 2C 2C 2C CB 2 C B2 C P22 ¼ B (113) @ 1 1 A@ 1 1 A ¼ @ 1 1 A ¼ P2    2 2 2 2 2 2 0 10 1 1 1 1 1 ! 0 0 B 2 2 CB 2  2 C CB C P1 P2 ¼ B (114) @ 1 1 A@ 1 1 A ¼ 0 0 ¼ 0  2 2 2 2 0 10 1 1 1 1 1    0 0 B2 CB 2 2 C 2 ¼0 (115) P 2 P1 ¼ @ 1 1 A @ 1 1 A ¼ 0 0  2 2 2 2

264

CHAPTER 6 Matrices 0

1 B2 P1 DP2 ¼ B @1 2

1 0 1 1 C B 2C þ B 2 1A @ 1  2 2

1 1  C 2C ¼ 1 A

1

0

0

1

! ¼1

(116)

2

As already said, matrices having these properties (idempotency, mutual exclusivity, completeness) are called projectors. In fact, acting on matrix C of Eqn (109) we see that: P1 C ¼ P1 c1 þ P1 c2 ¼ c1 since:

0

1 B2 P1 c1 ¼ B @1 2 0 1 B2 P1 c 2 ¼ B @1 2

1 0 1 0 1 0 1 1 1 1 1 1 1 p1ffiffiffi pffiffiffi þ pffiffiffi pffiffiffi B C B C 2C 2 CB C ¼ B 2 2 2 2 C ¼ B 2 C ¼ c1 AB A @ A @ @ 1 1 1 1 1 1 1 A pffiffiffi pffiffiffi þ pffiffiffi pffiffiffi 2 2 2 2 2 2 2 1 0 1 0 1 1 1 1 1 1 1    pffiffiffi þ pffiffiffi pffiffiffi B 2 2 2 2C 0 2C 2 CB C B C B ¼ ¼ ¼0 A 1 @ 1 A @ 1 1 1 1 A 0 pffiffiffi  pffiffiffi þ pffiffiffi 2 2 2 2 2 2

(117)

(118)

(119)

so that, acting on the complete matrix C of the eigenvectors, P1 selects its eigenvector c1 at the same time annihilating c2 : In the same way: (120) P2 C ¼ P2 c1 þ P2 c2 ¼ c2 This makes evident the projector properties of matrices P1 and P2 : Matrices P1 and P2 allow one to write matrix A in the canonical form: A ¼ l1 P1 þ l2 P2 Equation (121) is easily verified: 8 0 1 0 1 1 1 1 1 > >  > > B2 2C B > 2C > C þ ða  bÞB 2 C l1 P1 þ l2 P2 ¼ ða þ bÞB > > @ A @ > 1 1 1 1 A > >  < 2 2 2 2 0 1 > a þ b a  b a þ b a  b ! > >  > B 2 þ 2 C > a b > 2 2 C > ¼B > @a þ b a  b a þ b a  bA ¼ b a ¼ A > > > :  þ 2 2 2 2

(121)

(122)

The same holds true for any analytic function F of matrix A: FðAÞ ¼ Fðl1 ÞP1 þ Fðl2 ÞP2

(123)

6.6 Functions of Hermitian matrices

265

Therefore, it will be possible to calculate in an easy way, say, the inverse or the square root of the Hermitian matrix A. For instance, we obtain for the inverse matrix ðF ¼1 Þ: 8 0 1 0 1 > 1 1 1 1 > > > > B 2ða þ bÞ 2ða þ bÞ C B 2ða  bÞ 2ða  bÞ C > > B C B C 1 1 > > l P þ l P ¼ B CþB C 1 2 > 1 2 > @ A @ A 1 1 1 1 > > >  > > 2ða þ bÞ 2ða þ bÞ 2ða  bÞ 2ða  bÞ > > < ! ða  bÞ þ ða þ bÞ ða  bÞ  ða þ bÞ 1 > ¼  > > 2  b2 > 2 a ða  bÞ  ða þ bÞ ða  bÞ þ ða þ bÞ > > > > > > ! ! > > > a b 2a 2b > 1 1 > > ¼  ¼ A1 ¼ > > 2  b2 : 2 a2  b 2 a b a 2b 2a

(124)

1 1 and we obtain the usual resultpfor pffiffiffiffiffi matrix ðA A ¼ AA ¼ 1Þ ffiffiffiffiffi the inverse In the same way, provided l1 and l2 are positive, we can calculate the square root of matrix A pffi ðF ¼ Þ:

8 pffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi > > > A ¼ a þ b P 1 þ a  b P2 > > 1 0 > < AþB AB pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi ! aþbþ ab aþb ab B 2 1 2 C > C ¼ > pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi ¼ B > @A  B A þ BA > 2 > a þ b  a  b a þ b þ a  b > : 2 2 8 ! ! > > AþB AB pffiffiffiffipffiffiffiffi 1 A þ B A  B > > > A A¼ > > 4 AB AþB > AB AþB > > > >  ! > 2 2 > < 2 A 2  B2 1 ðA þ BÞ þ ðA  BÞ ¼  4 > > 2 A 2  B2 ðA  BÞ2 þðA þ BÞ2 > > >   ! > ! > > 2 A2 þ B2 2 A2  B2 4a 4b > 1 1 > > > ¼ ¼ ¼A   > > 4 2 A2  B2 2 A2 þ B2 4 4b 4a :

(125)

(126)

as it must be. In Eqns (125) and (126) we have put for brevity: A¼

pffiffiffiffiffiffiffiffiffiffiffiffi a þ b;

B¼

pffiffiffiffiffiffiffiffiffiffiffiffi ab

(127)

These examples show how far we can go when the eigenvalues and the eigenvectors of a symmetric (Hermitian) matrix are known.

266

CHAPTER 6 Matrices

6.7 THE MATRIX PSEUDOEIGENVALUE PROBLEM The eigenvalue Eqn (55) can be generalized to the case of a non-orthogonal metric: A0 c0 ¼ l0 M0 c0

(128)

M0 ¼ 1 þ S

(129)

where:

is the metric matrix and: Ss0

(130)

the matrix describing non-orthogonality. Matrix S is traceless. Eqn (128) is called the pseudoeigenvalue equation for the Hermitian matrix A0 , and gives the pseudosecular equation: 5

jA0  l0 M0 j ¼ 0

(131)

The full pseudoeigenvalue equation generalizing Eqn (65) is then: A0 C0 ¼ M0 C0 L0

(132)

and the problem is now to find a matrix of eigenvectors C0 such that matrices A0 and M0 are simultaneously diagonalized, with M0 brought to the identity matrix: C0y A0 C0 ¼ L0 ;

C0y M0 C0 ¼ 1

(133)

It is worth noting that matrix C0 is no longer unitary. It is immediately evident from the second of Eqn (133) that: C ¼ M0

1=2

C0

(134)

is a unitary matrix:

since:

Cy C ¼ CCy ¼ 1

(135)

 y ðM01=2 C0 Þ ðM01=2 C0 Þ ¼ C0y ðM01=2 M01=2 C0 ¼ C0y M0 C0 ¼ 1

(136)

The full pseudoeigenvalue Eqn (132) is hence equivalent to the ordinary eigenvalue equation: AC ¼ CL

(137)

A ¼ M01=2 A0 M01=2

(138)

L ¼ L0

(139)

for the symmetrically transformed matrix:

having:

Namely, tr S ¼

5

n X i¼1

Sii ¼ 0.

6.7 The matrix pseudoeigenvalue problem

267

the same eigenvalues, and: C ¼ M01=2 C0

(140)

the transformed eigenvectors. The complete pseudoeigenvalue problem for a Hermitian matrix A of order 2 is studied in Problem 6.13. Example: The Lo¨wdin’ symmetrical orthogonalization If: ^ 0 A0 ¼ c0y Ac

(141)

is the matrix representative of the operator A^ in the non-orthogonal basis

c0

with metric:

M0 ¼ c0y c0

(142)

Equation (137) can be interpreted as the full eigenvalue equation for the operator A^ in the symmetrically orthogonalized basis (Lo¨wdin, 1950): c ¼ c0 M01=2

(143)

We can easily check that: ^ ¼ M01=2 c0y Ac ^ 0 M01=2 ¼ M01=2 A0 M01=2 A ¼ cy Ac y

01=2 0y 0

M¼c c¼M c01

c cM

01=2

¼M

01=2

0

MM

01=2

¼1

(144) (145)

We now work out in detail the case of the symmetrical orthogonalization of two regular6 functions and c02 : We can write the set c0 as the ð1  2Þ row matrix:  c0 ¼ c01 c02 (146)

with: 0y

c ¼

c01 c02

! 

(147)

We assume the set to be normalized but non-orthogonal:7 hc01 jc01 i ¼ hc02 jc02 i ¼ 1; hc01 jc02 i ¼ hc02 jc01 i ¼ S Then the metric matrix of this basis is the ð2  2Þ Hermitian (positive definite8) matrix:   1 S 0 0y 0 M ¼c c ¼ S 1 6

(148)

(149)

We recall that (Section 1.1.1 of Chapter 1) a regular function is a mathematical function satisfying the three conditions of being (1) single-valued, (2) continuous with its first derivatives,R and (3) quadratically integrable, i.e. vanishing at infinity.  7 For the scalar products, we use the Dirac notation: hc0i jc0j i ¼ drc01 ðrÞc0j ðrÞ ¼ Sij s0 for jsi; ¼ 1 for j ¼ i. 8 A matrix is positive definite when its eigenvalues are real positive.

268

CHAPTER 6 Matrices

The Lo¨wdin symmetrically orthogonalized set is given by Eqn (143). Using the techniques described in Section 6.6 of this chapter it is easily seen that the inverse of the square root of matrix M0 is: 0 1 aþb ab 1=2 B 2 2 C (150) ¼ @a M0 b aþ bA 2 2 where:

(

b ¼ ð1  SÞ1=2   1 1 a2 þ b2 ¼ 2 1  S 2 ; a2  b2 ¼ 2S 1  S2 1=2

a ¼ ð1 þ SÞ

;

(151)

Then, the explicit form of the transformed functions will be: 8 0 a þ b þ c0 a  b > < c1 ¼ c1 2 2 2 > a þ b a  b : þ c01 c2 ¼ c02 2 2

(152)

the reason for the name symmetrical orthogonalization becoming now evident. We can easily check that the transformed functions Eqn (152) are normalized and orthogonal one to another. This technique of symmetrical orthogonalization was used by the author in valence bond calculations of the orientational dependence of the short-range interaction of two hydrogen molecules (Magnasco and Musso, 1967a,b).

6.8 THE LAGRANGE INTERPOLATION FORMULA The interpolation formula due to Lagrange: FðAÞ ¼

n X  Pnsm ðA  ln 1Þ  F lm Pnsm lm  ln m¼1

(153)

makes it possible to calculate any analytic function of a square matrix A. It is not limited to symmetric matrices and, at variance with the canonical form, makes use only of eigenvalues and positive powers (up to order n  1) of A. By comparing the canonical form of A with that of the Lagrange interpolation formula, it is seen that: Pm ¼ cm cym ¼

Pnsm ðA  ln 1Þ  Pnsm lm  ln

(154)

so that the matrix corresponding to the mth projector can be expressed in terms of a polynomial matrix of order (n1) in A with numerical coefficients which depend on the eigenvalues of A.

6.8 The Lagrange interpolation formula

269

Let A be the ð2  2Þ Hermitian matrix defined by Eqn (72). The Lagrange formula then reads: FðAÞ ¼ Fðl1 Þ

A  l2 1 A  l1 1 Fðl1 ÞðA  l2 1Þ  Fðl2 ÞðA  l1 1Þ þ Fðl2 Þ ¼ ¼ p1 þ qA l1  l2 l2  l1 l1  l2

(155)

with: p¼

l1 Fðl2 Þ  l2 Fðl1 Þ ; l1  l2

q¼

Fðl1 Þ  Fðl2 Þ l1  l2

(156)

Let us calculate the inverse of matrix A of Eqn (72) using the Lagrange interpolation formulae, Eqns (155) and (156). We have: p¼

1 l1 l1 l21  l22 l1 þ l2 2a 2  l2 l1 ¼ ¼ ¼ 2 l1  l2 l1 l2 ðl1  l2 Þ l1 l2 a  b2

(157)

1 l1 l2  l1 1 1 1  l2 ¼ ¼ ¼ 2 l1  l2 l1 l2 ðl1  l2 Þ l1 l2 a  b2

(158)

q¼ so that: A

1

2 1

¼ p1 þ qA [ ða  b 2



2a 0

0 2a



 þ

a b

b a





2 1



¼ a b 2

a b

b a

 (159)

as it must be. As a further example, let us calculate the square root of matrix A of Eqn (72) using the Lagrange interpolation formula. We have: p¼

l1 Fðl2 Þ  l2 Fðl1 Þ Bl1  Al2 Bða þ bÞ  Aða  bÞ aðA  BÞ þ bðA þ BÞ ¼ ¼ ¼ l1  l2 l1  l2 2b 2b q¼

Fðl1 Þ  Fðl2 Þ A  B ; ¼ l1  l2 2b

A¼

pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi a þ b; B ¼ a  b

(160) (161)

so that: pffiffiffiffi A ¼ p1 þ qA     1 aðA  BÞ þ bðA þ BÞ 0 aðA  BÞ bðA  BÞ ¼ þ 0 aðA  BÞ þ bðA þ BÞ bðA  BÞ aðA  BÞ 2b   1 aðA  BÞ þ bðA þ BÞ þ aðA  BÞ bðA  BÞ ¼ bðA  BÞ aðA  BÞ þ bðA þ BÞ þ aðA  BÞ 2b 0 1 AþB AB B 2 2 C ¼ @A B Aþ (162) BA 2 2 which coincides with the result of Eqn (125). Further examples for the general Hermitian matrix A of order 2 are given in Problems 6.10 and 6.11.

270

CHAPTER 6 Matrices

6.9 THE CAYLEY–HAMILTON THEOREM We give here without proof this theorem, which states that any square matrix A does satisfy its own characteristic equation:9 Pn ðAÞ ¼ a0 1 þ a1 A þ a2 A2 þ / þ an1 An1 þ an An ¼ 0 (163) The proof of this theorem is given in Problem 6.16. An important application of this theorem concerns the possibility of calculating positive or negative nth powers of matrix A, An, in terms of linear combinations of 1, A, A2 ; $$$, An1 . As an example, we evaluate A1 for the symmetric ð2  2Þ matrix (72). Matrix A satisfies its characteristic equation of degree n ¼ 2: P2 ðAÞ ¼ a0 1 þ a1 A þ a2 A2 ¼ 0

(164)

a0 ¼ det A ¼ a2  b2 s0; a1 ¼ 2a; a2 ¼ 1

(165)

where: 1

so that, multiplying through by A

; we obtain: a0 A1 þ a1 1 þ a2 A ¼ 0

" 1

A

1 1 ¼  ða1 1 þ a2 AÞ ¼ a0 a0

2a

0

0

2a

! þ

a

b

b

a

!#

(166) 

¼ a b 2

2 1

a

b

b

a

! (167)

as it must be.

6.10 THE EIGENVALUE PROBLEM IN HU¨CKEL’S THEORY OF THE p ELECTRONS OF BENZENE We now give a more chemical example of application of the matrix techniques, the Hu¨ckel theory of the p electrons in the benzene molecule (Hu¨ckel, 1931). More about Hu¨ckel theory will be said in Chapter 14.

6.10.1 General considerations The Hu¨ckel matrix for the closed chain (ring) of 0 a b Bb a B B0 b H¼B B0 0 B @0 0 b 0

the N ¼ 6 p electrons of the benzene molecule: 1 0 0 0 b b 0 0 0C C a b 0 0C C (168) b a b 0C C 0 b a bA 0 0 b a

where 1 and 6 are adjacent atoms (see Figure 14.10 of Chapter 14) gives the secular equation (169) jH  ε1j ¼ 0 9

See Eqns (58) and (59).

6.10 The eigenvalue problem in Hu¨ckel’s theory of the p electrons of benzene

which is best written explicitly as:

  x   1   0 D6 ¼   0   0  1

Here, we have posed:

1 0 x 1 1 x 0 1 0 0 0 0

0 0 1 x 1 0

 0 1   0 0   0 0  ¼0 1 0   x 1   1 x

8 > aε > > > < x ¼ b > > > > : ε ¼ a þ bx;

Dε ¼ ε  a ¼ xb;

Dε ¼ x b

271

(170)

(171)

so that x measures the bond energy of the p electron in units of b10 (x > 0 means bonding, x < 0 means antibonding). By expanding determinant Eqn (170) we obtain an algebraic equation of the sixth degree in x that can be easily factorized into the three quadratic equations:   2 (172) D6 ¼ x6  6x4 þ 9x2  4 ¼ x2  4 x2  1 ¼ 0 with the roots (the eigenvalues), written in ascending order: x ¼ 2; 1; 1; 1; 1; 2

(173)

as shown in Figure 14.9 of Chapter 14. Because of the high D6h symmetry of the molecule, two levels are doubly degenerate. The calculation of the molecular orbital (MO) coefficients (the eigenvectors) in the linear combination of the six carbon 2pz atomic orbitals in the basis c: ( 4 ¼ cC; C ¼ ð c c c c c c Þ 1 2 3 4 6 5 P (174) 4n [ cm cmn ; m; n ¼ 1; 2; /; 6 m

where cn is a ð6  1Þ column vector and C the ð6  6Þ matrix of the AO coefficients in the Hu¨ckel MOs, can be done using elementary algebraic methods by solving the linear homogeneous system corresponding to Eqn (170): 8 > xc1 þ c2 þ c6 ¼ 0 > > > c1  xc2 þ c3 ¼ 0 > > > < c  xc þ c ¼ 0 2 3 4 (175) >  xc þ c ¼0 c 3 4 5 > > > > > c4  xc5 þ c6 ¼ 0 > : c1 þ c5  xc6 ¼ 0

10

We recall that both a (the energy of the atomic 2pz electron) and b (the bond energy) are negative quantities.

272

CHAPTER 6 Matrices

The calculation of the coefficients for real MOs is given in detail elsewhere (Magnasco, 2010a). With reference to Figure 14.10 of Chapter 14, the calculation shows that the real MOs are: 8 1 > > 41 ¼ pffiffiffi ðc1 þ c2 þ c3 þ c4 þ c5 þ c6 Þ > > 6 > > > > > 1 > > > 42 ¼ ðc1  c3  c4 þ c6 Þfx > > 2 > > > > > 1 > > ðc1 þ 2c2 þ c3  c4  2c5  c6 Þfy > < 43 ¼ pffiffiffiffiffi 12 (176) > 1 > 2 2 > 44 ¼ pffiffiffiffiffi ðc1  2c2 þ c3 þ c4  2c5 þ c6 Þfx  y > > > 12 > > > > > 1 > > 4 ¼ ðc  c3 þ c4  c6 Þfxy > > > 5 2 1 > > > > > 1 > : 46 ¼ pffiffiffi ðc1  c2 þ c3  c4 þ c5  c6 Þ 6 The first degenerate MOs11 42 and 43 transform like (x, y) and are bonding MOs (highest occupied MOs), the second degenerate MOs 44 and 45 transform like ðx2  y2 ; xyÞ and are antibonding MOs (lowest unoccupied MOs). It is left as an easy exercise to the reader to verify that the MOs Eqn (176) are normalized and orthogonal and not interacting in pairs. Furthermore, even if their form is different, each pair of MOs belonging to the respective degenerate eigenvalue gives the same value for the corresponding orbital energy.

6.10.2 Unitary transformation diagonalizing the Hu¨ckel’s matrix It is easily shown that the coefficients of the AOs in the MOs Eqn (176) form a unitary matrix C diagonalizing the Hu¨ckel matrix H. From Eqn (176) we can construct the ð6  6Þ matrices C and Cy : 1 0 1 1 1 1 1 1 pffiffiffi pffiffiffiffiffi pffiffiffiffiffi pffiffiffi 2 12 12 6 C B 6 2 C B B 1 2 2 1 C B pffiffiffi 0 pffiffiffiffiffi pffiffiffiffiffi 0 pffiffiffi C C B 6 12 12 6C B C B 1 1 1 1 1 C B 1 pffiffiffiffiffi pffiffiffiffiffi  pffiffiffi C B pffiffiffi  C B 6 2 2 12 12 6 C (177) C¼B C B 1 1 1 C B pffiffiffi 1 p1ffiffiffiffiffi p1ffiffiffiffiffi C p ffiffi ffi  B 2 2 B 6 12 12 6C C B C B 1 2 2 1 B pffiffiffi 0 pffiffiffiffiffi pffiffiffiffiffi 0 pffiffiffi C C B 6 12 12 6 C B C @ 1 1 1 1 1 1 A pffiffiffi pffiffiffiffiffi pffiffiffiffiffi  pffiffiffi 2 6 2 12 12 6 11

Loosely speaking, we attribute to MOs a property (degeneracy) of energy levels.

6.11 Problems 6

which is best written as:

so that:

0 pffiffiffi 2 B pffiffi2ffi B B pffiffiffi 1 B 2 pffiffiffi ¼ pffiffiffiffiffi B B 12 B 2 B pffiffiffi @ 2 pffiffiffi 2

0 pffiffiffi 2 B pffiffi3ffi B B 1 B 1 Cy ¼ pffiffiffiffiffi B 12 B B 1 B pffiffiffi @ 3 pffiffiffi 2

satisfying:

pffiffiffi 3 1 0 2 pffiffiffi  3 1 pffiffiffi  3 1 0 2 pffiffiffi 3 1 pffiffiffi 2 0 2

pffiffiffi 1 pffiffiffi 2 3 pffiffiffi 0  2C C pffiffiffi pffiffiffi C 2 C  3 pffiffiffi C pffiffiffi 3  2C C pffiffiffi C 0 2 A pffiffiffi pffiffiffi  3  2

1 2 1 1 2 1

pffiffiffi pffiffiffi 2 2 pffiffiffi pffiffiffi  3  3 1

1

2 1 1 pffiffiffi pffiffiffi 0  3 3 pffiffiffi pffiffiffi pffiffiffi  2 2  2

pffiffiffi 2 0

pffiffiffi 1 2 pffiffiffi C 3 C C 2 1 C C 2 1 C C pffiffiffi C 0  3A pffiffiffi pffiffiffi 2  2

(178)

Cy C ¼ CCy ¼ 1

(179)

as can be proved by matrix multiplication of Eqn (178) by Eqn (177) and vice versa. Furthermore, direct matrix multiplication shows that:12 Cy HC ¼ ε namely: 0

a þ 2b B 0 B B 0 ε¼B B 0 B @ 0 0

0 aþb 0 0 0 0

0 0 aþb 0 0 0

0 0 0 ab 0 0

0 0 0 0 ab 0

273

1 0 0 ε1 B0 0 C C B B 0 C C¼B0 B 0 C C B0 0 A @0 0 a  2b

0 ε2 0 0 0 0

0 0 ε3 0 0 0

0 0 0 ε4 0 0

0 0 0 0 ε5 0

(180) 1 0 0C C 0C C (181) 0C C 0A ε6

and we obtain the diagonal matrix ε of the eigenvalues of H. So, Hu¨ckel’s matrix H is fully diagonalized by the unitary transformation with the complete matrix C of its real eigenvectors.

6.11 PROBLEMS 6 6.1. Show that a Hermitian matrix A has real eigenvalues. Answer: If Ay ¼ A then Ly ¼ L Hint: Use the mth eigenvalue equation for the Hermitian matrix A and its adjoint. 12

Here ε is the diagonal matrix of the eigenvalues ε of H.

274

CHAPTER 6 Matrices

6.2. Show that a Hermitian matrix A has orthonormal eigenvectors, i.e. that C is unitary. Answer: If Ay ¼ A

Cy AC ¼ L

Cy C ¼ CCy ¼ 1

Hint: Use the full eigenvalue equation for matrix A and its adjoint considering that Ay ¼ A and Ly ¼ L: 6.3. Solve the complete ð2  2Þ eigenvalue problem for the Hermitian matrix:   a1 b A¼ b a2 where, for the time being, we shall assume that a1 ; a2 ; b are all real negative quantities (as in the Hamiltonian matrix).13 Show that the complete matrix C of the eigenvectors is a unitary matrix. Answer: Let     D þ ða2  a1 Þ 1=2 D  ða2  a1 Þ 1=2 ; b¼ a¼ 2D 2D h i1=2 D ¼ ða2  a1 Þ2 þ4b2 >0 Then:

  a 2l1 ¼ ða1 þ a2 Þ  D; c1 ¼ b   b 2l2 ¼ ða1 þ a2 Þ þ D; c2 ¼ a 

so that: C ¼ ðc1 c2 Þ ¼

a b

 b ; Cy C ¼ Cy C ¼ 1 a

Hint: Solve first the quadratic secular equation for A; next the system of homogeneous linear equations for each eigenvalue in turn, taking into account the normalization condition for the coefficients. 6.4. Show by actual calculation that the Hermitian matrix A is diagonalized by the unitary transformation with the complete matrix of its eigenvectors. Answer: Cy AC ¼ L Hint: Use the properties of a; b found in Problem 6.3. If all matrix elements are positive (as in the metric matrix), we must change b into b in the expressions of c1 and c2, which implies the interchanging of signs in the off-diagonal elements of the projectors A1 and A2. 13

6.11 Problems 6

275

6.5. Find the projectors for the Hermitian matrix A ða1 ; a2 ; b < 0Þ and verify its canonical form. Answer:     a2 ab b2 ab ; A2 ¼ A1 ¼ ab b2 ab a2 l1 A1 þ l2 A2 ¼ A Hint: Follow the definitions and make use of the properties of a; b found in Problem 6.3. 6.6. Find the inverse of the Hermitian matrix A through its canonical form. Answer:   a2 b A1 ¼ ðdet AÞ1 b a1 Hint: Use: 1 A1 ¼ l1 1 A1 þ l2 A2

and some results of Problems 6.3 and 6.5. 6.7. Find the square root of the Hermitian matrix A, provided it is positive definite (positive eigenvalues):   a1 b a1 ; a2 ; b > 0; l1 ; l2 > 0 A¼ b a2 Answer: A1=2 ¼ where (Problem 6.3): 1=2

 1=2

A ¼ l1 ; B ¼ l2 ; a ¼

Aa2 þ Bb2

ðA  BÞab

ðA  BÞab

Ab2 þ Ba2

D  ða2  a1 Þ 2D

1=2 ; b¼

!

  D  ða2  a1 Þ 1=2 2D

Hint: Use the canonical form of matrix A taking into account previous footnote 13: 1=2

1=2

A1=2 ¼ l1 A1 þ l2 A2 6.8. Find the inverse of the square root of the positive definite Hermitian matrix A:   a1 b A¼ a1 ; a2 ; b > 0; l1 ; l2 > 0 b a2 Answer: A1=2 ¼

1 AB



Ab2 þ Ba2 ðA  BÞab

where A; B are defined in Problem 6.7.

ðA  BÞab Aa2 þ Bb2



276

CHAPTER 6 Matrices

Hint: Use the canonical form of matrix A taking into account Footnote 13. 6.9. Find expðAÞ and expðAÞ. Answer: expðAÞ ¼

expðAÞ ¼

a2 expðl1 Þ þ b2 expðl2 Þ

ab½expðl1 Þ  expðl2 Þ

ab½expðl1 Þ  expðl2 Þ

b2 expðl1 Þ þ a2 expðl2 Þ

!

a2 expðl1 Þ þ b2 expðl2 Þ

ab½expðl1 Þ  expðl2 Þ

ab½expðl1 Þ  expðl2 Þ

b2 expðl1 Þ þ a2 expðl2 Þ

!

where l1 ; l2 are the eigenvalues of A, and a,b are defined in Problem 6.7. 6.10. Find the inverse of matrix A according to the Lagrange formula. Answer:   b 1 a2 1 A ¼ ðdet AÞ b a1 Hint: Calculate coefficients p; q for F ¼ inverse. 6.11. Find the square root of the positive definite Hermitian matrix A according to the Lagrange formula. Answer: !  2 þ Bb2 ðA  BÞab Aa A1=2 ¼ ðA  BÞab Ab2 þ Ba2 where A; B; a; b were defined in Problem 6.7. Hint: Calculate coefficients p; q for F ¼ ð/Þ1=2. 6.12. Find the square root for the unsymmetrical matrix (Frazer et al., 1963):   3 4 A¼ 1 1 Answer: A

1=2

¼

2 1 2

2 0

0

! or A

1=2

2 ¼@ 1  2

2

1 A

0

Hint: Since matrix A cannot be diagonalized by the usual techniques (L turns out to be the identity matrix), we must solve the system of non-linear equations resulting from the definition: A1=2 A1=2 ¼ A

6.11 Problems 6

277

6.13. Solve the complete ð2  2Þ pseudoeigenvalue problem for the Hermitian matrices:     1 S a1 b ; M¼ H¼ S 1 b a2 where we assume that a1 ; a2 ; b < 0 and S > 0. Answer: Let: 1=2 1=2   ðl þ SÞ; B ¼ 1  S2 ð1 þ lSÞ A ¼  1  S2     D  ða2  a1 Þ 1=2 jb  a1 Sj 1=2 l¼ D þ ða2  a1 Þ jb  a2 Sj h

 i1=2 D ¼ ða2  a1 Þ2 þ 4 b  a1 S b  a2 S >0 Then:

  1=2 1 2 1  S ε1 ¼ ða1 þ a2  2bSÞ  D; c1 ¼ 1 þ l þ 2lS l 

2





2

   1=2 A  2 1  S2 ε2 ¼ ða1 þ a2  2bSÞ þ D; c2 ¼ 1 þ l2 þ 2lS B so that:

 1 1 C ¼ ðc1 c2 Þ ¼ 1 þ l þ 2lS l 

2

A B



Hint: Same as for Problem 6.3, taking into account non-orthogonality. 6.14. Show by direct matrix multiplication that, for the ð2  2Þ pseudoeigenvalue problem: Cy MC ¼ 1 so that C0 ¼ M1=2 C is unitary. Hint: Use results and properties found in Problem 6.14. 6.15. Show by direct matrix multiplication that, for the ð2  2Þ pseudoeigenvalue problem, matrix H is brought to diagonal form through a transformation with the non-unitary matrix C: Cy HC ¼ ε Hint: Use results and properties found in Problem 6.14 and the fact that best l diagonalizes the matrix representative of the Hermitian operator H^ over the MO basis resulting from the Ritz method. 6.16. Prove the Cayley–Hamilton theorem.

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CHAPTER 6 Matrices

6.12 SOLVED PROBLEMS 6.1. Real eigenvalues. Consider the mth eigenvalue equation for the Hermitian matrix A: Acm ¼ lm cm Multiply both members on the left by the adjoint eigenvector cym : cym Acm ¼ lm cym cm

Taking the adjoint of both members: cym Ay cm ¼ lm cym cm namely, since Ay ¼ A: cym Acm ¼ lm cym cm Subtracting the corresponding equations, we obtain:

 0 ¼ lm  lm cym cm so that, since cym cm s0,

lm  lm ¼ 0 0 lm ¼ lm

and the eigenvalues are real. Since this is true for all eigenvalues of A, we thereby obtain: Ly ¼ L If cym cm ¼ 1, the eigenvectors are normalized to 1. 6.2. Orthogonal eigenvectors. Consider the complete eigenvalue equation for the Hermitian matrix A: AC ¼ CL Multiplying both members on the left by C1, we obtain: C1 AC ¼ L Taking the adjoint of the last equation:  y Cy Ay C1 ¼ Ly where, since Ay ¼ A and Ly ¼ L:

 y Cy A C1 ¼ L

Comparing with the previous equation for L, we see that: C1 ¼ Cy ; Cy C ¼ CCy ¼ 1 and C is a unitary matrix of orthonormal eigenvectors.

6.12 Solved problems

279

6.3. The complete ð2  2Þ eigenvalue problem. Let the Hermitian (symmetric) matrix A be:   a1 b A¼ b a2 where we assume that a1 ; a2 ; b < 0; as useful in applications (e.g. Hu¨ckel’s theory). We construct the secular equation:    a1  l b   ¼ ða1  lÞða2  lÞ  b2 ¼ 0 P2 ðlÞ ¼  b a2  l  Solution of the resulting quadratic equation:  l2  ða1 þ a2 Þl þ a1 a2  b2 ¼ 0 gives two real roots: 2l1 ¼ ða1 þ a2 Þ  D; 2l2 ¼ ða1 þ a2 Þ þ D h i1=2 D ¼ ða2  a1 Þ2 þ4b2 >0 having the properties: l1 þ l2 ¼ a1 þ a2 ; l2  l1 ¼ D; l1 l2 ¼ a1 a2  b2 ¼ det A We now solve for the eigenvectors, by introducing each eigenvalue in turn in the original system of linear homogeneous equations. We see that the solution of the system gives only the ratio of the coefficients, so that we introduce the auxiliary condition of normalization which will give normalized eigenvectors. 1. First eigenvalue l1 ( ða1  l1 Þc1 þ bc2 ¼ 0 c21 þ c22 ¼ 1     c2 l1  a1 D  ða2  a1 Þ D  ða2  a1 Þ D  ða2  a1 Þ 1=2 ¼ ¼ ¼ >0 ¼ c1 1 b 2b 1jbj D þ ða2  a1 Þ since: D2 ¼ ða2  a1 Þ2 þ 4b2 ; 4b2 ¼ ½D  ða2  a1 Þ½D þ ða2  a1 Þ;   2b ¼ f½D  ða2  a1 Þ½D þ ða2  a1 Þg1=2 From the normalization condition for the coefficients it follows:   2  c2 2D ¼1 ¼ c21 c21 1 þ c1 1 D þ ða2  a1 Þ

280

CHAPTER 6 Matrices

Hence we have for the first eigenvector (in bolded type in the formulae):     D þ ða2  a1 Þ 1=2 D  ða2  a1 Þ 1=2 c11 ¼ ; c21 ¼ 2D 2D 2. Second eigenvalue l2 To get the eigenvector corresponding to the second eigenvalue, we simply interchange 1 / 2 into the homogeneous ( system: ða2  l2 Þc2 þ bc1 ¼ 0 c21 þ c22 ¼ 1   c1 l2  a2 D þ ða2  a1 Þ D þ ða1  a2 Þ D  ða2  a1 Þ ¼ ¼ ¼ ¼ c2 2 b 2b 2jbj 2jbj 1=2  D  ða2  a1 Þ

: pffiffiffiffiffi 12c2 þ ð6  lÞc3 ¼ 0 with the additional normalization condition c21 þ c22 þ c23 ¼ 1 For the eigenvector we are interested in, l ¼ 2, corresponding to L ¼ 1, we have rffiffiffiffiffi pffiffiffi 2 3 3 c2 ¼ c 1 ; c 1 ¼ c2 ¼ pffiffiffi c1 ; c3 ¼  10 2 3

(55)

(56)

430

CHAPTER 10 Angular momentum methods for atoms

so that the required singlet 1 P function will be rffiffiffiffiffi rffiffiffiffiffi 3 2 ðj1 þ j3 Þ  j jð1 PÞ ¼ 10 5 2

(57)

Using Eqn (50) and the Dirac’s formula for N ¼ 2, we can easily check that 2 L^ jð1 PÞ ¼ 1ð1 þ 1Þjð1 PÞ

(58)

2 S^ jð1 PÞ ¼ 0ð0 þ 1Þjð1 PÞ

as it must be. We then have for the 1 P-state arising from the 2p 3d configuration of two non-equivalent electrons the two alternative expressions 1. Complex form rffiffiffiffiffi rffiffiffi rffiffiffiffiffi  3 2 3   pþ1 d 1  ðj þ j3 Þ  j ¼ jð PÞ ¼ 10 1 5 2 20   1  þ k d1 pþ1 k þ  p1 d þ1  þ k dþ1 p1 k  pffiffiffi  p0 5 1

 d 0  þ k d0

p0 k



(59)

2. Real form rffiffiffiffiffi      3   2px 3dz x  þ k 3dz x 2px k þ  2py 3dyz  þ  3dyz 2py  jð PÞ ¼ 20  1  þ pffiffiffi  2pz 3dz2  þ k 3dz2 2pz k 5 (rffiffiffiffiffi  i 3h ¼ ð2px ðr1 Þ3dz x ðr2 Þ þ 3dz x ðr1 Þ2px ðr2 ÞÞ þ 2py ðr1 Þ3dyz ðr2 Þ þ 3dyz ðr1 Þ2py ðr2 Þ 20 ) 1 1  þ pffiffiffi 2pz ðr1 Þ3dz2 ðr2 Þ þ 3dz2 ðr1 Þ2pz ðr2 Þ pffiffiffi ½aðs1 Þbðs2 Þ  bðs1 Þaðs2 Þ 2 5 1

(60) after space-spin separation. States such as jð1 PÞ were used in CI studies of alkaline earth metals for calculating static dipole polarizabilities taking partial account of correlation effects in the valence shell (Magnasco and Amelio, 1978), and for accurate CI studies of static dipole, quadrupole and octupole polarizabilities for the ground states of H, He, Li, Be, Na, Mg (Figari et al., 1983). Correlation in the ground states of Be and Mg was introduced by allowing the ns2 configuration to interact with the nearly degenerate np2 configuration in a way determined by the variation theorem. Correlation in the excited states was introduced by allowing the 1 Lðs lÞ configuration to interact in an optimum way with the 1 Lðp l þ 1Þ configuration and by varying the non-spherical orbitals independently in

10.4 An outline of advanced methods for coupling angular momenta

431

the two configurations. The singlet excited configuration npn0 ðl þ 1Þ is described by the manydeterminant wavefunction:    i l þ 1 1=2 h 1 jð LÞ ¼ np0 n0 ðl þ 1Þ0 þ n0 ðl þ 1Þ0 np0 4l þ 6     l þ 2 1=2 h (61) þ npc n0 ðl þ 1Þc þ n0 ðl þ 1Þc npc 8l þ 12    i þ nps n0 ðl þ 1Þs þ n0 ðl þ 1Þs nps where ð/Þ denotes a normalized Slater determinant in real form with the doubly occupied core orbitals omitted for brevity, the bar refers as usual to b spin, and the orbitals subscripts to cosine (c), sine (s) or no (0) dependence on angle 4. For n ¼ 2, n0 ¼ 3, l ¼ 1, l0 ¼ 2, the general expression (61) reduces to the expression (60) derived before.

10.3.2 The projection operator method The projection operator method due to Lo¨wdin (see Chapter 9) can be used as well to construct atomic states belonging to a definite value of L. The operator projecting out of a function the state with L ¼ k is 2 Y L^  LðL þ 1Þ (62) O^k ¼ kðk þ 1Þ  LðL þ 1Þ LðskÞ where the product includes all Ls except the particular one k we want to construct. This operator acting on the general expansion of the function in terms of L eigenstates will annihilate all terms except that for which L ¼ k which remains unchanged. Simple applications can be found in Problems 10.8 and 10.9. The projection operator formula for the total angular momentum J in the LS coupling was given by Slater (1960).

10.4 AN OUTLINE OF ADVANCED METHODS FOR COUPLING ANGULAR MOMENTA In this section, we shall simply introduce some explanation for a few symbols that are frequently met in the literature when treating angular momentum problems, such as Clebsch–Gordan coefficients, Wigner 3-j and 9-j symbols, Gaunt coefficients, and coupling rules. Details can be found in excellent books on the subject (Rose, 1957; Brink and Satchler, 1993).

10.4.1 Clebsch–Gordan coefficients and Wigner 3-j and 9-j symbols Two angular momentum vectors, jl1 m1 i and jl2 m2 i, can be coupled to a resultant jLMi by the relation X jLMi ¼ jl1 m1 ijl2 m2 ihl1 m1 l2 m2 jLMi (63) m1 m2

where hl1 m1 l2 m2 jLMi is a real number called vector coupling or Clebsch–Gordan coefficient, and we used Dirac’s notation. The transformation (63) is unitary, its inverse being X (64) jl1 m1 ijl2 m2 i ¼ jLMihLMjl1 m1 l2 m2 i LM

432

CHAPTER 10 Angular momentum methods for atoms

where the coefficient hLMjl1 m1 l2 m2 i is the complex conjugate of the corresponding coefficient hl1 m1 l2 m2 jLMi. In recent literature, the Clebsch–Gordan coefficient is usually expressed in terms of the more symmetric Wigner 3-j symbol through the relation (Brink and Satchler, 1993)   pffiffiffiffiffiffiffiffiffiffiffiffiffiffi l1 l2 L (65) hl1 m1 l2 m2 jLMi ¼ ð1Þl1 l2 þM 2L þ 1 m1 m2 M 3- j Clebsch-Gordan ! l1 l2 l3 The Wigner 3-j symbol is non-zero provided m1 m2 m3 m1 þ m2 þ m3 ¼ 0





lm  ln  lg  lm þ ln and has the following general expression (Rose, 1957; Brink and Satchler, 1993)   l1 l2 l3 ¼ dm1 þm2 þm3 ;0  ð1Þl1 þm2 m3 m1 m2 m3 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðl1 þ l2  l3 Þ!ðl1  l2 þ l3 Þ!ð l1 þ l2 þ l3 Þ!ðl3 þ m3 Þ!ðl3  m3 Þ!  ðl1 þ l2 þ l3 þ 1Þ!ðl1 þ m1 Þ!ðl1  m1 Þ!ðl2 þ m2 Þ!ðl2  m2 Þ! X ðl1  m1 þ lÞ!ðl2 þ l3 þ m1  lÞ!  ð1Þl þ l þ l3  lÞ!ðl1  l2 þ m3 þ lÞ!ðl3  m3  lÞ! l!ð l 1 2 l where

maxð0; l2  l1  m3 Þ  l  minðl3  m3 ; l2  l1 þ l3 Þ

(66)

(67)

(68)

The summation over l is restricted to all integers giving non-negative factorials. The Wigner 3-j symbols are today available as standards on the Mathematica software (Wolfram, 2007), where they are evaluated in a sophisticated way in terms of the hypergeometric functions of Chapter 4 (Abramowitz and Stegun, 1965) for integer or half-integer (spin) values of (l, m). The vector coupling of three and four angular momenta implies use of Wigner 6-j or Wigner 9-j symbols, respectively. 8 9 A Wigner 9-j symbol is given as > < l1 l2 L > = (69) l01 l02 L0 > > : ; L1 L2 l It is invariant under interchange of rows and columns, and enjoys the property 9 8 l L L0 > ! ! > >  > = < 0 X L1 L2 L X X l1 l2 L l L L 0 L1 l1 l1 ¼  > 0 0 0 > M1 M2 0 > > m1 m2 m01 m02 m1 m2 0 M M 1 2 ; : L2 l2 l02  0    l1 l02 L0 l2 l02 L2 l1 l01 L1  m01 m02 0 m1 m01 M1 m2 m02 M2

(70)

10.4 An outline of advanced methods for coupling angular momenta

433

that reduces to summations over Wigner 3-j symbols, more easily calculable by Mathematica. Used backwards, relation (70) is said to express the contraction of summations of 3-j symbols to 9-j (Brink 1 1 and Satchler, 1993). Expression (70) is met in the spherical tensor expansion of the product $ ; r12 r10 20 which occurs in the second-order theory of long-range intermolecular forces (Wormer, 1975; Spelsberg et al., 1993; Ottonelli, 1998; Magnasco and Ottonelli, 1999b) in Chapter 17.

10.4.2 Gaunt coefficients and coupling rules The Gaunt coefficient arises from the integration of three complex spherical harmonics (with Condon– Shortley phase) of the same argument U ¼ q; 4:   Z L l1 l2 Gaunt coefficient dUYLM ðUÞ Yl1 m1 ðUÞ Yl2 m2 ðUÞ ¼ G m1 m2 M rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ð2l1 þ 1Þð2l2 þ 1Þð2L þ 1Þ l1 l2 L ¼ ð1ÞM m1 m2 M 4p   l l L  1 2 (71) 0 0 0 where M ¼ m1 þ m2

(72)

All these results take their most simple form in terms of modified spherical harmonics, defined as (Brink and Satchler, 1993) rffiffiffiffiffiffiffiffiffiffiffiffi 4p (73) Ylm ðUÞ Clm ðUÞ ¼ 2l þ 1 which have the orthonormality properties Z 4p dUClm dll0 dmm0 ðUÞCl0 m0 ðUÞ ¼ 2l þ 1

(74)

In terms of the Cs we can write 1. Coupling rule Cl1 m1 ðUÞCl2 m2 ðUÞ ¼ 2

X L

 CLM ðUÞð1ÞM ð2L þ 1Þ

l1 m1

l2 m2

L M

where M ¼ m1 þ m2, and the finite summation is in steps of 2. 2. Gaunt coefficient  Z  L l1 l2 ¼ dUCLM ðUÞCl1 m1 ðUÞCl2 m2 ðUÞ G m1 m2 M   l1 l l2 L ¼ 4pð1ÞM 1 m1 m2 M 0



l2 0

l2 0

l1 0

L 0

L 0

 (75)

 (76)

434

CHAPTER 10 Angular momentum methods for atoms

For m1 ¼ m2 ¼ M ¼ 0, l1 þ l2 þ L ¼ 2g (even), it is obtained as Z2p

Z1 d4

0

1



l dx Pl1 ðxÞPl2 ðxÞPL ðxÞ ¼ 4p 1 0

l2 0

L 0

2 ;

x ¼ cos q

(77)

which is the symmetrical Racah’s formula Z1 1

2  a b c dxPa ðxÞPb ðxÞPc ðxÞ ¼ 2 0 0 0 "

g! ¼ 2 ð1Þ ðg  aÞ!ðg  bÞ!ðg  cÞ! g

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi#2 ð2g  2aÞ!ð2g  2bÞ!ð2g  2cÞ! ð2g þ 1Þ!

(78)

where a, b, c are non-negative integers and a þ b þ c ¼ 2g (even).

10.5 PROBLEMS 10 10.1. Derive the Lande´ g-factor for LS coupling in the vector model (Herzberg, 1945). Answer: gJ ¼

3 SðS þ 1Þ  LðL þ 1Þ þ 2JðJ þ 1Þ 2

Hint: Use the triangle of Figure 10.8 and the Carnot’s theorem. 2 10.2. Find the effect of L^ on the three basis functions (38) having M ¼ MS ¼ 0. Answer: The result is given as Eqn (39). Hint: Make use of the Eqn (35) of the main text. 10.3. Find the eigenvectors of matrix (40) corresponding to l ¼ 0,2,6. Answer: l¼0

1 jð1 SÞ ¼ pffiffiffi ðj1  j2  j3 Þ 3

l¼2

1 jð3 PÞ ¼ pffiffiffi ðj2  j3 Þ 2

l¼6

1 jð1 DÞ ¼ pffiffiffi ð2j1 þ j2 þ j3 Þ 6

Hint: Insert each eigenvalue in turn into the system of linear homogeneous equations determining the secular Eqn (41).

10.5 Problems 10

435

10.4. Check Eqn (43). Hint: 2 2 Use Eqn (35) for L^ and the Dirac’s formula S^ ¼ I^ þ P^12 for spin. 2 1 10.5. Transform the jð SÞ arising from the p configuration of two equivalent electrons with l1 ¼ l2 ¼ 1 from complex to real form. Answer: The result is expression (45). Hint: Use the unitary transformation connecting complex to real functions and the elementary properties of determinants. 10.6. Write all microstates arising from the np3 configuration of three equivalent electrons with l1 ¼ l2 ¼ l3 ¼ 1, and construct the S quartet with MS ¼ 1/2. Answer: Putting for brevity pþ1 ¼ pþ, p1 ¼ p, the 20 Pauli allowed microstates are M MS    2 1=2 1: ð n 1 1 1=2 Þð n 1 0 1=2 Þ n 1 1 1=2  ¼ k pþ p0 pþ k    2:  n 1 1 1=2 n 1 0 1=2 ð n 1 1 1=2 Þ ¼ k pþ p0 pþ k 2 1=2     3: ð n 1 1 1=2 Þ n 1 1 1=2 n 1 1 1=2  ¼ k pþ p pþ k 1 1=2    4:  n 1 1 1=2 n 1 1 1=2 ð n 1 1 1=2 Þ ¼ k pþ p pþ k 1 1=2      5: ð n 1 1 1=2 Þð n 1 0 1=2 Þ n 1 0 1=2 ¼ k pþ p0 p0 k 1 1=2    6:  n 1 1 1=2 n 1 0 1=2 ð n 1 0 1=2 Þ ¼ k pþ p0 p0 k 1 1=2    7: ð n 1 1 1=2 Þ n 1 1 1=2 ð n 1 0 1=2 Þ ¼ k pþ p p0 k 0 3=2       8: ð n 1 1 1=2 Þ n 1 1 1=2 n 1 0 1=2 ¼ k pþ p p0 k 0 1=2    9: ð n 1 1 1=2 Þ n 1 1 1=2 ð n 1 0 1=2 Þ ¼ k pþ p p0 k 0 1=2      10: n 1 1 1=2 n 1 1 1=2 ð n 1 0 1=2 Þ ¼ k pþ p p0 k 0 1=2    11:  n 1 1 1=2 n 1 1 1=2 ð n 1 0 1=2 Þ ¼ k pþ p p0 k 0 1=2     12:  n 1 1 1=2 n 1 1 1=2 n 1 0 1=2  ¼ k pþ p p0 k 0 1=2       13: ð n 1 1 1=2 Þ n 1 1 1=2 n 1 0 1=2 ¼ k pþ p p0 k 0 1=2     14:  n 1 1 1=2 n 1 1 1=2 n 1 0 1=2  ¼ k pþ p p0 k 0 3=2    15:  n 1 1 1=2 n 1 0 1=2 ð n 1 0 1=2 Þ ¼ k p p0 p0 k 1 1=2    16:  n 1 1 1=2 ð n 1 0 1=2 Þ n 1 0 1=2  ¼ k p p0 p0 k 1 1=2     17:  n 1 1 1=2 n 1 1 1=2 n 1 1 1=2  ¼ k p pþ p k 1 1=2    18:  n 1 1 1=2 ð n 1 1 1=2 Þ n 1 1 1=2  ¼ k p pþ p k 1 1=2     19:  n 1 1 1=2 n 1 0 1=2 n 1 1 1=2  ¼ k p p0 p k 2 1=2    20:  n 1 1 1=2 ð n 1 0 1=2 Þ n 1 1 1=2  ¼ k p p0 p k 2 1=2

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CHAPTER 10 Angular momentum methods for atoms

The S quartet with MS ¼ 1/2 is 1 jð4 SÞ ¼ pffiffiffi ðj8 þ j9 þ j10 Þ 3 Hint: Use the techniques of the previous problems. 10.7. Construct the four 4 S functions out of the np3 configuration of three equivalent electrons (e.g. the ground-state configuration of the N atom). Answer: The four 4 S functions (M ¼ 0) are MS j1 ð4 SÞ ¼ j7 1 j2 ð4 SÞ ¼ pffiffiffi ðj8 þ j9 þ j10 Þ 3 1 j3 ð4 SÞ ¼ pffiffiffi ðj11 þ j12 þ j13 Þ 3 j4 ð4 SÞ ¼ j14

3 2 1 2 1  2 3  2

Hint: Use the techniques of the previous problems. 10.8. Construct the allowed L-states (L ¼ 0,1,2) out of the p2 configuration of two equivalent electrons by the projection method. Answer: 1 jð1 SÞ ¼ pffiffiffi ðj1  j2  j3 Þ 3 1 jð3 PÞ ¼ pffiffiffi ðj2  j3 Þ 2 1 jð1 DÞ ¼ pffiffiffi ð2j1 þ j2 þ j3 Þ 6 Hint: Act on j1 or j2 with the projector given by Eqn (62). 10.9. Construct the 1 P-state out of the 2p 3d configuration of two non-equivalent electrons by the projection method. Answer: rffiffiffiffiffi rffiffiffi 3 2 1 ðj þ j3 Þ  j jð PÞ ¼ 10 1 5 2 Hint: Act on j1 with the projector given by Eqn (62) which annihilates the components with L ¼ 2 and L ¼ 3.

10.6 Solved problems

437

10.6 SOLVED PROBLEMS 10.1. The Lande´ g-factor by the vector model. With reference to the left part of Figure 10.8, we first notice that mL,mS,mJ are in a direction opposite to L, S, J with magnitude pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mL ¼ be LðL þ 1Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mS ¼ 2be SðS þ 1Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mJ ¼ gJ be JðJ þ 1Þ where gJ is the Lande´ g-factor we want to calculate. From the Carnot’s theorem for the triangle pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi of sides LðL þ 1Þ; SðS þ 1Þ; JðJ þ 1Þ it follows pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi SðS þ 1Þ ¼ JðJ þ 1Þ þ LðL þ 1Þ  2 JðJ þ 1Þ LðL þ 1Þ cos ðL; JÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi LðL þ 1Þ ¼ JðJ þ 1Þ þ SðS þ 1Þ  2 JðJ þ 1Þ SðS þ 1Þ cos ðS; JÞ Then, the component of mJ in the J-direction is mJ ¼ mL cosðL; JÞ þ mS cosðS; JÞ gJ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi JðJ þ 1Þ ¼ LðL þ 1Þ cosðL; JÞ þ 2 SðS þ 1Þ cosðS; JÞ

where we divided both members by be. Then it follows pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi LðL þ 1Þ JðJ þ 1Þ þ LðL þ 1Þ  SðS þ 1Þ SðS þ 1Þ JðJ þ 1Þ þ SðS þ 1Þ  LðL þ 1Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi gJ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 JðJ þ 1Þ LðL þ 1Þ 2 JðJ þ 1Þ SðS þ 1Þ JðJ þ 1Þ JðJ þ 1Þ ¼

JðJ þ 1Þ þ LðL þ 1Þ  SðS þ 1Þ þ 2JðJ þ 1Þ þ 2SðS þ 1Þ  2LðL þ 1Þ 2JðJ þ 1Þ

¼1þ

JðJ þ 1Þ þ SðS þ 1Þ  LðL þ 1Þ 3 SðS þ 1Þ  LðL þ 1Þ ¼ þ 2JðJ þ 1Þ 2 2JðJ þ 1Þ

which is the Lande´ g-factor for the LS coupling. For a single s electron, L ¼ 0, S ¼ 1/2, J ¼ S, and therefore gJ ¼ 2. This spectroscopic value should not be confused with the intrinsic g-value of the electron (see Footnote 2 of Chapter 9). 10.2. We apply Eqn (35) in turn to j1 ; j2 ; j3 : 2 2 L^ j1 ¼ L^ k p0

¼ 4k p0

p0 k ¼ ½1ð1 þ 1Þ þ 1ð1 þ 1Þ þ 2$0$0k p0

p0 k þ 1ðkÞ2ðlÞ þ 2ðkÞ1ðlÞ

p0 k þ ½1ð1 þ 1Þ  0ð0 þ 1Þ1=2 ½1ð1 þ 1Þ  0ð0  1Þ1=2 k pþ1

þ ½1ð1 þ 1Þ  0ð0  1Þ ¼ 4j1 þ 2j2 þ 2j3

1=2

1=2

½1ð1 þ 1Þ  0ð0 þ 1Þ

k p1

pþ1 k

p1 k

438

CHAPTER 10 Angular momentum methods for atoms

2 2 L^ j2 ¼ L^ k pþ1

¼ 2k pþ1

p1 k ¼ ½1ð1 þ 1Þ þ 1ð1 þ 1Þ þ 2ðþ1Þð1Þk pþ1 p1 k þ ½1ð1 þ 1Þ  ð1Þð1 þ 1Þ

1=2

p1 k þ 1ðkÞ2ðlÞ þ 2ðkÞ1ðlÞ

½1ð1 þ 1Þ  1ð1  1Þ1=2 k p0

p0 k

¼ 2j2 þ 2j1 since 12 is zero because we cannot exceed the top or the bottom of the ladder with l ¼ 1. 2 2 L^ j3 ¼ L^ k p1

¼ 2k p1

pþ1 k ¼ ½1ð1 þ 1Þ þ 1ð1 þ 1Þ þ 2ð1Þðþ1Þk p1 pþ1 k þ ½1ð1 þ 1Þ  ð1Þð1 þ 1Þ

1=2

½1ð1 þ 1Þ  1ð1  1Þ1=2 k p0

¼ 2j3 þ 2j1 and we obtain Eqn (39) of the main text. 10.3. Eigenvectors of matrix Eqn (40) corresponding to l ¼ 0,2,6. The homogeneous system to be solved is 8 ð4  lÞc1 þ 2c2 þ 2c3 ¼ 0 > > < 2c1 þ ð2  lÞc2 ¼ 0 > > : 2c1 þ ð2  lÞc3 ¼ 0 with the normalization condition c21 þ c22 þ c23 ¼ 1 1. l ¼ 0 From Eqn 2 and Eqn 3 it follows immediately c2 ¼ c3 ¼ c1 1 c1 ¼ pffiffiffi ; 3

3c21 ¼ 1

1 c2 ¼ c3 ¼ pffiffiffi 3

2. l ¼ 2 c1 ¼ 0 c3 ¼ c2 2c22 ¼ 1 1 1 c2 ¼ pffiffiffi ; c3 ¼ pffiffiffi 2 2 3. l ¼ 6 c2 ¼ c3 ¼ 2 c1 ¼ pffiffiffi ; 6

1 c1 2

pþ1 k þ 1ðkÞ2ðlÞ þ 2ðkÞ1ðlÞ

2 c1 ¼ pffiffiffi 6 1 c2 ¼ c3 ¼ pffiffiffi 6

p0 k

10.6 Solved problems

439

10.4. Check Eqn (43). Using Eqn (35), it is obtained 1 2 L^ jð1 SÞ ¼ pffiffiffi ð4j1 þ 2j2 þ 2j3  2j2  2j1  2j3  2j1 Þ ¼ 0ð0 þ 1Þjð1 SÞ 3 1 2 L^ jð3 PÞ ¼ pffiffiffi ð2j2 þ 2j1  2j3  2j1 Þ ¼ 1ð1 þ 1Þjð3 PÞ 2 1 1 2 L^ jð1 DÞ ¼ pffiffiffi ð8j1 þ 4j2 þ 4j3 þ 2j2 þ 2j1 þ 2j3 þ 2j1 Þ ¼ pffiffiffi ð12j1 þ 6j2 þ 6j3 Þ 6 6 ¼ 2ð2 þ 1Þjð1 DÞ 2 so that functions (42) are the correct eigenfunctions of L^ belonging to the eigenvalues 2 2 L ¼ 0, 1, 2, respectively. Using Dirac’ S^ ¼ I^ þ P^12 , it is obtained for S^ :

1 2 S^ jð1 SÞ ¼ pffiffiffi ½k p0 3 þk p0

p0 k  k pþ1

p0 k  k pþ1

p1 k  k p1

p1 k  kp1

pþ1 k

pþ1 k  ¼ 0ð0 þ 1Þjð1 SÞ

1 2 S^ jð3 PÞ ¼ pffiffiffi ½k pþ1 p1 k  k p1 pþ1 k þ k pþ1 p1 k  k p1 2 1 ¼ 2 pffiffiffi ½k pþ1 p1 k  k p1 pþ1 k ¼ 1ð1 þ 1Þjð3 PÞ 2 1 2 S^ jð1 DÞ ¼ pffiffiffi ½2k p0 6 þ2k p0

p0 k þ k pþ1 p0 k þ k pþ1

p1 k þ k p1 p1 k þ k p1

pþ1 k

pþ1 k pþ1 k ¼ 0ð0 þ 1Þjð1 DÞ

as it must be. 10.5. Transform jð1 SÞ from complex to real form. In order to transform to real functions, we must take into account the Condon–Shortley phase for the complex spherical harmonics with m > 0: Ylm ¼ ð1Þm Pm l ðcos qÞ expðim4Þ;

Yl m ¼ P m l ðcos qÞ expðim4Þ

The phase factors are chosen so as to satisfy step-up and step-down equations and the phase of Yl0 is real and positive (Condon and Shortley, 1963). Then px þ ipy pþ1 ¼  pffiffiffi ; 2

p1 ¼

px  ipy pffiffiffi 2

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CHAPTER 10 Angular momentum methods for atoms

and, introducing into the complex determinants of Eqn (44), for the properties of determinants we obtain     px  ipy px  i py  px  ipy px  i py     pffiffiffi    pffiffiffi pffiffiffi  k pþ1 p1 k  k p1 pþ1 k ¼  pffiffiffi 2 2 2 2     1   px þ ipy px  i py  þ  px  ipy px þ i py  2      1  ¼  p x p x  i p y  þ i  p y p x  i p y  þ  px p x þ i p y  2    i py px þ i py        1 ¼ k p x p x k  i  p x p y  þ i  p y p x  þ  py p y  2       þ k px p x k þ i  p x p y   i  p y p x  þ  p y p y    ¼ k px px k þ  p y py  ¼

Hence, we obtain

i 1 h jð1 SÞ ¼ pffiffiffi k p0 p0 k  k pþ1 p1 k  k p1 pþ1 k complex 3   1  ¼ pffiffiffi k px px k þ  py py  þ k pz pz k real 3

10.6. According to the vector model three equivalent p electrons with l1 ¼ l2 ¼ l3 ¼ 1 can have L ¼ 0,1,2,3. The state with L ¼ 3 is, however, not allowed by the Pauli’s exclusion principle. The 20 are given as Slater determinants numbered from 1 to 20.  states  6 The ¼ 20 with L ¼ 0,1,2 Pauli’s allowed states are (Herzberg, 1944) 3 4

2

S

P

M[0

MS [ ±3/2, ±1/2

4 states

M¼

1 MS ¼

2 1

2 1

2

6 states

1 MS ¼

2 1

2 1

2 1

2 1

2

10 states

(1 0

(

1 2 1

2

D

M¼

0 1 2

10.6 Solved problems

441

The S quartet with MS ¼ 1/2 must be a linear combination of functions 8,9,10, the remaining two being the components of the doublets 2 P and 2 D. Proceeding as we did in the previous problems, we see that 2 L^ j8 ¼ 4j8  2j9  2j10 2 L^ j9 ¼ 4j9  2j8  2j9 ¼ 2j8 þ 2j9 2 L^ j10 ¼ 4j10  2j10  2j8 ¼ 2j8 þ 2j10 2 The matrix representative of L^ over these 0 4 L2 ¼ @ 2 2

basis functions will be 1 2 2 2 0 A 0 2

giving the secular equation

4  l



2

2

2 2l 0

2



0 ¼0

2  l

with the roots l ¼ 0; 2; 6 0 L ¼ 0; 1; 2 as it must be. The eigenvectors are 1 jð4 SÞ ¼ pffiffiffi ðj8 þ j9 þ j10 Þ 3 1 jð2 PÞ ¼ pffiffiffi ðj9  j10 Þ 2 1 jð2 DÞ ¼ pffiffiffi ð2j8  j9  j10 Þ 6 These results can be easily checked so that   3 3 2 þ 1 jð4 SÞ S^ jð4 SÞ ¼ 2 2   1 1 2 2 2 2 2 ^ ^ þ 1 jð2 PÞ L jð PÞ ¼ 1ð1 þ 1Þjð PÞ; S jð PÞ ¼ 2 2   1 1 2 2 L^ jð2 DÞ ¼ 2ð2 þ 1Þjð2 DÞ; S^ jð2 DÞ ¼ þ 1 jð2 DÞ 2 2

2 L^ jð4 SÞ ¼ 0ð0 þ 1Þjð4 SÞ;

442

CHAPTER 10 Angular momentum methods for atoms

10.7. The four 4 S functions out of the np3 configuration of three equivalent electrons. One of these functions, having L ¼ 0, MS ¼ 1/2, was already found in Problem 10.6. It is 1 j2 ð4 SÞ ¼ pffiffiffi ðj8 þ j9 þ j10 Þ 3 The third function, having L ¼ 0, Ms ¼ 1/2, is given immediately as 1 j3 ð4 SÞ ¼ pffiffiffi ðj11 þ j12 þ j13 Þ 3 after spin was systematically changed in the Slater determinants having MS¼1/2. For the remaining two, we start from the state of maximum spin multiplicity (for brevity: pþ1 ¼ pþ, p1 ¼ p): j7 ¼ k p þ

p0 k

p

2 having S ¼ MS ¼ 3/2. Acting with L^ we find

2 2 L^ j7 ¼ L^ k pþ p p0 k ¼ 4k pþ p p0 k þ 2k pþ p0 p k þ 2k p0 p ¼ 4k pþ p p0 k  2k pþ p p0 k  2k pþ p p0 k ¼ 0ð0 þ 1Þj7

pþ k

so that j1 ð4 SÞ ¼ j7 is the first function wanted. The like is true for j14 ; which differs from j7 for having all spins 1/2. Hence j4 ð4 SÞ ¼ j14 is the last function of the 4 S-state. We can note the one-to-one correspondence between the four 4 S functions written as Slater determinants of spin orbitals and the pure spin states arising from the N ¼ 3 problem in Chapter 9:

MS aaa 1 pffiffiffi ðaab þ aba þ baaÞ 3 1 pffiffiffi ðbba þ bab þ abbÞ 3 bbb

9 3=2 > > > > > > > > 1=2 > =

> > > 1=2 > > > > > > ; 3=2

S

3/2

10.6 Solved problems

443

10.8. Construction of the allowed L-states (L ¼ 0,1,2) out of the p2 configuration of two equivalent electrons by the projection method. We saw in Section 10.3.1 that the Pauli allowed states for the p2 configuration of two equivalent electrons are 1 S;

3 P;

1D

L ¼ 0;

1;

2

We saw that there are three states with ML ¼ MS ¼ 0: j 1 ¼ k p0

p0 k;

j 2 ¼ k pþ

p k;

j3 ¼ k p 

pþ k

In Problem 10.2 we found that 2 L^ j1 ¼ 4j1 þ 2j2 þ 2j3 2 L^ j2 ¼ 2j1 þ 2j2 2 L^ j3 ¼ 2j1 þ 2j3

Wanting the S-state (L ¼ 0), we must annihilate the components having L ¼ 1 and L ¼ 2. The corresponding projector will be O^0 ¼

Y L^2  LðL þ 1Þ L^2  1ð1 þ 1Þ L^2  2ð2 þ 1Þ 8 ^2 1 ^2 ^2 ¼ $ ¼1 L þ L $L LðL þ 1Þ 1ð1 þ 1Þ 2ð2 þ 1Þ 12 12 Lðs0Þ

Using the relations above it is easily found that  2   2   2   2  2 L^ L^ j1 ¼ 4 L^ j1 þ 2 L^ j2 þ 2 L^ j3 ¼ 4ð4j1 þ 2j2 þ 2j3 Þ þ 2ð2j1 þ 2j2 Þ þ 2ð2j1 þ 2j3 Þ ¼ 24j1 þ 12j2 þ 12j3 Then it follows 8  ^2  1 ^2  ^2  L j1 þ L L j1 O^0 j1 ¼ j1  12 12 8 1 ð4j1 þ 2j2 þ 2j3 Þ þ ð24j1 þ 12j2 þ 12j3 Þ 12 12 1 1 ¼ ðj1  j2  j3 Þfpffiffiffi ðj1  j2  j3 Þ S-state 3 3

¼ j1 

which coincides with the result (1) of Problem 10.3 after normalization.

444

CHAPTER 10 Angular momentum methods for atoms

We can proceed similarly for the remaining components. Wanting the P-state (L ¼ 1) Y L^2  LðL þ 1Þ L^2 L^2  6 6 2 1 2 2 ¼ $ ¼ L^  L^ $ L^ O^1 ¼ 2 26 2  LðL þ 1Þ 8 8 Lðs1Þ 6  ^2  1 ^2  ^2  6 1 O^1 j1 ¼ L j1  L L j1 ¼ ð4j1 þ 2j2 þ 2j3 Þ  ð24j1 þ 12j2 þ 12j3 Þ ¼ 0 8 8 8 8  2   2   2  2 L^ L^ j2 ¼ 2 L^ j1 þ 2 L^ j2 ¼ 2ð4j1 þ 2j2 þ 2j3 Þ þ 2ð2j1 þ 2j2 Þ ¼ 12j1 þ 8j2 þ 4j3 6  ^2  1 ^2  ^2  6 1 O^1 j2 ¼ L j2  L L j2 ¼ ð2j1 þ 2j2 Þ  ð12j1 þ 8j2 þ 4j3 Þ 8 8 8 8 1 1 ¼ ðj2  j3 Þfpffiffiffi ðj2  j3 Þ P-state 2 2 after normalization. Wanting the D-state (L ¼ 2) Y L^2  LðL þ 1Þ L^2 L^2  2 1 ^2 ^2 2 ^2 ¼ $ ¼ L $L  L O^2 ¼ 6 6  LðL þ 1Þ 4 24 24 Lðs2Þ Then

1 ^2  ^2  2  ^2  1 2 L L j1  L j1 ¼ ð24j1 þ 12j2 þ 12j3 Þ  ð4j1 þ 2j2 þ 2j3 Þ O^2 j1 ¼ 24 24 24 24 1 1 ¼ ð2j1 þ j2 þ j3 Þfpffiffiffi ð2j1 þ j2 þ j3 Þ D-state 3 6 after normalization. 2 Once the effect of L^ on the basis functions is known, use of the projector (62) is the simplest way of obtaining the correct combination having the desired value of L. 10.9. Construct the 1 P-state out of the 2p 3d configuration of two non-equivalent electrons by the projection method. We saw in Section 10.3.1 (2) what the possible allowed states arising from the 2p 3d 2 configuration are. The basis functions are given by (47), while the effect of L^ on these functions is given by (50), which we report below for convenience: pffiffiffiffiffi 2 L^ j1 ¼ 6j1 þ 12j2 pffiffiffiffiffi 2 L^ j2 ¼ 8j2 þ 12ðj1 þ j3 Þ pffiffiffiffiffi 2 L^ j3 ¼ 6j3 þ 12j2 To build the 1 P-state out of the 2p 3d configuration we must annihilate the D and F components. The projector for k ¼ 1 will be  Y L^2  LðL þ 1Þ L^2  6 L^2  12 1  2 2 2 ^ ¼ $ ¼ O1 ¼ 72  18L^ þ L^ $ L^ 2  LðL þ 1Þ 2  6 2  12 40 Lðs1Þ

10.6 Solved problems

445

Now   pffiffiffiffiffi  pffiffiffiffiffi 2 2 2 2 2 L^ L^ j1 ¼ L^ 6j1 þ 12j2 ¼ 6L^ j1 þ 12L^ j2  pffiffiffiffiffi  pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi  pffiffiffiffiffi ¼ 6 6j1 þ 12j2 þ 12 8j2 þ 12j1 þ 12j3 ¼ 48j1 þ 14 12j2 þ 12j3 

Acting with O^1 on the first function, it is hence obtained 72 18  ^2  1 ^2  ^2  j1  O^1 j1 ¼ L j1 þ L L j1 40 40 40  pffiffiffiffiffi  1  pffiffiffiffiffi 72 18  ¼ j1  6j1 þ 12j2 þ 48j1 þ 14 12j2 þ 12j3 40 40 40       p ffiffiffiffiffi 72 108 48 18 14 12  þ þ 12j2  þ þ j3 ¼ j1 40 40 40 40 40 40 rffiffiffiffiffi rffiffiffi pffiffiffi 3 3 2 3 ðj1 þ j3 Þ  j P-state j2 f ¼ ðj1 þ j3 Þ  10 10 5 2 5 after normalization. This result coincides with that of Eqn (57) found previously.

CHAPTER

The physical principles of quantum mechanics

11

CHAPTER OUTLINE 11.1 The Orbital Model....................................................................................................................... 449 11.2 The Fundamental Postulates of Quantum Mechanics ..................................................................... 450 11.2.1 Correspondence between observables and operators .................................................450 11.2.2 State function and average values of observables .....................................................454 11.2.3 Time evolution of state function..............................................................................455 11.3 The Physical Principles of Quantum Mechanics ........................................................................... 456 11.3.1 Wave-particle dualism............................................................................................456 11.3.2 Atomicity of matter ................................................................................................458 11.3.3 Schroedinger’s wave equation.................................................................................459 11.3.4 Born interpretation ................................................................................................460 11.3.5 Measure of observables ..........................................................................................461 11.4 Problems 11 .............................................................................................................................. 463 11.5 Solved Problems ........................................................................................................................ 464

11.1 THE ORBITAL MODEL The forces keeping together electrons and nuclei in atoms and molecules are essentially electrostatic in nature, and, at the microscopic level, satisfy the principles of quantum mechanics. Experimental evidence (Karplus and Porter, 1970) brings us to formulate a planetary model of the atom (Rutherford1) made by a point-like nucleus (with a diameter of 0.01–0.001 pm) carrying the whole mass and the whole positive charge þZe surrounded by electrons, each having a negative elementary charge e and a mass about 2000 times smaller than that of proton, carrying the whole negative charge Ne of the atom (N ¼ Z for neutral atoms) distributed as a charge cloud in an atomic volume with a diameter of about 100 pm. The distribution of the electrons is apparent from the density contours obtained from X-ray diffraction spectra of polycyclic hydrocarbons (Bacon, 1969). The electron density in atoms can be described in terms of atomic orbitals, which are one-electron functions jðrÞ depending on a single centre (the nucleus of the atom), while electron density in molecules can be described in terms of molecular orbitals, many-centre one-electron functions depending on the 1

Rutherford Ernest (Lord) 1871–1937, English physicist, Professor at the Universities of Montreal, Manchester, Cambridge and London. Nobel Prize for Chemistry in 1908.

Elementary Molecular Quantum Mechanics 2E. http://dx.doi.org/10.1016/B978-0-444-62647-9.00011-7 Ó 2013, 2007 Elsevier B.V. All rights reserved

449

450

CHAPTER 11 The physical principles of quantum mechanics

different nuclei of the molecule. This is the basis for the so-called orbital model, with which the majority of applications is concerned. The physical meaning of jðrÞ is such that jjðrÞj2 dr ¼ probability of finding in dr the electron in state jðrÞ provided we satisfy the normalization condition: Z drjjðrÞj2 ¼ 1

(1)

(2)

where integration is extended over all space, and j* is the complex conjugate to j : jjj2 ¼ j j. This implies some physical restrictions on the form of the mathematical functions jðrÞ, which must be regular (singly valued, continuous with their first derivatives, quadratically integrable), and are obtained as permissible solutions of a variety of eigenvalue equations which can be traced back to some space form of Schroedinger-type differential equations. For a single electron the equation can be written as ^ ¼ Ej Hj

(3)

where H^ is the total energy operator (Hamilton or Hamiltonian operator), sum of the kinetic energy and potential energy operators of the electron in the atom, and E is the value assumed by the electron energy in state j: For a deeper understanding of the physical foundations of the orbital model, we introduce once more in the simplest way the basic principles of quantum mechanics (Margenau, 1961), which are needed for a correct physical description of the subatomic world, ending the chapter with the physics underlying these principles and the usual problems.

11.2 THE FUNDAMENTAL POSTULATES OF QUANTUM MECHANICS The simplest formulation of quantum mechanical principles is in the form of three postulates.

11.2.1 Correspondence between observables and operators There is a correspondence between dynamical variables in classical physics (observables) and linear Hermitian operators in quantum mechanics. In coordinate space, the basic correspondences are x 0 x^ ¼ x$ px 0 p^x ¼

h v v ¼  iZ 2pi vx vx

(4)

where x is the position coordinate, px the x-component of the linear momentum, p^p x the ffiffiffiffiffiffiffi corresponding quantum mechanical operator, Z ¼ h=2p the reduced Planck’s constant, and i ¼ 1 the imaginary unit ði2 ¼ 1Þ. The like holds for the remaining y and z components. As usual, the caret symbol will be henceforth used to denote operators.

11.2 The fundamental postulates of quantum mechanics

451

In three dimensions p ¼ ipx þ jpy þ kpz ^ ¼ iZV p V¼i

a vector

(5)

a vector operator ða vector whose components are operatorsÞ

v v v the gradient operator þj þk vx vy vz in Cartesian coordinates

(6) (7)

Examples of further observables. p2 Z2 v2 x  component of the kinetic energy T ¼ x 0 T^ ¼  of a particle of mass m 2m 2m vx2 In three dimensions T¼

 1  2 Z2 px þ p2y þ p2z 0 T^ ¼  r2 2m 2m

where r2 ¼ V$V ¼

(8)

v2 v2 v2 þ þ vx2 vy2 vz2

(9)

is the Laplacian operator, and the dot stands for the scalar product. Total energy 0

Z2 2 Total energy operator ^ r þV H¼ ðHamiltonian operatorÞ 2m

(10)

where V is the potential energy characterizing a given physical system. V is usually a multiplicative operator function of the coordinates. It seems appropriate to give once more here a few examples. 1. One-dimensional free particle Z 2 d2 p^2 V ¼ 0 0 H^ ¼ T^ ¼ x ¼  2m 2m dx2

(11)

kinetic energy only. 2. One-dimensional harmonic oscillator V¼

kx2 Z 2 d2 kx2 0 H^ ¼  þ 2 2m dx2 2

(12)

where k is the force constant. 3. Angular momentum of a particle of linear momentum p and vector position r    i j k     L¼rp¼ x y z    p p p  x

y

z

¼ iðypz  zpy Þ þ jðzpx  xpz Þ þ kðxpy  ypx Þ Lx

Ly

Lz

(13)

452

CHAPTER 11 The physical principles of quantum mechanics

   i j k    x y z   ^ ¼ r  ðiZVÞ ¼ iZ L  ¼ iL^x þ jL^y þ kL^z v v v    vx vy vz        v v v v v v ^ ^ ^ ; Ly ¼ iZ z  x ; Lz ¼ iZ x  y Lx ¼ iZ y  z vz vy vx vz vy vx

(14)

(15)

where the cross stands for the vector product of two vectors, and the cyclic permutation ðx / y / zÞ of indices should be noted. We then have for the square of the angular momentum operator in Cartesian coordinates 2 ^ L ^ ¼ L^2 þ L^2 þ L^2 L^ ¼ L$ x y z

(16)

In spherical coordinates ðr; q; 4Þ L^ can be related to the Laplacian operator (Problem 1.5):    1 v v 1 v2 2 L^ =Z2 ¼ r 2 r2 þ r 2 r2r ¼  (17) sin q þ 2 sin q vq vq sin q v42 2

where r2r

  1 v v2 2 v 2v r ¼ 2þ ¼ 2 vr r vr vr r vr

(18)

is the radial Laplacian. The operator in square brackets on the right hand side of Eqn (17) depends only on angles and is called the Legendre2 operator (or Legendrian). 4. Particle of mass m and charge e in an electromagnetic field of scalar potential 4ðx; y; zÞ and vector potential Aðx; y; zÞ. The classical Hamiltonian is 1 e 2 1  e   e  p þ A  e4 ¼ p þ A $ p þ A  e4 2m c 2m c c   2 1 e e e p2 þ p$A þ A$p þ 2 A2  e4 ¼ c 2m c c

H¼

(19)

where c is the velocity of light. The corresponding quantum mechanical operator can be derived from this classical expression taking into consideration that the resulting operator must be Hermitian:   1 Ze Ze e2 2 2 2 ^ H¼  Z r  i V$A  i A$V þ 2 A  e4 c 2m c c (20) Z2 2 Ze Ze e2 2 ¼ r i ðV$AÞ  i A$V þ A 2m 2mc2 2mc mc Legendre Adrien Marie 1752–1833, French mathematician, Professor at the E´cole Militaire and the E´cole Polytechnique of Paris.

2

11.2 The fundamental postulates of quantum mechanics

453

where ðV$AÞ (the divergence of A) does no longer operate on the function J, since ðV$AÞJ ¼ JðV$AÞ þ A$VJ

(21)

5. The one-electron atomic system (the hydrogen-like system) It consists of a single electron moving in the field of þZe nuclear charges (Z ¼ 1 is the hydrogen atom). In the SI system Ze2 (22) V ¼ VðrÞ ¼  4pε0 r the Coulomb3 law of attraction of the electron by the nucleus. The Hamiltonian will be Z2 Ze2 H^ ¼  r2  2m 4pε0 r

(23)

Because of the spherical symmetry of the atom, it is convenient to use the spherical coordinates ðr; q; 4Þ of Chapter 2, where r2 becomes 2 L^ =Z2 r2      1 v 1 1 v v 1 v2 2v r þ 2 sin q þ 2 ¼ 2 r vr vr r sin q vq vq sin q v42

r2 ¼ r2r 

(24)

The advantage of using spherical coordinates lies in the fact that it allows to separate radial from angular coordinates, as we already saw in Chapter 3 of this book. We notice that all of the previous formulae involve universal physical constants ðe; Z; m; 4pε0 Þ; whose values are well known. To simplify notation we introduce the system of atomic units of Chapter 1, defined by posing e ¼ Z ¼ m ¼ 4pε0 ¼ 1

atomic units

(25)

The relation between atomic units and physical constants was given in Section 1.1.6. The hydrogenic Hamiltonian in atomic units will then simplify to 1 Z H^ ¼  r2  2 r

(26)

Comparison with the previous formula makes immediately clear the great advantage of using atomic units, as also shown by the next two examples. 6. The two-electron atomic system In atomic units, the Hamiltonian will be 1 1 Z Z 1 1 ¼ h^1 þ h^2 þ H^ ¼  r21  r22   þ 2 2 r1 r2 r12 r12 3

Coulomb Charles Augustin 1736–1806, French physicist, Member of the Acade´mie des Sciences.

(27)

454

CHAPTER 11 The physical principles of quantum mechanics

1 Z where h^ ¼  r2  is the hydrogen-like Hamiltonian (a one-electron operator), and 1=r12 is the 2 r two-electron operator describing electron repulsion. For Z ¼ 2 we have the He atom. 7. The hydrogen molecule H2 R is the internuclear distance, r12 the interelectronic distance, the remaining ones the electron–nucleus distances. The full Hamiltonian in the Born–Oppenheimer approximation (Chapter 20) looks rather complicated: Z2 2 Z2 2 rA  r H^ ¼  2MA 2MB B (28) Z2 2 Z2 2 e2 e2 e2 e2 e2 e2 r  r      þ þ 2m 1 2m 2 4pε0 r1 4pε0 r2 4pε0 rB1 4pε0 rA2 4pε0 r12 4pε0 R but in atomic units this expression simplifies to 1 2 1 2 1 2 1 2 r  r  r  r H^ ¼  2M A 2M B 2 1 2 2 1 1 1 1 1 1 1 2 1 2 ^     þ þ ¼ rA  r þ hA1 þ h^B2 þ V r1 r2 rB1 rA2 r12 R 2M 2M A where

M ¼ Mx =m

(29)

is the proton mass in units of the electron mass

1 1 is the Hamiltonian of each separate atom h^x ¼  72  2 r 1 1 1 1 V¼  þ þ is the interatomic potential rB1 rA2 r12 R We now turn to the second postulate of quantum mechanics.

11.2.2 State function and average values of observables There is a state function (or wavefunction) J which describes in a probabilistic way the dynamical state of a microscopic system. In coordinate space, J (generally, a complex function), is a function of coordinate x and time t such that probability at time t of finding in dx Jðx; tÞJ ðx; tÞdx ¼ (30) the system in state J provided J is normalized:

Z

dxJ ðx; tÞJðx; tÞ ¼ 1

the integration being extended over all space. As already said for the orbitals, J must be restricted by • Single-valuedness • Continuity conditions with its first derivatives • Square integrability, i.e. Jðx; tÞ must vanish at infinity.

(31)

11.2 The fundamental postulates of quantum mechanics

455

These are the conditions that must always be satisfied because a mathematical function J could describe a physical probability. We have called such a function a Q-class function (function belonging to the L2 Hilbert4 space, normalizable or regular function). As a consequence of the probabilistic meaning of J, the average (expectation) value of the physical ^ characterizing the system in state J, is given by quantity A (with quantum mechanical operator A), R ^ R dxJ ðx; tÞAJðx; tÞ Jðx; tÞJ ðx; tÞ

  ¼ dxA^ R ¼ R  (32) dxJ ðx; tÞJðx; tÞ dxJ x; t J x; t probability density function where the integration is extended over all space. Notice that A^ acts always upon J and not on J , and that it must be Hermitian. The state function is hence needed to evaluate the average values of physical observables, which are the only quantities that can be measured by experiment.

11.2.3 Time evolution of state function J is obtained by solving the time-dependent Schroedinger equation: _ ^ ¼ iZJ HJ

(33)

_ ¼ vJ=vt. This is a partial differential equation in the position coordinate x and the time t; where J which are treated on a different footing. If H^ does not depend in an explicit way on t (stationary state), the variables in Eqn (33) can be separated according to the methods explained in Chapter 3 by posing Jðx; tÞ ¼ jðxÞgðtÞ

(34)

^ Hj Z d ln g ¼ ¼E j i dt

(35)

and we obtain

In this way, we obtain two separate differential equations, one in x; the other in t (E is the separation constant): ^ HjðxÞ ¼ EjðxÞ

(36)

E d ln gðtÞ ¼ i dt Z

(37)

The latter equation is immediately integrated to   E gðtÞ ¼ g0 exp i t ¼ g0 expðiutÞ h

E ¼u Z

(38)

giving the time evolution of the stationary state (always the same). From a mathematical standpoint, the Schroedinger Eqn (36) determining jðxÞ has the form of an eigenvalue equation, namely, that particular differential equation where the operator acting upon 4

Hilbert David 1862–1943, German mathematician, Professor at the University of Go¨ttingen.

456

CHAPTER 11 The physical principles of quantum mechanics

the function gives the function itself (the eigenfunction of the operator) multiplied by a constant (the eigenvalue of the operator). When several eigenfunctions belong to the same eigenvalue, we speak of degeneracy of the eigenvalue (for instance, the first excited level of the H atom is four times degenerate, i.e. there are four different states belonging to this eigenvalue, i.e. 2s,2px ; 2py ;2pz ). We notice that, entering in both Schroedinger equations, the energy observable (hence, the Hamiltonian operator) plays a peculiar role among all physical observables of quantum mechanics. We now look to the physical plausibility of our postulates by investigating the nature of measurements at the subatomic level.

11.3 THE PHYSICAL PRINCIPLES OF QUANTUM MECHANICS 11.3.1 Wave-particle dualism Experimental observation shows that electromagnetic waves present both a wave-like and a particlelike character. For light and X-rays this is shown by interference and diffraction phenomena on the one hand, and by the photo-electric and Compton5 effects on the other. A typical matter particle such as the electron can give diffraction figures characteristic of a wave-like nature. We shall consider in some detail the Compton effect (1923), where a monochromatic X-ray beam impinging on a substance is scattered in all directions with a wavelength that increases with increasing scattering angle. The photon of frequency n impinging on the particle along the x-direction is scattered in the direction specified by the angle q with a frequency v0. The conservation of linear momentum gives hn hn0 cos q þ px ¼ c c

0¼

hn0 sin q  py c

(39)

from which, squaring and adding the two components, assuming n0 xn, p2 ¼

 h2 n 2

h2 n 2 h2 2 2 q þ sin q ¼ z 2 ð1  cos qÞ ¼ 2 ð1  cos qÞ 1  2 cos q þ cos c2 c2 l2

(40)

where p2 ¼ p2x þ p2y is the squared momentum of the particle and we have used n ¼ c=l. On the other hand, conservation of the total energy gives hn ¼ hn0 þ

p2 2m

hn0 < hn

(41)

where the particle, even at rest, acquires a kinetic energy given by the second term in the equation. Hence we get for p2:   1 1 l0  l l0  l p ¼ 2hmðn  n Þ ¼ 2hmc  0 ¼ 2hmc 0 z 2hmc l l ll l2 2

5

0

Compton Arthur Holly 1892–1962, U.S. physicist. Nobel Prize for Physics in 1927.

(42)

11.3 The physical principles of quantum mechanics

457

Table 11.1 Photon Mass in Different Spectral Regions Region

k (mL1)

m (kg)

MW

102

2:2  1040

IR

105

2:2  1037

VIS

10

6

2:2  1036

UV

107

2:2  1035

X

1010

2:2  1032

g

1011

2:2  1031

Equating to the previous expression for the momentum, we get immediately that the wavelength shift is a function of the scattering angle q given by Dl ¼ l0  l ¼

h ð1  cos qÞ mc

(43)

The Compton effect shows hence that, as a result of photon–particle interaction, the wavelength of the scattered photon increases with q. For the electron, the constant will be h 6:626  1034 J s ¼ 2:426  1012 m ¼ 0:0243  A ¼ 31 kg  2:998  108 m s1 mc 9:109  10 The Compton effect shows that the photon is a particle of light, with a mass given by the Einstein6 relation: m¼

E hn h h ¼ 2¼ ¼ k 2 c c lc c

(44)

where k is the wave number, l1 . So the photon mass depends on the spectral region as shown in Table 11.1. Recalling that h k ¼ 2:210  1042 k c

(45)

E 1:602  1019 J ¼ ¼ 8:065  105 m1 hc 6:626  1034 J s  2:998  108 m s1

(46)

m¼ for 1 eV we have k¼ and, for 1 MeV

1 MeV ¼ 106 eV ¼ 8:065  1011 m1 6

(47)

Einstein Albert 1879–1955, German mathematical physicist, Professor at the Universities of Zu¨rich, Berlin and Princeton. Nobel Prize for Physics in 1921.

458

CHAPTER 11 The physical principles of quantum mechanics

Table 11.2 Properties of g-rays from Different Atomic Sources Isotope

l (m)

k (mL1)

E (MeV)

m (10L31 kg)

298

2:50  1011 2:60  1012 1:03  1012 9:70  1013 1:90  1013

4:00  1010 3:85  1011 9:71  1011 1:03  1012 5:26  1012

5:08  102 4:77  101 1:20  100 1:28  100 6:52  100

0.88 8.51 21.4 22.7 116

U Be 60 Co 22 Na 14 N 7

We give in Table 11.2 some properties of g-rays obtained from different atomic sources. We see from the table that the isotope 7 Be emits g-rays having a mass comparable with that of the electron, m ¼ 9:109  1031 kg: Just after the discovery of the Compton effect, in his famous thesis (1924, 1925) de Broglie7 suggested the wave-like character of material particles, assuming that the relation p¼

hn h ¼ ¼ hk c l

(48)

true for photons (particles of light), should equally be valid for particles (e.g. electrons, particles of matter). In this way, a property of particles, the linear momentum, is proportional through Planck’s constant to a property of waves, the wave number k. Any moving microscopic body has associated a ‘wave’, whose wavelength is related to the momentum by the relation above. This hypothesis was later verified by experiments studying the diffraction pattern of electrons reflected by a nickel surface (Davisson and Germer, 1927) and by the formation of diffraction rings from cathode rays diffused through thin films of aluminium, gold and celluloid (Thomson, 1928).

11.3.2 Atomicity of matter A characteristic of atomic and molecular physics is the atomicity of matter (electron, e; proton, þe), energy (hn, Planck), linear momentum ðh=lÞ, and angular momentum (Z; Bohr8). This implies the peculiar character that any experimental measurement has in atomic physics (i.e. on a microscopic scale), particularly its limits that become apparent in the Heisenberg’s9 principle (1927) as a direct consequence of the interaction between the experimental apparatus and the object of measurement, which has a direct ineliminable effect on the physical property that must be measured at the microscopic level. • Heisenberg’s uncertainty principle 7

de Broglie Louis Victor 1892–1987, French physicist, Professor at the University of Paris, Member of the Acade´mie des Sciences. Nobel Prize for Physics in 1929. 8 Bohr Niels Heinrik David 1885–1962, Danish physicist, Professor at the University of Copenhagen. Nobel Prize for Physics in 1922. 9 Heisenberg Werner 1901–1976, German physicist, Professor at the Universities of Leipzig and Go¨ttingen. Nobel Prize for Physics in 1932.

11.3 The physical principles of quantum mechanics

459

Quantities that are canonically conjugate (in the sense of analytical mechanics) are related by the uncertainty relations: DxDpx wh

DEDtwh

(49)

where h is the Planck’s constant, Dx the uncertainty in the x-coordinate of the position of the particle, and Dpx the simultaneous uncertainty in the x-component of the linear momentum of the particle. In other words, the product of the uncertainties of two conjugate dynamical variables (like x and px) is of the order of the Planck’s constant, namely, the attempt to attain the exact measure of the coordinate position along x ðDx ¼ 0Þ implies the infinite uncertainty in the measure of the corresponding conjugate component of the linear momentum ðDpx ¼ NÞ: In quantum mechanical terms we may say that x p^x  p^x x ¼ ½x; p^x s0

(50)

namely, that the commutator of the corresponding operators is different from zero (we say that the two operators do not commute). The Heisenberg’s principle does not hold for non-conjugate components: i h (51) x; p^y ¼ 0 namely, quantities whose commutator vanishes can be measured at the same time with arbitrary accuracy. For a more general definition see Margenau (1961, pp. 46–47). The uncertainty principle must be considered as a law of nature, and stems directly from the interaction between experiment and object at the microscopic level, as the example in Problem 11.3 shows. As a consequence of the uncertainty principle, the only possible description of the dynamical state of a microscopic system is a probabilistic one, as contrasted with the deterministic description of classical mechanics. The problem is now to find the function which describes such a probability.

11.3.3 Schroedinger’s wave equation Schroedinger (1926) writes for a progressive wave the complex form: J ¼ A expðiaÞ ¼ A exp½2piðkx  ntÞ

(52)

where A is the amplitude and a the phase of a monochromatic plane wave of wave number k and frequency n which propagates along x. Taking into account the relations of de Broglie and Planck p k¼ ; h

E h

(53)

1 a ¼ ðpx  EtÞ Z

(54)

n¼

the phase of a matter wave can be written as

so that the wave equation for a matter particle will be  i J ¼ A exp ðpx  EtÞ ¼ Jðx; tÞ Z

(55)

460

CHAPTER 11 The physical principles of quantum mechanics

an equation which mathematically defines J as a function of x and t; with constant values of p and E: Taking the derivatives of J with respect to x and t; we have the correspondences vJ i ¼ pJ vx Z

hence

vJ i ¼  EJ vt Z

hence

p classical variable

0

E total energy

0

v iZ ¼ p^ vx quantum mechanical operator

(56)

v iZ ¼ H^ vt quantum mechanical operator ðHamiltonianÞ

(57)

These two relations give the basic correspondences we have seen before between physical variables (observables) in classical mechanics and linear Hermitian operators in quantum mechanics. The last equation is just the time-dependent Schroedinger equation giving the time evolution of J; and can be written in the usual form: ^ ¼ iZ vJ HJ vt

(58)

11.3.4 Born interpretation Born10 (1926) suggested that the intensity of de Broglie’s wave f jJj2 should be regarded as a probability density (probability per unit volume). In other words, Born’s interpretation of de Broglie and Schroedinger waves is such that Jðx; tÞJ ðx; tÞ dx ¼ jJðx; tÞj2 dx ¼

probability of finding the particle in the infinitesimal volume element dx at the point x at time t

For this interpretation to be correct, it must be Z dx J ðx; tÞJðx; tÞ ¼ 1

(59)

(60)

where integration is over all space. For a correct definition of probability, J must be normalized to one. For a stationary state the probability does not depend on time. In fact jJðx; tÞj2 dxfjjðxÞj2 jg0 j2 dx

(61)

is constant on time. We can conclude by saying that the wave-like character of particles is due to the fact that the probability function j does satisfy a wave equation. This is the explanation of the wave–particle duality which was at first so difficult to understand. 10

Born Max 1882–1970, German physicist, Professor at the Universities of Berlin, Frankfurt, Go¨ttingen and Edinburgh. Nobel Prize for Physics in 1954.

11.3 The physical principles of quantum mechanics

461

11.3.5 Measure of observables We recall briefly here a few mathematical definitions on regular functions, which were introduced in detail in Section 1.1.1. 1. The Dirac notation for the scalar product of two functions 4; j Z

dx4 ðxÞjðxÞ ¼ < 4jj >

(62)

2. The concept of orthonormal set f4k ðxÞg; implying < 4k j4k0 >¼ dkk0 ; where dkk0 is the Kronecker delta (¼1 for k0 ¼ k; ¼0 for k0 sk) 3. The expansion theorem for any function FðxÞ FðxÞ ¼

X

4k ðxÞCk ¼

k

X

j4k >< 4k jF >

(63)

k

where f4k ðxÞg is any suitable set of orthonormal basis functions. Now, let fAk g and f4k ðxÞg, with < 4k j4k0 >¼ dkk0 , be the set of eigenvalues and eigenfunctions of the Hermitian operator A^ corresponding to the physical observable A: The average value of A in state J is given by Z ^ ^ > tÞ ¼< JjAJ (64) ¼ dxJ ðx; tÞAJðx; provided the state function J is normalized: < JjJ >¼ 1

(65)

• If J is an eigenstate of A^ with eigenvalue Ak ; then < A> ¼ Ak

(66)

so that, by doing a measure of A at the time t, we shall certainly obtain for the observable A the value Ak . ^ we can expand J into the complete set of the eigenstates of A; ^ • If J is not an eigenstate of A; obtaining ¼

XX k

k0

^ k0 > ¼ Ck ðtÞCk0 ðtÞ < 4k jA4

X

jCk ðtÞj2 Ak

(67)

k

so that, by doing a measure of A at the time t, we shall have the probability Pk ðtÞ ¼ jCk ðtÞj2 of observing for A the value Ak. Probability distribution Pk ðtÞ ¼ jCk ðtÞj2 ¼ Ck ðtÞCk ðtÞ 



 Ck t ¼ < 4k jJ >; Ck t ¼

(68) (69)

462

CHAPTER 11 The physical principles of quantum mechanics

FIGURE 11.1 Definite value for the k-th eigenvalue

1. If at time t J is an eigenstate of A^ Jh4k Ck ðtÞ ¼ 1 Ck0 ðtÞ ¼ 0

for any k0 sk

(70)

we have a 100% probability of observing for A the value Ak (Figure 11.1). 2. Otherwise, we can expand J into stationary states: X X ak00 expðiuk00 tÞjk00 ðxÞ; J ¼ ak0 expðþiuk0 tÞjk0 ðxÞ; J¼ k0 0

Pk ðtÞ ¼

XX k0

k00

uk ¼ Ek =Z

(71)

k0

ak0 ak00 exp½iðuk00  uk0 Þt

where the last two integrals are space integrals. Then we have: • If A is an observable different from E, the probability Pk ðtÞ fluctuates in time (Figure 11.2).

FIGURE 11.2 Fluctuation in time of an eigenvalue distribution

(72)

11.4 Problems 11

463

FIGURE 11.3 Energy eigenvalue distribution

• If A ¼ E (the total energy) ^ 4k ðxÞ ¼ jk ðxÞ; ¼ dkk0 A^ ¼ H; Pk ðtÞ ¼

XX k0

k00

ak0 ak00 exp½iðuk00  uk0 Þdkk0 dkk00 ¼ jak j2

(73) (74)

since k0 ¼ k00 ¼ k is the only surviving term. In this case, we obtain for the energy a distribution constant in time (Figure 11.3): X (75) < H^ > ¼ E ¼ jak j2 Ek k

where Ek is the k-th energy eigenvalue (the energy level).

11.4 PROBLEMS 11 11.1 Show that for a macroscopic body (such as a cannon ball) to an accurate measurement of the position corresponds an unmeasurably small uncertainty in the corresponding value of the conjugate momentum. 11.2 Show that for a microscopic body (such as an electron in an electric field) to a rough measurement of the position corresponds an uncertainty in the value of the corresponding conjugate momentum which exceeds its calculated magnitude. 11.3 Explain the physics underlying the Heisenberg’s quantum microscope. Answer: The calculated uncertainties of position and conjugated momentum of the electron observed by the microscope are found to be Dx ¼

l h ; Dpx ¼ 2 sin ε 0 Dx Dpx wh 2 sin ε l

where h is the Planck’s constant, l the wavelength of the photon impinging the electron and 2ε the angular aperture of the lens of the objective of the microscope.

464

CHAPTER 11 The physical principles of quantum mechanics

11.5 SOLVED PROBLEMS 11.1 Macroscopic body (the cannon ball). Let us try to make an extremely accurate measurement of the position of this macroscopic body, say 1 mm, Dx ¼ 106 m. This corresponds to an unmeasurably small uncertainty in the corresponding value of the conjugate momentum: Dpx ¼

6:626  1034 J s ¼ 6:626  1028 kg m s1 106 m

11.2 Microscopic body (the electron in an electric field). Consider an electron accelerated through a potential difference of 50 V. Since 1 eV is the energy acquired by an electron in a potential difference of 1 V (Coulson, 1958) 1 eV ¼ 1:602177  1019 J the kinetic energy acquired by the electron in the field will be T¼

p2 ¼ 50 eV ¼ 8:011  1018 J 2m

with the momentum pffiffiffiffiffiffiffiffiffi

1=2 p ¼ 2mT ¼ 2  9:109  1031 kg  8:011  1018 kg m2 s2 ¼ 3:821  1024 kg m s1 For a specification of the electron within an atomic dimension ð1  A ¼ 1010 mÞ, not a very accurate requirement Dx ¼ 1010 m;

Dpx ¼

6:626  1034 J s ¼ 6:626  1024 kg m s1 1010 m

so that the uncertainty in the momentum exceeds its calculated magnitude. 11.3 The Heisenberg g-ray microscope. This is a typical ‘Gedanken experiment’, an experiment ideally devised by Heisenberg (1930) to show the uncertainty principle for a couple of conjugated dynamical variables (e.g. x and px) of a particle. Suppose we want to determine the position and the linear momentum of an electron impinged by a photon travelling along the x-direction. The optical axis of our microscope tube will be along the perpendicular y-coordinate. The way of proceeding is much along the same lines as those for the Compton effect. The resolving power of our microscope is Dx ¼

l 2 sin ε

where 2ε is the angular aperture of the lens of our objective. To improve precision in the determination of the electron position we must reduce Dx, namely, we must reduce the wavelength l or increase the frequency, i.e. the wave number k, of our incident photon. We saw in Table 11.2 that the weak g-rays

11.5 Solved problems

465

from the 7 Be source have a photon mass comparable to that of the electron. Hence, by illuminating our microscope with a photon of this energy we shall have the collision of particles of like mass, which will be scattered in different directions. If q is the scattering angle, the conservation of linear momentum along x gives hn hn0 ¼ cos q þ px c c h hn h px ¼ ðn  n0 cos qÞ z ð1  cos qÞ ¼ ð1  cos qÞ c c l Because the scattered photon becomes observable, it must be scattered inside the microscope tube, so that q must be restricted to  p  p  p  p ε q þε cos  ε  cos q  cos þ ε 2 2 2 2 h h sin ε  cos q   sin ε ð1  sin εÞ  px  ð1 þ sin εÞ l l The momentum of the electron, px, can hence be determined to within an uncertainty of h Dpx ¼ 2 sin ε l so that, using the expression found previously for Dx, follows the Heisenberg’s uncertainty principle for the x-component. On the other hand, the conservation of the linear momentum in the perpendicular y-direction gives 0¼

hn0 hn0 hn sin q  py 0 py ¼ sin q z sin q c c c

with cos ε  sin q  cos ε The only possible value for sin q is now cos ε; and therefore the transverse component of the momentum is exactly measurable: py ¼

h cos ε l

0 Dpy ¼ 0

This is in accord with the fact that Heisenberg’s uncertainty relation does not hold for dynamical quantities that are not conjugate, and which are described by commuting operators: h i x; p^y ¼ 0 ½x; p^x  ¼ iZ; From this follows Dirac’s observation that the greatest possible specification of a given physical system is to find the maximum set of commuting operators. In this case, it is possible to find a set of observables, such that simultaneous exact knowledge of all members of the set is possible (Troup, 1968).

CHAPTER

Atomic orbitals

12

CHAPTER OUTLINE 12.1 12.2 12.3 12.4 12.5 12.6

Introduction ............................................................................................................................... 467 Hydrogen-like Atomic Orbitals..................................................................................................... 468 Slater-type Orbitals..................................................................................................................... 473 Gaussian-type Orbitals ................................................................................................................ 475 Problems 12 .............................................................................................................................. 477 Solved Problems ........................................................................................................................ 478

12.1 INTRODUCTION We saw in the preceding chapter that the atomic orbitals (AOs) are the one-electron one-centre functions needed for describing the probability of finding the electron at any given point in space. Therefore, they are the building blocks of any theory that can be devised inside the orbital model, and, particularly, they must be the basis of all approximation methods based on the variation theorem. The Schroedinger eigenvalue equation for the one-electron atomic problem, the so-called hydrogen-like system, was fully solved in Section 3.7 of Chapter 3. The physically permissible solutions are the simplest example of AOs and their properties. The hydrogen-like atomic orbitals (HAOs) were considered in detail in Section 3.7.4 of Chapter 3. It was shown there that real HAOs have the same transformation properties of the spherical tensors in real form discussed in Section 4.11.2 of Chapter 4. The mathematical techniques of solution of the pertinent differential equations are typical of mathematical physics and completely general, and can also be applied to the hydrogen atom in electric or magnetic fields, as we did in Section 3.8 of Chapter 3, giving the possibility of exact evaluation of second-order properties such as electric polarizabilities or magnetic susceptibilities. The physical constraints imposed on the mathematical solutions do explain in a clear way the origin of quantum numbers, justifying the rather mysterious assumptions of the Bohr’s theory. However, the HAOs are of no interest in practice, except that they are of importance in giving relatively simple solutions in exact analytical form, what is essential for checking unequivocally approximate solutions such as those obtained by use of the variation theorem, as we shall do to some extent in the next chapter. Instead, the great majority of quantum chemical calculations on atoms and molecules is based on the use of a basis of AOs which have a radial dependence which is different from that of the hydrogenic HAOs. They can be distinguished into two classes based on whether their decay with the radial variable r is exponential (Slater-type orbitals (STOs), by far the best) or Gaussian (Gaussian-type orbitals (GTOs)). Elementary Molecular Quantum Mechanics 2E. http://dx.doi.org/10.1016/B978-0-444-62647-9.00012-9 Ó 2013, 2007 Elsevier B.V. All rights reserved

467

468

CHAPTER 12 Atomic orbitals

In this chapter, after a short resume´ of the form and properties of the HAOs, we shall introduce in some detail the STOs and the GTOs and their properties, and a few very simple one-centre oneelectron integrals which will be needed in the next chapter. A more detailed account of the molecular integrals over STOs and GTOs will be discussed to some extent in Chapter 18.

12.2 HYDROGEN-LIKE ATOMIC ORBITALS We saw in Chapter 3 that the HAOs in their complex form simultaneously satisfy the following three eigenvalue equations: ^ nlm ¼ En jnlm ; L^2 jnlm ¼ lðl þ 1Þjnlm ; L^z jnlm ¼ mjnlm Hj

(1)

with the energy eigenvalue in atomic units given by En ¼ 

Z2 2n2

(2)

while l(l þ 1) is the eigenvalue of the square angular momentum operator in atomic units. The corresponding eigensolutions (AOs) are characterized by the three quantum numbers (n, l, m), which take on the values n ¼ 1; 2; 3; / principal quantum number l ¼ 0; 1; 2; 3; /ðn  1Þ angular ðorbitalÞ quantum number

(3)

m ¼ 0; 1; 2; /  l magnetic quantum number The eigenfunctions (HAOs) are given by jnlm ðr; q; 4Þ ¼ Rnl ðrÞYlm ðq; 4Þ ¼ jnlmi

(4)

Ylm ðq; 4Þ ¼ Qlm ðqÞFm ð4Þ

(5)

where Y are the spherical harmonics. The HAOs depend on the three quantum numbers (n, l, m) characterizing the quantum states that the electron can assume in the absence of external perturbations (n ¼ 1, l ¼ m ¼ 0 is the spherical ground state, all others being excited states). The radial part R(r) depends on n, l; the angular part Y(q, 4) depends Z on l, m. We want to underline once more that the radial polynomial solution (a polynomial in x ¼ r of n degree n  l  1) is characteristic only of HAOs, while the angular part in complex form has the same form for all AOs, even not hydrogen-like. Other definitions used for the angular part in complex form are (Brink and Satchler, 1993; Stone, 1996) rffiffiffiffiffiffiffiffiffiffiffiffi 4p Ylm ð^rÞ (6) Clm ðq; 4Þ ¼ Clm ð^rÞ ¼ 2l þ 1 known as modified spherical harmonic; Rlm ðrÞ ¼ rl Clm ð^rÞ; Ilm ðrÞ ¼ rl1 Clm ð^rÞ

(7)

12.2 Hydrogen-like atomic orbitals

469

regular and irregular solid spherical harmonics, respectively. In these equations we use the notation r r ^r ¼ ¼ (8) U q; 4 It is important to stress once more that the AOs in complex form are also eigenfunctions of the ^ with eigenvalue m. In fact, operator L^z (the z-component of the angular momentum vector operator L) in spherical coordinates v (9) 0 L^z j ¼ mj L^z ¼ i v4 2 F f expðim4Þ travelling waves eigenfunctions of L^ and L^z

(10)

vF ¼ mðR Q FÞ ¼ mj L^z j ¼ ðR QÞL^z F ¼ iðR QÞ v4

(11)

The F-functions in complex form are usually given with the Condon–Shortley phase (Condon and Shortley, 1963; Brink and Satchler, 1993): Fþm ¼ ð1Þm

expðim4Þ pffiffiffiffiffiffi m > 0; 2p

Fm ¼

expðim4Þ pffiffiffiffiffiffi ¼ ð1Þm Fþm 2p

(12)

whereas valence theory mostly uses orbitals in real form. 2 The real Fs involve trigonometric functions which are still eigenfunctions of L^ but no longer of L^z . Real Fs are related to complex Fs by the unitary transformation cos m4 ð1Þm Fþm þ Fm sin m4 ð1Þm Fþm  Fm pffiffiffi pffiffiffi Fcm ¼ pffiffiffiffi ¼ ; Fsm ¼ pffiffiffiffi ¼ i p p 2 2

(13)

Real Fs can be visualized as standing waves. The inverse transformation (from real to complex form) is given by Fc þ iFs Fc  iFs (14) Fþm ¼ ð1Þm m pffiffiffi m ; Fm ¼ m pffiffiffi m 2 2 These transformations can be immediately verified by recalling Euler’s formulae for imaginary exponentials. In matrix form 0 1 m 1 m i B ð1Þ pffiffiffi ð1Þ pffiffiffi C B 2 2C (15) ðFcm Fsm Þ ¼ ðFþm Fm ÞB C ¼ ðFþm Fm ÞU i @ p1ffiffiffi pffiffiffi A 2 2 with the inverse transformation ðU1 ¼ Uy Þ

0

1 1 1 B ð1Þ pffiffiffi pffiffiffi C B 2 2C ðFþm Fm Þ ¼ ðFcm Fsm ÞB i C @ ð1Þm piffiffiffi p ffiffiffi A 2 2 m

(16)

470

CHAPTER 12 Atomic orbitals

The unitary transformation preserves normalization, so that cos m4 Fcm ¼ pffiffiffiffi p Fþm

expðim4Þ pffiffiffiffiffiffi ¼ ð1Þ 2p m

sin m4 Fsm ¼ pffiffiffiffi m > 0 p Fm

(17)

expðim4Þ pffiffiffiffiffiffi ¼ ð1Þm Fþm ¼ 2p

All F-functions, either real or complex, are normalized to one and mutually orthogonal. Spherical harmonics in real form are also known as tesseral harmonics (MacRobert, 1947), and are given in unnormalized form as Yl0 ;

c Ylm wQlm cos m4;

s Ylm wQlm sin m4 ðm > 0Þ

(18)

The explicit form of the first few (un-normalized) HAOs in real form up to n ¼ 3 was already given in Section 3.7.4 of Chapter 3 and will not be repeated here. The plots of the first few radial hydrogenlike functions up to n ¼ 3 are given in Figure 12.1. The real AOs are seen to have the same transformation properties of the (x, y, z)-coordinates or of their combinations. Replacing the radial polynomial part of the HAOs by rn1, the functional dependence of these real HAOs on (r, q, 4) is the same as that of STOs with the orbital exponent c considered as a variable parameter, which we shall consider in the next section. The first states of the one-electron atom are given in Table 12.1, while in Figure 12.2 is the diagram of the corresponding energy levels (orbital energies).

FIGURE 12.1 Plots of the first Rnl (x) radial functions

12.2 Hydrogen-like atomic orbitals

471

Table 12.1 First States of the Hydrogen-Like Electron n

l

m

En

jnlmi

1 2

0 0 1 1 1 0 1 1 1 2 2 2 2 2

0 0 1 0 1 0 1 0 1 2 1 0 1 2

E1 E2

j100i j200i j211i j210i j211i j300i j311i j310i j311i j322i j321i j320i j321i j322i

3

E3

1s 2s 2p 3s 3p

3d

FIGURE 12.2 Diagram of orbital energies for the hydrogen-like atom compared to that of the many-electron atom (not in scale)

472

CHAPTER 12 Atomic orbitals

• Comments • For the one-electron atom the energy eigenvalues depend only on n. For the many-electron atom, orbital energies do depend on n and l. ^ L^2 ; L^z Þ. A fundamental • Each jnlm AO is at a time eigenfunction of three different operators ðH; theorem of Quantum Mechanics (Eyring et al., 1944) then says that the three operators do commute with each other: i h i h i h ^ L^z ¼ L^2 ; L^z ¼ 0 ^ L^2 ¼ H; H; (19) which physically means that, in state jnlm ; energy, total angular momentum, and z-component of angular momentum, all have a definite value: in other words, we can measure each of these physical quantities with arbitrary precision, without altering the remaining two. Quantities commuting with the Hamiltonian are said to be constants of the motion. Quantities whose operators do not commute cannot be measured at the same time with arbitrary precision, because of Heisenberg’s uncertainty principle seen in the preceding chapter ðe:g: x; p^x or y; p^y or z; p^z ; L^x ; L^y or L^y ; L^z or L^z ; L^x Þ. • Since energy eigenvalues do depend only on n, the energy levels of the one-electron atom are strongly degenerate, the number g of different eigenstates for a given value of n being g¼

n1 X

ð2l þ 1Þ ¼ n þ 2

l¼0

n1 X

l¼nþ2

l¼0

nðn  1Þ ¼ n2 2

(20)

• Electron density distribution. We recall from the first principles that jjnlm ðrÞj2 dr ¼ ½Rnl ðrÞ2 jYlm ðq; 4Þj2 d4 sin q dq r 2 dr ¼ jYlm ðUÞj2 dU$½Rnl ðrÞ2 r 2 dr

(21)

probability of finding in dr the electron in state jnlm Here U is a shorthand for angles q; 4. Upon integration over all angles, we find the radial probability, namely, the probability of finding the electron in a spherical shell of thickness dr, independently of angles q; 4. If the spherical harmonics Ylm are normalized to 1, we are left with Z dUjYlm ðUÞj2 ½Rnl ðrÞ2 r 2 dr ¼ ½Rnl ðrÞ2 r 2 dr ¼ Pnl ðrÞdr (22) U

where

Pnl ðrÞ ¼ ½Rnl ðrÞ2 r 2

(23)

is the radial probability density. The radial probability densities for the first few states of the H atom (Z ¼ 1) are given in Figure 12.3. Further examples on the properties of ground and excited states of the oneelectron hydrogen-like atom are given as problems in Section 12.5.

12.3 Slater-type orbitals

473

FIGURE 12.3 The radial probability densities for the first few states of the H atom (Z ¼ 1)

12.3 SLATER-TYPE ORBITALS STOs were introduced long ago by Slater (1930) and Zener (1930), and extensively used by Roothaan (1951b) in developing his fundamental work on molecular integrals. Slater showed that for many purposes only the term with the highest power of r in the hydrogen-like Rnl(r) is of importance for practical calculations. Zener suggested replacing the hydrogenic orbital exponent c ¼ Z/n by an effective nuclear charge (Z  s) seen by the electron, and which is less than the true nuclear charge Z by a quantity s called the screening constant. This gives AOs which are more diffuse than the original HAOs. The Zener approach has today been replaced by the variational determination of the orbital exponents c, as we shall see in the next chapter. One of the major difficulties of STOs is that the excited STOs within a given angular symmetry are no longer orthogonal to their lowest terms. As we shall see, this is particularly troublesome for ns orbitals (n > 1), which lack the cusp which is characteristic of all s orbitals. Furthermore, multicentre integrals over STOs are difficult to evaluate. Retaining only the highest (n  l  1) power of r in the polynomial Eqn (170) of Chapter 3 defining Rnl(r), the dependence on l is lost and we obtain the general STO in real form as c;s c;s cnlm ðr; q; 4Þ ¼ jnlmi ¼ Nr n1 expðcrÞYlm ðq; 4Þ ¼ Rn ðrÞYlm ðq; 4Þ

(24)

where N is a normalization factor. Here Rn ðrÞ ¼ Nn r n1 expðcrÞðc > 0Þ

(25)

is the normalized radial part, and  c;s ðq; 4Þ Ylm

¼

NU Pm l ðxÞ

cos m4 m0 sin m4

(26)

is the normalized angular part in real form. As usual, we use U to denote the couple of angular variables q; 4.

474

CHAPTER 12 Atomic orbitals

Separate normalization of the radial and angular parts gives Nn ¼

ð2cÞ2nþ1 ð2nÞ!

!1=2

 ;

NU ¼

2l þ 1 2pð1 þ d0m Þ

ðl  mÞ! ðl þ mÞ!

1=2 (27)

so that the overall normalization factor for the general STO Eqn (24) will be "

#1=2 ð2cÞ2nþ1 1 2l þ 1 ðl  mÞ! $ $ N ¼ Nn NU ¼ ð2nÞ! pð1 þ d0m Þ 2 ðl þ mÞ!

(28)

We remember that ZN dr r n expðarÞ ¼

n!

(29)

anþ1

0

for n ¼ non-negative integer and a ¼ real positive number Z2p



 ð1 þ d0m Þ cos2 m4 d4 ¼p ð1  d0m Þ sin2 m4

(30)

0

Z1 1

 2 dx Pm l ðxÞ ¼

2 ðl þ mÞ! 2l þ 1 ðl  mÞ!

m ¼ jmj  0

(31)

The off-diagonal matrix element over STOs of the atomic one-electron hydrogen-like Hamiltonian is

 1 2 Z 0 0 hc0 c ¼ R Y  V  RY 2 r # " lðl þ 1Þ  nðn  1Þ ðc þ c0 Þ2 c þ c0 c2 þ ðnc  ZÞ (32)  ¼ Sn0 l0 m0 ;nlm 2 n þ n0 2 ðn þ n0 Þðn þ n0  1Þ where Sn0 l0 m0 ;nlm is the non-orthogonality integral given by Sn0 l0 m0 ;nlm

ðn þ n0 Þ!  c ¼ < n0 l0 m0 jnlm > ¼ dll0 dmm0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 ð2nÞ!ð2n0 Þ! c

nn0 2

2ðcc0 Þ1=2 c þ c0

The diagonal element of the Hamiltonian is

 1 2 Z c2 n þ 2lðl þ 1Þ Zc hcc ¼ RY  V  RY ¼  2 nð2n  1Þ 2 r n

!nþn0 þ1 (33)

(34)

12.4 Gaussian-type orbitals

475

For c ¼ Z/n (HAOs) Eqn (34) becomes Z 2 4n2  3n  2lðl þ 1Þ hcc ¼  2 2n nð2n  1Þ

(35)

where the last factor on the right is one only for the lowest AOs of each symmetry (l ¼ 0, 1, 2, /), since in this case STOs and HAOs coincide.

12.4 GAUSSIAN-TYPE ORBITALS GTOs are largely used today in atomic and molecular computations because of their greater simplicity in computing multicentre molecular integrals. GTOs were originally introduced by Boys (1950) and McWeeny (1950) mostly for computational reasons. In today’s molecular calculations it is customary to fit STOs in terms of GTOs, which requires rather lengthy expansions (see the variety of Pople’s bases in Chapter 14), or to minimize the deviation between STOs and GTOs as done by Huzinaga (1965). Some failures of GTOs with respect to STOs are briefly discussed in the next chapter. The most common Gaussian orbitals are given in the form of spherical or Cartesian functions. We shall give below some formulae which are of interest to us in the next chapter, while for details the reader is referred to Chapter 18 (Shavitt, 1963; Cook, 1974; Saunders 1975, 1983). 1. Spherical Gaussians For spherical GTOs it will be sufficient to consider the radial part of the orbital, since the angular part is the same as that for STOs. !1=2 1

 2nþ1 ð2cÞnþ2 n1 2 pffiffiffiffi (36) Rn ðrÞ ¼ Nn r exp cr ðc > 0Þ; Nn ¼ ð2n  1Þ!! p where the double factorial is defined by Eqn (138) of Chapter 4. The normalization factor in Eqn (36) is easily derived from the general integral: ZN

 ðn  1Þ!! dr r n exp ar 2 ¼ sðnÞ nþ1 ð2aÞ 2 0 where

rffiffiffiffi p sðnÞ ¼ 2

for n ¼ even;

sðnÞ ¼ 1

for n ¼ odd

(37)

(38)

The normalized spherical Gaussian orbital will then be written as c;s ðq; 4Þ jnlmi ¼ Gðnlm; cÞ ¼ Nr n1 expðcr 2 ÞYlm

(39)

where the normalization factor is N ¼ Nn NU with Nn given in Eqn (36) and NU in Eqn (27). The general matrix element of the atomic one-electron hydrogen-like Hamiltonian is

(40)

476

CHAPTER 12 Atomic orbitals

 hG 0 G ¼



Z  Gðnlm; cÞ 2 r " !nþn0 þ1 #1=2 nn0 ð2c þ 2c0 Þ1 2  c 2 2ðcc0 Þ1=2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c þ c0 ð2n  1Þ!!ð2n0  1Þ!! p c0

1 G0 ðn0 l0 m0 ; c0 Þ  72

¼ dll0 dmm0 nh

(41)

2ðlðl þ 1Þ  nðn  1ÞÞðn þ n0  3Þ!!ðc þ c0 Þ2 þ 2ð2n þ 1Þðn þ n0  1Þ!!cðc þ c0 Þ o pffiffiffi  2ðn þ n0 þ 1Þ!!c2 sðn þ n0 Þ  2 2 Zðn þ n0  2Þ!!ðc þ c0 Þ3=2 sðn þ n0  1Þ



a rather complicated unsymmetrical formula, with the diagonal element (c0 ¼ c)

 1 Z hGG ¼ Gðnlm; cÞ  72  Gðnlm; cÞ 2 r o n pffiffiffiffiffiffiffiffi 1 c ¼ pffiffiffiffiffiffi ½4ðlðl þ 1Þ  nðn  1ÞÞð2n  3Þ!! þ ð2n þ 1Þ!! p=2  4Zð2n  2Þ!!c1=2 2p ð2n  1Þ!! (42) For spherical GTOs there is no longer proportionality between matrix elements of the atomic one-electron hydrogen-like Hamiltonian h^ and the non-orthogonality integral S. For 1s spherical GTOs with different orbital exponents Eqn (41) gives n ¼ n0 ¼ 1 l ¼ m ¼ 0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D E 3 $32 ðcc0 Þ7=2 2 25 ðcc0 Þ3=2 hG 0 G ¼ G0 ð100; c0 Þ h^ Gð100; cÞ ¼  Z ðc þ c0 Þ2 p ðc þ c0 Þ5 and, if c0 ¼ c  hGG ¼



 1=2 1 2 Z 3 8c Gð100; cÞ  7  Gð100; cÞ ¼ c  Z 2 r 2 p

(43)

(44)

2. Cartesian Gaussians Un-normalized Cartesian Gaussians are best written as   Guvw ðcÞ ¼ xu yv zw expðcr 2 Þ ¼ xu yv zw exp c x2 þ y2 þ z2

(45)

where u, v, w are non-negative integers, the non-orthogonality integral between GTOs being   ð2U  1Þ!!ð2V  1Þ!!ð2W  1Þ!! p 3=2 Su0 v0 w0 ;uvw ¼ G0u0 v0 w0 ðc0 Þ Guvw ðcÞ ¼ c þ c0 ð2c þ 2c0 ÞUþVþW

(46)

where 2U ¼ u þ u0 ;

2V ¼ v þ v0 ;

2W ¼ w þ w0

(47)

12.5 Problems 12

The normalization factor of GTO Eqn (45) is then "  3=2 #1=2 ð4cÞUþVþW 2c N¼ ð2U  1Þ!!ð2V  1Þ!!ð2W  1Þ!! p

477

(48)

12.5 PROBLEMS 12 The problems in this chapter are limited to the HAOs, since applications to STOs and GTOs will form the object of the remaining part of this book. 12.1. Find the radial probability density of the 1s hydrogen-like ground state. Answer: P1s ðrÞ ¼ expð2crÞr 2 Hint: Follow definition Eqn (23) of the radial probability density for the 1s orbital. 12.2. Find the value of r for which the probability density for the 1s ground state is maximum. Answer: ðr1s Þmax ¼

1 c

Hint: Evaluate the first and second derivatives of the 1s STO. 12.3. Find the average distance of the 1s electron from its nucleus. Answer: < r >1s ¼

3 2c

Hint: Evaluate integral h1sjrj1si in spherical coordinates. 12.4. Find the radial probability density of the 2p excited state. Answer: PðrÞ ¼ expð2crÞr 4 Hint: Follow definition Eqn (23) of the radial probability density for the 2p excited state. 12.5. Find the average distance of the 2p electron from its nucleus. Answer: < r >2p ¼

5 2c

Hint: Evaluate integral h2pjrj2pi in spherical coordinates.

478

CHAPTER 12 Atomic orbitals

12.6. Evaluate some expectation values for ground and excited states of the H atom. Answer: The results are collected in Table 12.2 below. Table 12.2 Some Expectation Values (au) for the H Atom State 1s ¼ j100i 2s ¼ j200i 2p ¼ j210i

hri

hr 2 i

3 2Z 6 Z 5 Z

3 Z2 42 Z2 30 Z2

hr L1 i Z Z 4 Z 4

hVi Z 2 Z2 4 Z2  4 

hTi Z2 2 Z2 8 Z2 8

Hint: Evaluate the corresponding atomic integrals in spherical coordinates.

12.6 SOLVED PROBLEMS 12.1. The radial probability density of the 1s hydrogen-like ground state. The normalized HAO for the 1s ground state is  3 1=2 c j1s ¼ j100i ¼ 1s ¼ expðcrÞ c ¼ Z p According to Eqn (23) the radial probability density of state 1s will be P1s ðrÞ ¼ expð2crÞr 2 12.2. Maximum of the probability density for the 1s ground state. We evaluate the first and second derivatives of P1s(r):

 dP1s ðrÞ ¼ expð2crÞ 2r  2cr 2 dr 2 

d P1s ðrÞ ¼ expð2crÞ 2  8cr þ 4c2 r 2 dr2 Now, the necessary condition for the maximum of P1s(r) is dP1s 1 ¼ 0 0 cr ¼ 1 0 r ¼ dr c with

 2  d P1s ¼ expð2Þð2  8 þ 4Þ ¼ 2 expð2Þ < 0 dr 2 cr¼1

E Z2 2 Z2  8 Z2  8 

12.6 Solved problems

479

Since the second derivative is negative, cr ¼ 1 is a true maximum. For the H atom, c ¼ Z ¼ 1, rmax ¼ 1, which is the Bohr radius. 12.3. Average distance of the 1s electron from the nucleus. Evaluating the integral we find 1s ¼< 1sjrj1s > ¼

c3 $4p p

ZN dr r 2 $r expð2crÞ ¼

3 2c

0

In the ground-state H atom, the probability density has a maximum for a0 ¼ 1 (Bohr radius), while the average electron radius is larger, 1.5 a0. The radial probability density for H(1s) is plotted against r in Figure 12.4.

FIGURE 12.4 Radial probability density for H(1s)

12.4. Radial probability density for the excited 2p state. The normalized hydrogen-like excited 2p state is j2pz ¼ j210i ¼

 5 1=2 c expðcrÞr cos q p

2c ¼ Z

The radial probability density of state 2pz is then P2p ðrÞ ¼ expð2crÞr 4 dP2p ¼ expð2crÞð4r 3  2cr 4 Þ dr

 d2 P2p ¼ expð2crÞ 12r2  16cr 3 þ 4c2 r 4 2 dr dP2p 2 ¼ 0 0 cr ¼ 2 0 r ¼ dr c

480

CHAPTER 12 Atomic orbitals

The value of the second derivative for cr ¼ 2 is  2  d P2p ¼ expð4Þ 4 r 2 ð3  8 þ 4Þ ¼ expð4Þ 4 r2 < 0 dr 2 cr¼2 so that cr ¼ 2 is a true maximum. For the H atom, 2c ¼ Z ¼ 1, rmax ¼ 4, so that the greatest probability of finding the hydrogen-like electron in state 2p is at r ¼ 4a0, i.e. 4 times larger than the ground-state value. 12.5. Average distance of the 2p hydrogen-like electron from the nucleus. The average distance of the 2p electron from the nucleus is < r>2p

c5 ¼ < 2pjrj2p > ¼ $2p p ¼ 2c5 $

2 3

ZN 0

Z1

ZN dx x

1

dr r 2 $r$expð2crÞr2

2 0

4 5! 5 dr r 5 expð2crÞ ¼ c5 $ ¼ 6 3 ð2cÞ 2c

where we have put, as usual, cos q ¼ x in the integral over q. In the H atom, the average distance from the nucleus of the electron in the excited 2p state is 5 a0, larger than the most probable distance, 4 a0 (Figure 12.5).

FIGURE 12.5 Radial probability density for H(2p)

12.6. Expectation values for the ground and excited states of the H atom. The expectation values (in atomic units) for ground and excited states of the hydrogen-like atom collected in Table 12.2 are immediately obtained by performing all necessary integrals in spherical coordinates. A few comments to the table seem appropriate here. • Comments to Table 12.2. • In the ground state, the average distance of the electron from the nucleus is greater than its most probable distance (¼1/Z).

12.6 Solved problems

481

• In the excited states, the average distance of the electron from the nucleus is greater than that of the ground state. • On the average, the electron is nearer to the nucleus in the 2p rather than in the 2s state. • Increasing the nuclear charge Z, the electron is nearer to the nucleus for any state. • Diamagnetic susceptibility is proportional to < r 2 >. In the excited states it is hence much greater than in the ground state (e.g. in the 2p state it is 10 times larger than in the 1s state). • Nuclear attraction is proportional to the nuclear charge, and in the ground state it is four times larger than in the first excited state. • The average kinetic energy is repulsive, and in the ground state it is four times larger than in the first excited state. • The results of Table 12.2 show that the virial theorem holds: 1 E ¼ < T > ¼ < V > 2 • We notice that while average values of positive powers of r are different for excited states of different symmetries, the average value of the reciprocal of the distance of the electron from the nucleus is the same for states 2s and 2p. This is due to the fact that < r 1 > appears in the potential energy expression, whose average value is related to the total energy through the virial theorem, and the total energy is degenerate for n ¼ 2 (i.e. different states have the same energy, hence the same average nuclear attraction). • On the average, nuclear attraction overcomes repulsion due to kinetic energy (in a ratio determined by the virial theorem), and the electron is bound either in the ground state or in the first 2s and 2p excited states.

CHAPTER

Variational calculations

13

CHAPTER OUTLINE 13.1 Introduction ............................................................................................................................... 483 13.2 The Variational Method............................................................................................................... 484 13.2.1 Variational principles in first order...........................................................................484 13.2.2 Variational approximations .....................................................................................485 13.2.3 Basis functions and variational parameters ..............................................................486 13.3 Non-linear Parameters................................................................................................................ 487 13.3.1 The 1s ground state of the atomic one-electron system .............................................487 13.3.2 The first 2s, 2p excited states of the atomic one-electron system...............................490 13.3.3 The 1s2 ground state of the atomic two-electron system............................................493 13.4 linear Parameters and the Ritz Method ........................................................................................ 496 13.5 Atomic Applications of the Ritz Method ....................................................................................... 497 13.5.1 The first 1s2s excited state of the atomic two-electron system ...................................497 13.5.2 The first 1s2p excited state of the atomic two-electron system ...................................499 13.5.3 Results for hydrogen-like AOs .................................................................................502 13.6 Molecular Applications of the Ritz Method................................................................................... 503 13.6.1 The ground and first excited state of the Hþ 2 molecular ion ........................................503 13.6.2 The interaction energy and its components ..............................................................505 13.7 Variational Principles in Second Order ........................................................................................ 510 13.7.1 The dipole polarizability of the H atom ....................................................................510 13.7.2 The London attraction between two ground-state H atoms .........................................512 13.8 Problems 13 .............................................................................................................................. 514 13.9 Solved Problems ........................................................................................................................ 517

13.1 INTRODUCTION We have seen in Chapter 3 that the Schroedinger equation can be solved exactly only for a few physical systems, and we have studied in detail, among others, the cases of the particle in different boxes, the harmonic oscillator, and the one-electron atomic and molecular systems. For the majority of the remaining applications, we must resort to methods which enable us to evaluate approximations to the energy of some states of the system, as explained in Chapter 1. We saw there that they are essentially two: (1) the variational method due to Rayleigh, and (2) the perturbation methods introduced by Schroedinger and others. In this chapter, we shall be mostly concerned with applications of Elementary Molecular Quantum Mechanics 2E. http://dx.doi.org/10.1016/B978-0-444-62647-9.00013-0 Ó 2013, 2007 Elsevier B.V. All rights reserved

483

484

CHAPTER 13 Variational calculations

the variational method, which is extremely powerful in finding bounds to the energy of ground and excited states of atomic and molecular systems.

13.2 THE VARIATIONAL METHOD In the following, we shortly recall the basis of Rayleigh’s variational principles which allow us to find approximations to energy and wavefunction of ground and excited states of the system, either in the first or the second order. We stress once more that in the first order of perturbation theory we evaluate approximations to the total energy and to first-order properties, while in the second order we evaluate approximations to second-order energy and to second-order molecular properties such as polarizabilities.

13.2.1 Variational principles in first order Let 4 be a normalizable regular trial (or variational) function. The 4-dependent functional ε½4 ¼

^ h4jHj4i h4j4i

(1)

where H^ is the Hamiltonian of the system, is called the Rayleigh ratio. Then ε½4  E0

(2)

is the Rayleigh variational principle for the ground state, E0 being the true ground-state energy of the system, and ε½4  E1

provided hj0 j4i ¼ 0

(3)

is the Rayleigh variational principle for the first excited state (orthogonal to the true ground state j0), E1 being the true energy of the first excited state, and so on. So, evaluation of the integrals in Eqn (1) under suitable constraints gives upper bounds to the energy of the ground and excited states. Inequalities (2) and (3) were proved in Section 1.2 through the formal expansion of the trial function 4 ^ into eigenstates {jk} of H: We further recall the three properties of the variational solutions: (1) If 4 ¼ j0 0 ε½4 ¼ E0

(4)

^ and the equality sign holds (the true lowest eigenvalue of H). (2) If 4 ¼ j0 þ d

(5)

namely, if the variational function differs from the true j0 by a small first-order function d; then
 (6) ε  E0 ¼ hdjH^  E0 jdi ¼ O d2  0 i.e. the variational energy differs from the true energy by a second-order quantity, which means that, in variational approximations, energy is better approximated than function.

13.2 The variational method

485

(3) The variational method privileges the space regions near the nucleus, where the potential energy is larger (r small). Use of variationally optimized wavefunctions can give poor results for operators different from H^ (like the dipole moment operator m ¼ er, which takes large values far from the nucleus).

13.2.2 Variational approximations The functional ε½4 gives an upper bound to E0 or E1 (Figure 13.1). The variation method consists in applying the Rayleigh’s variational principles to the determination of approximations to the energy and the wavefunction. Even if in some cases (see Problems 13.1 and 13.2) it is possible to get bounds to the true energy simply by imposing the boundary conditions on the variational wavefunction, usually we shall introduce in the trial function 4 a number of variational parameters fcg and minimize the energy ε with respect to these parameters: Z ^ dx 4 ðx; cÞH4ðx; cÞ ^ h4jHj4i ¼ Z ¼ εðcÞ (7) ε½4 ¼ h4j4i dx 4 ðx; cÞ4ðx; cÞ In this way, by evaluating the integrals in Eqn (7), the functional of 4 is changed into an ordinary function of the variational parameters {c}. For a single parameter, we have a plot like that of Figure 13.2, εmin ¼ ε(cmin) being then the best variational approximation arising from the given form

FIGURE 13.1 Energy upper bounds to ground and excited states

486

CHAPTER 13 Variational calculations

FIGURE 13.2 Schematic plot of the variational energy around the minimum

of the trial function 4(c). The best value of the parameter c (hence, of ε and 4) is found by minimizing the variational energy, i.e. by solving the equation arising as a necessary condition for the minimum of ε:  2  vε v ε ¼ 0 0 cmin provided >0 (8) vc vc2 cmin Increasing the number of variational parameters increases the flexibility of the variational wavefunction, and so increases the accuracy of the variational approximation.

13.2.3 Basis functions and variational parameters In molecular theory the variational wavefunctions are usually expressed in terms of a basis of atomic orbitals (STOs or GTOs, see Sections 12.3 and 12.4), possibly orthonormal or anyway orthogonalizable by the Schmidt method: cnlm ðr; UÞ ¼ Rn ðrÞYlm ðUÞ where the radial part is

 Rn r fr n1 exp cr STO



  or Rn r fr n1 exp cr 2 GTO

(9)

c>0

(10)

13.3 Non-linear parameters

487

Flexibility in the variational wavefunction is introduced through the so-called variational parameters, which can be of the two types: 8 • Non-linear > > > > > ðorbital exponentsÞ > < Variational • Linear (11) parameters > > ðcoefficients of the linear combination > > > > : in the Ritz methodÞ The Ritz method is more systematic and yields typical secular equations which can be solved by the standard matrix techniques discussed in Chapter 6, while careful optimization of non-linear parameters is more difficult (see Problem 13.5 for the simple Ransil method applied to the case of a single parameter).

13.3 NON-LINEAR PARAMETERS We shall now apply the variational techniques for finding approximations to energy and wavefunctions of ground and excited states of the hydrogen-like system, for which exact solutions were found in Chapter 3. This allows us to compare the results with the true solutions, unequivocally judging for the accuracy of the numerical results. We start by considering approximate functions of non-linear parameters. Short applications to ground and excited states of the harmonic oscillator will be given later in this chapter as Problems 13.3 and 13.4.

13.3.1 The 1s ground state of the atomic one-electron system We look for the best variational energy and average distance of the electron from the nucleus for the ground state of the hydrogen-like system arising from the three (un-normalized) variational functions depending on the single non-linear parameter c: 9 1: 4 ¼ expðcrÞ 1s  STO > = 2: 4 ¼ expðcrÞr 2s  STO (12) 
> ; 2 1s  GTO 3: 4 ¼ exp cr where c > 0 is the adjustable orbital exponent. The necessary integrals are easily obtained from the general formulae given in Chapter 12, and we obtain the results given in Table 13.1. Putting Z ¼ 1 we find the numerical values for the H atom given in Table 13.2. Function one gives for c ¼ 1 the exact value for either the energy eigenvalue or the average distance of the electron from the nucleus (the function 41 is the exact eigenfunction, the energy corresponding to the equality sign in Rayleigh’s principle (2)). The reasons of the poor behaviour at the origin of functions two (4 ¼ 0) and three (d4/dr ¼ 0) become apparent from the plot (Figure 13.3) giving the dependence of the three (normalized) functions on the radial variable r. The numerical values were taken from Table 13.3 (functions two and three lack the requested cusp at the origin). It is important to notice how function 2G improves by about 12% the energy eigenvalue but not at all the eigenfunction (next paragraph).

488

CHAPTER 13 Variational calculations

Table 13.1 Variational Approximations to the Ground State of the Hydrogen-like System 4

hri4

ε(c)

cmin

εmin

1. exp(cr)

3 2c

c2  Zc 2

Z



2. exp(cr)r

5 2c

c2 Zc  6 2

3 Z 2

3  Z2 8

8 2 Z 9p



3. exp(cr2)

2 pc

!1=2

3 8c cZ 2 p

!1=2

Z2 2

4 2 Z 3p

Table 13.2 Variational Approximations to the H-atom Ground State 4

cmin

εðcÞ=Eh

hri4

1. exp(cr)

1

0.5

1.5

2. exp(cr)r

1.5

0.375

1.57

3. exp(cr2) ¼ 1G

0.2829  a ¼ 1:3296 b ¼ 0:2014

0.4244

1.5

0.4858

1.48

4. 2G

a

a

2G ¼ A exp(ar2) þ B exp(br2) with A ¼ 0.2425, B ¼ 0.1759.

FIGURE 13.3 Plot of the (normalized) variational 1s functions against r

13.3 Non-linear parameters

489

Table 13.3 Radial Dependence of the Three (Normalized) Variational Functions r/a0

a

b

0 0.25 0.5 0.75 1 2 3 4 5

0.564 0.439 0.432 0.267 0.208 0.0763 0.0281 0.0103 0.0038

0 0.154 0.212 0.219 0.200 0.0893 0.0299 0.0089 0.0025

41 [ N1exp(Lr)

42 [ N2exp(L1.5r)r

c

43 [ N3exp(L0.2829r2)

0.276 0.272 0.258 0.236 0.208 0.0892 0.0217 0.0030 0.0002

a

N1 ¼ p1/2. N2 ¼ ½ð1:5Þ5 =3p1=2 : c N3 ¼ (0.5658/p)3/4. b

Comments. 41 is the point-by-point exact function. 42 is zero at the origin, has a maximum of 0.220 at r ¼ 0.67a0, being always smaller than 41 in the region 0.251a0, greater in 23a0 then decreases. 43 has a zero derivative at the origin, changes slowly at small values of r being always smaller than 41, coincides with 41 at r ¼ 1a0 then decreases rapidly with r. Gaussian bases A more detailed comparison of different N-GTO approximations (Van Duijneveldt, 1971) to the exact 1s results for the H-atom ground state is given in Table 13.4 for different values of N. In the first column is the number N of optimized 1s GTOs, in the second the overlap with the exact 1s eigenfunction, in the third the error in the eigenvalue Dε ¼ εNG  E0 ; and in the last column ^ H4i ^  ε2 . the quadratic average error (the variance) defined as hðH^  εÞ4jðH^  εÞ4i ¼ hH4j We see from the table that, in going from N ¼ 4 to N ¼ 10, the error Dε reduces by a factor 1000 (from about 1mEh to 1mEh), whereas the quadratic error (last column) reduces by just 10 times. This means that the Van Duijneveldt N-GTO is still point-by-point very different from the true j0, even for large values of N. As already observed, this is due to the lack of cusp in the 1s

Table 13.4 Errors Arising from Optimized Gaussian 1s Functions N

hj0 j4i

Dε=10L3 Eh

2 ^ H4iLε ^ hH4j =Eh2

1 2 3 4 5 10

0.9782 0.9966 0.9994 0.9998 0.99997 0.9999999

75.58 14.19 3.02 0.72 0.19 0:001 ¼ 1mEh

0.2912 0.2857 0.2032 0.1347 0.0887 0.0134

490

CHAPTER 13 Variational calculations

Table 13.5 Comparison of Different Gaussian Functions with Cusped Gaussians 4

Variational Parameters

Dε=10L9 Eh

2 ^ H4iLε ^ hH4j =Eh2

VDa WMb K10c K15c

Linear/non-linear 10/10 32/2 10/1 (c ¼ 0.11181) 15/1 (c ¼ 0.08761)

755.3 0.2 216.2 0.0

1.3  102 3.2  104 7.8  106 3.4  109

a

Van Duijneveldt, 1971 (VD). Wheatley and Meath, 1993 (WM). c Magnasco et al., 1995 (KN). b

GTO, which persists even increasing the number of GTOs. To remove this error, Klopper and Kutzelnigg (KK, 1986) suggested representing the 1s H orbital by means of a linear combination of n Gaussian functions having different and systematically increasing principal quantum numbers: 1s, 2s, 3s,., even with a single orbital exponent. Then 

(13) 4 ¼ exp cr 2 1 þ c1 r þ c2 r 2 þ /
 d4 ¼ exp cr 2 ½c1 þ 2c2 r þ /  2crð1 þ c1 r þ /Þ dc d4 lim ¼ c1 s0 r/0 dc

(14) (15)

and 4 can now account for the cusp at r ¼ 0. In Table 13.5 we collect some results obtained from different Gaussian functions. KN denotes the Kutzelnigg-like cusped GTOs with a single orbital exponent. We may notice the improvement in the variance already obtained by K10 (which has the same number of linear parameters as VD, but only one optimized non-linear parameter). The cusped K15, still with a single non-linear parameter for a basis set whose dimensions are about half those of the uncusped WM, is seen to give an ε value exact to 10 decimal figures and a variance exceedingly small. This means that K15 with c ¼ 0.08761 is practically point-by-point coincident with the exact 1s eigenfunction.

13.3.2 The first 2s, 2p excited states of the atomic one-electron system We now look for the best variational approximations to the first excited state of the atomic one-electron system using STOs of the appropriate symmetry. (1) Excited 2s state The spherical function s ¼ (c5/3p)1/2r exp(cr) (normalized and nodeless 2s STO) cannot be used as such in a variational calculation for state 2s (the first excited state having the same spherical symmetry of the ground state j0 ¼ 1s) since it is not orthogonal to j0. We have already seen that optimizing such a function with respect to the orbital exponent c, we find a poor approximation to the true energy of the ground state. A convenient variational function for the

13.3 Non-linear parameters

491

first excited state of spherical symmetry will hence be obtained by Schmidt orthogonalizing s against j0: s  Sj0 4 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi 0 hj0 j4i ¼ 0 1  S2

(16)

where S ¼ hj0 jsis0: The variation theorem for excited states of the same symmetry can now be applied and gives

 1 2 Z 1

ε4 ¼ 4  V  4 ¼ 1  S2 hss þ S2 hj0 j0  2Shsj0 2 r (17)
 S2 hj0 j0 þ hss  2Shsj0 ¼ hss þ Epn  E2s ¼ hss þ 1  S2 where the term Epn ¼




 hsj0  Shj0 j0 þ hsj0  Shss S2 hj0 j0 þ hss  2Shsj0 ¼ S >0 1  S2 1  S2

(18)

which arises from the non-orthogonality between s and j0 (S s 0), gives a repulsive contribution avoiding the excited 2s electron to collapse onto the inner 1s state. We call this term penetration energy correction, and we shall see that it is of great importance whenever problems of nonorthogonality or overlap arise for interacting systems (Section 17.5). We see from Eqn (17) that the variational energy ε4 differs from hss just by this correction term (which is always positive, i.e. repulsive). We can evaluate the necessary integrals with the formulae given in Chapter 12, and we find 8 c20 c2 Zc > >  Zc  h ¼ ; h ¼ > j j ss 0 > < 0 0 2 6 2 ! (19)  1=2 3 > pffiffiffi S 3 c 2ðc cÞ > 0 2 > > : hsj0 ¼ 3 ðc0  ZÞðc0 þ cÞ  2 c0 ; S ¼ 3 c þ c c þ c 0 0 All matrix elements can now be evaluated as functions of the variational parameter c assuming c0 ¼ Z ¼ 1. The results (atomic units) are collected in Table 13.6 and plotted against c in Figure 13.4. Table 13.6 Variational Results (Eh) for H(2s) c

hss

Epn

ε4

0.2 0.3 0.4 0.5 0.6 0.7 0.8

0.0933 0.1350 0.1733 0.2083 0.2400 0.2683 0.2933

0.0059 0.0221 0.0502 0.08915 0.1385 0.1994 0.2750

0.0874 0.1129 0.1231 0.1192 0.1015 0.0689 0.01835

492

CHAPTER 13 Variational calculations

FIGURE 13.4 Variational energy components for H(2s) plotted vs c

The variational optimization of c by the Ransil method (Problem 13.5) yields c ¼ 0.4222 (Z ¼ 1), and the energy upper bound: ε4 ¼ 0:1234Eh

(20)

which is within 98.7% of the exact value (0.125Eh). For c ¼ 0.5 (unoptimized orbital exponent in the STO, corresponding to the hydrogenic value c ¼ Z/n), the Schmidt orthogonalized STO (16) gives the fair value ε4 ¼ 0.1192Eh, which is still within 95% of the correct value. This clearly illustrates the importance of orthogonalization in establishing correct upper bounds to the variational energy of the excited states.

13.3 Non-linear parameters

(2) Excited 2p state As a variational function of the appropriate symmetry we take the normalized 2pz-STO:  5 1=2 c expðcrÞr cos q 4¼ p

493

(21)

which is now orthogonal to j0 by symmetry hj0 j4i ¼ 0 Using the formulae in Chapter 12 for the necessary integrals we find

 1 2 Z  1
εðcÞ ¼ 4  V  4 ¼ c2  Zc 2 r 2  2  dε Z Z d ε ¼1>0 ¼c ¼0 0 c¼ ; dc 2 2 dc2 c¼Z=2   1 Z2 Z2 Z2  ¼ εmin ¼ 2 8 2 4

(22)

(23) (24) (25)

as it must be. The optimization of the orbital exponent c gives in this case the exact answer (Z ¼ 1 for the H atom).

13.3.3 The 1s2 ground state of the atomic two-electron system The two-electron He-like atom (Figure 13.5) has the Hamiltonian (atomic units) ^ 2Þ ¼ h^1 þ h^2 þ 1 Hð1; r12 1 Z one-electron hydrogen-like Hamiltonian h^ ¼  V2  2 r 1 electron repulsion ðtwo-electron operatorÞ r12

FIGURE 13.5 The atomic two-electron system

(26)

494

CHAPTER 13 Variational calculations

where, for different values of the nuclear charge Z Z ¼ 1; H

2; He

3; Liþ

4; Beþ2

/

(27)

we have the isoelectronic series of the atomic two-electron system. If we could not to take into account electronic repulsion, the two-electron Schroedinger equation would be separable into two one-electron hydrogen-like equations, one for each electron, by posing 4ð1; 2Þ ¼ 41 42

(28)

^ 2Þ4ð1; 2Þ ¼ E4ð1; 2Þ Hð1;

(29)

Then

ðh^1 41 Þ42 þ 41 ðh^2 42 Þ ¼ E41 42 h^1 41 h^2 42 ¼ E ¼ ε1 41 42

separation constant

(30)

giving h^1 41 ¼ ε1 41

h^2 42 ¼ ε2 42

ε2 ¼ E  ε1

(31)

The presence of the electron repulsion term in the complete Hamiltonian Eqn (26) gives, however, a non-separable two-electron Schroedinger Eqn (29), which does not admit an exact solution. Approximations can, however, be found using the variational method, the simplest being by choosing as trial function the product (28) of two 1s STOs, one for each electron, containing just a simple variational parameter c (the orbital exponent): 4ð1; 2Þ ¼ 41 ð1Þ42 ð2Þ ¼ N expðcr1 Þ $ expðcr2 Þ

(32)

N ¼ c3 =p

(33)

where

is a normalization factor. Then 

  2
2 2 1 c 5 ^ ^ εðcÞ ¼ 41 42 h1 þ h2 þ 41 42 ¼ 2h1s1s þ 1s 1s ¼ 2  Zc þ c 2 r12 8   5 ¼ c2  2 Z  c 16

(34)

where the new two-electron repulsion integral in charge density notation (1s2j1s2) is calculated in Section 18.5. We then get as stationarity condition   dε 5 ¼ 2c  2 Z  ¼0 (35) dc 16

13.3 Non-linear parameters

495

giving cmin ¼ Z 

5 16

(36)

The optimized orbital exponent is the nuclear charge Z diminished by the quantity s ¼ 5/16, the screening constant (Zener) arising from the effect of the second electron. For the best energy we find     5 5 2 ¼ Z (37) εmin ¼ ε cmin ¼ Z  16 16 and, for Z ¼ 2 (He atom) εmin ¼ 2:84765Eh

(38)

which is within 98% of the correct value 2.90372Eh (Pekeris, 1958). The hydrogen-like approximation (without the screening effect) with c ¼ Z ¼ 2 would give εðc ¼ 2Þ ¼ 2:75Eh

(39)

which is within 95% of the correct value. This justifies the hypothesis of the screening effect suggested by Zener (1930) in proposing reasonable, but not variational, values for the screening constant s (Chapter 12). It is worth noting that the effective potential felt by electron one in presence of the second electron is Z Zeff Veff ðrÞ ¼  þ J1s ðrÞ ¼  r r

(40)

where the first term is the bare nuclear attraction and the second the electrostatic repulsive potential at r due to the second electron (Section 18.4). The effective nuclear charge felt by the electron is then given by
 (41) Zeff r ¼ ðZ  1Þ þ expð2crÞð1 þ crÞ Hence (1) r ¼ small (electron near the nucleus)
 exp 2cr z 1  2cr;

Zeff z Z

For r/0 the electron sees the whole (unscreened) nuclear charge Z. (2) r ¼ large (electron far from the nucleus)
 exp 2cr z 0; Zeff z Z  1

(42)

(43)

For r /N the electron sees the nuclear charge as it were fully screened by the other electron acting as a point-like negative charge. Therefore, Zeff depends on r, being Z near the nucleus and (Z  1) far from it. The variation theorem averages between these two cases, with a larger weight for the regions near the nucleus (hence, Zeff z 1.7 closer to 2 rather than to 1).

496

CHAPTER 13 Variational calculations

Table 13.7 Variational Errors in Energy (ε) and Ionization Potential (I) for He (Z¼2) Approximation

Percentage Error in ε

Percentage Error in I

c ¼ Z ¼ 2 (Hydrogen-like) c ¼ 1.6875 (one non-linear parameter) c1 ¼ 2.183, c2 ¼ 1.188 (two non-linear parameters)

5 2

11 6

1

3

Using different orbital exponents for different electrons and symmetrizing the resulting function 4ð1; 2Þ ¼ N½expðc1 r1 Þ $ expðc2 r2 Þ þ expðc2 r1 Þ $ expðc1 r2 Þ

(44)

Eckart (1930b) obtained a sensibly better variational value for the energy, the value for He being ε ¼ 2.87566Eh (c1 ¼ 2.18, c2 ¼ 1.19), which differs by 1% from the accurate Pekeris’ value. This is due to the ‘splitting’ of the spherical shells of the electrons, which introduces some radial correlation into the wavefunction. The physics of the problem does suggest how to improve the wavefunction. Details of the calculation are given in Problem 13.6. By increasing the flexibility of the variational wavefunction we improve the accuracy of the variational result. Comparison with experimental results is possible through the calculation of the first ionization potential I. For He 
(45) I ¼ ε Heþ  εðHeÞ ¼ 2  εðHeÞ Accurate values for the ground-state He atom are ε ¼ 2.90372Eh and I ¼ 0.90372Eh ¼ 24.6 eV. The errors resulting for different variational wavefunctions are collected in Table 13.7 for He. The greater error in I is due to the fact that the ionization potential is smaller in absolute value than the corresponding energy.

13.4 LINEAR PARAMETERS AND THE RITZ METHOD In Section 1.2.2 we introduced the variational approximation due to Ritz (Ritz, 1909; Pauling and Wilson, 1935) consisting in the linear combination of a basis of given functions, where flexibility is introduced in the wavefunction through the coefficients of the linear combination. This method is more systematic than the previous one involving non-linear parameters, and optimization of the linear coefficients yields now to secular equations whose roots are upper bounds to ground and excited states of the system. Molecular orbital (MO) and valence bond (VB) approximations are typical applications of the Ritz method in valence theory, where the variational wavefunction is expressed in terms of a given basis of AOs or VB structures, respectively. We saw in Chapter 1 that the Ritz method is intimately connected with the problem of matrix diagonalization of Chapter 6. In the next two sections we shall give some applications of the Ritz method to both atomic and molecular problems.

13.5 Atomic applications of the Ritz method

497

13.5 ATOMIC APPLICATIONS OF THE RITZ METHOD In the following, we shall apply the Ritz method to the study of the first excited states of the twoelectron (He-like) atomic system, (1) the 1s2s state of spherical symmetry, and (2) the 1s2p state.

13.5.1 The first 1s2s excited state of the atomic two-electron system The two-electron basis functions are c1 ¼ 1s1 2s2 ;

c2 ¼ 2s1 1s2

(46)

which are orthonormal if the one-electron basis (1s2s) is orthonormal, what we assume. The secular equation is then H11  ε H12 ¼0 (47) H12 H22  ε with the matrix elements (electrons in dictionary order)

1 2
 1 1 2 ^ ^ H11 ¼ 1s 2s h1 þ h2 þ 1s 2s ¼ h1s1s þ h2s2s þ 1s2 2s2 ¼ E0 þ J ¼ H22 r12 H12 ¼ H21 ¼



1 2 1 2

1 2 1 1 1 2 1s 2s h^1 þ h^2 þ 2s 1s ¼ 1s 2s 2s 1s ¼ ð1s2sj1s2sÞ ¼ K r12 r12

(48)

(49)

where E0 ¼ h1s1s þ h2s2s

(50)


 J ¼ 1s2 2s2

(51)

K ¼ ð1s2sj1s2sÞ

(52)

is the one-electron energy;

is the two-electron Coulomb integral; and

is the two-electron exchange integral, both given in charge density notation (see Chapter 18). Roots are εþ ¼ H11 þ H12 ¼ E0 þ J þ K;

ε ¼ H11  H12 ¼ E0 þ J  K

(53)

1 4 ¼ pffiffiffi ð1s2s  2s1sÞ 2

(54)

with the corresponding eigenfunctions 1 4þ ¼ pffiffiffi ð1s2s þ 2s1sÞ ; 2

which are, respectively, symmetric and antisymmetric in the electron interchange. The schematic diagram of the energy levels for the excited 1s2s state is given in Figure 13.6. Electron repulsion couples the two physically identical states 1s2s and 2s1s (through the offdiagonal term H12), removing the double degeneracy of the atomic level H11 ¼ H22 ¼ E0 þ J, and

498

CHAPTER 13 Variational calculations

FIGURE 13.6 Schematic diagram of the energy levels for the excited S(1s2s) state of the atomic two-electron system

originates two distinct levels (a doublet) whose splitting is 2K. So, the experimental measurement of the splitting gives directly the value of the exchange integral K. We notice that the (un-optimized) hydrogen-like AOs give a wrong order of the excited levels because they overestimate electron repulsion. Using as 2s the function
1=2 ðs  SkÞ S ¼ hkjsi (55) 2s ¼ 1  S2  s¼

c5s 3p

1=2 expðcs rÞr;

 3 1=2 c k¼ 0 expðc0 rÞ p

(56)

variational optimization of εþ against cs (for c0 ¼ 1.6875)1 gives cs ¼ 0:4822 εþ ¼ 2:09374Eh ;

ε ¼ 2:12168Eh

(57) (58)

with excitation energies (from the ground state) which are within 99.7% of the experimentally observed values (Moore, 1949) Dε 0:7539 Exptl: 0:7560

0:7260 0:7282

(59)

and a splitting (2K) which is within 95.5% of the experiment 2K Exptl:

0:0279 0:0292

(60)

Optimization of the orbital exponent of the orthogonalized 2s function gives not only the correct order of the atomic levels, but also results that are quantitatively satisfactory in view of simplicity of our variational approximation. Notice that the best value of cs is less than half the hydrogen-like value c ¼ Z/n ¼ 1, so that electron repulsion is strongly reduced. 1

We optimize the root having the same symmetry of the ground state.

13.5 Atomic applications of the Ritz method

499

13.5.2 The first 1s2p excited state of the atomic two-electron system Because of space degeneracy there are now six states belonging to the 1s2p configuration: 1s2pz ; 2pz 1s; 1s2px ; 1

2

2px 1s;

3

1s2py ; 2py 1s

4

5

(61)

6

giving a basis of six functions. Functions belonging to different (x, y, z) components are orthogonal and not interacting with respect to H by symmetry. Functions belonging to the same symmetry can interact through electron repulsion. All functions are also orthogonal and not interacting with respect to all S states. The (6  6) secular equation has a block-diagonal form and factorizes into three (2  2) secular equations each corresponding to a given coordinate axis (x, y, z) ˇ

H11  ε H 21

wz

H12 H22  ε

0 H33  ε

H34

H43

H44  ε

0

wx H55  ε

H56

H65

H66  ε

¼0 wy

(62)

So, it will be sufficient to consider the (2  2) secular equation: H11  ε H12 ¼0 H H22  ε 12

(63)

whose roots are spatially triply degenerate (a degeneracy that can be removed only by including spin). Matrix elements are
 H11 ¼ H22 ¼ h1s1s þ h2p2p þ 1s2 2p2 ¼ E00 þ J 0 (64) H12 ¼ H21 ¼ ð1s2pj1s2pÞ ¼ K 0

(65)

εþ ¼ H11 þ H12 ¼ E00 þ J 0 þ K 0

(66)

ε ¼ H11  H12 ¼ E00 þ J 0  K 0

(67)

with the triply degenerate roots

and the corresponding eigenfunctions 1 4þ ¼ pffiffiffi ð1s2p þ 2p1sÞ; 2

1 4 ¼ pffiffiffi ð1s2p  2p1sÞ 2

(68)

which are, respectively, symmetric and antisymmetric in the electron interchange (2p ¼ 2pz, 2px, 2py). The schematic diagram of the energy levels for the 1s2p state is qualitatively similar to that given in Figure 13.6, but now each level is triply degenerate.

500

CHAPTER 13 Variational calculations

Table 13.8 Variational Results (Atomic Units) for the S(1s2s) and P(1s2p) Excited States of the He Atom State

c

E0

J

K

εD

εL

1s2s 1s2p

0.4822 0.4761

2.34801 2.31399

0.24029 0.23667

0.01395 0.00495

2.09374 2.07237

2.12168 2.08226

The numerical values of the atomic integrals occurring in these calculations for the He atom (Z ¼ 2) are collected in Table 13.8. Using a variational function 2pz ¼ ðc5p =pÞ1=2 expðcp rÞrcos q, orthogonal by symmetry to k ¼ 1s, optimization of εþ against the non-linear parameter cp gives cp ¼ 0:4761 εþ ¼ 2:07237Eh ;

ε ¼ 2:08226 Eh

(69) (70)

with excitation energies (from the ground state) which are within 99.4% of the experimentally observed values (Moore, 1949) Dε

0:7753

0:7654

Exptl: 0:7796

0:7703

(71)

and a splitting (2K0 ) which is 6% larger than experiment 2K 0

0:0099

Exptl:

0:0093

(72)

We can notice the satisfactory agreement with the experimental results for the calculated excitation energies Dε and, also, for the splittings 2K and 2K0 , which are much more difficult to evaluate. The variational orbital exponents give AOs which are sensibly more diffuse than the hydrogenic ones. The schematic diagram (not in scale) of the energy levels for the S(1s2s) and P(1s2p) excited states of the He-like atom is qualitatively depicted in Figure 13.7. In the left part of the figure is shown the splitting occurring between the two energy levels (strongly degenerate in the absence of electron repulsion and grossly exaggerated in the figure) which is just twice the value of the exchange integrals, 2K for S(1s2s) and 20 for P(1s2p). In the latter case the roots of the secular equations are still triply degenerate, so that the sixfold degeneracy occurring in the one-electron approximation, E00 is not completely removed when admitting electron repulsion. Complete removal of the residual degeneracy would occur only by introducing electron spin into the wavefunctions, and by allowing for spin-orbit coupling and the Zeeman effect in an external magnetic field B. Electron spin doubles the number of Pauli’s allowed states, and the now 16 energy levels will have different energies only in the presence of the external magnetic field after the spin-orbit (LS) coupling. These splittings yield the so-called multiplet structure of the He-like atom which is sketched in the right part of the figure. When spin-orbit coupling is allowed in the Russell-Saunders (LS) scheme of Chapter 10, the electronic states of the atom are denoted by 2Sþ1 X J S being the total spin and J ¼ L þ S the quantum number resulting in the

13.5 Atomic applications of the Ritz method

501

FIGURE 13.7 Schematic diagram of the multiplet structure of the S(1s2s) and P(1s2p) excited energy levels of the He-like atom in the presence of a magnetic field B

502

CHAPTER 13 Variational calculations

LS-coupling scheme. The energies of the J levels belonging to the P term (L ¼ S ¼ 1, J ¼ 0, 1, 2) of the 1s2p electron configuration are then given by EJ¼2 ¼ A;

EJ¼1 ¼ A;

EJ¼2 ¼ 2A

(73)

where A is a constant characteristic of the atom. These J energy levels are shown in the diagram of Figure 13.7. Each J-level then splits into (2J þ 1) sublevels according to the values of the magnetic quantum number MJ which takes the values MJ ¼ J, (J  1),.,(J  1), J. Switching on the magnetic field, B s 0, and the Zeeman effect splits all energy levels so that the energies of the resultant 16 Pauli’s allowed states are now different. Even if we are not directly interested in the theory of atomic spectra, we want to mention here the method of coefficients of fractional parentage introduced by Racah (1942a-b, 1943, 1949; Kaplan, 1975) for the tensor operator calculation of the spectra of N-electron configurations. This method entirely bypasses the previous calculations based on diagonalization of matrices constructed from the Slater determinants of Chapter 14.

13.5.3 Results for hydrogen-like AOs Using the normalized hydrogen-like AOs 1=2
expðc0 rÞ h j0 ðc0 ¼ ZÞ 1s ¼ c30 =p pffiffiffi
3 1=2 expðcrÞð1  crÞ ¼ k  3s ð2c ¼ ZÞ 2s ¼ c =p

(74)

where k and s are STOs with identical orbital exponents c
1=2 k ¼ c3 =p expðcrÞ;


1=2 s ¼ c5 =3p expðcrÞr;

S ¼ hkjsi ¼

pffiffiffi 3=2

(75)

the resulting order of energy levels is wrong and the splittings are much too large. Table 13.9 collects the results of such calculations. We observe the inversion in the order of the levels 1s2p(3P) and 1s2s(1S), the excitation energies are too small, the splittings too large, more than three and seven times, respectively. As we have already said, this is due to the overestimation of the electronic repulsion due to the sensible overestimation of the hydrogenic orbital exponents (c ¼ 1 instead of z0.5).

Table 13.9 Variational Calculation of Ground and Excited 1s2s and 1s2p States in He Using Hydrogen-like AOs Electron Configuration State ε/Eh Dε=Eh DK=103 Eh

1s2 1

1 S 2.75 0

1s2s

1s2p

1s2s

1s2p

3

3

1

1

S 2.12414 0.6259 87.79

P 2.04854 0.7015 68.28

S 2.03635 0.7136

P 1.98026 0.7697

13.6 Molecular applications of the Ritz method

503

13.6 MOLECULAR APPLICATIONS OF THE RITZ METHOD The Ritz method will now be applied to the study of the ground and first excited states of the hydrogen molecule-ion Hþ 2.

13.6.1 The ground and first excited state of the HD 2 molecular ion The electron is at point P, where in A,B are the two protons at the fixed distance R (Figure 13.8). In atomic units, the one-electron molecular Hamiltonian of Hþ 2 will be   1 1 2 1 1 1 ^ ^ H ¼hþ ¼  V   þ (76) R 2 rA rB R As a first approximation to the variational wavefunction we choose the one-electron MO arising from the linear combination of the two 1s AOs centred at A and B, respectively: 4 ¼ ac1 þ bc2 1 a ¼ 1sA ¼ pffiffiffiffi expðrA Þ; p hajai ¼ hbjbi ¼ 1;

(77)

1 b ¼ 1sB ¼ pffiffiffiffi expðrB Þ p

(78)

hajbi ¼ hbjai ¼ S

(79)

where S is the overlap between the AOs. At variance with the atomic cases considered so far, we observe that the two AOs onto different centres are now non-orthogonal. The Ritz method for two nonorthogonal functions gives the (2  2) pseudosecular equation: Haa  ε Hab  εS (80) H  εS H  ε ¼ 0 ab bb Because of the nuclear symmetry we must have Haa ¼ Hbb ;

FIGURE 13.8 Interparticle distances in the Hþ 2 molecular ion

Hba ¼ Hab

(81)

504

CHAPTER 13 Variational calculations

so that the expansion of the secular determinant gives ðHaa  εÞ2 ¼ ðHab  εSÞ2 Haa  ε ¼ ðHab  εSÞ εþ ¼

Haa þ Hab Hba  SHaa ¼ Haa þ ; 1þS 1þS

ε ¼

Haa  Hab Hba  SHaa ¼ Haa  1S 1S

(82)

(83)

the first corresponding to the ground state, the second to the first excited state of the Hþ 2 molecular ion. We now evaluate the best value for the coefficients of the linear combination for the ground state. From the homogeneous system, we have for the first eigenvalue εþ ¼

Haa þ Hba 1þS

   8 H þ Hba Haa þ Hba > < Haa  aa c1 þ Hba  S c2 ¼ 0 1þS 1þS > : 2 c1 þ c22 þ 2c1 c2 S ¼ 1 where the last equation is the normalization condition for non-orthogonal AOs. We obtain   c2 Hba  SHaa ¼ lþ ¼ ¼1 c1 þ Hba  SHaa  c21

 2   c2 c2 1þ þ2 S ¼ 1 0 c1 ¼ c2 ¼ ð2 þ 2SÞ1=2 c1 þ c1 þ

(84)

(85)

(86)

(87)

giving the bonding MO aþb 4þ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 þ 2S In the same way, we have for the remaining root   c1 Hba  SHaa ¼ l ¼  ¼ 1 c2  Hba  SHaa c2 ¼ ð2  2SÞ1=2 ;

c1 ¼ ð2  2SÞ1=2

(88)

(89) (90)

giving the antibonding MO ba 4 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2  2S

(91)

We notice that, while 4þ has no nodes, 4 has a nodal plane occurring at the midpoint of AB (it is like a 2pz AO). We notice, too, that in both cases the LCAO coefficients are completely determined by symmetry.

13.6 Molecular applications of the Ritz method

505

We now evaluate the matrix elements of H^ over the two basis functions:



  1 2
 1 1 1 1 1 1 ^ (92) Haa ¼ a h þ a ¼ haa þ ¼ a  V   a þ ¼ EA þ a2 rB1 þ R R 2 rA rB R R where



 1 2 1 E A ¼ a  V  a 2 rA

(93)

is the energy of the isolated A atom (¼1/2Eh), and we have used the charge density notation

 1
 a  a ¼ a2 rB1 (94) rB being the attraction by the nucleus of B of the electron described by the density a2 centred on A. In the same way



  1 2 1 1 1 1 1 ^ Hba ¼ b h þ a ¼ hba þ S ¼ b  V   a þ S R R 2 rA rB R
 1 ¼ EA S þ ab rB1 þ S (95) R We therefore obtain for the ground state
 
1 
2 1  1 ab rB  S a2 rB1 Hba  SHaa ¼ EA þ a rB þ εþ ¼ Haa þ þ 1þS 1þS R

(96)

and for the first excited state


 
1 
2 1  1 ab rB  S a2 rB1 Hba  SHaa ¼ EA þ a rB þ  ε ¼ Haa  1S 1S R

(97)

13.6.2 The interaction energy and its components Even if this matter will be discussed to a greater extent in Chapter 17, we see from Eqns (96) and (97) that the interaction energy, defined as the difference between εþ (or ε) and the energy EA of the isolated H atom, can be written as

þ

þ DE 2 Sg ¼ εþ  EA ¼ DEcb þ DEexchov 2 Sg (98)
þ
þ DE 2 Su ¼ ε  EA ¼ DEcb þ DEexchov 2 Su

(99)

The interaction energy is seen to depend on the electronic states of Hþ 2 , resulting from the sum of the two components:  1
(100) DEcb ¼ a2 rB1 þ R

506

CHAPTER 13 Variational calculations

the semiclassical electrostatic energy (which is the same for the two states), and


ab  Sa2  r 1  B exchov 2 þ Sg ¼ DE 1þS
þ DEexchov 2 Su

 ab  Sa2  rB1 ¼ 1S

(101)




(102)

the quantum mechanical components arising from the exchange-overlap density (abSa2), which has the property Z 


 dr a r b r  Sa2 r ¼ S  S ¼ 0 (103) As can be seen from Eqns (101) and (102), at variance with DEcb ; DEexchov depends on the symmetry of the wavefunction and is different for the ground (attractive) and excited (repulsive) states. These components and their corrections occurring in higher orders of perturbation theory will be examined in detail in Chapter 17. The one-electron two-centre integrals occurring in the present calculation are evaluated in Chapter 18. For completeness, however, we give here their analytic form as a function of the internuclear distance R (c0 ¼ 1):   R2 (104) S ¼ ðabj1Þ ¼ expðRÞ 1 þ R þ 3


1  ab rB ¼ exp R 1 þ R

(105)


2 1  1 expð2RÞ a rB ¼  ð1 þ RÞ R R

(106)

Their values to seven significant figures are collected in Table 13.10 as a function of the internuclear distance R. Even if in molecular computations the energy is usually evaluated in terms of molecular integrals (Chapter 18), in this simple case it is possible to obtain the analytic expressions for the components of the interaction energy as a function of R as expð2RÞ ð1 þ RÞ R   

þ expðRÞ 2 DEexchov 2 Sg ¼ ð1 þ SÞ1 1  R2 R 3   expð3RÞ 4 1 1 þ 2R þ R2 þ R3  R 3 3 DEcb ¼

(107)

(108)

These expressions allow the direct calculation of the interaction energy in Hþ 2 , avoiding the ‘roundoff errors’ which increase with increasing R (see in Table 13.10 the difference between R1 and ða2 jrB1 Þ; which differ by charge-overlap terms).

13.6 Molecular applications of the Ritz method

507

Table 13.10 Numerical Values of the Two-Centre Integrals (c0 ¼1) Occurring in the Hþ 2 Calculation as a Function of R (Energy Integrals in Eh) R/a0

RL1

S

ða2 jrBL1 Þ

ðabjrBL1 Þ

1.0 1.2 1.4 1.6 1.8 2.0 2.5 3.0 3.5 4.0 4.5 5.0 6.0 7.0 8.0

1 8.333333  101 7.142857  101 6.250  101 5.555556  101 5.0  101 4.0  101 3.333333  101 2.857143  101 2.5  101 2.222222  101 2.0  101 1.666667  101 1.428571  101 1.25  101

8.583854  101 8.072005  101 7.529427  101 6.972160  101 6.413597  101 5.864529  101 4.583079  101 3.485095  101 2.591942  101 1.892616  101 1.360852  101 9.657724  102 4.709629  102 2.218913  102 1.017570  102

7.293294  101 6.670171  101 6.100399  101 5.587614  101 5.130520  101 4.725265  101 3.905669  101 3.300283  101 2.845419  101 2.495807  101 2.220714  101 1.999455  101 1.666595  101 1.428562  101 1.249999  101

7.357589  101 6.626273  101 5.918327  101 5.249309  101 4.628369  101 4.060059  101 2.872975  101 1.991483  101 1.358882  101 9.157819  102 6.109948  102 4.042768  102 1.735127  102 7.295056  103 3.019164  103

Table 13.11 gives the calculated interaction energy and its components (103Eh) for the ground2 þ state H2 þ ð2 Sþ g Þ, while Table 13.12 gives the same quantities for the excited state ð Su Þ. Figure 13.9 gives the plots vs R of the interaction energies DEð103 Eh Þ for the two electronic states. Comments to Tables 13.11 and 13.12 are as follows. (1) 2 Sþ g ground state While DEcb is always repulsive (classically it is not possible to form any chemical bond between H and Hþ), the quantum mechanical component DEexchov ð2 Sþ g Þ is always attractive (going to zero as R/0) and appears as the main factor determining the formation of the one-electron bond exchov 2 þ ð Sg Þ has its minimum near R ¼ 2a0, the distance of the chemical bond. The in Hþ 2 . DE minimum in the potential energy curve for ground-state Hþ 2 occurs near R ¼ 2.5a0, i.e. for a value of R which is about 25% larger than the experimental value (2a0). The true minimum of the simple MO wavefunction of Table 13.11 occurs at Re ¼ 2:493a0 and is DE ¼ 64:84  103 Eh . Even if the simple MO wavefunction of Table 13.11 gives a qualitatively correct behaviour of the potential energy curve for ground state Hþ 2 the quantitative error is still very large, the calculated bond energy at R ¼ 2a0 being no more than 52% of the correct value. (2) 2 Sþ u excited state The components of DE are now both repulsive (Table 13.12) and, to this level of approximation (rigid AOs), the excited 2 Sþ u state is repulsive (a scattering state). The polarization of the H atom þ by the proton H (see Chapter 17) yields at large distances (R ¼ 12.5a0) a weak Van der Waals

508

CHAPTER 13 Variational calculations

þ

þ

Table 13.11 2 Sg Ground State of H2 . Interaction Energy and its Components (103Eh) for the Simple MO Wavefunction (c0 ¼1) D

R/a0

DEcb

DEexchLov ð2 SD g Þ

DEð2 Sg Þ

Accuratea

1.0 1.2 1.4 1.6 1.8 2.0 2.5 3.0 3.5 4.0 4.5 5.0 6.0 7.0 8.0

270.67 166.31 104.25 66.239 42.504 27.473 9.4331 3.3050 1.1724 4.193  101 1.508  101 5.448  102 7.168  103 9.503  104 1.266  104

59.037 68.731 75.591 79.750 81.509 81.245 74.2625 62.388 49.346 37.285 27.180 19.258 9.075 4.036 1.730

211.634 97.579 28.659 13.511 39.005 53.772 64.829 59.083 48.174 36.866 27.029 19.204 9.068 4.035 1.729

48.21

102.63 93.82 77.56 60.85 46.08 33.94 24.42 11.97 5.59 2.57

a

Peek, 1965.

3 þ Table 13.12 2 Sþ u Excited State of H2 . Interaction Energy and its Components (10 Eh) for the Simple MO Wavefunction (c0 ¼1) þ

þ

R/a0

DE exchLov ð2 Su Þ

DEð2 Su Þ

Accuratea

1.0 1.2 1.4 1.6 1.8 2.0 2.5 3.0 3.5 4.0 4.5 5.0 6.0 7.0 8.0

774.73 644.25 536.34 447.03 373.04 311.67 199.92 129.14 83.877 54.693 35.743 23.375 9.972 4.219 1.765

1045.4 810.56 640.59 513.27 415.54 339.15 209.36 132.44 85.049 55.112 35.894 23.430 9.979 4.220 1.765

935.19

a

Peek, 1965.

332.47 207.93 131.91 84.50 54.45 35.17 22.71 9.36 3.73 1.40

13.6 Molecular applications of the Ritz method

509

FIGURE 13.9 Plot of the MO interaction energies DE vs R for the two states of Hþ 2 (c0 ¼1). The points are the accurate results by Peek (1965)

bond with DE ¼ 60:8  106 Eh . It is interesting to note (Figure 13.9) that the simple MO results for the excited state are in better agreement with the accurate Peek’s values than those of the ground state. Systematically improving the quality of the basic AOs gives the improved results collected in Table 13.13. þ Table 13.13 2 Sþ g of Ground-State H2 . Effect of Improving the Basic AO a in the MO Wavefunction on the Bond Energy DE at R¼2a0, and Residual Error with Respect to the Accurate Value

a 1. 2. 3. 4. 5. a

1sA fexpðrA Þ ðc0 ¼ 1Þ 1sA fexpðc0 rA Þ ðc0 ¼ 1:2387Þ 1sA þ l2psA ðc0 ¼ 1:2458; cp ¼ 1:4824; l ¼ 0:1380Þ expðarA  brB Þ ða ¼ 1:1365; b ¼ 0:2174Þ Accurate

Pauling, 1928. Finkelstein and Horowitz, 1928. c Dickinson, 1933. d Guillemin Jr. and Zener, 1929 (GZ). e Peek, 1965. b

Basic AO

DE=10L3 Eh

Residual Error/10L3Eh

H 1sa STO 1sb (optimized) Dipole polarized AO (sp-hybrid)c two-centre GZ AOd

53.77 86.51 100.36

48.86 16.12 2.27

102.44

0.19

e

102.63

0

510

CHAPTER 13 Variational calculations

From the table it follows that we can make the following comments on the nature of the basic AOs. (1) The undistorted H atom AOs (c0 ¼ 1) do not allow finding the correct bond length (as already said, the bond is 25% too long, since the bond energy is insufficient). (2) Optimization of the orbital exponent c0 in the 1s STO functions allows us to introduce a great part of the spherical distortion (polarization) of the H atom by the proton Hþ: the bond length Re is now correct and the bond energy DE improved by over 30%. (3) Dipole distortion of 1s (sp hybridization) improves DE by a further 13%, giving a result within 2% of the accurate value (Peek, 1965). (4) Use of a two-centre AO (GZ, compare the Eckart split-shell for He) allows us to reproduce nearly entirely the polarization of the H atom by the proton Hþ, with a result which is now within 0.2% of the accurate value. The two-centre GZ orbital is tantamount to including in the basis functions onto A all polarization functions with l ¼ 0,1,2,3,4,. (spherical, dipole, quadrupole, octupole, hexadecapole,...). GZ is the simplest variational function containing two optimizable non-linear parameters.

13.7 VARIATIONAL PRINCIPLES IN SECOND ORDER We saw in Section 1.3.2.4 that the Ritz method can be applied as well to find variational approximations to the second-order energy in the context of the Hylleraas’ variational method. It was shown there that best second-order approximations to energy and perturbed wavefunctions can be obtained in terms of linear pseudostates. The examples of the dipole polarizability of the hydrogen atom and the C6 dispersion coefficient for the long-range interaction between two ground-state hydrogen atoms will show the power of the pseudostate method, even at its simplest level, the two-term approximation. Further applications will be given later in Chapter 17, where the relation between second-order properties and second-order energies will be discussed in the context of the study of the interaction between molecules.

13.7.1 The dipole polarizability of the H atom The static dipole polarizability a of the hydrogen atom in terms of pseudostate contributions is given as X a¼ ai (109) i

and the C6 dispersion coefficient for atoms by the so-called London formula (Eqn (95) of Chapter 17) εi ε j 1 XX ai aj (110) C6 ¼ 6C11 ¼ 6 $ εi þ εj 4 i j In the latter equation, besides the pseudostate components of the static dipole polarizabilities (ai) of atom A and (aj) of atom B, we must know also the respective excitation energies from the ground state, εi and εj. This originates what is known as the N-term dipole pseudospectrum of the atom (Magnasco, 2007): fai ; εi g

i ¼ 1; 2; /; N

(111)

Here we limit ourselves to the simple two-term approximation (N ¼ 2). For a hydrogen atom in a uniform electric field of strength F directed along z, in spherical coordinates, the first-order perturbation will be (112) H^1 ¼ Fz ¼ Fr cos q

13.7 Variational principles in second order

511

Now, let the first-order variational function be described by ~ 1 ¼ Fc0 C j

(113)

where the two-term dipole polarized basis c0 is chosen as c0 ¼ ðc01 c02 Þ with

c01

¼ zj0 ;

c02

rffiffiffiffiffi 2 zrj0 ; ¼ 15

(114) 1 j0 ¼ pffiffiffiffi expðrÞ p

(115)

Functions (115) are normalized and orthogonal to j0 by symmetry, but are not orthogonal in themselves, having the non-orthogonality integral rffiffiffi  0 0 5 S ¼ c1 c2 ¼ (116) 6 Schmidt orthogonalization then gives the orthonormal set: c ¼ ðc1 c2 Þ c1 ¼ c01 ;

c2 ¼

(117)

pffiffiffi 0 pffiffiffi 0 6 c2  5 c1

(118)

Next we construct the two fundamental matrices M and N, Eqns (204) and (205) of Chapter 1: pffiffiffi ! pffiffiffi !   9 5 1 1 1 5  5 1 pffiffiffi M¼ (119) ; N ¼ F ; M ¼ pffiffiffi 10  5 4 0 9 5 5 giving for E~2 9 E~2 ¼ Ny M1 N ¼  F 2 4

(120)

! d2 E~2 9 a¼ ¼ ¼ 4:5 dF 2 2

and for the dipole polarizability

(121)

which is the exact value ða30 Þ of the static polarizability of the ground-state hydrogen atom. The two pseudostates are obtained by diagonalizing matrix M, what can be done by solving the (2  2) secular equation: jM  ε1j ¼ 0 0 5ε2  7ε þ 2 ¼ 0 which has the real roots ε1 ¼ 1; and the corresponding eigenvectors 1 C1 ¼ pffiffiffi 6



 1 pffiffiffi ;  5

ε2 ¼

2 5

1 C2 ¼ pffiffiffi 6

(122)

(123) pffiffiffi ! 5 1

(124)

512

CHAPTER 13 Variational calculations

The two-term dipole pseudostates are then pffiffiffi 8 5 1 > > p ffiffi ffi pffiffiffi c2 ; > ¼  c j 1 > < 1 6 6 pffiffiffi > > 5 1 > > : j2 ¼ pffiffiffi c1 þ pffiffiffi c2 ; 6 6

ε1 ¼ 1 (125) 2 ε2 ¼ 5

and E~2 is given as the two-term sum-over-pseudostates

0 11 ! 1 pffiffiffi  1 0 pffiffiffi 5 @ 2A 0 5 5 1 !   0 1 25 9 2 1 A þ ¼  F2 pffiffiffi ¼ F 5 6 12 4 5 2

1
E~2 ðbestÞ ¼ Nj y ε1 Nj ¼ F 2 1 6 ¼ F 2

1
1 6

0

pffiffiffi  5 @

1 0

(126)

giving the dipole polarizability as the sum of the two pseudostate contributions a¼

2 25 27 9 þ ¼ ¼ ¼ 4:5 6 6 6 2

(127)

as it must be.

13.7.2 The London attraction between two ground-state H atoms The advantage of relations (127) and (123) is that they give the two-term dipole pseudospectrum of the ground-state hydrogen atom as shown in Table 13.14. Using the values from this table in Eqn (110), our two-term approximation for the London C6 dispersion coefficient of the H–H interaction gives   1 ε2 ε1 ε2 ε22 121 363 ¼ 6$ a12 1 þ 2a1 a2 ¼ ¼ 6:482 (128) þ a22 C6 ¼ 6 $ 2ε1 ε1 þ ε2 2ε2 4 112 56 a value which is slightly better than the C6 ¼ 6:480 Eh a60 obtained by Eisenschitz and London (1930) in their standard RS perturbative calculation using the complete set of eigenstates of H^0 following early work by Sugiura (1927). Adding to Eqn (114) a third function fz r 2 j0 , the three-term pseudospectrum gives a finer subdivision in pseudostates yielding always the exact polarizability for the hydrogen atom while improving the value of the C6 coefficient to 6.4984, which now differs by less than 0.01% from the Table 13.14 Two-term Dipole Pseudospectrum for the Ground-State Hydrogen Atom P

i

ai =a30

εi =Eh

1

2 6 25 6

1

2

2 5

i

ai =a30

9 ¼ 4:5 2

13.7 Variational principles in second order

513

Table 13.15 EisenschitzeLondon Eigenstate Contributions ðEh a06 Þ to the C6 Dispersion Coefficient for the HeH Interaction P

DiscreteeDiscrete

3.948

dE 00

DiscreteeContinuous

1.104

P

ContinuouseDiscrete

1.104

ContinuouseContinuous

0.324

Total

6.480

n0 n00

PR n0

R R

dE 0

n00

R dE 0 dE 00

accurate value 6.49902 which is reached for N ¼ 5. It must be admitted that, at least for this simple one-electron atomic system,2 the convergence of the sum-over-pseudostates is really extraordinary. We give in Table 13.15 the partial components of the sum-over-eigenstates calculation of Eisenschitz and London (1930) as quoted by Lennard-Jones (1930). In a modified form of Rayleigh–Schroedinger’s perturbation theory published at about the same time, Lennard-Jones (1930) gave a new derivation of the Eisenschitz–London result with a different partitioning of the components: 8 > Integral ¼ 6 > > P > >  < New Final term ¼ 0:612 n0 n00 Remainder ¼ 0:132 > > > > > : Total ¼ 6:480 (129) While Eisenschitz and London calculations involve series of slow convergence which can only be evaluated with great labour, the new derivation shows indeed faster convergence, but the difficulty related to the need of calculating both discrete and continuous contributions from the eigenstate spectrum is left intact. We notice that the largest initial value calculated by Lennard-Jones with the new method corresponds to taking just the first term (N ¼ 1) in our pseudospectral expansion. This shows without further comments the superiority of the expansion in pseudostates compared to the expansion in eigenstates. It is rather astonishing that consideration of just the further function fz r j0 in the basis set (114) in the context of the Ritz variational method for E~2 yields a result already better than the laborious and extraordinarily difficult classical eigenstate calculations by Eisenschitz–London and Lennard-Jones.

The same is true for the simplest one-electron molecular system (Magnasco and Ottonelli, 1999c), the Hþ 2 hydrogen molecular ion, while convergence is sensibly slower for the two-electron atomic (He) and molecular (H2) systems (Magnasco and Ottonelli, 1999a,b) either for dipole polarizabilities or C6 dispersion coefficients, which require in this case sensibly longer expansions. An efficient technique for obtaining accurate reduced pseudospectra from these longer expansions was recently derived in our laboratory (Magnasco and Figari, 2009) and some results are shown in Sections 17. 3.7 and 17.4.2. 2

514

CHAPTER 13 Variational calculations

13.8 PROBLEMS 13 13.1. Find a simple variational approximation to the ground state of the particle in a box of side a with impenetrable walls. Answer: 4 ¼ N xða  xÞ;

ε4 ¼

5 a2

Hint: Use a trial function satisfying the boundary conditions. 13.2. Find a variational approximation to the first excited state of the particle in a box of side a with impenetrable walls. Answer: 

a 21 4 ¼ Nxða  xÞ  x ; ε4 ¼ 2 2 a Hint: Follow the same suggestions of Problem 13.1. 13.3. Find a simple variational approximation to the ground state of the one-dimensional harmonic oscillator. Answer: 
c k 4 ¼ N exp cx2 ; εðcÞ ¼ þ c1 2 8 where k is the force constant. Optimization of the orbital exponent c gives as best variational values cmin ¼

1 pffiffiffi k; 2

εmin ¼

1 pffiffiffi k 2

Hint: Evaluate all necessary integrals over the Gaussian functions using the general formulae given in Section 12.4. 13.4. Find a variational approximation to the first excited state of the one-dimensional harmonic oscillator. Answer: 
c 3 4 ¼ Nx exp cx2 ; hj0 j4i ¼ 0; εðcÞ ¼ c þ kc1 2 8 Optimization of the orbital exponent c gives as best variational values cmin ¼

1 pffiffiffi k; 2

Hint: Follow the suggestions of Problem 13.3.

εmin ¼

3 pffiffiffi k 2

13.8 Problems 13

515

13.5. Find a simple method allowing minimization of the energy as a function of a single non-linear variable parameter c. Answer: The Ransil method. Hint: Expand the function ε(c) in powers of c according to Taylor. 13.6. Perform Eckart’s calculation for He(1s2) and find the best variational values for energy and orbital exponents. Answer: ε ¼ 2:875 661Eh ; c1 ¼ 2:183 171; c2 ¼ 1:188 531 Hint: c1 þ c2 and Evaluate the necessary matrix elements and integrals, transform to variables x ¼ 2 3=2 3 S ¼ x ðc1 c2 Þ , find the relation between x and S, then optimize ε against the single parameter S using Ransil’s method. 13.7. Find the dipole polarizability a of H(1s) from the expectation value of the induced dipole moment. Answer: 9 a¼ 2 Hint: Evaluate hjjzjji using j ¼ j0 þ j1 and expand the result to first order in the electric field F. ~ 1 ¼ CFc, where c is a normalized 13.8. Evaluate a for H(1s) using the single linear pseudostate j 2pz STO. Answer: m2 a ¼ 2 ; m ¼ hcjzjj0 i; ε ¼ hcjH^0  E0 jci ε Hint: ~ 1 into the Hylleraas functional E~2 and optimize the linear Introduce the variational function j coefficient C. 13.9. Evaluate the transition moment m and the excitation energy ε of Problem 13.8. Answer: !5  2c1=2 1
; ε ¼ c2  c þ 1 m¼ cþ1 2 Hint: Evaluate the integrals in spherical coordinates. 13.10. Optimize the single pseudostate result of Problem 13.8 with respect to the non-linear parameter c (the orbital exponent). Answer: The required value of c (¼0.7970) is the real root of the cubic equation: 7c3  9c2 þ 9c  5 ¼ 0 Hint: Find the stationarity condition for E~2 ðcÞ using the logarithmic derivatives of m and ε.

516

CHAPTER 13 Variational calculations

13.11. Find normalization factor, non-orthogonality integral, transition moment and element of the excitation energy matrix for the n-th power STO dipole function c0n ¼ Nn0 z r n1 j0 with c ¼ 1. Answer: Nn0

 ¼

3 $ 22nþ1 ð2n þ 2Þ!

m0n ¼ Nn0

1=2 ;

ðn þ 3Þ! ; 3 $ 2nþ2

Snm ¼ Nn0 Nm0

0 Mnm ¼ Nn0 Nm0

ðn þ m þ 2Þ! 3 $ 2nþmþ1

ðn þ mÞ! ð2 þ nmÞ 3 $ 2nþm

Hint: Use the basic integrals (p,q  0 integers):  2 p  ðp þ 4Þ! j0 z r j 0 ¼ 3 $ 2pþ3 ðp þ q þ 2Þ! hj0 z r p jH^0 jj0 z r q i ¼  ½pðp  q þ 3Þ þ qðq  p þ 3Þ 3 $ 2pþqþ4 13.12. Find the single normalized pseudostate equivalent of the exact j1 for H(1s). 13.13. Evaluate a for H(1s) using the three normalized STO functions 2pz, 3pz and 4pz (c ¼ 1), and construct the N ¼ 3 dipole pseudospectrum (atomic units). Answer: a ¼ a1 þ a2 þ a3 ¼

9 2

i

ai =a30

εi =Eh

1 2 3

3.488744  100 9.680101  101 4.324577  102

3.810911  101 6.165762  101 1.702333  100

Hint: Follow the technique suggested in Section 13.7.1. 13.14. Compare different pseudostate variational calculations of the dipole polarizability for the ground state of the H atom with the results obtained by the perturbation expansion in eigenstates of H^0 . Answer: Compare the results given in Table 13.16 with those in Table 13.17. Hint: Evaluate a for the H atom using the pseudostate techniques of Section 1.3.2.4. 13.15. Construct approximate four-term and two-term pseudospectra (atomic units) for the dipole polarizabilities of ground-state H2 ð1 Sþ g Þ at R ¼ 1.4a0.

13.9 Solved problems

517

Answer: ak k

at k

i

ai =a30

εi =Eh

at i =a0

εt i =Eh

1 2 3 4 P

4.567 1.481 0.319 0.011 6.378

0.473 0.645 0.973 1.701

2.852 1.350 0.335 0.022 4.559

0.494 0.699 1.157 2.207

i ai

Hint: Select, respectively, the four and two most important contributions out of the 34-term 1 pseudospectra of 1 Sþ u and Pu symmetry obtained by Magnasco and Ottonelli (1996) using as j0 the Ko1os-Wolniewicz 54-term wavefunction for ground-state H2.

13.9 SOLVED PROBLEMS 13.1 Variational approximation to the ground state of the particle in a box of side a. We see that the simple function 

4 ¼ Nx a  x ¼ N ax  x2 is a correct variational wavefunction not containing any adjustable parameter but satisfying the boundary conditions of the problem at x ¼ 0 and x ¼ a (4 ¼ 0). Normalization factor  1=2 Za
2 a5 30 h4j4i ¼ N 2 dx ax  x2 ¼ N 2 ¼ 1 0 N 5 30 a 0

Evaluation of the derivatives d4 d2 4 ¼ Nða  2xÞ; ¼ 2N dx dx2  2

Za d 4
 10 2 4 2 ¼ 2N dx ax  x2 ¼  2 dx a 0

thereby giving ε4 ¼ hTi4 ¼



 1 d 2 4 4 ¼ 5 4  2 dx2 a2

Since the exact value for the ground state is (Chapter 3) p2 1 E1 ¼ 2 ¼ 4:934 802 2 2a a the variational result exceeds the correct value by 1.32%.

518

CHAPTER 13 Variational calculations

13.2 Variational approximation to the first excited state of the particle in a box of side a. In this case, we must construct a variational function satisfying the boundary conditions at x ¼ 0, x ¼ a/2 and x ¼ a (4 ¼ 0) and orthogonal to the ground-state function. A trial function satisfying all these conditions is  N


a  a2 x  3ax2 þ 2x3 4 ¼ Nxða  xÞ  x ¼ 2 2 It can be easily shown that such a function is orthogonal to either 41 or to the true ground state j1. This is easily shown for 41, since Za 
  2 2 2 3 h41 j4if ax  x a x  3ax þ 2x ¼ dx a3 x2  4a2 x3 þ 5ax4  2x5 0

¼a

 6

1 2 1þ1 3 6

 ¼0

while a little longer calculation shows that the chosen variational wavefunction for the first excited state is orthogonal to the true ground-state wavefunction j1 f sin ax with a ¼ p/a for n ¼ 1. The skilled students may do this calculation by themselves using the general integral (Gradshteyn and Ryzhik, 1980, p. 183):   nk Z n

X p n x dx xn sin ax ¼  k! cos ax þ k k akþ1 2 k¼0 Turning to our starting function 4, we now evaluate all necessary integrals. Normalization factor N2 h4j4i ¼ 4

Za Za

  N2 2 2 2 2 dx x a  3ax þ 2x ¼ dx a4 x2  6a3 x3 þ 13a2 x4  12ax5 þ 4x6 4 0

¼ N2

0

    a7 1 3 13 4 a7 840 1=2 ¼1 0 N¼  þ 2þ ¼ N2 4 3 2 5 840 7 a7

Evaluation of the derivatives  d4 N
2 ¼ a  6ax þ 6x2 ; dx 2

d2 4 ¼ 3Nða þ 2xÞ dx2

 2

Za d 4
 3 2 42 4 2 ¼ N dx a3 x þ 5a2 x2  8ax3 þ 4x4 ¼  2 dx 2 a 0

thereby giving



 1 d2 4 4 ¼ 21 ε4 ¼ hTi4 ¼ 4  2 2 dx a2

13.9 Solved problems

519

Since the exact value for the first excited state is E2 ¼ 4

p2 2p2 1 ¼ 2 ¼ 19:739 209 2 2 2a a a

the variational result exceeds the exact value by 6.4%. So, the approximation for the excited state is worst than that for the ground state, but it must not be forgotten that 4 has no variational parameters in it. The following two problems illustrate the equality sign in Eqns (2) and (3) of Section 13.2.1 in the case of the first two states of the harmonic oscillator, which was treated by us in Section 3.6. We summarize here the main exact results for comparison. pffiffiffi b a ¼ ; 2a ¼ b ¼ k k ¼ force constant 2   1=4   1=4
 2a b b 2 1 1 pffiffiffi 2 j0 ðxÞ ¼ exp ax ¼ exp  x ; E0 ¼ b ¼ k p p 2 2 2 rffiffiffiffi!1=2 rffiffiffiffiffi!1=2  
 2a b b 2 3 3 pffiffiffi 2 xexp ax ¼ 2b xexp  x ; E1 ¼ b ¼ k j1 ðxÞ ¼ 4a p p 2 2 2 13.3 Variational calculation for the ground state of the harmonic oscillator in one dimension. As a convenient variational function, satisfying all regularity conditions, we choose the Gaussian 
4 ¼ N exp cx2 where c > 0 is a non-linear variational parameter. Using the general integral over Gaussian functions given in Section 12.4, the following results are easily obtained. Normalization factor ZN ZN

  2 2 2 dx exp 2cx ¼ 2N dx exp 2cx2 h4j4i ¼ N N

¼ N2

p 1=2 2c

0

 1=4 2c ¼1 0 N¼ p

Average kinetic energy
 d4 ¼ N exp cx2 ð2cxÞ; dx  2  4 x 4 ¼ N 2


d2 4 ¼ 2c 1  2cx2 4 2 dx

ZN ZN  

2 2 2 dx x exp 2cx ¼ 2N dx x2 exp 2cx2 N

¼ 2N 2

1 p 1=2 8c 2c

0

 1=2 2c 1 p 1=2 1 ¼ 2 $ ¼ p 8c 2c 4c

520

CHAPTER 13 Variational calculations

 2

 d 4  4 2 ¼ 2c 4 1  2cx2 4 dx  
 2  1 ¼ c ¼ 2c 1  2c 4 x 4 ¼ 2c 1  2c 4c

 1 d 2 4 4 ¼ c hTi ¼ 4  2 dx2 2 Average potential energy

 2

kx k 1 k hVi ¼ 4 4 ¼ $ ¼ 2 2 4c 8c

Total energy c k εðcÞ ¼ hTi þ hVi ¼ þ c1 2 8

pffiffiffi k dε 1 k 2 k 2 ¼  c ¼ 0 0 c ¼ 0 cmin ¼ 2 dc 2 8 4  2  d2 ε k 3 d ε 2 ¼ c 0 ¼ pffiffiffi > 0 p ffiffi 2 2 dc 4 dc c¼ k=2 k

ensuring the existence of a true minimum. Therefore pffiffiffi pffiffiffi k k 2 k 1 pffiffiffi þ pffiffiffi ¼ 2 ¼ k ¼ E0 εmin ¼ 4 4 8 k 2 4best

pffiffiffi !  1=4   k 2 b b x ¼ ¼ N exp  exp  x2 ¼ j0 2 p 2

which are, respectively, the exact energy and wavefunction for the ground state. So, in this case, the equality sign holds in the variation principle for the ground state. 13.4 Variational calculation for the first excited state of the harmonic oscillator in one dimension. We choose the function 
4 ¼ Nx exp cx2 where c > 0 is the non-linear variation parameter. This function satisfies the regularity conditions and is orthogonal to j0 by symmetry hj0 j4ifheven functionjodd functioni ¼ 0 as can be easily verified. In fact, consider the integral

13.9 Solved problems

ZN Z0 ZN


   2 2 dx x exp 2cx ¼ dx x exp 2cx þ dx x exp 2cx2 N

N

0

In the first integral of the right-hand side, change x/x obtaining Z0


 dx x exp 2cx2 ¼

N

Z0


 dx x exp 2cx2 ¼ 

N

ZN
 dx x exp 2cx2 0

so that ZN ZN ZN


   2 2 dx x exp 2cx ¼  dx x exp 2cx þ dx x exp 2cx2 ¼ 0 N

0

0

We then easily obtain the following results. Normalization factor h4j4i ¼

N2

ZN ZN

  2 2 2 dx x exp 2cx ¼ 2N dx x2 exp 2cx2 N

0

1 p 1=2 1 p 1=2 ¼ 2N 2 ¼ N2 ¼1 0 N¼ 8c 2c 4c 2c

rffiffiffiffiffi!1=2 2c 4c p

Average kinetic energy

 d4 ¼ N exp cx2 1  2cx2 ; dx  2  4 x 4 ¼ 2N 2


 d2 4 ¼ 2c 3  2cx2 4 2 dx

ZN 
dx x4 exp 2cx2 0

¼ 2N 2

 1=2 3 p 1=2 2c 3 p 1=2 3 ¼ 4c $ ¼ 5 2 p 16c2 2c 4c 2 c 2c

 2

d 4   4 2 ¼ 2c 4 3  2cx2 4 dx    2  3 ¼ 2c 3  2c 4 x 4 ¼ 2c 3  2c $ ¼ 3c 4c


521

522

CHAPTER 13 Variational calculations

hTi ¼



 1 d2 4 4 ¼ 3 c 4  2 dx2 2

Average potential energy  2

kx k 3 k hVi ¼ 4 4 ¼ $ ¼ 3 2 2 4c 8c Total energy εðcÞ ¼ hTi þ hVi ¼

3 3 c þ k c1 2 8

pffiffiffi k dε 3 3 k 2 2 ¼  k c ¼ 0 0 c ¼ 0 cmin ¼ 2 dc 2 8 4 d2 ε 3 ¼ k c3 0 dc2 4



d2 ε dc2

 pffiffi c¼ k=2

6 ¼ pffiffiffi > 0 k

ensuring the existence of a true minimum. Therefore pffiffiffi 3 k 3 2 3 pffiffiffi 3 pffiffiffi þ k pffiffiffi ¼ 2 εmin ¼ k¼ k ¼ E1 2 2 8 4 2 k 4best

pffiffiffi ! k 2 x ¼ ¼ Nx exp  2

rffiffiffiffi!1=2   b b 2 x exp  x ¼ j1 2b p 2

so that the best variational results coincide with the exact results for energy and wavefunction of the first excited state. In this case, again, the equality sign holds in the variation principle for the first excited state. 13.5 The Ransil method. Ransil (1960) suggested an elementary method that allows for the three-point numerical minimization of a function having a parabolic behaviour near the minimum. By restricting ourselves to a function of a single parameter c, we expand ε(c) around c0 in the Taylor power series     dε 1 d2 ε Dc þ ðDcÞ2 þ/ εðcÞ ¼ εðc0 Þ þ dc c0 2 dc2 c0 where Dc ¼ c  c0

13.9 Solved problems

523

with c0 a convenient starting point (better near to the minimum). At the minimum point we must have  2    dε dε d ε þ Dc ¼ 0 ¼ dDc dc c0 dc2 c0 so that the first correction will be   dε dc c0 Dc ¼  2  d ε dc2 c0 By choosing as a new starting point c0 þ Dc, the process can be iterated until a predetermined threshold is reached. This usually happens provided the iteration process converges if the starting point c0 is well chosen. The first and second derivatives can be evaluated numerically starting from the definitions   dε εðc0 þ DcÞ  εðc0  DcÞ ¼ dc c0 2Dc 

d2 ε dc2

 ¼

εðc0 þ DcÞ þ εðc0  DcÞ  2εðc0 Þ ðDcÞ2

c0

As a starting increment it is convenient to choose a small value for Dc, say Dc ¼ 0.01. For convenience, the second derivative is evaluated from the incremental ratio of the first derivatives Dc Dc taken in c0 þ and c0  ; respectively: 2 2     Dc Dc Dc Dc   þ  ε c0 þ  ε c0 þ dε εðc0 þ DcÞ  εðc0 Þ 2 2 2 2 ¼ ¼ Dc dc c0 þ Dc Dc 2$ 2 2 Similarly

  dε εðc0 Þ  εðc0  DcÞ ¼ dc c0  Dc Dc 2

so that 

d2 ε dc2

 ¼ c0

    dε dε  dc c0 þ Dc dc c0  Dc 2

2

Dc

¼

εðc0 þ DcÞ þ εðc0  DcÞ  2εðc0 Þ

which is the result given in the previous equation.

ðDcÞ2

524

CHAPTER 13 Variational calculations

13.6 Eckart’s calculation on He. Eckart’ split-shell approach to the ground state of the He-like atom was revisited by Figari (1991), who introduced a variable transformation allowing optimization of the variational energy with respect to just the non-orthogonality integral S between 41 and 42, obtaining in this way a highly accurate result. Let  3 1=2  3 1=2 c c expðc1 rÞ; 42 ¼ 2 expðc2 rÞ 41 ¼ 1 p p be the normalized 1s STOs having the non-orthogonality integral S ¼ h41 j42 i ¼

2ðc1 c2 Þ1=2 c1 þ c2

!3

The symmetrized two-electron trial function containing two non-linear variational parameters (an atomic ‘split-shell’ function) in its normalized form is
1=2 4ð1; 2Þ ¼ N½41 ð1Þ42 ð2Þ þ 42 ð1Þ41 ð2Þ; N ¼ 2 þ 2S2 and is reminiscent of the Heitler–London wavefunction for ground-state H2, the corresponding molecular ‘split-shell’ function. After integration, the energy functional ε[4] becomes a function of c1 and c2:  1 ε½4 ¼ 4 h^1 þ h^2 þ r

12


2 2

h þ h þ 2Sh þ 41 42 þ ð41 42 j41 42 Þ 11 22 12 4 ¼ ¼ εðc1 ; c2 Þ 1 þ S2

since all matrix elements depend only on c1 and c2 c22  Zc2 ; 2 " #
2 2 c c c c 1 2 1 2 41 42 ¼ 1þ ; c1 þ c2 ðc1 þ c2 Þ2 h11 ¼

c21  Zc1 ; 2

h22 ¼

S h12 ¼ ½c1 c2  Zðc1 þ c2 Þ 2 ð41 42 j41 42 Þ ¼ 20

ðc1 c2 Þ3 ðc1 þ c2 Þ5

Transforming from variables c1, c2 to the new variables x, S defined as x¼ the variational energy becomes

c1 þ c2 ; 2

S ¼ x3 ðc1 c2 Þ3=2

 
 2 1 2=3 1 4=3 5 2 2=3 2 1S S þ S þ S x x þ 2 8 8 εðx; SÞ ¼ 2Zx þ S2=3 x2 þ 2 1þS

13.9 Solved problems

525

The stationarity condition of ε against x


  4 1 vε ¼ 2Z þ 2S2=3 x þ vx S gives Z x¼

S2=3





1 2=3 1 4=3 5 2 S þ S þ S xþ 2 8 8 1 þ S2

 ¼0

  1 2=3 1 4=3 5 2 S  S þ Z S 4 16 16 2  S2=3 þ S8=3

a relation which allows us to optimize ε with respect to the single variable S. For any given value of S in the range 0  S  1, the best x is then obtained from the relation above. For S ¼ 1, c1 ¼ c2 5 and x ¼ Z  ; the well-known variational result for the single orbital exponent. When ε has 16 been minimized with respect to S (for instance, by the Ransil method of Problem 13.5), from the relations c1 þ c2 ¼ 2x

c1 c2 ¼ x2 S2=3

the best values of the orbital exponents c1 and c2 are obtained by the inverse relations: h

h

1=2 i 1=2 i c1 ¼ x 1 þ 1  S2=3 ; c2 ¼ x 1  1  S2=3 The best value of ε to nine decimal figures ε ¼ 2:875 661 331Eh is then obtained for S ¼ 0:872 348 108;

x ¼ 1:685 850 852

giving as best values for the orbital exponents of the optimized Eckart function c1 ¼ 2:183 170 865;

c2 ¼ 1:188 530 839

The ‘splitting’ of the orbital exponents from their average value x ¼ 1.6858 (not far from the best value for the single orbital exponent, 1.6875) accounts for some ‘radial’ correlation (l ¼ 0) between the electrons, yielding a lowering in the electronic energy of about 28  103Eh. Improvement upon Eckart’s result might be obtained by further introducing the ‘angular’ correlation (l s 0) between the electrons, i.e. using Eckart-like wavefunctions involving p, d, f, g,. optimized orbitals. 13.7 The expectation value of the induced dipole moment can be written as hjjzjji ¼

2hj0 jzjj1 i hj0 þ j1 jzjj0 þ j1 i ¼ 1 þ hj1 jj1 i 1 þ hj1 jj1 i

y 2hj0 jzjj1 i½1  hj1 jj1 i y 2hj0 jzjj1 i

526

CHAPTER 13 Variational calculations

to the first order in F (contained in j1). Since  

r r2 ; j1 ¼ Fj0 z 1 þ ¼ Fj0 cos q r þ 2 2

j0 ¼

expðrÞ pffiffiffiffi p



 2 r 3 2 j hj0 jzjj1 i ¼ F j0 cos q r þ 2 0 

4 ¼F 3

   ZN  r5 5 9 expð2rÞ ¼ F 1 þ ¼ F dr r 4 þ 2 4 4 0

giving for the expectation value of the induced dipole moment hjjzjji y 2hj0 jzjj1 i ¼

9 F 2

The expansion   1 1 mz ¼ m0 þ aF þ bF 2 þ / ¼ aF þ bF 2 þ / 2! 2! then gives a ¼ 9/2 as coefficient of the term linear in F, and we obtain the exact value for the static dipole polarizability of the hydrogen atom. 13.8 The single 2pz linear pseudostate (Figure 13.10).  5 1=2 c ~ expðcrÞr cos q j1 ¼ CFc; c ¼ 2pz ¼ p gives the second-order variational energy E D D E   2 2 ~ þ 2 jjFzjj ~ ~ 1 jH^0  E0 jj ^ E~2 ¼ j 0 ¼ F C hcjH 0  E0 jci  2Chcjzjj0 i dE~2 ¼0 dC

FIGURE 13.10 Dipole pseudostate transition from j0 to c for H(1s)

13.9 Solved problems

527

gives C¼

m hcjzjj0 i ¼ hcjH^0  E0 jci ε

as the best value for the linear coefficient C. Then   2 m m m2 E~2 ðbestÞ ¼ F 2 2 $ ε  2 $ m ¼ F 2 ε ε ε d2 E~2 m2  2 ¼2 ¼a dF ε 13.9 Evaluation of m and ε in spherical coordinates. c5=2 m ¼ hcjzjj0 i ¼ ðj0 cjzÞ ¼ 2p p ¼

ε ¼ hcjH^0  E0 jci ¼ ¼

4 5=2 4 $ 3 $ 2 ¼ c 3 ðc þ 1Þ5

Z1

ZN dx x dr r 4 exp½ðc þ 1Þr 2

1

2c1=2 cþ1

0

!5



   1 2c 2c  1 c2 c 1 c  V2  þ  E0 c ¼  þ ð2c  1Þ þ 2 2 r r 2 2  1
2 c cþ1 2

13.10 Optimization of the non-linear parameter c in the single pseudostate approximation. Since E~2 fm2 ε1 taking the first c-derivative dE~2 dm dε ¼ 2m ε1  m2 ε2 dc dc dc gives as stationarity condition 2

dln m dln ε ¼ dc dc

Now, it follows dln m 5 1  c ¼ dc 2 cðc þ 1Þ dln ε 2c  1 ¼ 2 dc c cþ1

528

CHAPTER 13 Variational calculations

Substituting in the previous equation then gives the cubic equation 5ð1  cÞ 2c  1 0 7c3  9c2 þ 9c  5 ¼ 0 ¼ 2 cðc þ 1Þ c  c þ 1 which has the real root c ¼ 0.7970. 13.11 Using the basic integrals given as hint in Problem 13.11, it is easily found (1) Normalization factor    0 0 ð2n þ 2Þ! ¼ 1 0 Nn0 ¼ cn cn ¼ Nn02 j0 z2 r 2n2 j0 ¼ Nn02 3 $ 22nþ1



3 $ 22nþ1 ð2n þ 2Þ!

1=2

(2) Non-orthogonality     ðn þ m þ 2Þ! Snm ¼ c0n c0m ¼ Nn0 Nm0 j0 z2 r nþm2 j0 ¼ Nn0 Nm0 3 $ 2nþmþ1

(3) Transition moment     ðn þ 3Þ! m0n ¼ c0n jzjj0 ¼ Nn0 j0 z2 r n1 j0 ¼ Nn0 3 $ 2nþ2 (4) Element of the excitation energy matrix     ðn þ mÞ! 0 0 0 n1 ^ 0 ¼ c0 jH ^ ð2 þ nmÞ jH 0  E0 jj0 zr m1 ¼ Nn0 Nm0 Mnm n 0  E0 jcm ¼ Nn Nm j0 zr 3 $ 2nþm We recall that the dipole functions c0n are normalized and orthogonal to j0 by symmetry, but not orthogonal to each other. 13.12 We take ~ 1 ¼ C Fj j where j is the exact first-order function normalized to one

r  j ¼ Nj0 z þ z 2 The normalization factor will be

 D r 2 E 1 hjjji ¼ N 2 j0 z þ z j0 ¼ N 2 j0 z2 þ z2 r þ z2 r 2 j0 2 4   1=2  5 15 43 2 8 ¼ N2 1 þ þ ¼ N ¼10N ¼ 2 8 8 43 The transition moment is

13.9 Solved problems

529



    2 1 2 1 5 9 9 2 1=2 ¼ N¼ m ¼ hjjzjj0 i ¼ N j0 z þ z r j0 ¼ N 1 þ $ 2 2 2 4 2 43 

The excitation energy is D

r  r E 1 ε ¼ hjjH^0  E0 jji ¼ N 2 j0 z þ z jH^0 jj0 z þ z þ 2 2 2  1 ¼ N 2 hj0 zjH^0 j j0 z i þ hj0 zrjH^0 j j0 zr i þ hj0 zjH^0 j j0 zr i 4 p¼q¼0

 ¼N

2

p¼q¼1

p¼0;q¼1

 1 4! 3! 1 7 1 18 0 $ $6  $ 4 þ ¼  N2 þ ¼ 4 3 $ 26 2 16 2 43 3 $ 25

where use was made of the basic integrals given as hint in Problem 13.11. 13.13 Three-term dipole pseudospectrum for H(1s). Take
 ~ 1 ¼ Fc0 C c0 ¼ c01 c02 c03 j where c01 ; c02 ; c03 are now normalized 2pz, 3pz, 4pz STOs with c ¼ 1 c01 ¼ N10 j0 z;

c02 ¼ N20 j0 zr;

c03 ¼ N30 j0 zr 2

Proceeding in the same way as we did analytically in Section 13.7.1, we must now do the calculation numerically, and we finally find a ¼ a1 þ a2 þ a3 yielding the three-term dipole pseudospectrum for H(1s) already given as answer to this problem. The exactP value of a for H(1s) is obtained this time as the sum of three pseudostate contributions, a ¼ 3i¼1 ai , while the three-term dipole pseudospectrum {ai,εi}i ¼ 1,2,3 will give an improved evaluation of the dispersion constant C11 for the biatomic system H–H. More refined five-term dipole pseudospectra for H(1s) are given in Table 17.6, while five-term quadrupole and octupole pseudospectra for H(1s) are given in Tables 17.8 and 17.9. 13.14 Compare the results of various pseudostate calculations of a for H (1s), given in Table 13.16, with those resulting from the expansion in eigenstates of H^0 , given in Table 13.17. The technique for these calculations are those described in the previous problems. We limit ourselves here to give the results with short comments of them. Comments to Table 13.16 are the following. In the last column of the table are given the percentages of the exact value of a ð4:5 a30 Þ obtained by the different approximations. It is immediately evident the enormous improvement in the results obtained by all pseudostate approximations (last four rows) in comparison with the extremely poor result found using the first eigenstate of H^0 : Moreover (1) Using c ¼ 1 (single STO pseudostate, Kirkwood) we see that the pseudostate function is much more contracted than the eigenstate and the a value improves by about 23%,

530

CHAPTER 13 Variational calculations

Table 13.16 Pseudostate Approximations to a for H (1s) 4

c=aL1 0

m=ea0

ε=Eh

a=a30

Percentage

2pza

1 ¼ 0:5 2

0.7449

3 ¼ 0:375 8

2.96

66

2pzb

1

1

1 ¼ 0:5 2

4.0

89

2pzc

0.7970

0.9684 pffiffiffi 6 ¼ 0:4082 6 pffiffiffiffiffiffi 30 ¼ 0:9129 6 rffiffiffiffiffiffi 9 2 ¼ 0:9705 2 43

0.4191

4.48

99.5

1

1 3

99.5

2 ¼ 0:4 5

25 , 4.5 6

Exact

18 ¼ 0:4186 43

4.5

Exact

1 2pz + 3pdz

Nj0 ðz þ

1

1 zrÞe 2

1

H-like eigenstate of H^0 One-term Kirkwood c One-term optimized. pffiffiffiffiffiffiffiffiffiffiffi d Two-term normalized STOs: 2pz ¼ j0z, 3pz ¼ 2=15j0 zr. e Two-term Kirkwood generalized 0 Single normalized pseudostate equivalent to the exact j1. a

b

Table 13.17 Expansion in Discrete Eigenstates of H^ 0 (npz Functions) n

c [ 1=n

m=ea0

ε=Eh

a=a30

Percentage

2 3 4 5 6 7 30

0.5 0.333 0.25 0.2 0.167 0.143 0.033

0.7449 0.2983 0.1758 0.1205 0.0896 0.0701 0.0076

0.375 0.4444 0.4687 0.48 0.4861 0.4898 0.4994

2.960 3.360 3.492 3.552 3.585 3.606 3.660

65.8 74.7 77.6 78.9 79.7 80.1 81.3

sensibly more than the limit of the discrete part of the eigenstates of H^0 (a ¼ 3.66 with N ¼ 30 terms). (2) Optimization of the orbital exponent in the single pseudostate gives an a value which is within 0.5% of the exact value. (3) Two linear pseudostates (2pz þ 3pz STOs with c ¼ 1) give the exact value of a (as expected, since exact a has two radial components). P (4) Using N > 2 we get always the correct value of a ¼ i ai , which now results from an increasingly sophisticated excited pseudospectrum {ai,εi}i ¼ 1,2,., N (compare the

13.9 Solved problems

531

Table 13.18 N-term Dipole Pseudospectra of H(1s) for Increasing Values of N i

ai =a30

εi =Eh

1 1 2 1 2 3 1 2 3 4 1 2 3 4 5

4.000000  100 4.166667  100 3.333333  101 3.488744  100 9.680101  101 4.324577  102 3.144142  100 1.091451  100 2.564244  101 7.982236  103 3.013959  100 9.536869  101 4.556475  101 7.479674  102 1.910219  103

5.000000  101 4.000000  101 1.000000  100 3.810911  101 6.165762  101 1.702333  100 3.764634  101 5.171051  101 9.014629  101 2.604969  100 3.753256  101 4.785249  101 6.834311  101 1.255892  100 3.706827  100

P

3 i ai =a0

4.000000 4.5

4.5

4.5

4.5

results in Table 13.18) useful for gradually improving the calculations of the corresponding dispersion coefficients. (5) The single normalized pseudostate equivalent to the exact j1 (last row of Table 13.16, see Problem 13.12) gives a transition moment m and an excitation energy ε differing very little from those of the one-term optimized pseudostate (third row). The results of the expansion in eigenstates of H^0 are given in Table 13.17. In this last case, the transition moments are far too weak and the excitation energies too large, since hydrogenlike npz (c ¼ 1/n) are too diffuse with increasing the principal quantum number n. The expansion in discrete eigenstates of H^0 converges to the asymptotic value a ¼ 3.66 which is only 81.3% of the exact value 4.5. The remaining 18.7% is due to the contribution of the continuous part of the spectrum, which is necessary in order to make the expansion complete. 13.15 Dipole polarizabilities of ground-state H2. Magnasco and Ottonelli (1996) gave a pseudostate decomposition of the accurate Ko1os–Wolniewicz (1967, KW) static dipole polarizabilities of ground-state H2. They chose as unperturbed j0 the 54-term 1 Sþ g KW wavefunction (Ko1os and Wolniewicz, 1964) giving E0 ¼ 1:174470Eh at R ¼ 1.4a0 (the bottom of the potential energy curve), and for the 1 excited states the 34-term 1 Sþ u and Pu functions selected by KW as a basis for their polarizability calculation. The polarizability data reported in Table I of Ko1os-Wolniewicz work cannot, however, be used as they stand for computing the C6 dispersion coefficients in the homodimer, because of the lack of the excitation energies corresponding to each polarizability contribution. Therefore, Magnasco and Ottonelli independently developed explicit expressions for all necessary matrix elements, following the original James–Coolidge work (1933), and using the basic integrals developed in their previous work on H2 (Magnasco

532

CHAPTER 13 Variational calculations

et al., 1993). The linear coefficients in j0 were obtained by minimization of the molecular energy by the Ostrowski’s method (Ko1os and Wolniewicz, 1964), while the coefficients in the excited pseudostates were obtained by the Givens–Householder diagonalization of ðH^0  E0 Þ after Schmidt orthogonalization of the basis functions. The results are collected in Table 2 of their 1996 paper. Of these very accurate results, the four-term pseudospectra given as answer in Problem 13.15 were obtained by the simple selection of the most important four contributions to the dipole polarizabilities. The results of these reduced four-term pseudospectra are remarkably good, the calculated value for the parallel component, ak ¼ 6:378, being 99.9% of the accurate value (6.383), and that for the perpendicular component, reported in the original paper. As at ¼ 4:559; 99.6% of the accurate value (4.577) P a completeness test, the sum-rule gives Sð0Þ ¼ i ai ε2i ¼ 1:972 for ak ; and 1.911 for at ; instead of N ¼ 2 as obtained for the complete pseudospectrum. The reduced two-term pseudospectra are obtained by just taking the first two rows in the previous four-term values. The dipole polarizabilities for H2 obtained in this simple way are still reasonable, at least in a first approximation, being about 95% for ak (6.048 instead of 6.378) and 92% for at (4.202 instead of 4.577). The loss in accuracy in the two-term calculation is greater for the perpendicular component, where the pseudostate contributions are more ‘disperse’. We observe that both reduced spectra were not optimized. Highly accurate four-term reduced dipole pseudospectra for the ground state of the H2 molecule were recently derived by Magnasco and Figari (2009) using an efficient interpolation procedure, P and are fully reported in Chapter 17. The completeness test gives in this case Sð0Þ ¼ i ai ε2i ¼ 1:9956 for ak ; and 1.9938 for at ; showing the great improvement in the accuracy of the calculation.

CHAPTER

Many-electron wavefunctions and model Hamiltonians

14

CHAPTER OUTLINE 14.1 Introduction ............................................................................................................................. 534 14.2 Antisymmetry of the Electronic Wavefunction and the Pauli’s Principle........................................ 534 14.2.1 Two-electron wavefunctions ..................................................................................534 14.2.2 Many-electron wavefunctions and the Slater method...............................................535 14.3 Electron Distribution Functions.................................................................................................. 539 14.3.1 One-electron distribution functions: general definitions ...........................................539 14.3.2 Electron density and spin density ..........................................................................540 14.3.3 Two-electron distribution functions: general definitions ...........................................543 14.4 Average Values of One- and Two-Electron Operators ................................................................... 544 14.4.1 Symmetrical sums of one-electron operators ..........................................................544 14.4.2 Symmetrical sums of two-electron operators...........................................................545 14.4.3 Average value of the electronic energy ...................................................................546 14.5 The Slater’s Rules .................................................................................................................... 546 14.6 Pople’s Two-Dimensional Chart of Quantum Chemistry ................................................................ 548 14.7 Hartree–Fock Theory for Closed Shells ...................................................................................... 550 14.7.1 Basic theory and properties of the fundamental invariant r......................................550 14.7.2 Electronic energy for the HF wavefunction .............................................................552 14.7.3 Roothaan’s variational derivation of the HF equations .............................................553 14.7.4 Hall–Roothaan’s formulation of the LCAO-MO-SCF equations ..................................555 14.7.5 Mulliken population analysis.................................................................................558 14.7.6 Atomic bases in quantum chemical calculations.....................................................561 14.7.7 Localization of molecular orbitals ..........................................................................565 14.8 Hu¨ckel’s Theory ....................................................................................................................... 567 14.8.1 Recurrence relation for the linear chain .................................................................568 14.8.2 General solution for the linear chain ......................................................................569 14.8.3 General solution for the closed chain .....................................................................570 14.8.4 Alternant hydrocarbons ........................................................................................572 14.8.5 An introduction to band theory of solids.................................................................577 14.9 Semiempirical MO Methods ...................................................................................................... 579 14.9.1 Extended Hu¨ckel’s theory .....................................................................................580 14.9.2 The CNDO method ...............................................................................................580

Elementary Molecular Quantum Mechanics 2E. http://dx.doi.org/10.1016/B978-0-444-62647-9.00014-2 Ó 2013, 2007 Elsevier B.V. All rights reserved

533

534

CHAPTER 14 Many-electron wavefunctions and model Hamiltonians

14.9.3 The INDO method................................................................................................584 14.9.4 The ZINDO method ..............................................................................................584 14.10 Problems 14 ............................................................................................................................ 585 14.11 Solved Problems ...................................................................................................................... 588

14.1 INTRODUCTION In this chapter we shall examine how a many-electron wavefunction satisfying the Pauli’s principle can be constructed starting from an orbital basis, and how appropriate model Hamiltonians can be introduced to find definite MO approximations to the molecular energy. We shall follow here the classical work by Slater (1929), where such a wavefunction is built in terms of a determinant (Slater det) of orthonormal spin-orbitals (SOs). Reduction from the 4N-dimensional configuration space of the wavefunction to the ordinary three-dimensional space þ spin is then considered in terms of one- and two-particle density matrices, whose diagonal elements determine the probability of finding clusters of one and two particles. This allows us to discuss electron and spin densities in Section 14.3 and, in Section 14.4, the average values of one- and two-electron operators, while the problem of electron correlation will be considered in a later chapter. In Section 14.7 we introduce Hartree–Fock (HF) theory as the best independent particle model (IPM) to treat many-electron systems, considered as the central step of an ideal ladder having uncorrelated approaches below it, from Hu¨ckel’s topological theory to the linear combination of atomic orbitals (LCAO)-molecular orbital (MO)-self-consistent field (SCF) approach by Hall (1951) and Roothaan (1951a), and correlated approaches above it. While configurational interaction (CI) techniques, multiconfigurational (MC)-SCF, and a variety of many-body perturbation approaches will be treated later in Chapter 16, Hu¨ckel and semiempirical methods will be discussed to some extent in Sections 14.8 and 14.9, while many problems will be presented and discussed, as usual, at the end of the chapter.

14.2 ANTISYMMETRY OF THE ELECTRONIC WAVEFUNCTION AND THE PAULI’S PRINCIPLE 14.2.1 Two-electron wavefunctions Let x1 and x2 be two fixed points in the space–spin space, and Jðx1 ; x2 Þ a normalized two-electron wavefunction. Then, because of the indistinguishability of the electrons jJðx1 ; x2 Þj2 dx1 dx2 ¼ probability of finding electron one at dx1 and electron two at dx2 jJðx2 ; x1 Þj2 dx1 dx2 ¼ probability of finding electron two at dx1 and electron one at dx2 Therefore it follows: jJðx2 ; x1 Þj2 ¼ jJðx1 ; x2 Þj2 0 Jðx2 ; x1 Þ ¼ Jðx1 ; x2 Þ

(1)

and the wavefunction must be symmetric (þ sign) or antisymmetric ( sign) in the interchange of the space–spin coordinates of the two electrons.

14.2 Antisymmetry of the Electronic wavefunction and the Pauli’s principle

535

Pauli’s principle states that, in nature, electrons are described only by antisymmetric wavefunctions: Jðx2 ; x1 Þ ¼ Jðx1 ; x2 Þ

(2)

which is the Pauli’s antisymmetry principle in the form given by Dirac. This formulation includes the exclusion principle for electrons in the same orbital with the same spin: Jðx1 ; x2 Þ ¼ jl ðx1 Þjl ðx2 Þ  jl ðx1 Þjl ðx2 Þ ¼ 0

(3)

where we always take electrons in dictionary order, and interchange SOs only. Instead, it is allowable to put two electrons in the same orbital with a different spin: Jðx1 ; x2 Þ ¼ jl ðx1 Þjl ðx2 Þ  jl ðx1 Þjl ðx2 Þ



j ðx1 Þ j ðx1 Þ

l l
¼
jl ðx1 Þjl ðx2 Þ

¼
jl ðx2 Þ jl ðx2 Þ


(4)

where the wavefunction is written as a (2  2) Slater determinant of the occupied SOs (recall the notation where jl is associated to spin a, and jl to spin b). Hence, the antisymmetry requirement of the Pauli’s principle for a pair of different SOs is automatically met by writing the two-electron wavefunction as a determinant having electrons as rows and SOs as columns. Such a determinant is called a Slater det, since Slater (1929, 1931) was the first who suggested this approach, avoiding in this way the difficulties connected with older work which was mostly based on the use of group theoretical techniques. If the SOs are orthonormal, pffiffiffithen hjl jjl i ¼ hjl jjl i ¼ 1 and hjl jjl i ¼ hjl jjl i ¼ 0, and the det is normalized by the factor 1= 2, and it is usually assumed to represent the normalized det as
  1
(5) Jðx1 ; x2 Þ ¼ pffiffiffi
jl jl
¼  jl ðx1 Þ jl ðx2 Þ  2

14.2.2 Many-electron wavefunctions and the Slater method The three-electron Slater determinant was already derived in Section 8.12 of Chapter 8 in the study of the symmetric group of N ¼ 3 electrons. For an N-electron system (atom or molecule) the wavefunction J is antisymmetric if it is left unaltered by an even number of permutations of the electrons among the SOs, while it changes sign by an odd number of permutations. As an example P^12 Jðx1 ; x2 ; x3 /; xN Þ ¼ Jðx2 ; x1 ; x3 /; xN Þ ¼ Jðx1 ; x2 ; x3 /; xN Þ

(6)

The N-electron wavefunctions can be constructed from a set, in principle complete, of one-electron functions including spin (atomic or molecular SOs). If fji ðxÞg i ¼ 1; 2; / x ¼ rs Z 
 ji
jj ¼ dxj ðxÞjj ðxÞ ¼ dij i

(7) (8)

536

CHAPTER 14 Many-electron wavefunctions and model Hamiltonians

we can write Jðx1 Þ ¼ Jðx1 ; x2 Þ ¼

X

X

ji ðx1 ÞCi

i

ji ðx1 ÞCi ðx2 Þ ¼

i

Jðx1 ; x2 ; /; xN Þ ¼

X

Ci ¼ hji jJi XX

//

i

ji ðx1 Þjj ðx2 ÞCji

j

ji ðx1 Þjj ðx2 Þ/ jN ðxN ÞCN/ ji

(9)

(10) (11)

ij/N

The coefficients of the linear combination of the products of SOs are determined by the fact that the resultant J must be antisymmetric with respect to the interchange of any pair of electrons (Pauli). The simplest way of writing such a function was introduced by Slater (1929, 1931), as we have seen, and consists in writing J in the form of a determinant of order N (Slater det) having N rows and N columns:




j1 ð1Þ j2 ð1Þ / jN ð1Þ




1

j1 ð2Þ j2 ð2Þ / jN ð2Þ

Jðx1 ; x2 ; /; xN Þ ¼ pffiffiffiffiffi

¼ k j1 ð1Þ j2 ð2Þ / jN ðNÞ k (12) N!
/ / / /



j ðNÞ j ðNÞ / j ðNÞ


1 2 N where rows ¼ spaceespin electron coordinates columns ¼ spin-orbital functions If the SOs are orthonormal

(13)


 ji
jj ¼ dij

then the many-electron wavefunction (12) is normalized to 1 hJjJi ¼ 1

(14)

1. Properties of the Slater determinants a. The J written in the form of a Slater det is antisymmetric with respect to the interchange of any pair of electrons:




j1 ð2Þ j2 ð2Þ / jN ð2Þ


1

j1 ð1Þ j2 ð1Þ / jN ð1Þ

(15) Jðx2 ; x1 ; /; xN Þ ¼ pffiffiffiffiffi

¼ Jðx1 ; x2 ; / xN Þ / / /

N!

/


j1 ðNÞ j2 ðNÞ / jN ðNÞ


14.2 Antisymmetry of the Electronic wavefunction and the Pauli’s principle

537

since this is equivalent to interchanging two rows of the determinant, and this changes sign to the function. b. If two SOs are equal, the determinant has two columns equal and therefore vanishes identically:




j1 ð1Þ j1 ð1Þ / jN ð1Þ


1

j1 ð2Þ j1 ð2Þ / jN ð2Þ

(16) Jðx1 ; x2 ; /; xN Þ ¼ pffiffiffiffiffi

h0 / / /

N!

/


j1 ðNÞ j1 ðNÞ / jN ðNÞ
A function of this kind cannot exist. This is nothing but the Pauli’s principle in its exclusion form: two atomic SOs cannot have equal the four quantum numbers jnlmms i. The same is true for molecular SOs, since we cannot have two spatial MOs identical with the same spin. c. The Slater det is unchanged if we orderly interchange rows with columns:




j1 ð1Þ j1 ð2Þ / j1 ðNÞ


1

j2 ð1Þ j2 ð2Þ / j2 ðNÞ

(17) Jðx1 ; x2 ; /; xN Þ ¼ pffiffiffiffiffi

/ / /

N!

/
j ð1Þ j ð2Þ / j ðNÞ

N
N N where now rows ¼ spin-orbital functions columns ¼ spaceespin electron coordinates

(18)

d. The probability density (and the Hamiltonian operator) is left unchanged after any number of interchanges among the electrons: jJðx2 ; x1 ; /; xN Þj2 ¼ jJðx1 ; x2 ; /; xN Þj2

(19)

e. A single Slater det is sufficient as a first approximation in the case of closed shells (with spin S ¼ MS ¼ 0) or open shells with jMS j ¼ S (all spins parallel or antiparallel), while in the general 2 open-shell case to get an eigenstate of S^ with eigenvalue S (state of definite spin) it is necessary to take a linear combination of Slater dets. The effect of the spin operator P^sk sl on a Slater det is equivalent to interchanging the two spin functions hk and hl among the two columns of the original det, leaving unaltered the orbital part (which is purely spatial). In such a way, the Dirac’s formula of Chapter 9 can be equally well applied to the Slater dets to verify or to construct determinants which are eigenstates of spin. f. Examples. i. The ground and the first excited states of the He atom are described in terms of Slater dets by   J 1s2 ; 1 S ¼ k1s1sk S ¼ MS ¼ 0

538

CHAPTER 14 Many-electron wavefunctions and model Hamiltonians

8 > > k1s 2sk > > > > > < 1 h   i Jð1s 2s; 3 SÞ ¼ pffiffiffi 1s 2s þ 1s 2s > 2 > > > >   > > : 1s 2s

S ¼ 1; MS ¼ 1 0 1

  i 1 h Jð1s 2s; 1 SÞ ¼ pffiffiffi 1s 2s  1s 2s S ¼ MS ¼ 0 2 8 > > S ¼ 1; MS ¼ 1 k1s 2pk > > > > > < 1 h   i 0 Jð1s 2p; 3 PÞ ¼ pffiffiffi 1s 2p þ 1s 2p > 2 > > >   > > > 1 : 1s 2p nine states ( p ¼ x, y, z),

  i 1 h Jð1s 2p; 1 PÞ ¼ pffiffiffi 1s 2p  1s 2p 2

three states ( p ¼ x, y, z). Other examples. 8 > > > < k1s 1s 2sk 2 2 ii. Lið1s 2s; SÞ ¼ > > > : k1s 1s 2sk

S ¼ MS ¼ 0

1 1 S ¼ ; MS ¼ 2 2 

1 2

The ground state of the Li atom is a doublet S. iii. Beð1s2 2s2 ; 1 SÞ ¼ k1s1s 2s 2sk

S ¼ MS ¼ 0

The ground state of the Be atom is a singlet S. iv. H2 ðsg2 ; 1 Sgþ Þ ¼ ksg sg k S ¼ MS ¼ 0 aþb The ground state of the H2 molecule is a singlet 1 Sgþ (the bonding MO sg ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi is doubly 2 þ 2S occupied by electrons with opposite spin. Here, S is the overlap integral between orbitals a and b). 8 > S ¼ 1; MS ¼ 1 ksg su k > > > > < 1 0 v. H2 ðsg su ; 3 Suþ Þ ¼ pffiffiffi ½ksg su k þ ksg su k > 2 > > > > : ksg su k 1

14.3 Electron distribution functions

The 3 Suþ excited triplet state of the

ba p ffiffiffiffiffiffiffiffiffiffiffiffiffi ffi su ¼ is the antibonding MO . 2  2S

H2

molecule

has

vi. H2 Oð1a21 2a21 1b22 3a21 1b21 ; 1 A1 Þ ¼ k1a1 1a1 2a1 2a1 / 1b1 1b1 k

singly

occupied

539

MOs

S ¼ MS ¼ 0

The ground state of the H2O molecule (N ¼ 10) is described by five doubly occupied nondegenerate MOs (see Chapter 7 on symmetry and Chapter 8 on group theory), where a1, b1, b2,/ are MOs having the symmetry of the C2v point group (the symmetry group to which the H2O molecule belongs). We now verify using Dirac’s rule that the ground-state wavefunction for H2 is a singlet (S ¼ 0), and that for the excited state is a triplet (S ¼ 1).    2 J s2 ; 1 Sgþ ¼ sg sg ; S^ ¼ I^ þ P^12 g

        2 S^ J ¼ sg sg  þ sg sg  ¼ sg sg   sg sg  ¼ 0ð0 þ 1ÞJ

S ¼ 0; singlet

  i   1 h J sg su ; 3 Suþ ¼ pffiffiffi sg su  þ sg su  2 h        i 1 2 S^ J ¼ pffiffiffi sg su  þ sg su  þ sg su  þ sg su  2   i 1 h ¼ 2 $ pffiffiffi sg su  þ sg su  ¼ 1ð1 þ 1ÞJ 2

S ¼ 1; triplet

Other examples for Lið2 SÞ and Beð1 SÞ are given as Problems 14.1 and 14.2.

14.3 ELECTRON DISTRIBUTION FUNCTIONS The distribution functions determine the distribution of electron ‘clusters’, and allow us to pass from the abstract 4N-dimensional space of the N-electron wavefunction to the three-dimensional þ spin physical space where experiments are done (McWeeny, 1960).

14.3.1 One-electron distribution functions: general definitions Let Jðx1 ; x2 ; /; xN Þ with hJjJi ¼ 1

(20)

be a normalized N-electron wavefunction satisfying the antisymmetry requirement. Then the first principles state that Jðx1 ; x2 ; /; xN ÞJ ðx1 ; x2 ; /; xN Þdx1 dx2 /dxN ¼ probability of finding electron ðor particleÞ 1 at dx1 ; 2 at dx2 ; /; N at dxN

(21)

where dx1 ; dx2 ; /; dxN are fixed infinitesimal space–spin volume elements in configuration space.

540

CHAPTER 14 Many-electron wavefunctions and model Hamiltonians

The probability of finding electron 1 at dx1 independently of the remaining (N  1) electrons will be given by 

Z  dx2 / dxN JJ dx1 (22) For the indistinguishability of the electrons the probability of finding any one unspecified electron at dx1 will be N times this quantity:

Z  N dx2 / dxN Jðx1 ; x2 ; /; xN ÞJ ðx1 ; x2 ; /; xN Þ dx1 ¼ r1 ðx1 ; x1 Þdx1 (23) where the bilinear function

Z

r1 ðx1 ; x1 Þ ¼ N

dx2 / dxN Jðx1 ; x2 ; /; xN ÞJ ðx1 ; x2 ; /; xN Þ

(24)

from J J is called the one-electron distribution function. It is the diagonal element ðx01 ¼ x1 Þ of the more general mathematical quantity:1 Z 0 r1 ðx1 ; x1 Þ ¼ N dx2 / dxN Jðx1 ; x2 ; /; xN ÞJ ðx01 ; x2 ; /; xN Þ (25) which is called the one-electron density matrix. It is important to stress that Eqn (25) has no physical meaning, while its diagonal element (24) is the function determining the physical distribution of the electrons. The one-electron distribution function (24) satisfies the conservation relation: Z (26) dx1 r1 ðx1 ; x1 Þ ¼ N where N is the total number of electrons.

14.3.2 Electron density and spin density The most general expression for the one-electron distribution function as a bilinear function of space– spin variables is r1 ðx1 ; x1 Þ ¼ r1 ðr1 s1 ; r1 s1 Þ ¼ P1 ðra1 ; ra1 Þaðs1 Þa ðs1 Þ þ P1 ðrb1 ; rb1 Þbðs1 Þb ðs1 Þ þ P1 ðra1 ; rb1 Þaðs1 Þb ðs1 Þ þ P1 ðrb1 ; ra1 Þbðs1 Þa ðs1 Þ

(27)

where the P1s are purely spatial functions, and we shall sometimes make use of the short notation P1 ðra1 ; ra1 Þ ¼ Pa1 etc 1

(28)

As far as notation is concerned, we shall use r for the density (or density matrix) with spin, P for the density (or density matrix) without spin.

14.3 Electron distribution functions

Integrating over the spin variable Z ds1 r1 ðr1 s1 ; r1 s1 Þ ¼ P1 ðra1 ; ra1 Þ þ P1 ðrb1 ; rb1 Þ

541

(29)

where P1 ðra1 ; ra1 Þdr1 ¼ probability of finding at dr1 an electron with spin a

(30)

P1 ðrb1 ; rb1 Þdr1

(31)

¼ probability of finding at dr1 an electron with spin b

The two remaining integrals in Eqn (27) vanish because of the orthogonality of the spin functions. In terms of the two spinless components Pa1 and Pb1 we define Pðr1 ; r1 Þ ¼ P1 ðra1 ; ra1 Þ þ P1 ðrb1 ; rb1 Þ ¼ Pa1 þ Pb1

(32)

the electron density (as measured from experiment), with Pðr1 ; r1 Þdr1 ¼ probability of finding at dr1 an electron with either spin Qðr1 ; r1 Þ ¼ P1 ðra1 ; ra1 Þ 

P1 ðrb1 ; rb1 Þ

¼ Pa1 

Pb1

(33) (34)

the spin density, with Qðr1 ; r1 Þdr1 ¼ probability of finding at dr1 an excess of spin a over spin b

(35)

We have the conservation relations: Z dr1 P1 ðra1 ; ra1 Þ ¼ Na

(36)

dr1 P1 ðrb1 ; rb1 Þ ¼ Nb

(37)

dr1 Pðr1 ; r1 Þ ¼ Na þ Nb ¼ N

(38)

dr1 Qðr1 ; r1 Þ ¼ Na  Nb ¼ 2MS

(39)

the number of electrons with a spin;

Z

the number of electrons with b spin; Z

the total number of electrons;

Z

^ spin operator where MS is the eigenvalue of the z-component of the S S^z h ¼ MS h

(40)

As an example, consider the doubly occupied (normalized) MO 4(r). The normalized two-electron wavefunction will be

1

4að1Þ 4bð1Þ

1 (41) J ¼ k44k ¼ pffiffiffi

¼ 4ðr1 Þ4ðr2 Þ pffiffiffi ½aðs1 Þbðs2 Þ  bðs1 Þaðs2 Þ 2
4að2Þ 4bð2Þ
2

542

CHAPTER 14 Many-electron wavefunctions and model Hamiltonians Z

dx2 Jðx1 ; x2 ÞJ ðx1 ; x2 Þ 8 1 > > > pffiffiffi > Z < 4ðr1 Þ4ðr2 Þ 2 ½aðs1 Þbðs2 Þ  bðs1 Þaðs2 Þ ¼ 2 dr2 ds2 > 1  >      > > : 4 ðr1 Þ4 ðr2 Þ pffiffiffi ½a ðs1 Þb ðs2 Þ  b ðs1 Þa ðs2 Þ 2

r1 ðx1 ; x1 Þ ¼ 2

¼ 4ðr1 Þ4 ðr1 Þ½aðs1 Þa ðs1 Þ þ bðs1 Þb ðs1 Þ

(42)

P1 ðra1 ; ra1 Þ ¼ P1 ðr1b ; r1b Þ ¼ 4ðr1 Þ4 ðr1 Þ ¼ Rðr1 ; r1 Þ

(43)

Pðr1 ; r1 Þ ¼ Pa1 þ Pb1 ¼ 24ðr1 Þ4 ðr1 Þ ¼ 2j4ðr1 Þj2

(44)

so that

Qðr1 ; r1 Þ ¼ Pa1 

Pb1

¼0

(45)

provided we remember that, under the action of a differential operator (like 72 ), the operator acts only on 4 and not on 4* (this is one of the main reasons for using density matrices, where there is a distinction between r and r0 ). In terms of an atomic basis ðcA cB Þ, where cA and cB are normalized non-orthogonal atomic orbitals (AOs) with overlap S ¼ hcA jcB is0, the MO 4(r) can be written as cA þ lcB 4ðrÞ ¼ cA ðrÞcA þ cB ðrÞcB ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ l2 þ 2lS

(46)

where l¼

cB cA

(47)

is the polarity parameter of the MO, and  1=2 cA ¼ 1 þ l2 þ 2lS

(48)

the normalization factor. The electron density can be analyzed into elementary densities coming from Eqn (46) as PðrÞ ¼ 2j4ðrÞj2 ¼ qA c2A ðrÞ þ qB c2B ðrÞ þ qAB where c2A ðrÞ; c2B ðrÞ are atomic densities, ized to 1, while the coefficients qA ¼

cA ðrÞcB ðrÞ c ðrÞcA ðrÞ þ qBA B S S

(49)

cA ðrÞcB ðrÞ c ðrÞcA ðrÞ and B overlap densities, all normalS S

2 ; 2 1 þ l þ 2lS

qB ¼

2l2 1 þ l2 þ 2lS

(50)

are atomic charges, and qAB ¼ qBA ¼

2lS 1 þ l2 þ 2lS

(51)

14.3 Electron distribution functions

543

are overlap charges. The charges are normalized so that qA þ qB þ qAB þ qBA ¼

2 þ 2l2 þ 4lS ¼2 1 þ l2 þ 2lS

(52)

the total number of electrons in the bond orbital (BO) 4(r). For a homopolar bond, l ¼ 1 qA ¼ qB ¼

1 ; 1þS

qAB ¼ qBA ¼

S 1þS

(53)

For a heteropolar bond, l s 1, and we can define gross charges on A and B as QA ¼ qA þ qAB ¼

2 þ 2lS 1 þ l2 þ 2lS

(54)

QB ¼ qB þ qBA ¼

2l2 þ 2lS 1 þ l2 þ 2lS

(55)

and formal charges on A and B as dA ¼ 1  Q A ¼

l2  1 1 þ l2 þ 2lS

dB ¼ 1  QB ¼ 

l2  1 1 þ l2 þ 2lS

(56) (57)

If l > 1; dA ¼ d > 0; dB ¼ dA ¼ d < 0, and we have the dipole Aþd Bd (e.g. LiH). þ þ Further examples for the 1 Sg ground state and the 3 Su excited triplet state of H2 are given as Problems 14.3 and 14.4.

14.3.3 Two-electron distribution functions: general definitions The two-electron distribution function determines the distribution of the electrons in pairs, and is defined as Z r2 ðx1 ; x2 ; x1 ; x2 Þ ¼ NðN  1Þ dx3 / dxN Jðx1 ; x2 ; x3 ; /; xN ÞJ ðx1 ; x2 ; x3 ; /; xN Þ (58) from J J It is the diagonal element of the two-electron density matrix: Z 0 0 r2 ðx1 ; x2 ; x1 ; x2 Þ ¼ NðN  1Þ dx3 / dxN Jðx1 ; x2 ; x3 ; /; xN ÞJ ðx01 ; x02 ; x3 ; /; xN Þ

(59)

Its physical meaning is r2 ðx1 ; x2 ; x1 ; x2 Þdx1 dx2 ¼ probability of finding an electron at dx1 and; simultaneously; another electron at dx2 (60)

544

CHAPTER 14 Many-electron wavefunctions and model Hamiltonians

From the definition (58) of r2 and the normalization of J, it follows that Z ZZ dx2 r2 ðx1 ; x2 ; x1 ; x2 Þ ¼ ðN  1Þr1 ðx1 ; x1 Þ; dx1 dx2 r2 ðx1 ; x2 ; x1 ; x2 Þ ¼ NðN  1Þ

(61)

which is the total number of indistinct pairs of electrons. The second of Eqn (61) is the conservation relation for r2, while the first shows that r1 can be obtained from r2 by integration, while, in general, the opposite is not true. As we shall see later in this chapter, the only exception is the HF theory where r1 determines r2.

14.4 AVERAGE VALUES OF ONE- AND TWO-ELECTRON OPERATORS The electron distribution functions we saw so far (better, their corresponding density matrices) allow to compute easily the average values of symmetrical sums of one- and two-electron operators, such as those occurring in evaluating the expectation value for the electronic energy in atomic and molecular systems.

14.4.1 Symmetrical sums of one-electron operators
+ *


X N XZ

^ h^i Jðx1 ; x2 ; /; xN ÞJ ðx1 ; x2 ; /; xN Þdx1 dx2 /dxN J
h
J ¼
i¼1 i
i Z Z ¼ dx1 h^1 dx2 /dxN Jðx1 ; x2 ; /; xN Þ ðx1 ; x2 ; /; xN Þ Z Z þ dx2 h^2 dx1 /dxN Jðx2 ; x1 ; /; xN ÞJ ðx2 ; x1 ; /; xN Þ þ /N terms identical to the first one

Z  Z ¼ dx1 h^1 N dx2 /dxN Jðx1 ; x2 ; /; xN ÞJ ðx1 ; x2 ; /; xN Þ Z ¼

dx1 h^1 r1 ðx1 ; x1 Þ

(62)

where we must take care that h^1 acts only upon the first set of variables (those coming from J) and not on the second (those coming from J ). It is more correct to write
+ Z *

X


J
(63) h^
J ¼ dx1 h^1 r1 ðx1 ; x01 Þjx0 ¼x1 1
i i
where r1 ðx1 ; x01 Þ is the one-electron density matrix, and the prime is removed before the final integration over dx1. In this way, the 4N-dimensional integration over the N-electron wavefunction J is replaced by a 4-dimensional integration over the one-electron density matrix. It must be stressed here that, for a J in

14.4 Average values of one- and two-Electron operators

545

the form of a single Slater det of order N such as that occurring in the HF theory for closed shells, the matrix element (62) would involve a 4N-dimensional integration of (N!)2 terms.

14.4.2 Symmetrical sums of two-electron operators In the following, we make explicit reference to the electron repulsion operator, but the same holds for any other two-electron operators, such as those involving spin operators.2
+ *

X 0 0 Z X 1

1
Jðx1 ; x2 ; /; xN ÞJ ðx1 ; x2 ; /; xN Þdx1 dx2 /dxN J

J ¼
i; j rij
r ij ij Z ZZ 1 dx3 /dxN Jðx1 ; x2 ; x3 ; /; xN ÞJ ðx1 ; x2 ; x3 ; /; xN Þ ¼ dx1 dx2 r12 ZZ Z 1 þ dx1 dx3 dx2 /dxN Jðx1 ; x3 ; x2 ; /; xN ÞJ ðx1 ; x3 ; x2 ; /; xN Þ r13 þ / NðN  1Þ terms identical to the first one

 Z ZZ 1 NðN  1Þ dx3 /dxN Jðx1 ; x2 ; x3 ; /; xN ÞJ ðx1 ; x2 ; x3 ; /; xN Þ ¼ dx1 dx2 r12 ZZ 1 r ðx1 ; x2 ; x1 ; x2 Þ ¼ dx1 dx2 r12 2 (64) where the two-electron operator 1/r12 is now a simple multiplier. Hence, we conclude that, in general
+ *

X
XZ

^ O i
J ¼ O^i Jðx1 ; x2 ; /; xN ÞJ ðx1 ; x2 ; /; xN Þdx1 dx2 /dxN J


i i Z ¼ dx1 O^1 r1 ðx1 ; x01 Þjx0 ¼x1 1

(65)

D
X0
E X0 Z

O^ij Jðx1 ; x2 ; x3 ; /; xN ÞJ ðx1 ; x2 ; x3 ; /; xN Þdx1 dx2 dx3 /dxN O^ij
J ¼ J
i; j

i; j

ZZ ¼

dx1 dx2 O^12 r2 ðx1 x2 ; x1 x2 Þ

(66)

The dash on the double sum means that the terms j ¼ i are excluded from the summation. There are altogether N(N  1) indistinct terms.

2

546

CHAPTER 14 Many-electron wavefunctions and model Hamiltonians

14.4.3 Average value of the electronic energy The electronic Hamiltonian in the Born–Oppenheimer approximation (Chapter 20) contains two of such symmetrical sums. Therefore, its average value over the (normalized) many-electron wavefunction J can be written as
+ *

X D

E X0 1

1
h^ þ Ee ¼ J
H^e
J ¼ J

J
i i 2 i;j rij
Z ZZ 1 1 ¼ dx1 h^1 r1 ðx1 ; x01 Þjx0 ¼x1 þ r ðx1 ; x2 ; x1 ; x2 Þ (67) dx1 dx2 1 2 r12 2

 Z 1 2 0 (68) ¼ dx1  V1 r1 ðx1 ; x1 Þ 2 x01 ¼x1 Z (69) þ dx1 V1 r1 ðx1 ; x1 Þ 1 þ 2

ZZ dx1 dx2

1 r ðx1 ; x2 ; x1 ; x2 Þ r12 2

where 1 h^ ¼  V2 þ V 2

V¼

(70)

X Za a

ra

(71)

is the one-electron bare-nuclei Hamiltonian, and V is the nuclear attraction. Equation (68) is the average kinetic energy of the electron distribution r1 (it is the only term which implies the one-electron density matrix because of the presence of the differential operator V21 which acts on J and not on J ). Equation (69) expresses the average potential energy of the charge distribution r1 in the field provided by all nuclei in the molecule. Equation (70) expresses the average electronic repulsion of an electron pair described by the pair function r2. In this way, the molecular energy E in the Born–Oppenheimer approximation takes on its simplest physically transparent form as 1 X0 Za Zb (72) E ¼ Ee þ EN ¼ E e þ 2 a;b Rab where the electronic energy Ee is evaluated in a given fixed nuclear configuration, and we added nuclear repulsion as the last term.

14.5 THE SLATER’S RULES Well before density matrix techniques were introduced in molecular quantum mechanics (Lo¨wdin, 1955a,b; McWeeny, 1960; see, however, Lennard-Jones, 1931), Slater (1929) gave some simple rules for the evaluation of the matrix elements of any symmetrical sum of one-electron and two-electron operators between any pair of Slater dets built from orthonormal SOs. These rules were later extended by Lo¨wdin (1955a) to include the case of Slater dets built from non-orthogonal SOs, while in the latter

14.5 The Slater’s rules

547

case Figari and Magnasco (1985) suggested convenient mathematical techniques to deal with the cofactors arising from n-substituted Slater dets. At variance with Lo¨wdin’s formulae, which go into trouble if the actual overlap matrix becomes singular, Figari and Magnasco’s formulae go smoothly into the Slater’s formulae when the overlap goes to zero. Slater’s rules are as follows. Let 
 J ¼ kj1 j2 / jN k ji
jj ¼ dij hJjJi ¼ 1   D 0

0 E J0 ¼ j01 j02 / j0N  j2
jj ¼ dij hJ0 jJ0 i ¼ 1

(73) (74)

be a pair of normalized N-electron Slater dets built from orthonormal SOs. Then 1. One-electron operators a. Zero SO differences

*

+ XD



E
X
O^i

J ¼ ji
O^1
ji J

i

b. One SO difference

(75)

i



+ D

E
X


O^i

J ¼ j0i
O^1
ji J


*

0


(76)

i

if J0 differs from J for j0i s ji. c. Two or more SO differences *



+
X
^ Oi

J ¼ 0 J
0


(77)

i

2. Two-electron operators a. Zero SO differences *

+





E D E
X
1 XX D
^

^


^ Oij
J ¼ J
ji jj
O12
ji jj  jj ji
O12
ji jj 2 i i > > > > / > < cm1  xcm þ cmþ1 > > >/ > > > > > :c N1  xcN

¼

0

¼

0

¼

0

(177)

The general equation is cm1  xcm þ cmþ1 ¼ 0

m ¼ 1; 2; /N

(178)

with the boundary conditions c0 ¼ cNþ1 ¼ 0

(179)

The general solution is the ‘standing’ wave: cm ¼ A expðimqÞ þ B expðimqÞ

(180)

x ¼ 2 cos q

(181)

provided

1. From the first boundary condition it is obtained that c0 ¼ A þ B ¼ 0 0 B ¼ A

(182)

cm ¼ A½expðimqÞ þ expðimqÞ ¼ 2iA sin mq ¼ C sin mq

(183)

where C ¼ 2iA is a normalization factor.

8

The secular equations for linear and closed polyene chains, even with different bs for single and double bonds, were first solved by Lennard-Jones (1937a,b). See also Hu¨ckel (1931a).

570

CHAPTER 14 Many-electron wavefunctions and model Hamiltonians

The general equation gives Afexp½iðm  1Þq  x expðimqÞ þ exp½iðm þ 1Þqg þ Bfexp½  iðm  1Þq  x expðimqÞ þ exp½  iðm þ 1Þqg ¼ A expðimqÞ½expðiqÞ  x þ expðiqÞ þ B expðimqÞ½expðiqÞ  x þ expðiqÞ ¼ ½A expðimqÞ þ B expðimqÞ½expðiqÞ  x þ expðiqÞ ¼ cm ð2 cos q  xÞ ¼ 0

(184)

so that, for cm s 0 2 cos q  x ¼ 0 0 x ¼ 2 cos q as required. 2. From the second boundary condition it follows that cNþ1 ¼ C sinðN þ 1Þq ¼ 0 ðN þ 1Þq ¼ kp

k ¼ 1; 2; 3; /N

(185)

(186) (187)

with k a ‘quantum number’ qk ¼

kp Nþ1

Angle q is ‘quantized’. Therefore, the general solution for the linear chain will be kp xk ¼ 2 cos Nþ1 kp cmk ¼ ck sin m Nþ1

(188)

(189) (190)

the first being the p bond energy of level k (in units of b), the second the coefficient of the m-th AO in the k-th MO. Problem 14.12 gives the application of the general formulae (189) and (190) to the case of the allyl radical (N ¼ 3).

14.8.3 General solution for the closed chain In the same paper, Coulson (1938) also gave the general solution for the closed chain (ring) of N atoms: 8 xc1 þ c2 þ / þ cN ¼ 0 > > > > > >

> > / > > > : c1 þ / þ cN1  xcN ¼ 0 The general equation for the coefficients is the same as that for the linear chain: cm1  xcm þ cmþ1 ¼ 0 m ¼ 1; 2; /N

(192)

14.8 Hu¨ckel’s theory

571

but the boundary conditions are now different: c0 ¼ cN ;

c1 ¼ cNþ1 0 cm ¼ cmþN

(193)

the last being a periodic boundary condition. The general solution is now the ‘progressive’ wave: cm ¼ A expðimqÞ

(194)

Afexp½iðm  1Þq  x expðimqÞ þ exp½iðm þ 1Þqg ¼ 0

(195)

A expðimqÞ½expðiqÞ  x þ expðiqÞ ¼ cm ð2 cos q  xÞ ¼ 0

(196)

x ¼ 2 cos q

(197)

and the general Eqn (192) gives

namely, for cm s 0 as before. From the periodic boundary condition it follows A expðimqÞ ¼ A exp½iðm þ NÞq

(198)

expðiNqÞ ¼ cos Nq þ i sin Nq ¼ 1

(199)

Nq ¼ k2p

(200)

k ¼ 0; 1; 2; /

8 > N > > >

> N1 > > : 2

N ¼ even (201) N ¼ odd

where k is the ‘quantum number’ for the ring. In this case, all energy levels are doubly degenerate except those for k ¼ 0 and k ¼ N/2 for N ¼ even. The general solution for the N-ring will hence be

cmk

2p xk ¼ 2 cos qk ¼ 2 cos k N

2pk ¼ Ak exp im N

(202)

þ 0 anticlockwise ðpositive rotationÞ  0 clockwise ðnegative rotationÞ

(203)

572

CHAPTER 14 Many-electron wavefunctions and model Hamiltonians

The general MO in complex form will be

X 2pk cm exp im 4 k ¼ Ak N m with



2pk cmk ¼ Ak exp im ; N

(204)



2pk cmk ¼ Ak exp im N

(205)

The coefficients can be expressed in real form through the transformation: cmk  cmk 2pk ¼ Ak sin m ¼ amk 2i N cmk þ cmk 2pk ¼ Ak cos m ¼ bmk 2 N giving the MOs in real form as 4sk ¼

X

cm amk ; 4ck ¼

X

m

(206) (207)

cm bmk

(208)

m

Problem 14.13 gives the application of the general formulae in real form to the benzene case (N ¼ 6).

14.8.4 Alternant hydrocarbons For the allyl radical (linear chain with N ¼ 3), the Hu¨ckel’ secular equation gives

with the ordered roots x1 ¼

  D3 ¼ x x2  2 ¼ 0

(209)

pffiffiffi 2;

(210)

x2 ¼ 0;

pffiffiffi x3 ¼  2

The corresponding MOs are (Problem 14.12) pffiffiffi c þ 2c 2 þ c 3 c c ; 42 ¼ 1pffiffiffi 3 ; 41 ¼ 1 2 2

43 ¼

c1 

pffiffiffi 2c2 þ c3 2

(211)

Figure 14.6 gives the diagram of the MO levels for the allyl radical and their occupation by the electrons in the ground state, Figure 14.7 a sketch of the resulting MOs. The electron configuration of the radical is 421 42 , giving for the electron density PðrÞ ¼ Pa1 þ Pb1 ¼ c21 þ c22 þ c23

(212)

a uniform charge distribution (one electron onto each atom), and for the spin density of the doublet (S ¼ 1/2) with MS ¼ 1/2  1 2 QðrÞ ¼ Pa1  Pb1 ¼ (213) c þ c23 2 1

14.8 Hu¨ckel’s theory

573

FIGURE 14.6 MO diagram for the allyl radical (N ¼ 3)

FIGURE 14.7 The three MOs of the allyl radical (N ¼ 3)

FIGURE 14.8 Electron (a) and spin density (b) MO distributions in the allyl radical (N ¼ 3)

According to the present HT, the unpaired electron (spin a) is 1/2 onto atom one and 1/2 onto atom three, and zero at atom two. This MO result is, however, incorrect, and we shall see in the next chapter that the VB calculation of the spin density in C3 H$5 shows that there is a negative spin density at the central atom, in agreement with the results from experimental electron spin resonance (ESR) spectra. The error in the MO result is due to the lack of any electron correlation in the wavefunction. Figure 14.8 gives the electron and spin density MO distributions in the allyl radical. The p bond energy (units of b) of the allyl radical is pffiffiffi (214) DEp ðallylÞ ¼ 2 2 ¼ 2:828

574

CHAPTER 14 Many-electron wavefunctions and model Hamiltonians

while that of an ethylenic double bond is DEp ðethyleneÞ ¼ 2

(215)

The difference 0.828 (an attractive stabilizing energy) is called delocalization energy of the double bond in allyl. For the p electron system of the benzene molecule (N ¼ 6), the Hu¨ckel’ secular equation gives   2 (216) D6 ¼ x6  6x4 þ 9x2  4 ¼ x2  4 x2  1 ¼ 0 with the ordered roots x1 ¼ 2;

x2 ¼ x3 ¼ 1;

x4 ¼ x5 ¼ 1;

x6 ¼ 2

(217)

The corresponding MOs in real form are (Problems 14.13 and 8.10) 1 41 ¼ pffiffiffi ðc1 þ c2 þ c3 þ c4 þ c5 þ c6 Þ 6 1 42 ¼ ðc1  c3  c4 þ c6 Þ w x 2 1 43 ¼ pffiffiffiffiffi ðc1 þ 2c2 þ c3  c4  2c5  c6 Þ w y 12 1 44 ¼ pffiffiffiffiffi ðc1  2c2 þ c3 þ c4  2c5 þ c6 Þ w x2  y2 12

(218)

1 45 ¼ ðc1  c3 þ c4  c6 Þ w xy 2 1 46 ¼ pffiffiffi ðc1  c2 þ c3  c4 þ c5  c6 Þ 6 A direct explicit solution of the homogeneous system giving the coefficients of the real MOs for benzene was given elsewhere (Magnasco, 2010a). Figure 14.9 gives the diagram of the MO levels for the p electrons in benzene and their occupation by the electrons in the ground state, while Figure 14.10 gives a sketch of the real MOs. In the drawings, we have reported only the signs of the upper lobes of the 2pz AOs. To find the charge distribution resulting from the ground-state electron configuration 421 422 423 in benzene we must add up the contributions from a and b spin: PðrÞ ¼ Pa1 ðrÞ þ Pb1 ðrÞ ¼ ra ðrÞ þ rb ðrÞ The r and r components of the p electron distribution function for benzene are equal:  1 2 c1 þ c22 þ c23 þ c24 þ c25 þ c26 ra ¼ rb ¼ 2 a

(219)

b

(220)

so that we obtain for the density of the p electrons PðrÞ ¼ ra ðrÞ þ rb ðrÞ ¼ c21 þ c22 þ c23 þ c24 þ c25 þ c26

(221)

14.8 Hu¨ckel’s theory

575

FIGURE 14.9 MO diagram for benzene (N ¼ 6)

FIGURE 14.10 The six real MOs for the benzene ring (N ¼ 6)

and the charge distribution of the p electrons in benzene is uniform (one electron onto each carbon atom), as expected for an alternant hydrocarbon, whereas the spin density is zero: QðrÞ ¼ ra ðrÞ  rb ðrÞ ¼ 0 as it must be for a singlet (S ¼ 0, 2S þ 1 ¼ 1) state.

(222)

576

CHAPTER 14 Many-electron wavefunctions and model Hamiltonians

As far as the delocalization energy is concerned, we have for benzene DE p ðbenzeneÞ ¼ 2  ðþ2Þ þ 4  ðþ1Þ ¼ 8 p

(223)

DE ð3 ethylenesÞ ¼ 3  ðþ2Þ ¼ 6

(224)

DE p ðbenzeneÞ  DE p ð3 ethylenesÞ ¼ 2

(225)

so that is the delocalization energy (units of b) for the p system of the benzene molecule, while the stabilization energy due to the closure of the ring is (see Figure 14.11) DE p ðbenzeneÞ  DE p ðhexatrieneÞ ¼ 8  6:988 ¼ 1:012

(226)

So, the closure of the chain to the ring with N ¼ 6 (no tensions in the s skeleton) is energetically favoured, while delocalization of the p bonds is the largest. This explains the great stability of the benzene ring, where the three delocalized p bonds have a nature completely different from that of three ethylenic double bonds. Figure 14.11 shows the Hu¨ckel’s results for the MO levels resulting for N ¼ 4, 6, 8, 10 in the case of the open chain and the ring. As a matter of fact, the rings for N ¼ 8 and N ¼ 10 are not planar, N ¼ 8 having a ‘tube’ conformation (the system is not aromatic), N ¼ 10 being unstable in view of the strong overcrowding involving trans H-atoms inside the ring, what favours its isomerization to di-hydro-naphthalene (Figure 14.12).

FIGURE 14.11 Hu¨ckel’s MO levels for the open chain (above) and the ring (bottom) with N ¼ 4, 6, 8, 10

14.8 Hu¨ckel’s theory

577

FIGURE 14.12 The rings with N ¼ 8 and N ¼ 10

We note from Figure 14.11 that the antibonding levels are all symmetrical about zero, and correspond to the bonding levels changed in sign. So, energy levels occur in pairs, with a p bond energy  x, and the coefficients of the paired orbitals are either the same or simply change sign. These are the properties of alternant hydrocarbons, which are conjugated molecules in which the carbon atoms can be divided into two sets, crossed and circled, such that no two members of the same set are bonded together (Figure 14.13a). They are characterized (as we have already seen for allyl and benzene) by a uniform charge distribution, and do not present any dipole moment in the ground state. In non-alternant hydrocarbons (Figure 14.13b), two circles (or two crosses) are close together, and these properties are lost. Ground-state azulene has a dipole moment of about 0.4ea0 directed from ring 5 to ring 7.

14.8.5 An introduction to band theory of solids Increasing the number of interacting AOs increases the number of resulting MOs. For the polyene chain CNHNþ2 the MO levels, which always range between a þ 2b and a  2b, become closer and closer up to transforming in bands (a continuous succession of molecular levels) which are characteristic of solids. Using the general formula derived by Coulson (1938) for the orbital energy of the k-th MO in the N-atom linear polyene chain p k k ¼ 1; 2; /N (227) εk ¼ a þ 2b cos Nþ1 the MO levels for N ¼ 2,/,12 were calculated and reported in the upper diagram of Figure 14.14. The limiting values a þ 2b and a  2b are reached asymptotically when N / N. In this case, the energy difference between two successive levels tends to zero, and we have the formation of electronic bands where the MO levels form the continuum depicted in the lower part of the figure.

578

CHAPTER 14 Many-electron wavefunctions and model Hamiltonians

FIGURE 14.13 Alternant (a) and non-alternant (b) hydrocarbons

These results, apparent from the figure, are easily derived from the formula above. 1. First level (k ¼ 1) ε1 ¼ a þ 2b cos

p 0 lim ε1 ¼ a þ 2b N/N N þ1

(228)

2. Last level (k ¼ N) εN ¼ a þ 2b cos

pN ¼ a þ 2b cos Nþ1

p 1þ

1 N

0 lim εN ¼ a  2b N/N

(229)

3. Difference between two successive levels

p p p 2k þ 1 p 1 Dε ¼ εkþ1  εk ¼ 2b cos ðk þ 1Þ  cos k ¼ 4b sin sin Nþ1 Nþ1 2 Nþ1 2 Nþ1 (230) where use was made of the trigonometric identity cos a  cos b ¼ 2 sin

aþb ab sin 2 2

Hence, for N / N Dε / 0, and we have formation of a continuous band of molecular levels. 4. For N / N, therefore, the polyene chain becomes the model for the one-dimensional crystal. We have a bonding band with energy ranging from a þ 2b to a, and an antibonding band with energy ranging from a to a  2b, which are separated by the so-called Fermi level, the top of

14.9 Semiempirical MO methods

(i) MO levels of linear N–atom Polyene chain

579

α – 2β Antibonding levels Non-bonding levels α

2

3

4

5

6

7

8

9

10 11 12

Bonding levels α + 2β N

(ii) Electronic bands in the infinite polyene chain

α – βd – βs Antibonding band (unocc)

α – 2β Antibonding band (unocc)

α – βd + βs – 4β > 0

Single β

2 ⎜β – β ⎜ }Δ = Band gap

ε F Fermi level α εF

α + βd – βs

Bonding band (occ) α + 2β

Bonding band (occ) α + βd + βs

d

s

βd > β s

FIGURE 14.14 Origin of the electronic bands in solids as a limiting case of the infinite linear polyene chain

the bonding band occupied by electrons. It is important to notice that using just one b, equal for single and double bonds, there is no band gap between bonding and antibonding levels (bottom left in Figure 14.14). If we admit jbd j > jbs j, as reasonable, we have the formation of a band gap D ¼ 2jbd  bs j (bottom right in the figure), which is of great importance in the properties of solids. Metals and covalent solids, conductors and insulators, semiconductors, all can be traced back to the model of the infinite polyene chain extended to three dimensions (McWeeny, 1979).

14.9 SEMIEMPIRICAL MO METHODS The semiempirical MO methods are derived, at different levels of sophistication, from the LCAO-MO approach by making approximations in terms of parameters whose value is mostly determined by comparison with experimental results.

580

CHAPTER 14 Many-electron wavefunctions and model Hamiltonians

14.9.1 Extended Hu¨ckel’s theory The extended Hu¨ckel’s theory (EHT) is an MO theory for hydrocarbons (even saturated) proposed by Hoffmann (1963). It makes use of an atomic basis formed by the valence s and p AOs of carbon and h AOs of the hydrogen atoms. The matrix elements include now the interactions even between atoms which are not neighbours, and are given as Hmm ¼ am Hmn ¼ bmn

m ¼ s; p; h

(231)

am þ an Smn ¼K 2

(232)

where Smn is the overlap between AOs m and n. The as for the C atom are taken as the negative of the ionization potentials of the valence state of C(sp3), namely, as ¼ 21:4 eV;

ap ¼ 11:4 eV;

ah ¼ 13:6 eV

(233)

The expression (232) for the off-diagonal elements of the Hu¨ckel Hamiltonian H is called the Wolfsberg–Helmholtz approximation, where K is a constant whose value is assumed empirically (Hoffmann uses K ¼ 1.75). At variance with Hu¨ckel’s original approach, all off-diagonal terms as well as all overlaps are taken into account. As in HT, the total energy ε and the population matrix P are given by ε¼2

occ X

εi

(234)

i

P ¼ 2R ¼ 2CCy

Pmn ¼ 2

occ X

cmi cni

(235)

i

Calculations were done on many simple saturated and conjugated hydrocarbons, especially for what concerns molecular geometry, ionization energies, and torsional barriers. The method cannot be used as such for molecules containing heteroatoms.

14.9.2 The CNDO method The CNDO (Complete Neglect of Differential Overlap) method was developed by Pople’s group (Pople et al., 1965). It is an LCAO-MO-SCF theory limited to valence electrons and an STO minimal basis. It includes part of the electron repulsion, so that, at variance of Hu¨ckel’s theory, can be used also with molecules containing heteroatoms. In the Roothaan form of the SCF equations

X X 1 Fmn  Gmn Pmn (236) Ee ¼ 2 m n Pmn ¼ 2

occ X

cmi cni

(237)

i


E D

E D


Fmn ¼ cm
F^
cn ¼ cm
h^ þ G^
cn ¼ hmn þ Gmn

(238)

14.9 Semiempirical MO methods

 hmn ¼

Gmn




1
cm

 V2 þ V

cn 2

 D

E XX 1

^ ¼ cm G cn ¼ Pls ðnmjlsÞ  ðlmjnsÞ 2 s l

581

(239)

(240)

the following approximations are introduced: 1. Two-electron integrals ðlmjnsÞ ¼ gmn dml dns

(241)

It is assumed to neglect completely the differential overlap in the charge density cl ðr1 Þcm ðr1 Þ in such a way that only the Coulomb integrals survive: 8
 
< s2A
s2A ¼ gAA
(242) gmn ¼ c2m
c2n ¼ 
 : s2
s 2 ¼ g AB A

B

where the spherical integrals are evaluated analytically using STOs. To obtain a theory invariant under rotation of axes onto each centre, it is assumed that the integrals g depend only on atoms A and B. The matrix elements of G then become

 1 Pls ðmmjlsÞ  ðmljmsÞ 2 s l

 XX 1 Pls ðmmjlsÞ  gmm dml dms ¼ 2 s l

Gmm ¼

XX

¼

m on A; ls on A or B

XX 1 Pmm gAA þ Pls ðmmjllÞdls 2 s l

m on A; ll on A or B

ðAÞ ðBÞ X X X 1 ¼  Pmm gAA þ Pll gAA þ Pll gAB 2 l BðsAÞ l

¼

X 1 Pmm gAA þ PAA gAA þ PBB gAB 2 BðsAÞ

(243)

where PAA ¼

ðAÞ X

Pll ;

PBB ¼

l

are the total electron populations on atom A or B.

ðBÞ X l

Pll

(244)

582

CHAPTER 14 Many-electron wavefunctions and model Hamiltonians

 1 Pls ðmm0 jlsÞ  ðmljm0 sÞ m0 s m; both on A 2 s l 1 XX 1 Pls ðmmjm0 m0 Þdml dm0 s ¼  Pmm0 gAA ¼ 2 l s 2

 PP 1 m on A s n on B Pls ðmnjlsÞ  ðmljnÞs Gmn ¼ 2 l s 1 XX 1 Pls gmn dml dns ¼  Pmn gAB ¼ 2 l s 2

Gmm0 ¼

PP

(245)

(246)

2. One-electron integrals  hmm ¼



 X
1
cm

 V2 þ V

cn  Umm þ VAB 2 BðsAÞ Umm ¼ 

m on A

 1 1 Im þ Am  ZA  gAA 2 2

where Im ¼ ionization potential (we extract an electron from cm ) Am ¼ electron affinity (we add an electron to cm ) ZA ¼ ‘core’ charge 



VAB ¼ c2m
VB ¼ ZB c2m
c2n 
 ¼ ZB s2A
rB1 ¼ ZB gAB hmm0 ¼ 0

hmn ¼ b0AB Smn ;

CNDO=1

CNDO=2

if m0 s m both on A

b0AB ¼

b0A þ b0B 2

m on A; n on B

(247)

(248)

(249)

(250)

(251)

(252)

(253)

CNDO/2 parameters for atoms of the first row are given in Table 14.4. CNDO/2 turns out to be a better approximation than CNDO/1, and was next extended to the atoms of the second row (Na, Mg, Al, Si, P, S, Cl) using rather contracted 3d STOs (Santry and Segal, 1967).

14.9 Semiempirical MO methods

583

Table 14.4 CNDO/2 Parameters (eV) for First-Row Atoms Atom

H

Li

(i) STO orbital exponents/a1 0 1s 1.2 2s, 2p 0.65

Be

B

C

N

O

F

0.975

1.30

1.625

1.95

2.275

2.60

(ii) Ionization potentials and electron affinities/eV 1 ðIs þ As Þ 2 1 ðIp þ Ap Þ 2

7.18

3.11

5.95

9.59

14.05

19.32

25.39

32.37

1.26

2.56

4.00

5.57

7.27

9.11

11.08

13

17

21

25

31

39

(iii) Atomic bond parameters/eV 9 9 b0 A

3. Matrix elements With the previous approximations, the elements of the Fock matrix become 

X   1 1 ðPBB  ZB ÞgAB Fmm ¼  Im þ Am þ ðPAA  ZA Þ  Pmm  1 gAA þ 2 2 BðsAÞ

m on A (254)

1 Fmm0 ¼ Gmm0 ¼  Pmm0 gAA 2 1 Fmn ¼ b0AB Smn  Pmn gAB 2

m0 s m both on A

(255)

m on A; n on B

(256)

with the secular equation, assuming orthonormal AOs jF  ε1j ¼ 0 0 MO-SCF

(257)

CNDO/2 gives reasonable values for molecular geometries, valence angles, dipole moments and bending force constants. Torsional barriers are usually underestimated, and the method is not appropriate either for the calculation of spin densities (see INDO below) or for conjugated molecules. 4. Molecular energy The Born–Oppenheimer molecular energy in the CNDO/2 approximation can be written as X X EA þ EAB (258) E ¼ E e þ EN ¼ A

where EA ¼

X m

Pmm Umm þ

A <  lr ¼0 vxi vxi r > : 4r ðx1 ; x2 ; /; xN Þ ¼ 0

i ¼ 1; 2; /; N r ¼ 1; 2; /; m

14.10 Problems 14

587

Hint: It is convenient to subtract from the original function f the m constraints, each one multiplied by a suitable Lagrange multiplier l, and to write then the extremum conditions for the resulting unconstrained N-variable function (Problem 14.8). 14.8. Show that the unconstrained variation of the energy functional ε ¼ HM1 is fully equivalent to the result obtained by the method of Lagrange multipliers. Answer: dH  εdM ¼ 0 Hint: Take an infinitesimal variation of the energy functional ε taking into account the normalization condition M ¼ 1. 14.9. Show the identity between the first and last part in the right-hand side of Eqn (110) for the Coulomb operator J. Answer: For the Coulomb integral J, we have X X X hdji jJ1 jji i þ hji jJ1 jdji i ¼ hji jdJ1 jji i i

i

i

Hint: Use the definitions (92) and (101) of r and J, and interchange in the integrals summation indices i; i0 and electron labels 1, 2. 14.10. Eliminate spin from the Fock operator for closed shells. Answer: ^ 1 Þ ¼ 1 V21 þ V1 þ 2Jðr1 Þ  Kðr ^ 1Þ Fðr 2 Z

where: Jðr1 Þ ¼

Rðr2 ; r2 Þ dx2 ; r12

^ 1Þ ¼ Kðr

Z dx2

Rðr1 ; r2 Þ ^ P r1 r2 r12

are spinless Coulomb and exchange potentials, and R(r1;r2) the spinless density matrix for closed shells. Hint: ^ 1 s1 Þ act on the SO function 4i ðr1 Þaðs1 Þ, say, and integrate over the spin ^ 1 Þ ¼ Fðr Make Fðx variable of electron 2. 14.11. Find the projector properties of the density matrix R over the non-orthogonal AO basis c with metric M. Answer: RMR ¼ RðidempotencyÞ; tr RM ¼ nðconservationÞ where n is the number of the doubly occupied MOs. Hint Use the properties of the density matrix Rðr; r0 Þ over the orthogonal MOs and the LCAO expression for the latter.

588

CHAPTER 14 Many-electron wavefunctions and model Hamiltonians

14.12. Find roots and coefficients of the AOs in the MOs for the allyl radical (N ¼ 3) using the general formulae for the linear chain derived in Section 14.8.2. Answer: pffiffiffi pffiffiffi x1 ¼ 2; x2 ¼ 0; x3 ¼  2 pffiffiffi pffiffiffi c 1 þ 2c 2 þ c 3 c1  c3 c1  2c2 þ c3 ; 42 ¼ pffiffiffi ; 43 ¼ 41 ¼ 2 2 2 Hint: Use Eqns (189) and (190) of Coulson’s general solution for the linear chain with N ¼ 3. 14.13. Find roots and coefficients of the AOs in the MOs in real form for the p electrons in benzene (N ¼ 6) using the general formulae for the closed chain derived in Section 14.8.3. Answer: The results are given in Eqns (217) and (218). Hint: Use Eqns (202,206–208) of Coulson’s general solution for the closed chain with N ¼ 6.

14.11 SOLVED PROBLEMS 14.1. Using Dirac’s formula for N ¼ 3 and the properties of determinants, we find for the component with MS ¼ þ1/2

   3^ ^ 2 I þ P12 þ P^13 þ P^23 1s 1s2s S^ 1s1s 2s ¼ 4         3 ¼ 1s 1s2s þ 1s 1s2s þ 1s 1s2s þ 1s 1s2s 4

 1 1   3   ¼ 1s 1s2s ¼ þ 1 1s1s 2s 4 2 2 where the second and third determinants cancel altogether, and the last one vanishes because of Pauli’s principle. 14.2. Using Dirac’s formula for N ¼ 4 and the properties of determinants, we find     2 S^ 1s 1s 2s 2s ¼ P^12 þ P^13 þ P^14 þ P^23 þ P^24 þ P^34 1s1s 2s2s         ¼ 1s1s 2s2s þ 1s 1s2s 2s þ 1s 1s2s 2s þ 1s1s2s 2s       þ 1s1s 2s2s þ 1s 1s 2s2s ¼ 0ð0 þ 1Þ1s1s 2s2s where the first and sixth determinants are nothing but the second and fifth with minus sign, while the third and fourth are identically zero for the exclusion principle. 14.3. The MO wavefunction for ground-state H2 ð1 Sþ g Þ is    1 2 1 þ J sg ; Sg ¼ sg sg  ¼ sg ðr1 Þsg ðr2 Þ pffiffiffi ½aðs1 Þbðs2 Þ  bðs1 Þaðs2 Þ 2 where we notice that the space part is symmetric and the spin part antisymmetric.

14.11 Solved problems

Then

589

8 1 > > > Z < sg ðr1 Þsg ðr2 Þ pffiffi2ffi ½aðs1 Þbðs2 Þ  bðs1 Þaðs2 Þ r1 ðx1 ; x1 Þ ¼ 2 dx2 >  1 > > : sg ðr1 Þsg ðr2 Þ pffiffiffi ½a ðsÞb ðs2 Þ  b ðs1 Þa ðs2 Þ 2 ¼ sg ðr1 Þsg ðr1 Þ½aðs1 Þa ðs1 Þ þ bðs1 Þb ðs1 Þ

P1 ðra1 ; ra1 Þ ¼ Pa1 ðr1 ; r1 Þ ¼ P1 ðrb1 ; rb1 Þ ¼ Pb1 ðr1 ; r1 Þ ¼ Rðr1 ; r1 Þ ¼ sg ðr1 Þsg ðr1 Þ where R is the notation usual in the MO-LCAO theory. Then



2 Pðr1 ; r1 Þ ¼ Pa1 ðr1 ; r1 Þ þ Pb1 ðr1 ; r1 Þ ¼ 2sg ðr1 Þsg ðr1 Þ ¼ 2
sg ðr1 Þ


is the electron density, and Qðr1 ; r1 Þ ¼ Pa1 ðr1 ; r1 Þ  Pb1 ðr1 ; r1 Þ ¼ 0 the spin density. The electron density integrates to the total number of electrons: Z Z 2  dr1 Pðr1 ; r1 Þ ¼ 2 dr1 sg ðr1 Þ ¼ 2 aþb since the bonding MO sg ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi is normalized to 1. The electron density distribution in the 2 þ 2S H2 molecule can be further analyzed into its atomic and overlap contributions, Eqns (50–53): Pðr1 ; r1 Þ ¼ 2

ða þ bÞ2 a2 ðr1 Þ þ b2 ðr1 Þ þ aðr1 Þbðr1 Þ þ bðr1 Þaðr1 Þ ¼ 2 þ 2S 1þS

¼ qA a2 ðr1 Þ þ qB b2 ðr1 Þ þ qAB

aðr1 Þbðr1 Þ bðr1 Þaðr1 Þ þ qBA S S

where q A ¼ qB ¼

1 1þS

is the fraction of the electronic charge on A or B distributed with the normalized atomic densities a2(r1) or b2(r1); S qAB ¼ qBA ¼ 1þS is the fraction of the electronic charge distributed with the normalized overlap densities a(r1) b(r1)/S or b(r1)a(r1)/S. We can check that qA þ qB þ qAB þ qBA ¼ 2 the total number of electrons in the bonding MO sg. At the equilibrium bond distance Re ¼ 1.4a0, for 1s AOs: 3 4 3 S z : qA ¼ qB ¼ z 0:57; qAB ¼ qBA ¼ z 0:43 4 7 7

590

CHAPTER 14 Many-electron wavefunctions and model Hamiltonians

FIGURE 14.15 Origin of the quadrupole moment in H2

showing that in the bound H2 molecule the charge on atoms is less than that in the free H atoms (qA ¼ qB ¼ 1): 0.43 electrons are shifted from each atom to the interbond region, screening internuclear repulsion. The two electrons are distributed in the molecule according to the scheme of Figure 14.15. Such a distribution of the electronic charge (the dot means the midpoint of the bond) determines the first non-zero electric moment of the centrosymmetric H2 molecule, its quadrupole moment. 14.4. The three MO wavefunctions of the triplet 3 ðSþ u Þ state of H2 are   sg su  S ¼ 1; MS ¼ 1    1  pffiffiffi sg su  þ sg su  2   sg su 

0 1

and can be written by separating space from spin part as 8 > aðs1 Þaðs2 Þ > > > > <  1 1  pffiffiffi sg ðr1 Þsu ðr2 Þ  su ðr1 Þsg ðr2 Þ pffiffiffi ½aðs1 Þbðs2 Þ þ bðs1 Þaðs2 Þ > 2 2 > > > > : bðs1 Þbðs2 Þ Proceeding as we did before, we then have for the different spin components of the wavefunction • S ¼ 1, MS ¼ 1 (all electrons have a spin) h i r1 ðx1 ; x1 Þ ¼ sg ðr1 Þsg ðr1 Þ þ su ðr1 Þsu ðr1 Þ aðs1 Þa ðs1 Þ P1 ðra1 ; ra1 Þ ¼ Pa1 ðr1 ; r1 Þ ¼ sg ðr1 Þsg ðr1 Þ þ su ðr1 Þsu ðr1 Þ;

P1 ðrb1 ; rb1 Þ ¼ Pb1 ðr1 ; r1 Þ ¼ 0

Pðr1 ; r1 Þ ¼ Pa1 ðr1 ; r1 Þ ¼ s2g ðr1 Þ þ s2u ðr1 Þ ¼ Qðr1 ; r1 Þ In this case, the spin density coincides with the electron density. • S ¼ 1, MS ¼ 0 (electrons have both spins) i 1h r1 ðx1 ; x1 Þ ¼ sg ðr1 Þsg ðr1 Þ þ su ðr1 Þsu ðr1 Þ ½aðs1 Þa ðs1 Þ þ bðs1 Þb ðs1 Þ 2 i 1h Pa1 ðr1 ; r1 Þ ¼ Pb1 ðr1 ; r1 Þ ¼ sg ðr1 Þðr1 Þsg þ su ðr1 Þsu ðr1 Þ 2

14.11 Solved problems

591

Pðr1 ; r1 Þ ¼ Pa1 ðr1 ; r1 Þ þ Pb1 ðr1 ; r1 Þ ¼ s2g ðr1 Þ þ s2u ðr1 Þ Qðr1 ; r1 Þ ¼ Pa1 ðr1 ; r1 Þ  Pb1 ðr1 ; r1 Þ ¼ 0 The spin density is zero, but the electron density is the same as before. • S ¼ 1, MS ¼ 1 (all electrons have b spin) h i r1 ðx1 ; x1 Þ ¼ sg ðr1 Þsg ðr1 Þ þ su ðr1 Þsu ðr1 Þ bðs1 Þb ðs1 Þ Pb1 ðr1 ; r1 Þ ¼ sg ðr1 Þsg ðr1 Þ þ su ðr1 Þsu ðr1 Þ;

Pa1 ðr1 ; r1 Þ ¼ 0

Pðr1 ; r1 Þ ¼ Pb1 ðr1 ; r1 Þ ¼ s2g ðr1 Þ þ s2u ðr1 Þ h i Qðr1 ; r1 Þ ¼ Pb1 ðr1 ; r1 Þ ¼  s2g ðr1 Þ þ s2u ðr1 Þ ¼ Pðr1 ; r1 Þ The spin density is the negative of the electron density. The distribution of the electronic charge in the triplet state of H2 gives qA ¼ q B ¼

1 > 1; 1  S2

qAB ¼ qBA ¼ 

S2

<  lr ¼0 i ¼ 1; 2; /; N vxi vxi r > : 4r ðx1 ; x2 ; /; xN Þ ¼ 0 r ¼ 1; 2; /; m which, together with the equations defining the m constraints, form a system of N þ m equations allowing the determination of the extremum points ðx01 ; x02 ; /; x0N Þ and the Lagrange multipliers ðl1 ; l2 ; /; lm Þ. In our case, f and 4r are functionals, and the variables xi are the SOs ji ðxÞ. 14.8. Unconstrained variation of the energy functional ε ¼ HM 1 Taking an infinitesimal arbitrary variation of ε, we have dε ¼ M 1 dH  HM 11 dM ¼ M 1 ðdH  εdMÞ ¼ 0 namely, dH  εdM ¼ 0 So, the unconstrained variation of the energy functional ε is fully equivalent to the separate variation of H minus the separate variation of the constraint M (here, the normalization integral) multiplied by the Lagrange multiplier ε. ε plays hence the role of the Lagrange multiplier of Problem 14.7. 14.9. According to definitions (92) and (101) we have Z Z rðx2 ; x2 Þ X j 0 ðx2 Þji0 ðx2 Þ dx2 i ¼ J1 ¼ dx2 r12 r12 i0

14.11 Solved problems

Z dJ1 ¼ X

dx2

drðx2 ; x2 Þ X ¼ r12 i0

Z dx2

595

ji0 ðx2 Þdji0 ðx2 Þ þ dji0 ðx2 Þji0 ðx2 Þ r12

hji jdJ1 jji i

i

¼

XZ i

" dx1 ji ðx1 Þ

XZ i0

j 0 ðx2 Þdji0 ðx2 Þ þ dji0 ðx2 Þji0 ðx2 Þ dx2 i r12

#

¼ interchange in the integrals summation indices i; i0 and electron labels 1; 2 " # XZ X ji ðx1 Þdji ðx1 Þ þ dji ðx1 Þji ðx1 Þ  ¼ dx1 dx2 ji0 ðx2 Þ ji0 ðx2 Þ r12 i i0 " #  XZ X j 0 ðx2 Þj 0 ðx2 Þ i i dx2 dx1 dji ðx1 Þ ¼ ji ðx1 Þ r12 i i0 " # XZ X X X ji0 ðx2 Þji0 ðx2 Þ  dx2 dx1 ji 0 ðx1 Þ dji ðx1 Þ ¼ þ hdji jJ1 jji i þ hji jJ1 jdji i r12 i i i i0 which is the required result. The same holds for the variation of the exchange integral dK^1 . ^ 1 Þ for closed shells. 14.10. Elimination of spin from Fðx Z Z 1 2 rðr2 s2 ; r2 s2 Þ rðr1 s1 ; r2 s2 Þ ^ ^ ^ Pr1 s1 ;r2 s2 Fðx1 Þ ¼ Fðr1 s1 Þ ¼  V1 þ V1 þ dr2 ds2  dr2 ds2 2 r12 r12

Z ^ 1 s1 Þ4i ðr1 Þaðs1 Þ ¼  1 V21 þ V1 4i ðr1 Þaðs1 Þ þ dr2 ds2 Rðr2 ; r2 Þ½aðs2 Þa ðs2 Þ Fðr 2 r12 Z Rðr1 ; r2 Þ ½aðs1 Þa ðs2 Þ þ bðs2 Þb ðs2 Þ4i ðr1 Þaðs1 Þ  dr2 ds2 r12 þ bðs1 Þb ðs2 Þ4i ðr2 Þaðs2 Þ

Z 1 Rðr2 ; r2 Þ ¼  V21 þ V1 4i ðr1 Þaðs1 Þ þ 2 dr2 4i ðr1 Þaðs1 Þ 2 r12 Z Rðr1 ; r2 Þ ^ Pr1 r2 4i ðr2 Þaðs1 Þ  dr2 r12

1 2 ^ ¼  V1 þ V1 þ 2Jðr1 Þ  Kðr1 Þ 4i ðr1 Þaðs1 Þ 2 from which follows Eqn (117) of the main text. 14.11. We start from the definitions: XX Rðr; r0 Þ ¼ cðrÞRcy ðr0 Þ ¼ cm ðrÞRmn cn ðr0 Þ; m

n

cy c ¼ M

596

CHAPTER 14 Many-electron wavefunctions and model Hamiltonians

Then i. Idempotency

Z

Z

dr00 Rðr; r00 ÞRðr00 ; r0 Þ ¼ Rðr; r0 Þ

dr00 cðrÞRcy ðr00 Þ $ cðr00 ÞRcy ðr0 Þ ¼ cðrÞRMRcy ðr0 Þ ¼ cðrÞRcy ðr0 Þ

whence it follows RMR ¼ R ii. Conservation Z Z XX XX drRðr; rÞ ¼ Rmn drcn ðrÞcm ðrÞ ¼ Rmn Mnm ¼ tr RM ¼ n m

n

m

n

where n ¼ N/2 is the number of doubly occupied MOs. 14.12. Allyl radical calculation from the general formulae for the linear chain with N ¼ 3. We recall that Hu¨ckel’s roots and coefficients are nothing but the eigenvalues and eigenvectors of the Hu¨ckel matrix H. We have p k ¼ 1; 2; 3 N ¼ 3 qk ¼ k 4 i. Roots xk ¼ 2 cos k x2 ¼ 2 cos

p 4

x1 ¼ 2 cos

2p ¼ 0; 4

pffiffiffi p 2 ¼ pffiffiffi ¼ 2; 4 2

x3 ¼ 2 cos

pffiffiffi  3p 2 ¼ 2 cos 135 ¼ pffiffiffi ¼  2 4 2

ii. Coefficients cmk ¼ sin m

kp 4

If C is a normalization factor



p p 2p 3p ¼ C c1 sin þ c2 sin þ c3 sin 4 4 4 4 m¼1 pffiffiffi

1 1 c þ 2c2 þ c3 ¼ C c1 pffiffiffi þ c2 þ c3 pffiffiffi ¼ 1 2 2 2

41 ¼ C

3 P

3 X

cm sin m

p c1  c3 ¼ pffiffiffi 2 2 m¼1 pffiffiffi 3 X 3p c1  2c2 þ c3 43 ¼ C ¼ cm sin m 2 4 m¼1

42 ¼ C

cm sin m

14.11 Solved problems

597

14.13. p electron benzene calculation from the general formulae (real form) for the closed chain with N ¼ 6. This application is of some interest since it illustrates well some arbitrariness connected to the numbering of the atoms (hence, of the AOs) along the benzene ring, and to the choice of coordinate axes in the molecular plane. Our reference is that of Figure 14.9, where atoms are numbered in an anticlockwise sense with atom one placed in the positive xy-plane (the same reference chosen in Problem 8.10 where benzene is treated according to C6v symmetry). This choice is, however, arbitrary, different equivalent descriptions being connected by unitary transformations among the MOs. We have 2p p ¼k 6 3

N¼6

qk ¼ k

p 3

x0 ¼ 2;

k ¼ 0; 1; 2; 3

i. Roots xk ¼ 2 cos k

x2 ¼ x2 ¼ 2 cos

x1 ¼ x1 ¼ 2 cos

2p ¼ 1; 3

x3 ¼ 2 cos

p ¼ 1; 3

3p ¼ 2 cos p ¼ 2 3

ii. Real coefficients amk ¼ A sin m

kp ; 3

bmk ¼ A cos m

kp 3

where A is a normalization factor; 40 ¼ 4c0 ¼ A 4s1 ¼ A 4c1 ¼ A 4s2 ¼ A 4c2 ¼ A

6 P

6 P

1 cm cos 0 ¼ pffiffiffi ðc1 þ c2 þ c3 þ c4 þ c5 þ c6 Þ 6 m¼1

cm sin m

p 1 ¼ ðc þ c2  c4  c5 Þ 3 2 1

cm cos m

p 1 ¼ pffiffiffiffiffi ðc1  c2  2c3  c4 þ c5 þ 2c6 Þ 3 12

cm sin m

2p 1 ¼ ðc1  c2 þ c4  c5 Þ 3 2

cm cos m

2p 1 ¼ pffiffiffiffiffi ð  c1  c2 þ 2c3  c4  c5 þ 2c6 Þ 3 12

m¼1 6 P m¼1 6 P m¼1 6 P m¼1

43 ¼ 4c3 ¼ A

6 P m¼1

cm cos m

3p 1 ¼ pffiffiffi ð  c1 þ c2  c3 þ c4  c5 þ c6 Þ 3 6

The MOs obtained in this way differ from those of Figure 14.10 by a unitary transformation U. Matrix U is the result of a two-step transformation on the Coulson set, first a rotation of 60 of

598

CHAPTER 14 Many-electron wavefunctions and model Hamiltonians

the coordinate axes about z (passive transformation, see Chapter 8), so that atom one is replaced by two, two by three, and so on; second a transformation restoring the order of the basic vectors of E1 and E2 symmetry as defined in Chapter 8. So, the transformation between the two sets is   ð41 42 43 44 45 46 Þ ¼ 40 4s1 4c1 4s2 4c2 43 U 0 1 1 pffiffiffi B C 1 3 B C B C 0 B C 2 2 B C pffiffiffi B C 1 3 B C B C  B C 2 2 C U¼B pffiffiffi B C 1 3 B C B C B C 2 2 B C pffiffiffi B C 1 3 B C B C  0 @ A 2 2 1 We can check this result for the E1 degenerates set: pffiffiffi pffiffiffi pffiffiffi pffiffiffi



1 s 1 1 1 1 2 3 c 3 3 3 4 þ þ  4 ¼ c1 $ pffiffiffi þ c2 $ pffiffiffi þ c3 0  $ pffiffiffi 2 1 4 4 2 1 2 2 2 2 3 2 3 2 3 pffiffiffi pffiffiffi pffiffiffi



1 1 1 1 2 3 3 3 þ c4   $ pffiffiffi þ c5  þ $ pffiffiffi þ c6 0 þ $ pffiffiffi 4 4 2 2 2 2 3 2 3 2 3 1 ¼ ðc1  c3  c4 þ c6 Þ ¼ 42 wx 2 pffiffiffi



pffiffiffi pffiffiffi 1 1 1 2 3 s 1 c 3 1 1 3 1 1 4  4 ¼ c1 $  $ pffiffiffi þ c2 $ þ $ pffiffiffi þ c3 0 þ $ pffiffiffi 2 2 2 3 2 2 2 3 2 2 3 2 1 2 1 2 2 pffiffiffi pffiffiffi



3 1 1 3 1 1 1 1 1 2 þ c4  $ þ $ pffiffiffi þ c5  $  $ pffiffiffi þ c6 0  $ pffiffiffi 2 2 2 2 2 3 2 2 2 3 2 2 3 pffiffiffi

4 2 2 4 2 3 2 ¼ c þ c þ c  c  c  c 4 3 1 3 2 3 3 3 4 3 5 3 6 1 ¼ pffiffiffiffiffi ðc1 þ 2c2 þ c3  c4  2c5  c6 Þ ¼ 43 wy 12

CHAPTER

Valence bond theory and the chemical bond

15

CHAPTER OUTLINE 15.1 Introduction ............................................................................................................................... 600 15.2 The Chemical Bond in H2 ............................................................................................................ 601 15.2.1 Failure of the MO theory for ground-state H2 ............................................................602 15.2.2 The Heitler–London theory for H2 ............................................................................608 15.2.3 Equivalence between MO-CI and full VB for ground-state H2 and improvements in the wavefunction .............................................................................................. 611 15.2.4 The orthogonality catastrophe in the covalent VB theory for ground-state H2 ...............618 15.3 Elementary VB Methods .............................................................................................................. 623 15.3.1 General formulation of VB theory ............................................................................623 15.3.2 Construction of VB structures for multiple bonds ......................................................626 15.3.3 The allyl radical (N ¼ 3) .........................................................................................627 15.3.4 Cyclobutadiene (N ¼ 4) ..........................................................................................630 15.3.5 VB description of simple molecules.........................................................................631 15.4 Pauling’s VB Theory for Conjugated and Aromatic Hydrocarbons ................................................... 641 15.4.1 Pauling’s formula for the matrix elements of singlet covalent VB structures ................642 15.4.2 Cyclobutadiene......................................................................................................644 15.4.3 Butadiene .............................................................................................................645 15.4.4 Allyl radical...........................................................................................................646 15.4.5 Benzene ...............................................................................................................647 15.4.6 Naphthalene .........................................................................................................655 15.4.7 Derivation of The Pauling’s formula for H2 and cyclobutadiene ..................................657 15.5 Hybridization and Directed Valency in Polyatomic Molecules........................................................ 661 15.5.1 sp2 Hybridization in H2O........................................................................................661 15.5.2 VB description of H2O............................................................................................662 15.5.3 Properties of hybridization......................................................................................664 15.5.4 The principle of maximum overlap in VB theory........................................................667 15.6 Problems 15 .............................................................................................................................. 669 15.7 Solved Problems ........................................................................................................................ 671

Elementary Molecular Quantum Mechanics 2E. http://dx.doi.org/10.1016/B978-0-444-62647-9.00015-4 Ó 2013, 2007 Elsevier B.V. All rights reserved

599

600

CHAPTER 15 Valence bond theory and the chemical bond

15.1 INTRODUCTION We shall see in Chapter 16 how increasingly sophisticated wavefunctions can be constructed for molecules starting from the Hartree–Fock approximation. However, the independent particle model, of which Hartree–Fock is the most refined expression, treats the molecule as a sea of uncorrelated electrons moving in the field of nuclei of given symmetry. The one-configuration approach yields at its best a single determinant wavefunction of delocalized molecular orbitals, whose energies bear some relation to the negative of the experimentally detectable ionization potentials (the so-called Koopmans’ theorem), but are otherwise the expression of the nature and limitations of the basis set. Molecular orbitals are many-centre one-electron functions reflecting molecular symmetry, and may change their form depending on unitary transformations among the basic orbitals. We saw that all physical properties of the system are embodied in the fundamental invariant r, the Fock–Dirac density matrix, which has semiempirically been extended to include correlation effects in density functional theory. What is lacking in the molecular orbital (MO) approach is any direct relation with the chemical bond and its stereochemistry. Even if molecular geometries can be obtained theoretically, within the correlation error, by optimizing bond lengths and bond angles, MO theory in its first approximation fails to describe bond dissociation, even in the simplest case of the two-electron bond in H2. The conventional chemical idea of a molecule as made by inner shells, chemical bonds and lone pairs is absent in an MO description of the molecule, although chemical intuition might help in avoiding brute force calculations. Even if the chemical bond is difficult to be defined exactly, we can measure experimentally the length of the O–H bonds in H2O and the angle they make in the molecule. It is certainly more familiar to chemists and molecular physicists the idea that a molecule is made by atoms that are bound together by some kind of electric forces. An interesting discussion on the origin of the chemical bond was given by Kutzelnigg (1990). The idea of the covalent bond stems directly from the pioneering work by Heitler and London (1927), where they describe in a correct qualitative way bond dissociation in the ground state of the H2 molecule. The Heitler–London (HL) theory appears as the first step in a possible perturbative improvement to the wavefunction, where polarization and correlation corrections can be accounted for in second order, yielding results that are almost in perfect agreement with the most advanced theoretical calculations and with experimental results (see Chapter 17). HL theory can be considered as the elementary formulation of the so-called valence bond (VB) theory in terms of covalent VB structures. These ideas were next extended by Pauling1 (1933) to the VB description of the p electron bonds in aromatic and conjugated hydrocarbons. The ‘resonance’ between Kekule´ structures, a great intuition of an experimental organic chemist,2 stems directly from the quantum mechanical treatment of the interaction between VB structures describing localized p bonds in benzene. One of the greatest problems of the VB theory, the preparation of suitably directed hybrids which should then be involved in the chemical bond, was solved in recent advances of the theory (Cooper et al., 1987), allowing for optimization of large basis sets. The other problem is connected with the 1

Pauling Linus Carl 1901–1994, U.S. quantum chemist, Professor at the Caltech; 1954 Nobel Prize for Chemistry and 1963 Nobel Prize for Peace. 2 Kekule´ Friedrich August von Stradonitz 1829–1896, German organic chemist, Professor at the Universities of Gand (Belgium), Heidelberg and Bonn.

15.2 The chemical bond in H2

601

non-orthogonality of VB structures, and this still remains a problem, especially in the evaluation of the matrix elements of the Hamiltonian. Emphasis in this chapter will always be on elementary VB methods and on how they can qualitatively help in studying electronic molecular structure, in a strict correspondence between quantum mechanical VB structures and chemical formulae. The content of this chapter will be the following. We first examine the possibility of forming the two-electron bond in H2, showing the failure of the simple MO approach in describing dissociation of the molecule into atoms. HL theory is then introduced as the simplest way of solving the dissociation problem, and the equivalence between MO-configuration interaction (CI) and full VB (covalent þ ionic) wavefunctions for H2 is fully discussed. The orthogonality catastrophe occurring in the covalent VB theory of H2 is examined, and the way of overcoming it suggested in detail. After this introductory part, the general formulation of elementary VB theory is presented, comparing VB and MO methods, then giving a qualitative VB description of many simple molecules. Pauling VB theory of p electron systems is presented in Section 15.4, with applications to a few important conjugated and aromatic molecules, and its theoretical failures and possible corrections are discussed. Finally, the problem of hybridization and of directed valency is briefly discussed, particularly with reference to the H2O molecule. An outline of the most recent advances in the ab initio VB theory involving optimization of large basis sets of atomic orbitals (generalized valence bond theory) will be discussed later in Section 16.4.3. Problems and solved problems conclude the chapter as usual.

15.2 THE CHEMICAL BOND IN H2 We shall now examine the formation of the two-electron chemical bond in the H2 molecule in terms of elementary MO and VB theories using a minimal basis set of atomic orbitals (AOs) centred at the two nuclei A and B. With reference to Figure 15.1, the molecular Hamiltonian for H2 in the Born– Oppenheimer approximation of Chapter 20 will be 1 1 1 þ H^ ¼ H^e þ ¼ h^1 þ h^2 þ R r12 R

FIGURE 15.1 Reference system for the H2 molecule

(1)

602

CHAPTER 15 Valence bond theory and the chemical bond

FIGURE 15.2 The overlap between a(r) and b(r) is the dashed area

where H^e is the electronic Hamiltonian, 1 1 1 h^ ¼  V2   2 rA rB

(2)

the one-electron molecular Hamiltonian, and atomic units (a.u.) are used throughout. In the first approximation, the two AOs are spherical 1s orbitals a(r1) and b(r2) centred at A and B, respectively, having an overlap S (the dashed area of Figure 15.2): 1 aðr1 Þ ¼ pffiffiffiffi expðrA1 Þ; p

1 bðr2 Þ ¼ pffiffiffiffi expðrB2 Þ p

(3)

Z S ¼ hajbi ¼

draðrÞbðrÞ ¼ SðRÞ

(4)

The orbitals in Eqn (3) are normalized 1s Slater-type orbitals (STOs) with orbital exponent c0 ¼ 1.

15.2.1 Failure of the MO theory for ground-state H2 The one-configuration MO description of ground-state H2 is given by the two-electron (normalized) singlet Slater determinant of doubly occupied sg MOs:     1 þ (5) J MO; 1 Sg ¼ sg sg  ¼ sg ðr1 Þsg ðr2 Þ pffiffiffi ½aðs1 Þbðs2 Þ  bðs1 Þaðs2 Þ 2 where sg is the (normalized) bonding MO aþb sg ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 þ 2S

(6)

The corresponding (normalized) antibonding su MO ba su ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2  2S

(7)

is empty in the ground state (Figure 15.3). We observe from the figure that the splitting of the doubly degenerate atomic level under the interaction is non-symmetric for S s 0, the antibonding level being more repulsive and the bonding less attractive than the symmetric case occurring for S ¼ 0. We further notice that the LCAO (linear

15.2 The chemical bond in H2

603

FIGURE 15.3 MO diagram of ground-state H2

combination of atomic orbital) coefficients are in this case completely determined by symmetry. The MO energy for the ground state will be     D    E    1 1  1 þ 1 þ 1 þ  ^ ^ ^ þ sg sg ¼ s g s g  h1 þ h 2 þ E MO; Sg ¼ J MO; Sg jHjJ MO; Sg r12 R    1  (8) ¼ 2hsg sg þ s2g s2g þ R where 2hsg sg

2 2

a VB þ b VA þ ðabjVB Þ þ ðbajVA Þ haa þ hbb þ hba þ hab ¼ 2EH þ ¼ 1þS 1þS

 

1  2  2 2  2 1 2  2

   a a þ b b þ a b þ ðabjabÞ þ a2 ba þ b2 ab  2 s2g s2g ¼ 4 ð1 þ SÞ2

(9)

(10)

We have used for the two-electron integrals the charge density notation: ZZ 1 2

½aðr2 Þb ðr2 Þ  dr1 dr2 ½aðr1 Þb ðr1 Þ ab a b ¼ r12

(11)

The different terms in Eqns (9) and (10) have the following physical meaning: 2  2  1

a VB ¼ a  rB

(12) 2

is the attraction by the B nucleus of electron one distributed with the one-centre density a (r1); 

(13) ðabjVB Þ ¼ ab rB1 is the attraction by the B nucleus of electron one distributed with the two-centre density a(r1) b(r1); 2 2

(14) a a

604

CHAPTER 15 Valence bond theory and the chemical bond

is the one-centre electrostatic repulsion between the densities a2(r1) and a2(r2) both on A; 2 2

a b

(15)

the two-centre Coulomb integral, describing the electrostatic repulsion between the one-centre densities a2(r1) on A and b2(r2) on B; ðabjabÞ (16) the two-centre exchange integral, describing the electrostatic interaction between the two-centre densities a(r1)b(r1) and a(r2)b(r2) shared between A and B; 2

(17) a ab the two-centre ionic (or hybrid) integral, describing the electrostatic repulsion between densities a2(r1) and a(r2)b(r2). The two-centre integrals are evaluated in spheroidal coordinates in Chapter 18. We give here, for completeness, their analytic expression for c0 ¼ 1 as a function of the internuclear distance R: 

R2 ¼ SðRÞ (18) S ¼ hajbi ¼ hbjai ¼ ðabj1Þ ¼ expðRÞ 1 þ R þ 3 2  1 1 expð2RÞ ð1 þ RÞ a rB ¼  R R  1

abrB ¼ expðRÞð1 þ RÞ

ðabjabÞ ¼

(19) (20)



2  2 1 expð2RÞ 11 3 2 1 3  1þ Rþ R þ R a b ¼  R R 8 4 6

(21)



 2  expðRÞ 5 1 expð3RÞ 5 1 þ R þ R2  þ R a ab ¼ R 16 8 R 16 8

(22)

 



1 25 23 1 6 expð2RÞ  R  3R2  R3 þ S2 ðg þ ln RÞ þ S02 Ei 4R 5 8 4 3 R   2SS0 Eið2RÞ

where

is the Euler constant, and

(23)



R2 S0 ¼ SðRÞ ¼ expðRÞ 1  R þ 3

(24)

g ¼ 0:577 215 664 9/

(25)

ZN t e ¼ E1 ðxÞ EiðxÞ ¼  dt t

(26)

x

the exponential integral function (Abramowitz and Stegun, 1965; see also Chapter 4) defined for x > 0.

15.2 The chemical bond in H2

605

Table 15.1 Numerical Values (Eh) of the Two-Electron Two-Centre Integrals Occurring in the H2 Calculation as a Function of R(c0 ¼ 1) (a2 jb2 )

R/a0

(a2 jab) 1

1.0 1.2 1.4 1.6 1.8 2.0 3.0 4.0 5.0 6.0 8.0

5.545214  10 5.295794  101 5.035210  101 4.771690  101 4.511638  101 4.259743  101 3.198035  101 2.475539  101 1.995691  101 1.665926  101 1.249979  101

(abjab) 1

5.070449  10 4.669873  101 4.258827  101 3.850683  101 3.455494  101 3.080365  101 1.607424  101 7.698167  102 3.495304  102 1.531146  102 2.738738  103

4.366526  101 3.789989  101 3.232912  101 2.715583  101 2.250001  101 1.841565  101 5.850796  102 1.562720  102 3.717029  103 8.140232  104 3.289596  105

A few values to seven significant figures of the two-electron two-centre integrals (21)–(23) are given in Table 15.1 as a function of R. The remaining one-electron two-centre integrals (18)–(20) were already given for Hþ 2 in Table 13.11. The interaction energy is obtained by subtracting from the molecular energy Eqn (8) the energy of the two ground-state H atoms  þ  þ  þ (27) DE 1 Sg ¼ E 1 Sg  2EH ¼ DEcb þ DEexchov 1 Sg where

   1 DEcb ¼ a2 VB þ b2 VA þ a2 b2 þ R

(28)

is the semiclassical Coulombic interaction, and 

þ DEexchov 1 Sg



 

ab  Sa2 VB þ ba  Sb2 VA ¼ 1þS

  



  

1 1 a2 a2 þ b2 b2  þ 2S þ S2 a2 b2 þ ðabjabÞ þ a2 ab þ b2 ba 4 2 þ ð1 þ SÞ2 (29)

the quantum mechanical component describing the exchange–overlap interaction between the charge distributions of the two H atoms, with its one-electron (upper part) and two-electron (lower part) contributions. The exchange–overlap (or penetration) component is a purely electronic quantum mechanical term arising from the Pauli’s principle: it depends on the nature of the spin coupling and is seen to give the largest contribution to the bond energy (the molecular energy at its minimum, in this case the energy of the chemical bond in the MO approximation). The MO description of this term is,

606

CHAPTER 15 Valence bond theory and the chemical bond

þ

Table 15.2 Numerical Results (Eh) for MO Calculations on Ground-State H2ð1 Sg Þ 2hss 2

2

ðs js Þ DEe 1 R DE

c0 [ 1, Re [ 1.6 a0a

c0 [ 1.1695, Re [ 1.4 a0b

HF, Re [ 1.4 a0c

1  1.27702

1  1.48585

0.55294

0.64378

0.72408

0.84207

0.84747

0.62500

0.71429

0.71429

0.09908

0.12778

0.13318

a

Hellmann, 1937. Coulson, 1937. c Coulson, 1938b. b

however, affected by a large correlation error, which becomes evident at large distances, and is such þ that the simple MO wavefunction (5) cannot describe correctly the dissociation of H2ð1 Sg Þ into two neutral H atoms in their ground state. Expression (27) for the MO interaction energy gives in fact   1  5 þ (30) Eh lim DE MO; 1 Sg ¼ a2 a2 ¼ R/N 2 16 corresponding to the erroneous dissociation  þ 1 H2 1 Sg / Hð2 SÞ þ H ð1 SÞ 2

(31)

In Eqn (30), ða2 ja2 Þ is the one-centre two-electron repulsion integral between the one-centre charge distributions a2(r1) and a2(r2) (both electrons on atom A) arising from the two-electron part þ of DEexchov ð1 Sg Þ. We shall see later in this chapter that this large correlation error, which is typical of the single determinant description of doubly occupied MOs, can be removed by CI between the ground state configuration s2g and the doubly excited one s2u . þ In Table 15.2 are collected some numerical results for MO calculations on the ground state 1 Sg of the H2 molecule in the bond region. The accurate theoretical value for the bond energy at Re ¼ 1.4 a0 from Ko1os and Wolniewicz þ (1965), is DEð1 Sg Þ ¼ 0:174474 Eh , as seen in Chapter 16. The first column of the table gives the MO results corresponding to the MO wavefunction (5), a calculation first done by Hellmann (1937). It can be seen that the resulting bond is too long (þ14%) and the bond energy too small, no more than 57% of the correct one. So, the MO description in terms of undistorted 1s AOs (c0 ¼ 1) is largely insufficient even in the bond region. The second column gives the MO results by Coulson (1937), where the orbital exponent c0 was variationally optimized at the different values at R. At the correct bond distance, Re ¼ 1.4a0, the AOs in the molecule are sensibly contracted (c0 z 1.17, spherical polarization), the bond energy being improved to about 73% of the true value. Coulson’s best variational values are c0 ¼ 1.197, Re ¼ 1.38 a0, DE ¼ 0.128184 Eh, respectively, 98.6% and 73.4% of the correct values. The third column gives the nearly HF values obtained by Coulson (1938b) using a five-term expansion of the MOs in spheroidal coordinates. Coulson’s calculation was later improved by

15.2 The chemical bond in H2

607

Table 15.3 MO Interaction Energy and Its Components (10e3 Eh) for GroundState H2 (c0 ¼ 1) D

R/a0

DEcb

DEexcheov

DE(1 Sg )

1.0 1.2 1.4 1.6 1.8 2.0 3.0 4.0 5.0 6.0 8.0

95.863 28.879 2.273 15.354 19.385 19.079 6.920 1.6074 0.3220 0.0596 0.00179

81.761 88.267 88.670 83.726 74.410 61.725 24.162 106.25 166.00 204.92 244.425

14.102 59.388 90.943 99.079 93.794 80.803 17.243 104.64 165.68 204.86 244.23

Goodisman (1963), who used nine terms for each MO expansion, getting Re ¼ 1.4a0 and DE ¼ 0.13340 Eh. This value is only DE ¼ 0.00023 Eh above the correct HF/2D one given by Pyykko¨ and coworkers (Sundholm et al., 1985), DE ¼ 0.13363 Eh. Table 15.3 gives the dependence on the internuclear distance R of the interaction energy and its þ Coulombic and exchange–overlap components (10–3 Eh) for the 1 Sg ground state of H2 according to the MO wavefunction of Eqn (5) with c0 ¼ 1 in the 1s AOs Eqn (3). Introducing the values for the two-centre integrals, it can be seen that the Coulombic interaction energy has the analytic expression: 

expð2RÞ 5 3 1 (32) 1 þ R  R2  R 3 DEcb ¼ R 8 4 6 which has a minimum of DEcb ¼ 19.6106  103 Eh at Re ¼ 1.8725 a0, that can be found analytically by solving the quartic equation 2R4 þ 7R3  12R2  12R  6 ¼ 0

(33)

In the limit of the united atom, He ð1 SÞ, B / A, b / a, S / 1, and the two-centre integrals tend to their one-centre counterpart (Section 18.7.3). The exchange–overlap component of the interaction energy tends to zero, so that the electronic energy becomes    þ þ lim Ee MO; 1 Sg ¼ 2EH þ lim De Ecb þ lim De Eexchov 1 Sg R/0 R/0 R/0 

 2  1 2  2 1 2  2 2 7     þ3 þ a a ¼ 2EH þ 2 a rA þ a a 4 4 2 5 5 19 ¼ 1  2 þ ¼ 3 þ ¼  8 8 8

(34)

608

CHAPTER 15 Valence bond theory and the chemical bond

On the other hand, for the united atom Heð1 SÞ Ee ðHe; 1 SÞ ¼ Z 2 þ

5 5 22 Z ¼ 4 þ ¼  8 4 8

(35)

3 so that Ee(R / 0) is in error by  ¼ 0:375 Eh. The variational optimization of c0 (c0 ¼ 1.6875) 8 removes this considerable error.

15.2.2 The Heitler–London theory for H2 The HL theory for H2 is the simplest example of VB theory applied to the covalent part of the wavefunction for H2. In VB theory, derived from the original work by Heitler and London (1927), the formation of a covalent bond between two atoms is possible if the atoms have, in their valence shell, orbitals containing unpaired electrons: the pairing to a singlet coupled state of two electrons with opposite spin yields the formation of a chemical bond between the two atoms. A basic requirement, which must always be satisfied, is that the resultant wavefunction must satisfy Pauli’s exclusion principle or, in other words, it must be antisymmetric with respect to electron interchange. þ For the 1 Sg ground state of H2 in the minimum basis set (ab) of 1s AOs, these requirements are met by the HL wavefunction written as a linear combination of the two Slater determinants   aðr1 Þbðr2 Þ þ bðr1 Þaðr2 Þ 1 þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi ½aðs1 Þbðs2 Þ  bðs1 Þaðs2 Þ (36) J HL; 1 Sg ¼ N½kabk  kabk ¼ 2 2 þ 2S2 The HL energy for the ground state will be    D   E þ þ þ ^ E HL; 1 Sg ¼ J HL; 1 Sg jHjJ HL; 1 Sg     ab þ ba  1 1  ab þ ba ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h^1 þ h^2 þ þ  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r12 R 2 þ 2S2 2 þ 2S2 2 2

haa þ hbb þ Sðhba þ hab Þ þ a b þ ðabjabÞ 1 ¼ þ 1 þ S2 R But



haa ¼ EA þ a2 VB ; hba ¼ EA S þ ðabjVB Þ;



hbb ¼ EB þ b2 VA hab ¼ EB S þ ðbajVA Þ

(37)

(38)

so that 

2 2

  a VB þ b VA þ S½ðabjVB Þ þ ðbajVA Þ þ a2 b2 þ ðabjabÞ 1 1 þ E HL; Sg ¼ EA þ EB þ þ (39) 1 þ S2 R where the first term is the sum of the energies of the individual ground-state H atoms, and the second the interatomic energy for the ground state. All terms in Eqn (39) have the same meaning as for the MO expression. The HL interaction energy is      þ þ þ DE HL; 1 Sg ¼ E HL; 1 Sg  2EH ¼ DEcb þ DEexchov 1 Sg (40)

15.2 The chemical bond in H2

609

where DEcb is the same as that for the MO expression (28), but where DEexch–ov now simplifies to  þ  S½ðabjV Þ þ ðbajV Þ þ ðabjabÞ  DEcb S2 B A e DEexcheov 1 Sg ¼ 1 þ S2  



ðabjabÞ  S2 a2 b2 ab  Sa2 VB þ ba  Sb2 VA ¼S þ 1 þ S2 1 þ S2  þ  þ ¼ DE1exchov 1 Sg þ DE2exchov 1 Sg (41) The exchange–overlap component (41) differs from the corresponding MO counterpart (29) in two þ respects: (1) the one-electron part, DE1exchov ð1 Sg Þ, differs from its MO counterpart by the factor S(1 þ S)(1 þ S2)1, which shows the greater importance of overlap in the HL theory (generally, in VB þ theory); and (2) the two-electron part, DE1exchov ð1 Sg Þ, is now remarkably simpler than its MO counterpart, and is characterized by the disappearance of the ionic and the atomic two-electron integrals. At variance with the MO wavefunction (5), the HL wavefunction (36) allows now for a correct dissociation of the H2 molecule into neutral ground-state atoms:   þ (42) lim DE HL; 1 Sg ¼ 0 R/N

 H2

1 þ Sg



/ 2Hð2 SÞ

(43)

þ

In the one-electron part, DE1exchov ð1 Sg Þ, appears, as already seen in the MO result (Eqn (29)), the exchange–overlap density a(r1)b(r1)Sa2(r1), which has the interesting property of giving a zero contribution to the electronic charge: Z 

dr1 a r1 b r1  Sa2 r1 ¼ S  S ¼ 0 (44) even if contributing in a relevant way to the exchange–overlap energy and, therefore, to the bond energy. þ The HL function for the excited triplet state, 3 Su , is given by three functions having the same spatial part and differing in the spin part: 8 S ¼ 1; MS ¼ 1 > > kabk

< 3 þ (45) 0 J HL; Su ¼ N½kabk þ kabk > > : 1 kabk 8 aðs1 Þaðs2 Þ > > > > < 1 aðr1 Þbðr2 Þ  bðr1 Þaðr2 Þ pffiffiffi ½aðs1 Þbðs2 Þ þ bðs1 Þaðs2 Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 2 > 2 2  2S > > > : bðs1 Þbðs2 Þ

(46)

610

CHAPTER 15 Valence bond theory and the chemical bond

FIGURE 15.4 Schematic R-dependence of the HL interaction energies for H2 for (i) left, singlet ground state, and (ii) right, excited triplet state

The exchange–overlap component of the interaction energy for the triplet is different from that of the singlet ground state, and is given by þ

S½ðabjVB Þ þ ðbajVA Þ þ ðabjabÞ  DEecb S2 >0 DEexchov 3 Su ¼  1  S2 þ

(47)

so that it is repulsive for any R. Since DEexchov ð3 Su Þ is larger than DEcb, the HL triplet interaction energy is always repulsive, describing a scattering (non-bonded) state. The qualitative behaviour of the two states for H2 is sketched in Figure 15.4. Table 15.4 gives the dependence on the internuclear distance R of the interaction energy and its þ þ exchange–overlap component (10–3 Eh) for the 1 Sg and 3 Su states of H2 according to the HL wavefunctions in the minimum basis with c0 ¼ 1. The DEcb component is the same as that for the MO wavefunction. In parenthesis are given the accurate theoretical values of Ko1os and Wolniewicz (1965) obtained using an 80-term wavefunction expanded in spheroidal coordinates of the two electrons and containing explicitly r12 and the appropriate dependence on the hyperbolic functions of the v-variables (see Chapter 16). It can be seen from Table 15.4 that the HL results for both states of H2 are only in qualitative þ agreement with the accurate theoretical results of Ko1os and Wolniewicz (KW). For the 1 Sg ground state, the HL value at the correct Re ¼ 1.4 a0 is only 60.5% of the KW value, while at Re ¼ 8a0 the HL þ result is only 33% of KW. For the 3 Su excited state, all the HL values severely overestimate the accurate KW results. This is not surprising, however, since we have already said that the HL wavefunction can be considered only as the first approximation describing the interaction between undistorted H atoms. Accounting for polarization and dispersion (correlation) effects, what can be þ done in second order of Rayleigh-Schroedinger perturbation theory for the 1 Sg ground state of H2 (Chapter 1), greatly improves agreement with the accurate KW results, as shown by us in Section 17.7.

15.2 The chemical bond in H2

611

Table 15.4 HL Interaction Energies and Their ExchangeeOverlap Components (10e3 Eh) for 1 Sþ and 3 Sþ States of H (c ¼ 1) Compared with Accurate Resultsa 2 0 g u DEexcheov R/a0 1.0 1.2 1.4 1.6 1.8 2.0 3.0 4.0 5.0 6.0 8.0 a

D Su

D Sg

3

92.286 100.865 103.201 100.356 93.664 84.473 34.754 9.678 2.203 0.4495 0.0156

609.045 478.110 373.392 290.217 224.566 173.037 44.364 10.397 2.244 0.452 0.0156

1

DE 1

D Sg

3.576(124.54)a 71.99(164.93) 105.47(174.47) 115.71(168.58) 113.05(155.07) 103.55(138.13) 41.674(57.31) 11.285(16.37) 2.525(3.763) 0.5092(0.815) 0.0174(0.053)

3

D

Su

704.91(378.48)a 506.99(281.04) 371.12(215.85) 274.86(168.28) 205.18(131.71) 153.96(102.94) 37.44(27.99) 8.790(6.622) 1.922(1.315) 0.392(0.1875) 0.0138(0.0196)

Kołos and Wolniewicz, 1965. þ

The results of Tables 15.3 and 15.4 for the 1 Sg ground state of H2 are plotted against R in Figure 15.5. The qualitatively correct behaviour of the HL calculation during dissociation is evident from the figure, as is the incorrect MO behaviour in the same region of internuclear distance, with its exceedingly large correlation error, which asymptotically reaches the value of 312.5  103 Eh (horizontal dashed line in the figure). þ At variance with the 1 Sg ground state, Problem 15.1 shows the complete equivalence between MO þ and HL wavefunctions for the 3 Su excited state.

15.2.3 Equivalence between MO-CI and full VB for ground-state H2 and improvements in the wavefunction In the minimal basis set (ab), it is possible to improve the HL covalent wavefunction for ground-state H2 by variational mixing with the ionic wavefunction having the same symmetry:   aðr1 Þaðr2 Þ þ bðr1 Þbðr2 Þ 1 þ pffiffiffi ½aðs1 Þbðs2 Þ  bðs1 Þaðs2 Þ (48) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi J ION; 1 Sg ¼ N½kaak þ kbbk ¼ 2 2 þ 2S2 which describes the two equivalent ionic VB structures H–Hþ and HþH–. The complete VB wavefunction will be in general       þ þ þ J VB; 1 Sg ¼ c1 J HL; 1 Sg þ c2 J ION; 1 Sg (49) where c1 and c2 are variational coefficients to be determined by the Ritz method. At R ¼ 1.6 a0, c1/c2 z 0.138, and the covalent (HL) structure gives the main contribution to the energy of the chemical bond.

612

CHAPTER 15 Valence bond theory and the chemical bond

300

ΔE / 10–3 Eh CORRELATION ERROR

H2 (1∑+g) 200

100 MO

0

HL KW

–100

–200 1

2

3

R ao

4

FIGURE 15.5 Plot vs R of MO, HL and KW interaction energies for H2

It is possible to obviate the MO error by means of CI. The possible electron configurations for H2 are given in Figure 15.6. We can mix only functions having the same symmetry, so that the interconfigurational wavefunction for ground-state H2 will be       þ þ þ J MO  CI; 1 Sg ¼ c1 J s2g ; 1 Sg þ c2 J s2u ; 1 Sg (50)

FIGURE 15.6 Ground-state and excited electron configurations in H2

15.2 The chemical bond in H2

613

ΔE 10–3 Eh H2

300 1 + ∑g

3 + ∑u

200

100 MO,VB 0 – 0.2

MO - CI VB λ

–100

– 0.4 – 0.6 – 0.8 – 1.0

–200 1

2

3

4 R ao

FIGURE 15.7 Plots vs R of interaction energies DE and mixing parameter l (right scale) for ground and excited states of H2

This function describes correctly the dissociation of the H2 molecule, and is completely equivalent to the full VB (HL þ ION) function (49), provided all variational parameters are completely optimized (Problem 15.2):     ð1  SÞ þ lð1 þ SÞ 1 1 þ ðaa þ bbÞ pffiffiffi ðab  baÞ (51) J MO  CI; Sg ¼ N ðab þ baÞ þ ð1  SÞ  lð1 þ SÞ 2 For large R, l ¼ c2/c1 / 1 (see Figure 15.7), S z 0 and JðMO  CIÞ / JðHLÞ, the covalent HL structure which dissociates correctly. For intermediate values of R, the interconfigurational function JðMO  CIÞ reduces the weight of the ionic structures, which is one in the MO wavefunction. Further improvements can be found by optimizing the molecular energy with respect to the nonlinear parameter c0, the orbital exponent of the 1s AOs (c0 ¼ 1 in the original HL theory). In such a way, it is possible to account for part of the spherical distortion of the AOs during the formation of the bond. The HL wavefunction with optimized c0 satisfies the virial theorem: 2hTi ¼ hVi  R

dE dR

(52)

614

CHAPTER 15 Valence bond theory and the chemical bond

co a–1 o

1.8

1.6

1.4

Re

1.2

1.0

1

2

3

4

5

6 R ao

FIGURE 15.8 Dependence on R of optimized c0 for ground-state H2

guaranteeing the correct partition of the expectation value of the molecular energy into its kinetic hTi and potential energy hVi components, which is not true for the original HL wavefunction with c0 ¼ 1. The dependence of c0 on R is given in Figure 15.8. The effect of improving the quality of the basic AOs (ab) on the bond energy DE at Re ¼ 1.4 a0 for þ the 1 Sg ground state of H2, and the residual error with respect to the accurate value (Ko1os and Wolniewicz, 1965), is shown in Table 15.5 for the covalent HL and in Table 15.6 for the full VB (HL þ ION) wavefunctions. We see from Table 15.5 that admitting part of the spherical distortion of the H orbitals (second row) reduces the error by 33.58 mEh (1 mEh ¼ 10–3 Eh), while the inclusion of some ps polarization (third row) improves the result by a further 8.69 mEh. The best that can be done at the HL level with two nonlinear parameters is the Inui result (fourth row of Table 15.5), where the error is reduced by another 1.41mEh. The full VB results (Table 15.6) show a sensible improvement of 8.84 mEh for the Weinbaum wavefunction (second row), which means that admitting ionic structures partly accounts for higher polarizations. It is worth noting that the error with the second function of Table 15.6 is quite close to that of the third function of Table 15.5, where partial dipole polarization of the orbitals is admitted. The

15.2 The chemical bond in H2

615

Table 15.5 Effect of Improving the Basic AOs in the Covalent HL Wavefunction on the Bond Energy DE at Re ¼ 1.4a0 for Ground-State H2, and Residual Error with Respect to the Accurate Value a

Basic AO

DE/10L3 Eh

Error/10e3 Eh

1. 1sA fexpðrA Þðc0 ¼ 1Þ

H1sa

105.47

69.00

2. 1sA fexpðc0 rA Þðc0 ¼ 1:1695Þ

STO 1sb

139.05

35.42

3: 1sA þ l0 2psA ðc0 ¼ 1:19; cp ¼ 2:38; l0 ¼ 0:105Þ

Dipole polarized AO (sp hybrid)c 2-centre GZd AOe

147.74

26.73

149.15

25.32

f

174.47

0

4. expðarA  brB Þða ¼ 1:0889; b ¼ 0:1287Þ 5. Accurate a

Heitler and London, 1927. Wang, 1928. c Rosen, 1931. d Guillemin and Zener (GZ), 1929. e Inui, 1938. f Kołos and Wolniewicz, 1965. b

Table 15.6 Effect of Improving the Basic AOs in the Full VB (HL þ ION)a Wavefunction on the Bond Energy DE at Re ¼ 1.4a0 for Ground-State H2, and Residual Error with Respect to the Accurate Value a

Basic AO

DE/10L3 Eh

Error/10e3 Eh

1. sA fexpðrA Þ (c0 ¼ 1, l ¼ 0.1174)a

H1sb

106.56

67.91

2. 1sA fexpðc0 rA Þ (c0 ¼ 1.193, l ¼ 0.2564)

STO 1s

147.89

26.58

Dipole polarized AO (sp hybrid)d 2-centre GZ AOe

151.49

22.98

153.33

21.14

0

3. 1sA þ l 2psA ðc0 ¼ 1:19; cp ¼ 2:38; l0 ¼ 0:07; l ¼ 0:1754Þ 4. expðarA  brB Þ (a ¼ 1.0889, b ¼ 0.1287, l z 1) 5. Accurate

c

f

174.47

0

a

l is the relative weight Ion/Cov. Figari, 1985. c Figari, 1985. d Weinbaum, 1933. e Ottonelli and Magnasco, 1995. f Kołos and Wolniewicz, 1965. b

best bond energy value obtained by admitting the ionic structures in the two-parameter Inui wavefunction is, however, still 21.14 mEh above the correct value. For a further comparison we give in Table 15.7 the SDCI bond energy results for ground-state H2 at Re ¼ 1.4 a0, taken from Wright and Barclay (1987), who gradually added GTO polarization

616

CHAPTER 15 Valence bond theory and the chemical bond

Table 15.7 SDCI Bond Energya of Ground-State H2 at Re ¼ 1.4a0, and Residual Error with Respect to the Accurate Valueb for Various GTO Basis Sets GTO Basis Set

Number of Functions

DE/10L3 Eh

Error/10e3 Eh

4s 4s3p 4s3p2d 4s3p2d1f Accurateb

8 26 50 70 80

154.32 171.83 173.75 173.97 174.47

20.15 2.64 0.72 0.50 0

a

Wright and Barclay, 1987. Kołos and Wolniewicz, 1965.

b

functions centred at the two nuclei to a starting (9s) / [4s] contracted GTO basis on each H atom. It can be seen that the SDCI value with only spherical functions is not far from the full VB Inui value (fourth row of Table 15.6), while convergence towards the correct value when including polarization functions with l ¼ 1,2,3 is rather slow. Nonetheless, the final 4s3p2d1f result is within 0.5 mEh of the accurate KW result, and even better than the best 13-term result including r12 quoted in the classical paper by James and Coolidge (1933), DE ¼ 173.45  103 Eh. In this last case, however, some inaccuracy is expected in the hand-evaluated integral values. We end this section by quoting the interesting work by Coulson and Fischer (1949). They introduced the semilocalized AOs: a þ lb a0 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi; 1 þ l2 þ 2lS

b þ la b0 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ l2 þ 2lS

(53)

that are normalized ha0 ja0 i ¼ hb0 jb0 i ¼ 1

(54)

and have a non-orthogonality S0 S0 ¼ ha0 jb0 i ¼ hb0 ja0 i ¼

Sl2 þ 2l þ S 1 þ l2 þ 2lS

(55)

If l < 1, a0 is localized near nucleus A, and b0 near nucleus B. We note that the Coulson–Fischer semilocalized AOs are two-centre AOs containing a single linear variational parameter, while Inui AOs (Section 16.4.5) are two-centre Guillemin–Zener AOs containing two non-linear variational parameters. If we construct an HL symmetrical space function with Coulson–Fischer AOs (53), then the suitably normalized space part (omitting spin for short) will be J0 ¼ where S0 is given by Eqn (55). Then

a0 ðr1 Þb0 ðr2 Þ þ b0 ðr1 Þa0 ðr2 Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 þ 2S02

(56)

15.2 The chemical bond in H2

617

1. l ¼ 0 gives the ordinary HL wavefunction: J0 ðl ¼ 0Þ ¼

aðr1 Þbðr2 Þ þ bðr1 Þaðr2 Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 þ 2S2

(57)

where no ionic structures are present; 2. l ¼ 1 gives the ordinary MO wavefunction: J0 ðl ¼ 1Þ ¼

½aðr1 Þ þ bðr1 Þ½bðr2 Þ þ aðr2 Þ 2 þ 2S

(58)

where the ionic structures have now the same weight of the covalent structure, which is the origin of the correlation error; 3. The variational optimization of l at any given R gives the correct mixing between covalent and ionic structures, the same result that can be obtained by optimizing the linear parameter in Eqn (49). So, optimization of l removes the correlation error and improves upon the HL result. It can be shown (Magnasco and Peverati, 2004) that full optimization of c0 and l in the Heitler–London–Coulson–Fischer wavefunction (56) at Re ¼ 1.4 a0 (c0 ¼ 1.2, l ¼ 0.135, S ¼ 0.6744) enhances interorbital overlap by over 18%, yielding a bond energy DE ¼ 0.1478 Eh which is practically coincident with that resulting from the full-VB calculation with the original basis set. This is better than Rosen’s dipole polarized result of Table 15.5 and within 85% of the accurate bond energy of Ko1os and Wolniewicz (1965). The resulting Coulson–Fischer orbital a0 ¼ 0:91281sA þ 0:12311sB

(59)

is plotted in the three-dimensional graph of Figure 15.9 with its section in the zx-plane. The cusp at nucleus A (origin of the coordinate system) and the small cusp at nucleus B due to delocalization are apparent from the figure.

–2

–1 0 –2

1 2

–1

0

1

2

R/a0

3

FIGURE 15.9 Coulson–Fischer optimized AO for ground-state H2 at Re ¼ 1.4 a0 (left) and its section in the zx-plane (right)

618

CHAPTER 15 Valence bond theory and the chemical bond

15.2.4 The orthogonality catastrophe in the covalent VB theory for ground-state H2 In 1951 Slater pointed out that the orthogonal atomic orbitals (OAOs), first introduced by Wannier (1937) in solid state physics and next by Lo¨wdin (1950) in molecular problems, do not give a simple way of overcoming the non-orthogonality problem in the HL method. An HL calculation of the H2 molecule ground state using these OAOs shows that no bond can be formed between H atoms since the molecular energy has no minimum. We saw before (first column of Table 15.4) that the HL exchange–overlap þ component DEexch–ov of the bond energy is always attractive for the 1 Sg ground state of H2, this being due to the one-electron part of this quantum component. If we set S ¼ 0 in Eqn (41), we see that   þ (60) DEexchov 1 Sg ; S ¼ 0 ¼ ðabjabÞ the two-electron exchange integral, which is always positive. At the expected bond length of Re ¼ 1.4 a0, DEcb ¼ 2.273  103 Eh (Table 15.3), ðabjabÞ ¼ 323:3  103 Eh (Table 15.1), so that þ DEð1 Sg Þz321  103 Eh , and we have a strong repulsion between the H atoms. This orthogonality catastrophe can, however, be overcome by admitting with a substantial weight the ionic structures in a complete VB calculation, as we shall see below. Let us now, in fact, investigate in greater detail the mixing of covalent (HL) and ionic VB structures for ground-state H2, starting either from ordinary (non-orthogonal) AOs or from Lo¨wdin’s OAOs. For a correct comparison, it will be convenient to use values of the molecular integrals correct to nine significant figures. 1. Normalized non-orthogonal basis For c0 ¼ 1, Re ¼ 1.4 a0, the one-electron two-centre integrals can be taken from Table 13.11 and the two-electron ones from Table 15.1, but are now given below with nine-figure accuracy. We then have S ¼ 0:752 942 730 haa ¼ hbb ¼ 1:110 039 890; hba ¼ hab ¼ 0:968 304 078 2

2 2



a ab ¼ 0:425 882 670 a b ¼ 0:503 520 926; abab ¼ 0:323 291 175;

(61)

Let (space part only) ab þ ba j1 ðHLÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi; 2 þ 2S2

aa þ bb j2 ðIONÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 þ 2S2

(62)

be the covalent (HL) and ionic VB wavefunctions for H2, respectively. They are strongly nonorthogonal: S12 ¼ hj1 jj2 i ¼

2S ¼ 0:961 046 392 1 þ S2

(63)

showing that there is a strong linear dependence between them (at R ¼ 0, they become identical). The matrix elements (Eh) of the molecular Hamiltonian H^ are

 

^ 1 i ¼ 1 þ S2 1 ½haa þ hbb þ Sðhba þ hab Þ þ 1 þ S2 1 a2 b2 þ ðabjabÞ þ 1 H11 ¼ hj1 jHjj R ¼ 0:347 425 747 þ 0:527 665 403 þ 0:714 285 714 ¼ 1:105 473 880 (64)

15.2 The chemical bond in H2

619

which is the result of the third column of Table 15.4 when truncated to the fifth decimal place. ^ 2i H22 ¼ hj2 jHjj

1

1 1  2  2 2  2

1 ½haa þ hbb þ Sðhba þ hab Þ þ 1 þ S2 a a þ b b þ 2ðabjabÞ þ ¼ 1 þ S2 2 R ¼ 2:347 425 747 þ 0:605 192 564 þ 0:714 285 714 ¼ 1:027 947 468 (65) so that the energy of the ionic state (HHþ þ HþH) is higher than that of H–H. ^ 2i H12 ¼ hj1 jHjj

1

1  2  2  1 a ab þ b ba þ S12 ½Sðhaa þ hbb Þ þ ðhba þ hab Þ þ 1 þ S2 ¼ 1 þ S2 R ¼ 2:302 730 674 þ 0:543 591 149 þ 0:686 461 708 ¼ 1:072 677 817

(66)

so that covalent and ionic structures are also strongly interacting. The pseudosecular equation for the (non-orthogonal) ionic-covalent ‘resonance’ in ground-state H2 will hence be    H11  E H12  ES12   ¼0 (67)  H  ES H22  E  12 12 which can be expanded to the quadratic equation in E



2 1  S212 E2  ðH11 þ H22  2S12 H12 ÞE þ H11 H22  H12 ¼0

(68)

The lowest root E1 will be E1 ¼

H11 þ H22  2S12 H12 D



 2 1  S212 2 1  S212

(69)

where h i1=2 D ¼ ðH22  H11 Þ2 þ 4ðH12  H11 S12 ÞðH12  H22 S12 Þ >0

(70)

The numerical value of E1 is E1 ¼ 0:468 873 311  0:637 683 279 ¼ 1:106 556 590

(71)

corresponding to a bond energy of DE1 ¼ E1  2EH ¼ 0:106 556 590 Eh in agreement with the first row of Table 15.6. The mixing coefficients in the non-orthogonal basis are calculated as usual from ( ðH11  E1 Þc1 þ ðH12  E1 S12 Þc2 ¼ 0 c21 þ c22 þ 2c1 c2 S12 ¼ 1

(72)

(73)

620

CHAPTER 15 Valence bond theory and the chemical bond

giving l¼

 c2 E1  H11 ¼ ¼ 0:117 359 290 c1 1 H12  E1 S12

so that we finally obtain   j1 þ lj2 þ ffi ¼ 0:898 262 463 j1 þ 0:105 419 445 j2 J VB; 1 Sg ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ l2 þ 2lS12

(74)

(75)

The relative weights of the non-orthogonal structures are then %HL ¼ c21 þ c1 c2 S12 ¼ 89:79%;

%ION ¼ c1 c2 S12 þ c22 ¼ 10:21%

(76)

Over the non-orthogonal basis the contribution of the two ionic structures is about 10%, as small as physically expected. It is worth noting that the single HL wavefunction built from the overlap-enhanced Coulson–Fischer AOs (54) with the optimum value l ¼ 0.058 883 153 gives a molecular energy E ¼ 1.106 556 590 Eh, which coincides exactly to all figures with the fullVB result (72). As a general rule, enhancing atomic overlap reduces the importance of the ionic structures. For the minimal basis set, we have therefore the complete equivalence:  þ (77) E 1 Sg ¼  1:106 556 590 Eh ¼  1:106 556 590 Eh Full  VBð3 structures Þ Coulson-Fischer HL with l ¼ 0:058 883 153

So, in the fully optimized Coulson–Fischer covalent wavefunction the ionic structures disappear. 2. Normalized orthogonalized basis To get the normalized orthogonal (OAO) basis, we do a Lo¨wdin symmetrical orthogonalization (Chapter 1) of the original basis set, which gives a ¼ 1:383 585 021 a  0:628 290 845 b b ¼ 1:383 585 021 b  0:628 290 845 a

(78)

We note (Slater, 1963) that Lo¨wdin OAOs (78) are nothing but the Coulson–Fischer semilocalized AOs (54) when l is chosen to be a solution of the quadratic equation:3 Sl2 þ 2l þ S ¼ 0 l1 ¼ 



1=2 1  1  S2 ; S

l2 ¼ 

(79)

1=2 1 þ 1  S2 S

(80)

Choosing the first root, l1, we obtain the relations (that can be checked either analytically or numerically):

1=2 A þ B

1=2 A  B ¼ ¼ ; l1 1 þ l21 þ 2l1 S (81) 1 þ l21 þ 2l1 S 2 2 where (Lo¨wdin)

3

A ¼ ð1 þ SÞ1=2 ;

B ¼ ð1  SÞ1=2

We discovered a wrong sign in the denominator of Eqn (4.13) in Slater’s (1963) book.

(82)

15.2 The chemical bond in H2

621

We now transform all integrals (61) to the OAO basis, obtaining S ¼ 0;

haa ¼ 0:879 663 802;

2

ða2 jb Þ ¼ 0:426 039 142; ða2 jabÞ ¼ 0:005 066 793;

hba ¼ 0:305 967 614

ðabjabÞ ¼ 0:009 878 391

(83)

ða2 ja2 Þ ¼ 0:706 541 235

Lo¨wdin’s orthogonalization has the effect of reducing to some extent the one-electron integrals, while drastically reducing the two-electron integrals involving the two-centre charge density aðrÞbðrÞ. The covalent (HL) and ionic VB structures in the OAO basis will be j1 ¼

ab þ ba pffiffiffi ; 2

j2 ¼

aa þ bb pffiffiffi 2

(84)

with the matrix elements ^ 1 i ¼ haa þ h þ ða2 jb2 Þþ ðabjabÞ þ 1 H 11 ¼ hj1 jHjj bb R ¼ 1:759 327 605 þ 0:435 917 533 þ 0:714 285 714 ¼ 0:609 124 357

(85)

H 11  2EH ¼ 0:390 875 649

(86)

so that

describes strong repulsion. The HL structure over the OAO basis does not allow describing the formation of any chemical bond in H2. i h ^ 2 i ¼ haa þ h þ 1 ða2 ja2 Þ þ ðb2 jb2 Þ þ 2ðabjabÞ þ 1 H 22 ¼ hj2 jHjj bb 2 R (87) ¼ 1:759 327 605 þ 0:716 419 626 þ 0:714 285 714 ¼ 0:328 622 265 H 22  2EH ¼ 0:671 377 735

(88)

so that we get an even greater repulsion for the ionic state. ^ 2 i ¼ h þ h þ ½ða2 jabÞ þ ðb jbaÞ þ H 12 ¼ hj1 jHjj ba ab 2

1 R

¼ 0:611 935 227  0:010 133 586 ¼ 0:622 068 813

(89)

which is sensibly smaller than the corresponding term (66) over the non-orthogonal basis. Hence, the secular equation for the (orthogonalized) ionic-covalent ‘resonance’ in ground-state H2 will be   H  E H 12   11 (90)  ¼0  H 12 H 22  E  which expands to the quadratic equation in E 2

E2  ðH 11 þ H 22 ÞE þ H 11 H 22  H 12 ¼ 0

(91)

622

CHAPTER 15 Valence bond theory and the chemical bond

The lowest root E1 will be E1 ¼

H 11 þ H 22 D  2 2

(92)

where h i 2 1=2 D ¼ ðH 22  H 11 Þ2 þ4H 12 >0

(93)

The numerical value of E1 is now E1 ¼ 0:468 873 311  0:637 683 279 ¼ 1:106 556 590   DE1 1 Sþ ¼ 0:106 556 590 g

(94) (95)

in perfect agreement with the value (72) obtained with the non-orthogonal basis. In this way, the chemical bond in H2 has been restored through a strong CI with the ionic state. The CI coefficients in the orthogonal basis are calculated from ( H 11  E1 Þc1 þ H 12 c2 ¼ 0 (96) c21 þ c22 ¼ 1 giving l¼

 c2 E1  H 11 ¼ ¼ 0:799 641 812 c1 1 H 12

(97)

a value which is about seven times larger than the corresponding non-orthogonal value (74). The ‘resonance’ between ionic and covalent VB structures in the OAO basis will be described by the wavefunction: j1 þ lj2 JðVB; 1 Sþ g Þ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 0:781 005 253 j1 þ 0:624 524 455 j2 2 1þl

(98)

where the structures have now the relative weights %HL ¼ c21 ¼ 61%;

%ION ¼ c22 ¼ 39%

(99)

In the orthogonalized basis, the contribution of the two ionic structures is about four times larger than the true value, a mathematical effect of the variational principle which has to restore the physical reality in the formation of the bond in H2, namely, its dependence on the effective overlap between the two H atoms.4 Exactly the same effects were observed in an ab initio OAO calculation of the short-range interaction in the H2–H2 system (Magnasco and Musso, 1968), where a small amount (less than þ þ  1%) of the charge-transfer states H 1 H2 and H2 H2 was seen to give interaction energies which 4

It is worth noting that exactly the same result can be obtained in terms of just the single covalent (HL) structure from the Coulson–Fischer AOs built from Lo¨wdin OAOs (Magnasco and Peverati, 2004) for l ¼ 0:499 627 110. In the Coulson–Fischer description, for this value of ~l the ionic structures disappear.

15.3 Elementary VB methods

623

are in substantial agreement with those obtained from the complete VB treatment (Magnasco and Musso, 1967b).

15.3 ELEMENTARY VB METHODS In this section we shall introduce elementary VB methods as an extension of the HL theory of H2, comparing their feasibility with the corresponding MO formulation of the same problems. The theory will be mostly used in a qualitative way, just to outline how chemical intuition can be used to construct ad hoc VB wavefunctions for some simple representative molecules. Deviation from the so-called perfect-pairing approximation (Coulson, 1961) will be discussed in terms of ‘resonance’ between different structures. In a few cases (allyl radical, XeF2, and p electron system of the benzene molecule) symmetry arguments will enable us to draw conclusions on bonding and electron distribution without doing any effective calculation. The different nature of the multiple bonds in N2, CO and O2 will be evident from the VB description of their p systems, as well as the difference between Pauli repulsion in He2 and the relatively strong s bond in Heþ 2 . The general aim of this section will be to show how brute force calculations can be avoided if chemical intuition can be used from the outset to concentrate effort on the physically relevant part of the electron bonding in molecules.

15.3.1 General formulation of VB theory Originating from the work of Heitler and London (1927) on H2 we have thoroughly discussed in the previous section, VB theory was further developed by Slater and Pauling (1930–1940), McWeeny (1954), Goddard III (1967, 1968) up to the recent advances in the theory by Cooper et al. (1987) presented in Chapter 16. A modern group theoretical approach was also given by Gallup (1973, 2002). The most useful formulation for us is based on the use of Slater determinants (dets). As we have already said at the beginning of the previous section, for describing the formation of a covalent bond between atoms A and B, each having orbitals a and b singly occupied by electrons with opposite spin, we start by writing the Slater det kabk, called parent det,5 then interchanging spin between the orbitals forming the bond, obtaining in this way a second Slater det kabk, which must be added to the first with the minus sign (singlet).6 It can be easily verified that in this way we obtain a molecular state where the total spin takes a definite value, and which is therefore eigenstate of S^z and S^2 with eigenvalues MS and S, respectively (covalent VB structure, S ¼ 0 for the singlet). The corresponding VB structures will be Jð1 Sþ g Þ ¼ N½kabk  kabk

(100)

0 Jð3 Sþ u Þ ¼ N ½kabk þ kabk

(101)

where N and N0 are normalization factors. The first is the singlet (S ¼ MS ¼ 0) VB structure describing a s chemical bond (attraction between A and B), the second the triplet state (S ¼ 1, MS ¼ 0) describing 5 6

The parent det is a Slater det from which a VB wavefunction is obtained by doing all necessary spin interchanges. The plus sign will give the corresponding triplet state (as for H2).

624

CHAPTER 15 Valence bond theory and the chemical bond

repulsion between A and B. By expanding the dets we see that the singlet and triplet functions can be written in the original HL form   ab þ ba 1 (102) J 1 Sþ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi ðab  baÞ S ¼ MS ¼ 0 g 2 þ 2S2 2   ab  ba 1 (103) J 3 Sþ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi ðab þ baÞ S ¼ 1; MS ¼ 0 u 2  2S2 2 with the corresponding components with MS ¼ 1 for the latter (Eqn (46) of the previous section). Functions (100)–(103) are fully antisymmetric in the interchange of the space-spin coordinates of the two electrons and therefore satisfy the Pauli exclusion principle. To each VB structure is usually assigned a Rumer diagram, which describes in a visible way the formation of a covalent bond between the two atoms A and B: Jð1 Sþ gÞ 0 AB

(104)

the shared Lewis electron pair. If both electrons are on A, or B, we have the ionic structures, in which doubly occupied AOs appear, and which are described by single Slater dets, which are already eigenfunctions of S^2 with S ¼ 0: 1  þ Jð1 Sþ g Þ ¼ kaak ¼ aa pffiffiffi ðab  baÞ A B 2

(105)

1 þ  Jð1 Sþ g Þ ¼ kbbk ¼ bb pffiffiffi ðab  baÞ A B 2

(106)

The relative weight of the different structures having the same symmetry is determined by the Ritz method by solving the appropriate secular equations arising from the linear combination of the VB structures. This may be not easy because of the possible non-orthogonalities between the structures themselves. It is important to note, and we shall see later in this section, that often VB structures are non-orthogonal even if they are built from orthonormal spin-orbitals. The nearer the molecular energies pertaining to each individual VB structure are, the nearer the mixing coefficients will tend to become to each other, becoming identical in the case of ‘full resonance’ between equivalent structures. In the calculation of the relative weights we must correctly take into account the non-orthogonality between the structures, as we did for H2 in the previous section. We now schematize the elementary VB method and compare it with the corresponding schematization of the MO approach. 1. Schematization of the VB method Basis of (spatial) AOs / atomic spin-orbitals (ASOs) / antisymmetrized products, i.e. Slater dets of order equal to the number of electrons (defined MS) / VB structures, eigenstates of S^2 , defined S / symmetry combination of VB structures / multideterminant functions describing at the VB level the electronic states of the molecule. 2. Schematization of the MO method Basis of AOs / MO by the LCAO method, classified according to symmetry-defined types / single Slater det of doubly occupied MOs (for singlet states), or combination of

15.3 Elementary VB methods

625

different dets with even singly occupied MOs (for non-singlet states) / MO-CI among all dets with the same symmetry / multideterminant functions describing at the MO-CI level the electronic states of the molecule. Starting from the same basis of AOs, VB and MO methods are entirely equivalent at the end of each process, but may be deeply different at the early stages of each approach. The MO method has been more widely used, compared to the VB method, because for closed-shell molecules (S ¼ 0, singlet) a single Slater det may be often sufficient as a first approximation, and for the further reasons we shall indicate below. 3. Advantages of the VB method a. The VB structures are related to the existence of chemical bonds in the molecule, the most important ones corresponding to the rule of ‘perfect pairing’. b. The principle of maximum overlap, better the minimum of the exchange–overlap bond energy (Magnasco and Costa, 2005), determines the stereochemistry of the bonds in a polyatomic molecule, hence the directed valency. c. It allows for a correct dissociation of the chemical bonds, what is of paramount importance in chemical reactions. d. It gives a sufficiently accurate description of spin densities, even at the most elementary level. e. For small molecules, it is possible to account today for about 80% of electronic correlation and to get bond distances within 0.02 a0. 4. Disadvantages of the VB method a. Non-orthogonal basic AOs / non-orthogonal VB structures / difficulties in the evaluation of the matrix elements of the Hamiltonian (Slater rules are no longer valid). b. The number of covalent VB structures of given S increases rapidly with the number n (2n ¼ N) of the bonds, according to the Wigner’s formula:

  ð2S þ 1Þð2nÞ! 2n 2n N  ¼ (107) fS ¼ nS nS1 ðn þ S þ 1Þ!ðn  SÞ! An example for the p electron systems of a few polycyclic hydrocarbons is given in Table 15.8. The table makes immediately evident the striking difference between the numbers of the second (Hu¨ckel) and the last column (VB). The situation is even worse if, besides covalent structures, we take ionic structures into account. In this case, the total number of structures, Table 15.8 Comparison between the Order of Hu¨ckel and VB Secular Equations for the Singlet State of Some Polycyclic Hydrocarbons Molecule

2n

n

f0N

Benzene Naphthalene Anthracene Coronene

6 10 14 24

3 5 7 12

5 42 429 208012

626

CHAPTER 15 Valence bond theory and the chemical bond

covalent plus all possible ionic, is given by the Weyl’s formula (Weyl, 1931; Mulder, 1966; McWeeny and Jorge, 1988): 0 10 1 mþ1 m þ 1 2S þ 1 @ A@ N A (108) f ðN; m; SÞ ¼ N þSþ1 mþ1 S 2 2 where N is the number of electrons, m the number of basic AOs, and S the total spin. As an example, for the p electron system of benzene N ¼ 6; m ¼ 6; S ¼ 0

f ð6; 6; 0Þ ¼ 175

singlet (covalent þ ionic) VB structures. N ¼ 6; m ¼ 6; S ¼ 1

f ð6; 6; 1Þ ¼ 189

triplet (covalent þ ionic) VB structures. With a DZ basis set (m ¼ 12), for N ¼ 6 the singlet structures are N ¼ 6; m ¼ 12; S ¼ 0

f ð6; 12; 0Þ ¼ 15730

The total number of all possible VB structures is hence seen to increase very rapidly with the size of the basic AOs. 5. Advantages of the MO method a. As a first approximation, singlet molecular states can be described in terms of a single Slater det of doubly occupied MOs. b. The non-orthogonality of the basic AOs does not make any problem. c. MOs are always orthogonal, even inside the same symmetry. d. Electron configurations of molecules are treated on the same foot as are those for atoms. 6. Disadvantages of the MO method a. MOs are delocalized over the different nuclei in the molecule and are not suitable for direct chemical interpretation. b. The single det cannot describe correctly the dissociation of the two-electron bond. c. The ionic part of the wavefunction is overestimated. d. The single det of doubly occupied MOs does not describe whatever correlation between electrons with opposite spin (see Section 16.3). e. MO spin densities are often inaccurate (see the case of the allyl radical).

15.3.2 Construction of VB structures for multiple bonds It is convenient to use for the parent det (that describes which bonds are formed in the molecule) a shorthand notation, as the following example shows for the description of the triple bond in N2: Jð1 Sþ g Þ ¼ ðsA sB xA xB yA yB Þ

(109)

Equation (109) specifies the formation of a s bond between orbitals 2pzA and 2pzB (possibly allowing for some sp hybridization, the z-axis being directed from A to B along the bond), and two p bonds (perpendicular to the z-axis) between 2pxA, 2pxB, and 2pyA, 2pyB. We have omitted for short the remaining eight electrons which are assumed to make a generalized ‘core’ which, in the first

15.3 Elementary VB methods

627

approximation, is assumed to be ‘frozen’ during the formation of the bond. In such a way, attention is focussed on the physically relevant part of the triple bond. The advantage of this ‘short’ notation (109) may be appreciated when compared with the ‘full’ notation, which should involve the normalized Slater det of order 14 (the number of electrons in N2):    J ¼  1sA 1sA 1sB 1sB 2sA 2sA 2sB 2sB  sA sB xA xB yA yB  (110) |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl} core

triple bond

For the moment, we shall not take into consideration hybridization, that is, the mixing between 2s and 2ps AO onto the same centre. The parent (109) is a normalized Slater det which is the eigenstate of S^z with MS ¼ 0, but as yet not an eigenstate of S^2 . To obtain the singlet VB structure (eigenstate of S^2 with S ¼ 0) we must do in the parent det all possible spin interchanges between the AOs forming the bonds (s or p), taking into account that for each interchange we must change the sign of the det which is being added (minus sign for an odd number of interchanges, plus sign for an even number). The complete covalent VB function for ground-state N2 is hence made by the linear combination of the following eight Slater dets: 1 Jð1 Sþ g Þ ¼ pffiffiffi ½ðsA sB xA xB yA yB Þ  ðsA sB xA xB yA yB Þ  ðsA sB xA xB yA yB Þ  ðsA sB xA xB yA yB Þ 8 þ ðsA sB xA xB yA yB Þ þ ðsA sB xA xB yA yB Þ þ ðsA sB xA xB yA yB Þ  ðsA sB xA xB yA yB Þ (111) which belongs to the eigenvalue S ¼ 0 as can be easily shown by using Dirac’s formula for S^2. It is worth noting that all doubly occupied orbitals do not contribute to S, and can therefore be omitted in the calculation of the total spin. The normalization factor in Eqn (111) assumes orthonormal Slater dets. We give in the following other simple examples.

15.3.3 The allyl radical (N [ 3) Consider the p system of the allyl radical C3 H$5 as given in Figure 15.10, where attention has to be focussed only on the p electron system (top), and where below each chemical structure we write the parent det.

FIGURE 15.10 The two resonant chemical structures of the p electrons in allyl radical

628

CHAPTER 15 Valence bond theory and the chemical bond

We see that the parent is the same in the two resonant VB structures (which are fully equivalent), but the linear combination of Slater dets is different, since in j1 we have a bond between a and b, in j2 between b and c: 1 1 (112) j1 ¼ pffiffiffi ½ðabcÞ  ðabcÞ S ¼ MS ¼ 2 2 1 j2 ¼ pffiffiffi ½ðabcÞ  ðabcÞ 2

S ¼ MS ¼

1 2

(113)

As said before, the notation omits the description of the 20 electrons of the ‘frozen core’, formed by the electrons of the three inner shells of the C atoms, the two C–C bonds and the five C–H bonds. Even assuming orthonormal spin-orbitals, as we shall do, the two covalent structures describing the doublet state of the radical are non-orthogonal: S12 ¼ hj1 jj2 i ¼

1 2

(114)

It will be shown in Problem 15.3 that if we Schmidt orthogonalize j1 to j2 , the resulting orthogonalized set is in one-to-one correspondence with the two spin doublets found in Chapter 9 using pure spin functions. In other words the two VB structures j1 and j2 constructed using the physically appealing elementary method are not linearly independent, while Schmidt orthogonalization gives two linearly independent functions which, however, have lost their simple graphical representation. The electronic structure of the p system in the allyl radical is hence given by the resonant VB function: J ¼ j1 c1 þ j2 c2

(115)

1 c1 ¼ c2 ¼ pffiffiffi 3

(116)

hJjJi ¼ c21 þ c22 þ 2c1 c2 S12 ¼ 3c21 ¼ 1

(117)

where, because of symmetry

since

Knowing J from Eqn (115), we can immediately calculate the p electron and spin density distributions in the allyl radical, an interesting example of population analysis in multideterminant wavefunctions. The one-electron distribution function from Eqn (115) will be ZZ dx2 dx3 JJ r1 ¼ 3  ZZ   ZZ   ZZ 

2  2    dx2 dx3 j1 j1 þ c2 3 dx2 dx3 j2 j2 þ c1 c2 3 dx2 dx3 j1 j2 þ j2 j1 ¼ c1 3   



1 2 1 1 1 1 2 a þ c2 aa þ b2 bb þ a2 bb þ b2 þ c2 aa þ c22 a þ c2 aa ¼ c21 2 2 2 2 2    



1 1 1 1 2 1 (118) þ b2 bb þ a2 þ b2 aa þ c2 bb þ c1 c2 2 a þ c2 aa þ b2 bb 2 2 2 2 2 where we used Slater’s rules for orthonormal determinants.

15.3 Elementary VB methods

629

We then have ra1 ¼ coefficient of aa in r1





 1 2 1 2 2 1 2 2 2 1 2 2 2 c þc þ c1 c2 þ b c þ c þ c c1 þ c2 þ c1 c2 ¼a 2 1 2 2 1 2 2 2 rb1 ¼ coefficient of bb in r1 ¼ a2





 1 2 1 1 1 c1 þ b2 c21 þ c22 þ c1 c2 þ c2 c22 2 2 2 2





PðrÞ ¼ ra1 ðrÞ þ rb1 ðrÞ ¼ c21 þ c22 þ c1 c2 a2 þ b2 þ c2 ¼ a2 þ b2 þ c2



2 1 2 QðrÞ ¼ ra1 ðrÞ  rb1 ðrÞ ¼ a2 c22 þ c1 c2  b2 ðc1 c2 Þ þ c2 c21 þ c1 c2 ¼ a2  b2 þ c2 3 3 3

(119)

(120) (121) (122)

Equation (121) shows that the allyl radical has a uniform charge distribution of its p electrons (as it must be for an alternant hydrocarbon, as seen in Chapter 14), while if atoms a and c have a spin there must be some b spin at the central atom b. At variance with MO theory (Section 14.8.4), VB theory correctly predicts the existence of some b spin at the central atom whenever some a spin is present at the external atoms, a result which agrees with the experimental electron spin resonance spectra of the radical. The p electron and spin density VB distributions in the allyl radical are shown in Figure 15.11. The relative weights of the two resonant (non-orthogonal) structures in the ground state of the allyl radical will be %j1 ¼ c21 þ c1 c2 S12 ¼

1 1 1 1 þ $ ¼ 3 3 2 2

(123)

%j2 ¼ c22 þ c2 c1 S12 ¼

1 1 1 1 þ $ ¼ 3 3 2 2

(124)

as expected on symmetry grounds.

FIGURE 15.11 p electron (top) and spin density (bottom) VB distributions in allyl radical

630

CHAPTER 15 Valence bond theory and the chemical bond

FIGURE 15.12 The two resonant chemical structures of the p electrons in cyclobutadiene

15.3.4 Cyclobutadiene (N [ 4) Consider now the p system of cyclobutadiene as given in Figure 15.12. Even in this case, the parent det is the same in either resonant VB structures but the p bonds are different, so that the singlet (S ¼ MS ¼ 0) structures are i 1h (125) j1 ¼ ðabcdÞ  ðabcdÞ  ðabcdÞ þ ðabcdÞ 2 i 1h j2 ¼ ðabcdÞ  ðabcdÞ  ðabcdÞ þ ðabcdÞ (126) 2 Even these VB structures are non-orthogonal, since the first and the fourth det in each structure are equal: 1 (127) S12 ¼ hj1 jj2 i ¼ 2 The electronic structure of the p system in cyclobutadiene is given by the resonant VB function: J ¼ j1 c1 þ j2 c2

(128)

1 c1 ¼ c2 ¼ pffiffiffi 3

(129)

hJjJi ¼ c21 þ c22 þ 2c1 c2 S12 ¼ 3c21 ¼ 1

(130)

where again, by symmetry:

since A calculation similar to that in the allyl radical shows that now 

c21 c21 c22 c22 2 2 a 2 2 2 þ þ þ þ c1 c2 r1 ¼ a þ b þ c þ d 4 4 4 4 4 2



1 2 1 ¼ a þ b2 þ c 2 þ d 2 c1 þ c22 þ c1 c2 ¼ a2 þ b2 þ c2 þ d 2 ¼ rb1 2 2 so that

PðrÞ ¼ ra1 ðrÞ þ rb1 ðrÞ ¼ a2 þ b2 þ c2 þ d2 QðrÞ ¼

ra1 ðrÞ  rb1 ðrÞ

¼0

(131) (132) (133)

15.3 Elementary VB methods

631

and the electron charge distribution of the p electrons in cyclobutadiene is uniform, as expected for an alternant hydrocarbon, whereas the spin density is zero (singlet state).

15.3.5 VB description of simple molecules We shall now give a few further examples of the VB description of chemical bonds in simple molecules, assuming in the first approximation that (1) only valence electrons are considered, (2) all electrons not directly involved in the formation of the bond of interest are assumed ‘frozen’, and (3) hybridization is not taken into account. This allows us for the qualitative VB description of a few diatomic molecules and for the study of the electronic structure and charge distribution of XeF2 and O3, while for the H2O molecule hybridization becomes crucial in the VB study of directed valency. Quantitative calculations would imply the evaluation of the matrix elements of the Hamiltonian between these structures and the solution of the related secular equations. 1. LiH ð1 Sþ Þ Lið2 SÞ : 1s2Li s

Hð2 SÞ : h

Covalent structure LiH

Ionic structures Liþ H Li Hþ

ðshÞ

ðhhÞ

(134)

ðssÞ

where ðshÞ ¼ k1sLi 1sLi s hk ðhhÞ ¼ k1sLi 1sLi h hk

(135)

ðssÞ ¼ k1sLi 1sLi s sk are the short notations for the four-electron normalized Slater dets. ðshÞ is the parent for the covalent function for Li–H. The full singlet VB structure describing resonance between covalent and ionic structures in Li–H will be i

h (136) J 1 Sþ f ðshÞ  ðshÞ þ l1 ðhhÞ þ l2 ðssÞ where l1 and l2 must be determined by the Ritz method. Because of the different electronegativities of Li and H, it is expected that l1 [l2 . 2. FH ð1 Sþ Þ Fð2 PÞ : 1s2F 2s2F 2pp4F 2psF

Hð2 SÞ : h

Covalent structure FH

Ionic structures F Hþ Fþ H

ðsF hÞ

ðsF sF Þ

ðhhÞ

i

h J 1 Sþ f ðsF hÞ  ðsF hÞ þ l1 ðsF sF Þ þ l2 ðhhÞ with l1 > l2 .

(137)

(138)

632

CHAPTER 15 Valence bond theory and the chemical bond

FIGURE 15.13 Covalent s and px bonds in N2

3. N2 ð1 Sþ g Þ N^N triple bond NA ð4 SÞ : 1s2NA 2s2NA sA xA yA

s ¼ 2ps ¼ 2pz

(139)

NB ð4 SÞ : 1s2NB 2s2NB sB xB yB

x ¼ 2ppx ¼ 2px

(140)

The triple bond in N2 will be described by the parent: ðsA sB xA xB yA yB Þ

(141)

while the full singlet (S ¼ MS ¼ 0) VB covalent structure will be given by the combination of the eight Slater dets of Eqn (111). We can likely introduce ionic VB structures NNþ, NþN in terms of ðsA sA Þ, ðxA xA Þ, ðsB sB Þ, ðxB xB Þ and so on (Figure 15.13). 4. CO ð1 Sþ Þ Cð3 PÞ : 1s2C 2s2C sC xC y0C Oð3 PÞ : 1s2O 2s2O sO xO y2O |fflfflfflffl{zfflfflfflffl} |fflfflfflffl{zfflfflfflffl} core

or x0C yC or x2O yO

(142) (143)

valence

The three most important covalent VB structures in CO ð1 Sþ Þ all have a heteropolar s bond between sC and sO. The peculiarity of CO (isoelectronic with N2) comes from its p system, as shown in Figure 15.14. In (a), we have a heteropolar px bond between xC and xO, not shown in the figure, a lone pair y2O, while y0C is empty. In (b), a heteropolar py bond between yC and yO,

15.3 Elementary VB methods

633

FIGURE 15.14 Parent dets in ground-state CO

a lone pair x2O, and x0C is empty. This will induce electron transfer from the doubly occupied lone pair orbitals to the empty orbitals of the other atom, giving (c) as the most probable structure showing that a ionic triple bond is formed in CO. The bond is ionic, with polarity COþ, since now seven electrons are on carbon, seven electrons on oxygen. Of course, the truth will be given by the variational determination of the mixing coefficients between the three VB structures associated with the parents of Figure 15.14. This will reduce strongly the molecular dipole moment of CO, being in the opposite sense of the s and p heteropolar effects both going in the sense CþO. This is experimentally observed, and the result of accurate theoretical calculations (Maroulis, 1996) gives m(COþ) ¼ 0.04 ea0. Therefore, the VB function describing the mixing of the covalent structures of Figure 15.14, will be

(144) J1 COV; 1 Sþ ¼ j1 c1 þ j2 c2 þ j3 c3 where

i 1h ðsC sO xC xO yO yO Þ  ðsC sO xC xO yO yO Þ  ðsC sO xC xO yO yO Þ þ ðsC sO xC xO yO yO Þ (145) 2 i 1h j2 ¼ ðsC sO xO xO yC yO Þ  ðsC sO xO xO yC yO Þ  ðsC sO xO xO yC yO Þ þ ðsC sO xO xO yC yO Þ (146) 2

j1 ¼

while j3 is given by the linear combination of eight Slater dets as in Eqn (111) with A ¼ C, B ¼ O. To J1 we must add variationally the corresponding ionic function J2 ðION; 1 Sþ Þ describing the polarity of s and p bonds in CO. 5. Pauli’s repulsion between closed shells A typical example is the interaction between two ground-state He atoms which, each having the closed-shell electron configuration 1s2, cannot give any chemical bond and therefore, in the first

634

CHAPTER 15 Valence bond theory and the chemical bond

approximation7 (one-determinant, no correlation), must repel each other. In this case, VB and MO descriptions coincide. He2 ð1 Sþ gÞ

1sA ¼ a; 1sB ¼ b   1 þ   JðVB; 1 Sþ g Þ ¼ kaabbk ¼ sg sg su su ¼ JðMO; Sg Þ

(147)

as can be seen immediately from the properties of determinants and the Pauli’s principle (Problem 15.4). The electron density contributed by the four electrons of the two atoms is



2 2 2S ab ba 2 þ b þ a  PðrÞ ¼ 2s2g ðrÞ þ 2s2u ðrÞ ¼ 1  S2 1  S2 S S aðrÞbðrÞ bðrÞaðrÞ þ qBA S S

(148)

2S2 qAB ¼ qBA ¼  2; 1  S2

3 þ a result similar to that observed for triplet H2 (Problem 14.4). In He2 ð1 Sþ g Þ, as in H2 ð Su Þ, electrons escape from the bond region originating repulsion (the overlap charge is negative). Recent ab initio calculations of Pauli’s repulsion in He2 (Magnasco and Peverati, 2004) show that a simple optimized 1s basis set gives fair values at R ¼ 3a0 (c0 ¼ 1.691, DE ¼ 12.964  103 Eh) and R ¼ 4a0 (c0 ¼ 1.688, DE ¼ 1.073  103 Eh), which compare favourably either with the accurate theoretical SCF calculations by Liu and McLean (1973) (DE ¼ 13.52  103 Eh, DE ¼ 1.355  103 Eh, respectively) or even better with the experimental results by Feltgen et al. (1982). In this case, however, it is expected that our actual one-determinant underestimation of the interaction will compensate, in part, for the effect of the attractive London forces. A like repulsion is observed between closed-shell molecules, or between pairs of saturated bonds or electron lone pairs in molecules (Pauli repulsion). Pauli repulsion between C–H bonds is at the origin of the torsional barrier in ethane C2H6, as shown by Musso and Magnasco (1982) using an improved bond orbital wavefunction supplemented by small correction terms accounting for electron delocalization. The theory was then analysed in terms of localized singlet VB structures revealing that bonding and charge transfer occur between the four electrons involved in each excitation (Magnasco and Musso, 1982), and was successfully extended to the study of 19 flexible molecules possessing a single internal rotation angle about a B–N, C–C, C–N, C–O, N–N, N–O, O–O central bond (Musso and Magnasco, 1984). The molecules possess 16 to 34 electrons and a variety of functional groups differing in their chemical structure (CH3, NH2, OH, NO, CHO, CH]CH2, NH], and some of their F-derivatives). 6. Three-electron bonds While the bond in Hþ 2 can be considered as the prototype of the one-electron bond, we saw that the great part of s or p chemical bonds is made by two-electron bonds, agreeing with Lewis’ idea of the bond electron pair. There are, however, cases where we observe the formation of three-electron bonds like, for example, in Heþ 2 and O2. 7

Attraction forces due to interatomic electron correlation are described at the multiconfiguration level.

15.3 Elementary VB methods

635

a. The three-electron s bond in Heþ 2 At variance with Pauli’s repulsion in He2 ð1 Sþ g Þ; it is possible to form a rather strong chemical 2 þ bond between a neutral Heð1s2 Þ atom and a Heþ(1s) ion: Heþ 2 ð Su Þ exists and is fairly stable 3 (Huber and Herzberg, 1979): DE ¼ 90.78  10 Eh at Re ¼ 2.04 a0 (compare with the value þ DE ¼ 102.6  103Eh observed at Re ¼ 2a0 for ground-state Hþ 2 ). For the VB theory, He2 is the prototype of the three-electron s bond. 2 þ Heþ 2 ð Su Þ

Heð1 SÞ : 1s2

Heþ ð2 SÞ : 1s

1sA ¼ a; 1sB ¼ b

(150)

In this case, the parents coincide with the VB structures (they are eigenfunctions of S^2 with S ¼ MS ¼ 1/2) (Figure 15.15). By symmetry, the two structures must have equal weight, so that the complete VB function with the correct symmetry (u under inversion) will be 1 1 1 S ¼ ; MS ¼ J 1 ð2 S þ u Þ ¼ pffiffiffi ½kaabk  kbbak 2 2 2 1 1 1 S ¼ ; MS ¼  J2 ð2 Sþ u Þ ¼ pffiffiffi ½kaabk  kbbak 2 2 2

(151) (152)

It is clear that for each component of the doublet 1 i^J1 ¼ pffiffiffi ½kbbak  kaabk ¼ J1 2

1 i^J2 ¼ pffiffiffi ½kbbak  kaabk ¼ J2 2

so that J is odd (u) under inversion. As for He2 it might be shown that     J1 ðVB; 2 Sþ u Þ ¼ sg sg su     J2 ðVB; 2 Sþ u Þ ¼ sg sg su

(153)

(154) (155)

so that, even in this case, VB and MO descriptions coincide. 2 þ Recent ab initio calculations on Heþ 2 ð Su Þ (Magnasco and Peverati, 2004) show that the simple optimized 1s basis set gives a fair representation of the potential energy curve in the bond region, with a calculated bond energy of DE ¼ 90.50  103 Eh at R ¼ 2.06 a0 (c0 ¼ 1.832), in excellent agreement with the experimental results quoted above (Huber and Herzberg, 1979) and the results of the accurate theoretical calculations by Liu (1971).

FIGURE 15.15 Parent dets in ground-state Heþ 2

636

CHAPTER 15 Valence bond theory and the chemical bond

b. The three-electron p bonds in O2 O2 ð3 S g Þ One two-electron s bond

Two three-electron p bonds

OA ð3 PÞ : 1s2A 2s2A sA xA y2A OB ð3 PÞ : 1s2B 2s2B sB xB y2B |fflfflffl{zfflfflffl} |fflfflfflffl{zfflfflfflffl} core

or x2A yA or

x2B yB

(156) (157)

valence

There are four possible covalent VB structures for ground-state O2. i. Two equivalent singlet (S ¼ MS ¼ 0) VB structures The singlet VB structures associated with the parents of Figure 15.16 (a px bond with two y-lone pairs, a py bond with two x-lone pairs) cannot be very stable because of the strong Pauli repulsion between electron lone pairs having the same symmetry (the lone pairs lie in the same yz- or zx-plane). ii. Two equivalent triplet (S ¼ 1) VB structures The triplet VB structures associated with the parents of Figure 15.17 are expected to be much more stable, since the two lone pairs now lie in perpendicular planes with a strong

FIGURE 15.16 Parent dets in singlet ground-state O2

FIGURE 15.17 Parent dets in triplet ground-state O2

15.3 Elementary VB methods

637

reduction in repulsive effects and, what is more interesting, the possibility of forming two equivalent three-electron p bonds (Wheland, 1937). Omitting the doubly occupied orbitals, in an ultrashort notation, the VB function describing the covalent triplet state of groundstate O2 will be 8 1 > > pffiffiffi ½ðxA yB Þ þ ðxB yA Þ S ¼ 1; MS ¼ 1 > > > 2 > > < 1 (158) JðVB; 3 S 0 g Þ ¼ > ½ðxA yB Þ þ ðxA yB Þ þ ðxB yA Þ þ ðxB yA Þ 2 > > > > > > p1ffiffiffi ½ðxA yB Þ þ ðxB yA Þ 1 : 2 It can be verified (Problem 15.5) that J simultaneously satisfies the following eigenvalue equations: S^ 2 J ¼ 1ð1 þ 1ÞJ;

L^z J ¼ 0 $ J

^J ¼ J; s

i^J ¼ J

(159)

so that J correctly describes a 3 S g state (s is chosen as the symmetry zx-plane). At variance with the MO wavefunction, the VB function (158) correctly describes the disso3 ciation of O2 ð3 S g Þ into two neutral Oð PÞ atoms. c. Ionic structures in triplet O2 For the triplet ground state of the O2 molecule we can have the OOþ and OþO ionic structures shown in Figure 15.18. These ionic structures, which we saw to have erroneously a weight equal to that of the covalent structure in the MO wavefunction of H2, are expected to have some importance in O2 in view of the acceptable energetic cost needed to form OOþ (and OþO) from the neutral atoms. In fact Oð3 PÞ2p4 / Oþ ð4 SÞ2p3 þ e

I:P: ¼ 13:6 eV

(160)

Oð3 PÞ2p4 þ e / O ð2 PÞ2p5

E:A: ¼ 1:5 eV

(161)

so that, at the bond distance of R ¼ 2.28 a0, we have in a first approximation 1 27:21 ¼ 13:6  1:5  ¼ 0:17 eV ¼ 6:1  103 Eh I:P:  E:A:  Re 2:28

FIGURE 15.18 Parent dets for ionic structures in triplet O2

(162)

638

CHAPTER 15 Valence bond theory and the chemical bond

where the last term is surely overestimated at this value of R since we do not take into account charge-overlap effects damping the Coulombic attraction between the ions. A VB calculation at Re (McWeeny, 1990) with a DZ-GTO basis set yields: J ¼ 0:59 jcov  0:23 jion

(163)

with a rather large ionic contribution to the VB wavefunction. However, this may be not completely unexpected since it is well known that DZ basis sets tend to overestimate the polarity of the molecule. The Pauli’s repulsion still existing between electron lone pairs even in orthogonal planes, but on the same atom, does not certainly contribute to the stability of the ionic structures in triplet O2. The symmetry of the ionic VB structures associated with the parents of Figure 15.18 can be easily studied using the same techniques used for the covalent structure. 7. XeF2 (DNh) XeF2 is a centrosymmetric linear molecule (symmetry DNh ) with an experimental XeF ¼ 4 a0 bond length, which was studied by Coulson (1964). Xeð1 SÞ : 5s2 5pp4 5ps2 Fð2 PÞ : 2s2 2pp4 2ps

N ¼ 54 electrons

(164)

N ¼ 9 electrons

(165)

The heavy rare gas Xe is easily ionizable, while F atom has a high electron affinity: Xeð1 SÞ / Xeþ ð2 PÞ þ e Fð2 PÞ þ e / F ð1 SÞ

I:P: ¼ 12:1 eV E:A: ¼ 3:5 eV

(166) (167)

As we have seen for O2, in a first approximation, the electrostatic energy needed to form XeþF at Re ¼ 4a0, will be I:P:ðXeÞ  E:A:ðFÞ 

1 27:21 ¼ 1:8 eV ¼ 66:1  103 Eh ¼ 12:1  3:5  Re 4

(168)

This rather large energy will be recovered by the formation of a covalent Xeþ–F bond on the other part of the molecule (Figure 15.19). We hence have complete resonance between the two VB structures: F  Xeþ F 5 F Xeþ  F: The two structures, normalized and orthogonal, are 1 j1 ¼ pffiffiffi ½ðabccÞ  ðabccÞ 2

(169)

1 j2 ¼ pffiffiffi ½ðaabcÞ  ðaabcÞ 2

(170)

hj1 jj2 i ¼ 0

(171)

with

15.3 Elementary VB methods

639

FIGURE 15.19 2ps (F) and 5ps (Xe) AOs and VB structures involved in covalent bonding in XeF2

The function describing complete resonance between the two covalent structures will therefore be 1 J ¼ pffiffiffi ðj1 þ j2 Þ (172) 2 It is now possible to calculate the electron charge distribution in XeF2 as we did for the allyl radical. Using Slater’s rules for orthonormal det, it is easily obtained:    



1 2

1 1 2 1 2 1 1 2 2 2 2 2 a þc þ b þc þ a þb þ a þc (173) ra1 ¼ 2 2 2 2 2 2  from j2 j2 from j1 j1 rb1 ¼

   

1

1

1 1 2 1 1 2 b þ c 2 þ a2 þ c 2 þ a þ c2 þ a2 þ b2 2 2 2 2 2 2

(174)

namely, ra1 ¼ rb1 ¼

1 2 3a þ 2b2 þ 3c2 4

(175)

Therefore, according to this VB description, XeF2 has the following electron charge distribution: 3 3 (176) PðrÞ ¼ ra1 ðrÞ þ rb1 ðrÞ ¼ a2 ðrÞ þ b2 ðrÞ þ c2 ðrÞ 2 2 Z 3 2 3 (177) tr PðrÞ ¼ dr PðrÞ ¼ þ þ ¼ 4 2 2 2 The corresponding electron and formal charges on the atoms in the molecule are given in Figure 15.20. In a first approximation, the formal charge d ¼ 1/2 originates a linear quadrupole moment equal to 1 m2 ¼  jdjð2RXeF Þ2 ¼ 16 ea20 ¼ 2:1  1025 esu cm2 2

(178)

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CHAPTER 15 Valence bond theory and the chemical bond

FIGURE 15.20 Electron (top) and formal charges (bottom) in XeF2

which is within 10% of the experimentally observed value of 1.9  1025 esu cm2. This value of the quadrupole moment is very large when compared to that of other molecules (for instance, CO2 has m2 ¼ 0.3  1025 esu cm2), and is due to the fact that, in the molecule, the fluorine atoms carry a substantial net negative charge. A similar calculation can be done for the p system of the triangular O3 ð1 A1 Þ molecule (2q ¼ 117 , ROO ¼ 2.41a0). The resonance between the equivalent ionic structures (each containing a covalent bond) (Figure 15.21) yields a p formal charge d ¼ 1/2, which gives a p contribution to the dipole moment of mp ¼ ½2m2B ð1  cos 2 qÞ1=2 ¼ 2:05 ea0 . Since the accurate m value is sensibly lower than this (Xie et al., 2000), m ¼ ms þ mp ¼ 0.22 ea0 we can reasonably expect an even larger, and of opposite sign, contribution by the s-skeleton of the molecule. Simple ab initio VB calculations were based on the direct use of Lo¨wdin’s rules (1955a) for the evaluation of matrix elements between Slater determinants of non-orthogonal orbitals. This was the approach mostly followed by Simonetta’s group in Milan in the 1970s. An a priori VB theory, which could be applied to states of any multiplicity, was formulated in a general form including both spin and orbital degeneracies (Simonetta et al., 1968). Structures were related to products of spin functions found by means of extended Rumer diagrams, the spin functions corresponding to each diagram being the product of (N  2S) spins coupled in pairs and 2S parallel coupled (‘leading term’). All possible ways of coupling electrons in pairs were considered. Calculations in the s  p approximation using STO bases were done for simple hydrocarbons, molecules, radicals and ions. An application to benzyl radical with inclusion of an increasing number of ionic structures was found to give excellent results for the hyperfine coupling constants of the radical calculated from VB spin densities (Raimondi et al., 1972).

FIGURE 15.21 Resonance between ionic structures in the p electron system of ozone

15.4 Pauling’s VB theory for conjugated and aromatic hydrocarbons

641

The theory was next applied at the ab initio level to small molecules and radicals, using minimal basis sets of STOs. Calculations were done on ground-state (2 P) energy, proton and 13 C hyperfine splittings of the CH radical (Tantardini and Simonetta, 1972, 1973), and on different electronic states of CH2 (Tantardini et al., 1973). The quality of the orbitals in the basis set (Slater, Slater with optimized orbital exponents, Hartree–Fock AOs) and the number of structures included in the calculation was then examined in minimum basis set ab initio VB calculations on LiH, CH, CH2, CH3, NH3, and H2O by Raimondi et al. (1974). Confirming previous results on H2 and LiH by Yokoyama (1972), it was shown that a relatively small number of structures could give fair results if the basic AOs were carefully chosen. The problems arising in the VB treatment when attempting to extend the basis set were discussed, for the same set of molecules, by Raimondi et al. (1975). They came to the conclusion that there was the need for a general method for going beyond the minimal basis set, then making a selection of the resulting VB structures according to chemical intuition. In this way, connection with Gerratt’s work (1971), where use is made of ‘best’ orbitals to build spin-coupled VB functions (Section 16.4.3), arose quite naturally.

15.4 PAULING’S VB THEORY FOR CONJUGATED AND AROMATIC HYDROCARBONS We have used so far VB theory mostly for a qualitative description of the chemical bond in molecules. Use of symmetry arguments for the resonance of equivalent structures has allowed us, in some cases, to determine the electron and spin density distributions in simple molecules without doing any effective calculation of the relative weight of the structures. However, to do effective, even if approximate, energy calculations we are required to evaluate the matrix elements between structures. This is the main object of any VB theory once the proper wavefunctions have been prepared according to the general rules given in the preceding sections. The general formulation of this problem is rather difficult because of the non-orthogonality of the atomic basis set and non-orthogonality of the VB structures themselves. A great simplification might be reached if we assume that the basic AOs are orthonormal, even if this may be not true for structures. This is a very delicate point, however, since we saw in Section 15.2.4 that assumed, or forced, orthogonalization of the basic AOs yields no bonding at all for atoms described by covalent wavefunctions. It is mostly for this reason that the interesting approach introduced long ago by Pauling (1933) for the VB theory of the p electrons of conjugated and aromatic hydrocarbons has been criticized first and then fully dismissed. From a theoretical point of view, we saw in the simple case of H2 how the orthogonality drawback can be removed from the treatment of the covalent bond by admitting with a substantial weight ionic structures which restore the correct charge distribution between the interacting atoms. This way was followed by McWeeny (1954b) in his rigorous mathematical reformulation of the conventional VB theory based on orthogonalized AOs, where he extended Pauling’s rules to include ionic structures. This rigorous VB treatment was also used by the author and coworkers (Magnasco and Musso, 1967a,b) in an ab initio study of the short-range interaction of two H2 molecules in their ground state. It was shown there (Magnasco and Musso, 1967b) that the orthogonality constraint disappears when the VB treatment is complete and all possible ionic structures are included in the calculation.

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CHAPTER 15 Valence bond theory and the chemical bond

On the other hand, VB theory can be applied at its lowest semiempirical level much in the same way as was Hu¨ckel’s theory in the case of MO theory. In this case, all criticism is inappropriate, since the results are parametrized in terms of Coulomb and exchange integrals, Q and K, which are treated as fully empirical negative parameters, much in the same way as were the as and bs of Hu¨ckel’s theory. Even at this level, the theory is seen to give some interesting insights into the electronic structure of p electron systems, and of s systems as well. Furthermore, the rigorous derivation of Pauling’s rules under their restrictive assumptions is an interesting introduction to the evaluation of Hamiltonian matrix elements between covalent structures for the more advanced theory. For all these reasons, and for their historical importance, we shall give some space in the following to the Pauling’s rules for the evaluation of the matrix elements between singlet covalent VB structures and to their application to the p electron system of some conjugated and aromatic hydrocarbons. The section will end with a short derivation of Pauling’s formula in the case of H2 (N ¼ 2) and cyclobutadiene (N ¼ 4).

15.4.1 Pauling’s formula for the matrix elements of singlet covalent VB structures Let us restrict ourselves to the singlet (S ¼ MS ¼ 0) states of molecules with spin degeneracy only (Pauling and Wilson, 1935). Among the different ways in which valence bonds can be drawn between pairs of orbitals, the number of independent singlet covalent bond structures (the so-called canonical structures) which can be constructed from 2n singly occupied orbitals is given by the Wigner’s formula Number of singlet covalent VB structures ¼

ð2nÞ! n!ðn þ 1Þ!

(179)

where n is the number of bonds. For the case of four orbitals a,b,c,d arranged as a square ring (cyclobutadiene), the bonds can be drawn by lines in three ways, as Figure 15.22 shows, but only j1 and j2 ; having not intersecting lines, are canonical structures.

FIGURE 15.22 Possible covalent bonds in the four-orbital problem

15.4 Pauling’s VB theory for conjugated and aromatic hydrocarbons

643

FIGURE 15.23 Superposition patterns for the four-orbital problem

The lines drawn in Figure 15.22, denoting single covalent p bonds between singly occupied AOs, are called Rumer diagrams. From them, it is possible to construct the so-called superposition patterns of Figure 15.23: The superposition patterns consist of closed polygons or islands, each formed by an even number of bonds. Based on such premises, and on the following assumptions • Orthonormality of the basic AOs • Consideration of singlet covalent VB structures only • Consideration of single interchanges between adjacent orbitals only Pauling (1933) derived simple graphical rules for the evaluation of the general matrix element between structures jr and js ; which are embodied by the formula # " X

1X

1 Hrs  ESrs ¼ ni Q  E þ (180) Kij bonded  Kij non-bonded 2 2 i;j i;j where n is the number of bonds, i the number of islands in each superposition pattern (Figure 15.23), Q(0 2 2 dq ð1 þ S2 Þ q¼0

(293)

(294)

In the equations above, a and b are positive constants involving one- and two-electron integrals independent of angle q. Equations (293) and (294) then say that the HL exchange–overlap strength of the covalent bond formed between the AOs pA and sB has a maximum (a minimum of negative energy) for the straight bond (q ¼ 0 ), which has the physically appealing expression

   S 2b  S ap þ as þ ðsA sB jsA sB Þ  S2 s2A s2B exchov ðq ¼ 0Þ ¼ (295) E 1 þ S2 8

We recall that the bond strength is the negative of the bond energy De at the bond length Re.

15.6 Problems 15

669

the exact counterpart of Eqn (41) for ground-state H2. Here, sA and pA (involved in the constants a,b) are the components of pA along the bond direction and that perpendicular to it. We may conclude that the principle of maximum overlap of elementary valence theory (Coulson, 1961) should more appropriately be replaced by the principle of maximum exchange–overlap in the formation of the covalent bond. A similar conclusion was obtained in elementary Hu¨ckel’s theory including overlap (Magnasco, 2005).

15.6 PROBLEMS 15 15.1. Show the complete equivalence between MO and HL wavefunctions for the 3 Sþ u excited state of H2. Answer: 8 aa > > > >

< 1 1 pffiffiffi ðab þ baÞ p ffiffi ffi Þ ¼ s  s s s JðMO; 3 Sþ g u u g u > 2 2 > > > : bb 8 aa > > > < 1 ab  ba ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi ðab þ baÞ ¼ JðHL; 3 Sþ uÞ 2 > 2 2  2S > > : bb Hint: Use the definitions, expanding sg and su according to expressions (6) and (7) of the main text. 15.2. Show the equivalence between the MO-CI wavefunction (50) and the full VB (HL þ ION) (49). Answer: h   2 1 þ i 2 1 þ JðMO  CI; 1 Sþ g Þ ¼ N J sg ; Sg þ l J su ; Sg   ð1  SÞ þ lð1 þ SÞ 1 ¼ N ðab þ baÞ þ ðaa þ bbÞ pffiffiffi ðab þ baÞ ð1  SÞ  lð1 þ SÞ 2 1 þ ¼ c1 JðHL; 1 Sþ g Þ þ c2 JðION; Sg Þ

Where l ¼ c2/c1. Hint: Use the definitions, expanding sg and su according to expressions (6) and (7) of the main text. 15.3. Show that the Schmidt orthogonalization of j1 to j2 in the case of the allyl radical yields two functions which are in one-to-one correspondence with the two pure spin doublets found in Problem 9.1.

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CHAPTER 15 Valence bond theory and the chemical bond

Answer: j1  S12 j2 1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ pffiffiffi ½ðabcÞ þ ðabcÞ  2ðabcÞ j01 ¼ q 6 1  S212 1 j02 ¼ j2 ¼ pffiffiffi ½ðabcÞ  ðabcÞ 2 to be compared with the two pure spin doublet functions of Chapter 9 1 h1 ¼ pffiffiffi ðaab þ aba  2baaÞ 6 1 h3 ¼ pffiffiffi ðaab  abaÞ 2 Hint: Evaluate the Schmidt-transformed VB function j01 . 15.4. Show the equivalence between MO and VB wavefunctions for He2 ð1 Sþ g Þ. Answer:   1 þ   JðMO; 1 Sþ g Þ ¼ sg sg su su ¼ kaabbk ¼ JðVB; Sg Þ Hint: Use the definitions (6) and (7) for sg and su, and the elementary properties of determinants. 15.5. Write the VB wavefunction for ground-state O2 and classify its electronic state. Answer: In the ultrashort notation, the three components of the triplet VB wavefunction for groundstate O2 are 1 J1 ¼ pffiffiffi ½ðxA yB Þ þ ðxB yA Þ 2 1 ½ðxA yB Þ þ ðxA yB Þ þ ðxB yA Þ þ ðxB yA Þ 2 1 J3 ¼ pffiffiffi ½ðxA yB Þ þ ðxB yA Þ: 2 J2 ¼

S ¼ 1; MS ¼ 1 0 1

Hint: Write the covalent VB wavefunction for ground-state O2 in the ultrashort notation specifying only the two unpaired p electrons, and verify its symmetry properties using the transformation table of Problem 7.2. 15.6. Evaluate the matrix elements between the covalent VB structures for cyclobutadiene. Answer: Equations (266) and (272) of the main text. Hint: Use Slater’s rules for orthonormal dets and the elementary properties of determinants.

15.7 Solved problems

671

15.7. Construct the three sp2 hybrids of C2v symmetry for H2O. Answer: If the molecule is chosen to lie in the yz-plane, the three hybrids are hy1 ¼ as þ bp1 ;

hy2 ¼ as þ bp2 ;

where, if 2q is the interhybrid angle sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 sin2 q  1 1 ; b¼ ; a¼ 2 2 sin q 2 sin2 q

hy3 ¼ cs  dz

c ¼ cot q;

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 sin2 q  1 d¼ sin2 q

Hint: Use equivalence and orthonormality relations between the three hybrids. 15.8. Prove that the angle made by two equivalent orthogonal hybrids is greater than 90 . Answer: If 2q is the angle between the two equivalent orthogonal hybrids bi ¼ s cos u þ pi sin u and bj ¼ s cos u þ pj sin u (u is the hybridization parameter), then cos u2

> > > < 1 1  JðMO; 3 Sþ u Þ ¼ pffiffiffi sg ðr1 Þsu ðr2 Þ  su ðr1 Þsg ðr2 Þ > pffiffiffi ½aðs1 Þbðs2 Þ þ bðs1 Þaðs2 Þ 2 2 > > > : bðs1 Þbðs2 Þ Considering only the spatial part of the wavefunction, we have ½aðr1 Þ þ bðr1 Þ½bðr2 Þ  aðr2 Þ  ½bðr1 Þ  aðr1 Þ½aðr2 Þ þ bðr2 Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2  2S2 ðab þ bb  aa  baÞ  ðba  aa þ bb  abÞ ab  ba pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ JðHL; 3 Sþ uÞ 2 2  2S2 2 2  2S

JðMO; 3 Sþ uÞ¼

which is Eqn (46) of the main text.

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CHAPTER 15 Valence bond theory and the chemical bond

15.2. Equivalence between MO-CI and full VB (HL þ ION) wavefunctions for ground-state H2. Let l ¼ c2/c1 be the ratio between the linear coefficients in Eqn (50). Then, using Eqns (6) and (7) of the main text, we obtain h   2 1 þ i 2 1 þ JðMO-CI; 1 Sþ g Þ ¼ c1 J sg ; Sg þ lJ su ; Sg   ða þ bÞða þ bÞ ðb  aÞðb  aÞ ¼ c1 þl  SPIN 2 þ 2S 2  2S

 c1 aa þ ba þ ab þ bb bb  ab  ba þ aa þl  SPIN ¼ 2 1þS 1S 



 c1 1 l 1 l ðab þ baÞ ¼  þ ðaa þ bbÞ þ  SPIN 2 1þS 1S 1þS 1S 2 1 l 3 þ 6 7 ¼ N 4ðab þ baÞ þ ðaa þ bbÞ 1 þ S 1  S 5 SPIN 1 l  1þS 1S   ð1  SÞ þ lð1 þ SÞ ðaa þ bbÞ  SPIN ¼ N ðab þ baÞ þ ð1  SÞ  lð1 þ SÞ which is Eqn (51) of the main text, N being a normalization factor. 15.3. Schmidt orthogonalization of j1 to j2 in the case of the allyl radical. 8 j1  S12 j2 1 0 > > < j1 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi S12 ¼ 2 1  S212 > > : 0 j2 ¼ j2 gives

  2 1 1 j01 ¼ pffiffiffi $ pffiffiffi ½ðabcÞ  ðabcÞ  ½ðabcÞ  ðabcÞ 2 3 2 1 1 ¼ pffiffiffi ½2ðabcÞ  2ðabcÞ  ðabcÞ þ ðabcÞ ¼ pffiffiffi ½ðabcÞ þ ðabcÞ  2ðabcÞ 6 6

S ¼ MS ¼

1 2

whose associated spin structure is into a one-to-one correspondence with that previously found in Problem 9.1 for the first doublet pure spin function 1 1 h1 ¼ pffiffiffi ðaab þ aba  2baaÞ S ¼ MS ¼ 2 6 The second VB structure 1 j02 ¼ j2 ¼ pffiffiffi ½ðabcÞ  ðabcÞ 2

S ¼ MS ¼

1 2

15.7 Solved problems

673

differs by an irrelevant (1) phase factor from that for the second doublet pure spin function 1 h3 ¼ pffiffiffi ðaab  abaÞ 2

S ¼ MS ¼

1 2

The doublet nature of the two VB structures for the allyl radical ground state can be verified by applying S^2 in the Dirac’s form for N ¼ 3: 3 S^2 ¼ I^ þ P^12 þ P^13 þ P^23 4 We have   1 3 3 S^2 j1 ¼ pffiffiffi ðabcÞ þ ðabcÞ þ ðabcÞ þ ðabcÞ  ðabcÞ  ðabcÞ  ðabcÞ  ðabcÞ 4 2 4 

  3 1 1 1 pffiffiffi ½ðabcÞ  ðabcÞ ¼ þ 1 j1 ¼ 4 2 2 2   1 3 3 2 ^ p ffiffi ffi S j2 ¼ ðabcÞ þ ðabcÞ þ ðabcÞ þ ðabcÞ  ðabcÞ  ðabcÞ  ðabcÞ  ðabcÞ 4 2 4 

  3 1 1 1 pffiffiffi ½ðabcÞ  ðabcÞ ¼ þ 1 j2 ¼ 4 2 2 2 as it must be for doublet S ¼ 1/2 states. 15.4. Equivalence between MO and VB wavefunctions for He2ð1 Sþ g Þ. The following elegant proof is due to Ottonelli (1997). We start from the single determinant MO wavefunction: JðMO; 1 Sþ g Þ ¼ jj sg sg su su jj ¼ jj sg su sg su jj    a þ b ba aþb b  a    ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 þ 2S 2  2S 2 þ 2S 2  2S 

1 ¼  4 1  S2 ja þ b

b  a a þ b b  aj    jDj 0    

1  $ja b a bj ¼  4 1  S2    0 jDj  

1 ¼  4 1  S2 jdet Dj2 $ j a b

1 ¼  1  S2 ja

b

a b j ¼ jj a

bj

a a

b

b jj ¼ JðVB; 1 Sþ gÞ

674

CHAPTER 15 Valence bond theory and the chemical bond

since

D¼

1 1

 1 ; 1

 1 jDj ¼ det D ¼  1

 1  ¼2 1 

2

jdet Dj ¼ 4 ð a þ b

1 b  aÞ ¼ ða bÞ 1

1 1



D being the matrix of a linear transformation between the basic AOs. 15.5. It was shown in this chapter that the MS ¼ 1 component of the covalent triplet VB wavefunction describing the two three-electron p bonds in ground-state O2 can be written, using the usual short notation, as 1 J1 ¼ pffiffiffi ½ðxA yA yA yB xB xB Þ þ ðxB yB yB yA xA xA Þ 2 which can be further contracted into the ultrashort notation specifying only the two unpaired p electrons which are of interest to us 1 J1 ¼ pffiffiffi ½ðxA yB Þ þ ðxB yA Þ 2 Using the transformation table for real AOs of Problem 7.2, under the different symmetry operations of DNh we obtain the following results: •

1 L^z J1 ¼ pffiffiffi ½ðL^z1 xA yB Þ þ ðL^z1 xB yA Þ þ ðxA L^z2 yB Þ þ ðxB L^z2 yA Þ 2 1 ¼ pffiffiffi ½iðyA yB Þ þ iðyB yA Þ  iðxA xB Þ  iðxB xA Þ 2 1 ¼ pffiffiffi ½iðyA yB Þ  iðyA yB Þ  iðxA xB Þ þ iðxA xB Þ ¼ 0 $ J1 2 (S state)

•

1 1 ^yB Þ þ ð^ ^yA Þ ¼ pffiffiffi ½ðxA  yB Þ þ ðxB  yA Þ ^J1 ¼ pffiffiffi ½ð^ sxA s sxB s s 2 2 1 ¼ pffiffiffi ½ðxA yB Þ  ðxB yA Þ ¼ J1 2 (S state)

•

1 ^ ^ ^ A Þ ¼ p1ffiffiffi ½ðxB  yA Þ þ ðxA  yB Þ ^ B iy i^J1 ¼ pffiffiffi ½ðix A iyB Þ þ ðix 2 2 1 ¼ pffiffiffi ½ðxB yA Þ þ ðxA yB Þ ¼ J1 2 (S g state).

15.7 Solved problems

675

Taking into account spin, we see that J1 properly describes the MS ¼ 1 component of the triplet 3 S g characteristic of the ground state of the O2 molecule. Similarly we can proceed with the remaining triplet components J2 (MS ¼ 0) and J3 (MS ¼ 1) of the covalent VB wavefunction of O2. Lastly, we shall show the equivalence, with respect to the symmetry operations of DNh ; of the simple two-electron VB wavefunction in the ultrashort notation with the full six-electron VB wavefunction describing in the short notation the two three-electron p bonds in O2. We have •

  1 L^zk pffiffiffi ½ðxA yA yA yB xB xB Þ þ ðxB yB yB yA xA xA Þ 2 k¼1 h 1 ¼ pffiffiffi ið yA yA /Þ þ ið yB yB /Þ 2 |ffl{zffl} |ffl{zffl}

L^ z J1 ¼

6 X

k¼1

ið xA xA /Þ  ið xB xB /Þ |ffl{zffl} |ffl{zffl}

2

iðxA yA xA yB xB xB Þ  iðxB yB xB yA xA xA Þ

3

iðxA yA yA xB xB xB Þ  iðxB yB yB xA xA xA Þ |ffl{zffl} |ffl{zffl}

4

þiðxA yA yA yB yB xB Þ þ iðxB yB yB yA yA xA Þ |ffl{zffl} |ffl{zffl} i þiðxA yA yA yB xB yB Þ þ iðxB yB yB yA xA yA Þ ¼ 0 $ J1

5 6

Of the 12 terms resulting by the action of the one-electron operator L^z on J1, terms 1,2,3,4,7,8,9,10 vanish because of the exclusion principle (determinants with two rows or columns equal), the 6-th and the 12-th term being reduced, after three interchanges, to the 5-th and 11-th term with opposite sign, so that the whole expression in square brackets vanishes as it must be. In fact ðxB yB xB yA xA xA Þ / ðxA yB xB yA xB xA Þ / ðxA yA xB yB xB xA Þ / ðxA yA xA yB xB xB Þ 3  interchanges ðxB yB yB yA xA yA Þ / ðxA yB yB yA xB yA Þ / ðxA yA yB yB xB yA Þ / ðxA yA yA yB xB yB Þ 3  interchanges •

1 ^ sJ1 ¼ pffiffiffi ½ðxA  yA  yA  yB xB xB Þ þ ðxB  yB  yB  yA xA xA Þ 2 1 ¼ pffiffiffi ½ðxA yA yA yB xB xB Þ þ ðxB yB yB yA xA xA Þ ¼ J1 2

676

CHAPTER 15 Valence bond theory and the chemical bond

since (1)3 can be factored out from each determinant. •

i h ^ 1 ¼ p1ffiffiffi ð xB  yB  yB  yA  xA  xA Þ þ ðxA  yA  yA  yB  xB  xB Þ iJ 2 i 1 h ¼ pffiffiffi ðxB yB yB yA xA xA Þ þ ðxA yA yA yB xB xB Þ ¼ J1 2

since (1)6 can now be factored out from each determinant. Hence, we see that manipulation of the full six-electron VB wavefunction brings to the same results as the much simpler, and much easily tractable, two-electron VB wavefunction. All doubly occupied AOs (or MOs) have no effect on the symmetry operations of the point group. 15.6. VB matrix elements for cyclobutadiene. With reference to Eqn (264) expressing j1 and j2 in terms of Slater dets, we have ^ 1i ¼ H11 ¼ hj1 jHjj

1h ^ ^ þ hðabcdÞjHð22ÞjðabcdÞi hðabcdÞjHð11ÞjðabcdÞi 4

i ^ ^ þ hðabcdÞjHð44ÞjðabcdÞi þ hðabcdÞjHð33ÞjðabcdÞi

þ

2h ^ ^ ^ ^ hðabcdÞjHð12Þ þ Hð21ÞjðabcdÞi  hðabcdÞjHð13Þ þ Hð31ÞjðabcdÞi 4 ^ ^ ^ ^ þ hðabcdÞjHð14Þ þ Hð41ÞjðabcdÞi þ hðabcdÞjHð23Þ þ Hð32ÞjðabcdÞi ^ ^ ^ ^  hðabcdÞjHð24Þ þ Hð42ÞjðabcdÞi  hðabcdÞjHð34Þ þ Hð43ÞjðabcdÞi

i

¼ use Slaters rules for orthonormal dets; considering only single interchanges and omitting for short 1=r12 in the Dirac’s notation ( i h i 1 h Q  hacjcai  hbdjdbi þ Q  hadjdai  hbcjcbi ¼ 4 ) i h i h þ Q  hadjdai  hbcjcbi þ Q  hacjcai  hbdjdbi þ

i 2h habjbai þ hcdjd ci þ hcdjd ci þ habjbai 4

¼ eliminate spin using charge density notation ( ) h i h i 1 4Q  2 ðacjacÞ þ ðadjadÞ þ ðbcjbcÞ þ ðbdjbdÞ þ 4 ðabjabÞ þ ðcdjcdÞ ¼ 4

where Q is the Coulomb integral arising from the product of orbital functions (zero interchanges)  

^ Q ¼ habcdjHjabcdi ¼ Vnn þ haa þ hbb þ hcc þ hdd þ a2 b2 þ a2 c2    

þ a2 d 2 þ b2 c2 þ b2 d2 þ c2 d2

15.7 Solved problems

677

Since, for ‘non-adjacent’ orbitals ðacjacÞ ¼ ðbdjbdÞ ¼ 0 and all other exchange integrals are equal by symmetry, we finally obtain 1 H11 ¼ Q þ 2K  ð2KÞ ¼ Q þ K 2 which is Eqn (269) of the main text. For the off-diagonal element, we have h ^ 2 i ¼ 1 hðabcdÞjHð11ÞjðabcdÞi ^ ^ H12 ¼ hj1 jHjj þ hðabcdÞjHð22ÞjðabcdÞi 4 ^ ^ þ hðabcdÞjHð44ÞjðabcdÞi þ hðabcdÞjHð33ÞjðabcdÞi þ

i

1h ^ ^  hðabcdÞjHð13ÞjðabcdÞi hðabcdÞjHð12ÞjðabcdÞi 4 ^ ^  hðabcdÞjHð21ÞjðabcdÞi þ hðabcdÞjHð14ÞjðabcdÞi ^ ^  hðabcdÞjHð24ÞjðabcdÞi þ hðabcdÞjHð23ÞjðabcdÞi ^ ^ þ hðabcdÞjHð32ÞjðabcdÞi  hðabcdÞjHð31ÞjðabcdÞi ^ ^ þ hðabcdÞjHð41ÞjðabcdÞi  hðabcdÞjHð34ÞjðabcdÞi i ^ ^  hðabcdÞjHð42ÞjðabcdÞi  hðabcdÞjHð43ÞjðabcdÞi

( 1 h ¼ Q  hacjcai  hbdjdb i  hbdjdbi  hdbjbdi þ ½Q  hac j cai 4 ) i  hbdjdbi þ

1h hadjdai þ hcbjbci þ hbajabi  hcajaci 4

i þ hcdjd ci þ hd cjcdi  hacjcai þ habjbai þ hbcjcbi þ hdajadi

( 1 h 2 Q  ðacjacÞ  ðbdjbdÞ  2ðbdjbdÞ þ 2½ðabjabÞ  ðacjacÞ ¼ 4 ) i þ ðadjadÞ þ ðbcjbcÞ þ ðcdjcdÞ ¼

1h Q þ ðabjabÞ  2ðacjacÞ þ ðadjadÞ 2 i 1 þ ðbcjbcÞ  2ðbdjbdÞ þ ðcdjcdÞ ¼ ðQ þ 4KÞ 2

which is Eqn (272) of the main text.

678

CHAPTER 15 Valence bond theory and the chemical bond

FIGURE 15.37 Resolution of a pq AO into orthogonal components

15.7. We first study the resolution of a pq orbital, making an angle q with the positive z-axis in the yz-plane, into its orthogonal components py and pz (Figure 15.37). We have

k and j can be replaced by pz ¼ z and py ¼ y. Therefore pq ¼ z cos q þ y sin q where cos q and sin q are the direction cosines of pq in the yz-plane. Similarly, changing q into q pq ¼ z cos q  y sin q and we have the linear transformation p1 ¼ z cos q þ y sin q;

p2 ¼ z cos q  y sin q

ðp1 p2 Þ ¼ ðyzÞL where

L¼

sin q cos q

sin q cos q



The two transformed AOs, p1 and p2, are not orthogonal: h p1 jp2 i ¼ cos2 q  sin2 q ¼ cos 2q The transformation matrix L has det L ¼ sin 2q, and is therefore unitary only for q ¼ p/4. The inverse transformation is 0 1 1 1 B 2 sin q 2 cos q C C L1 ¼ B @ 1 1 A  2 sin q 2 cos q

15.7 Solved problems

679

giving y¼

1 ðp1  p2 Þ; 2 sin q

z¼

1 ðp1 þ p2 Þ 2 cos q

We can now pass to the construction of the three sp2 hybrids equivalent under C2v symmetry. We can write hy1 ¼ as þ bp1 ¼ as þ bz cos q þ by sin q hy2 ¼ as þ bp2 ¼ as þ bz cos q  by sin q hy3 ¼ cs  dz where the coefficients a,b,c,d must satisfy the orthonormality conditions ( a2 þ b2 ¼ 1 a2 þ b2 cos 2q ¼ 0 c2 þ d 2 ¼ 1

ac  bd cos q ¼ 0

By solving it is found a2 ¼

cos 2q 1 2 sin2 q  1 ¼ ; ¼1 cos 2q  1 2 sin2 q 2 sin2 q

c2 ¼

2 b2 cos2 q 2 ¼ cot2 q; 1  c cos q ¼ a2 sin2 q

whence, by choosing the positive roots sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 sin2 q  1 1 ; b¼ a¼ ; 2 2 sin q 2 sin2 q

b 2 ¼ 1  a2 ¼

d 2 ¼ 1  c2 ¼

1 2 sin2 q

2 sin2 q  1 sin2 q

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 sin2 q  1 d¼ sin2 q

c ¼ cot q;

The unitary matrix U doing the sp2(C2v) hybridization is then ð hy1 where

hy2 0

hy3 Þ ¼ ð s z

a

B U ¼ @ b cos q b sin q

y ÞU 1

a

c

b cos q

C d A

b sin q

0

It is possible to express U as a function of coefficients c and d only, using the relations 1 a ¼ pffiffiffi d; 2

1 b cos q ¼ pffiffiffi c; 2

1 b sin q ¼ pffiffiffi 2

so that we obtain for U the simplified form 0 1 1 1 pffiffiffi d pffiffiffi d c B 2 C 2 B C B 1 C B pffiffiffi c p1ffiffiffi c d C U¼B C B 2 C 2 B C @ 1 A 1 pffiffiffi pffiffiffi 0 2 2

680

CHAPTER 15 Valence bond theory and the chemical bond

By direct matrix multiplication it is immediately verified that UUy ¼ Uy U ¼ 1 15.8. Angle between equivalent orthogonal hybrids. With reference to Figure 15.34, let bi, bj be the two s,p hybrids pointing in the directions i and j and making an interhybrid angle 2q. If we denote by u the single hybridization parameter (equivalent hybrids), we can write the hybrids as bi ¼ s cos u þ pi sin u bj ¼ s cos u þ pj sin u We now resolve pj into its components in the direction of the i-th bond and in the direction perpendicular to it: pj ¼ pi cos 2q þ pt sin 2q The condition of orthogonality between the two hybrids then gives hbi jbj i ¼ cos 2 u þ sin2 uh pi jpj i ¼ cos 2 u þ sin2 u cos 2q ¼ 0 and we therefore obtain

cos u2 cos 2q ¼ 

uk ðx1 Þdetu ðx1 jkÞ Uðx1 ; x2 ; /; xN Þ ¼ ðN!Þ1=2 > < k X 1=2 > > vl ðx1 Þdetv ðx1 jlÞ : Vðx1 ; x2 ; /; xN Þ ¼ ðN!Þ

(14)

l

Taking the normalization of U and V into account, the one-electron transition density matrix for non-orthogonal spin-orbitals is then given by r1 ðUVjx1 ; x01 Þ

Z

dx2 dx3 /dxN Uðx1 ; x2 ; /; xN ÞV  ðx01 ; x2 ; /; xN Þ X ¼ ðDUU DVV Þ1=2 uk ðx1 Þvl ðx01 ÞDVU ðljkÞ ¼N

(15)

k;l

Likely, the two-electron transition density matrix for non-orthogonal spin-orbitals is given by r2 ðUVjx1 ; x2 ; x01 ; x02 Þ

Z

dx3 /dxN Uðx1 ; x2 ; x3 ; /; xN ÞV  ðx01 ; x02 ; x3 ; /; xN Þ X uk1 ðx1 Þuk2 ðx2 Þvl1 ðx01 Þvl2 ðx02 ÞDVU ðl1 l2 jk1 k2 Þ ¼ ðDUU DVV Þ1=2

¼ NðN  1Þ

k1 k2 ;l1 l2

2

See Section 6.3.

(16)

16.3 Spinless pair functions and the correlation problem

685

The matrix element of the electronic Hamiltonian H^e between the two N-electron Slater determinants U and V built from non-orthogonal spin-orbitals will then be given by  + 8 *   N N D   E > X X   > 1 1 > U    ^ ^ > V He U ¼ V  hi þ >  > 2 r > ij  > i¼1 i;jðjsiÞ  > > < Z ZZ 1 1 ^1 r1 ðUVjx1 ; x0 Þj 0 (17) ¼ dx dx1 dx2 þ r ðUVjx1 ; x2 ; x1 ; x2 Þ h 1 1 x1 ¼x1 > 2 r12 2 > > > # " > > X > 1 X > > hlk DVU ðljkÞ þ gl l ;k k DVU ðl1 l2 jk1 k2 Þ ¼ ðDUU DVV Þ1=2 > : 2 k k ;l l 1 2 1 2 k;l 1 2 1 2

the matrix elements of the one- and two-electron operators being given as D   E   hmn ¼ mh^n ; gmn;ss ¼ hmnjgjssi

(18)

16.3 SPINLESS PAIR FUNCTIONS AND THE CORRELATION PROBLEM The most general expression for the two-electron distribution function is r2 ðx1 ; x2 ; x1 ; x2 Þ ¼ P2 ðra1 ; ra2 ; ra1 ; ra2 Þaðs1 Þaðs2 Þa ðs1 Þa ðs2 Þ þ P2 ðrb1 ; rb2 ; rb1 ; rb2 Þbðs1 Þbðs2 Þb ðs1 Þb ðs2 Þ þ P2 ðra1 ; rb2 ; ra1 ; rb2 Þaðs1 Þbðs2 Þa ðs1 Þb ðs2 Þ þ P2 ðrb1 ; ra2 ; rb1 ; ra2 Þbðs1 Þaðs2 Þb ðs1 Þa ðs2 Þ þ /

(19)

By integrating over spin we obtain the spinless pair function, which can be written in short as P2 ðr1 ; r2 Þ ¼ P2 ðra1 ; ra2 Þ þ P2 ðrb1 ; rb2 Þ þ P2 ðra1 ; rb2 Þ þ P2 ðrb1 ; ra2 Þ

(20)

where only the diagonal elements appear in the short notation. The physical meaning of each component is self-evident, that for the third component being P2 ðra1 ; rb2 Þdr1 dr2 ¼ probability of finding an electron at dr1 with spin a and; simultaneously; another electron at dr2 with spin b; and so on

(21)

It may be useful to introduce a correlation factor f (McWeeny, 1960): P2 ðra1 ; rb2 Þ ¼ P1 ðra1 ÞP1 ðrb1 Þ½1 þ f ðra1 ; rb2 Þ

(22)

where f ðra1 ; rb2 Þ describes the deviation from the independence ( f ¼ 0) in the motion of the two electrons due to electron correlation. For antisymmetric functions, if r2 / r1 f ðra1 ; ra2 Þ ¼ f ðrb1 ; rb2 Þ ¼ 1

(23)

686

CHAPTER 16 Post-Hartree–Fock methods

namely, there is 100% negative correlation, so that electrons with the same spin cannot be found in the same point of space. This is called ‘Fermi correlation’, and it represents the most general formulation of the Pauli’s principle, being completely independent of the form of the wavefunction. Examples for the components with MS ¼ 1 and MS ¼ 0 of the 1s2sð3 SÞ excited state of the He atom are given as Problems 16.1 and 16.2, respectively, at the end of this chapter. The conclusion is that, in both cases (parallel spins), the correlation factor tends to 1 in the limit r2 / r1. The situation is different for electrons with different spins occupying the same (atomic or molecular) orbital, and one of the typical problems of advanced quantum mechanics is that of calculating as exactly as possible the correlation factor for different spins f ðra1 ; rb2 Þ. In the simple one-configuration molecular orbital (MO) theory (J approximated as a single Slater det of doubly occupied MOs), this correlation factor is equal to zero (Problem 16.3): P2 ðra1 ; rb2 ; ra1 ; rb2 Þ ¼ P2 ðra1 ; rb2 Þ ¼ P1 ðra1 ÞP1 ðrb2 Þ ¼ P1 ðrb1 ÞP1 ðra2 Þ

(24)

so that the distribution function of electron pairs with different spins is simply the product of the distribution functions of the single electrons (no correlation). This means that the two electrons can approach each other in a completely arbitrary way, at variance with the physical reality of the Coulomb repulsion of the two electrons which would tend to N as r12 / 0. In other words, the single det MO wavefunction does not correlate at all the motion of the electrons (independent particle model (IPM)). Instead, if we consider the Heitler–London (HL) wavefunction for ground state H2    1 (25) JHL ðx1 ; x2 Þ ¼ N ab þ kbak ¼ Nðab þ baÞ pffiffiffi ðab  baÞ 2 we have a two-configuration wavefunction of atomic SOs. Then, calculation shows that (Problem 16.4) P2 ðra1 ; rb2 ; ra1 ; rb2 Þ ¼ P2 ðrb1 ; ra2 ; rb1 ; ra2 Þ ¼ N 2 ½aðr1 Þbðr2 Þa ðr1 Þb ðr2 Þ

þ bðr1 Þaðr2 Þb ðr1 Þa ðr2 Þ þ aðr1 Þbðr2 Þb ðr1 Þa ðr2 Þ

þ bðr1 Þaðr2 Þa ðr1 Þb ðr2 ÞsP1 ðra1 ÞP1 ðrb2 Þ where P1 ðra1 Þ ¼

Z

  dr2 P2 ðra1 ; rb2 ; ra1 ; rb2 Þ ¼ N 2 a2 þ b2 þ 2 Sab ¼ P1 ðrb2 Þ ¼

(26)

Z dr1 P2 ðra1 ; rb2 ; ra1 ; rb2 Þ (27)

so that JHL introduces some sort of correlation between electrons with different spins. This is due to its molecular ‘split-shell’ structure, which partially prevents the electrons from doubly occupying the same orbital with different spin.

16.4 CONFIGURATIONAL INTERACTION METHODS From a basis set of one-electron orbitals it is possible to obtain interconfigurational many-electron wavefunctions improving either MO or valence bond (VB) wavefunctions, giving results which become identical when full use of the basis set is accomplished even if they may be different in their starting approximations.

16.4 Configurational interaction methods

687

In this section, we present first the ordinary configuration interaction (CI) approximation, next have a glance at the large-scale CI method and at the GVB method developed in recent years, ending with the study of the cusp effects on the convergence of the interconfigurational expansion showing how it is possible to improve convergence by explicitly introducing the interelectronic distance in the wavefunction.

16.4.1 Configuration interaction For a given basis of atomic or molecular spin-orbitals, we construct a linear combination of electron configurations in the form of many-electron Slater determinants {Jk}, with coefficients Ck determined by the Ritz method, to give the CI wavefunction: X Jk ðx1 ; x2 ; /; xN ÞCk (28) Jðx1 ; x2 ; /; xN Þ ¼ k

When all possible configurations arising from a given basis set are included, we speak of fullconfiguration interaction (FCI) wavefunction. It should be recalled that only configurations of given S, MS belonging to a given molecular symmetry have non-zero matrix elements of the molecular Hamiltonian. Even if CI is ‘in principle’ exact for a complete basis set, giving convergence to the exact (nonrelativistic) wavefunction and energy, convergence becomes rather slow if one strives at ‘chemical accuracy’,3 and ‘spectroscopic accuracy’4 is not accessible to ordinary CI techniques. Difficulties are due not to the large dimensions of the secular equation,5 but rather to the numerical problems arising from the near-linear dependencies that are unavoidable for extremely large basis sets. The traditional approach, where matrix H is computed and eigenvalues and eigenvectors are obtained by standard diagonalization techniques, is unwieldy to deal with such a situation, mostly for problems of computer storage.

16.4.2 Large-scale CI methods In the 1970s, new strategies were presented for overcoming these problems (McWeeny, 1992). These techniques are ‘global’, in the sense that they consider all interconfigurational functions (CFs) of any desired symmetry that can be constructed from a given set of spin-orbitals, allowing rapid construction of all types of matrix elements without setting up the full matrix, and the iterative refinement of all expansion coefficients. The problems in handling expansions of, say, millions terms, arise from (1) how to label in the most efficient way each one of the CFs for all orbital configurations and all spin couplings, (2) how to associate one- and two-electron integrals with pairs of CFs, and hence with matrix elements Hmn, so that all matrix elements can be generated simultaneously, and, finally, (3) how to construct from any estimated eigenvector Cm an improved estimate Cm0 . The CFs employed in an FCI calculation are constructed from an orbital basis set fci g of m functions by setting up orbital products, adding spin factors, antisymmetrizing so as to obtain Slater Errors within 1mEh ¼ 10–3 Eh. Errors of the order of 1mEh ¼ 10–6 Eh. 5 Secular equations involving millions of determinants can be managed using special matrix techniques (Roos, 1972). 3 4

688

CHAPTER 16 Post-Hartree–Fock methods

determinants, and combining the resultant Slater determinants according to symmetry requirements. Provided the full CI space is used, we can use as well any new linear combination fc0i g of the basis functions. A transformation of orbitals ci / c0i (one-electron basis functions) induces a transformation of CFs (many-electron wavefunctions), an arbitrary function in the full CI space being left invariant with suitably modified expansion coefficients. These transformations can be related to two groups, which were already examined in Chapter 8, (1) the symmetric group SN of permutations of the N electronic variables, and (2) the unitary group U(m) of the transformations leading from one orthonormal set to the other. It is the relationship between these two groups that has been exploited operatively in recent years, giving what has been called the large-scale CI and the unitary group approaches. Much more about this can be found in McWeeny (1992).

16.4.3 Generalized valence bond methods The classical VB method, considered as a simple extension of the HL calculations for the hydrogen molecule (1927), becomes quickly impracticable if use of large basis sets is needed to get highly accurate results. Gallup (1973; Gallup et al., 1982) uses group theoretical techniques with Young idempotents to project out functions of appropriate symmetry from a set of arbitrary orbital products, but not optimizing the orbitals. The theory was presented in some detail by Gallup (2002) in a recent textbook on VB methods, and fully implemented in a commercial program package (CRUNCH) developed by the author and his students. McWeeny (1988b, 1990) recasted classical VB theory in a spin-free form which seems to provide a practicable route to ab initio calculations of molecular electronic structure even for extended basis sets. The formalism is based on the construction of symmetry-adapted functions obtained by applying Wigner projectors to suitable products of space orbitals, and coupling them with the associated ‘dual’ spin functions carrying an fSN dimensional irrep of the SN group. To maintain contact with classical VB theory, however, the spin functions are chosen to be of the Weyl–Rumer form and not of the standard Kotani’s form as those used by Gerratt (1971) and by Cooper et al. (1987). The Weyl–Rumer functions for jMS j ¼ S correspond to the lowest path in the standard branching diagram, where (N  2S)/2 spins are coupled in pairs while all remaining spins are coupled in parallel (see also Serber, 1934a,b). A standard variational approach allows one to determine the best expansion coefficients and orbitals in the expansion of the wavefunction over all possible symmetry-adapted orbital products. The matrix elements are easily evaluated (even within the memory of a fast personal computer) provided efficient algorithms are used for systematically generating permutations and for handling Rumer diagrams. The method should be sufficient for dealing with molecular systems containing up to 10 electrons outside a closed-shell core, and simple calculations on the H2–H2 system (McWeeny, 1988a), and on the H2O and O2 molecules (McWeeny and Jorge, 1988; McWeeny 1990) seem to confirm the conclusions of Cooper et al. (1987) that, using strongly overlapping optimized orbitals, a small number of ‘classical’ covalent structures can give results close to those occurring in a FCI calculation with the same basis set. Goddard (1967a,b), (1968a,b) and Gerratt (1971) independently proposed a general theory where use of group theoretical arguments allows one to obtain an energy expansion corresponding to a linear combination of ‘structures’6 for a single orbital configuration, then optimizing both orbitals 2 A VB ‘structure’ is an eigenfunction of the spin operator S^ belonging to the eigenvalue S.

6

16.4 Configurational interaction methods

689

and expansion coefficients to obtain the best possible one-configuration approximation. The nonorthogonal orbitals resulting therefrom were called unrestricted generalized valence bond (GVB) orbitals by Goddard and spin-coupled SCF orbitals by Gerratt. As far as the numerical results are concerned, Goddard made a preliminary calculation on the magnetic hyperfine structure of the Li atom (Goddard III, 1967c), followed by GVB calculations for several molecules, including H2 (Goddard III, 1967b), Li2, CH3, CH4 (Goddard III, 1968b), LiH (Palke and Goddard III, 1969), LiH, BH, H3, H2O, C2H6, O2 (Hunt et al., 1972) and O2 (Moss et al., 1975). The calculations gave a fairly correct torsional barrier in ethane, and an accurate description 1 þ of the relative position of the 3 S g and Sg electronic states of O2. Particular attention was paid to the shape of the optimized GVB orbitals, which show enhanced overlap over the entire internuclear distances. Gerratt did calculations using an extension of his original approach on H2 (Wilson and Gerratt, 1975), LiH, BH, Li2, HF (Pyper and Gerratt, 1977), the potential energy curves of different electronic states of BeH (Gerratt and Raimondi, 1980), the dipole moment of ground-state LiH (Cooper et al., 1985), the reaction ðB þ H2 Þþ (Cooper et al., 1986a), and a review (Cooper et al., 1987) where the results on small molecular systems, containing up to 10 electrons, were presented. Again it was found that the orbitals resulting from the optimized procedure are of the Coulson–Fischer enhanced-overlap type, and that relatively short expansions usually lead to wavefunctions of high quality. The spincoupled functions typically yield 85% of the observed binding energies and equilibrium bond ˚ . About 200–700 structures are usually sufficient to reproduce the first 10 distances accurate to 0.01 A states of a given symmetry to an accuracy of about 0.01 eV. The explicit introduction of r12 into the Coulson–Fischer wavefunction was examined by Clarke et al. (1994). Similar VB-SCF and VB-CI methods were later proposed by Van Lenthe and Balint-Kurti (1983). An extension to the many-configuration theory, the so called spin-coupled VB theory will be outlined in what follows. This theory was first put forward by Gerratt (1971, 1976) and then mostly developed by Gerratt, Cooper and Raimondi (Gerratt and Raimondi, 1980; Cooper et al., 1987). In Gerratt’s approach, the most general approximate function obtainable from an N-electron spatial function FN given in the form of a product of N spatial orbitals FN ðr1 ; r2 ; / rN Þ ¼ c1 ðr1 Þc2 ðr2 Þ / cN ðrN Þ

(29)

is written as a linear combination of all possible spin couplings k N

JSMS ¼

fS X

JSMS ;k CSk

(30)

k¼1

where JSMS ; k is an un-normalized spin-coupled function !1=2 f N

S X 1 S Y^lk FN QNSMS ; l JSMS ; k ¼ N fS k¼1

(31)

and S Y^lk ¼

 N 1=2 X fS r UlkS P^ N! P

(32)

690

CHAPTER 16 Post-Hartree–Fock methods

is the Young–Yamanouchi–Wigner operator7 projecting out of FN a basis transforming in an irreducible way under the operations of the symmetric group SN. fSN is the dimension of the irreducible S representation [S]. QNSM ;l is the spin function ‘dual’ to ðY^lk FN Þ in the Wigner’ sense (Chapter 8), and is S

constructed by the synthetic method due to Kotani et al. (1963) starting from the spin functions for a single electron and building up the N-electron function by successively coupling the spins according to the usual quantum mechanical rules for angular momentum (Chapter 10). All QNSMS ;l are eigen2 functions of S^ and S^z with eigenvalues S and MS, and form an orthonormal set of spin functions making a basis for the irreducible representations of SN ‘dual’ to the projected spatial part and s transforming under a permutation P^ of the spin variables according to N

P^

s

QNSMS ; k

¼ εP

fS X

QNSMS ; l UlkS ðPÞ

(33)

l¼1

where εP is the parity of the permutation. The spatial part ‘dual’ to the spin function (33) in the Wigner’ sense of Chapter 8 has the fundamental permutational symmetry property: N

P^ FSk ¼ r

fS X

S FN Sl Ulk ðPÞ

(34)

l¼1

where r is a spatial variable. Each individual function FSk is a solution of the Schroedinger equation: ^ Sk ¼ EFSk HF

(35)

fSN -fold

so that the energy level E has an spin degeneracy. The energy corresponding to function (30) is given by " # N N X N N X N X X 1 X DðkjlÞhkl þ DðkljssÞgkl;ss  DðkljssÞgkl;ss ES ¼ D k;l¼1 k> r1, r2) in a power series in Rn up to R3 8 1=2  h i1=2   > 1 1 z1 r12 > 2 2 2 2 2 1=2 > ¼ R  2z1 R þ r1 ¼ 12 þ 2 > < rB1 ¼ x1 þ y1 þ ðz1  RÞ R R R (82)   > z 2 2 2  r2 >   r 3z 1 z 3 1 z > 1 1 1 > : z 1 þ  12 þ 2 þ / ¼ þ 2 þ 1 3 1 þ O R4 R 2R R 2R R 8 R R where use was made of the Taylor expansion (41) up to x2 for x ¼ small. In the same way 8 1=2  h i1=2   > 1 z2 r22 > 2 2 1=2 > 1 ¼ x2 þ y2 þ ðz2 þ RÞ2 þ ¼ R þ 2z R þ r ¼ 1 þ 2 > 2 2 2 2 < rA2 R R2 R   > >   r22 1 z2 3 z 2 2 1 z2 3z2  r 2 > > : z 1  2þ 2 þ / ¼  2 þ 2 3 2 þ O R4 R 2R 2R R 8 R R R For the two-electron repulsion, we have 8 h i1=2 1 2 2 2 > > ¼ ðx  x Þ þ ðy  y Þ þ ðz  z  RÞ 1 2 2 2 1 2 > > r12 > > <  1=2 ¼ R2  2z1 R þ 2z2 R þ r12 þ r22  2x1 x2  2y1 y2  2z1 z2 > >  1=2 > > 1 z1 z2 r 2 r 2 2 > > : ¼ 1  2 þ 2 þ 12 þ 22  2 ðx1 x2 þ y1 y2 þ z1 z2 Þ R R R R R R

(83)

(84)

17.3 Interatomic potentials

so that, expanding according to Taylor to the same order 8  r2 r2 1 1 z1 z2 1 > > z 1 þ   12  22 þ 2 ðx1 x2 þ y1 y2 þ z1 z2 Þ > > > R R 2R 2R r12 R R > > > >  < 3 z 1 2 3 z 2 2 3 z 1 z 2 þ 2 þ 8 2 þ / þ 2 > 8 8 R 8 R R > > > > > > >   1 z1  z2 3z21  r12 3z22  r22 x1 x2 þ y1 y2  2z1 z2 > : þ þ þ þ O R4 ¼ þ 2 3 3 3 R 2R 2R R R

743

(85)

Adding all terms altogether with the appropriate signs, many terms do cancel finally giving   1 1 V z 3 ðx1 x2 þ y1 y2  2z1 z2 Þ ¼ 3 V11 þ O R4 (86) R R which is the leading term, the dipole–dipole interaction, of the expanded form of the interatomic potential V for neutral H atoms. It corresponds to the classical electrostatic interaction of two pointlike dipoles8 located at the nuclei of A and B (Coulson, 1958). Expansion (86) is the first term of what is known as the multipole expansion of the interatomic potential in long range. It is then easily seen that with such an expanded V es E cb 1 ¼ E1 ¼ 0

(87)

ind;A ind;B E~ 2 ¼ E~ 2 ¼0

(88)

so that the only surviving term in long range is the London dispersion attraction 8   2 > 1 X X ai bj jx1 x2 þ y1 y2  2z1 z2 ja0 b0 disp > > ~ ¼ 6 E > > < 2 εi þ εj R i j  2  2 > > 6 X X jða0 ai jz1 Þj b0 bj z2 > > ¼ 6 > : εi þ εj R i j where we have taken into account the spherical symmetry of atoms A and B, giving ( ða0 ai jx1 Þ ¼ ða0 ai jy1 Þ ¼ ða0 ai jz1 Þ on A       b0 bj x2 ¼ b0 bj y2 ¼ b0 bj z2 on B

(89)

(90)

Since aA i ¼2

8

In atomic units.

m2 jða0 ai jz1 Þj2 ¼ 2 i; εi εi

aA ¼

X i

aA i

(91)

744

CHAPTER 17 Atomic and molecular interactions

is the i-th pseudostate contribution to aA, the dipole polarizability of atom A;   b0 bj z2 2 X m2j B ¼ 2 ; aB ¼ aBj aj ¼ 2 εj εj j

(92)

the j-th pseudostate contribution to aB, the dipole polarizability of atom B, the formula for the leading term of the expanded dispersion can be written in the so-called London form ! 8 X X m2i m2j X X2m2  2m2j > εi εj 6 6 1 disp > i > E~ 2 ¼  ¼ 6 > 6 > εi þ εj εi εj εi þ εj R i R 4 i < j j (93) > XX > ε ε 6 1 6 i j > > ¼ 6 ai aj ¼  6 C11 > : εi þ εj R 4 i R j where C11 ¼

εi εj 1XX ai aj ε þ εj 4 i i j

(94)

is the dipole dispersion constant, the typical quantum mechanical part of the calculation of the dispersion coefficient, while six is a geometrical factor.9 Therefore C6 ¼ 6C11

(95)

is the C6 London dispersion coefficient for the long-range interaction between two ground-state H atoms. In this way, the calculated dipole pseudospectra fai ; εi g i ¼ 1; 2; .; N for each H atom can be used to obtain better values for the C6 London dispersion coefficient for the H–H interaction: a molecular (two-centre) quantity, C6, can be evaluated in terms of atomic (one-centre), non-observable quantities, ai (a alone is useless). The coupling between the different components of the polarizabilities occurs through the denominator in the London formula (94), so that we cannot sum over i or j to get the full, observable10 aA or aB. An alternative, yet equivalent, formula for the dispersion constant is due to Casimir and Polder (1948) in terms of the FDPs at imaginary frequencies of A and B: 8 ZN >     > 1 > > du aA iu aB iu C11 ¼ > < 2p 0 (96) > > X 2mð0kÞmðk0Þ > A A A A > > εk ; a ðstaticÞ ¼ a ð0Þ ¼ lim a ðiuÞ : a ðiuÞ ¼ u/0 ε2k þ u2 k where u is a real quantity. In this case, we must know the dependence of the FDPs on the real frequency u, and the coupling occurs now via the integration over the frequencies. When the necessary data are available, however, London formula (94) is preferable because use of the Casimir–Polder formula (96) presents some problems in the accurate evaluation of the integral through numerical quadrature techniques (Figari and Magnasco, 2003). 9

Depending on the spherical symmetry of the ground-state H atoms. That is, measurable.

10

17.3 Interatomic potentials

745

Table 17.6 Five-Term Dipole (l ¼ 1) Pseudospectrum for the H Atom Ground State i 1 2 3 4 5

ai =a30

εi =Eh 0

3.013959  10 9.536869  101 4.556475  101 7.479674  102 1.910219  103

3.753256  101 4.785249  101 6.834311  101 1.255892  100 3.706827  100

The N-term pseudospectra are evaluated using the Ritz–Hylleraas method for E~2 (Section 1.3.2.4) using a convenient set of N basis functions. The functions must be orthogonal to j0 and must be previously Schmidt-orthogonalized among themselves. A five-term dipole pseudospectrum for the hydrogen atom, taken form the PhD thesis of Massimo Ottonelli (1998) is given with seven significant figures in Table 17.6. Using the London formula (94) and different N-term pseudospectra from Ottonelli’s thesis, we obtain for the leading term of the H–H interaction the results collected in Table 17.7. The table shows that convergence with N is very rapid for the H–H interaction (unfortunately, this is not so for the many-electron atoms). The calculation for N ¼ 2 is carried out explicitly in Problem 17.3, and gives 121 ¼ 1:080357 112 363 ¼ 6:482142 C6 ðtwo-termÞ ¼ 6C11 ðtwo-termÞ ¼ 56 C11 ðtwo-termÞ ¼

a result which is over 99.7% of the exact value. The London C6 dispersion coefficient for the longrange H–H interaction is today one of the best known ‘benchmarks’ in the literature, its best value exact to 20 decimal digits having been given by Yan et al. (1996): C6 ¼ 6:499 026 705 405 839 313 13.

(97)

A rather good value, accurate to 15 decimal digits, was given independently by Koga and Matsumoto (1985; see also Koga and Ujiie, 1986), using a non-variational technique, and by Magnasco et al. (1998) using the Ritz–Hylleraas method with a basis of 25 STOs of appropriate symmetry: C6 ¼ 6:499 026 705 405 839 218. Table 17.7 N-Term Results for the Dipole Dispersion Constant C11 and the C6 London Dispersion Coefficient for the HeH Interaction N

C11 =Eh a60

C6 =Eh a60

Percentage Accurate

1 2 3 4 5

1 1.080357 1.083067 1.083167 1.083170

6 6.4821 6.4984 6.49900 6.49902

92.3 99.7 99.99 99.999 100

(98)

746

CHAPTER 17 Atomic and molecular interactions

17.3.6 Higher-order terms in the H–H long-range dispersion interaction The elementary expansion of the interatomic potential V for the H–H interaction in long range discussed in the previous section can be extended to the higher multipoles, giving 1 1 1 V ¼ 3 V11 þ 4 ðV12 þ V21 Þ þ 5 ðV13 þ V31 þ V22 Þ þ . (99) R R R where the dipole–dipole term V11 (la ¼ 1,lb ¼ 1) was already given by Eqn (86), while  3 V12 ¼  z1 r22 þ ð2x1 x2 þ 2y1 y2  3z1 z2 Þz2 2 is the dipole–quadrupole (la ¼ 1,lb ¼ 2) term;  3 2 V21 ¼ r1 z2 þ z1 ð2x1 x2 þ 2y1 y2  3z1 z2 Þ 2 is the quadrupole–dipole (la ¼ 2,lb ¼ 1) term;  1 ð3x1 x2 þ 3y1 y2  4z1 z2 Þ5z22  ðx1 x2 þ y1 y2  4z1 z2 Þ3r22 V13 ¼ 2 is the dipole–octupole (la ¼ 1,lb ¼ 3) term;  1 2 5z1 ð3x1 x2 þ 3y1 y2  4z1 z2 Þ  3r12 ðx1 x2 þ y1 y2  4z1 z2 Þ V31 ¼ 2 is the octupole–dipole (la ¼ 3,lb ¼ 1) term, and i   3h 2 2 V22 ¼ r1 r2  5 z21 r22 þ r12 z22  15z21 z22 þ 2ðx1 x2 þ y1 y2  4z1 z2 Þ2 4 is the quadrupole–quadrupole (la ¼ 2, lb ¼ 2) term, and so on. The second-order dispersion energy (59) has the long-range expansion: N X N X disp C2n ðla ; lb ÞR2n n ¼ la þ lb þ 1 E~ 2 ¼ 

(100)

(101)

(102)

(103)

(104)

(105)

la ¼0 lb ¼0

The higher order dispersion coefficients C2n(la ,lb) are evaluated from the corresponding N-term 2l-pole pseudospectra of the ground-state H atom. The five-term quadrupole and octupole pseudospectra, taken from Ottonelli’s PhD thesis (1998), are given in Tables 17.8 and 17.9. The C2n (la,lb) dispersion coefficients are then given by     εi εj 1 2la þ 2la X X 2la þ 2la Cab C2n ðla ; lb Þ ¼ ai aj ¼ (106) 2la 2la 4 εi þ εj i

where Cab ¼ are the corresponding dispersion constants.

j

εi εj 1XX ai aj εi þ εj 4 i j

(107)

17.3 Interatomic potentials

747

Table 17.8 Five-Term Quadrupole (l ¼ 2) Pseudospectrum for the H Atom Ground State i 1 2 3 4 5

ai6 a50

εi6 Eh 0

5.465443  101 4.517894  101 7.610195  101 1.291357  100 3.139766  100

6.313528  10 4.585391  100 3.484548  100 6.004247  101 1.610783  102

Table 17.9 Five-Term Octupole (l ¼ 3) Pseudospectrum for the H Atom Ground State i 1 2 3 4 5

ai6 a70

εi6 Eh 1

5.930732  101 8.080625  101 4.905187  101 1.296404  100 2.784163  100

5.948006  10 4.062757  101 2.332499  101 7.603273  100 2.140986  101

Table 17.10 gives in atomic units ðEh a2n 0 Þ the five-term long-range dispersion coefficients up to 2n ¼ 10 for the H–H interaction derived from these pseudospectra. The calculated H–H expanded dispersion energy components (105) obtained from such values of the dispersion coefficients for R ¼ 15 a0 agree to all figures given with the previous Kreek and Meath (1969) multipole results of Table 17.5. Table 17.11 gives the five-term long-range dispersion coefficients up to 2n ¼ 14 for the H–H interaction obtained using similar techniques by Magnasco et al. (1998). All digits reported in these tables are correct. Table 17.10 Higher Order Long-Range Dispersion Coefficients (Atomic Units) up to 2n ¼ 10 for the HeH Interaction from Five-Term Pseudospectra of the H Atom Ground State 2C8(1,2) 1.243990  102

C10(1,3) 1.075307  103

C10(2,2) 1.135214  103

2C10(1,3) þ C10(2,2) 3.285828  103

Table 17.11 Higher Order Long-Range Dispersion Coefficients (Atomic Units) up to 2n ¼ 14 for the HeH Interaction from Five-Term Pseudospectra of the H Atom Ground State 2C12(1,4) 5.77992  104

2C12(2,3) 6.36867  104

2C14(1,5) 2.22079  106

2C14(2,4) 2.52427  106

C14(3,3) 1.31570  106

748

CHAPTER 17 Atomic and molecular interactions

17.3.7 The expanded dispersion interaction for many-electron atoms The previous formulae for the H–H interaction can be generalized to the case of the interaction between two many-electron atoms A and B of nuclear charge ZA and ZB having, respectively, NA and NB electrons. The interatomic potential V V¼

NA NB NA X NB ZA ZB X ZB X ZA X 1   þ R r r r i¼1 Bi j¼1 Aj i¼1 j¼1 ij

(108)

is expanded in the asymptotically convergent Taylor series (Dalgarno and Davison, 1966; see Footnote 36 of Chapter 1) V¼

N X N X

Vla lb Rðla þlb þ1Þ

(109)

la ¼0 lb ¼0

where the term (la,lb) describes the interaction between the 2la -pole electric moment of atom A with components X l ria Yla ma ðqi ; 4i Þ (110) Qla ma ¼ i

and the

2lb -pole

electric moment of atom B with components X l   rjb Ylb mb qj ; 4j Qlb mb ¼

(111)

j

where R is the internuclear separation and the Ys are the spherical harmonics in complex form (Rose, 1957). The form of Vlalb depends on the choice of the coordinate system and reference to Figure 17.6 gives (Fontana, 1961) Vla lb ¼

l X m¼l ½ð2la

ð1Þlb 4pðla þ lb Þ!Qla m Qlb m þ 1Þðla þ mÞ!ðla  mÞ!ð2lb þ 1Þðlb þ mÞ!ðlb  mÞ!1=2

FIGURE 17.6 Interatomic reference system for the atom–atom interaction

(112)

17.3 Interatomic potentials

749

where l ¼ minðla ; lb Þ

(113)

So, the expanded form of potential (112) can be written as Vla lb ¼

l X

  ð1Þlb 4pðla þ lb Þ!ð1  dla 0 ZA =NA Þ 1  dlb0 ZB =NB

m¼l ½ð2la

þ 1Þðla þ mÞ!ðla  mÞ!ð2lb þ 1Þðlb þ mÞ!ðlb  mÞ!

r la Y 1=2 i la m

$ rjlb Ylb m

For spherical atoms in S states, Eqn (114) can be further simplified by noting that   l X 2la þ 2lb 2 1 ½ðla þ mÞ!ðla  mÞ!ðlb þ mÞ!ðlb  mÞ! ¼ ½ðla þ lb Þ! 2la

(114)

(115)

m¼l

In fact   l  X ½ðla þ lb Þ!2 la þ lb la þ lb ¼ la þ m lb þ m ðl þ mÞ!ðla  mÞ!ðlb þ mÞ!ðlb  mÞ! m¼l a m¼l l X

(116)

Now, if we change index m into the new index k, where k ¼ la þ m; then

0  k  2la

  X   2la  l  X la þ lb la þ lb la þ lb la þ lb ¼ la þ m lb þ m k 2la  k

(117)

(118)

k¼0

m¼l

Since for the binomial coefficients the following relation holds (Problem 17.1)    q   X u v uþv ¼ p qp q

(119)

p¼0

substituting in Eqn (118), the appropriate change of symbols gives relation (115). In Cartesian form (Margenau, 1939), the first few coefficients Vlalb of the multipole expansion (109), generalizing Eqns (86), (100)–(104) found in the previous section for the H–H interaction, are

X X xi xj þ yi yj  2zi zj (120) V11 ¼ i

j

the dipole–dipole (la ¼ lb ¼ 1) term;

i 3 X Xh 2 V12 ¼  zi rj þ 2xi xj þ 2yi yj  3zi zj zj 2 i j the dipole–quadrupole (la ¼ 1,lb ¼ 2) term;

i 3 X Xh 2 V21 ¼ ri zj þ zi 2xi xj þ 2yi yj  3zi zj 2 i j

(121)

(122)

750

CHAPTER 17 Atomic and molecular interactions

the quadrupole–dipole (la ¼ 2,lb ¼ 1) term; V13 ¼

i 1 X Xh ð3xi xj þ 3yi yj  4zi zj Þ5z2j  ðxi xj þ yi yj  4zi zj Þ3rj2 2 i j

(123)

the dipole–octupole (la ¼ 1,lb ¼ 3) term; V31 ¼

i 1 X Xh 2 5zi ð3xi xj þ 3yi yj  4zi zj Þ  3ri ðxi xj þ yi yj  4zi zj Þ 2 i j

(124)

the octupole–dipole (la ¼ 3,lb ¼ 1) term; and V22 ¼



2 i 3 X Xh 2 2 ri rj  5 z2i rj2 þ ri2 z2j  15z2i z2j þ 2 xi xj þ yi yj  4zi zj 4 i j

(125)

the quadrupole–quadrupole (la ¼ lb ¼ 2) term. The RS perturbative equation of the first order ðH^0  E0 Þj1 þ Vj0 ¼ 0

(126)

hj0 jj1 i ¼ 0

(127)

with

can be separated into an equation for each multipole value (la,lb) by expanding j1 in the form of Eqn (72) j1 ¼

N X N X la ¼0 lb ¼0

j1la lb Rðla þlb þ1Þ

where jlalb satisfies the differential equation ðH^0  E0 Þj1la lb þ Vla lb j0 ¼ 0



 j0 j1la lb ¼ 0

(128)

(129)

and Vlalb is given by Eqn (114) and its explicit forms, Eqns (120)–(125). In these equations, in an obvious notation, H^0 and E0 denote the unperturbed Hamiltonian and energy, while j0 is the product of the individually antisymmetrized functions describing the unperturbed ground state of A and B, respectively:  X 1 ZA 1X 1 A B A 2 ^ ^ ^ ^  Vi  þ (130) H0 ¼ H0 þ H0 ; H0 ¼ rAi 2 2 isi0 rii0 i E0 ¼ E0A þ E0B ;

B j 0 ¼ jA 0 j0

(131)

If we expand j1lalb into the finite set of the excited pseudostate products {AalaBblb} j1la lb ¼

XX a

b

ðl ;lb Þ

Aala Bblb Caba

(132)

17.3 Interatomic potentials

the only long-range surviving term in second order is the expanded dispersion energy XX disp E2 ¼ C2n ðla ; lb ÞR2n ; n ¼ la þ lb þ 1 la

(133)

lb

where C2n ðla ; lb Þ ¼

751

XX a

b

  Aal Bbl Vl l A0 B0 2 a b a b D E Aala Bblb H^0  E0 Aala Bblb

(134)

If we introduce the one-electron transition density matrices (McWeeny, 1960) for atoms A and B, that for atom A being Z A 0 r1 ð0ajr1 ; r1 Þ ¼ NA d s1 dx2 .dxNA A0 ðr1 s1 ;x2 .; xNA ÞAala ðr01 s1 ;x2 .; xNA Þ (135) for atoms in S states we can write the dispersion coefficients in the London form   2la þ 2lb C2n ðla ; lb Þ ¼ 2la R 2 R 2 l lb B X X dr1 r1a Pla ðcos q1 ÞrA 1 ð0ajr1 ; r1 Þ dr2 r2 Plb ðcos q2 Þr1 ð0bjr2 ; r2 Þ  Dεala þ Dεbla a b (136) The first integral at numerator is the average value of the 2la -pole moment of atom A evaluated with the la density rA 1 ð0ajr1 ; r1 Þ describing the 2 -pole electric transition from the ground state A0 to the excited pseudostate Aala with excitation energy Dεala: D E A Dεala ¼ Aala H^0  E0A Aala (137)

The theoretical calculation of the dispersion coefficients therefore requires the determination of the excitation energies and the transition moments of the two interacting atoms. For many-electron systems these quantities depend on the intra-atomic correlation energy, this effect being particularly important for atoms having more than one electron in the valence shell (Magnasco et al., 1977; Magnasco and Amelio, 1978; Figari et al., 1983). An accurate 40-term dipole pseudospectrum for the He ground state, explicitly including the interelectron distance r12 into the wavefunction, was obtained variationally by Magnasco and Ottonelli (1999a) using a set of Riley–Dalgarno (1971) basis functions with m þ n þ p  9: 4i ¼ 4mnp ¼

1 n p exp½ dðr1 þ r2 Þr m 1 r 2 r 12 2p

(138)

Sixty functions were chosen for the unperturbed 1S ground state of the He atom, 40 basis functions for the 1P excited state. Accurate pseudospectra for many-electron systems usually involve a rather large number of terms (Yan et al., 1996; Magnasco and Ottonelli, 1999a). However, they can be replaced by accurate n-term reduced pseudospectra obtained by a recently developed efficient interpolation procedure (Magnasco

752

CHAPTER 17 Atomic and molecular interactions

Table 17.12 Four-Term Reduced Dipole (l ¼ 1) Pseudospectrum for the He Atom Ground State i 1 2 3 4

ai =a30

εi =Eh 1

8.022210  101 1.118638  100 1.910251  100 4.259569  100

7.071107  10 4.977223  101 1.613181  101 1.675926  102

and Figari, 2009). The four-term reduced dipole pseudospectrum for He is given in Table 17.12. The calculated dipole polarizability for the ground-state He atom is aHe ¼

4 X

ai ¼ 1:383 a30

(139)

i¼1

in perfect agreement with the extremely accurate ‘benchmark’ value of 1:383 192 a30 obtained by Yan et al. (1996) using fully optimized variational wavefunctions containing 504 terms for j0(1S) and 728 terms for j1 ð1 PÞ. The calculated C6 dispersion coefficient for the long-range He–He interaction resulting from the four-term reduced pseudospectrum of Table 17.12 is 4 εi εj 1 X ai aj ¼ 1:460 Eh a60 (140) C6 ¼ 6C11 ¼ 6  εi þ εj 4 i; j¼1 which differs by just one digit in the third decimal place from the reference value of 1:461 Eh a60 obtained by Yan et al. (1996).

17.4 MOLECULAR INTERACTIONS The non-expanded intermolecular potential V arises from the Coulombic interactions between all pairs i, j of charged particles (electrons þ nuclei) in the molecules (Figure 17.7): X X qi qj (141) V¼ rij i j

FIGURE 17.7 Interparticle distances in the intermolecular potential V

17.4 Molecular interactions

753

where qi and qj are the charges of particles i (belonging to A) and j (belonging to B) interacting at the distance rij .

17.4.1 Non-expanded molecular energy corrections up to second order If A0, B0 are the unperturbed wavefunctions of molecules A (NA electrons) and B (NB electrons), and Ai, Bj a pair of excited pseudostates describing single excitations on A and B, all fully antisymmetrized within the space of A and B, we have to second order of RS perturbation theory es E cb 1 ¼ hA0 B0 jVjA0 B0 i ¼ E 1

(142)

the semiclassical electrostatic energy arising in first order from the interactions between undistorted A and B;   X jhAi B0 jVjA0 B0 ij2 X A0 Ai jUB 2 ind;A ~ E2 ¼ ¼ (143) εi εi i i the polarization (distortion) of A by the static field of B, described by UB U B ¼ hB0 jVjB0 i the MEP of B; E~ ind;B 2

¼

  X A0 Bj jVjA0 B0 2 j

εj

¼

(144)   X B0 Bj U A 2 j

εj

the polarization (distortion) of B by the static field of A, described by the MEP UA;

 2 X 1 ri0 j0 A0 B0 Ai Bj   2 X X Ai Bj jVjA0 B0 XX i0 0

(157)

We shall refer to Eqn (155) and (156) as of generalizations of the London (1930) formula and the Casimir–Polder (1948) formula, respectively. In Section 17.8.3 we shall give further details on the orientation dependence of the molecular interaction in the case of these generalized expanded dispersion energies. In Eqn (155), PA ðr1 ; r01 jiÞ is the i-th pseudostate component of the limit to zero frequency of the dynamic propagator (154): X (158) PA ðr1 ; r01 jiÞ lim PA ðr1 ; r01 jiuÞ ¼ PA ðr1 ; r01 Þ ¼ u/0

i

756

CHAPTER 17 Atomic and molecular interactions

FIGURE 17.9 The five angles specifying in general the relative orientation of two polyatomic molecules

In analogy with what happens for polarizabilities, we shall call PA ðr1 ; r01 Þ the ‘static propagator’ of molecule A, a quantity that should replace polarizabilities in non-expanded interactions. The general polarization propagator (154), and the corresponding FDPs that it determines, can be calculated by time-dependent perturbation techniques (time-dependent Hartree–Fock (TDHF) or MC-TDHF, see McWeeny, 1989). General formulae in terms of two-electron integrals between elementary charge distributions on A and B, and the corresponding elements of the transition density matrices have been developed in terms of an atomic orbital basis and can be found elsewhere (Magnasco and McWeeny, 1991).

17.4.2 Expanded molecular energy corrections up to second order In molecules, the interaction depends on the distance R between their centres of mass (c.o.m.) as well as on the relative orientation of the interacting partners, which can be specified in terms of the five independent angles (qA, qB, 4, cA, cB)11 shown in Figure 17.9. The first three angles describe the orientation of the principal symmetry axes of the two molecules, the last two the two rotations about these axes. In what follows, we shall limit ourselves mostly to consideration of the long-range dispersion interaction between (1) two linear molecules A and B (left of Figure 17.10), and (2) an atom A, at the origin of the intermolecular coordinate system, and a linear molecule B, whose orientation with respect to the z-axis is specified by the single angle q (right of Figure 17.10). We already observed that a linear molecule has two dipole polarizabilities, ak, the parallel or longitudinal component directed along the intermolecular axis, and at, the perpendicular or transverse component perpendicular to the intermolecular axis (McLean and Yoshimine, 1967). The molecular isotropic polarizability, defined by Eqn (9), can be compared to that of atoms, and the polarizability anisotropy, which is zero for ak ¼ at, is defined by Eqn (10).

11

These angles are simply related to the Euler’s angles describing the rotation of a rigid body (Brink and Satchler, 1993).

17.4 Molecular interactions

757

FIGURE 17.10 The angles specifying the relative orientation of two linear molecules (left), and the system atom A-linear molecule B (right) (Magnasco, V., 2011, Models of Bonding in Chemistry, Wiley. Reprinted with permission from John Wiley and Sons).

The composite system of two different linear molecules has hence four independent elementary dipole dispersion constants, which in London form can be written as 8 jj jj jj t > 1 X X jj jj εi εj 1 X X jj t εi εj > > A¼ ai aj jj ; B¼ ai aj jj > > jj > 4 i 4 i < εi þ εj εi þ εt j j j (159) > t εjj t εt > X X X X ε ε > 1 1 jj i j > t i j > at ; D¼ at > i aj t i aj t :C ¼ 4 jj 4 εi þ εt ε þ ε j i j i j j i For two identical linear molecules, there are only three independent dispersion constants since C ¼ B. 1 $ r 1 in Eqn (153) (Wormer, 1975; Spelsberg et al., Spherical tensor expansion of the product r12 10 2 0 1993; Magnasco and Ottonelli, 1999b) gives X X pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1  ¼ Rn dLþL0 þ2;n ð2L þ 1Þð2L0 þ 1Þ r12 r10 20 n LL0 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! !ffi u 0 X X 2L 2L 0 u  dla þlb ;L dl0a þl0b ;L0 ð1Þlb þlb t 2la 2l0a 0 0 la lb la lb X pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ð2LA þ 1Þð2LB þ 1ÞCLA M ð^rA ÞCLB M ð^rB ÞTðla l0 ÞLA ;ðlb l0 ÞLB a b LA LB M ! ! ! ! X la lb L la lb l0b LB l0a l0b L0 l0a LA (160)  m0 m0 0 m m0 M m m0 M mm0 m m 0     n l1 l2 l3 is a Wigner 3-j symbol (Section 10.4.1), a binomial coefficient, and we where m1 m2 m3 m have posed n ¼ la þ lb þ l0a þ l0b þ 2

(161)

758

CHAPTER 17 Atomic and molecular interactions

T AB is the product of the irreducible spherical tensors: T AB ¼ Tðla l0 ÞLA ;ðlb l0 ÞLB ¼ Tðla l0 ÞLA  Tðlb l0 ÞLB a a b b with ll0

Tðll0 ÞL ¼ ð1Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi X ð1Þm 2L þ 1 m



l l0 m m

(162)

 L Rlm ðrÞRl0 m ðr0 Þ 0

(163)

where Rlm ðrÞ ¼ rl Clm ð^rÞ

(164)

is a spherical tensor (regular solid harmonic) with rffiffiffiffiffiffiffiffiffiffiffiffi 4p r r Ylm ð^rÞ ^r ¼ ¼ Clm ð^rÞ ¼ 2l þ 1 U q; 4

(165)

a modified spherical harmonic in complex form with Condon–Shortley phase (Racah spherical harmonic). In formula (160), the elementary contributions ðla ; l0a Þ on A and ðlb ; l0b Þ on B have been coupled to resultant LA on A and LB on B according to la  l0  LA  la þ l0 ; lb  l0  LB  lb þ l0 (166) a

a

b

b

in steps of two, with 0  M  minðLA ; LB Þ

(167)

Expansion (160) explicates the dependence of the interaction on the intermolecular distance R and on the molecular orientations UA and UB. Introducing the expansion into the dispersion energy, Eqn (155), and turning to real spherical harmonics, we can write the leading (dipole–dipole) term of the long-range dispersion interaction between two linear molecules in the form disp E~2 ¼ R6 C6 ðqA ; qB ; 4Þ

(168)

C6(qA,qB,4) being an angle-dependent dipole dispersion coefficient, which can be expressed (Meyer, 1976) in terms of associated Legendre polynomials on A and B as12 X L LM M C6 ðqA ; qB ; 4Þ ¼ C6 g6 A B PM (169) LA ðcos qA ÞPLB ðcos qB Þ LA LB M

where LA, LB ¼ 0, 2 and M ¼ jMj ¼ 0, 1, 2. In (169), C6 is the isotropic coefficient and g6 an anisotropy coefficient defined as gL6 A LB M ¼

12

C6LA LB M C6

(170)

The expression for the angle-dependent C6 dispersion coefficient for two linear molecules was first given by Hirschfelder et al. (1954).

17.4 Molecular interactions

759

Table 17.13 LALBM-Components of C6 dispersion coefficients for (i) two linear molecules, and (ii) an atom and a linear molecule LA

LB

M

(i)

(ii) 2A þ 4B 2A  2B

0

0

0

2 ðA þ 2B þ 2C þ 4DÞ 3

0

2

0

2 ðA  B þ 2C  2DÞ 3

2

0

0

2 ðA þ 2B  C  2DÞ 3

2

2

0

2 (A  B  C þ D)

2

2

1

4  ðA  B  C þ DÞ 9

2

2

2

1 ðA  B  C þ DÞ 18

The different components of the C6 dispersion coefficients in the LALBM-scheme for (1) two different linear molecules, and (2) an atom and a linear molecule, are given in Table 17.13 (Magnasco and Ottonelli, 1999b) in terms of the symmetry-adapted combinations of the elementary dispersion constants (159). The coefficients with M s 0 are not independent, but are related to those with M ¼ 0 by the relations 2 C6221 ¼  C6220 ; 9

C6222 ¼

1 220 C 36 6

(171)

For identical molecules, C ¼ B in Table 17.13, and the (020) and (200) coefficients are equal. The determination of the elementary dispersion constants allows for a detailed analysis of the angle-dependent dispersion coefficients between molecules. An equivalent, yet explicit, expression of the C6 angle-dependent dispersion coefficient for the homodimer of two linear molecules as a function of the three independent dispersion constants was given by Briggs et al. (1971) in their attempt to determine the dispersion coefficients of two H2 1 molecules in terms of non-linear 1 Sþ u and Pu pseudostates: (      C6 ðqA ; qB ; 4Þ ¼ 2B þ 4D þ 3 B  D cos2 qA þ cos2 qB   (172) þ A  2B þ D ðsin qA sin qB cos 4  2 cos qA cos qB Þ2 Averaging Eqn (172) over the angles and noting that only squared terms contribute to the average, we obtain for the isotropic C6 dispersion coefficient:     1 1 2 2 1 1 1 þ ðA  2B þ DÞ $ $ þ4$ $ hC6 i ¼ ð2B þ 4DÞ þ 3ðB  DÞ þ 3 3 3 3 2 3 3 (173) 2 ¼ ðA þ 4B þ 4DÞ ¼ C6 3

760

CHAPTER 17 Atomic and molecular interactions

in accord with the result of the first row of the preceding table. Magnasco et al. (1990) gave an alternative interesting expression for C6(qA,qB,4)in terms of frequency-dependent isotropic polarizabilities a(iu) and polarizability anisotropies Da(iu) of the two linear molecules: 8 ZN  >           > 1 > > C6 ðqA ; qB ; 4Þ ¼ du aA iu aB iu þ 3 cos2 qB  1 aA iu DaB iu > > 2p > > > 0 > > >       < þ 3 cos2 qA  1 DaA iu aB iu (174) >  > > 2 2 2 2 > þ 4 cos qA cos qB  cos qA  cos qB  sin 2qA sin 2qB cos 4 > > > >  > >      > > : þ sin2 qA sin2 qB cos2 4 DaA iu DaB iu Averaging over angles, all coefficients involving polarizability anisotropies are zero, giving the isotropic C6 coefficient in the Casimir–Polder form (96). We give in Table 7.14 the four-term reduced pseudospectrum for the dipole polarizabilities of the 1 Sþ g ground state of the H2 molecule at R ¼ 1.4 a0 obtained by Magnasco and Figari (2009) from the accurate calculations of Magnasco and Ottonelli (1996), who used the 54-term Ko1os– Wolniewicz (1965) wavefunction for the 1 Sþ g unperturbed ground state and the 34-term wave1 functions for either the 1 Sþ u or the Pu excited states of the molecule, yielding for the dipole polarizabilities ða30 Þ of H2: at ¼ 4:577 0 a ¼ 5:179;

ajj ¼ 6:383;

Da ¼ 1:806

(175)

values which are remarkably good for both polarizabilities, improving Ko1os–Wolniewicz (1967) results (ak ¼ 6.380,at ¼ 4.578), and not far from the best variational result (a ¼ 5.1815) obtained by Bishop et al. (1991) using an extended set of 249 Ko1os–Wolniewicz like wavefunctions for the 1 Sþ g 1 unperturbed ground state, 113 terms for the 1 Sþ u and 190 terms for the Pu excited states of the molecule. We can then calculate the four-term approximation to the three independent elementary dispersion constants (159) for the homodimer H2–H2, obtaining from the reduced dipole pseudospectrum of Table 17.14 the following numerical results: k

k

k

B ¼ H2  H2t ¼ 2:032;

A ¼ H2  H2 ¼ 2:689;

D ¼ H2t  H2t ¼ 1:542

(176)

From these values, we obtain for the isotropic C6 dispersion coefficient for H2–H2 2 C6 ¼ C6000 ¼ ðA þ 4C þ 4DÞ ¼ 11:324 Eh a60 3

(177)

Table 17.14 Four-Term Reduced Dipole Pseudospectrum for H2 ð1 Sþ g Þ at R ¼ 1.4a0 k

k

i

ai =a30

εi =Eh

3 at i =a0

εt i =Eh

1 2 3 4

4.656280  100 1.501906  100 2.224207  101 2.073253  103

4.737530  101 6.657903  101 1.083145  100 3.394450  100

2.909189  100 1.368299  100 2.866886  101 1.335458  102

4.954656  101 7.171300  101 1.269471  100 2.948926  100

17.4 Molecular interactions

761

Table 17.15 Four-Term Reduced Dipole Pseudospectrum for the H Atom Ground State i 1 2 3 4

ai =a30

εi/Eh 0

3.794676  101 5.683764  101 1.106618  100 3.322886  100

3.359341  10 9.981466  101 1.378226  101 4.689923  103

and for the dipole anisotropies g020 6 ¼

C6020 2 ¼ ðA þ B  2DÞ ¼ 0:096 ¼ g200 6 C6 3

(178)

C6220 ¼ 2ðA  2B þ DÞ ¼ 0:029 C6

(179)

g220 6 ¼

As a second example, illustrating a heterodimer calculation, consider the C6 dispersion coefficient of the H–H2 system (atom–linear molecule interaction), where, for homogeneity, we use the four-term reduced dipole pseudospectrum for the hydrogen atom taken from Magnasco and Figari (2009) and given in Table 17.15 (compare with Ottonelli’s values given in Table 17.6). For the dispersion constants we have jj

A ¼ H  H2 ¼ 1:698;

B ¼ H  H2t ¼ 1:276

(180)

so that we finally obtain for the H–H2 dispersion interaction C6 ¼ C6000 ¼ 2A þ 4B ¼ 8:502 Eh a60 g020 6 ¼

C6020 AB ¼ ¼ 0:099 C6 A þ 2B

(181) (182)

Lastly, we give in Table 17.16 some results of the convergence with the dimension N of the pseudospectrum of the ground-state isotropic dipole polarizabilities a of a few simple atoms and linear molecules, and in Table 17.17 the like results for the isotropic C6 dispersion coefficients of the corresponding homodimers. It is seen that the convergence rate of either a or C6 is remarkably slower for either the atomic or molecular two-electron systems (He,H2). N ¼ 20 can be considered a sufficient first approximation for not too accurate calculations.

17.4.3 Multipole expansion of the first-order electrostatic energy in (HF)2 So far we have considered the dispersion interaction between molecules, whose accurate calculation requires knowledge of accurate pseudospectra of the appropriate symmetry, quantities that are not observable and that can be obtained by the Ritz–Hylleraas method for static polarizabilities in the London case (94) or by the numerical integration of the Casimir–Polder integral (96) for FDPs.

762

CHAPTER 17 Atomic and molecular interactions

Table 17.16 N-Term Convergence of Ground-State Isotropic Dipole Polarizabilities aða30 Þ for Simple Atoms and Molecules N

H

He

a HD 2

H2b

1 2 5 10 15 20 Accurate

4 4.5

0.694 1.042 1.135 1.364 1.378 1.382c 1.383g

1.485 2.614 2.837 2.855 2.864 2.864d 2.864h

2.523 3.481 4.452 4.999 5.145 5.165e 5.181i

4.5f

Hþ 2 : Re ¼ 2a0. H2: Re ¼ 1.4a0. c Magnasco and Ottonelli, 1999a. d Magnasco and Ottonelli, 1999c. e Magnasco and Ottonelli, 1996. f Exact value. g Yan et al., 1996. h Bishop and Cheung, 1978b; Magnasco and Ottonelli, 1999c. i Bishop et al., 1991. a

b

As a typical example, we shall examine in the following the multipole expansion of the first-order electrostatic energy (142), (147) for the long-range interaction of two HF molecules which involves the interaction between the permanent electric moments of the molecules, quantities that are observable and that can therefore, at least in principle, be determined by experiment (see Section 17.2.2). Table 17.17 N-Term Convergence of Isotropic C6 Dispersion Coefficients ðEh a60 Þ for the Homodimers of Simple Atoms and Molecules N

HeH

HeeHe

Da HD 2 eH2

H2 eH2 b

1 2 5 10 15 20 Accurate

6 6.482 6.499

0.695 1.142 1.242 1.443 1.455 1.459c 1.461g

1.449 2.965 3.234 3.265 3.284 3.284d 3.284h

4.729 7.325 9.645 10.965 11.263 11.289e 11.324e

6.499f

Hþ 2 : Re ¼ 2a0. H2: Re ¼ 1.4a0. c Magnasco and Ottonelli, 1999a. d Magnasco and Ottonelli, 1999c. e Magnasco and Ottonelli, 1996. f Exact value. g Yan et al., 1996. h Babb, 1994; Magnasco and Ottonelli, 1999c. a

b

17.4 Molecular interactions

763

Table 17.18 Equilibrium Bond Distances and Electric Multipole Moments (Atomic Units) for the Ground State of a Few Polar Linear Molecules Molecule

Re/a0

m1/ea0

m2 =ea20

m3 =ea30

OC FH ClH LiH H2 N2 O2 F2 CO2 C2H 2

2.132 1.733 2.409 3.015 1.40 2.074 2.282 2.71 2.192 6.213

0.044 0.704 0.4 2.294

1.47 1.71 2.8 3.097 0.44 1.04 0.30 0.536 3.18 4.03

3.46 2.50 6.326

Equilibrium bond distances and permanent electric moments up to l ¼ 3 (the octupole moment) for a few polar linear molecules, taken from Magnasco et al. (2006), are given in Table 17.18. We remind that the origin of the moments is taken at the heavy atom, and that for centrosymmetric molecules the first non-vanishing moment is the quadrupole moment. The first-order electrostatic interaction goes as Rn with n ¼ l þ l0 þ 1, and l,l0 ¼ 1,2,3 for the dipole, quadrupole and octupole moments, respectively. The explicit expressions for the electrostatic interaction for two linear molecules A and B expanded up to the R5 terms are (Magnasco et al., 1990) B mA 1 m1 ðsin qA sin qB cos 4  2 cos qA cos qB Þ R3

(183)

E112 ðesÞ ¼

B     mA 1 m2 3 cos qA 3 cos2 qB  1  sin qA sin 2qB cos 4Þ 4 R 2

(184)

E121 ðesÞ ¼

B    mA 2 m1 3 1  3 cos2 qA cos qB þ sin 2qA sin qB cos 4 4 R 2

(185)

E111 ðesÞ ¼

E113 ðesÞ ¼

B       mA 1 m3 1 (186) 4 cos qA 3  5 cos2 qB cos qB þ 3 sin qA sin qB 5 cos2 qB  1 cos 4 R5 2

E131 ðesÞ ¼

B       mA 3 m1 1 4 cos qA 3  5 cos2 qA cos qB þ 3 sin qA 5 cos2 qA  1 sin qB cos 4 (187) 5 R 2

E122 ðesÞ ¼

B    mA 2 m2 3 1  5 cos2 qA þ cos2 qB þ 17 cos2 qA cos2 qB 5 R 4  þ 2 sin2 qA sin2 qB cos2 4  4 sin 2qA sin 2qB cos 4

where spherical tensor notation is used for the multipole moments of the linear molecules.

(188)

764

CHAPTER 17 Atomic and molecular interactions

FIGURE 17.11 Angular dependence on qA of the first three terms of the expanded electrostatic interaction in linear (HF)2 (Magnasco, V., 2011, Models of Bonding in Chemistry, Wiley. Reprinted with permission from John Wiley and Sons).

FIGURE 17.12

Experimental structure (q z 60 ) of the (HF)2 linear dimer (Angle q in this figure is the supplement of angle qA of Figure 11.7d.) (Magnasco, V., 2009, Methods of Molecular Quantum Mechanics: An introduction to Electronic Molecular Structure, Wiley. Reprinted with permission from John Wiley and Sons).

17.5 The Pauli repulsion between closed shells

Choosing qB ¼ 180o, for (HF)2 the above formulae simplify to  HF 2 m 11 E1 ðesÞ ¼ 2 1 3 cos qA R E112 ðesÞ þ E121 ðesÞ ¼ E113 ðesÞ þ E131 ðesÞ þ E122 ðesÞ ¼

HF    3 mHF 1 m2 2 cos qA þ 3 cos2 qA  1 4 2 R

 HF 2  i 1 h HF HF 2 m cos q þ 3 m q  1 3 cos 8m A A 1 3 2 R5

765

(189)

(190)

(191)

The angular dependence on qA of the first three terms (n ¼ 3,4,5) of the expanded electrostatic interaction in linear (HF)2 is sketched in the drawings of Figure 17.11. In all the plots there, molecule B is kept fixed at qB ¼ 180 . It is seen that, while the dipole–dipole term would favour the head-to-tail shape of the dimer with formation of a collinear hydrogen bond (HF/HF, top left part a of the figure with qA ¼ qB ¼ 180 ), the higher multipole interactions lead to the final L-shape of the dimer depicted in the right bottom part d of the figure, which agrees with the structure of the dimer experimentally observed by molecular beams techniques (Howard et al., 1984) and given in Figure 17.12.

17.5 THE PAULI REPULSION BETWEEN CLOSED SHELLS The molecular interactions we have considered so far are all attractive. For closed-shell molecules, however, there is a repulsive contribution due to the first-order exchange-overlap energy which can be considered in the context of the MS-MA symmetry-adapted perturbation theory already introduced in Section 1.3.5. The exchange overlap-energy is a quantum mechanical contribution which depends on the nature of the spin coupling of the interacting molecules (for a general discussion see Dacre and McWeeny, 1970). For closed-shell molecules the resultant total spin is zero, and the first-order contribution ^ 0 to the exchange-overlap component of the interaction can be expressed in closed form if Aj is approximated as a single determinant of Hartree–Fock spin-orbitals of the individuals molecules. First-order exchange overlap can then be expressed in terms of two contributions of opposite sign. 1. An attractive contribution due to pure two-electron exchange between A and B Z ZZ 1 1 PB ð00jr1 ; r2 Þ A B A 0 ^ P ð00jr2 ; r1 Þ dr1 K ð00jr1 ÞP ð00jr1 ; r1 Þ ¼  dr1 dr2 K¼ 2 2 r12

(192)

where the prime on r01 must be removed after the action of the operator and before integration, and B K^ is the undistorted exchange potential operator Z B K^ ð00jr1 Þ ¼ dr2 K B ð00jr1 ; r2 ÞP^r1 r2 (193) an integral operator with kernel K B ð00jr1 ; r2 Þ ¼

PB ð00jr1 ; r2 Þ r12

(194)

766

CHAPTER 17 Atomic and molecular interactions

2. A repulsive contribution due to the Pauli’s repulsion between the overlapping static electron distributions of the two molecules, and given by   Z 1 ^B B 0 Eov ¼ dr1  U ðr1 Þ  K ð00jr1 Þ PA ov ð00jr1 ; r1 Þ 2 1 þ 2

ZZ dr1 dr2

PBov ð00jr2 ; r2 Þ 

1 B P ð00jr1 ; r2 ÞP^r1 r2 0 2 ov PA ov ð00jr1 ; r1 Þ r12

(195)

þ ðA 4 B; 1 4 2Þ where

      U B r ¼ V B r  J B 00 r

(196)

is the Coulomb potential (MEP) at r due to nuclei and undistorted electrons of molecule B, and 0 B 0 Pov ð00jr; r0 Þ ¼ PA ov ð00jr; r Þ þ Pov ð00jr; r Þ

(197)

is the overlap density, whose partition between A and B is only apparently additive, since each component contains the effect of the whole intermolecular overlap. The two overlap components are defined as 0 PA ov ð00jr; r Þ ¼ 2

ðAÞ X all X i

PBov ð00jr; r0 Þ ¼ 2

(198)

Fj ðrÞDjq Fq ðr0 Þ

(199)

p

ðBÞ X all X j

Fi ðrÞDip Fp ðr0 Þ

q

and have diagonal elements with the property13 (Problem 17.6) Z Z     r; r ¼ dr PB 00 r; r ¼ 0 dr PA 00 ov ov

(200)

In Eqn (198), index i runs over the occupied molecular orbitals (MOs) of A, index p over all occupied MOs of A and B, and Dip is an element of the matrix: D ¼ Sð1 þ SÞ1

(201)

which depends in a complicated way on the intermolecular overlap between the occupied MOs of A and B (the same being true for PBov ). The overlap energy, Eov of Eqn (195), expresses the interaction of the overlap density of one molecule with the Coulomb-exchange potential of the other, plus the Coulomb-exchange interaction of the overlap densities of the two molecules (Magnasco, 1982). All terms in Eov are rigorously zero for non-overlapping molecules. For clusters of many interacting molecules the overlap energy is the source of first-order non-additivity observed for intermolecular forces. 13

A generalization of what is found in the Heitler-London theory of H2.

17.6 The Van der Waals bond

767

Partition (197) allows for the precise definition of (1) the additional density which must supplement the ordinary and exchange electron densities of each molecule when overlap occurs, and (2) of the error D occurring when A0 and B0 are not the exact eigenstates of H^0 , but rather approximations satisfying the eigenvalue equation for some model Hamiltonian H^0 (e.g. the one constructed in terms A of the usual one-electron Fock operators of the isolated molecule). It can be shown that, if F^ ð00jrÞ is A the Fock operator for A, the D contribution to D is given by Z A A 0 D ¼ dr1 F^ ð00jr1 ÞPA ov ð00jr1 ; r1 Þ 1 þ 2

ZZ dr1 dr2

PA ov ð00jr2 ; r2 Þ 

1 A P ð00jr1 ; r2 ÞP^r1 r2 0 2 ov PA ov ð00jr1 ; r1 Þ r12

(202)

2 2 Since PA ov is at least of order O(S ), it can be seen that the first term in (202) is of order O(S ), and vanishes for Hartree–Fock A0 (Magnasco et al., 1990; see Problem 17.7), while the second term is of order O(S4) and survives in any case. When both the repulsive first-order and the attractive second-order components of the interaction are appropriately considered, a minimum is expected in the potential energy curve for the interaction between the two closed-shell molecules originating what is known as a weak VdW bond.

17.6 THE VAN DER WAALS BOND Weakly bound complexes with large-amplitude vibrational structures were called by Buckingham (1982) VdW molecules. Complexes of the heavier rare gases, such as Ar2, Kr2, Xe2,or weak complexes between centrosymmetrical molecules like (H2)2 or (N2)2, fit well into this definition, but complexes between proton donor and proton acceptor molecules, like (HF)2 or (H2O)2, which involve hydrogen bonding, are in the border-line between VdW molecules and ‘good’ molecules. In the latter complexes, bonding is essentially electrostatic in nature. However, all complexes above are characterized by having closed-shell monomers14 which are held together by weak forces, say with a binding energy comparable to kT ¼ 0.95  103Eh at T ¼ 300 K. The nature of the VdW bond has been discussed at different times by the author and his group (Magnasco and McWeeny, 1991; Magnasco, 2004a,b). Attraction due to electrostatic, induction and dispersion energies offsets in long range the weak Pauli’s repulsion due to exchange-overlap of the closed shells (see the previous section). For spherical atoms in S states (ground-state rare gases or H and Li dimers in excited ð3 Sþ u Þ states), only dispersion can offset, in second order, Pauli’s repulsion, leading to the typical R6 attraction first postulated by London (1930). This is shown in Figure 17.13 for the He–He interaction in the VdW region, where the bottom curve, which results from adding to first-order E1 (mostly exchange-overlap repulsion at these distances) second-order E2 (mostly dispersion), fits well with accurate data from experiment (Feltgen et al., 1982). A weak VdW bond with De ¼ 33.4  106Eh is observed at the rather large interatomic

14

With each monomer maintaining its original structure.

768

CHAPTER 17 Atomic and molecular interactions

FIGURE 17.13 VdW bond in He21 Sþ g

distance Re ¼ 5.6 a0, at the bottom of the potential energy curve, as the result of the balance in long range of the weak repulsive E1 with the weak attractive E2. The situation is quite different for the long-range interaction of two H2O molecules (Figure 17.14), since now the dipolar monomers already attract each other in first order ðE1 ¼ E1cb þ E1exchov Þ mostly with an R3 interaction, and the resultant minimum is deepened by second-order induction and dispersion. The minimum is now much deeper (about 7  103Eh at R ¼ 5.50 a0, roughly 200 times larger than that of He2 at about the same distance), which means that the hydrogen bond is essentially electrostatic in nature. The structures of VdW dimers are studied at low temperatures by far infrared spectra, highresolution rotational spectroscopy or molecular beams techniques. Distances Re between c.o.m. and bond strengths jDej at the VdW minimum for some homodimers of atoms and molecules are given in Table 17.19. Notice that the energy units chosen for molecules (last column of the table) are 103Eh while those chosen for atoms (third column) are 106Eh. Structures for molecular homodimers (Pople, 1982) as well as complete references can be found elsewhere (Magnasco, 2004b). We notice from the table how large Re and how small jDej values characterize VdW dimers with respect to the values occurring for ordinary ‘chemically bonded’ atoms.

17.6 The Van der Waals bond

769

FIGURE 17.14 H-bond in (H2O)2 at q ¼ 0 (not the absolute minimum)

Table 17.19 Bond Distances Re and Bond Strengths jDej (Atomic Units) at the Minimum of the Potential Energy Surface for Some Homodimers of Atoms and Molecules Atom

Re/a0

De/10L6Eh

Molecule

Re/a0

De/10L3Eh

H2 ð3 Sþ uÞ He2 Ne2 Ar2 Kr2 Xe2

7.8 5.6 5.8 7.1 7.6 8.2 8.0 4.7

20.1 33.4 133 449 633 894 1332 2964

(H2)2 (N2)2 (CH4)2 (NH3)2 (H2O)2 (HF)2 (BeH2)2 (LiH)2

6.5 8.0 7.3 6.2 5.4 5.1

0.12 0.39 0.69 6.47 10.3 11.4 52.2 75.8

Li2 ð3 Sþ uÞ Be2

4.0

770

CHAPTER 17 Atomic and molecular interactions

Ending this section, we notice that the distortion (induction) energy is zero for atoms, which do not have permanent moments, and mostly always smaller than the dispersion energy for molecules, with the exception of (LiH)2, where the isotropic C6 induction coefficient is 297 Eh a60 compared to a C6 dispersion coefficient of 125Eha0 (Bendazzoli et al., 2000). This large value of the former coefficient (C6 ¼ 2am2) is due to the combined large values of m and a for LiH ð1 Sþ Þ, 2.29ea0 and 28.3a0, respectively (Tunega and Noga, 1998). Lastly, we must say that much before London work, Keesom (1921) pointed out that if two molecules possessing a permanent dipole moment undergo thermal motions, they will on the average assume orientations leading to attraction, with a T-dependent C6 coefficient given by C6 ðTÞ ¼

2m2A m2B 3kT

(203)

where mA, mB are the strengths of the dipoles, and k is the Boltzmann’s constant. The corresponding attractive energies are the isotropic electrostatic contributions to the interaction energy and are temperature dependent. If U ¼ (qA, qB, 4) are the angles describing the orientation of the dipoles mA and mB, the long-range (electrostatic) interaction between the dipoles at a distance R between their centres is given by (Coulson, 1958)     m m (204) V U; R ¼ A 3 B F U R FðUÞ ¼ sin qA sin qB cos 4  2 cos qA cos qB

(205)

As shown in Problem 17.8, when averaged over all possible free orientations U assumed by the dipoles, hViU ¼ 0 (the same being true also for all higher permanent multipole moments of the molecules), but its thermal average is not zero and leads to attraction. Averaging the quantity Vexp(V/kT) over all possible orientations U R dU FðUÞexp½aFðUÞ mA mB d m m ¼ ln KðaÞ (206) hV expðV=kTÞiU ¼ A 3 B U R R R3 da U dU exp½aFðUÞ where

m m a ¼  A3 B < 0 R kT

is a dimensionless parameter depending on R, T, mA, mB, and the quantity Z KðaÞ ¼ dU exp½aFðUÞ

(207)

(208)

U

is called the Keesom integral. Evaluation of the Keesom integral for small values of a is straightforward (Keesom, 1921), and gives (Problem 17.9)   a2 (209) KðaÞ y K1 ðaÞ ¼ 8p 1 þ 3

17.6 The Van der Waals bond

771

Table 17.20 Comparison Between Isotropic Dispersion and Induction Coefficients and Keesom C6(T ) Coefficients for Some Homodimers in the Gas Phase at T ¼ 293K C6 =Eh a60 Molecule

m/ea0

a=a30 a

Keesom

Dispersiona

Induction

CO NO N2O NH3 HF H 2O LiH

0.04 0.06 0.07 0.58 0.70 0.73 2.29

13.1 11.5 19.7 14.6 5.60 9.64 28.3b

0.002 0.009 0.017 81.30 172.5 204.0 8436

81.4 69.8 184.9 89.1 19.0 45.4 125c

0.04 0.08 0.19 9.82 5.49 10.3 297

a

Buckingham et al., 1988. Tunega and Noga, 1998. c Bendazzoli et al., 2000. b

which are the first two terms (n ¼ 0, 1) in the expansion of the exponential in even powers of a 1  dln K1 ðaÞ 2 a2 2 ¼ a 1þ y a L1 ðaÞ ¼ 3 da 3 3

(210)

yielding the C6(T) Keesom coefficient of Eqn (203). Deviations occurring for large values of jaj can be accounted for by including higher terms in the expansion of the exponential. Recent work (Magnasco et al., 2006), where Keesom calculations were extended up to the R10 term, shows that deviations of the Keesom approximation from the full series expansion are less important than consideration of the higher order terms in the R2n expansion of the intermolecular potential. An asymptotic two-term expansion in inverse powers of (a) for very large values of jaj was recently derived by Battezzati and Magnasco (2004) in the simple form:15   4p expð2aÞ 2 1 KN ðaÞ y (211) 3 a2 3a The three long-range C6 isotropic coefficients for some homodimers at T ¼ 293K are compared in Table 17.20. It is seen that Keesom C6(T ) is negligible compared to dispersion and induction only for the first three homodimers, while for (NH3)2, (HF)2, (H2O)2 dipole orientation forces become increasingly dominant at room temperature, and cannot be neglected in assessing collective gas properties such as the equation of state for real gases and virial coefficients. For (HF)2 and (H2O)2 small corrections over the original Keesom formula (203) are needed to avoid overestimation of C6(T), while (LiH)2 is conveniently treated by the asymptotic formula (211). 15

This formula works particularly well for the halides of alkaline metals.

772

CHAPTER 17 Atomic and molecular interactions

17.7 ACCURATE THEORETICAL RESULTS FOR SIMPLE DIATOMIC SYSTEMS In the theory of atomic or molecular interactions it is often assumed as a first reasonable approximation (see point (5) of Section 1.3.5, and Magnasco and McWeeny, 1991) to add to E1cb ¼ E1es the first-order exchange-overlap (penetration) E1exch-ov term, which can be evaluated using just A0 and B0, and adding further the E2cb term (without second-order exchange), which introduces the main Coulombic secondorder effects describing distortion and correlation between the atomic (or molecular) charge distributions. As an example, the main contributions to the energies of the chemical bond and to the VdW bond in Hþ 2 and H2 are reported in Table 17.21. The table deserves the following comments. 1. Hþ 2 molecule-ion The main correction to the attractive HL bond energy for the 2 Sþ g ground state is the (attractive) distortion energy (mostly dipole) of the charge distribution. The main correction to the repulsive Table 17.21 Main Contributions to the Energies of the Chemical Bond (103 Eh) and the VdW Bond (106 Eh) in Hþ 2 and H2 Hþ 2 molecule-ion 1.

R ¼ 2 a0 E1 ð

2.

2.

Sþ gÞ

49.0

Total

102.8

R ¼ 12.5 a0

VdW bond

2

Sþ uÞ

Accurate 102.6

þ31

ind; a E21 Total

92

R ¼ 1.4 a0

Chemical bond

E1 ð1 Sþ gÞ

105.5

E2ind ,b

68.0

Total

173.5

R ¼ 8 a0

VdW bond þ14

E1 ð3 Sþ uÞ

a

Chemical bond 53.8

ind; a E21

E1 ð

H2 molecule 1.

2

61

E2ind ,c

34

Total

20

Accurate 60.8

Accurate 174.5

Accurate 20.1

Dipole contribution. Assuming cancellation between the remaining attractive multipole contributions and the repulsive second-order exchange effects. b Exact induction. Assuming cancellation between attractive dispersion (11.8) and repulsive second-order exchange effects. c Non-expanded dispersion.

17.8 A generalized multipole expansion for molecular interactions

773

HL energy for the 2 Sþ u excited state is the (attractive) distortion energy (mostly dipole) of the charge distribution. At variance with H2 (neutral molecule), induction is a long-range effect in Hþ 2 (charged molecule-ion). 2. H2 molecule The main correction to the attractive HL bond energy for the 1 Sþ g ground state is still the (attractive) distortion energy of the charge distribution. The main correction to the repulsive HL energy for the 3 Sþ u excited state of H2 is the (attractive) dispersion due to the interatomic electron correlation. At the distance of the VdW minimum, induction is negligible for the neutral molecule. The values reported as accurate in Table 17.21 are taken from Peek (1965) for Hþ 2 , Ko1os and 3 þ Wolniewicz (1965) for H2 ð1 Sþ g Þ, and Ko1os and Rychlewski (1990) for H2 Su . We notice that the expanded dispersion for H2 up to 2n ¼ 10 gives at R ¼ 8 a0 the results of Table 17.5 (Kreek and Meath, 1969): wdisp

E2

24.4

6.9

1.6

0.9

Multipoles

1,1

1,2 þ 2,1

1,3 þ 3,1

2,2

S ¼ 33.8  106 Eh

a value which is in very good agreement with the corresponding non-expanded value of 34  106 Eh.

17.8 A GENERALIZED MULTIPOLE EXPANSION FOR MOLECULAR INTERACTIONS Following early work by Koide (1976) for atoms, a generalized multipole expansion of the intermolecular potential V converging for all intermolecular separations R and allowing for separation of 1 angular from R-dependent factors was based on the complex FT of the interparticle distance rab (Knowles and Meath, 1987; Magnasco and Figari, 1989).

17.8.1 Generalized expansion of the intermolecular potential Using Eqn (365) of Chapter 4, we obtain for the inverse of the interparticle distance rab Z 1 1 1 dk ¼ 2 ¼ expðik $ rab Þ rab jR þ rb  ra j 2p k2 1 ¼ 2 2p

ZN

d4k expði k $ RÞ expðik $ rb Þ expðik $ ra Þ

dqk sin qk

dk 0

Z2p

Zp 0

(212)

0

where the transformation vector k (dimensions, length1) has spherical components ðk; qk ; 4k Þ. Each plane wave can be expanded in spherical waves (Rayleigh expansion: Rose, 1957): N l X   X il jl r k Ylm ðqk ; 4k ÞYlm ðqr ; 4r Þ (213) expði k $ rÞ ¼ 4p l¼0

m¼l

774

CHAPTER 17 Atomic and molecular interactions

where the Ys are real spherical harmonics (normalized tesseral harmonics) referred to the vectors k and r, and jl is a spherical Bessel function of order l (Section 4.6.3). Substituting in Eqn (212), we have Z la N X N X N l0 X X X 1 la l 0 þla þlb ¼ 32p ð1Þ i dk jl 0 ðRkÞjla ðra kÞjlb ðrb kÞ  rab l 0 ¼0 l ¼0 l ¼0 m 0 ¼l 0 m ¼l N

a

lb X



b

a

0

a

Yl0 m ðqR ; 4R ÞYla ma ðqa ; 4a ÞYlb mb ðqb ; 4b Þ

mb ¼lb

Z2p

Zp 

d4k Yl 0 m 0 ðqk ; 4k ÞYla ma ðqk ; 4k ÞYlb mb ðqk ; 4k Þ

dqk sin qk 0

(214)

0

The choice of axes of Figure 4.6 allows for some simplifications. Since R lies along the intermolecular axis z, qR ¼ 0 and rffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2l0 þ 1 Yl 0 m 0 ðqR ; 4R Þ ¼ (215) d0m0 4p so that the last integral in Eqn (214) will be different from zero only for ma ¼ mb ¼ m. Integrating over (qk, 4k), we obtain for real spherical harmonics (Rose, 1957) Zp

Z2p d4k Yl 0 0 ðqk ; 4k ÞYla m ðqk ; 4k ÞYlb m ðqk ; 4k Þ

dqk sin qk 0

0

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l ab X dl 0 L m ð2la þ 1Þð2lb þ 1Þ pffiffiffiffiffiffiffiffiffiffiffiffiffi ffi Cðla ; lb ; L; 0; 0ÞCðla ; lb ; L; jmj; jmjÞ ¼ ð1Þ 4p s¼0 2L þ 1

(216)

where the Cs are Clebsch–Gordan coefficients (Section 10.4.1), and we put L ¼ jla  lb j þ 2s 8 1 > lab ¼ minðla ; lb Þ ¼ ðla þ lb  jla  lb jÞ > > > 2 > < 1 Lab ¼ maxðla ; lb Þ ¼ ðla þ lb þ jla  lb jÞ; > > > 2 > > : la þ lb ¼ Lab þ lab ; jla  lb j ¼ Lab  lab

(217)

(218)

Inserting these results into Eqn (214), we finally obtain N X N X pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ¼8 ð1Þla þlab ð2la þ 1Þð2lb þ 1Þ rab l ¼0 l ¼0 a



b

lab X m¼lab

ð1Þm

lab X

ð1Þs Cðla ; lb ; L; 0; 0ÞCðla ; lb ; L; jmj; jmjÞ

s¼0

ZN  Yla m ðqa ; 4a Þ Ylb m ðqb ; 4b Þ 0

      dk jL R k jla ra k jlb rb k

(219)

17.8 A generalized multipole expansion for molecular interactions

775

In this way is obtained a generalized unified expansion of 1/rab, valid for any intermolecular separation R, where the coordinates of the particles, referred to their respective molecular centres, are conveniently factorized. Since Rayleigh expansion (213) converges for all values of its argument, the multipole expansion (219) converges for any value of R, including R ¼ 0. Specific cases are conveniently treated by evaluating the integral over three spherical Bessel functions in Eqn (219) by contour integration in the complex plane (Problem 5.4), the integration being simple provided one of the (R,ra,rb) is greater than the sum of the remaining two. Three typical asymptotic cases, corresponding to regions I (R  ra þ rb), III (rb  R þ ra), IV (ra  rb þ R) of Buheler and Hirschfelder (1951, 1952), are easily obtained in a unified way. The first case corresponds to the usual Rn long-range expansion, while the latter two altogether give the expansion valid when R ¼ 0, which coincides with the usual one-centre Neumann’s expansion for the inverse of the interparticle distance. The overlap case (region II of Buheler and Hirschfelder) might be dealt with as well, but it is rather more involved (Buheler and Hirschfelder, 1952). Radial expressions for the Buheler and Hirschfelder coefficients in the four regions were also given and discussed by Koide et al. (1986). Expansion (219) was also used by Figari et al. (1990) for the analytical evaluation in Roothaan’s closed form of the general two-electron two-centre Coulomb integral over Slater-type orbitals using the finite representation of spherical Bessel functions of Section 4.6.3 followed by contour integration in the complex plane. We now introduce the generalized (k-dependent) multipole moment operator for molecule A: QA la m ðkÞ ¼

X

q a RA la m ðra ; kÞ ¼

a

la X ka ¼la

Lklaam Q~ka m ðkÞ A

(220)

and the like for molecule B, where Rlm(r,k) is a generalized spherical tensor in real form (Section 4.11.3) rffiffiffiffiffiffiffiffiffiffiffiffi 4p (221) Ylm ðq; 4Þ Rlm ðr; kÞ ¼ ð2l þ 1Þ!!jl ðr; kÞ 2l þ 1 The real spherical harmonics Y are normalized and defined as in Magnasco et al. (1988). The last relation on the right-hand side of Eqn (220) expresses the transformation from space-fixed ~ axes, the latter differing by the former (intermolecular, Qs) to molecule-fixed (intramolecular, Qs) by rotations related to the Euler’s angles (Rose, 1957; Brink and Satchler, 1993; Magnasco et al., 1988). All information about this transformation is contained in the elements Llm k of matrix L in Eqn (224) below, which describes unambiguously the relative geometric configuration of the two molecules. In the generalized multipole moment operators (2l þ 1)!!jl(r k) replaces rl. Their relation to the usual multipole moment operators of Section 17.2.1 can be visualized using Eqn (263) of Chapter 4, so that Qlm ¼ lim

k/0

  Qlm ðkÞ 1 vl ðkÞ ¼ Q lm kl l! vkl k¼0

(222)

776

CHAPTER 17 Atomic and molecular interactions

In k-space, the intermolecular potential V in the body-fixed reference system then factorizes as la N X N X 2 X V¼ Kðla ; lb Þ p l ¼0 l ¼0 k ¼l a

b

a

Kðla ; lb Þ ¼ where Tklaalkbb ðR

kÞ ¼

lab X m¼lab

" ð1Þ

m

ZN lb X a

kb ¼lb

dk Tklaalkbb ðR kÞQ~la ka ðkÞQ~lb kb ðkÞ (223)

0

lb þlab

ð1Þ ð2la  1Þ!!ð2lb  1Þ!!

lab X

# s

ð1Þ Cðla ; lb ; L; 0; 0ÞCðla ; lb ; L; jmj; jmjÞjL ðR kÞ Lklaam Lklbbm (224)

s¼0

is a function collecting all geometrical parameters determining the relative orientation of the two molecules. The R-dependence of the intermolecular potential (223) is implicit in the argument of the spherical Bessel functions, and can be exploited by performing the integration over k.

17.8.2 Generalized molecular moments and polarizabilities k-Dependent electric moments and polarizabilities, generalizing the ordinary moments and polarizabilities of Section 17.2.1, are introduced via the generalized multipole moment operators (220). To simplify notation, we shall always refer to space-fixed coordinates, so avoiding the tilde on the relevant expressions. The generalized transition moment for the system undergoing a transition from state 0 to state k is defined by E Z D A ðokjkÞ ¼ A ðkÞ (225) mA Q A0 ¼ dr RA k pa qa pa qa pa qa ðr; kÞgð0kjrÞ where gA ð0kjrÞ ¼ PA ð0kjr; rÞ þ

X

Za dðr  ra Þdk0

(226)

a

is the (electron þ nuclear) distribution function of molecule A (Longuet-Higgins, 1956), the summation being over all nuclei a of charge Za. PA(0kjr;r) is the one-electron transition density (McWeeny, 1960) at point r on A (see Problem 17.5). When k ¼ 0 in Eqn (225), we have the generalized permanent moments of molecule A. The multipole polarizability can be given in terms of either a pseudospectral decomposition of the London type or in the form of a frequency-dependent Casimir-Polder integral. We have for the static polarizability X 0 0 aA (227) aA pa qa ; pa0 qa0 ðk; k Þ ¼ pa qa ;pa0 qa0 ðkjk; k Þ k

where 0 aA pa qa ;pa0 qa0 ðkjk; k Þ ¼

A 0 A 0 A mA pa qa ð0kjkÞmpa0 qa0 ð0kjk Þ þ mpa0 qa0 ð0kjk Þmpa qa ð0kjkÞ

εA ðkÞ

(228)

εA(k) being the excitation energy from the unperturbed ground-state function A0 to the k-th excited pseudostate Ak.

17.8 A generalized multipole expansion for molecular interactions

777

The generalized dynamic polarizability is given as A 0 A 0 A   mA pa qa ð0kjkÞmpa0 qa0 ð0kjk Þ þ mpa0 qa0 ð0kjk Þmpa qa ð0kjkÞ εA k 2 ½εA ðkÞ þ u2 k Z Z 0 0 ¼ dr1 RA dr10 PA ðr1 ; r01 jiuÞ RA pa qa ðr1 ; kÞ pa0 qa0 ðr1 ; k Þ

0 aA pa qa ;pa0 qa0 ðk; k jiuÞ ¼

X

(229)

where i is the imaginary unit, u is a real frequency, and PA ðr1 ; r01 jiuÞ ¼

X k

  PA ð0kjr1 ; r1 ÞPA ðk0jr01 ; r01 Þ þ PA ð0kjr01 ; r01 ÞPA ðk0jr1 ; r1 Þ εA k 2 ½εA ðkÞ þ u2

(230)

is the dynamic propagator which determines the linear response of the electron density at r1 to a perturbation acting at point r01 (McWeeny, 1984; Magnasco and McWeeny, 1991). Replacing the generalized spherical tensors (221) by their usual multipole expression, we recover the usual definitions of moments and polarizabilities of Section 17.2.1. Upper bounds to their exact values can be obtained either by the appropriate pseudostate expansions of the previous sections (Magnasco and Figari, 2009) or by TDHF or MC-TDHF (Langhoff et al., 1972; Jaszunski and McWeeny, 1982; McWeeny, 1983; Visser et al., 1983, 1985; Amos et al., 1985; Wormer and Rijks, 1986; Knowles and Meath, 1986, 1987) or Full-CI techniques (Bendazzoli et al., 2000, 2005, 2008).

17.8.3 Molecular interaction energies Since the formulae for the molecular interaction energies of the various orders resulting from the theory above are rather cumbersome, we limit ourselves here to consideration of the generalized dispersion energy which involves simultaneous double excitations, one on A and the other on B. Complete reference to generalized electrostatic and induction energies can be found elsewhere (Magnasco and Figari, 1989). The London type formula for the generalized dispersion energy is X X X XZ 1 XXXX ¼ 2 Kðla ; lb Þ Kðla0 ; lb0 Þ dk Tqlaalqbb ðRkÞ p l qa qb q 0 q 0 l l 0 l 0 N

disp E2

a

ZN  0

b

a

a

b

b

0

    X X ε A k εB l 0 la0 lb0 0 ~A ~ Blb qb ;lb0 qb0 ðljk; k0 Þ a dk Tqa0 qb0 ðRk Þ  ðkjk; k0 Þ a A ðkÞ þ εB ðlÞ la qa ;la0 qa0 ε k l

(231)

where we must sum over all pseudostate contributions {k} of A and {l} of B to the k-dependent static polarizabilities. The corresponding Casimir–Polder formula in terms of generalized FDPs is XXXX 2 XXXX disp ¼ 3 Kðla ; lb Þ Kðla 0 ; lb 0 Þ E2 p l qa qb q 0 q 0 l l 0 l 0 a

b

a

ZN 

dk Tqlaalqbb ðRkÞ 0

a

b

ZN 0

dk0 Tqlaa00lqbb0 0 ðRk0 Þ 

ZN 0

b

(232) 0 ~ Blb qb ;lb 0 qb 0 ðk; k0 jiuÞ ~A du a la qa ;la 0 qa 0 ðk; k jiuÞ a

778

CHAPTER 17 Atomic and molecular interactions

Expressions (231) and (232) are entirely equivalent, provided the polarizabilities are evaluated in the same set of basis functions. In all expressions above, the l-expansions are over the generalized multipoles, while the summations over the qs transform the multipole moments from intermolecular (space-fixed) to intramolecular (body-fixed) axes.

17.8.4 The damping of dispersion in the (HF)2 homodimer As an example of application of the above formulae for dispersion, we give some results obtained by Knowles and Meath (1987) for the damping of the dispersion interaction in the (HF)2 homodimer. The damping functions measure the deviation of the non-expanded radial coefficients from their asymptotic long-range limit, as we saw in Section 17.3.4 for the H–H interaction. The dispersion damping functions X are defined as the ratio of each term of the generalized multipole expansion with respect to its corresponding long-range expanded term. Their value is one at large intermolecular separations R, and decreases below this value at shorter distances. Knowles and Meath results were based on a sequence of Pade´ approximants to TDHF calculations of polarizabilities. Details can be found in the original paper. We give here (Table 17.22) a few values of the dispersion damping function X6 for the C6 isotropic dispersion coefficient at the intermediate distance of R ¼ 5 a0 as a function of the order N of the approximant. The table shows that the X6 value oscillates to some extent for small values of N but stabilizes for larger values, reaching a satisfactory convergence (within 1%) for N ¼ 9. The situation is worse for the higher terms describing the anisotropy of the dispersion interaction. For the most important anisotropic term at R10, the uncertainty is about 7% at N ¼ 9. In general, convergence is better for the lower Rn terms, and improves as R increases. Knowles and Meath estimate that the final uncertainties in their results are probably within the errors expected in the calculations due to the incompleteness of the basis set at the TDHF level. The last row of the table shows that the damping effect in the dispersion interaction of the two HF molecules is still rather important at R ¼ 5 a0. These considerations show how difficult is to obtain reliable values of non-expanded dispersion coefficients for molecules. Table 17.22 Convergence of the Dispersion Damping Function X6 for (HF)2 at R ¼ 5 a0 According to the Order N of the Approximant N

X6

0 1 2 3 4 5 6 7 8 9

1.0000 0.8246 0.7538 0.8349 0.8226 0.8347 0.8355 0.8351 0.8357 0.8360

17.9 Problems 17

779

17.9 PROBLEMS 17 17.1. Prove the general formula for the binomial coefficients.    q   X u v uþv ¼ p qp q p¼0

Hint: Use the well-known binomial theorem and an algebraic power relation. 17.2. Derive by elementary integration the integral transform (157). Answer: Evaluate the integral ZN I¼

du

1 1 p 1 $ ¼ a2 þ u2 b2 þ u2 2 abða þ bÞ

0

with a,b > 0, where a ¼ εi, b ¼ εj. Hint: Use elementary integration techniques and change of the integration variable. 17.3. Evaluate the C6 dispersion coefficient for the H–H interaction from the two-term pseudospectrum of the dipole polarizability of H(1s). Answer: C6 ¼

363 56

Hint: Use the two-term dipole pseudospectrum given in Table 13.14. 17.4. Evaluate the C6 dispersion coefficients for the H2–H2 interaction. Answer: The dispersion constants are A ¼ 2:683; B ¼ C ¼ 2:018; D ¼ 1:522 giving 1. Isotropic dispersion coefficient C6 ¼ 11:23 2. Anisotropic coefficients 220 g020 6 ¼ 0:098; g6 0:030

Hint: Use the four-term dipole pseudospectrum for H2 ð1 Sþ g Þ at R ¼ 1.4a0 found in Problem 13.15. 17.5. Verify the expression for the general transition matrix element and derive the Coulombic interaction energy expressions up to the second order in the intermolecular potential V.

780

CHAPTER 17 Atomic and molecular interactions

Hint: Use the very properties of the Dirac d-function given in Section 4.8. 17.6. Show that the exchange-overlap density matrix of molecule A vanishes when integrated over all space. Answer: Z   dr PA 00 r; r ¼ 0 ov

which shows that, even contributing to the first-order interaction energy, PA ov does not contribute to the integral of the total electron density, a property shared with the transition densities occurring in higher orders of perturbation theory. Hint: Use definitions (198) and (201) and interchange summation indices, remembering that the overlap matrix S is traceless. 17.7. Show that for the Hartree–Fock wavefunction of molecule A Z A 0 dr F^ 00 r PA ov ð00jr; r Þ r 0 ¼r ¼ 0 Hint: Use the same suggestions of Problem 17.6 and notice that all diagonal elements of matrix S are individually zero. 17.8. Prove that the first-order temperature-dependent electrostatic C3(T) coefficient is zero when averaged over all equiprobable molecular orientations. Answer: Z Z C3 ðTÞ ¼ mA mB dU FðUÞ= dU ¼ 0 U

U

Hint: Evaluate the integral over the angles. 17.9. Derive the formula for the C6(T) Keesom coefficient in the case a ¼ small. Answer: 2m2 m2 C6 ðTÞ ¼ A B 3kT Hint: Evaluate in spherical coordinates all necessary integrals occurring in the Keesom integral.

17.10 SOLVED PROBLEMS 17.1. An important formula for binomial coefficients. Using the well-known binomial theorem and the algebraic power relation u   X u r u x ; ð1 þ xÞu ð1 þ xÞv ¼ ð1 þ xÞuþv ð1 þ xÞ ¼ r r¼0

17.10 Solved problems

we can write ð1 þ xÞ

uþv

 u   X v   uþv  X u r v s X uþv q x x ¼ x ¼ r s q r¼0

s¼0

q¼0

The coefficient of the general power xq in the above expression is given by           X q   u v u v u v u v þ þ/þ ¼ p qp 0 q 1 q1 q 0 p¼0

so that by comparison it follows (Paci, 1985)    q   X u v uþv ¼ p qp q p¼0

which is the required formula. 17.2. Elementary derivation of the integral transform for expanded dispersion. According to a well-known technique of the integral calculus, in the integral ZN I¼

du

a2

1 1 $ 2 2 þ u b þ u2

a; b > 0

0

we decompose the product in the integrand according to     A b2 þ u2 þ B a2 þ u2 A B þ ¼ a2 þ u 2 b 2 þ u 2 ða2 þ u2 Þðb2 þ u2 Þ with the conditions that

    A b2 þ u2 þ B a2 þ u2 ¼ 1     2 Ab þ Ba2 þ A þ B u2 ¼ 1

giving

  Ab2 þ Ba2 ¼ 1 0 A b2  a2 ¼ 1

A þ B ¼ 0 0 B ¼ A; We therefore obtain

 1 A ¼ b2  a 2 ;

1  B ¼  b2  a 2    2  1 1 1 2 1 ¼ b  a  ða2 þ u2 Þðb2 þ u2 Þ a2 þ u 2 b 2 þ u 2 For the first of the two resulting integrals we have ZN 0

1 1 du 2 ¼ a þ u2 a 2

ZN 0

1

1 du ¼ 2 a 1 þ ðu=aÞ

ZN 0

1 1 1 N p dx ¼ tan x ¼ 1 þ x2 a 2a 0

781

782

CHAPTER 17 Atomic and molecular interactions

where we have posed u/a ¼ x, du ¼ a dx, the integration limits being unchanged. Proceeding similarly with the remaining integral, we finally obtain ZN I¼

 1 1 1 du 2 $ 2 ¼ b2  a 2 2 2 a þu a þb

0



ZN du

1 1  a2 þ u 2 b 2 þ u 2



0

 ¼ b2  a 2

1 p 2a



p 2b

1 p b  a p  1 ¼ ¼ b2  a2 2 ab 2 abða þ bÞ

which is the required value. Hence, we get the integral transform 1 2 ¼ aþb p

ZN du

a b $ a2 þ u2 b2 þ u2

a; b > 0

0

An elegant derivation of this transform was also given in Section 5.3.4 using the integration techniques in the complex plane. 17.3. C6 dispersion coefficient for H(1s)–H(1s) from the two-term dipole polarizability pseudospectrum of H(1s). We evaluate first the dipole dispersion constant: C11 ¼

2 X 2 εi εj 1X 1 1 ε1 ε2 1 ai aj ¼ a21 ε1 þ a1 a2 þ a22 ε2 εi þ εj 8 ε1 þ ε2 8 4 i¼1 j¼1 2

Then, using the pseudospectrum given in Table 13.14, we obtain 1 4 1 50 2 5 1 625 2 1 25 125 121 $ ¼ þ þ ¼ C11 ¼ $ $ 1 þ $ $ $ þ $ 8 36 2 36 5 7 8 36 5 72 126 144 112 and the C6 dispersion coefficient will be C6 ¼ 6C11 ¼

363 ¼ 6:482 142 857 142 86/ 56

The C6 dispersion coefficient is thus obtained as a fraction of simple not divisible integers. Using a non-variational technique in momentum space, Koga and Matsumoto (1985; see also Koga and Ujiie, 1986) gave the three-term C6 for H–H as the ratio of two not divisible integers as 12529 C6 ¼ ¼ 6:498 443 983 402 49/ 1928 and, for the four-term C6 ¼

6313807 ¼ 6:499 002 577 446 93/ 971504

The last value is accurate to four decimal figures, a better C6 value, accurate to 15 decimal figures, being (Ottonelli, 1998, N ¼ 25; Magnasco et al., 1998) C6 ¼ 6:499 026 705 405 839/

17.10 Solved problems

783

17.4 C6 Dispersion coefficients for H2–H2. An accurate evaluation of C6 dispersion coefficients for H2–H2, based on the pseudostate decomposition of Kołos and Wolniewicz (1967) static dipole polarizabilities for ground-state H2 can be found in a paper by Magnasco and Ottonelli (1996). We shall be content here with a less accurate evaluation based on the four-term dipole pseudospectra of H2 ð1 Sþ g Þ at R ¼ 1.4a0 given in Problem 13.15. We first calculate the three independent dipole–dipole dispersion constants A, B ¼ C, D given by Eqn (159), which in London form are A¼

jj jj 1 X X jj jj εi εj ai aj jj ; jj 4 i ε þε j i

B¼C¼

D¼

j

t t 1 X X t t εi εj ai aj t 4 i εi þ ε t j j

jj t 1 X X jj t εi εj ai aj jj 4 i ε þ εt j i

for the homodimer

j

The four-term pseudospectrum then gives C11 ¼

4 X 4 εi εj 1X ai aj εi þ εj 4 i¼1 j¼1

Using the four-term values of Problem 13.15, we obtain for H2–H2 the following numerical results: A ¼ 1:422 þ 1:261 ¼ 2:683 0 99:8% of 2:689 diagonal cross-term B ¼ C ¼ 0:969 þ 0:629 þ 0:420 ¼ 0:969 þ 1:049 ¼ 2:018 0 99:3% of 2:032 i¼j ij diagonal cross-terms D ¼ 0:677 þ 0:845 ¼ 1:522 0 98:7% ¼ of 1:542 diagonal cross-term The results for the three dispersion constants are excellent, all being within 99% of the accurate values (1996) or more. Using these results, we obtain for the LALBM-components of the C6 dispersion coefficients for H2–H2 (R ¼ 1.4 a0) the numerical results collected in Table 17.23. Table 17.23 LALBM-Components of the C6 Dispersion Coefficients for H2eH2 at R ¼ 1.4 a0 LA

LB

M

C6

g6

0 0 2 2 2 2

0 2 0 2 2 2

0 0 0 0 1 2

11.228 1.105 1.105 0.338 0.075 0.0094

1 0.098 0.098 0.030 0.0067 0.00084

784

CHAPTER 17 Atomic and molecular interactions

The four-term results of Table 17.23 compare favourably with the results of the accurate calculations (Magnasco and Ottonelli, 1996; Ottonelli, 1998) reported in Eqns (176)–(179) of the main text: 1. The isotropic dispersion coefficient, C6000 ¼ C6 ¼ 11:23, is within 99.2% of the accurate value 11.32; 2. The anisotropy coefficients, defined as C LA LB M gL6 A LB M ¼ 6 C6 are also in good agreement with the accurate data g020 6 ¼ 0:098 instead of 0:096;

g220 6 ¼ 0:030 instead of 0:029

17.5. We first recall from Chapter 4 the very property of the Dirac d-function Z dx0 dðx  x0 Þf ðx0 Þ ¼ f ðxÞ The general matrix element can then be written as ZZ   gA ð0ijr1 ; r1 ÞgB ð0jjr2 ; r2 Þ Ai Bj jVjA0 B0 ¼ dr1 dr2 r12 A X Z X Z P ð0ijr1 ; r1 Þ PB ð0jjr2 ; r2 Þ ¼ d0j Zb dr1  d0i Za dr2 r1b r2a a b ZZ X X Za Zb PA ð0ijr1 ; r1 ÞPB ð0jjr2 ; r2 Þ þ dr1 dr2 þ d0i d0j rab r12 a b where

gA ð0ijrÞ ¼ d0i

X

Za dðr  ra Þ  PA ð0ijr; rÞ

a

is the transition charge density (nuclei þ electrons) operator at r associated to the transition 0 / i on molecule A (Longuet-Higgins, 1956) (Compare with Eqn (226) of Section 17.8.2). Then, we obtain all possible energy contributions up to the second order in the intermolecular potential V: 1. i ¼ 0, j ¼ 0 First-order electrostatic energy Z  ZZ Z gA ð00jr1 ÞgB ð00jr2 Þ gB ð00jr2 Þ A dr2 g ð00jr1 Þ dr1 dr2 ¼ dr1 E1es ¼ r12 r12 # " #" Z X Zb Z PB ð00jr2 ; r2 Þ X A  dr2 Za dðr1  ra Þ  P ð00jr1 ; r1 Þ ¼ dr1 r12 r1b a b Z X X Za Zb X Z PA ð00jr1 ; r1 Þ X PB ð00jr2 ; r2 Þ ¼  Zb dr1  Za dr2 rab r1b r2a a a b b ZZ PA ð00jr1 ; r1 ÞPB ð00jr2 ; r2 Þ þ dr1 dr2 r12

17.10 Solved problems

785

2. i s 0, j ¼ 0 Second-order induction energy 0 B polarizes A   P Zb d r2  rb  PB ð00jr2 ; r2 Þ Z Z B g ð00jr2 Þ b ¼ dr2 U B ð00jr1 Þ ¼ dr2 r12 r12 Z B X 1 P ð00jr2 ; r2 Þ ¼ Zb  dr2 r r12 1b b

ind;A E~2 ¼

2 A B RR dr1 dr2 g ð0ijr1 ; r1 Þg ð00jr2 ; r2 Þ X r 12

εi

i

R X dr1 U B ð00jr1 ÞPA ð0ijr1 ; r1 Þ 2 ¼ εi i and the like for A 4 B with i ¼ 0, j s 0 (A polarizes B). 3. i s 0, j s 0 Second-order dispersion energy 0 Mutual polarization of A and B

disp ¼ E~2

¼

2 A B RR dr1 dr2 g ð0ijr1 ; r1 Þg ð0jjr2 ; r2 Þ X X r12 i

j

εi þ εj

i

j

εi þ εj

2 A B RR dr1 dr2 P ð0ijr1 ; r1 ÞP ð0jjr2 ; r2 Þ X X r12

a purely electronic term. 17.6. Recall that the metric matrix M of the non-orthogonal basis 4 of occupied MOs is M¼1þS so that its inverse can be written as M1 ¼ ð1 þ SÞ1 ¼ 1  Sð1 þ SÞ1 ¼ 1 þ D where the overlap matrix S is traceless and has each diagonal element individually zero. Using definitions (198) and (201) we find that Z

ðAÞ X ðAÞ X all all X all X X   Dip ð1 þ SÞpi ¼ 2 Siq ð1 þ SÞ1 dr PA ov 00 r; r ¼ 2 qp ð1 þ SÞpi p

i

¼ 2

i

ðAÞ X all X i

q

Siq

all X p

ð1

þ SÞ1 qp ð1

p

q

þ SÞpi

! ¼ 2

ðAÞ X all X i

q

Siq dqi ¼ 2

ðAÞ X i

Sii ¼ 0

786

CHAPTER 17 Atomic and molecular interactions

17.7. We proceed much in the same way as in Problem 17.6: Z Z ðAÞ X all X A A 0 dr 4p ðrÞF^ ð00jrÞ4i ðrÞ ð00jr; r Þ ¼ 2 D dr F^ ð00jrÞPA ip 0 ov r ¼r p

i

ðAÞ X all X

¼ 2

ðAÞ X

¼ 2

εi

all X all X p

i ðAÞ X

¼ 2

Dip εi ð1 þ SÞpi

p

i

Siq ð1 þ SÞ1 qp ð1 þ SÞpi  2

q

ðAÞ X

εi

all X

i

Siq dqi

q

εi Sii ¼ 0

i

17.8. The first-order T-dependent electrostatic C3(T) coefficient is zero for all multipoles when averaged over equiprobable molecular orientations. We must evaluate the integral: Z Z mA mB m m dU FðUÞ= dU ¼ A 3 B I hViU ¼ R3 R U

with

Z I¼

U

Z dU ¼

dU FðUÞ= U

U

Z

I2 ; I1

I1 ¼

Z dU; I2 ¼

U

dU FðUÞ U

In these expressions U ¼ qA ; qB ; 4;

FðUÞ ¼ sin qA sin qB cos 4  2 cos qA cos qB

and V¼

mA mB ðsin qA sin qB cos 4  2 cos qA cos qB Þ R3

where 4 ¼ 4A  4B is the dihedral angle between the planes specified by mA, mB and R. Putting xA ¼ cos qA ; xB ¼ cos qB we have Z I1 ¼

Z2p dU ¼

U

I2 ¼

0

Z2p dU FðUÞ ¼

U

0

d4 cos 44

Z1



dx 1  x 1

 2 1=2

dxB ¼ 8p

dxA 1

0

2

Z1

Z1 dqB sin qB ¼ 2p

dqA sin qA

d4 0

Z

Zp

Zp

32 5 2

Z2p 0

1

2 d44

Z1 1

32 dx x5 ¼ 0

17.10 Solved problems

787

so that I ¼ 0. This is true for all odd powers of F(U), so that, in first order, the average potential energy for free orientations vanishes for all multipoles (dipoles, quadrupoles, octupoles, hexadecapoles, etc.). 17.9. The Keesom C6(T) coefficient. We have to evaluate the temperature-dependent Boltzmann’s integral: R dUFðUÞexp½aFðUÞ m m hV expðV=kTÞiU ¼ A 3 B U R R dU exp½aFðUÞ U

where the angles have been defined before and m m V ¼ A 3 B ðsin qA sin qB cos 4  2 cos qA cos qB Þ R m m The parameter a is given by Eqn (207) as a ¼  A3 B < 0. R kT Introducing the Keesom integral (208) Z KðaÞ ¼ dU exp½aFðUÞ U

we have hV expðV=kTÞiU ¼

mA mB d ln KðaÞ R3 da

We now evaluate this expression for a ¼ small (high temperatures and large distances between the dipoles), by expanding the exponential in the Keesom integral:   Z Z a2 FðUÞ2 þ / dU exp½aFðUÞ z dU 1 þ a FðUÞ þ 2 U

U

where the second term in the expansion vanishes being an odd power of F(U), so that only the quadratic term can contribute to the integral. We have Z2p

Z 2

dU FðUÞ ¼ U

Zp

Zp dqA sin qA

d4 0

0

  dqB sin qB sin2 qA sin2 qB cos2 4 þ 4 cos2 qA cos2 qB

0

2 ¼ 8p 3 so that

Z dU U

a2 a2 FðUÞ2 ¼ 8p 2 3

788

CHAPTER 17 Atomic and molecular interactions

Then

Z U

    a2 a2 FðUÞ2 ¼ 8p 1 þ dU 1 þ 2 3

   d a2 ln 8p 1 þ ¼ da 3

1 1þ

a2

$

2 2 az a 3 3

3

for a ¼ small. Hence, we obtain the final result for the T-dependent average attraction energy between the dipoles: hV expðV=kTÞiU z

mA mB 2 2 m2A m2B a ¼  R3 3 3kT R6

giving the Keesom coefficient as C6 ðTÞ ¼ which is Eqn (203) of the main text.

2m2A m2B 3kT

CHAPTER

Evaluation of molecular integrals

18

CHAPTER OUTLINE 18.1 Introduction ............................................................................................................................... 790 18.2 The Basic Integrals .................................................................................................................... 790 18.2.1 The indefinite integral ............................................................................................790 18.2.2 Definite integrals and auxiliary functions .................................................................791 18.3 One-centre Integrals ................................................................................................................... 793 18.3.1 One-electron integrals ............................................................................................793 18.3.2 Two-electron integrals ............................................................................................795 18.4 Evaluation of the Electrostatic Potential J1S .................................................................................. 795 18.4.1 Spherical coordinates ............................................................................................795 18.4.2 Spheroidal coordinates...........................................................................................797 18.5 The (1S2j1S2) Electron Repulsion Integral .................................................................................... 798 18.5.1 Same orbital exponent ...........................................................................................798 18.5.2 Different orbital exponents .....................................................................................799 18.6 General Formula for One-centre Two-electron Integrals................................................................. 799 18.7 Two-centre Integrals Over 1S STOS............................................................................................... 800 18.7.1 One-electron integrals ............................................................................................801 18.7.2 Two-electron integrals ............................................................................................803 18.7.3 Limiting values of two-centre integrals ....................................................................809 18.8 On the General Formulae for Two-centre Integrals ........................................................................ 812 18.8.1 Spheroidal coordinates...........................................................................................812 18.8.2 Spherical coordinates ............................................................................................813 18.9 A Short Note on Multicentre Integrals.......................................................................................... 814 18.9.1 Three-centre one-electron integral over 1s STOs .......................................................814 18.9.2 Four-centre two-electron integral over 1s STOs.........................................................815 18.10 Molecular Integrals Over GTOS .................................................................................................. 817 18.10.1 Some properties of Gaussian functions ................................................................817 18.10.2 Integrals of Gaussian functions ...........................................................................819 18.10.3 Integral transforms ............................................................................................820 18.10.4 Molecular integrals ............................................................................................822 18.11 Problems 18 ............................................................................................................................ 824 18.12 Solved Problems ...................................................................................................................... 825

Elementary Molecular Quantum Mechanics 2E. http://dx.doi.org/10.1016/B978-0-444-62647-9.00018-X
2013, 2007 Elsevier B.V. All rights reserved

789

790

CHAPTER 18 Evaluation of molecular integrals

18.1 INTRODUCTION A few one-centre one-electron integrals over Slater-type orbitals (STOs) or Gaussian-type orbitals (GTOs) were already introduced by us in Chapter 12. In this chapter, we shall take into consideration also two-electron integrals, mostly two-centre integrals over STOs. From a mathematical standpoint the two-electron integrals are multiple integrals over six variables, the position coordinates in space of the two interacting electrons. They are best dealt with by finding first the electrostatic potential due to one charge distribution, say that of the second electron, followed by integration of the resulting potential with the charge distribution of the first electron. In this way, all two-electron integrals can be reduced to just one-electron integrals in three variables. While one-electron integrals involve at most three centres, the molecular two-electron integrals may be classified into one-, two-, three-, and four-centre integrals, according to the number of nuclei to which the atomic orbitals (AOs) are referred. The difficulty with their analytical evaluation greatly increases with the number of centres involved, three- and four-centre integrals being necessarily evaluated by numerical techniques. In the context of some particularly refined molecular energy calculations, which involve the interelectronic distance r12 directly into the wavefunction (Kutzelnigg and Klopper, 1991), we saw in Chapter 16 that still more difficult three- and fourelectron many-centre integrals may occur. In this chapter, we shall be mostly concerned with an elementary approach to the evaluation of some one- and two-centre two-electron integrals over STOs when the integrand is expressed in spherical or spheroidal coordinates, respectively. In particular, we shall derive in Section 18.7 the explicit expressions for all two-centre molecular integrals over 1s STOs occurring in the study of the H2 molecule, while Section 18.8 illustrates two different strategies for the evaluation of twocentre integrals over general STOs (an alternative way is touched upon in Section 17.8.1). A short outline of a possible way of evaluating multicentre integrals over 1s STOs is then given in Section 18.9, while Section 18.10 gives an outlook of the calculation of multicentre integrals over GTOs. Problems and solved problems conclude the chapter as usual.

18.2 THE BASIC INTEGRALS 18.2.1 The indefinite integral The basic indefinite integral occurring in all atomic or molecular calculations involving STOs with exponential decay is (Gradshteyn and Ryzhik, 1980) Z dx expðaxÞxn ¼ expðaxÞ

n X

ð1Þk

k¼0

n! xnk ðn  kÞ! akþ1

(1)

with n ¼ non-negative integer and a ¼ real number, a result that can be obtained by repeated integration by parts. The case of interest in molecular quantum mechanics is a ¼ r

ReðrÞ > 0

(2)

18.2 The basic integrals

Z dx expðrxÞxn ¼ expðrxÞ

n X

ð1Þk

k¼0

¼ ¼

"

n! xnk ðn  kÞ! ðrÞkþ1

ðrxÞn ðrxÞn1 ðrxÞ2 þ þ/þ þ rx þ 1 expðrxÞ n! ðn  1Þ! 2!

n! rnþ1 n!

expðrxÞ

rnþ1

791

#

n X ðrxÞk ¼ Fn ðxÞ k! k¼0

(3)

where Fn(x) is the primitive function. This result can be checked by taking the first derivative of Fn(x). Then ( " # dFn n! 2ðrxÞ 3ðrxÞ2 nðrxÞn1 ¼  nþ1 expðrxÞ r 1 þ þ/þ þ dx 3! n! r 2! #) (" # " ðrxÞ2 ðrxÞn n! ðrxÞ2 ðrxÞn1 þ/þ ¼  n expðrxÞ 1 þ rx þ þ/þ (4) r 1 þ rx þ ðn  1Þ! r 2! n! 2! " #)   ðrxÞ2 ðrxÞn1 ðrxÞn n! ðrxÞn  1 þ rx þ þ/þ þ ¼  n expðrxÞ  ¼ expðrxÞxn 2! ðn  1Þ! n! n! r There is a term-by-term cancellation with the exception of the last term in the second sum. We now turn to the definite integrals of interest to us.

18.2.2 Definite integrals and auxiliary functions For atomic (one-centre) problems ZN dx expðrxÞxn ¼ 0

0

(5)

"

Zu dx expðrxÞxn ¼

n! rnþ1

n X ðruÞk 1  expðruÞ k! k¼0

n! rnþ1

ZN dx expðrxÞxn ¼ u

n! rnþ1

# (6)

n X ðruÞk k! k¼0

(7)

n X rk ¼ An ðrÞ k! k¼0

(8)

expðruÞ

Adding Eqn (6) to Eqn (7) gives Eqn (5). For molecular (two-centre) problems ZN dx expðrxÞxn ¼ 1

n! rnþ1

expðrÞ

792

CHAPTER 18 Evaluation of molecular integrals

Zu dx expðrxÞxn ¼ An ðrÞ  1

ZN dx expðrxÞxn ¼ u

n! rnþ1

n! rnþ1

expðruÞ

expðruÞ

n X ðruÞk k! k¼0

n X ðruÞk k! k¼0

(9)

(10)

Z1 dx expðrxÞxn ¼ Bn ðrÞ ¼ ð1Þnþ1 An ðrÞ  An ðrÞ

(11)

1

Adding Eqn (9) to Eqn (10) gives Eqn (8). In the calculation of two-centre molecular integrals, the two integrals (8) and (11) are known as auxiliary functions (Rosen, 1931; Roothaan, 1951b). It should be noted that Bn ðrÞ ¼ ð1Þn Bn ðrÞ;

Bn ð0Þ ¼

2 den nþ1

e ¼ even

(12)

The explicit form for the first few auxiliary functions is A0 ðrÞ ¼

expðrÞ ; r

A1 ðrÞ ¼

expðrÞ ð1 þ rÞ; r2

  6 expðrÞ r 2 r3 1þrþ þ ; A3 ðrÞ ¼ 2 6 r4

  24 expðrÞ r2 r3 r4 1þrþ þ þ A4 ðrÞ ¼ 2 6 24 r5

(13)

(14)

(15)

expðrÞ expðrÞ expðrÞ ð1  rÞ  ð1 þ rÞ ¼ ½ð1  rÞ  expð2rÞð1 þ rÞ 2 2 r r r2

(16)

2 expðrÞ B2 ðrÞ ¼ r3 B3 ðrÞ ¼

B4 ðrÞ ¼

  2 expðrÞ r2 1 þ r þ 2 r3

expðrÞ expðrÞ expðrÞ  ¼ ½1  expð2rÞ r r r

B0 ðrÞ ¼

B1 ðrÞ ¼

A2 ðrÞ ¼

6 expðrÞ r4

24 expðrÞ r5



r2 1rþ 2

 1rþ

 1rþ

r2 r3  2 6





  r2  expð2rÞ 1 þ r þ 2

(17)

  r2 r3  expð2rÞ 1 þ r þ þ 2 6

(18)

   r2 r 3 r4 r2 r3 r4  þ  expð2rÞ 1 þ r þ þ þ 2 6 24 2 6 24

(19)

18.3 One-centre integrals

793

Recurrence relations are often used in numerical calculations. We give as an example those for An(r). From the definition (8) it follows An1 ðrÞ ¼

n1 k X ðn  1Þ! r expðrÞ n k! r k¼0

(20)

Hence follows the recurrence relation An ðrÞ ¼

n! rnþ1

expðrÞ

n1 k X r n! rn þ nþ1 expðrÞ k! r n! k¼0

# " n1 k X n ðn  1Þ! r expðrÞ 1 ¼ þ expðrÞ ¼ ½nAn1 ðrÞ þ rA0 ðrÞ n k! r r r r k¼0

(21)

18.3 ONE-CENTRE INTEGRALS 18.3.1 One-electron integrals Non-orthogonality, Coulomb, and Laplacian integrals are calculated directly in spherical coordinates starting from the general definition of STO orbitals in real form, Eqn (24) of Chapter 12. We obtain the following. (1) Non-orthogonality Sn0 l0 m0 ;nl0 m

ðn þ n0 Þ!  c  ¼ hn0 l0 m0 jnlmi ¼ dll0 dmm0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 ð2nÞ!ð2n0 Þ! c

nn0 2

2ðcc0 Þ1=2 c þ c0

!nþn0 þ1 (22)

Since the spherical harmonics are orthonormal, it will be sufficient to integrate over the radial part: ZN 0 dr rnþn exp½ðc þ c0 Þr hRn0 ðrÞjRn ðrÞi ¼ Nn Nn0 0 0

¼

ð2cÞ2nþ1 ð2c0 Þ2n þ1 ð2nÞ!ð2n0 Þ!

!1=2

ðn þ n0 Þ! ðc þ c0 Þnþn þ1 0

ðn þ n0 Þ!  c  ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 ð2nÞ!ð2n0 Þ! c

nn0 2

2ðcc0 Þ1=2 c þ c0

!nþn0 þ1

(23)

as can be easily verified. In fact 0

2nþ2n0 þ2

0

¼ 2nþn þ1 ; ð22nþ1þ2n þ1 Þ1=2 ¼ 2 2  1=2 0 ¼ c2nþ1 $c02n þ1

0

0

0

0

ðcnn þnþn þ1 $c0n nþnþn þ1 Þ1=2

as it must be. In the following, we put for brevity R ¼ Rn, Y ¼ Ylm, R0 ¼ Rn0 , Y 0 ¼ Yl0 m0 .

(24)

794

CHAPTER 18 Evaluation of molecular integrals

(2) Coulomb

  0 0 0 1

 n l m r nlm ¼ R0 Y 0 r 1 RY ¼ dll0 dmm0 R0 r 1 R ZN ¼ dll0 dmm0 Nn Nn0

0

dr r nþn 1 exp½ðc þ c0 Þr ¼ dll0 dmm0 Nn Nn0

0

¼

dll0 dmm0 Nn Nn0

!

ðn þ n0 Þ! ðc þ c0 Þ

nþn0 þ1

ðn þ n0  1Þ! ðc þ c0 Þnþn

0

c þ c0 c þ c0 ¼ Sn0 l0 m0 ;nlm n þ n0 n þ n0

(25)

So, the one-centre Coulomb integral over STOs is proportional to the non-orthogonality integral S, and vanishes for orthogonal STOs. The nuclear attraction integral is obtained by multiplying Eqn (25) by minus the nuclear charge Z. (3) Laplacian

+ * 2

^  0 0 0 2

L



(26) n l m V nlm ¼ R0 Y 0 V2r  2 RY

r

but 2 L^ Ylm ¼ lðl þ 1ÞYlm ;

hYl0 m0 jYlm i ¼ dll0 dmm0 ;

V2r ¼

d2 2 d þ ; dr 2 r dr

RðrÞ ¼ r n1 expðcrÞ



dR ¼ expðcrÞ ðn  1Þr n2  cr n1 dr

(27)



d2 R ¼ expðcrÞ ðn  1Þðn  2Þrn3  2cðn  1Þrn2 þ c2 r n1 2 dr

V2r R ¼ expðcrÞ nðn  1Þr n3  2ncr n2 þ c2 r n1 so that

  V2 ðRYÞ ¼ Y V2r R  lðl þ 1Þr 2 ðRYÞ   ¼ Y expðcrÞ ½nðn  1Þ  lðl þ 1Þrn3  2ncr n2 þ c2 r n1

 R0 Y 0 V2 RY ¼ dll0 dmm0 Nn Nn0



ZN

(28)

  0 dr rn þ1 exp½  ðc þ c0 Þr ½nðn  1Þ  lðl þ 1Þr n3  2ncr n2 þ c2 rn1

0

( ¼ dll0 dmm0 Nn Nn0 ½nðn  1Þ  lðl þ 1Þ

ðn þ n0  2Þ!

 2nc

þc

2

ðn þ n0 Þ!

)

ðc þ c0 Þnþn þ1  ðn þ n0 Þ! nðn  1Þ  lðl þ 1Þ 2nc 0 2 0 2 ¼ dll0 dmm0 Nn Nn0 ðc þ c Þ  ðc þ c Þ þ c 0 n þ n0 ðc þ c0 Þnþn þ1 ðn þ n0 Þðn þ n0  1Þ   nðn  1Þ  lðl þ 1Þ 2nc 0 2 0 2 ðc þ c Þ  ðc þ c Þ þ c ¼ Sn0 l0 m0 ;nlm ðn þ n0 Þðn þ n0  1Þ n þ n0 (29) 

ðc þ c0 Þnþn 1

ðn þ n0  1Þ!

0

0

ðc þ c0 Þnþn

0

18.4 Evaluation of the electrostatic potential J1S

795

so that even the off-diagonal matrix element of the one-centre Laplacian operator over STOs is proportional to the non-orthogonality integral S. This is not true for GTOs. For the diagonal element we have n0 ¼ n; l0 ¼ l; m0 ¼ m; c0 ¼ c; S ¼ 1



 n þ 2lðl þ 1Þ 2 nlm V2 nlm ¼  c nð2n  1Þ

(30)

18.3.2 Two-electron integrals Putting jai ¼ jnlmi; we recall the two equivalent notations:

!  1 2 1 2  1 2

00 0 000 1

0 00 000 a a a a ¼ a a a a

r12 Dirac

(31)

Charge density

The charge density notation is the most used in molecular calculations. The general two-electron repulsion integral (31) is reduced to a one-electron integral by evaluating first the electrostatic potential Ja00 a000 at point r1 due to electron 2 of density ða00 ðr2 Þa000 ðr2 ÞÞ: Z ½a00 ðr2 Þa000 ðr2 Þ Ja00 a000 ðr1 Þ ¼ dr2 (32) r12 Hence 0

00 000

ðaa ja a Þ ¼

Z

dr1 Ja00 a000 ðr1 Þ½aðr1 Þa0 ðr1 Þ

(33)

where either the potential or the final integral is evaluated in spherical coordinates. We take as two simple examples the evaluation of the potential J1s due to a spherical (1s2) charge distribution, and the two-electron repulsion integral (1s2j1s2).

18.4 EVALUATION OF THE ELECTROSTATIC POTENTIAL J1S 18.4.1 Spherical coordinates With reference to Figure 18.1, [1s(r2)]2dr2 is the element of electronic charge (in atomic units) at dr2 due to electron 2 in state 1s. Then (electric potential ¼ charge/distance): ½1sðr2 Þ2 dr2 r12 is the element of electrostatic potential at space point r1 due to the electron charge at r2; Z ½1sðr2 Þ2 J1s ðr1 Þ ¼ dr2 r12

(34)

the resultant of all elementary electrostatic potential contributions at r1 due to the continuously varying charge distribution of electron 2, that is the electrostatic potential at the space point r1.

796

CHAPTER 18 Evaluation of molecular integrals

FIGURE 18.1 Infinitesimal volume elements for electrostatic potential calculation

To evaluate the electrostatic potential (34) in a general way, it is convenient to use the one-centre Neumann’s expansion (Eyring et al., 1944) for the inverse of the interelectronic distance r12: N l l X X 1 4p r< ¼ Ylm ðU1 ÞYlm ðU2 Þ lþ1 r12 2l þ 1 r> l¼0 m¼l

(35)

where r< ¼ minðr1 ; r2 Þ;

r> ¼ maxðr1 ; r2 Þ

and the Ylm are spherical harmonics in real form (tesseral harmonics), having the properties Z dUYlm ðUÞYl0 m0 ðUÞ ¼ dll0 dmm0

(36)

(37)

Here U stands for the angular variables q; 4: Taking into account integrals (6) and (7), we can easily evaluate the electrostatic potential J1s(r1) in spherical coordinates. The interval of variation of r2 must be divided into the two regions of Figure 18.2: It is convenient to choose 1s AOs separately normalized in the form  1=2 1 expðcrÞY00 Y00 ¼ pffiffiffiffiffiffi (38) 1sðrÞ ¼ 4c3 4p

FIGURE 18.2 The two regions occurring in the integration over r2

18.4 Evaluation of the electrostatic potential J1S

797

Introducing Neumann’s expansion into the expression for the potential gives Z

¼

XX m

l

" 

¼

l X X 4p r< Ylm ðU1 ÞYlm ðU2 Þ lþ1 r> m 2l þ 1 l

dr2 r22 dU2 4c3 expð2cr2 ÞðY00 Þ2

J1s ðr1 Þ ¼

4p 4c Ylm ðU1 ÞY00 ðU1 Þ 2l þ 1

Zr1

1 r1lþ1

dU2 Ylm ðU2 ÞY00 ðU2 Þ #

ZN dr2 r22

expð2cr2 Þr2l

þ

r1l

dr2 r22

expð2cr2 Þ

r1

0

4c3 $ 4p

Z

3

1

(39)

r2lþ1

  Zr1   ZN 1 2 1 2 pffiffiffiffiffiffi dr2 expð2cr2 Þr2 þ dr2 expð2cr2 Þr2 r1 4p r1

0

since only the term l ¼ m ¼ 0 survives in the expansion because of the spherical symmetry of the density 1s2. Evaluating the integrals over r2 with the aid of Eqns (6) and (7) gives J1s ðr1 Þ ¼

4c3

( " !!# 1 2! ð2cr1 Þ2 1  expð2cr1 Þ 1 þ 2cr1 þ 2! r1 ð2cÞ3

þ

expð2cr1 Þ ð2cÞ2

(40)

) ð1 þ 2cr1 Þ

¼

1 expð2cr1 Þ  ð1 þ cr1 Þ ¼ Jðr1 Þ r1 r1

showing that the potential J1s due to the spherical density 1s2 has only a radial dependence on the distance r1 of the electron from its nucleus.

18.4.2 Spheroidal coordinates We can evaluate the potential J1s in spheroidal coordinates as well. Since the distance of electron one from the nucleus is fixed during the evaluation of the potential, we may use a system of spheroidal coordinates with r replacing R, in the form (Figure 18.3) m¼

r2 þ r12 ; r

n¼

r2  r12 ; r

4

r r r2 ¼ ðm þ nÞ; r12 ¼ ðm  nÞ 2 2  r 3   dr2 ¼ m2  n2 dm dn d4 2

(41)

798

CHAPTER 18 Evaluation of molecular integrals

FIGURE 18.3 A confocal spheroidal coordinate system for the evaluation of J1s

By posing r ¼ cr, we find (r1 ¼ r): Z J1s ðr1 Þ ¼

½1sðr2 Þ2 c3 dr2 ¼ r12 p

Z

expð2cr2 Þ c3  r 3 ¼ dr2 p 2 r12

Z1

Z2p d4

1

0

ZN   exp½rðm þ nÞ dn dm m2  n2 rðm  nÞ=2 1

3 2   Z1 ZN Z1 ZN   2p r 3 2 6 7 dn expðrnÞ dm expðrmÞm þ dn expðrnÞn dm expðrmÞ5 ¼ 4 p 2 r 1

r2

¼ 2c þ ¼

2

1

1

1

r2 expðrÞ

½A1 ðrÞB0 ðrÞ þ A0 ðrÞB1 ðrÞ ¼ 2c

r2

2

expðrÞ expðrÞ ½ð1  rÞ  expð2rÞð1 þ rÞ r r2

ð1 þ rÞ

expðrÞ ½1  expð2rÞ r



c 1 expð2crÞ ½2  2 expð2rÞð1 þ rÞ ¼  ð1 þ crÞ 2r r r (42)

which is the result Eqn (40) found previously.

18.5 THE (1S2j1S2) ELECTRON REPULSION INTEGRAL 18.5.1 Same orbital exponent The two-electron integral (1s2j1s2) in charge density notation can then be written as  2 2 1s 1s ¼

ZZ dr1 dr2

½1sðr2 Þ2 ½1sðr1 Þ2 ¼ r12

Z dr1 J1s ðr1 Þ½1sðr1 Þ2

(43)

18.6 General formula for one-centre two-electron integrals

799

The integral is easily calculated using the expression just found for the potential J1s(r1). We immediately obtain   ZN  2 2  c3 1 expð2crÞ $ 4p dr r 2  ð1 þ crÞ expð2crÞ 1s 1s ¼ p r r 2 ¼ 4c3 4 " ¼

4c3

0

ZN

ZN dr expð2crÞr 

0

3   dr expð4crÞ r þ cr 2 5

(44)

0

#

1



ð2cÞ2

1 ð4cÞ2



2c

5 ¼ c 8

ð4cÞ3

18.5.2 Different orbital exponents 1

2

Write the two-electron integral as ð41 41 j42 42 Þ, where 41 ¼ 1s1 (orbital exponent c1) and 42 ¼ 1s2 (orbital exponent c2). The electrostatic potential due to electron two is now (r1 ¼ r, r1 ¼ r): Z ½4 ðr2 Þ2 1 expð2c2 rÞ ð1 þ c2 rÞ J42 42 ¼ dr2 2 ¼  (45) r12 r r giving for the two-electron integral 



421 422



Z ¼

ZN 2

drJ42 42 ðrÞ½41 ðrÞ ¼

4c31

  dr expð2c1 rÞ r  expð2c2 rÞ r þ c2 r 2

0

"

1

1

2c2

#

"

c21 þ c22 þ 2c1 c2  c21

c2

#

(46) ¼  ð2c1 þ 2c2 Þ3 c21 ðc1 þ c2 Þ2 ðc1 þ c2 Þ3 " # " # c31 c2 c1 c22 þ 2c21 c2 c21 c1 c2 2c1 þ c2 c1 c2 c1 c2 ¼ 1þ  ¼  ¼ c1 þ c 2 ðc1 þ c2 Þ2 ðc1 þ c2 Þ3 c1 þ c2 c1 þ c2 ðc1 þ c2 Þ2 ðc1 þ c2 Þ2 ¼

4c31

ð2c1 Þ2



ð2c1 þ 2c2 Þ2



c31

For c1 ¼ c2 ¼ c, we recover our previous expression (44).

18.6 GENERAL FORMULA FOR ONE-CENTRE TWO-ELECTRON INTEGRALS The product of two STOs onto the same centre originates a density, say ½aðrÞa0 ðrÞ, that can be reduced to a finite linear combination of elementary (or basic) charge distributions DNLM (r) (Roothaan, 1951b): rffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2L þ 1 DNLM ðrÞ ¼ (47) RNL ðrÞYLM ðq; 4Þ 4p RNL ðrÞ ¼

2L ð2zÞNþ2 N1 r expð2zrÞ ðN þ L þ 1Þ!

(48)

800

CHAPTER 18 Evaluation of molecular integrals

where z¼

c a þ c a0 2

(49)

N ¼ na þ na0  1 jla  la0 j  L  la þ la0

(50)

ðL þ la þ la0 ¼ evenÞ

(51)

DNLM (r) behaves as a single STO of quantum numbers (NLM) and orbital exponent 2z acting at large distances as a multipole of order 2L and magnitude zL. The complete expression of ½aðrÞa0 ðrÞ in terms of the DNLMs involves the Clebsch–Gordan vector coupling coefficients of Chapter 10 (Brink and Satchler, 1993) arising from the coupling of the angular momenta of the individual AOs (Guidotti et al., 1962; Figari et al., 1990). A compact and elegant general formula for the two-electron integral between any two of such basic charge distributions was given by Gianinetti et al. (1959) as 1 2 ðz z ÞLþ1 22Lþ1 ðDNLM jDN 0 L0 M0Þ ¼ dLL0 dMM 0 1 2 NþN 0 þ1 ðz1 þ z2 Þ " ! 0 2L NL1 X N þ N 0 þ 1 z1  ðN  LÞ!zNþN 2  ðN þ L þ 1Þ! z2 k k¼0 0

2L ðN 0  LÞ!zNþN 1 þ ðN 0 þ L þ 1Þ!

! N þ N 0 þ 1 z  2 z 1 k

N 0X L1 k¼0 0

ðN þ N 0 þ 1Þ!zNL z2N L 1 þ ðN þ L þ 1Þ!ðN 0 þ L þ 1Þ!

(52)

#

The two-electron integrals ðaa0 ja00 a000 Þ are then easily obtained once the decomposition of the densities ½aðr1 Þa0 ðr1 Þ and ½a00 ðr2 Þa000 ðr2 Þ in terms of the DNLMs is known. Gianinetti et al. (1959) gave tables containing the explicit coefficients for all expansions of s, p, d Slater’s AOs in terms of such fundamental charge distributions.

18.7 TWO-CENTRE INTEGRALS OVER 1S STOS These are the integrals occurring in the elementary MO and HL theories for the H2 molecule. They can be evaluated using the system of spheroidal coordinates described in Chapter 2 (Figure 18.4) and the auxiliary functions An and Bn of Section 18.2. For the sake of simplicity, we shall take identical orbital exponents (c0 ¼ c) onto the two centres. We have rA þ rB rA  rB ; n¼ ; 4 m¼ R R  3  R R R  2 rA ¼ ðm þ nÞ; rB ¼ ðm  nÞ; dr ¼ m  n2 dm dn d4 (53) 2 2 2

18.7 Two-centre integrals over 1S STOS

801

FIGURE 18.4 The system of confocal spheroidal coordinates for the evaluation of two-centre integrals

18.7.1 One-electron integrals (1) Overlap Sba ¼ hbjai ¼ ðabj1Þ ¼

c3 p

Z dr exp½ cðrA þ rB Þr ¼ cR

  Z2p Z1 ZN   c3 R 3 d4 dn dm m2  n2 expðrmÞ ¼ p 2 1

0

1

2

3 ZN ZN Z1 r3 Z1 4 dn dm expðrmÞm2  dnn2 dm expðrmÞ5 ¼2 2 1

1

1

(54)

1

 r3  2 4r3 2A2 ðrÞ  A0 ðrÞ ¼ ¼2 ð3A2  A0 Þ 2 3 3 2 ¼ (2) Exchange 

   4r3 expðrÞ  r2 2 2  r 6 þ 6r þ 3r ¼ expðrÞ 1 þ r þ 3 3 2 r3

 ab r 1 ¼ B

¼

Z

aðrÞbðrÞ c3 ¼ dr rB p

Z dr

1 exp½  cðrA þ rB Þ rB

  Z1 ZN c3 R 3 m2  n2 expðrmÞ 2p dn dm p 2 Rðm  nÞ=2 1

1

2

3 ZN Z1 ZN r2 Z1 4 ¼ 2c dn dm expðrmÞm þ dnn dm expðrmÞ5 2 1

1

r2 ½2A1 ðrÞ ¼ c expðrÞð1 þ rÞ ¼ 2c 2

1

1

(55)

802

CHAPTER 18 Evaluation of molecular integrals

(3) Coulomb This integral was already calculated in Section 18.4, Eqn (42), if we take r2 ¼ rA, r12 ¼ rB, r1 ¼ R. The result is  2 1  c (56) a rB ¼ ½1  expð2rÞð1 þ rÞ r (4) Laplacian



V2

 ba

   ¼ b V2 a ¼ ab V2

 3 1=2 i h r c exp  ðm þ nÞ ; p 2

a¼

V2 ¼

b¼

(57)

 3 1=2 i h r c exp  ðm  nÞ p 2

(58)

  c2 m2  n 2 v2 2 2 V þ V þ n r2 ðm2  n2 Þ=4 m ðm2  1Þð1  n2 Þ v42

  v2 v V2m ¼ m2  1 þ 2m ; 2 vm vm

(59)

  v2 v  2n V2n ¼ 1  n2 2 vn vn

(60)

where the last factor involving v2/v42 can be omitted for spherical AOs. Evaluating the derivatives gives    r   r2 r2 r  m2  rm  exp  m (61) V2m exp  m ¼ 4 4 2 2    r   r2 r2 r  exp  n V2n exp  n ¼  n2 þ rn þ 4 4 2 2

(62)

h i 2 exp  r ðm þ nÞ   c  r r2  2 2 2 2 V exp  ðm þ nÞ ¼ m  n  rðm  nÞ r2 ðm2  n2 Þ=4 4 2 h

i

(63)

Hence       Z1 ZN 2  c3 R 3  c2 m2  n2 r2  2 2



m 2p dn dm 2 2  n bV a ¼  rðm  nÞ expðrmÞ r ðm  n2 Þ=4 4 p 2 1

1

8 2 9 3 > > ZN Z1 Z1 ZN ZN

> :4 ; 1

1

1

1

 2  

r 2 1 ¼ c2 r 2A2  A0  2rA1 ¼ c2 r r2 ð3A2  A0 Þ  12rA1 4 3 6

1

1

(64)

18.7 Two-centre integrals over 1S STOS

803

Evaluating the terms in brackets gives r2 ð3A2  A0 Þ  12rA1 ¼ r2

 expðrÞ  expðrÞ ð1 þ rÞ 6 þ 6r þ 3r2  r2  12r 3 r r2 2

 expðrÞ r ¼6 1  r þ 3 r

(65)

so that the integral results in

  2  r2 2



b V a ¼ c expðrÞ 1  r þ 3

(66)

which coincides with the result given by Roothaan (1951b).

18.7.2 Two-electron integrals These integrals are evaluated in spheroidals by first finding the potential due to the charge distribution of one electron, say electron 2. (1) Coulomb  2 2 a b ¼

Z dr1 JB ðr1 Þa2 ðr1 Þ

(67)

Z

b2 ðr2 Þ 1 ¼ ½1  ð1 þ crB1 Þexpð2crB1 Þ r12 rB1  o n  c r ¼ 1  1 þ ðm  nÞ exp½rðm  nÞ rðm  nÞ=2 2

JB ðr1 Þ ¼

dr2

(68)

so that     Z1 ZN  o   2 2  c3 R 3 c m 2  n2 n r

a b ¼ 2p dn dm 1  1 þ ðm  nÞ exp½rðm  nÞ exp½rðm þ nÞ rðm  nÞ=2 p 2 2 1

1

n r2 Z1 ZN o r dn dmðm þ nÞ exp½rðm þ nÞ  expð2rmÞ  ðm  nÞexpð2rmÞ ¼ 2c 2 2 1

1

  r2  1 ¼ 2c ½A1 ðrÞB0 ðrÞ þ A0 ðrÞB1 ðrÞ  2A1 ð2rÞ þ rA2 ð2rÞ  rA0 ð2rÞ 2 3 (69)

804

CHAPTER 18 Evaluation of molecular integrals

Evaluating the terms in brackets gives A1 ðrÞB0 ðrÞ þ A0 ðrÞB1 ðrÞ expðrÞ expðrÞ expðrÞ expðrÞ ð1 þ rÞ ½ð1  rÞ  expð2rÞð1 þ rÞ ½1  expð2rÞ þ 2 r r r r2 20 1 0 13 1þr 1 4@ 1 þ r A A5 ¼ 2 ½1  expð2rÞð1 þ rÞ  expð2rÞ@ ¼ 3 r r3 1r 1þr

¼

2A1 ð2rÞ þ rA2 ð2rÞ 

1 1 rA0 ð2rÞ ¼ ½6A1 ð2rÞ þ 3rA2 ð2rÞ  rA0 ð2rÞ 3 3

# "  1 expð2rÞ 2r 2 expð2rÞ  expð2rÞ 2 ð1 þ 2rÞ þ 3 ð2rÞ ¼ 1 þ 2r þ 2r  r 6 3 2 ð2rÞ2 ð2rÞ3 ð2rÞ2 0

6 þ 12r

1

 C 3 expð2rÞ  1 expð2rÞ 2 B 4 2 B C 2 1 þ 2r þ r ¼ 6r B þ3 þ 6r þ 6r C ¼ @ A 4 3 ð2rÞ2 r2 9 2r2 so that   2 expð2rÞ 11 3 2 1 3 2þ rþ r þ r f/g ¼ 3  r r3 4 2 3 finally giving     2 2 c 11 3 1 a b ¼ 1  expð2rÞ 1 þ r þ r2 þ r3 r 8 4 6

(70)

For c ¼ 1, r ¼ R:    2 2  1 expð2RÞ 11 3 2 1 3

a b ¼  1þ Rþ R þ R R R 8 4 6

(71)

the first term being the Coulomb part of the integral, the second the charge-overlap part decreasing exponentially.

18.7 Two-centre integrals over 1S STOS

805

(2) Hybrid (or ionic) 

 ab b2 ¼

Z

    Z1 ZN c m2  n2 c3 R 3 dr1 JB ðr1 Þ½aðr1 Þbðr1 Þ ¼ 2p dn dm p 2 rðm  nÞ=2 1

1

 o n  r  1  1 þ ðm  nÞ exp½rðm  nÞ expðrmÞ 2 r2 Z1 ZN dn dmðm þ nÞ½expðrmÞ  expð2rmÞexpðrnÞ ¼ 2c 2 1

1

i r2 n r  ðm  nÞexpð2rmÞexpðrnÞ ¼ 2c ½2A1 ðrÞ  A1 ð2rÞB0 ðrÞ  A0 ð2rÞB1 ðrÞ 2 2 o r  ½A2 ð2rÞB0 ðrÞ  A0 ð2rÞB2 ðrÞ 2 (72) Taking into account the parity Eqn (12) of the auxiliary functions Bn(r), evaluating the terms in brackets gives 2

expðrÞ expð2rÞ expðrÞ ð1 þ rÞ  ð1 þ 2rÞ ½1  expð2rÞ 2 r2 r ð2rÞ þ

expð2rÞ expðrÞ ½ð1  rÞ  expð2rÞð1 þ rÞ 2r r2

 expðrÞ r 2 expð2rÞ  ½1  expð2rÞ 1 þ 2r þ 2r2 3 2 r ð2rÞ     r expð2rÞ 2 expðrÞ r2 r2 1rþ  expð2rÞ 1 þ r þ þ 2 2 2 2r r3     expðrÞ 5 1 expð3rÞ 5 1 2 ¼ þ  þ r þ 2r  r r3 8 4 r3 8 4 

so that we finally obtain       2 c 5 1 5 1 þ r þ r2  expð3rÞ þ r ab b ¼ expðrÞ r 16 8 16 8 For c ¼ 1, r ¼ R:      2  expðRÞ 5 1 expð3RÞ 5 1 2

þ RþR  þ R ab b ¼ R 16 8 R 16 8 The hybrid integral over 1s STOs is a pure charge-overlap term.

(73)

(74)

806

CHAPTER 18 Evaluation of molecular integrals

(3) Exchange This rather difficult integral can be evaluated in two steps: (1) by finding first the two-centre exchange potential at r1 due to the density [a(r2)b(r2)], which requires calculation of three characteristic parametric integrals depending on the variable m (Tauber, 1958), (2) then followed by a repeated partial integration over this variable to get the final result. Therefore Z 1 2 (75) ab ab ¼ dr1 Kab ðr1 Þ½aðr1 Þbðr1 Þ where

Z Kab ðr1 Þ ¼

dr2

aðr2 Þbðr2 Þ r12

(76)

is the two-centre exchange potential at r1. Expanding 1=r12 in spheroidal coordinates (real form) according to Neumann (1887) gives N X l 1 2X m m m ¼ Dlm Qm l ðm> ÞPl ðm< ÞPl ðn1 ÞPl ðn2 Þcos mð41  42 Þ r12 R l¼0 m¼0   ðl  mÞ! 2 m Dlm ¼ ð1Þ 2ð2l þ 1Þ m > 0; Dl0 ¼ 2l þ 1 m ¼ 0 ðl þ mÞ!

(77)

(78)

m where Pm l ;Ql are associated Legendre functions of the first and the second kind (Hobson, 1965), respectively. For spherical 1s AOs with equal orbital exponent, only the terms l ¼ 0, 2 survive upon integration over 42 (that gives 2p for m ¼ 0, zero otherwise), so that after integration over n2 the series truncates to 2   ZN   4 r 36 2 2 Kab ðr1 Þ ¼ 4 dm2 expðrm2 Þ 2m2  Q0 ðm> Þ R 2 3 1



4 P2 ðn1 Þ 3

ZN

3

(79)

7 dm2 expðrm2 ÞQ2 ðm> ÞP2 ðm< Þ5

1

where (Chapter 4) Q0 ðxÞ ¼

1 xþ1 ln ; 2 x1

Q2 ðxÞ ¼ P2 ðxÞQ0 ðxÞ 

P1 ðxÞ ¼ x;

P2 ðxÞ ¼

3x2  1 2

3 P1 ðxÞ 2

(80) (81)

Splitting the integration range over m2 in the two regions of Figure 18.5, the exchange potential (which has cylindrical symmetry) can be written as (m1 ¼ m, n1 ¼ n) 16 r3 Kab ðm; nÞ ¼ ½AðmÞ  P2 ðnÞBðmÞ (82) 3R 2

18.7 Two-centre integrals over 1S STOS

807

FIGURE 18.5 The two regions occurring in the integration over m2

where AðmÞ ¼ Q0 ðmÞI2 ðmÞ þ K2 ðmÞ BðmÞ ¼ Q2 ðmÞI2 ðmÞ þ P2 ðmÞK2 ðmÞ 

3 J1 ðmÞP2 ðmÞ 2

(83) (84)

having defined (Tauber, 1958) Zm In ðmÞ ¼

dx expðrxÞPn ðxÞ

(85)

dx expðrxÞPn ðxÞ

(86)

dx expðrxÞPn ðxÞQ0 ðxÞ

(87)

1

ZN Jn ðxÞ ¼ m

ZN Kn ðxÞ ¼ m

Integration by parts gives for the integrals of interest I2 ðmÞ ¼ 

  expðrmÞ 3 3 3m2  1 3 þ m þ þ 3 S r r2 r 2 r

(88)

  expðrmÞ 1 þm r r

(89)

3 3 3 expðrmÞ  3 S0 Ei½rðm þ 1Þ þ 3 SEi½rðm  1Þ 2r2 2r 2r   expðrmÞ 3 3 3m2  1 Q0 ðmÞ 2 þ m þ þ r r r 2

(90)

J1 ðmÞ ¼ K2 ðmÞ ¼ 

where 

 r2 S ¼ SðrÞ ¼ expðrÞ 1 þ r þ ; 3

  r2 S ¼ SðrÞ ¼ expðrÞ 1  r þ 3 0

(91)

808

CHAPTER 18 Evaluation of molecular integrals

and

ZN EiðxÞ ¼ 

dt

expðtÞ ¼ E1 ðxÞ ; t

dEiðxÞ expðxÞ ¼ dx x

(92)

x

is the exponential integral function (Abramowitz and Stegun, 1965) discussed in Chapter 4, and g ¼ lim ½ ln x þ EiðxÞ ¼ 0:577 215 665/ x/0

is the Euler’s constant. Introducing the expression for the exchange potential into the exchange integral (75) we obtain, R1 upon integration over angle 4 and noting that 1 dnP2 ðnÞ ¼ 0   Z1 ZN   c3 R 3 16 r3 2p dn dm m2  n2 ðabjabÞ ¼ p 2 3R 2 1

1

2 r6 6  ½AðmÞ  P2 ðnÞBðmÞexpðrmÞ ¼ 4 6R ZN

Z1 

dnn 1

ZN

1

1

ZN dnP2 ðnÞ

1

1

dm expðrmÞm2 AðmÞ

dn

Z1 dm expðrmÞAðmÞ 

2

Z1

dm expðrmÞm2 BðmÞ 1

8 >   Z1 ZN < ZN 6 1 7 r 2 2 2 dm expðrmÞ m  AðmÞ þ dnP2 ðnÞn dm expðrmÞBðmÞ5 ¼ 6R > 3 : 3

1

1



Z1 þ

dnP2 ðnÞ 1

2 ¼

r6 3R

6 4

1

2 1 P2 ðnÞ þ P0 ðnÞ 3 3

dm expðrmÞBðmÞ 1

ZN dm expðrmÞ

2 1 2 P2 ðmÞAðmÞ þ $ 3 3 5

1

2 r6 ¼ 45 R

ZN

> ; 3

7 dm expðrmÞBðmÞ5

1

ZN dm expðrmÞ½5P2 ðmÞAðmÞ þ BðmÞ 1

2 r6 ¼ 45 R

ZN 1

(93)

9 > =

 ZN

  3 dm expðrmÞ 6P2 ðK2 þ I2 Q0 Þ  ðP1 I2 þ J1 P2 Þ 2

18.7 Two-centre integrals over 1S STOS

809

Repeated integration by parts then gives ZN dm expðrmÞðP1 I2 þ J1 P2 Þ ¼ 1

  expð2rÞ 15 15 2 3 þ r þ r þ 3r 8 4 r5

ZN dm expðrmÞP2 ðK2 þ I2 Q0 Þ ¼ 1

þ

  expð2rÞ 45 27 3 2  r  r 16 8 2 r5

9 2 S ðg þ ln rÞ þ S02 Eið4rÞ  2SS0 Eið2rÞ 6 2r



  3 factor  2

factor þ6



(94)

  logarithmic part

Collecting all terms altogether we finally obtain    c 25 23 1 expð2rÞ  r  3r2  r3 ðabjabÞ ¼ 5 8 4 3

6 þ S2 ðg þ ln rÞ þ S02 Eið4rÞ  2SS0 Eið2rÞ r



(95)

which, for c ¼ 1, r ¼ R, coincides with the result first given by Sugiura (1927). An alternative, more general, way of evaluating the exchange integral was suggested by Rosen (1931) in terms of the generalized double integral: ZN

ZN Hl ðm; n; rÞ ¼ Hl ðn; m; rÞ ¼

n dm2 exp½ rðm1 þ m2 Þmm 1 m2 Ql ðm> ÞPl ðm< Þ

dm1 1

(96)

1

This method can be easily extended to calculations involving James–Coolidge or Ko1os– Wolniewicz wavefunctions and is suitable for implementation on electronic computers. A further generalization is due to Ruedenberg (1951), and will be recalled later on in Section 18.8.

18.7.3 Limiting values of two-centre integrals All two-centre integrals for H2 we saw so far go to zero as r / N: In the limit of the united atom (r / 0), we shall give for completeness the coefficients of the non-vanishing terms going to zero as r2. We shall use the small-r expansions for exponential and exponential integral functions (Abramowitz and Stegun, 1965): 1 1 1 expðxÞ z 1  x þ x2  x3 þ x4 2 6 24   1 1 1 EiðxÞ z ðg þ ln xÞ þ x þ x2  x3 þ x4 4 18 96

(97) (98)

To illustrate how the calculation must be organized, we shall give below the example of the more difficult two-electron two-centre exchange integral ðabjabÞ; while the similar treatment for the simpler two-electron two-centre Coulomb integral ða2 jb2 Þ can be found in Chapter 13 of Magnasco (2007).

810

CHAPTER 18 Evaluation of molecular integrals

In the following tables, the upper line is the expansion of expð2rÞ in powers of r up to r4, the second line the polynomial part of the expression. For the analytic part of the two-electron two-centre exchange integral (95), we have the multiplication table: 1

2r

þ 2r2

4  r3 3

25 8

23  r 4

3r2

1  r3 3

25 8

25  r 4

þ

25 2 r 4

25  r3 6

þ

25 4 r 12

23  r 4

þ

23 2 r 2

23  r3 2

þ

23 4 r 3

þ6r3

6r4

1  r3 3

2 þ r4 3

12r3

þ

3r2

25 8

12r

þ

59 2 r 4

2 þ r4 3

53 4 r 12

For the logarithmic part (non-analytic at R ¼ 0), we now expand each individual term inside the square bracket of Eqn (95). We give below the expansion for S2, which is seen to be equal to the expansions for S02 and SS0 :

1

2r

þ2r2

4  r3 3

2 þ r4 3

1

þ2r

5 þ r2 3

2 þ r3 3

1 þ r4 9

1

2r

þ2r2

4  r3 3

þ2r

4r2

þ 4 r3

5 þ r2 3

10  r3 3 2 3 þ r 3

2 þ r4 3 8 4  r 3 10 þ r4 3 4 4  r 3 1 þ r4 9

1

•

1  r2 3

•

1 þ r4 9

18.7 Two-centre integrals over 1S STOS

Next, using expansion (98) for the exponential integral functions   4 1 Eið2rÞ z ðg þ ln 2rÞ þ 2r þ r2  r3 þ r4 9 6   32 3 8 4 2 Eið4rÞ z ðg þ ln 4rÞ þ 4r þ 4r  r þ r 9 3

811

(99) (100)

we obtain

  1 2 1 4 S ðg þ ln rÞ z ðg þ ln rÞ 1  r þ r 3 9     1 1 20 4 S02 Eið4rÞ z ðg þ ln 4rÞ 1  r2 þ r4 þ  4r þ 4r2  r3 þ r4 3 9 9 3     2 2 2 4 4 3 2 4 0 2 2SS Eið2rÞ z ðg þ ln 2rÞ  2 þ r  r þ 4r  2r  r þ r 3 9 9 3 2

(101) (102) (103)

By adding all such terms we see that the logarithmic parts cancel altogether: S2 ðg þ ln rÞ þ S02 Eið4rÞ  2SS0 Eið2rÞ z ðg þ ln rÞ þ ðg þ ln r þ ln 4Þ  2ðg þ ln r þ ln 2Þ     8 4 þ 2r2  r3 þ 2r4 ¼ 2r2 1  r þ r2 3 3 so that we obtain for the terms in brackets in Eqn (95)     25 59 53 4 25 5  12r þ r2  10r3 þ r4 þ 12r 1  r þ r2 ¼  r2 þ 2r3 f/g z 8 4 12 3 8 4

(104)

(105)

finally giving as limiting value for the two-electron exchange integral  ðabjabÞ z c

5 1 2  r 8 4

 (106)

For completeness, we collect below the limiting values for r / 0 resulting for all two-centre integrals involving 1s AOs to order r2:   1 2 

1  1 2 ab rB z c 1  r SðrÞ ¼ SðrÞ z 1  r ; 6 2         2 2 5 2 2 2 1 2



ab V z c 1  r a rB z c 1  r ; 3 6 (107)         5 1 5 7 2 2 2 2 2 a ab z c  r a b z c  r ; 8 12 8 48     5 1 abjab z c  r2 8 4

812

CHAPTER 18 Evaluation of molecular integrals

18.8 ON THE GENERAL FORMULAE FOR TWO-CENTRE INTEGRALS General formulae for one- and two-electron two-centre integrals over STOs were obtained by us by generalizing the techniques described in the previous sections. All expressions were tested (1) with the similar results by Roothaan (1951b), and (2) by numerical computations based on use of the Mathematica software (Wolfram, 2007) or by the QCPE program DERIC.1 Figure 18.6 gives the coordinate systems used by us (Roothaan) and by DERIC (James–Coolidge). Roothaan uses two Cartesian coordinate systems, one right-handed centred at A, the other left-handed centred at B, while DERIC uses a unique reference system centred at A (James and Coolidge, 1933). Great care is needed in choosing the correct sign resulting for integrals involving 2ps functions when comparing results from different sources.

18.8.1 Spheroidal coordinates Compact analytical formulae for one- and two-electron two-centre integrals over complex STOs were recently derived by our group (Casanova, 1997; Magnasco et al., 1998) after expressing in spheroidals the charge distributions on the two centres (see Wahl et al., 1964). One-electron integrals can all be expressed in terms of the auxiliary functions Tjm and Gjm, while Coulomb, hybrid and exchange

FIGURE 18.6 Reference systems for the calculation of two-centre molecular integrals

1

DERIC (Diatomic Electron Repulsion Integral Code), N. 252 of the Quantum Chemistry Program Exchange (QCPE) of the Indiana University (Hagstrom, 1974). The program evaluates one-electron integrals as well.

18.8 On the general formulae for two-centre integrals

813

two-electron integrals are described by a unified formula containing the two functions Bm lj and Hlmpq introduced by Ruedenberg (1951). The accuracy in the value of these integrals depends on the accuracy with which such auxiliary functions are calculated. The generalized auxiliary functions occurring in these calculations were fully defined in Section 4.7.6 through Eqn (203) for Tjm(r), (204) for Gjm(r), (205) for Bm lj ðrÞ; and (206) for Hlmpq(r1,r2). It was seen there that Tjm(r) and Gjm(r) can be written as finite sums involving An and Bn functions, and that stability in the calculation of Gjm, Bm lj ðrÞ and Hlmpq with high numerical accuracy (up to 14–15 figures) can be obtained by use of a series expansion of the exponential in (204), (205), and (206). The functions Hlmpq represent the main time-consuming factor in the calculation of the two-centre integrals, owing to the considerable difficulty of their accurate evaluation. We shall not give here the final formulae for the two-centre integrals, full details being found in the original paper.2 We simply outline here that exchange (resonance) and Laplacian one-electron integrals are all expressed in terms of suitably scaled overlap integrals, while using the complex form of the Neumann’s expansion (1887) for 1/r12: N X l 1 4X ðl  mÞ! m Q ðm ÞPm ðm ÞPm ðn1 ÞPm ¼ ð1Þm l ðn2 Þexp½imð41  42 Þ r12 R l¼0 m¼l ðl þ mÞ! l > l < l

(108)

a unified formula was derived for the general two-electron integral. Similar formulae for real STOs were derived by Yasui and Saika (1982) and Guseinov and Yassen (1995). As a simple example, we give in Problem 18.3 the calculation of the overlap integral (54) from the general formula.

18.8.2 Spherical coordinates If an STO on centre B is expressed in spherical coordinates (r, q, 4) referred to centre A (taken as origin of the coordinate system), one-centre integration over these variables would be straightforward either for one-electron or two-electron integrals. This can be done by an exact translation of the regular solid harmonic part of the orbital (Barnett and Coulson, 1951; Lo¨wdin, 1956) followed by the series expansion of the residual spherical part in powers of the radial variable. Using complex STOs with their angular part expressed in terms of normalized Racah spherical harmonics with Condon–Shortley phase, this method was successfully tested in high-accuracy calculations of overlap (Rapallo, 1997; Magnasco et al., 1999) and two-centre two-electron molecular integrals (Magnasco and Rapallo, 2000). Good rate of convergence in the expansion and great numerical stability under wide changes in the molecular parameters (orbital exponents and internuclear distances) were obtained in both cases. The calculation of Coulomb and hybrid integrals is carried out by means of suitable one-centre one-electron potentials, while the exchange integral requires translation of two spherical residues, related to two displaced orbitals referred to the different electrons, and use of two series expansions. At variance with what we did for H2 following Tauber, one-electron potentials were not used in this last case. Angular coefficients, unique to all two-centre two-electron integrals, arise from integration over angular variables, and are expressed in terms of Gaunt coefficients related in turn to Clebsch–Gordan coupling coefficients (Section 10.4; Brink and Satchler, 1993). They can be stored and re-used during the integral evaluation. 2

We regret that the CPL (1998) Letter has plenty of typographical misprints.

814

CHAPTER 18 Evaluation of molecular integrals

Radial coefficients arise from integration over the radial variable, and are particularly involved for the exchange integral. They need calculation of further auxiliary functions and of En, the generalized exponential integral function of order n (Abramowitz and Stegun, 1965): ZN En ðrÞ ¼

dx expðrxÞxn

(109)

1

with n a non-negative integer and Re(r) > 0. High accuracy (10–12 digits) in the final numerical results can be achieved through multiple precision arithmetic calculations using recurrence relations and accurate Gaussian integration techniques (Ralston, 1965) to get reliable values for the starting terms of the recursion. This spherical approach seems quite promising for its possible extension to the calculation of multicentre integrals. An outlook to an alternative way involving spheroidal coordinates is presented in the next section for 1s STOs.

18.9 A SHORT NOTE ON MULTICENTRE INTEGRALS Multicentre integrals over STOs are very difficult to evaluate for more than two centres. They are the three-centre one-electron ðabjrC1 Þ Coulomb integral, the three-centre two-electron Coulomb (bcja2) and exchange (abjac) integrals, and the four-centre two-electron (abjcd) integral. They are still today the bottleneck of any ab initio calculation in terms of accurate functions showing correct cusp behaviour and exponential decay. Rather than giving a general survey of the subject, that in recent years enjoyed increased popularity among quantum chemists, we shall treat briefly here in some detail the evaluation of the three-centre one-electron integral ðabjrC1 Þ and of the four-centre two-electron integral (abjcd) when a, b, c, d are 1s STOs centred at different nuclei in the molecule. These integrals were evaluated in the early 1960s by the pioneering work of the author’s group (Magnasco and Dellepiane, 1963, 1964; Musso and Magnasco, 1971) using three-dimensional numerical integration techniques in spheroidal coordinates following previous work by Magnusson and Zauli (1961). This avoids any convergence problem in the series expansions over the radial variables which are typical of all approaches involving one-centre expansions in spherical coordinates.

18.9.1 Three-centre one-electron integral over 1s STOs With reference to the right-handed Cartesian system centred at the midpoint of AB (Figure 18.7), centre C is always chosen in the zx-plane, with its position specified by the two coordinates XC, ZC. In the system of confocal spheroidal coordinates m, n, 4 with foci in A, B, the three-centre oneelectron integral for two 1s STOs with different orbital exponents ca and cb can be written as     ðc c Þ3=2 R2 ZN Z1   R R a b 2 2 1

ab rC ¼ dm dn m  n exp  ðca þ cb Þm  ðca  cb Þn p 2 2 2



1

1

þ ZC2

1þ



Z2p d4

XC2

m2

þ n2

 2ZC mn  2XC



m2

1

1=2 

1=2 1  n2

1=2 cos 4

0

(110)

18.9 A short note on multicentre integrals

815

FIGURE 18.7 Coordinate system for the evaluation of the three-centre one-electron integral

Since the integration variables are not separable, a three-dimensional numerical integration by the Gauss–Legendre method (Kopal, 1961) was used to get the numerical value of the integral by conveniently choosing the integration points in those regions where the integrand is larger. This allows for a reduction in the number of integration points. Since three-dimensional integration is today standard with Mathematica software (Wolfram, 2007), this technique seems quite interesting even today.

18.9.2 Four-centre two-electron integral over 1s STOs With reference to the right-handed Cartesian system centred at the midpoint of AB (Figure 18.8), centre C was chosen in the zx-plane, the geometry of the system being completely specified by giving (in atomic units) the distance R between centres A and B, and the coordinates (units of R/2) XC, ZC of centre C, and XD,YD, ZD of centre D. The four-centre two-electron integral between 1s STOs is then given by Z ðabjcdÞ ¼ dr2 Kab ðr2 Þ½cðr2 Þdðr2 Þ (111) where Kab(r2) is the two-centre potential at r2 Z Kab ðr2 Þ ¼

dr1

aðr1 Þbðr1 Þ r12

(112)

816

CHAPTER 18 Evaluation of molecular integrals

FIGURE 18.8 Coordinate system for the evaluation of the four-centre two-electron integral

When the orbital exponents on centres A and B are the same (cb ¼ ca), this two-centre potential was already calculated in the case of the two-centre exchange integral ðabjabÞ using Tauber’s method in Eqns (79)–(90) of Section 18.7.2. We thereby obtain for the four-centre integral ðabjcdÞ ¼ ðcc cd Þ

3=2 R

3

ZN

Z1 dm

8p 1

  dn m2  n2 Kab ðm; nÞ

1

8 Z2p < R h d4 exp  cc XC2 þ ZC2  1 þ m2 þ n2  2ZC mn  : 2 0

i1=2  1=2 1=2  1  n2 cos 4  2XC m2  1 

R h 2 cd XD þ YD2 þ ZD2  1 þ m2 þ n2  2ZD mn 2 

 2 m2  1

1=2 

1n

 2 1=2

ðXD cos 4 þ YD sin 4Þ

9 i1=2 = ;

(113)

This integral is again evaluated using three-dimensional Gauss–Legendre numerical integration, and is obviously more time consuming than integral (110). When the orbital exponents on A and B are different (cb sca ), Tauber’s method originates for Kab an infinite series expansion in terms of associated Legendre functions of the first and second kinds (Hobson, 1965) multiplied by suitable coefficients. In this last case, the four-centre integral was fully

18.10 Molecular integrals over GTOS

817

evaluated by Musso and Magnasco (1971) using appropriate recursion formulae for the auxiliary functions. The evaluation of the integral implies now a four-dimensional numerical integration (a three-dimensional integration for each term of the series). The convergence of the series was found to be satisfactory as far as accuracy and computing time were concerned. Examples and further details can be found in the original paper.

18.10 MOLECULAR INTEGRALS OVER GTOS Even if their decay with the radial distance r is completely wrong (Chapter 13), nonetheless GTOs are still largely used today in molecular calculations (see Chapter 14). The reason is that it was early discovered (Shavitt and Karplus, 1962) that multicentre molecular integrals over GTOs are much more easily tractable than their STO counterpart. The unique feature of GTOs in the calculation of manycentre molecular integrals is their dependence on the square of the argument rather than on the argument itself, which avoids the appearance of square roots when the argument implies the distance between two particles. Since we already considered in Chapter 12 a few simple one-centre one-electron integrals over GTOs, we shall limit ourselves here to consideration of the main properties of Gaussian functions then giving a few examples of applications of the Gaussian transform techniques in the evaluation of multicentre molecular integrals (Shavitt, 1963; Cook, 1974; Saunders, 1975, 1982) mostly following Shavitt’s work.

18.10.1 Some properties of Gaussian functions The three-dimensional Gaussian functions are defined in un-normalized form as   2 Gi ðrA Þ ¼ Gðci ; rA Þ ¼ exp ci rA

(114)

where ci is a positive orbital exponent (non-linear variational parameter) and rA is the distance of point A from a given reference system. The fixed point A of Cartesian coordinates Ax, Ay, Az will be referred to as the centre of the Gaussian. Gi(rA) can be written as a function of the Cartesian coordinates of the point P(x, y, z) as   h i  (115) Gi ðrA Þ ¼ exp ci x2A þ y2A þ z2A where xA ¼ x  Ax ;

yA ¼ y  A y ;

zA ¼ z  Az

(116)

In some applications both end points of vector r are variable, and if the points are denoted by 1 and 2, the Gaussian is written G(r12) which in Cartesian coordinates means  h i (117) Gi ðr12 Þ ¼ exp ci ðx1  x2 Þ2 þ ðy1  y2 Þ2 þ ðz1  z2 Þ2 We note that both Eqns (115) and (117) can be factorized into three simple one-dimensional Gaussians, one in each of x, y, and z.

818

CHAPTER 18 Evaluation of molecular integrals

Finally, we speak of range of a Gaussian as the maximumpdistance r for which G(c,r) cannot be ffiffiffi considered negligible. The range is inversely proportional to c: We then have the following properties for the GTOs. (1) Gaussians with fixed centres The product of two Gaussians having different centres A and B is itself a Gaussian whose centre lies somewhere on the segment AB:

where

Cx ¼

Gi ðrA ÞGj ðrB Þ ¼ KGk ðrC Þ

(118)

  ci cj 2 K ¼ exp  AB ci þ cj

(119)

ck ¼ ci þ cj

(120)

ci Ax þ cj Bx ; ci þ cj

Cy ¼

ci Ay þ cj By ; ci þ cj

Cz ¼

ci Az þ cj Bz ci þ cj

(121)

(2) Gaussians with variable centres Let Pijk be a product of Gaussians: Pijk ðr1 ; r2 Þ ¼ Gi ðr1 ÞGj ðr2 ÞGk ðr12 Þ

(122)

where r1 and r2 are the distances from the origin. Using Eqn (117) it is possible to factor out the three components involving x, y and z: Pijk ðr1 ; r2 Þ ¼ Pijk ðx1 ; x2 ÞPijk ðy1 ; y2 ÞPijk ðz1 ; z2 Þ where

h i Pijk ðx1 ; x2 Þ ¼ exp ci x21  cj x22  ck ðx1  x2 Þ2  h i  2  2 ¼ exp  ðci þ ck Þx1 þ cj þ ck x2  2ck x1 x2

(123)

(124)

with corresponding equations for the y and z components. The expression in square brackets is a positive definite quadratic form which can be diagonalized by an orthogonal transformation leading to a new set of coordinates x~1 and x~2 : h  i (125) Pijk ðx1 ; x2 Þ ¼ exp  c~1 x~21 þ c~2 x~22 where c~1 and c~2 are linear combinations of ci, cj and ck. In some applications only the product c~1 c~2 is required, so that



c þ ck ck

¼ ci cj þ ci ck þ ck cj (126) c~1 c~2 ¼

i ck cj þ ck

because the determinant of a quadratic form is invariant against an orthogonal transformation.

18.10 Molecular integrals over GTOS

819

In this way, function Pijk ðr1 ; r2 Þ of Eqn (123) can be factorized into the six simple factors:         Pijk ðr1 ; r2 Þ ¼ exp ~ c1 x~21 exp ~ c2 x~22 exp ~ c1 y~21 / exp ~ c2 ~z22

(127)

This is true also for the product of more than two Gaussians involving distances between any number of points.

18.10.2 Integrals of Gaussian functions In the applications we shall require a few definite integrals involving Gaussian functions, some of which were already seen in Chapter 12. We list below without proof some formulae which are important for us, which can anyway be found in Gradshteyn and Ryzhik (1980). Some proofs are given in Shavitt (1963). rffiffiffiffi ZN   1 p 2 dx exp cx ¼ c>0 (128) 2 c 0

ZN



dxr exp cx k

2



  1 ðkþ1Þ=2 k þ 1 G ¼ c 2 2

(129)

0

where G is the gamma function discussed in Section 4.7.1. If the integrand is the product of a Gaussian and a simple exponential rffiffiffiffi  2 ZN   p d 2 c>0 exp dx exp cx þ 2dx ¼ c c

(130)

N

ZN

  1 dx exp cx2  d=x2 ¼ 2

rffiffiffiffi  pffiffiffiffiffi p exp 2 cd c; d > 0 c

(131)

0

A more general form can be derived from an integral representation of the modified Bessel function of the second kind Kn ðxÞ introduced in Section 4.6.4: ZN

 pffiffiffiffiffi    c ðn1Þ=4 dx xn exp cx2  d=x2 ¼ Kðn1Þ=2 2 cd d

c; d > 0

(132)

0

Z1 Fm ðtÞ ¼

  dx x2m exp tx2 ¼

1 2t mþ1=2

  1 g m þ ;t 2

t > 0; m ¼ 0; 1; 2; /

(133)

0

where g is the incomplete gamma function discussed in Section 4.7.2. We have the recurrence relations: Fm ðtÞ ¼

1 ½2tFmþ1 þ expðtÞ 2m þ 1

(134)

820

CHAPTER 18 Evaluation of molecular integrals

yielding by iteration the series expansion for Fm(t) N X ð2tÞk   Fm ðtÞ ¼ expðtÞ 3 k¼0 ð2m þ 1Þð2m þ 3Þ/ 2m þ 2k þ 2

(135)

which can be alternatively expressed in terms of gamma functions of half-integral argument   N X 1 1 tk   Fm ðtÞ ¼ G m þ expðtÞ 3 2 2 k¼0 G m þ k þ 2

(136)

It is useful to define the function Fm ðtÞ complementary to Fm(t)   1 N Z G mþ   2 dx x2m exp tx2 ¼  Fm ðtÞ Fm ðtÞ ¼ 2tmþ1=2 1

     1 1 ¼ mþ1=2 G m þ g mþ 2 2 2t 1

which for large values of t has the asymptotic expansion   ½N 1 expðtÞ X 1   Fm ðtÞ z G m þ 1 k 2 2t k¼0 G m  k þ t 2

(137)

(138)

ensuring that the error in the function is less than the first neglected term provided we compute at least m terms.

18.10.3 Integral transforms We saw in Chapter 14 that a regular function f (r) can be expanded into a finite discrete set of GTOs: X   ai exp ai r 2 (139) f ðrÞ ¼ i

In some cases it may be useful as a continuous expansion of the form ZN f ðrÞ ¼

  daCðaÞexp ar 2

(140)

0

An example, useful in the evaluation of some molecular integrals, is ZN   1 1=2 da a1=2 exp ar 2 ¼p r 0

(141)

18.10 Molecular integrals over GTOS

821

with its generalization 1 1 ¼ r l Gðl=2Þ

ZN

  da al=21 exp ar 2

(142)

0

Many other transforms can be generated by substituting t ¼ r2 in the Laplace transforms of Section 4.10: ZN ds gðsÞexpðtsÞ (143) f ðtÞ ¼ 0

A useful one is the following: ZN ds sn1 expða=4sÞexpðtsÞ ¼ 2

 a n=2 4t

Kn

pffiffiffiffi at

(144)

0

where Kn ðxÞ is the modified Bessel function of the second kind discussed in Section 4.6.4 (Watson, 1952). Since Kn ðxÞ ¼ Kn ðxÞ

(145)

t ¼ r 2 ; a ¼ a2 ; l ¼ n

(146)

putting

we find kl ðarÞ ¼ ðarÞl Kl ðarÞ ¼

1 2  l a =2 2

ZN

    ds sl1 exp  a2 =4s exp sr 2

(147)

0

For l > 0 the reduced Bessel functions kl ðarÞ are regular at the origin and decrease exponentially as r / N: For half-integral values of l they reduce to polynomials in ar multiplied by exp(ar). The first two of them are pffiffiffiffiffiffiffiffiffi (148) k1=2 ðarÞ ¼ p=2 expðarÞ k3=2 ðarÞ ¼ and, generally knþ1=2 ðarÞ ¼

pffiffiffiffiffiffiffiffiffi p=2ð1 þ arÞexpðarÞ pffiffiffiffi p

2nþ1=2

expðarÞ

n X ð2n  kÞ! k¼0

k!ðn  kÞ!

(149)

(150)

Thus, the functions kl ðarÞ; with variable r, may be used as the radial part of a generalized type of orbital in atomic and molecular calculations.

822

CHAPTER 18 Evaluation of molecular integrals

The special case of Eqn (147) with l ¼ 1/2 is of particular interest in the evaluation of molecular integrals between ordinary exponential-type orbitals. Using Eqn (148), we obtain the transform of the exponential function:  ZN      a pffiffiffiffi ds s3=2 exp a2 =4s exp sr 2 (151) expðarÞ ¼ 2 p 0

18.10.4 Molecular integrals While a more complete list can be found in Shavitt’s paper (1963), we shall limit ourselves here to consideration of the GTO transform technique to the case of the four-centre two-electron integral between 1s STOs (Shavitt and Karplus, 1962). Consider the integral Z Z 1 expðc1 r1A  c2 r1B  c3 r2C  c4 r2D Þ (152) I¼ dr1 dr2 r12 The application of the transform Eqn (151) to each of the exponential terms of Eqn (152) gives ZN ZN ZN ZN   2 I ¼ c1 c2 c3 c4 =16p ds1 ds2 ds3 ds4 ðs1 s2 s3 s4 Þ3=2 0

0

0

0

(153)

   2

1 c1 c22 c23 c24 s1 As2 Bjs3 Cs4 D  exp  þ þ þ 4 s1 s2 s3 s4

where s1 As2 Bjs3 Cs4 D stands for the four-centre exchange integral between Gaussians Z Z

  1 2 2 2 2 dr1 dr2 exp c1 r1A  c2 r1B  c3 r2C  c4 r2D s1 As2 Bjs3 Cs4 D ¼ r12   2p5=2 ðc1 þ c2 Þðc3 þ c4 Þ 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi F0 PQ ¼ ðc1 þ c2 Þðc3 þ c4 Þ c1 þ c2 þ c3 þ c4 c1 þ c2 þ c3 þ c4   c1 c2 c3 c4 2 2 AB  CD  exp  c1 þ c2 c3 þ c4

(154)

The points P and Q lie on AB and CD; respectively, their Cartesian coordinates being Px ¼

c1 Ax þ c2 Bx ; c1 þ c2

Qx ¼

c3 Cx þ c4 Dx ; c3 þ c4

etc:

(155)

Function F0 is defined through Eqn (133) with m ¼ 0. We now transform our variables to z ¼ s1 þ s2 þ s3 þ s4 ;

u ¼ s1 =ðs1 þ s2 Þ;

v ¼ s3 =ðs3 þ s4 Þ;

w ¼ ðs1 þ s2 Þ=ðs1 þ s2 þ s3 þ s4 Þ

(156)

with the inverse transformation s1 ¼ uwz;

s2 ¼ ð1  uÞwz;

s3 ¼ vð1  wÞz;

s4 ¼ ð1  vÞð1  wÞz

(157)

18.10 Molecular integrals over GTOS

with the Jacobian determinant (Section 2.3)   s1 ; s2 ; s3 ; s4 ¼ wð1  wÞz 3 J z; u; v; w

823

(158)

The Cartesian coordinates of points P and Q are then given by s1 A x þ s2 B x ¼ Bx þ uax ; etc: Px ¼ s1 þ s2 Qx ¼

s3 Cx þ s4 Dx ¼ Dx þ vcx ; s3 þ s4

px ¼ uax  vcx þ ex ;

(159)

etc:

etc:

where ax ¼ Ax  Bx ;

cx ¼ Cx  Dx ;

ex ¼ Bx  Dx ;

px ¼ P x  Q x ;

a2

2

¼ AB ;

c2

2

¼ CD ;

p2

etc:

¼ PQ

(160)

2

Using the definitions f ¼ uð1  uÞwa2 þ vð1  vÞð1  wÞc2   c23 c22 c24 1 c21 þ þ þ g¼ 4 uw ð1  uÞw vð1  wÞ ð1  uÞð1  wÞ

(161)

q ¼ wð1  wÞp2 Equation (153) then transforms to pffiffiffiffi Z1 Z1 Z1 ZN p F0 ðqzÞexpðfz  g=zÞ c1 c2 c3 c4 du dv dw dz I¼ 8 ½uð1  uÞvð1  vÞ3=2 ½wð1  wÞ3 z11=2 0

0

0

0

pffiffiffiffi Z1 Z1 Z1 p 3=2 3=2 ¼ dv½vð1  vÞ dw½wð1  wÞ3  J50 ðq; f ; gÞ c1 c2 c3 c4 du½uð1  uÞ 8 0

0

(162)

0

with ZN Jnm ðq; f ; gÞ ¼

dz zn1=2 Fm ðqzÞexpðfz  g=zÞ

(163)

0

¼ qn1=2 Jnm ð1; f =q; gqÞ 1 Jnm ð0; f ; gÞ ¼ 2m þ 1

ZN

dz zn1=2 expðfz  g=zÞ

0

¼

qs0

1 ðm þ 1=2Þð2gÞ

n1=2

 pffiffiffiffi kn1=2 2 fg

q¼0

(164)

824

CHAPTER 18 Evaluation of molecular integrals

Further calculation yields for integral (162) the final expression: Z1 Z1 Z1 p 3=2 3=2 9 dv½vð1  vÞ dw½wð1  wÞ3=2 T50 ðs; sÞ I ¼ c1 c2 c3 c4 du½uð1  uÞ jpj 8 0

0

(165)

0

which is computed by three-dimensional numerical integration. In Eqn (165) we have posed f uð1  uÞ a2 vð1  vÞ c2 þ s¼ ¼ p2 q 1  w p2 w   p2 1  w 2 1  w 2 w 2 w 2 c1 þ c þ c þ c s ¼ gq ¼ 4 u 1u 2 v 3 1v 4 pffiffiffiffiffi r ¼ 1 þ s; k ¼ 2 rs

(166)

Function Tlm ðs; sÞ is related to the reduced Bessel functions seen previously through the expressions lm1 N X X Sli ðs; sÞ þ Rli ðs; sÞ (167) Tlm ðs; sÞ ¼ i¼0

Rli ðs; sÞ ¼ Sli ðs; sÞ ¼

i¼0

1 Gðl þ i þ 3=2Þð2rÞiþ1=2 1 Gðl  i þ 1=2Þð2rÞiþ1=2

kiþ1=2 ðkÞ

(168)

kiþ1=2 ðkÞ

(169)

where r and k are given in Eqn (166). The series in Tlm(s,s) is evaluated using recurrence relations between its terms such as i 1h (170) Tlþ1;m ðs; sÞ ¼ lTlm ðs; sÞ þ sTl1;m ðs; sÞ  Sl1;lm2 ðs; sÞ s where there is no round-off problem since the negative term in Eqn (170) is sensibly smaller in magnitude than the first positive term. Since this calculation of the four-centre integral involves a final three-dimensional numerical integration, it seems to us that, at least for 1s GTOs, our method introduced in Section 18.9.2 is sensibly simpler than that just described in this section. However, Eqn (165) can be used as it stands for two-, three-, and four-centre exchange integrals, while formulae for multicentre integrals involving higher orbitals (namely, 2s, 2p, etc.) can be obtained simply by application of differential operators to the basic 1s formulae. Details are left to Shavitt’s paper (1963).

18.11 PROBLEMS 18 18.1. Give an alternative way of evaluating two-centre Laplacian integrals over STOs. Answer: The result is the same as that obtained by direct use of the Laplacian operator in spheroidals. Hint: Use matrix elements of the hydrogenic Hamiltonian and the fact that 1s and 2ps are eigenfunctions when Z ¼ c and Z ¼ 2c respectively.

18.12 Solved problems

825

18.2. Apply the same technique to the evaluation of the two-centre Laplacian integral over s STOs. Answer: pffiffiffi

2

1  2 3 

1    2



c 1sA sB rA for c0 sc sA sB 7 ¼ c ðsA sB j1Þ  4c sA sB rA þ 3 Hint: Use a suitable linear combination of sA and 1sA (same orbital exponent c) which is eigenfunction of the hydrogenic Hamiltonian with Z ¼ 2c and eigenvalue c2 =2: 18.3. Calculate the overlap integral ð1sA 1sB j1Þ for c0 ¼ c from the general formula given in Magnasco et al. (1998). Answer: The same as in Eqn (54). Hint: Identify all necessary parameters, finding the explicit expressions for coefficients, summation limits and integrals Tjm(r) and Gjm(0) for 1s STOs as given in Appendix B of that paper.

18.12 SOLVED PROBLEMS 18.1. An alternative way of evaluating two-centre Laplacian integrals. An interesting simple way of calculating two-centre Laplacian integrals over STOs, alternative to the direct calculation of the Laplacian operator in spheroidals, uses the matrix element of the hydrogenic Hamiltonian between AOs which are eigenfunctions of this Hamiltonian with the appropriate nuclear charge Z, say 1s and 2ps STOs with Z ¼ c and Z ¼ 2c respectively. As the simplest example, let us take the integral

  1sA ðcÞ1sB ðc0 Þ 72 We start from the matrix element

 

1 Z

1sB

 72 

1sA 2 rA and use the fact that 1sA is eigenfunction of the hydrogenic Hamiltonian having Z ¼ c with eigenvalue c2 =2

  

  

1

Z

1 c 1

þ ðc  ZÞ

1sA 1sB

 72 

1sA ¼ 1sB

 72  2 rA 2 rA rA ¼

1   c2 ð1sA 1sB j1Þ þ ðc  ZÞ 1sA 1sB rA 2

giving, after simplification and multiplication by 2



1    1sA 1sB 72 ¼ c2 ð1sA 1sB j1Þ  2c 1sA 1sB rA The Laplacian integral is hence expressible in terms of the overlap and the Coulomb integrals involving the same density ½1sA ðr1 Þ1sB ðr1 Þ. If c0 ¼ c   1 ð1sA 1sB j1Þ ¼ expðrÞ 1 þ r þ r2 3

826

CHAPTER 18 Evaluation of molecular integrals



1  1sA 1sB rA ¼ c expðrÞð1 þ rÞ

giving 

1 0 1 2  

 r 1 þ r þ 1 2 0 2 2 2 A @ 3 ¼ c expðrÞ 1  r þ r 1sA ðcÞ1sB ðc Þ 7 ¼ c expðrÞ 3 2  2r

which coincides with our previous result (66). The same can be easily done for the integral ð2psA 2psB j72 Þ, obtaining the result 

c0 sc

2

1   2psA 2psB 7 ¼ c2 ð2psA 2psB j1Þ  4c 2psA 2psB rA

c0 ¼ c   1 2 2 3 1 4 ð2psA 2psB j1Þ ¼ expðrÞ 1 þ r þ r  r  r 5 15 15  

1   c 1 2psA 2psB rA ¼  expðrÞ 1 þ r  r3 2 3 giving 

 

 1 8 1 2psA 2psB 72 ¼ c2 expðrÞ 1 þ r  r2  r3 þ r4 5 15 15

FIGURE 18.9 Plot of ð2psA 2psB j1Þ vs R for c ¼ 0.5

18.12 Solved problems

827

The overlap integral ð2psA 2psB j1Þ is plotted vs R in Figure 18.9 for c ¼ 0.5 It is seen that S is negative up to Rz5a0 , then becomes positive with a maximum at about R ¼ 9a0, then goes asymptotically to zero from above. 18.2. Evaluation of ðsA sB j72 Þ by the alternative method. The same method cannot be applied directly to the integral:

  sA ðcÞsB ðc0 Þ 72 since the 2s STO sA is not an eigenfunction of the appropriate hydrogenic Hamiltonian. We notice, however, that the normalized linear combination (same orbital exponent c) pffiffiffi 2sA ¼ 1sA  3sA 2 is an eigenfunction pffiffiffi of the hydrogenic Hamiltonian with Z ¼ 2c and eigenvalue c =2: In fact (same c, S ¼ 3=2) + *



pffiffiffi pffiffiffi pffiffiffi 1 2 Z

h2s2s ¼ 1s  3s

 7 

1s  3s ¼ h1s1s þ 3hss  2 3hs1s 2 r pffiffiffi    2    2  c c 1 2 3 3 1  Zc þ 3  Zc  S 2cðc  ZÞ  c2 ¼ c2  Zc ¼ 2 6 2 3 2 2

which, for Z ¼ 2c gives h2s2s ¼

 1 2 c2 c  2c2 ¼  2 2

Therefore, we can express the sA STO through the inverse transformation: 1 sA ¼ pffiffiffi ð1sA  2sA Þ 3 as a linear combination of two eigenfunctions of the hydrogenic Hamiltonian. We then have * sB





ðc0 Þ 

+ *

+



1 2 Z 1sA  2sA 1 2 Z



pffiffiffi 7  sA ðcÞ ¼ sB  7 

2 rA 2 rA 3 "* 

+ 



1 1 c 1 2 þ ðc  ZÞ

1sA sB

 7  ¼ pffiffiffi 2 rA rA 3 * 

+# 



1 2c 1 2 þ ð2c  ZÞ

2sA  sB

 7  2 rA rA  2

1   1 c ¼ pffiffiffi  ð1sA sB j1Þ þ ðc  ZÞ 1sA sB rA 2 3  

1   c2   ð2sA sB j1Þ þ ð2c  ZÞ 2sA sB rA 2

828

CHAPTER 18 Evaluation of molecular integrals

The last term can be rearranged to i h pffiffiffi

1  pffiffiffi

1 i c2 h  3 sA sB rA ð1sA sB j1Þ  3ðsA sB j1Þ  ð2c  ZÞ 1sA sB rA 2 giving, upon substitution and simplification pffiffiffi

 

1

1 

1   Z

c2 3  sB

 72 

sA ¼  ðsA sB j1Þ þ ð2c  ZÞ sA sB rA  c 1sA sB rA 2 2 r 3 A

from which it follows pffiffiffi

 

1 

1 

1 2  c2 3 



sA sB  7 ¼  ðsA sB j1Þ þ 2c sA sB rA  c 1sA sB rA 2 2 3 Hence we finally obtain pffiffiffi



1  2 3 

1    c 1sA sB rA þ sA ðcÞsB ðc0 Þ 72 ¼ c2 ðsA sB j1Þ  4c sA sB rA 3 For c0 ¼ c

  4 1 1 ðsA sB j1Þ ¼ expðrÞ 1 þ r þ r2 þ r3 þ r4 9 9 45  

1  c  4 2 1 3

sA sB rA ¼ expðrÞ 1 þ r þ r þ r 2 9 9 pffiffiffi  

1   2 2 3

1sA sB rA ¼ c expðrÞ 1 þ r þ r 3 3

and, adding the three terms altogether 0

1 4 1 1 3 þ 3r þ r2 þ r3 þ r4 B 3 3 15 C C B C B C B

2 1 2 C B  8 2 2 3

C B sA sB 7 ¼ c expðrÞB  6  6r  r  r C 3 3 3 C B C B C B A @ 4 2 þ 2 þ 2r þ r 3   1 2 1 3 1 4 ¼ c expðrÞ  1  r  r þ r 3 3 15

  which is the result given by Roothaan (1951b). c þ c0 R To save space, the direct evaluation of the integral s ¼ 2 



2psA 2ps0A rB1



24 ¼ 7 s



1 1 þ s2 3



  7 2 11 3 1 5 4  expð2sÞ 1 þ 2s þ s þ s þ s þ s 3 6 3

18.12 Solved problems

829

which is of some interest to see how to organize hand-computations in a rather heavy case, is omitted here but can be found in Chapter 13 of Magnasco (2007). 18.3. Calculation of the overlap ð1sA 1sB j1Þ from the general formulae. According to Magnasco et al. (1998) the general two-centre overlap integral between STOs is given by  Nþ1 R a mb N na N nb a m Sba ¼ ðna la ma ; nb lb mb j1Þ ¼ dma mb la alb 2 XX  aab pj Gjm ðsab ÞTNL2pj;m ðsab Þ p

j

where "

N ¼ n a þ nb ;

L ¼ la þ lb ;



am l

2l þ 1 ðl  mÞ! ¼ 2 ðl þ mÞ!

sab ¼

R ðca þ cb Þ; 2

ð2cÞ2nþ1 Nn ¼ ð2nÞ!

#1=2

1=2

sab ¼

m ¼ jmj ¼ ma þ mb R ðca  cb Þ 2

and explicit expressions for the coefficients aab pj and the summation limits are given in Appendix B of that paper. For 1s STOs na ¼ nb ¼ 1;

la ¼ lb ¼ ma ¼ mb ¼ m ¼ 0; N ¼ 2;  1=2 Na ¼ Nb ¼ 4c3 ; a00 ¼ ð1=2Þ1=2

p˛½0; L  M ¼ 0;

L¼0

j˛½maxð0; 0; 0Þ; minð2; 2; 2Þ ¼ ½0; 2 ¼ 0; 1; 2

and we obtain, for ca ¼ cb ¼ c, cR ¼ r  3 2  2 X R ðNa Þ2 a00 aab Sba ¼ ð100; 100j1Þ ¼ 0j Gj0 ð0ÞT2j;0 ðrÞ 2 j¼0 ¼

r3 ab ab a00 G00 ð0ÞT20 ðrÞ þ aab 01 G01 ð0ÞT10 ðrÞ þ a02 G20 ð0ÞT00 ðrÞ 4

Coefficients and integrals in this expression are seen to be j

aab 0j

G integrals

0 1 2

1 0 1

G00 G10 G20

2 0 2/3

T integrals T00 T10 T20

exp(r)r1 exp(r)r2(1 þ r) exp(r)r3(2 þ 2 þ r2)

830

CHAPTER 18 Evaluation of molecular integrals

so that we finally obtain Sba ¼ ð100; 100j1Þ ¼ h1sA j1sB i    2 expðrÞ r3 expðrÞ  2 2 ¼ 2 þ 2r þ r  4 r3 3 r 0 1 4 þ 4r þ 2r2 r3 expðrÞ @ A ¼ 2 4 r3  r2 3     expðrÞ 4 2 1 2 ¼ 4 þ 4r þ r ¼ expðrÞ 1 þ r þ r 4 3 3 as it must be.

CHAPTER

Relativistic molecular quantum mechanics

19

CHAPTER OUTLINE 19.1 19.2 19.3 19.4 19.5 19.6

Introduction ............................................................................................................................... 831 The Schroedinger’s Relativistic Equation ..................................................................................... 832 The Klein–Gordon Relativistic Equation........................................................................................ 833 Dirac’s Relativistic Equation for the Electron................................................................................ 834 Spinors: Small and Large Components ......................................................................................... 835 Dirac’s Equation for a Central Field ............................................................................................. 838 19.6.1 Separation of the radial equation ............................................................................841 19.6.2 The hydrogen-like atom ..........................................................................................843 D2 19.7 One-Electron Molecular Systems: HD ........................................................................ 846 2 and HHe 19.8 Two-Electron Atomic System: The He Atom .................................................................................. 847 19.9 Two-Electron Molecular Systems: H2 and HHeD ........................................................................... 850 19.10 Many-Electron Atoms and Molecules ......................................................................................... 854 19.11 Problems 19 ............................................................................................................................ 857 19.12 Solved Problems ...................................................................................................................... 858

19.1 INTRODUCTION In this chapter, we extend the study of the non-relativistic Schroedinger’s equation to the case of a particle that has a speed approaching that of light,1 in a way that must be consistent with the Lorentz transformation equations of the special theory of relativity. A characteristic feature of relativistic wave equations is that spin arises naturally from the beginning and must not be added afterwards to the non-relativistic Schroedinger’s equation as was in Pauli’s theory. After an introduction on the Schroedinger’s attempt of reconsidering his wave equation in a relativistic way and on the related Klein–Gordon equation, we turn to the Dirac’s quantum theory of the electron, next extending our considerations to the relativistic theory of an electron in a central field, then to the one-electron atomic and molecular systems, followed by the two-electron atomic and molecular systems, and, finally, by a glance at the relativistic calculations on many-electron atoms and molecules. In the following Light velocity in vacuum c ¼ 2.99792458  108 m s1 (Mohr et al., 2008). In atomic units (au), if a is the fine-structure constant, c ¼ 137.035999679(94) ¼ 1/a.

1

Elementary Molecular Quantum Mechanics 2E. http://dx.doi.org/10.1016/B978-0-444-62647-9.00019-1 Ó 2013, 2007 Elsevier B.V. All rights reserved

831

832

CHAPTER 19 Relativistic molecular quantum mechanics

discussion, we shall refer mostly to the books of Mott and Sneddon (1948), Schiff (1955), and Grant (2010) and to the Helsinki lecture notes by Pyykko¨ (2001).

19.2 THE SCHROEDINGER’S RELATIVISTIC EQUATION A relativistic equation for a free particle satisfying the requirements of special relativity was derived by Schroedinger (1926b) in the transition from the classical non-relativistic equation E¼

p2 2m

(1)

to the relativistic equation for the energy E2 ¼ c2 p2 þ m2 c4

(2)

where p is the momentum vector of the free particle of mass m, and the energy E includes the rest mass energy mc2. Using the correspondences ^ ¼ iZV; p 0 p

v E 0 H^ ¼ iZ vt

(3)

we obtain from Eqn (2) Z2

v2 j ¼ Z2 c2 72 j þ m2 c4 j vt2

where Z ¼ h=2p is the reduced Planck’s constant. The plane waves jðr; tÞ ¼ exp½iðk$r  utÞ

(4)

(5)

^ with eigenvalues Zu and Zk, respectively. with u ¼ E=Z, are eigenfunctions of the operators H^ and p The plane waves in Eqn (5) satisfy Eqn (4) if 1=2
(6) Zu ¼  Z2 c2 k2 þ m2 c4 Taking for the moment the positive square root in Eqn (6), it could be seen that, if we define the two real quantities   iZ vj Z  vj ; Sðr; tÞ ¼ j  j ðj Vj  jVj Þ (7) Pðr; tÞ ¼ vt 2mc2 vt 2im the conservation equation v Pðr; tÞ þ div Sðr; tÞ ¼ 0 vt

(8)

turns out to be invariant against a Lorentz transformation.2 The expression for S(r, t) is identical with that of its non-relativistic counterpart, while the expression for P(r, t) reduces to jjðr; tÞj2 in the A Lorentz transformation, shifting the origin of a Cartesian system centred in O to O0 along the coordinate axis z, satisfies the relations: z0 ¼ ðz  vtÞð1  v2 =c2 Þ1=2 ; t0 ¼ ðt  vz=c2 Þð1  v2 =c2 Þ1=2 .

2

19.3 The Klein–Gordon relativistic equation

833

non-relativistic limit. It should be noted that P given in Eqn (7) is not necessarily positive, so that it could not be interpreted as a space probability density. Multiplied by the charge e of the particle, it could, however, be interpreted as an electric charge density since the charge density can have either sign as long it is real. However, it is not possible to include the Pauli’ spin matrices into the previous equations without destroying the invariance of the theory. So, the Schroedinger’s relativistic equation represents a particle that has no spin.

19.3 THE KLEIN–GORDON RELATIVISTIC EQUATION The Hamiltonian for a single electron of rest mass m and charge e in an arbitrary electromagnetic (EM) field of scalar potential 4 and vector potential A, moving at a speed comparable to that of light c can be written as (Mott and Sneddon, 1948) H¼c

h

pþ

i1=2 e 2  e4 A þ m2 c2 c

(9)

and, for a conservative system where H equals the total energy E of the system, as i1=2 e E h e 2  4 ¼ p þ A þ m 2 c2 c c c

(10)

Squaring both members and writing E/c ¼ p0, we obtain the symmetrical equation  e 2  e 2  p0 þ 4 þ p þ A þ m 2 c 2 ¼ 0 c c

(11)

Using the correspondences of Eqn (3) we obtain the operator 

iZ v e þ 4  c vt c

2  e 2 þ  iZV þ A þ m2 c2 c

(12)

which acting on a wave function j gives the so-called Klein–Gordon equation 2    iZ v e e 2 2 2 þ 4 þ  iZV þ A þ m c j ¼ 0  c vt c c

(13)

Equation (13) suffers for some difficulties, firstly because it is not linear in the time differential operator v/vt, so requiring as initial conditions the knowledge not only of j but also of vj/vt, and, secondly, because the resulting expressions for the charge density and the current density are of difficult physical interpretation. In the absence of an EM field both 4 and A are zero, and Eqn (13) reduces to the Klein–Gordon equation for a free electron, which we can write as   (14)  p^20 þ p^2x þ p^2y þ p^2z þ m2 c2 j ¼ 0

834

CHAPTER 19 Relativistic molecular quantum mechanics

19.4 DIRAC’S RELATIVISTIC EQUATION FOR THE ELECTRON The considerations of the previous sections show that the correct relativistic equations must be (1) linear in the time differential operator v/vt and (2) must be invariant under a Lorentz transformation. In the theory of relativity, there must be a complete symmetry between the space coordinates x, y, z and the time coordinate ict, so that if the resulting differential equation is linear in v/vt, it must also be linear in v/vx, v/vy, v/vz. For this reason, Dirac (1928a) approached the problem of finding a relativistic wave equation for the free electron by starting from the usual time-dependent Schroedinger equation v ^ (15) Hjðr; tÞ ¼ iZ jðr; tÞ vt and modified the Hamiltonian, linear in the time coordinate, in such a way so as to make it linear in the space derivatives. The simplest Hamiltonian satisfying such requirements is H^ ¼ ca$p  bmc2

(16)

 E þ ca$p þ bmc2 j ¼ 0

(17)

  v iZ  iZc a$V þ bmc2 j ¼ 0 vt

(18)

Substituting into Eqn (15) gives

namely,




which is Dirac’s wave equation for the free particle. It is evident that Eqn (18) treats on the same foot time and space derivatives (entering the gradient operator V). Considering the four quantities ax, ay, az, and b, one can conclude that a and b are independent of r, t, p, and E, so that they commute with all of them, but not necessarily commute with each other, so that they are not numbers. If we impose that any solution j of Eqn (18) should also be a solution of Schroedinger’s relativistic equation, Eqn (4), multiplying Eqn (18) on the left by 
(19) E  ca$p  b mc2 we obtain h 
i n  
E2  c2 a2x p2x þ a2y p2y þ a2z p2z þ ax ay þ ay ax px py þ ay az þ az ay py pz þ ðaz ax þ ax az Þpz px h io
 þ m2 c4 b2  mc3 ðax b þ bax Þpx þ ay b þ bay py þ ðaz b þ baz Þpz j ¼ 0 (20) Equation (20) agrees with Eqn (4) if the quantities a and b satisfy the following equations: 8 < ax ay þ ay ax ¼ ay az þ az ay ¼ az ax þ ax az ¼ 0 ax b þ bax ¼ ay b þ bay ¼ az b þ baz ¼ 0 (21) : 2 2 2 2 ax ¼ ay ¼ az ¼ b ¼ 1

19.5 Spinors: small and large components

namely, satisfy the anti-commutation relations:

(

  ax ; ay þ ¼ ay ; az þ ¼ ½az ; ax þ ¼ dkl

k; l ¼ x; y; z

835

(22)

½ak ; bþ ¼ 0

where dkl is the Kronecker delta. It can also be seen that, in this case, Eqn (20) reduces to the Klein– Gordon Eqn (14) for a free particle. The four quantities a and b anti-commute in pairs and their squares are unity. It can be shown (Schiff, 1955) that the explicit representation of the four (4  4) matrices a and b is given by 8 0 1 0 1 0 0 0 1 0 0 0 i > > > > B C B C > > B0 0 1 0C B0 0 i 0 C > > B C B C ¼ ¼ ; a a > x y > B0 1 0 0C B 0 i 0 0 C > > @ A @ A > > > < 1 0 0 0 i 0 0 0 (23) 0 1 0 1 > 1 0 0 0 0 0 1 0 > > > B C B C > > B0 1 0 B 0 0 0 1 C > 0 C > B C B C > ; b ¼ ¼ a > z > B 0 0 1 0 C B1 0 0 0 C > > @ A @ A > > : 0 0 0 1 0 1 0 0 These matrices are evidently Hermitian, and, when partitioned according to   0 s a¼ s 0   1 0 b¼ 0 1 show that the (2  2) submatrices:   0 1 ax ¼ ¼ sx ; 1 0

 ay ¼

0 i

i 0



 ¼ sy ;

az ¼

1 0

0 1

(24) (25)

 ¼ sz

(26)

are proportional to the Pauli’s matrices for spin 1/2. All s and b submatrices satisfy the relations Eqn (21).

19.5 SPINORS: SMALL AND LARGE COMPONENTS Since a and b are represented by (4  4) matrices, Eqn (18) has no meaning unless the wave function j is itself a vector with four rows and one column 0 1 j1 ðr; tÞ B j ðr; tÞ C B C jðr; tÞ ¼ B 2 (27) C @ j3 ðr; tÞ A j4 ðr; tÞ

836

CHAPTER 19 Relativistic molecular quantum mechanics

Thus, Eqn (18) is equivalent to four simultaneous first-order partial differential equations that are linear and homogeneous in the four j0 s. Plane-wave solutions of the form Eqn (5)



 j ¼ 1; 2; 3; 4 (28) jj r; t ¼ uj exp i k$r  ut can then be found, where the uj are numbers. These are eigenfunctions of the momentum and energy operators Eqn (3) with eigenvalues Zk, and Zu, respectively. Substitution of Eqn (28) and Eqns (24) and (25) into Eqn (18) gives a set of algebraic equations for uj where E ¼ Zu and the components of p ¼ Zk are now numbers 8
 > E þ mc2 u1 þ cpz u3 þ cð px  ipy Þu4 ¼ 0 > >
>  > < E þ mc2 u2 þ cð px þ ipy Þu3  cpz u4 ¼ 0 (29)
 > cpz u1 þ cð px  ipy Þu2 þ E  mc2 u3 ¼ 0 > > > > : cð p þ ip Þu  cp u þ
E  mc2 u ¼ 0 x

namely, in matrix form 0 E þ mc2 B B 0 B B cp z @ cð px þ ipy Þ

y

z 2

1

0 E þ mc2 cð px  ipy Þ cpz

4

cpz cð px þ ipy Þ E  mc2 0

10 1 u1 cð px  ipy Þ CB C CB u2 C cpz CB C ¼ 0 C@ u3 A 0 A 2 u4 E  mc

(30)

Since these equations are homogeneous in uj, they have non-vanishing solutions if and only if the determinant of the coefficients in Eqn (30) is zero. This determinant is E þ mc2 0 D¼ cpz cð px þ ipy Þ

0 E þ mc2 cð px  ipy Þ cpz

cð px  ipy Þ cð px þ ipy Þ cpz ¼0 2 E  mc 0 2 0 E  mc cpz

Expanding the determinant (Problem 19.1) 2
D ¼ E 2  m 2 c 4  c 2 p2 ¼ 0

(31)

(32)

is obtained, where p2 ¼ p$p ¼ p2x þ p2y þ p2z

(33)

Explicit solutions can be obtained for any value of the momentum p by choosing a sign for the energy, say
1=2 (34) Eþ ¼ þ c2 p2 þ m2 c4

19.5 Spinors: small and large components

837

Then there are two linearly independent solutions, which can be written as u1 ¼ 

cð px þ ipy Þ u2 ¼  ; Eþ þ mc2

cpz ; Eþ þ mc2

u1 ¼ 

cð px  ipy Þ ; Eþ þ mc2

u2 ¼

cpz ; Eþ þ mc2

u3 ¼ 1; u3 ¼ 0;

u4 ¼ 0 u4 ¼ 1

Similarly, the two independent solutions for 1=2
E ¼  c 2 p 2 þ m 2 c 4

(35) (36)

(37)

are u1 ¼ 1; u1 ¼ 0;

u2 ¼ 0; u2 ¼ 1;

u3 ¼ u3 ¼

cpz ; E þ mc2

cð px  ipy Þ ; E þ mc2

u4 ¼

cð px þ ipy Þ E þ mc2

cpz u4 ¼  E þ mc2

(38) (39)

Each of these four solutions can be normalized to 1 by multiplying it by the factor f1 þ ½c2 p2 = ðEþ þ mc2 Þ2 g1=2 . The complex conjugate of vector (27) j, j ; is a vector with one row and four columns. The solutions Eqns (35) and (36) correspond to the positive energy, and the solutions Eqns (38) and (39) to the negative energy. If v is the speed of the particle, in the non-relativistic limit in which Eþ ¼ E is close to mc2 and large in comparison with cjpj, the two solutions u1 and u2 are of the order of v/c times u3 or u4 for the positive-energy solution, the opposite being true for the negativeenergy solutions. The two solutions for each sign of the energy can be distinguished by defining three new spin matrices s0x ; s0y ; s0z , each of which has four rows and columns:   s 0 (40) s0 ¼ 0 s The operator Zs0 =2 can be interpreted as the operator that represents the intrinsic spin angular momentum of the particle. The wave functions ui are called spinors, and, in atomic units, if we denote 1=2
¼ Ep ; Ep þ c2 ¼ A (41) E ¼ c p2 þ c2 the unnormalized spinors become those given in Table 19.1, Table 19.1 Spinors for Spin 1/2 ui

(Ep, 1/2)

(Ep, L1/2)

(LEp, 1/2)

(LEp, L1/2)

u1 u2 u3 u4

1 0 cpz/A c( px þ ipy)/A

0 1 c(px  ipy)/A cpz/A

cpz/A c(px þ ipy)/A 1 0

c( px  ipy)/A cpz/A 0 1

838

CHAPTER 19 Relativistic molecular quantum mechanics

The normalized spinors become

0   1 a sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1$ B C 2 2 
b Ep þ c j 1 Ep þ c B C ¼ u p; Ep ¼  C B j2 2Ep 2Ep @ cðs$^ ^ pÞ a A Ep þ c 2 b

0  1 ^ pÞ a sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  cðs$^ 2
 b C Ep þ c 2 j 1 Ep þ c 2 B B Ep þ c C ¼ u p; Ep ¼   B C j2 A 2Ep 2Ep @ a 1$ b       a 1 0 where is either or . b 0 1

(42)

(43)

For Ep > 0, the ratio of the norms for small v/c is
S 2 j j2 2 c2 p2 c 2 p2 1 p 2 1v2 ¼ y ¼ z ¼
L 2 j1 2 ðE þ c2 Þ2 ð2c2 Þ2 4 mc 4 c j

(44)

Therefore, j1 ¼ jL and j2 ¼ jS are called the large and small components, respectively. For electrons in light atoms, v z 1 au, and this plane-wave estimate gives   1v2 1 1 2 z z 105 4 c 4 137

(45)

19.6 DIRAC’S EQUATION FOR A CENTRAL FIELD In this section, we separate Dirac’s equation for a general central field and find the energy levels of the hydrogen atom in the relativistic approximation. In a second paper, Dirac (1928b) extended his theory to the consideration of an electron in an external magnetic field discussing the Zeeman effect. We start from Dirac’s relativistic equation in the presence of the EM potentials A(r, t) and 4(r, t) that, for a particle of electric charge e, we write as 

(46) E  e 4 þ a$ðc p  e AÞ þ bmc2 j ¼ 0 With A(r, t) ¼ 0 and 4(r, t) ¼ 4(r), this equation becomes   v 2 iZ þ ca$p þ bmc þ V j ¼ 0 vt

(47)

where V ¼ e 4

(48)

19.6 Dirac’s equation for a central field

839

^ is still a constant of the motion in such We should ask if the orbital angular momentum ^l ¼ r  p a central field even in the relativistic approximation where an intrinsic spin is present. The answer is no, the quantity commuting with the Hamiltonian H^ being instead ^l þ 1 Zs ^0 (49) 2 The quantity Eqn (49) can then be taken as the appropriate quantity describing the total angular momentum. It can be shown (Schiff, 1955) that the spin-orbital energy xðrÞð^l$^sÞ

(50)

where xðrÞ ¼

1 1 dV 2m2 c2 r dr

(51)

is a natural consequence of Dirac’s relativistic equation. This term can be shown to be of the order (v/c)2 times the potential energy 1 1 1 V v2 pah w 2 (52) xðrÞð^l$^sÞ ¼ 2 2 2 c V Vm c a where a represents the linear dimensions of the system, and h w p w mv a

(53)

In order to give a consistent approximation, we must introduce the now familiar two-component wave functions, replacing in Eqn (47) j by j1 and j2, which now represent the fist two and the last two components of j, assuming that j1 and j2 together make a non-relativistic energy eigenfunction, so that (54) E ¼ E0 þ mc2 can be regarded as a number and not an operator. Since E0 and V are assumed to be small in comparison with mc2, the wave equation can be written as (

 E0 þ 2mc2  V j1 þ c s$p j2 ¼ 0 (55) ðE0  VÞj2 þ cðs$pÞj1 ¼ 0 ^ p ^. where s, p are both operators, so that henceforth they will be denoted by s; The first of these equations shows that j1 is of the order of v/c times j2, so that we eliminate it, obtaining an equation in terms of j2 alone. The substitution
1 ^ pÞj2 cðs$^ (56) j1 ¼  E0 þ 2mc2  V gives

  1 E0  V 1 ^ ^ ðs$^ pÞ 1 þ E j2 ¼ ðs$^ pÞ þ V j2 2m 2mc2 0



where no approximation has been made so far.

(57)

840

CHAPTER 19 Relativistic molecular quantum mechanics

We now make the approximation of taking only the lowest terms in the expansion in powers of

We then obtain

E0  V 2mc2

(58)

  E0  V 1 E0  V 1þ x1  2 2mc 2mc2

(59)

^V ¼ V p ^  iZVV p

(60)

^ ^ pÞ ¼ ðVVÞ$^ ^ ^ ðs$VVÞð s$^ p þ is$½ðVVÞ p

(61)

Using relations Eqns (59) and (60), Eqn (57) becomes  2  ^ E0  V p Z2 Z2 ^ þ V j2  2 2 ðVVÞ$ðVj2 Þ þ 2 2 s$½ðVVÞ 1 E 0 j2 ¼  ð^ pj2 Þ 2 2m 4m c 4m c 2mc

(62)

For a spherically symmetric central field, we have the further simplifications ðVVÞ$V ¼

VV ¼

dV v dr vr

(63)

1 dV r r dr

(64)

^2 p 2m

(65)

Noting that E0  Vx

^4 =2m. Equation (62) can it will be accurate enough to replace the second-order term in Eqn (62) by p then be re-written as  2 ^ ^4 Z2 dV v 1 1 dV ^ p p 0 þV  3 2 2 2 þ ð^s$lÞ j2 (66) E j2 ¼ 2m 8m c 4m c dr vr 2m2 c2 r dr where ^s ¼

1 ^ Zs; 2

^l ¼ r  p ^

(67)

The first two terms on the right-hand side of Eqn (66) give the non-relativistic Schroedinger’s wave equation. The third term has the form of the classical relativistic mass correction, which can be obtained by expanding the square root of Eqn (2): 1=2
^2 ^4 p p  3 2 x E0 ¼ E  mc2 ¼ c2 p2 þ m2 c4 2m 8m c

(68)

19.6 Dirac’s equation for a central field

841

The last term in Eqn (66) gives the spin–orbit energy, which now appears as an automatic consequence of Dirac’s equation. The fourth term in Eqn (66) is the relativistic correction to the potential energy and has no classic analogue and cannot be easily demonstrated experimentally.

19.6.1 Separation of the radial equation Dirac’s equation for the hydrogen atom can be separated without approximations in spherical coordinates. The procedure is more complicated than that seen in Chapter 3 of this book, because of the interdependence of orbital and spin angular momenta. We begin by defining radial momentum and velocity operators, both of which are Hermitian, as
 ^ r ¼ r 1 a$r p  iZÞ; a (69) p^r ¼ r 1 ðr$^ We also define a new operator k^ related to the total angular momentum ^ 0 $^l þ ZÞ Zk^ ¼ bðs

(70)

^l ¼ r  p ^

(71)

where

is the orbital angular momentum of the electron. Direct substitution shows that a$^ p ¼ ar pr þ iZr 1 ar bk

(72)

H^ ¼ ca$^ p  bmc2 þ V

(73)

iZc H^ ¼ car pr  ar bk  bmc2 þ V r

(74)

The Hamiltonian

becomes

It can be shown that k^ commutes with either ar, pr, and b, so that it also commutes with the Hamiltonian Eqn (74) and is therefore a constant of the motion. The eigenvalues of k^ can be inferred by squaring Eqn (70), so obtaining 2  1 1 ^ 2 þ 2Zðs ^ 0 þ Z2 ^ 0 $lÞ ^ 0 $^lÞ þ Z2 ¼ ^l þ Zs (75) Z2 k^2 ¼ ðs 2 4  2 1 ^0 The quantity ^l þ Zs is the square of the total angular momentum and has eigenvalues 2 2 jðj þ 1ÞZ , where j is half a positive odd integer. Thus, k^2 has eigenvalues  k ¼ 2

1 jþ 2

2 k ¼ 1; 2; /

(76)

842

CHAPTER 19 Relativistic molecular quantum mechanics

If we now choose a representation in which H^ and k^ are diagonal and represented by numbers E and k, respectively, then ar and b can be represented by any Hermitian matrix satisfying a2r ¼ b2 ¼ 1;

ar b þ bar ¼ ½ar ; bþ [ 0

Such matrices have two rows and columns, and we put    0 i 1 ; b¼ ar ¼ i 0 0

0 1

(77)

 (78)

The angular and spin parts of the wave function are fixed by the requirement that j is an eigenfunction of the k^ operator Eqn (70). For the computation of the energy levels, we are concerned only with the radial part, which has two components which, according to Eqn (78), we write as the two-row vector ! r 1 FðrÞ (79) r 1 GðrÞ Substitution of Eqns (78) and (79) into the wave equation with the Hamiltonian Eqn (74) gives the radial equations for the electron moving in a central field. Using the relation   v 1 þ (80) p^r ¼ iZ vr r we obtain the two coupled equations 8
 dG Zck 2 > > < E þ mc  V F  Zc dr  r G ¼ 0 >
> : E  mc2  V G þ Zc dF  Zck F ¼ 0 dr r

(81)

Making the substitutions mc2 þ E a1 ¼ ; Zc Equation (81) become

mc2  E a2 ¼ ; Zc




r ¼ ar;

a ¼ þða1 a2 Þ

8    d k a1 V > > >  þ G F¼0 > < dr r a Zca     > > d k a2 V > > þ  F G¼0 : a Zca dr r

1=2

m2 c4  E2 ¼ Zc

1=2 (82)

(83)

All previous equations are valid for a generic central field V(r). The exact solution of Dirac’s differential equation for an electron in a central Coulomb field was first given by Darwin (1928) who presented the theory in terms of differential equations rather than in terms of non-commutative algebra.

19.6 Dirac’s equation for a central field

843

19.6.2 The hydrogen-like atom For the hydrogen-like atom, an electron moving in the field of þZe nuclear charges, the potential energy V(r) is VðrÞ ¼ 

Ze2 r

(84)

Defining a constant g by g¼

Ze2 Zc

(85)

we write the potential energy as V g ¼ Zca r

(86)

From now on, we follow for both components F and G a procedure like the one already introduced in Section 3.7.1 of Chapter 3 of this book. If we put FðrÞ ¼ f ðrÞexpð rÞ; Equation (83) become

GðrÞ ¼ gðrÞexpð rÞ

(87)

  8 > 0  g þ kg  a1 þ g f ¼ 0 > g > > a r r < (88)

  > > kf a2 g > > :f0  f    g¼0 a r r We now expand both f and g in the power series of r 8 N P > s > ak rk ; > < f ðrÞ ¼ r

f 0 ðrÞ ¼ rs

N > P > > bk rk ; : gðrÞ ¼ rs

g0 ðrÞ

k¼0

k¼0

N P k¼0

¼

rs

N P k¼0

kak rk1

a0 s0 (89)

kbk

rk1

b0 s0

For the finiteness of the two-valued vector components Eqn (79) at r ¼ 0, we expect that s be greater than or equal to 1. Proceeding in the usual way and putting equal to zero the coefficient of rsþk1, we find the coupled recurrence relations 8 a1 > > < ðs þ k þ kÞbk  bk1  gak  a ak1 ¼ 0 k ¼ 1; 2; / (90) > > : ðs þ k  kÞak  ak1 þ gbk  a2 bk1 ¼ 0 a

844

CHAPTER 19 Relativistic molecular quantum mechanics

For k ¼ 0 the indicial equations give (

ðs þ kÞb0  ga0 ¼ 0

(91)

ðs  kÞa0 þ gb0 ¼ 0

These are two homogeneous linear equations in a0, b0, that have non-vanishing solutions if the determinant of the coefficients is zero g s þ k ¼0 (92) s  k g giving the two possible values of s: 1=2
s ¼  k 2  g2

(93)

For the regularity condition at the origin, we take for s the positive sign in Eqn (93), and the square root is real.3 A relation between ak and bk can be found by multiplying the first of Eqn (90) by a, the second by a1, and subtracting, thereby obtaining bk ½aðs þ k þ kÞ  a1 g ¼ ak ½a1 ðs þ k  kÞ þ ag

(94)

We now study the convergence of the series Eqn (89). Unless both series terminate, the behaviour of both at large values of r is determined by their higher terms, so that we can neglect constant factors in comparison with k. We then obtain from Eqns (90) and (94) ak 2 z ; ak1 k

bk 2 z bk1 k

(95)

This means that both series behave asymptotically as exp(2r), which is not admissible for us when r / N. So, regular solutions are obtained only if both series terminate, say at a value k ¼ n0 , so that: an0 þk ¼ bn0 þk ¼ 0 k  1

(96)

Then, from Eqn (94) follows the relation a1 an0 ¼ abn0

n0 ¼ 0; 1; 2; /

(97)

The relativistic energy levels of the hydrogen-like atom are obtained by posing k ¼ n0 in Eqn (94) and, using Eqn (97), we find 2aðs þ n0 Þ ¼ gða1  a2 Þ ¼ Since jkj  1 and g ¼ Zðe2 =ZcÞ ¼ Z=137 < 1.

3

2Eg Zc

(98)

19.6 Dirac’s equation for a central field

845

Putting n ¼ n0 þ jkj; the total quantum number, and taking into account Eqns (76) and (93), by squaring Eqn (98) we find the formula first derived by Sommerfeld (1916a,b) on the basis of the old quantum theory, which accounts quite well for the observed atomic spectrum of the hydrogen atom (Z ¼ 1) " #1=2 n h 1=2 i2 o1=2
g2 2 ¼ mc2 1 þ g2 n  jkj þ k2  g2 (99) E ¼ mc 1 þ 2 ðs þ n0 Þ Expanding the square roots for g ¼ small, to the order g2 we find (Problem 19.2):   g2 2 E x mc 1  2 2n

(100)

and, by subtracting the rest mass of the electron and substituting Eqn (85), we find the usual nonrelativistic formula for the hydrogen-like system. To the order g4 we find (Problem 19.3)    g2 g4 n 3 2  E ¼ mc 1  2  4 (101) 2n 2n jkj 4 To the order g4 it is found that the spread of the fine-structure levels for a given value of n mc2 g4 n  1 n3 2n

(102)

is sensibly smaller than the value obtained from Schroedinger’s relativistic equation and agrees with experiment. For n0 > 0, all positive or negative values of k are permissible, but, to avoid contradiction, for 0 n ¼ 0, k must only assume negative integer values. In order to connect l with the level, we make the non-relativistic approximation that the orbital angular momentum is well defined. In this case, G in ^ in Eqn (70), obtaining in this ^ 0 by s Eqn (79) is much larger than F and we can replace b by 1 and s approximation  2  3 2 ^l þ 1 Zs ^ ^lÞ ¼ jð j þ 1ÞZ2 ^ ¼ lðl þ 1Þ þ Z þ Zðs$ 2 4 In this way we obtain

8 > > < l  1;

1 k ¼ lðl þ 1Þ  jð j þ 1Þ  ¼ 4 > > : l;

1 2 1 j¼l 2

(103)

j¼lþ

(104)

As an example, consider the relativistic energy levels in the hydrogen atom for n ¼ 3. Then, n0 ¼ 0, 1, 2 with k ¼ (3  n0 ) for n0 ¼ 1, 2 and k ¼ 3 for n0 ¼ 0. The levels with their non-relativistic classifications resulting from Eqns (101) and (104) are given in Table 19.2. According to Eqns (94) and

846

CHAPTER 19 Relativistic molecular quantum mechanics

Table 19.2 Multiplet Structure of the Hydrogen Atom Level with n ¼ 3 0

n

k

0

l

3

1

2

2 2

j

Term

5/2

2

D5=2

3/2

2

D3=2 P3=2

1

2

1

3/2

2

2

1

1

1/2

2

P1=2

2

1

0

1/2

2

S1=2

(99), states with the same value of jkj or j have the same energy, whereas Eqn (99), expanded to the order g4, shows that the energy increases on increasing jkj.

19.7 ONE-ELECTRON MOLECULAR SYSTEMS: H2D AND HHeD2 For molecular systems of light atoms, relativistic corrections are small, so they are of interest only in calculations of high precision. Highly accurate relativistic calculations for the paradigmatic oneþ2 were done by Laaksonen and Grant (1984a) using an electron linear molecules Hþ 2 and HHe extension of the fully numerical HF/2D technique developed by the Pyykko¨’s group (Sundholm, 1985) and explained in Chapter 14. Using atomic units, Dirac’s equation for a single electron in the potential V is written as  L   L  ^ $^ V p  J J
cs ¼ ε (105) ^ p cs$^ ^ p  2c2 cs$^ JS JS where V¼

ZA ZB  rA rB

(106)

is the potential attracting the electron by the two nuclei of charge þZA and þZB, respectively, rA and rB ^ ¼ iV is the momentum operator of the electron, are the distances of the electron from the nuclei, p ^ are the Pauli spin matrices, JL and JS are the large and small components of the wave function, s respectively, and c is the speed of light. In a study on the conditions of convergence of variational solutions of Dirac’s equation in finite basis sets, Grant (1982) has shown that it is possible to eliminate the small component, so obtaining the large-component equation in the form    c2 ^ pÞ þ V JL ¼ εJL ^ pÞ 2 ðs$^ (107) ðs$^ 2c þ ε  V Since it is true that ^ ^ pÞf ðrÞðs$^ ^ pÞ ¼ f ðrÞ^ ^Þg pÞ þ iðs$½Vf ðrÞ  p ðs$^ p2  ifðVf ðrÞ$^

(108)

19.8 Two-electron atomic system: the He atom

847

þ2 Table 19.3 HF/2D Relativistic Energies (Eh) for Hþ at R ¼ 2a0 2 and HHe

Molecule

State

E (rel)

E (non-rel)

DE

Hþ 2

1sg

1.1026415709

1.1026342145

7.3564  106

Hþ 2

1su

0.6675527640

0.6675343923

1.8372  105

HHeþ2

1s

2.512296099

2.512193020

1.0308  104

where f(r) is a function of the variables, Eqn (107) can be written as nh
nh
1 2 i 1 i ^  ic2 V$ 2c2 þ ε  V ^ p p c2 2c2 þ ε  V h
io o 1 ^ V 2c2 þ ε  V ^ þ V JL ¼ εJL þis$ p

(109)

2 þ ε  VÞ1  p ^ ^ is the spin-orbit coupling which is zero for a S state, while the The term s$½Vð2c 1 2 term Vð2c þ ε  VÞ can be calculated analytically. So, the equation to be solved becomes

nh
o h
1 i 2 1 i  c2 2c2 þ ε  V V þ V JL ¼ εJL V  c2 V$ 2c2 þ ε  V

(110)

Equation (110) is solved directly through a numerical two-dimensional integration in the spheroidal coordinates m, n of Section 2.6 of Chapter 2, the angular variable 4 factorizing out in the integration. Using a grid of (97  93) integration points, the results given in Table 19.3 are obtained, in the last column being reported the difference DE ¼ E (rel)  E (non-rel). The results show that the relativistic correction is one order of magnitude larger for the first excited state of the homonuclear þ2 molecule Hþ 2 and two orders larger for the heteronuclear molecule HHe .

19.8 TWO-ELECTRON ATOMIC SYSTEM: THE He ATOM For the two-electron systems, one is faced with the problem of evaluating relativistic corrections to the electron repulsion 1/r12 term. In a series of papers, Breit (1929, 1930, 1932) gave an approximate wave equation taking into account terms which are of the order (v/c)2 in the interaction of the two electrons in the form " #   2 2 e e ða $r Þða $r Þ 1 12 2 12 E U ¼ a1 $a1 þ (111) h^1 þ h^2 þ 2 r12 2r12 r12 where ^1 þ e Aðr1 Þ h^1 ¼ e4ðr1 Þ þ b1 mc2 þ a1 $½c p

(112)

In Eqn (111), E is the total energy; 4 and A are, respectively, the scalar and vector potentials of the external EM field, including the nuclear Coulomb potential; r12 ¼ jr1  r2 j is the interelectronic

848

CHAPTER 19 Relativistic molecular quantum mechanics

^ ¼ iZV is the distance; the a0 s and b are the one-electron Dirac matrix operators Eqn (23); and p momentum operator of the electron. The wave function U depends on the positions r1 and r2 of the two electrons and has 16 spinor components, four each for electrons 1 and 2. However, Breit’s Eqn (111), unlike the Dirac’s equation, is not fully invariant with respect to a Lorentz transformation, and is approximate in the sense that its right-hand side is only an approximation to the relativistic interaction between the two electrons which is prescribed in quantum electrodynamics (QED). In a correct field theoretical treatment of pair theory, processes involving the creation of virtual electron–positron pairs occur, and Eqn (111) needs to be modified as follows (Bethe and Salpeter, 1957): ^ ¼ EJ; HJ

H^ ¼ H^0 þ H^1 þ H^2 þ H^3 þ H^4 þ H^5 þ H^6

(113)

the Hamiltonian H^ in atomic units being given as the sum of the following seven terms. 1 H^0 ¼ h^1 þ h^2 þ ; r12

1 Z h^1 ¼  721  ; 2 r1

1 Z h^2 ¼  722  2 r2

(114)

is the ordinary non-relativistic Hamiltonian for a two-electron atom of nuclear charge þZ, h^i ði ¼ 1; 2Þ is the one-electron bare nuclei Hamiltonian;  1
(115) H^1 ¼  a2 741 þ 742 8 is the classical relativistic correction due to the variation of mass with velocity, independent of the electron spin; " # 2 a r $ðr $V ÞV 12 12 1 2 V1 $V2 þ (116) H^2 ¼  2 2r12 r12 is the classical relativistic correction to the interaction between the electrons due to the retardation of the EM field generated by an electron, an orbit–orbit term; 2

a H^3 ¼ i 2

("

" # # ) 2ðr12  V2 Þ 2ðr12  V1 Þ ðV1 VÞ  V1 þ $s1 þ ðV2 VÞ  V2 þ $ s2 3 3 r12 r12

(117)

is the interaction between the spin magnetic moment and the orbital magnetic moment of the electrons, the spin-orbit coupling; 1 H^4 ¼ a2 ½V1 $ðV1 VÞ þ V2 $ðV2 VÞ 4

(118)

is a term characteristic of the Dirac’s theory, called the Darwin term, also present in the Hamiltonian for a single electron in an electric field; ( " #0 ) 8 1 3ðs $r Þðs $r Þ 1 12 2 12 (119) H^5 ¼ a2  pðs1 $ s2 Þ dð3Þ ðr12 Þ þ 3 s1 $ s2  3 3 r12 r12

19.8 Two-electron atomic system: the He atom

849

Table 19.4 DiraceFock (DF) 2D Relativistic Results (Atomic Units) for the Ground State of the He Atom Property E ε

a

Non-relativistic 2.8616799956122 0.91799075 0.927223 1.8477192

Relativistic a

2.8618132 0.9179907 0.927223 1.8477193

Ga´zquez and Silverstone, 1977.

is the interaction between the spin magnetic moments of the two electrons; and 1 H^6 ¼ ðB1 $ s1 þ B2 $ s2 Þ  iðA1 $V1 þ A2 $V2 Þ 2

(120)

is the interaction of the electron spins with an external magnetic field of vector potential A. This term is zero in absence of the external field. A Dirac–Fock (DF) relativistic calculation for the He atom (Z ¼ 2) was done by Laaksonen and Grant (1984b) using the fully numerical 2D integration of the relevant equation for the large component as they did for the one-electron systems. Their results, reported in Table 19.4, give the relativistic correction to the Hartree-Fock value for He. The relativistic 2D calculated values for energy (E), orbital energy (ε), average distance of an electron from the nucleus , and average value of the square of the distance of the last column of the table are compared with the nonrelativistic Hartree-Fock values given in the second column and taken from Ga´zquez and Silverstone (1977). Even if the accuracy is smaller than that of the one-electron cases, it is apparent from the table that only the total energy is affected by the relativistic corrections, giving a lowering in energy of DE ¼ 1.33205  104Eh. This calculation does not consider the correlation energy for the atomic electron pair, which is by far more important, lowering the energy by 0.0420443814208Eh.4 Multiconfigurational DF (MC-DF) calculations were performed by Hata and Grant (1983b) in an attempt to estimate the relativity corrections to the 1s2sð3 S1 Þ  1s2pð3 P0;1;2 Þ splittings of the twoelectron ions, following a method described in a previous paper (Hata and Grant, 1983a). Their results showed small but systematic differences from previous estimates based on the Breit-Pauli method, giving high-precision estimates of the 23P fine-structure splittings in two-electron ions having Z ¼ 14, 16, 17. Hata and Grant also evaluated higher order QED corrections for the same ions and for He (Hata and Grant, 1983c). A careful analysis shows that theory and experiment agree to within the experimental limits (5  104 cm1) as shown in Table 19.5 for He.

4

This value is the difference between the Frankowski and Pekeris (1966) value and the Ga´zquez and Silverstone (1977) value, both of which accurate to 1 picohartree pEh ¼ 1012Eh.

850

CHAPTER 19 Relativistic molecular quantum mechanics

Table 19.5 Comparison with Experiment of Accurate MC-DF Relativistic Estimates (cm1)5 of Fine-Structure Triplet Splittings in He Transition 3

3

2 S  2 P0 23S  23P1 23S  23P2

Corrected Total

Observed

9231.8562 9230.8681 9230.7918

9231.85640(50) 9230.86830(50) 9230.79200(50)

19.9 TWO-ELECTRON MOLECULAR SYSTEMS: H2 AND HHeD For the biatomic two-electron molecule in the Born-Oppenheimer approximation, the relativistic Hamiltonian is the same as that for the two-electron atom, provided the bare nuclei one-electron Hamiltonian h^i in Eqn (114) is replaced by its two-centre counterpart, that for the hydrogen molecule being 1 1 1 ; h^1 ¼  721   2 r1 rB1

1 1 1 h^2 ¼  722   2 r2 rA2

(121)

A fully numerical two-dimensional approach for the uncorrelated electronic DF equation of two-electron diatomics was developed by Laaksonen and Grant (1984b) and tested on the ground states of H2 and HHeþ, following their previous work on the one-electron diatomics considered in Section 19.7. Using atomic units, the DF equation for a two-electron case in a potential V is written as 

V ^ p cs$^

^ p 
cs$^ V  2c2



JL ðrÞ JS ðrÞ

! ¼ε


! JL r JS ðrÞ

(122)

where V ¼

Z1 Z2  þ Ve ðrÞ r1 r2

(123)

is the Coulomb potential for the diatomic molecule, and Ve(r) is the interelectronic Coulomb potential obtained by solving Eqn (126). As usual, p^ ¼ iV is the linear momentum operator in atomic units; ^ are the Pauli spin matrices; JL and JS are the large and small components of the wave function, s respectively; and c is the speed of light. The total electronic energy for the ground state of the two-electron system is calculated from the typical independent-particle form E ¼ 2ε  hre ðrÞVe ðrÞi 1 cm1 ¼ 4:55633525  106 Eh .

5

(124)

19.9 Two-electron molecular systems: H2 and HHeþ

851

Elimination of the small component as seen in the previous section gives h
n h
1 i 2 2 i o L $ V þV J ðrÞ ¼ εJL ðrÞ 7  c2 V 2c2 þ ε  V  c2 2c2 þ ε  V

(125)

For the two-electron case, the interelectronic Coulomb potential Ve(r) is found through use of the Poisson’s equation 72 Ve ðrÞ ¼ 4pre ðrÞ

(126)

as in the non-relativistic case, but where now the electron density re(r) has the two components




 re ðrÞ ¼ JL r $ JL ðrÞ þ JS ðrÞ $JS ðrÞ

(127)

where the asterisk denotes complex conjugation and the dot the scalar product. The best point distribution for wave function and potentials was found using an elliptical coordinate system ðx; hÞ through the following substitution: 

x ¼ cosh m; h ¼ cos n;

0 > n 2 g4 n > > n 2 g4 n : ; 2 2n2  2 g þ 2 1 8 n þ 1  g 2 2k jkj jkj 4k jkj jkj 8 9 > > > > < = g2 3g4 2 ¼ mc 1  þ       2 4 2 > > g g n > > g2 g4 n : ; 2n2 1  1 þ 8n4 1  þ 2 2 1 njkj 4n2 k2 jkj njkj 4n k jkj         g2 g2 g4 n 3g4 2g2 g4 n ¼ mc2 1  2 1 þ  2 2 1 þ 2 2 1 þ 4 1þ 2n njkj 4n k 8n njkj 2n k jkj jkj      g2 g4 3 g4 g2 g4 n 3 ¼ mc2 1  2  3 þ 4 ¼ mc2 1  2  4  2n 2n jkj 8 n 2n 2n jkj 4

E ¼ mc2 1 þ h

g2

i2

which is the required Eqn (101) of the main text. Equation (101) should be compared with Sommerfeld’s result obtained using the old quantum theory 13 2 0 g2 g4 B n 3C7 6 E ¼ mc2 41  2  4 @  1 4A5 2n 2n lþ 2     1 1 rather than j þ . The first term on the right-hand side of where jkj is replaced by l þ 2 2 Sommerfeld’s result is the rest energy of the electron. The second term is seen to be equal to 

mc2 g2 mZ 2 e4 Z2 ¼  ¼  2n2 2n2 2Z2 n2

in atomic units, the energy of the hydrogen atom as resulting from the usual Schroedinger’s theory. The third term is the fine-structure energy, which removes the degeneracy between

862

CHAPTER 19 Relativistic molecular quantum mechanics

energy levels having the same value of n and different values of l. The total spread of the finestructure levels for a given value of n obtained by Sommerfeld is mc2 g4 n  1 1 n3 n 2 a result which is much larger than that observed by experiment for the hydrogen spectrum, while Dirac’s result (102) mc2 g4 n  1 n3 2n is substantially less than Sommerfeld’s result and agrees with experiment.

CHAPTER

Molecular vibrations

20

CHAPTER OUTLINE 20.1 20.2 20.3 20.4 20.5 20.6 20.7 20.8 20.9 20.10

Introduction ............................................................................................................................. 863 Separation of Translational and Rotational Motions.................................................................... 863 Normal Coordinates in Classical and Quantum Mechanics .......................................................... 864 The Born–Oppenheimer Approximation....................................................................................... 867 Electronically Degenerate States and the Renner’s Effect in NH2 ................................................. 872 The Jahn–Teller Effect in CHD 4 .................................................................................................. 874 The Von Neumann–Wigner Non-crossing Rule in Diatomics ........................................................ 878 Conical Intersections in Polyatomic Molecules .......................................................................... 878 Problems 20 ............................................................................................................................ 883 Solved Problems ...................................................................................................................... 884

20.1 INTRODUCTION In this last chapter, we shall give a brief outlook of the field of molecular vibrations. Since the subject is well covered by many treatises (see, for instance, Pauling and Wilson, 1935; Wilson et al., 1955; Herzberg, 1956, 1966; Califano, 1976), we shall limit ourselves to a short introduction to the fundamental problem of separating translational and rotational motions from the internal (vibrational) motions of the nuclei, and to some consideration of the normal coordinates in classical and quantum mechanics, which are introduced in Sections 20.2 and 20.3. Most attention will be given, instead, to the interaction of the electronic with the nuclear motions, by considering the so-called vibronic terms in the molecular Hamiltonian, since these small correction terms are responsible for important structural effects in linear (Renner) and polyatomic (Jahn–Teller) molecules, in the non-crossing rule in diatomics and the conical intersections in polyatomics. Much of the following is due to the excellent papers of Longuet-Higgins (1961, 1975) and to the lecture notes delivered by Persico (1989) at the Theoretical Chemistry School in Villa Gualino, Turin (Italy), during 14–23 September 1989. Problems and solved problems conclude the chapter as usual.

20.2 SEPARATION OF TRANSLATIONAL AND ROTATIONAL MOTIONS Even in quantum mechanics, the dynamical study of a system of interacting particles (nuclei þ electrons) in a molecule is much simplified by the use of an appropriate reference system allowing to separate translational and rotational motions from the internal motions of the molecule. Elementary Molecular Quantum Mechanics 2E. http://dx.doi.org/10.1016/B978-0-444-62647-9.00020-8 Ó 2013, 2007 Elsevier B.V. All rights reserved

863

864

CHAPTER 20 Molecular vibrations

The translational motion of a molecule as a whole is easily separated from rotational and internal motions, since we can always write the total wavefunction for the N particles as
 ^ $ R Jrotint ðx1 ; x2 ; /; x3N3 Þ (1) Jðr1 ; r2 ; /; rN Þ ¼ exp iK ^ the where ri denotes the position of the i-th particle, R the position of the molecular centre of mass, K total momentum vector operator, which commutes with the Hamiltonian and is hence a constant of the motion. Therefore, Jrotint depends on 3N3 coordinates xj which express the relative positions of the N particles. It is possible to choose the xj coordinates in a way such that the kinetic energy operator in the barycentric system is expressible as a sum of terms quadratic in the conjugate momenta1 ^j ¼ iv=vxj : p 1 T^ ¼ 2

3N3 X p^2j j¼1

(2)

mj

where mj is the reduced mass for the j-th particle. This is possible in the so-called Jacobi coordinate system, where the coordinates of the second atom are defined in relation to the coordinates of the first one, those of the third atom in relation to the centre of mass of the first two, and so on, those of the j-th atom being in relation to the centre of mass of the first j  1 atoms. The Jacobi coordinate system is rigorously applied to the nuclear coordinates, whereas the electronic coordinates are referred just to the centre of mass of the nuclei, what implies a small correction to the kinetic energy term Eqn (2) of the order of m/M, the ratio between the electron mass m and the total nuclear mass M. The separation of the rotational motion is more complicated. For the biatomic molecule it is possible to define a rotating system with an axis coinciding with the internuclear separation so that the nuclear angular momentum is zero in this system. For the polyatomic molecule, it is possible to use either (1) the Eckart system (Eckart, 1935), where the angular momentum of the rotating system is zero at the equilibrium position of the nuclei in the molecule, or (2) the principal axis system of Meyer and Gu¨nthard (1968), where the rotating axes are chosen to be the principal axes of inertia of the molecule.

20.3 NORMAL COORDINATES IN CLASSICAL AND QUANTUM MECHANICS Assuming we have separated translational and rotational motions of the molecule, we shall refer in the following only to the internal degrees of freedom (the vibrational terms) which are 3N  6 for a polyatomic molecule and 3N  5 for the linear molecule, so omitting for brevity the specification of the upper term in all summations. We follow here a simplified approach, which is a particular case of the Eckart’s one, namely, we define the nuclear coordinates as a linear combination with constant coefficients of the displacements ri of the nuclei from their equilibrium positions, ri being a vector of Cartesian components xi, yi, zi. It is customary to introduce mass-weighted coordinates:2 1=2

1=2

1=2

xi ¼ mi xi ; hi ¼ mi yi ; 2i ¼ mi zi 1

Whenever possible we use the atomic units (au) discussed in Chapter 1. Our coordinates are assumed to be real.

2

i ¼ 1; 2; /; N

(3)

20.3 Normal coordinates in classical and quantum mechanics

865

where we put for convenience x1 ; h1 ; 21 ; x2 ; h2 ; 22 ; /; 2N ¼ q1 ; q2 ; q3 ; /; q3N

(4)

and internal coordinates s obtained from the qs by the transformation X Bij qj si ¼

(5)

j

giving the kinetic and potential energies of the system as X X   s_i G1 ij s_j ; 2V ¼ si fij sj 2T ¼ i; j

(6)

i; j

where P s_i ¼ dsi =dt is the first-time derivative of si, fij are the elements of the force constant matrix, and  Gij ¼ r Bir m1 r Brj . Putting these equations in matrix form, we can write 2T ¼ s_ y G1 s_ ;

2V ¼ sy F s

(7)

where s_ ¼ ds=dt. We now introduce a transformation L which simultaneously reduces our matrices in Eqn (7) to diagonal form:3 L

s / Q;

s ¼ LQ;

sy ¼ Qy Ly

(8)

so that   _ y Ly G1 L Q; _ 2T ¼ Q

  2V ¼ Qy Ly F L Q

(9)

If matrix L is such that Ly G1 L ¼ 1 we obtain _ _ y Q; 2T ¼ Q

2V ¼ Qy L Q 0 2T ¼

X

2 Q_ i ;

2V ¼

i

X

li Q2i

(10)

i

where L ¼ Ly F L is a diagonal matrix. In this way we obtain quadratic forms for both kinetic and potential energies where the cross-terms are absent. The Qs are known as normal coordinates of the system. In normal coordinates, the classical equations of motion in Lagrangian form become   v vT vV vV þ ¼ 0; ¼ li Qi (11) vt vQ_ i vQi vQi giving Q€i þ li Qi ¼ 0 i ¼ 1; 2; /; 3N  6 namely, 3N  6 differential equations of the second order whose solutions in real form are
pffiffiffiffi  Qi ¼ Q0i sin li t þ di i ¼ 1; 2; /; 3N  6 3

This is the basis of the so-called GF method (Wilson et al., 1955) as shown in Problem 20.1.

(12)

(13)

866

CHAPTER 20 Molecular vibrations

where the Q0i s are amplitude constants and the di s phase factors of integration. In this special case, p then, ffiffiffiffi each one of the normal coordinates Qi undergoes a harmonic oscillation whose frequency, ni ¼ li =2p; is determined by the constant li. In this way, we obtain 3N  6 particular solutions of the equations of motion, one for each root of the secular equation arising from the diagonalization of matrix F. A general solution can be obtained by adding all Eqn (13) together, yielding
pffiffiffiffiffi  X lk t þ dk Q0k Bik sin (14) qi ¼ k

This solution of the equations of motions contains 2(3N  6) arbitrary constants (the amplitudes Q0k and the phase factors dk ) whose value in each particular case is determined from the knowledge of the initial positions and velocities of the nuclei in the molecule. Turning to quantum mechanics, the wave equation for the nuclear motion in terms of the mass-weighted coordinates qi will be X v2 J i

vq2i

which in normal coordinates becomes X v2 J i

vQ2i

þ

þ ðW  VÞJ ¼ 0

! 1X 2 W li Qi J ¼ 0 2 i

(15)

(16)

a partial differential equation in 3N  6 variables which is separable into 3N  6 one-dimensional equations if we put

obtaining the equations

J ¼ j1 ðQ1 Þj2 ðQ2 Þ/j3N6 ðQ3N6 Þ

(17)

  d2 ji 1 2 þ Wi  li Qi ji ¼ 0 i ¼ 1; 2; /; 3N  6 2 dQ2i

(18)

each of which is identical with the equation for the one-dimensional harmonic oscillator of Chapter 3. The total molecular energy W is then the sum of the energies Wi associated with each normal coordinate, namely,  X X 1 vi þ hni ni ¼ 0; 1; 2; / Wi ¼ (19) W¼ 2 i i pffiffiffiffi where vi is the vibrational quantum number and ni ¼ li =2p is the classical frequency of the i-th normal mode of vibration. Symmetry considerations can be applied to simplify the solution of the secular equation, yielding symmetry combinations of the normal coordinates for which hold all what was said in Chapter 8 on group theory. The applications of group theory to the analysis of molecular vibrations is discussed in great detail in Chapter 6 of Wilson et al. (1955). A detailed example of the normal coordinate analysis of the vibrational problem in HCN in the valence field approximation is given as Problem 20.2 (Dixon, 1965).

20.4 The Born–Oppenheimer approximation

867

20.4 THE BORN–OPPENHEIMER APPROXIMATION The Born–Oppenheimer approximation concerns the separation, in molecules, of the slow motion of the nuclei from the fast motion of the electrons. Let the wave equation for the molecular motion be ^ ¼ WJ HJ where H^ is the molecular Hamiltonian in atomic units ! X 1 X X 1 1 2 2 ^  Va þ  Vi þ Ven þ Vee þ Vnn ¼  V2a þ H^e þ Vnn H¼ 2M 2 2M a a a a i

(20)

4

(21)

and H^e the electronic Hamiltonian H^e ¼

X 1  V2i þ Ven þ Vee 2 i

(22)

The molecular wavefunction J ¼ Jðx; QÞ

(23)

is a function of the electronic coordinates x and of the nuclear coordinates Q. According to Born–Oppenheimer (1927), the heavier nuclei move so slowly that, in the average, electrons see only the position of nuclei and not their velocity. Therefore, Born and Oppenheimer assumed that the electronic wavefunction je will depend on the electron coordinates x, being only parametric in the Qs, which describe the nuclear configuration: J z je ðx; QÞcn ðQÞ

(24)

The electronic wavefunction je is assumed normalized and satisfying the electronic wave equation:   H^e je ¼ Ee Q je (25) Z (26) hje jje i ¼ dx je je ¼ 1 where Ee(Q) is the electronic energy, which depends on the configuration Q of the nuclei. Considering je cn as a nuclear variation function with je ¼ fixed; Longuet-Higgins (1961) showed that, for non-degenerate electronic states, the best nuclear function satisfies the eigenvalue equation (Problem 20.3): ! X 1  V2 þ U^e ðQÞ cn ðQÞ ¼ Wcn ðQÞ (27) 2Ma a a

4

Ma is the mass of nucleus a in units of the electron mass.

868

CHAPTER 20 Molecular vibrations

where U^e ðQÞ is a potential energy operator for the motion of the nuclei in the electron distribution of the molecule X 1 Z X 1 Z  2 ^ d xje 7a je  dx je Va je $ Va U e ðQÞ ¼ Ee ðQÞ þ Vnn  a 2Ma a Ma Z X 1 Z 1 X 2 d x je P^a je þ dx je P^a je $ P^a P^a ¼ iVa ¼ Ue ðQÞ þ (28) 2Ma a M a a The last two terms in Eqn (28) describe the coupling between the motion of the nuclei and that of the electrons and are called vibronic terms. Since they are small in comparison with the first two terms in Eqn (28) (being of the order of 1=Ma z103 ), they can be overlooked in a first approximation. In this way, it is possible to define a potential energy surface for the motion of the nuclei in the field provided by the nuclei themselves and by the molecular electron charge distribution: Ue ðQÞ z Ee ðQÞ þ Vnn ¼ EðQÞ

(29)

We shall refer to Eqn (29) as to the molecular energy in the Born–Oppenheimer approximation. In this approximation, the nuclear wave equation is ! X 1 2  V þ Ue ðQÞ cn ðQÞ ¼ Wcn ðQÞ (30) 2Ma a a where Ue(Q) acts as an ‘effective’ potential for the nuclei. Equation (30) determines the nuclear motion (e.g. molecular vibrations) in the Born–Oppenheimer approximation, which is so familiar in spectroscopy. The adiabatic approximation includes the third term in Eqn (28), which describes the effect on je of the nuclear Laplacian 72a ; while the last term in Eqn (28) gives the effect on je of the nuclear linear momentum ðP^a ¼ iVa Þ. When this term, which involves the gradient operator Va, is zero or very small we speak of diabatic approximation. The electronic Hamiltonian Eqn (22) is a function of the nuclear coordinates, even if, in the Born–Oppenheimer approximation Eqn (29), it is simply a multiplier with respect to them, so that we can write for the K-th electronic state jK H^e ðQÞjjk ðQÞi ¼ UK ðQÞjjK ðQÞi

(31)

where the eigenvalues UK and the eigenfunctions jK both depend on the Qns. The vibronic matrix elements of the molecular Hamiltonian Eqn (21) then become " ! D E Z X v2 1  jK cu H^ jL cv ¼ /dQs /cu UK ðQÞ  dKL 2 r vQ2r  # 1X ðrÞ v ðrÞ  2gKL þ tKL cv 2 r vQr where we have put ðrÞ gKL

¼

v jK vQ

r

 jL ;

ðrÞ tKL

¼

2  v jK 2 jL vQ r

(32)

(33)

20.4 The Born–Oppenheimer approximation

869

We notice that, if the electronic wavefunctions jK and jL are orthonormal, matrix t(r) is Hermitian, while matrix g(r) is anti-Hermitian (Problem 20.4). For real electronic wavefunctions this implies ðrÞ that the diagonal elements of matrix g(r) are zero, gKK ¼ 0. Here, the cu s are eigenfunctions of the vibrational Hamiltonian: 1 H^vib ¼ UK ðQÞ  2

3N6 X r¼1

v2 vQ2r

(34)

where the electronic energy UK(Q) plays the role of an ‘effective’ potential for the nuclear motion. Since the potential UK(Q) does depend on the electronic state K, even the vibrational functions will carry an index K, and will be different according to the electronic state to which they belong: H^vib jcKu i ¼ EKu jcKu i

EKu ¼ WKu

(35)

The relative minima of the function UK(Q) are equilibrium positions for the nuclei, and so determine the shape of the molecule. In fact, to every sufficiently deep minimum will correspond a fundamental vibrational state with the maximum of jcj2 close to the equilibrium point. The saddle points correspond to the transition states for the chemical reaction. Over 99% of the chemical and physico-chemical phenomena depend on the lowest part of the potential hypersurface U0(Q), while only in electronic spectroscopy and photochemistry it is necessary to consider the wavefunctions besides the electronic energies. ðrÞ We can add the diagonal elements tKK to the potential UK in the Schroedinger’s vibrational Eqn (35) without any essential complication. If the electronic and vibrational wavefunctions satisfy Eqns (31) and (35), the diagonal matrix elements of the molecular Hamiltonian Eqn (21) reduce to the vibronic energies EKu: E D (36) jK cKu H^ jK cKu ¼ EKu Off-diagonal elements of H^ occur only for K s L, not for K ¼ L, u s v:  D E X

ðrÞ v 1 ðrÞ cKu gKL ðQÞ þ tKL ðQÞ cLv jK cu H^ jL cv ¼  vQr 2 r

(37)

In this context, we can say that the Born–Oppenheimer approximation consists of neglecting the matrix elements of Eqn (37), so that matrix H is diagonal in the basis of the product functions jjK cKu i: ðrÞ In the strict Born–Oppenheimer approximation the diagonal terms tKK are neglected, while they are included in the so-called adiabatic approximation, as already said before. In any case, the validity of the Born–Oppenheimer approximation depends on the ratio between the matrix elements Eqn (37) and a typical energy difference DEKu,Lv ¼ EKu  ELv. (1) Electronic ground state As an example, consider a not too excited vibrational state of the electronic ground state (K ¼ 0). The vibrational state interacts only with excited electronic states (L > 0). For a molecule near the equilibrium configuration, the first excited electronic state lies at least 0.1 Eh above the ground state, so that DEKu,Lv > 0.1 Eh. This order of magnitude depends in part on the fact that in the

870

CHAPTER 20 Molecular vibrations

Schroedinger equation for the electrons compares the electron mass m, and in part on the shape of ðrÞ ðrÞ the Coulombic potential V(x,Q). The matrix elements gKL v=vQr and tKL imply, respectively, first derivatives of jL and cLv ; and second derivatives of jL . The gradient operator is a linear combination of nuclear momentum operators, with matrix elements between the cKu functions pffiffiffiffiffiffiffi ffi of the order of Tvib ; the square root of the nuclear kinetic energy: since Tvib z102 w103 Eh ; we shall have hcKu jv=vQr jcLv iz0:1Eh . Usually, the electronic wave functions jL ðx; QÞ depend on the Qs much less than the cLv ðQÞ; which can change many times in a range of 0.1w1a0. Therefore, the derivatives vjL =vQr are very small, usually ðrÞ gKL >  > ^ aP ^ a cn0 ¼ Ecn ^ Pa =2Ma cn þ U ðQÞ þ O > > < e a a (47) " #
 > X X > 2 > ^ ^ a cn þ U 0 ðQÞ þ > P^a =2Ma cn0 ¼ Ecn0 Oa P > e : a

a

where7 Ue ðQÞ ¼ We ðQÞ þ

X a

Z ð1=2Ma Þ

dxje P^a je 2

(48) ðaÞ

^ a is the coupling gradient operator (see g 0 ): with a similar equation for Ue0 ðQÞ; and O ee Z
 ^ ^ a ¼ dxje P ^ a =Ma je0 ¼ Oa O

(49)

The appearance of the coupled equations for the nuclear functions cn and cn0 is an essential feature of the doubly degenerate electronic state. The off-diagonal operators are important because, since ðje ; je0 Þ are nearly degenerate, a very small change in certain nuclear coordinates will induce a very heavy mixing of je with je0 , so that some terms in the neighbourhood of the nuclear configuration Q0 are very large. Another effect of this coupling is that, in Eqn (47), some of the terms in the sum will also be very large when a symmetrical configuration is approached. Hence, although We and We0 are continuous functions of Q, Ue and Ue0 may not be, so that particular care must be used in handling these two functions. Longuet-Higgins (1961) used Eqn (47) in a study of the Renner’s vibronic energy levels of a linear triatomic molecule, the NH2 radical in a P electronic state. Leaving the details of the calculation to Longuet-Higgins’ paper (1961; see also Pople and Longuet-Higgins, 1958), we can say that out of the 7

The * symbol has no operational significance.

874

CHAPTER 20 Molecular vibrations

four vibrational coordinates, the relevant ones in this case are the doubly degenerate bending modes (Qx, Qy), which when expressed in polar coordinates depend on the amplitude r and azimuth 4 of the bending deformation. When r is small, the functions ðje ; je0 Þ are found to be much more sensitive to changes in 4 rather than to changes in the non-degenerate coordinates Q1, Q2 and r. To a certain degree of approximation, it is possible to separate the non-degenerate vibrations from the others. When thirdorder terms are neglected, cn and cn0 can then be factorized as     cn ¼ h1 ðQ1 Þh2 ðQ2 Þh3 r; 4 ; cn0 ¼ h1 ðQ1 Þh2 ðQ2 Þh03 r; 4 (50) where h1 and h2 satisfy harmonic oscillator equations with eigenvalues E1 and E2, with harmonic force constants f1 and f2, respectively, whereas the pair of bending functions h3 and h03 satisfy two coupled differential equations similar to Eqn (47), which are given in matrix form in terms of the (2  2) effective Hamiltonian: 1 0 i 2   2 1  h
2 1 rP ^4 ^r þ P^24 þ 1 U þ 2Mr P i Mr C B C (51) Heff ¼ B i h
 A @ 2  2 1  1 2 0 2 ^ ^ ^ P4 i Mr U þ 2Mr rP r þ P 4 þ 1 as

0



 ^ B H eff 11  E3 @   H^eff 21

 

H^eff 

H^eff

1



22

C h3 A h03  E3 12

! ¼0

(52)

where U ¼ Ue’(r) and U 0 ¼ Ue ðrÞ are obtained from Ue(Q1, Q2, r) and Ue’(Q1, Q2, r) by minimizing with respect to Q1 and Q2, and subtracting U0, and E ¼ U 0 þ E1 þ E 2 þ E3

(53)

In the case of the lowest vibronic quantum number, K ¼ 0, the vibronic wavefunctions take a relatively simple form when one of the potential functions, U 0 say, has a deep minimum at some finite value of r, and the minimum of U lies sufficiently far from it, as found experimentally by Dressler and Ramsay (1959) for the NH2 radical. The molecule is then markedly bent in its lowest vibronic levels, and in these levels the contribution of je to the wavefunction may be neglected in comparison to that of je0 . In this case, the total wavefunction is a simple product je0 cn0 and the problem of determining the energy levels can be handled by the standard theory of the rotating-vibrating molecule. The problem is much more difficult when K s 0, but details are left to the Pople and Longuet-Higgins’ (1958) paper. These theoretical calculations of the Renner’s vibronic term values in the NH2 radical are seen to be in good agreement with those observed experimentally by Dressler and Ramsay (1959).

20.6 THE JAHN–TELLER EFFECT IN CHD 4 Dixon (1971) did some ab initio calculations of the vibronic potential energy surface for CHþ 4 with the aim of studying its photoelectron spectroscopy. In the following, after a short introduction to the theory of the Jahn–Teller effect, we shall summarize Dixon’s most important energy conclusions for this molecule-ion.

20.6 The Jahn–Teller effect in CHþ 4

875

1. Theory The potential energy surface for an orbitally degenerate electronic state exhibiting Jahn–Teller instability is usually discussed in terms of an expansion in powers of nuclear displacements. The electronic Hamiltonian for an arbitrary fixed nuclear configuration is developed in a Taylor expansion in nuclear displacements from a reference configuration of high symmetry. Integration over the electronic coordinates then gives a vibronic Hamiltonian which is a function of the nuclear coordinates only. This vibronic Hamiltonian contains not only totally symmetric terms but also terms which belong to non-totally symmetric representations of the reference molecular point group. 2 The electronic ground state of tetrahedral CHþ 4 is a T 2 state, whose orbital degeneracy may be removed by vibronic perturbations of species e and t2 (Herzberg, 1966). Choosing a basis of real 2 T 2 component functions which transform as translations along cubic axes (x, y, z), the electronic energies of the three Cartesian components of the 2 T 2 state for the distorted nuclear configurations are the eigenvalues of the (3  3) effective Hamiltonian matrix containing all coupling terms: 0

Heff


pffiffiffi ^0 þ 1 H^0e;a  3H^0e;b H B 2 B B 0 B ¼B H^t2 ;z B @ 0 H^t2 ;y

0 H^t2 ;z

0 H^t2 ;y

pffiffiffi 0  1
0 H^0 þ H^e;a þ 3H^e;b 2

0 H^t2 ;x

0 H^t2 ;x

0 H^0  H^e;a

1 C C C C C C A

(54)

0 0 where H^0 includes all terms of irreducible representations a1, H^e;a and H^e;b are the components of 0 0 0 the perturbations belonging to the irreducible representation e, and H^t2 ;x ,H^t2 ;y ,H^t2 ;z are the components of the perturbations belonging to the irreducible representation t2. The nine (3N  6) vibrational displacement coordinates for CH4 transform as a1 þ e þ 2t2 and are given in Eqn (2) of Dixon’s paper. H^0 contains the same harmonic and anharmonic force constants as for a 1 A1 state, 0 0 while the non-totally symmetric perturbation terms H^e and H^t complete to second order in the 2

displacements coordinates, are given in Eqn (3) of Dixon’s paper. There are 3 first-order Jahn– Teller splitting parameters and 12 second-order parameters. The absolute minimum in the electronic energy of the lowest component of the 2 T 2 state corresponds to a tetragonal distortion to the 0 point group D2d if H^e is the dominant perturbation, or to a trigonal distortion to the point group C3v

0 if H^t2 is the dominant perturbation, these two energies being not additive. Dixon’s work was based on separate calculations of the total electronic energy as a function of the symmetry coordinates si(i ¼ 1,2,3,4) for nuclear configurations of symmetry Td, D2d, C3v and C3v, respectively. 2. Results The calculations of the potential energy surface were carried out in the linear combination of atomic orbitals (LCAO)-molecular orbital (MO)-self-consistent field (SCF) approximation using a minimal basis (2,3/2,3) of appropriate linear combinations of Gaussian-type orbitals (GTOs) to approximate Slater-type orbitals (STOs), three Gaussians for each 1s STO, and two GTOs for each 2s and 2p (54 Gaussian functions contracted to 18 STOs). The orbital basis consisted of Clementi’s recommended double-zeta basis for carbon 1s,

876

CHAPTER 20 Molecular vibrations

Table 20.1 Calculated MO-SCF Stationary Energies for Different Structures of CH4 and CHþ 4 Point group distortion Td Td D2d(s2)

D4h(s2) Td(C3v) C3v(s3)

C3v(s4)

Structure

CH4 1 A1

2 CHD B2 or 2 A1 4

2 2 CHD 4 ð T2 Þ E

r ¼ 1.07 A˚ (4) r ¼ 1.147 A˚ (4)

40.1655

39.6472

39.6472

40.1529

39.6584

39.6584

40.0867

39.7101

39.5512

39.8879

39.6765

39.3539

39.6659

39.6588

40.1311

39.6829

39.6226

40.1320

39.6864

39.6221

r ¼ 1.147 A˚ (4) :HCH ¼ 141.2 (2) :HCH ¼ 96.3 (4) r ¼ 1.147 A˚ (4) r ¼ 1.147 A˚ (4) :HCH ¼ 109.5 (6) r ¼ 1.345 A˚ (1) r ¼ 1.081 A˚ (3) :HCH ¼ 109.5 (6) r ¼ 1.147 A˚ (4) :H1CHi ¼ 96.9 (3) :HCH ¼ 118.6 (3) :HCH ¼ 141.2 (2) :HCH ¼ 96.3 (4)

2s and 2p orbitals (1964) with orbital exponents taken from Arrighini et al. (1968) for the ground state of CH4. The reliability of the calculations was checked by comparing the calculated structure and force constants of neutral CH4 with experiment. For each open-shell calculation on CHþ 4 use was made of Roothaan’s (1960) formulation of the symmetry restricted Hamiltonian for the appropriate point group and component state. Some energy results for the different nuclear configurations of CH4 and CHþ 4 are given in Table 20.1, while in Table 20.2 the calculated force constants are compared with experimental results whenever possible. The minimum energy for the 1 A1 ground state of CH4 was found to be 40.1655Eh at a bond ˚ . The experimental equilibrium CH bond length is 1.085 A ˚ (Kuchitsu and length of 1.079 A Bartell, 1962), the calculated symmetric stretching force constant being about 10% higher than the experimental value, what is not unexpected for uncorrelated calculations (see the results for the H2O molecule in Table 16.4 of Chapter 16). The theory given in (1) shows that the minimum energy occurs for a tetragonal distortion by s2 for one of the three equivalent D2d point groups, for instance, by excitation of s2,a. Only 8 of the 24 symmetry operations of Td are retained in D2d, and the orbital degeneracy of the state 2 T2 of CHþ 4 is now split, leading to 2 B2 and 2 E states. The minimum energy was attained for the 2 B2 state at a strongly distorted nuclear configuration, with two HCH angles of 141.2 and four of 96.3 , corresponding to an energy stabilization of 0.0517Eh (1.41 eV). The lowering in energy for the C3v distortions is sensibly less than that for the D2d distortion. These results do not confirm the semiempirical findings of a significant trigonal distortion to C3v symmetry obtained by Coulson

20.6 The Jahn–Teller effect in CHþ 4

877

Table 20.2 Calculated MO-SCF and Observed Force Constants for CH4 and CHþ 4 CH4 ð1 A1 Þ Constant

Calculated

Observed

f11 f22 f33 f44 l2 l3 l4 l22 l33(t) l44(t)

6.45 0.505 4.56 0.513

5.84 0.486 5.38 0.458

2 CHD 4 ð T2 Þ Calculated

5.05 0.429 4.12 0.422 0.432 0.577 0.261 0.004 0.145 0.022

Units mdyn/A˚ mdyn/A˚ mdyn/A˚ mdyn/A˚ mdyn mdyn mdyn mdyn/A˚ mdyn/A˚ mdyn/A˚

and Strauss (1962), who used the Hellmann–Feynman theorem8 with simple assumptions about location and distribution of the total electronic charge density, taking from spectroscopic work some values of the necessary force constants. Distortion to the square planar configuration of D4h symmetry turns out to be 0.0336Eh (0.91 eV) higher than the corresponding 2 B2 state, and as much as 0.1973Eh (5.36 V) higher than the corresponding 2 E state of D2d symmetry. Dixon ended his paper with a fairly extended discussion of the resulting photoelectron spectrum of CH4. A rigorous calculation of the expected vibrational intensity distribution in the photoelectron spectrum of methane using the potential functions resulting from Table 20.2 would require the knowledge of the vibronic energy levels and wavefunctions, a formidable task involving three strongly coupled Schroedinger’s equations in eight vibrational displacements coordinates. However, a meaningful comparison with experiment may be done using simplified models. We shall not insist more on this argument here, leaving the interested reader of spectroscopy to details of the original Dixon’s paper. The Jahn–Teller effect in the benzene cation C6 Hþ 6 was studied by Salem (1966) using perturbation theory within the Hu¨ckel’s model, and by Ko¨ppel et al. (1988) who estimated the Jahn–Teller coupling constants for the electronic 2 E2g and 2 A2u electronic states of C6 Hþ 6 using ab initio SCF and semiempirical CNDO/S calculations. The latter authors showed that the model exhibits a variety of conical intersections which dominate the vibronic dynamics. Lastly, the topological aspects of the conformational stability problem in degenerate electronic states exhibiting Jahn–Teller distortions were examined in great detail by Liehr (1963) for a variety of regular polygons and regular polyhedra, as well as for a number of selected highly symmetric irregular polygons and polyhedra. 8

Clinton and Rice (1959) first reformulated the Jahn–Teller theorem in terms of the Hellmann–Feynman theorem, giving a discussion of the Jahn–Teller effect in the BH3 molecule and the NHþ 3 cation.

878

CHAPTER 20 Molecular vibrations

20.7 THE VON NEUMANN–WIGNER NON-CROSSING RULE IN DIATOMICS With reference to the last part of the discussion in Section 20.4, consider now the case of a diatomic molecule in which the electronic energies of two different states having the same symmetry, say j1 and j2 ; coincide. The electronic Hamiltonian in this subspace, orthogonal to all other electronic states K, is given by the (2  2) Hermitian matrix H with elements:   H11 H12 (55) H[ H12 H22 whose eigenvalues are the solutions of the quadratic secular equation H11  U H12 ¼0 H12 H22  U h i1=2 1 2 U1;2 ¼ ðH11 þ H22 ÞH ðH22  H11 Þ2 þ 4H12 2

(56) (57)

In order that the two solutions become identical, U1 ¼ U2, we must have H22 ¼ H11 and H12 ¼ 0. Suppose that we can vary a parameter contained in H^e ; for instance one or more values of the nuclear coordinates Qr. It is well possible that we can find one or more values of Qr for which H22 ¼ H11 and H12 ¼ 0, but in general the two equations will have different solutions. Therefore, in no point the energies of the two electronic states j1 and j2 of the same symmetry can be coincident or can cross, thus giving the famous non-crossing rule of Von Neumann and Wigner (1929) (see also Longuet-Higgins, 1975). A classical example is given by the first two singlet states of alkali halides such as NaCl (Spiegelmann and Malrieu, 1984a,b). At short internuclear distances its ground state is essentially ionic (NaþCl). At large distances, however, the neutral state ðNa; 2 S þ Cl; 2 PÞ is lower in energy, since the electron affinity of the chlorine atom (0.134Eh) is less than the ionization potential of the sodium atom (0.189Eh). This behaviour is essentially due to the Coulombic attraction between the two ions, Naþ and Cl. At intermediate distances, say R > 10a0, if we neglect the mutual polarization of the ions, the ionic state has an energy of about Uion xI:P:ðNaÞ  E:A:ðClÞ 

1 R

(58)

At the crossing point, the off-diagonal term is about 1.5  104Eh ¼ 30 cm1, so that the two ‘adiabatic’ curves, namely, the two eigenvalues of H as a function of R, should have a DU ¼ 2H12 x 60 cm1, after which the two curves detach themselves again. A schematic drawing of the avoided crossing in NaCl is given in Figure 20.2.

20.8 CONICAL INTERSECTIONS IN POLYATOMIC MOLECULES The crossing of potential surfaces in polyatomic molecules was first studied by Teller (1937). In a polyatomic molecule we have to consider more than a single nuclear coordinate Qr. If we change in

20.8 Conical intersections in polyatomic molecules

879

FIGURE 20.2 Schematic drawing of the avoided crossing in NaCl

H^e more than one parameter, it will be possible to find a set of values of the Qrs for which U2 ¼ U1; in general, in an N-dimensional space, two variables will be determined, as a function of the remaining (N  2), by the conditions H22 ¼ H11 and H12 ¼ 0. Therefore, in a plane Q1, Q2 we shall have one (or no one) point where U2 ¼ U1; in a three-dimensional space Q1, Q2, Q3 we shall have a curve, etc. Such a situation is said a conical intersection (Herzberg and Longuet-Higgins, 1963; Ohrendorf et al., 1988), such as the one shown schematically in Figure 20.3. Strictly speaking, the non-crossing rule is valid only in the case of just one vibrational coordinate, namely, for the diatomic molecule. For states of definite spin, states with different values of S or MS will always have H12 ¼ 0, as long as the magnetic terms in the Hamiltonian are not considered, and in these cases the non-crossing rule does not hold. In the same way, in a molecule possessing some symmetry, the crossing between states belonging to different irreducible representations (different types of symmetry) is allowed. If we define the Qr as symmetry coordinates (see the CH4 case of Section 20.6), we see that the totally symmetric coordinate will leave the molecular symmetry unchanged, while the others will lower the symmetry of the system. Consider, for instance, two coordinates, QS symmetric and QA antisymmetric with respect to a given symmetry elements of the point group G. Suppose that two states, jS symmetric and jA antisymmetric with respect to G, be degenerate for QS ¼ X0 and QA ¼ 0, and that along the QS coordinate the crossing is possible. If QA s 0, however, the symmetry element characterizing jS does no longer exist, H12 ¼ 0, and the avoided crossing occurs. We have now a conical intersection due to symmetry breaking. Another possible case of symmetry breaking occurs when an external perturbation modifies the energy levels of a molecule so that the crossing along a totally symmetric coordinate

880

CHAPTER 20 Molecular vibrations

FIGURE 20.3 Conical intersection of two potential energy surfaces

between the potential energy curves of two states of different symmetry becomes an avoided crossing, and the non-crossing rule holds. In all such cases, the adiabatic coupling terms may become very large, and we can ask if the adiabatic basis is the best for describing the stationary states of the system. An answer can be obtained from a deeper analysis of the time evolution of a molecule near an avoided crossing (see the small insert in Figure 20.2). Consider the time-dependent Schroedinger equation for the electrons only, when H^e depends on t through the nuclear coordinates: i

v jj ðtÞi ¼ H^e ½QðtÞjje ðtÞi vt e

We can expand je ðtÞ either on an adiabatic basis jjK i X jjL ðQÞicL ðtÞ jje ðtÞi ¼

(59)

(60)

L

or on a diabatic basis jhK i as jje ðtÞi ¼

X jhL ðQÞicL ðtÞ

(61)

L

Substituting Eqn (60) into Eqn (59) we have

! X vcL X X vra i þ cL $ Va jjL i ¼ H^e jjL icL vt a vt L L

(62)

20.8 Conical intersections in polyatomic molecules

881

with a like expression for Eqn (61). Multiplying Eqn (62) on the left by hjK j and integrating over the electronic coordinates we obtain ! X vcK X ðaÞ ¼ cL ðtÞ iHKL  va $ gKL (63) vt a L where HKL are the matrix elements of H^e in the given basis: they are diagonal only and equal to UK for ðaÞ

ðrÞ

the adiabatic basis, and gKL ¼ hjK jVa jjL i is the Cartesian coordinate equivalent of gKL ; in the ðaÞ gKL

diabatic basis, instead, ¼ hhK jVa jhL i is zero or very negligible. With just a single state, Eqn (63) becomes dcK ðtÞ ¼ iHKK cK ðtÞ dt which immediately integrates to



Zt

cK ðtÞ ¼ cK ð0Þexp i

(64)

dt0 HKK ðt0 Þ

(65)

0

To simplify the coupled equations, for the expansion in adiabatic functions we pose

Zt

cK ðtÞ ¼ aK ðtÞexp i

0

0

dt UK ðt Þ

(66)

0

so that, substituting in Eqn (63), we obtain

! Zt

X X vaK ðaÞ ¼ aL ðtÞ va $ gKL exp i dt0 ðUK  UL Þðt0 Þ vt a LsK

(67)

0

In the same way, for the expansion in diabatic functions, we can pose

Zt 0 0 cK ðtÞ ¼ dK ðtÞexp  i dt HKK ðt Þ

(68)

0

obtaining

Z

X vdK 0 0 ¼ i dL ðtÞHKL ðtÞexp i dt ðHKK  HLL Þðt Þ vt LsK t

(69)

0

The aK and dK coefficients are equal in modulus to the cK ones, so that jaK j2 and jdK j2 give the weight of the corresponding adiabatic or diabatic wavefunction in the time-dependent electronic state. Equations (67) and (69) show that the transitions between adiabatic states are due to the coupling terms ðaÞ gKL and are more probable for a high velocity of the nuclei, whereas the transitions between diabatic states are due to the electronic coupling terms HKL and do not depend on the velocity of the nuclei. Furthermore, the oscillating time factor tends to reduce the value of the integral in a way which is greater as greater is the difference in energy. The physical comprehension of physical facts is easier

882

CHAPTER 20 Molecular vibrations

when a basis is chosen that minimizes the coupling and the probability transitions. In molecular physics, this is very often the case of the adiabatic basis, when the Born–Oppenheimer approximation ðaÞ retains its validity. However, in the case of avoided crossings (small H12 and 6U, large values of gKL ) for diatomics or conical intersections in polyatomics, the use of a diabatic basis may be more convenient. This is the case, for instance, of the atomic or molecular collisions in molecular beam experiments, where usually high velocities of the colliding beams occur. Consider now a trajectory passing through an avoided crossing or near a conical intersection. In a diabatic description, if we use a two-state model such as the one already seen in Section 20.7, it is convenient to define a complex parameter a(t) such that d1 ðtÞ ¼ cos a;

d2 ðtÞ ¼ sin a

(70)

Since vd1 va ¼ sin a ; vt vt Equation (69) reduces to

vd2 va ¼ cos a vt vt

2 t 3 Z va ¼ iH12 exp4i dtðH11  H22 Þ5 vt

(71)

(72)

0

an equation that can be easily integrated with some approximations, valid for the strictly avoided crossing where the time of crossing is very short. Let Q be the coordinate along the trajectory and let Q ¼ Q0 be the crossing point, where H11 ¼ H22. We approximate the matrix elements of H^e in the vicinity of Q0 by posing   H11  H22 ¼ F Q  Q0 ; H12 ¼ constant (73) We notice that, in this case, the function

 X v ðaÞ va g12 ðQÞ ¼ j1 j2 ¼ gKL $ vQ jvj a

(74)

is a simple Lorentzian.9 We further assume that the velocity remains constant during the crossing, so that QðtÞ ¼ Q0 þ vt; the time scale being arbitrarily chosen so that Q ¼ Q0 at t ¼ 0. Then 2 t 3   Z va t2 0 05 4 ¼ iH12 exp i dt Fvt ¼ iH12 exp iFv (75) vt 2 0

If we integrate Eqn (75) from a time much before the crossing, N for the sake of simplicity, to a time much after the crossing, þN say, we shall have   ZN t2 aðfinalÞ  aðinitialÞ ¼ iH12 dt exp iFv (76) 2 N

9

After Lorentz Hendrik Antoon, 1853–1928, Dutch physicist, Professor of Mathematical Physics at the University of Leiden (The Netherlands); 1902 Nobel Prize for Physics.

20.9 Problems 20

883

A change of variables gives  aðfinalÞ  aðinitialÞ ¼ iH12

2 Fv

1=2 ZN   dx exp ix2

(77)

N

The integral in Eqn (77) can be evaluated by the technique of integration in the complex plane discussed in Chapter 5 (Problem 20.6) giving rffiffiffiffi ZN ZN  2  2 p dx exp ix ¼ 2 dz exp iz ¼ ð1 þ iÞ 2

N

(78)

0

so that we finally obtain aðfinalÞ  aðinitialÞ ¼ ði  1ÞH12


p 1=2 Fv

(79)

If the system is initially in state jh1 i, we shall have d1 (initial) ¼ 1, d2 (initial) ¼ 0, a (initial) ¼ 0. In our model, where we have a small transition probability from jh1 i to jh2 i, a is not very different from zero, so that we can put sin a x a, obtaining the final result: jd2 ðfinalÞj2 ¼

2p 2 H jFvj 12

(80)

which is the Landau–Zener’s rule (Landau, 1932; Zener, 1932) giving the transition probability between two diabatic states during a single passage through an avoided crossing.

20.9 PROBLEMS 20 20.1. Derive the equations of the GF matrix method mentioned in Section 20.3. Answer: jGF  l1j ¼ 0 Hint: Use the relations Eqn (8) defining matrix L. 20.2. A valence force field (VFF) calculation of the vibrational frequencies of HCN. Answer: The results are given in the section on Solved Problems. Hint: Use the normal coordinate analysis of Section 20.3 following Dixon’s (1965) suggestions. 20.3. Give a variational derivation of the Born–Oppenheimer Eqn (28) (Longuet-Higgins, 1961). Answer: Equation (28) of the main text. Hint: Use F ¼ je ðx; qÞcn ðqÞ as nuclear variational wavefunction and optimize cn subject to the normalization condition.

884

CHAPTER 20 Molecular vibrations

ðrÞ

20.4. Prove that matrix g(r) of elements gKL is anti-Hermitian. Answer: h i ðrÞ ðrÞ  gKL ¼  gLK Hint: Use the definition of anti-Hermitian operator given in Chapter 1 and the orthonormality of jK and jL : 20.5. Give a variational derivation of the coupled Eqn (47) (Longuet-Higgins, 1961) in the case of a doubly degenerate electronic state. Answer: Equation (47) of the main text. Hint: Use F ¼ je ðx; QÞcn ðQÞ þ je0 ðx; QÞcn0 ðQÞ as nuclear variational wavefunction and optimize F with respect to simultaneous variations in cn and cn0 subject to the normalization and orthogonality conditions. 20.6. Evaluate integral Eqn (78) of Section 20.8. Answer:  rffiffiffiffi ZN   p dz exp iz2 ¼ 1 þ i 2 N

Hint: Use the techniques of integration in the complex plane of Section 5.3.4 in Chapter 5.

20.10 SOLVED PROBLEMS 20.1. The GF method. From Eqns (8) and (10), we have Ly G1 L ¼ 1;

Ly FL ¼ L

so that, solving the first equation for Ly ¼ L1G and substituting in the second, we obtain after multiplying by L on the left: L1 GFL ¼ L 0 GFL ¼ LL a set of simultaneous equations which determine matrix L, whose elements are obtained from the eigenvectors of the secular equation jGF  l1j ¼ 0 pffiffiffiffi The eigenvalues li give instead the frequencies ni ¼ li =2p of the harmonic oscillations associated to each normal coordinate. 20.2. A VFF calculation of the vibrational frequencies of HCN. HCN is a linear triatomic molecule, and has therefore 3N  5 ¼ 4 degrees of vibrational freedom. A convenient set of four internal displacement coordinates would be one each for the increase in

20.10 Solved problems

885

FIGURE 20.4 The four internal coordinates si for HCN

the length of the bonds (stretching coordinates), and one each for the departure from linearity in two orthogonal directions perpendicular to the molecular axis (bending coordinates) as shown in Figure 20.4. In order that these internal coordinates si shall form a set from which to construct normal coordinates they must satisfy the stationarity conditions for the centre of mass and the rotational axes. The nuclear displacement vectors corresponding to small changes in each internal coordinate si ðs1 ¼ dr1 ; s2 ¼ dr2 ; s3 ¼ ðr1 r2 Þ1=2 dqÞ satisfying these conditions are those schematically drawn in the figure (Dixon, 1965) and their expressions are best written in matrix form. For the stretching of the m1  m2 and m2  m3 bonds along the z-axis we can set the following equations for the z-coordinates: 8 0 1 0 10 1 0 1 z1 m1 m2 m3 0 z1 > < m1 z1 þ m2 z2 þ m3 z3 ¼ 0 B C B CB C B C 0 @0A¼@ 0 1 1 A@ z2 A ¼ L1 @ z2 A z2  z3 ¼ 0 > : z3 z3 z2  z1 ¼ dr1 ¼ s1 s1 1 1 0 0 1 0 1 z1 0 B C C 1 B 0 @ z2 A ¼ L1 @ 0 A z3 s1 8 0 1 0 10 1 0 1 z1 m1 m2 m3 0 z1 > < m1 z1 þ m2 z2 þ m3 z3 ¼ 0 B C B CB C B C 0 @ 0 A ¼ @ 1 1 0 A@ z2 A ¼ L2 @ z2 A z1  z2 ¼ 0 > : z3 z3 z3  z2 ¼ dr2 ¼ s2 s2 0 1 1 0 1 0 1 z1 0 B C B C 1 0 @ z2 A ¼ L2 @ 0 A z3

s2

886

CHAPTER 20 Molecular vibrations

while in the zx-plane, the three equations for the change in the x-coordinates, the first one arising from geometrical requirements, the remaining two from the physical constraints,10 are 8x  x x2  x3 2 1 > þ xp  q ¼ jdqj > > r2 > r1 > < m1 x 1 þ m2 x 2 þ m3 x 3 ¼ 0 > > > > > : m1 r1 x1 ¼ m3 r2 x3

0

1 jdqj B r1 B C B C B 0 B @ 0 A¼B B m1 @ 0 m 1 r1 0 1 x1 B C 1 C 0 B @ x2 A ¼ L3 jdqj x3 0

1

1 1 þ r1 r2 m2 0

1 1 0 1 0 1  x1 x1 r2 C CB C B C CB C B x2 ¼ L3 @ x2 C A m3 C C@ A A x3 x3 m3 r2

Calculating the inverse matrices, we then obtain for the four displacement vectors: m2 þ m3 m1 1 s1 z1 ¼ ðL1 s1 ; z2 ¼ ðL1 s1 1 Þ13 dr1 ¼  1 Þ23 dr1 ¼ z3 ¼ ðL1 Þ33 dr1 ¼ M M s2 s3x

m3 m1 þ m2 1 s2 ; z3 ¼ ðL1 s2 z1 ¼ ðL1 2 Þ33 dr2 ¼ 2 Þ13 dr2 ¼ z2 ¼ ðL2 Þ23 dr2 ¼  M M   m2 m3 r2 1=2 m1 m3 r1 þ r2 ¼  x1 ¼ ðL1 Þ s3x ; x2 ¼  s3x ; jdqj 3 11 N r1 N ðr1 r2 Þ1=2   m1 m2 r1 1=2 s3x x3 ¼  N r2

s3y where

    m2 m3 r2 1=2 m1 m3 r1 þ r2 m1 m2 r1 1=2 y1 ¼  s3y ; y2 ¼  s3y ; y3 ¼  s3y N r1 N ðr1 r2 Þ1=2 N r2 M ¼ det L1 ¼ det L2 ¼ m1 þ m2 þ m3 ; N ¼ det L3 h i. ¼ m1 m2 r12 þ m1 m3 ðr1 þ r2 Þ2 þ m2 m3 r22 r1 r2

The total kinetic energy associated with simultaneous changes in all four displacement coordinates can then be expressed by  1X  2 1X mi x_i þ y_ 2i þ z2i ¼ s_r mrr0 s_r0 T¼ 2 i 2 r;r0 If we substitute the results from the previous expressions, putting for convenience s23 ¼ s23x þ s23y The last equation arises from the condition for rotational invariance, m1 x_1 r1 ¼ m3 x_3 r2 ; after multiplying both members by dt and integrating.

10

20.10 Solved problems

887

little calculation shows that the reduced masses mrr0 are given by m11 ¼ m1 ðm2 þ m3 Þ=M; m33 ¼ m1 m2 m3 =N;

m12 ¼ m21 ¼ m1 m3 =M;

m22 ¼ m3 ðm1 þ m2 Þ=M

m13 ¼ m31 ¼ m23 ¼ m32 ¼ 0

The general harmonic potential for the molecule is X sr frr0 sr0 V¼ r;r 0

where frr0 is the force constant involving atom r and atom r 0 ; and where, because of the symmetry of the linear triatomic molecule f13 ¼ f31 ¼ f23 ¼ f32 ¼ 0 The classical equations of motion in Lagrangian form are then given by   X X v vT vV þ ¼0 0 mrr0 s€r þ frr0 sr r; r 0 ¼ 1; 2; 3 vt vs_r0 vsr0 r r Putting equal to zero the phase factors dr ; in each normal mode of vibration each internal coordinate sr vibrates with a constant amplitude Ark at a frequency nk: pffiffiffiffiffi  pffiffiffiffiffi pffiffiffiffiffi sr ¼ Ark sin lk t ; nk ¼ lk =2p 0 lk ¼ 2pnk ¼ uk pffiffiffiffiffi  s€r ¼ lk Ark sin lk t ¼ lk sr Substituting in the equations of motion a set of three simultaneous equations in the amplitudes Ark is obtained: 8 ðf11  lk m11 ÞA1k þ ðf12  lk m12 ÞA2k ¼ 0 > > < ðf12  lk m12 ÞA1k þ ðf22  lk m22 ÞA2k ¼ 0 > > : ðf33  lk m33 ÞA3k ¼ 0 This set of simultaneous equations has non-trivial solutions, different from zero, if and only if the determinant of the coefficients is zero: f11  lk m11 f12  lk m12 0 f12  lk m12 f22  lk m22 ¼0 0 0 0 f33  lk m33 It is seen that this secular determinant factorizes into a quadratic and a linear equation, so that f33 is immediately obtained. However, the two stretching frequencies u1 and u2 are not sufficient to determine the three force constants f11, f22 and f12, so that we shall make the further simplifying assumption that the interaction force constant be f12 ^ aP ^ a cn0 ¼ Ecn P^a =2Ma cn þ O > Ue Q þ > < a a " #  > X^ X
2 > >  ^ a cn þ U 0 ðQÞ þ > P^a =2Ma cn0 ¼ Ecn0 Oa P > e : a

a

where

Z X 2 ð1=2Ma Þ dx je P^a je

Ue ðQÞ ¼ We ðQÞ þ

a

with a similar equation for

^ a is the coupling operator and O
 ^ ^ a =Ma je0 ¼ O dx je P a

Ue0 ðQÞ;

^a ¼ O

Z

In this way, we recover Eqns (47)–(49) of the main text. 20.6. Evaluation of integral Eqn (78) of Section 20.8. The integral ZN



2

dz exp iz N



ZN   ¼ 2 dz exp iz2 0

can be evaluated by the methods explained in Section 5.3.4 of Chapter 5 as follows. Choose the integration path in the upper half-plane of Figure 20.6, where y is the imaginary axis and x is the real axis.

FIGURE 20.6 Integration path for integral Eqn (78)

20.10 Solved problems

893

The linear paths A and C are of length R, whereas the path B is a circular section having the polar axis 4 ¼ p=4: Since there are no poles, the integral along the closed path is zero: Z Z Z Z  2  2  2   dz exp iz ¼ dz exp iz þ dz exp iz þ dz exp iz2 ¼ 0 ABC

B

A

C

The integral we must evaluate is that along the real x axis when A / N Z   lim dz exp iz2 A/N

A

The integral along path B is zero, since Z Zp=4  2    lim dz exp iz ¼ lim dq exp iR2 expð2iqÞiR expðiqÞ R/N R/N B

 lim

R/N

j j

0

Zp=4

j

j

  dqR exp R2 sin 2q ¼ 0

0

The integral along path C is easily evaluated as Z



2

dz exp iz



Z0 ¼ N

C

rffiffiffiffi ZN
 h i   1þi p 2 2 dx 1 þ i exp iðx þ ixÞ ¼ ð1 þ iÞ dx exp 2x ¼  2 2 0

so that Z



2

dz exp iz A



Z ¼



dz exp iz C

which is Eqn (78) of the main text.

2



ZN ¼ 2



2

dz exp iz 0



rffiffiffiffi p ¼ 1þi 2 



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Author Index Note: Page numbers followed by “f” denote figures; “t” tables.

A Abramowitz, M., 108, 114, 127, 129, 152, 154, 156, 168, 172, 174, 178, 180, 181, 189, 190, 195, 287, 432, 604, 808, 809, 814 Adams, W.H., 49 Aitken, A.C., 247, 250 Alexander, M.H., 70, 76, 96, 98, 124, 125, 131 Alliluev, S.P., 77, 130–131 Allnatt, A.R., 775. See Koide, A. Altmann, S.L., 297, 311, 326, 331, 338, 341, 353, 358, 362, 364 Amelio, M., 430, 731, 751. See Magnasco, V. Amemiya, A., 397, 690. See Kotani, M. Amos, R.D., 727, 777 Amos, R.D., 45, 48. See Musher, J.I. Arrighini, G.P., 876 Atkins, P.W., 383 Austin, E.J., 55

B Babb, J.F., 762t Babb, J.F., 730, 745, 751, 752, 762t. See Yan, Z.C. Bachvalov, N.S., 181 Bacon, G.E., 449 Baker Jr., G.A., 197, 716 Balint-Kurti, G.G., 689. See Van Lenthe, J.H. Ballinger, R.A., 727t Banerjea, E., 131. See Nguyen, H. Barclay, V.J., 615, 616t. See Wright, J.S. Bardo, R.D., 562. See Ruedenberg, K. Barnett, M.P., 813 Bartell, L.S., 876. See Kuchitsu, K. Bartlett, R.J., 698, 698t, 707 Bartlett, R.J., 711, 712. See Noga, J. Bates, D.R., 93, 121, 123 Battezzati, M., 36, 49, 77, 93, 121, 771 Battezzati, M., 36, 751, 763. See Magnasco, V. Bauer, E., 297. See Meijer, P.H.E. Becke, A.D., 714, 716 Bendazzoli, G.L., 197, 731, 770, 771t, 777 Bessis, D., 53 Bethe, H., 848, 855, 856 Beveridge, G.L., 584. See Pople, J.A. Bird, R.B., 758. See Hirschfelder, J.O. Bishop, D.M., 694t, 760, 762t Blanchard, C.H., 185. See Prosser, F.P.

Bobrowicz, F.W., 689. See Moss, B.J. Born, M., 89, 460, 867 Bowman, J.D., 51, 732. See Chipman, D.M. Boys, S.F., 475 Brandow, B.H., 707 Breit, G., 847 Briggs, M.P., 759 Brillouin, L., 37, 39, 40, 55, 254 Brink, D.M., 115, 190, 361, 397, 418, 431, 432, 433, 468, 469, 756, 775, 800, 813 Brueckner, K.A., 40, 707 Buckingham, A.D., 724, 767, 771t Buheler, R.J., 775 Bukowski, R., 52, 700 Burkill, J.C., 56 Byers Brown, W., 77, 93, 121, 123 Byers Brown, W., 35. See Hirschfelder, J.O. Byers-Brown, W., 692. See Pack, R.T. Byron, F.W., 202

C Cade, P.E., 727t Cade, P.E., 812. See Wahl, A.C. Califano, S., 863 Campion, W., 663. See Raimondi, M. Carrà, S., 800. See Gianinetti, E. Casanova, M., 812 Casanova, M., 181, 812, 825, 829. See Magnasco, V. Casimir, H.B.G., 744, 755 Castro, E.A., 53. See Fernández, F.M. Cederbaum, L.S., 877. See Köppel, H. Cederbaum, L.S., 879. See Ohrendorf, E. Cencek, W., 52. See Bukowski, R. Cencek, W., 52. See Jeziorska, M. Cha1asinski, G., 48, 177, 202, 737, 741 Chen, A.C., 53 Cheung, L.M., 694t, 762t. See Bishop, D.M. Chipman, D.M., 51, 732  zek, J., 711 Ci Clarke, N.J., 689 Claverie, P., 42 Clayton, M.M., 49. See Adams, W.H. Clementi, E., 563, 564, 727t Clementi, E., 695. See Largo-Cabrerizo, A. Clementi, E., 695. See Urdaneta, C. Clinton, W.L., 877

911

912

Author Index

Cloney, R.D., 185. See Todd, H.D. Cohen, H.D., 729 Coleman, A.J., 348 Collar, A.L., 247, 276. See Frazer, R.A. Collins, J.R., 688. See Gallup, G.A. Compton, A.H., 456 Condon, E.U., 76, 96, 128, 130, 148, 417, 439, 469 Cook, D.B., 475, 817 Coolidge, A.S., 616, 692, 694t, 812. See James, H.M. Cooper, D.L., 600, 623, 654, 688, 689, 691 Cooper, D.L., 689. See Clarke, N.J. Corongiu, G., 691 Corongiu, G., 563. See Clementi, E. Costa, C., 751, 777. See Bendazzoli, G.L. Costa, C., 775, 800. See Figari, G. Costa, C., 7, 45, 117, 202, 490t, 565, 625, 667, 745, 747, 760, 763, 767, 771, 775, 782. See Magnasco, V. Coulson, C.A., 31, 77, 121–123, 293, 464, 559, 567–570, 577, 606, 606t, 616, 623, 638, 663, 664, 667, 669, 743, 770, 876–877 Coulson, C.A., 813. See Barnett, M.P. Courant, R., 15, 16, 32, 199–200, 202 Cross, P.C., 863, 865, 866. See Wilson Jr., E.B. Császár, A.G., 857 Császár, A.G., 856, 857. See Pyykkö, P. Curtiss, C.F., 758. See Hirschfelder, J.O. Cybulski, S.M., 760, 762t. See Bishop, D.M.

D Dacre, P.D., 765 Daiker, K.C., 185. See Todd, H.D. Dalgarno, A., 118, 120, 748 Dalgarno, A., 751. See Riley, M.E. Dalgarno, A., 730, 745, 751, 752, 762t. See Yan, Z.C. Darwin, C.G., 842 Das, G., 695. See Wahl, A.C. Davies, D.W., 25, 724 Davison, W.D., 748. See Dalgarno, A. Davisson, C., 458 De Broglie, L.V., 458 Debye, P.J.W., 39 Decius, J.C., 863, 865, 866. See Wilson Jr., E.B. Dellepiane, G., 814. See Magnasco, V. Demidovic, B.P., 181 Dickinson, B.N., 509t Diner, S., 42. See Claverie, P. Dirac, P.A.M., 395, 834, 838 Dixon, R.N., 385, 866, 872, 874, 883, 885 Dobosh, P.A., 584. Pople, A.J. Doi, S., 130 Domcke, W., 877. See Köppel, H. Drake, G.W.F., 730, 745, 751, 752, 762t. See Yan, Z.C.

Drawin, H.-W., 131. See Nguyen, H. Dressler, K., 874 Duncan, W.J., 247, 276. See Frazer, R.A. Dunham, J.L., 57, 59 Dunning Jr., T.H., 565 Dunning Jr., T.H., 565. See Woon, D.E. Dyall, K.G., 856, 857. See Pyykkö, P. Dyke, T.R., 564t Dyke, T.R., 765. See Howard, B.J.

E Eckart, C., 190, 496, 864 Edmiston, C., 567 Eisenschitz, R., 45, 512, 513 Ellison, F.O., 664 Epstein, P.S., 42, 76, 124 Epstein, S.T., 53 Epstein, S.T., 35. See Hirschfelder, J.O. Epstein, S.T., 777. See Langhoff, P.W. Erdèly, A., 56 Ermler, W.C., 698t. See Rosenberg, B.J. Eyring, H., 107, 108, 110, 127, 334, 351, 353, 383, 417, 472, 796

F Farrell, R.A., 197, 711. See Wilson, S. Feltgen, R., 634, 767 Fernández, F.M., 53, 54 Feshbach, H., 7, 199. See Morse, P.M. Feynman, R.P., 707, 728, 871 Figari, G., 161, 430, 524, 547, 615t, 691, 731, 744, 751, 775, 800 Figari, G., 731, 770, 771t, 777. See Bendazzoli, G.L. Figari, G., 7, 42, 45, 117, 168, 185, 195, 202, 241, 490t, 513, 531–532, 563, 694t, 731, 734t, 745, 747, 751–752, 760, 761, 767, 773, 775, 777, 782. See Magnasco, V. Filter, E., 185 Finkelstein, B.N., 509t Fischer, I., 616. See Coulson, C.A. Fontana, P.R., 748 Fowler, P.W., 724 Fowler, P.W., 771t. See Buckingham, A.D. Frankowski, K., 692, 849 Frazer, R.A., 247, 276 Frisch, M.J., 562 Fromm, D.M., 699

G Galindo, A., 55 Gallup, G.A., 349, 623, 688 Gammel, J.L., 197, 716. See Baker Jr., G.A.

Author Index

Gázquez, J.L., 849, 849t Gel’fand, I.M., 213 Germer, L.H., 458. See Davisson, C. Gerratt, J., 398, 641, 654, 688, 689 Gerratt, J., 654, 689. See Clarke, N.J. Gerratt, J., 600, 623, 654, 688, 689, 691. See Cooper, D.L. Gerratt, J., 689. See Pyper, N.C. Gerratt, J., 689. See Wilson, S. Gianinetti, E., 800 Gianinetti, E., 691. See Raimondi, M. Gianinetti, E., 640. See Simonetta, M. Goddard III, W.A., 623, 688, 689 Goddard III, W.A., 689. See Hunt, W.J. Goddard III, W.A., 689. See Moss, B.J. Goddard III, W.A., 689. See Palke, W.E. Goidenko, I., 856. See Labzowsky, L. Gold, L.P., 726. See Wharton, L. Goldstone, J., 40, 707, 710 Goodisman, J., 607 Goscinski, O., 710 Goscinski, O., 197. See Bendazzoli, G.L. Goudsmit, S., 381. See Uhlenbeck, G.E. Gradshteyn, I.S., 142, 518, 790, 819 Grant, I.P., 832, 846 Grant, I.P., 849. See Hata, J. Grant, I.P., 846, 849, 850. See Laaksonen, L. Gray, C.G., 117 Griffiths, P.R., 185 Grotendorst, J., 185. See Weniger, E.J. Grozdanov, T.P., 53 Guidotti, C., 800 Guidotti, C., 876. See Arrighini, G.P. Guillemin Jr., V., 509t, 615t Günthard, Hs.H., 864. See Meyer, R. Guo, H., 640. See Xie, D. Guseinov, I.I., 813

H Hagstrom, S.A., 812 Hall, G.G., 297, 326, 331, 351, 356, 358, 361, 371, 534, 555 Hameka, H.F., 202, 738 Hameka, H.F., 202, 738. See Pan, Y.H. Handy, C.R., 53. See Bessis, D. Handy, N.C., 777. See Amos, R.D. Handy, N.C., 714. See King, R.A. Harrison, R.J., 730. See Koch, H. Hartree, D.R., 14 Hata, J., 849 Hay, P.J., 689. See Hunt, W.J. Hayes, I.C., 691 Heisenberg, W., 458, 464, 688 Heitler, W., 45, 180, 600, 608, 615t, 623, 688, 694t

913

Hellmann, H., 606, 606t, 728, 871 Herman, L., 131. See Nguyen, H. Herold, H., 53. See Ruder, H. Herzberg, G., 316, 383, 417, 418, 434, 440, 563, 564t, 863, 872, 875, 879 Herzberg, G., 635, 727. See Huber, K.P. Hilbert, D., 15, 16, 32, 199–200, 202. See Courant, R. Hill, R.N., 699. See Fromm, D.M. Hirschfelder, J.O., 7, 35, 45, 46, 47, 758 Hirschfelder, J.O., 775. See Buheler, R.J. Hirschfelder, J.O., 51, 732. See Chipman, D.M. Hobson, E.W., 114, 115, 152, 153, 155, 157, 806, 816 Hoffmann, R., 580 Hohenberg, P., 713 Hohn, F.E., 247, 253 Horowitz, G.E., 509t. See Finkelstein, B.N. Howard, B.J., 765 Hubac, I., 40, 202 Hubac, I., 40. See Papp, P. Hubbard, J., 40 Huber, K.P., 635, 727 Hückel, E., 270, 567, 569, 667 Hunt, W.J., 689 Huo, W.H., 727t. See Cade, P.E. Hutson , J.M., 771t. See Buckingham, A.D. Huzinaga, S., 475 Hylleraas, E.A., 33, 53, 121, 123

I Ince, E.L., 56, 88, 100 Inui, T., 615t, 694, 694t Ishiguro, E., 397, 690. See Kotani, M.

J Jacobs, W.P.J.H., 777. See Visser, F. Jahn, H.A., 872 James, H.M., 616, 692, 694t, 812 Jankowski, P., 52. See Bukowski, R. Jaszunski, M., 777 Jeziorska, M., 52 Jeziorski, B., 48, 51, 52 Jeziorski, B., 700. See Bukowski, R. Jeziorski, B., 48, 177, 202, 737, 741. See Cha1asinski, G. Jeziorski, B., 52. See Jeziorska, M. Jeziorski, B., 52. See Patkowski, K. Joachain, C.J., 202. See Byron, F.W. Jorge, F.E., 626, 664, 688. See McWeeny, R.

K Kain, J.S., 857. See Császár, A.G. Kaplan, I.G., 297, 334, 349, 502

914

Author Index

Karplus, M., 449, 568 Karplus, M., 34. See Kolker, H.J. Karplus, M., 777. See Langhoff, P.W. Karplus, M., 663. See Raimondi, M. Karplus, M., 817, 822. See Shavitt, I. Kato, T., 692 Kauzmann, W., 140, 144, 147 Keesom, W.H., 770 Kelly, H.P., 707 Kettle, S.F., 568. See Murrell, J.N. Kimball, G.E., 107, 108, 110, 127, 334, 351, 353, 383, 417, 472, 796. See Eyring, H. Kimura, T., 397, 689. See Kotani, M. King, R.A., 714 Kirchner, R.F., 584. See Zerner, M.C. Kirkwood, J.G., 33 Kirst, H., 634, 767. See Feltgen, R. Kiselev, A.A., 872. See Petelin, A.N. Klemperer, W., 765. See Howard, B.J. Klemperer, W., 727. See Muenter, J.S. Klemperer, W., 726. See Wharton, L. Klessinger, M., 664 Klopper, W., 490, 699 Klopper, W., 699, 712, 790. See Kutzelnigg, W. Klopper, W., 700, 701, 712, 713. See Noga, J. Klopper, W., 699. See Termath, V. Klopper, W., 699, 700, 730. See Tunega, D. Knowles, P.J., 185, 775, 777, 778 Knowles, P.J., 777. See Amos, R.D. Koch, H., 730 Koga, T., 745, 782 Köhler, K.A., 634, 767. See Feltgen, R. Kohn, W., 713, 715 Kohn, W., 713. See Hohenberg, P. Koide, A., 185, 213, 773, 775 Kolker, H.J., 34 Ko1os, W., 531, 532, 606, 610, 611t, 614, 615t, 616t, 617, 694, 694t, 695, 739, 760, 773, 783, 852 Ko1os, W., 48, 49. See Jeziorski, B. Komasa, J., 855. See Pachuki, K. Kopal, Z., 815 Köppel, H., 877 Köppel, H., 879. See Ohrendorf, E. Kotani, M., 397, 690 Kramers, H.A., 55 Kreek, H., 740, 741, 747, 773 Kreek, H., 202, 738. See Singh, T.R. Kuchitsu, K., 876 Kutzelnigg, W., 600, 691, 692, 699, 712, 790 Kutzelnigg, W., 490, 699. See Klopper, W. Kutzelnigg, W., 699, 700, 712. See Noga, J. Kutzelnigg, W., 699. See Termath, V.

L Laaksonen, L., 846, 849, 850 Laaksonen, L., 564, 607, 726. See Sundholm, D. Labzowsky, L., 856 Landau, L.D., 129, 297, 883 Langhoff, P.W., 777 Largo-Cabrerizo, A., 695 Largo-Cabrerizo, A., 695. See Urdaneta, C. Lazzeretti, P., 562, 563, 564t, 727t Lebeda, C.F., 41 Ledermann, W., 326 Ledsham, K., 93, 121, 123. See Bates, D.R. Lee, C., 714, 716 Lennard-Jones, J.E., 31, 37, 39, 254, 513, 546, 550, 569, 586, 592, 705 Lewis, J.T., 118, 120. See Dalgarno, A. Lie, G.C., 695. See Largo-Cabrerizo, A. Lie, G.C., 695. See Urdaneta, C. Lievin, J., 695. See Urdaneta, C. Liehr, A.D., 877 Lifshitz, E.M., 129, 297. See Landau, L.D. Lischka, H., 727 Liu, B., 634, 635 Lo, B.W.N., 117. See Gray, C.G. Loew, G.H., 584. See Zerner, M.C. London, F., 755, 767 London, F., 45, 512, 513. See Eisenschitz, R. London, F., 45, 180, 600, 615t, 623, 688, 694t. See Heitler, W. Longuet-Higgins, H.C., 89, 353, 702, 753, 754, 776, 784, 863, 867, 872, 873, 878, 883, 884, 889, 891 Longuet-Higgins, H.C., 879. See Herzberg, G. Longuet-Higgins, H.C., 873, 874. See Pople, J.A. Lorenz, T., 433. See Spelsberg, D. Löwdin, P.-O., 7, 8, 42, 43, 44, 267, 395, 398, 546, 550, 551, 618, 640, 682, 683, 691, 705, 739, 740, 813 Löwdin, P.-O., 7. See Hirschfelder, J.O. Löwdin, P.-O., 7, 739, 740. See Shull, H.

M MacDonald, J.K.L., 23 Mach, P., 40, 202. See Hubac, I. Mach, P., 40. See Papp, P. Maclagan, R.G.A.R., 664 MacRobert, T.M., 114, 116, 152, 191, 470 Maestro, M., 876. See Arrighini, G.P. Maestro, M., 800. See Guidotti, C. Magnasco, V., 7, 36, 42, 45, 51, 62, 71f, 72f, 101, 117, 124, 168, 181, 185, 195, 202, 241, 268, 272, 323, 326, 330, 334f, 342f, 379, 430, 433, 490t, 510, 513, 517, 531–532, 548, 558, 563, 566, 567, 574, 617, 622, 623, 625, 634, 635, 641, 669, 680, 694t, 724, 725f, 727, 731, 734t, 745, 747,

Author Index

751–752, 754, 756, 757f, 759, 760, 761, 762t, 763, 764f, 766, 767, 768, 771, 772, 773, 775, 777, 782, 783, 784, 809, 812, 813, 814, 825, 829, 857 Magnasco, V., 36, 49, 77, 93, 121, 771. See Battezzati, M. Magnasco, V., 731, 770, 771t, 775, 800. See Bendazzoli, G.L. Magnasco, V., 430, 547, 691, 731, 744, 751, 775, 800. See Figari, G. Magnasco, V., 45, 181, 634, 666, 814, 817. See Musso, G.F. Magnasco, V., 615t. See Ottonelli, M. Magnus, W., 177 Magnusson, E.A., 814 Malkin, I.A., 77, 130–131. See Alliluev, S.P. Malrieu, J.-P., 42. See Claverie, P. Malrieu, J.-P., 878. See Spiegelman, F. Margenau, H., 45, 69, 76, 100, 247, 450, 459, 749 Maron, I.A., 181. See Demidovic, B.P. Maroulis, G., 633 Mathews, J., 200 Matsen, F.A., 351 Matsumoto, S., 745, 782. See Koga, T. McLean, A.D., 724, 756 McLean, A.D., 634. See Liu, B. McWeeny, R., 31, 42, 44, 200, 349, 356, 475, 539, 546, 568, 579, 623, 626, 638, 641, 645, 655, 664, 682, 683, 685, 687, 688, 696, 751, 754, 756, 776, 777 McWeeny, R., 765. See Dacre, D.P. McWeeny, R., 777. See Jaszunski, M. McWeeny, R., 51, 754, 756, 767, 772, 777. See Magnasco, V. Meath, W.J., 185, 773, 777, 778. See Knowles, P.J. Meath, W.J., 775. See Koide, A. Meath, W.J., 740, 741, 747, 775. See Kreek, H. Meath, W.J., 202, 738. See Singh, T.R. Meath, W.J., 490t. See Wheatley, R.J. Meijer, P.H.E., 297 Meixner, J., 202 Merrifield, D.P., 558, 559, 563, 564, 564t. See Pitzer, R.M. Meyer, R., 864 Meyer, W., 758 Meyer, W., 433, 757. See Spelsberg, D. Meyer, W., 729, 730. See Werner, H.-J. Michels, H., 34. See Kolker, H.J. Moats, R.K., 185. See Todd, H.D. Moccia, R., 876. See Arrighini, G.P. Mohr, P.J., 14, 831 Møller, C., 696 Monari, A., 731, 777. See Bendazzoli, G.L. Monkhorst, H.J., 694t, 695. See Ko1os, W. Moore, C.E., 498, 500 Morales, J.A., 53, 54. See Fernández, F.M. Morse, P.M., 7, 199 Moss, B.J., 689 Moszynski, R., 52. See Jeziorski, B.

915

Mott, N.F., 832, 833 Mueller-Westerhoff, U.T., 584. See Zerner, M.C. Muenter, J.S., 564t. See Dyke, T.R. Mulder, J.J.C., 626 Mulliken, R.S., 298, 558, 665 Murphy, G.M., 69, 76, 100. See Margenau, H. Murphy, G.M., 298. See Rosenthal, J.E. Murrell, J.N., 45, 48, 313, 568 Murrell, J.N., 759. See Briggs, M.P. Musher, J.I., 45, 48 Musso, G.F., 45, 181, 634, 666, 814, 817 Musso, G.F., 430, 731, 751. See Figari, G. Musso, G.F., 45, 268, 565, 622, 623, 634, 641. See Magnasco, V.

N Nesbet, R.K., 42 Neumann, F., 806, 813 Newell, D.B., 14, 831. See Mohr, P.J. Nguyen, H., 131 Noga, J., 699, 700, 701, 711, 712, 713 Noga, J., 700, 726, 730, 731, 770, 771t. See Tunega, D. Norbeck, J.M., 638. See Gallup, G.A. Nordheim-Pöschl, G., 691 Nusair, M., 197, 716. See Vosko, S.H.

O Oberhettinger, F., 177. See Magnus, W. Ogilvie, J.F., 53. See Fernández, F.M. Ohno, K.A., 664. See McWeeny, R. Ohrendorf, E., 879 Öpik, U., 872 Öpik, U., 872. See Longuet-Higgins, H.C. Oppenheimer, R., 89, 867. See Born, M. Orlandi, G., 197. See Bendazzoli, G.L. Ostlund, N.S., 696. See Szabo, A. Ottonelli, M., 433, 615t, 673, 745, 746, 782, 784 Ottonelli, M., 433, 513, 517, 531, 745, 747, 751, 759, 760, 762t, 782, 783, 784. See Magnasco, V.

P Pachuki, K., 855 Paci, D., 781 Pack, R.T., 692 Paldus, J., 349 Palke, W.E., 689 Pan, Y.H., 202, 738 Papadopoulos, M.G., 731 Papp, P., 40 Parr, R.G., 714, 716 Parr, R.G., 714, 716. See Lee, C.

916

Author Index

Partridge, H., 857 Pascual, P., 53. See Galindo, A. Patkowski, K., 52 Patkowski, K., 52. See Jeziorska, M. Paul, R., 197 Pauli, W., 381, 383 Pauling, L., 21, 91, 140, 496, 509t, 600, 642, 643, 655, 657t, 667, 863 Pauly, H., 634, 767. See Feltgen, R. Peek, J.M., 121, 508t, 509f, 509t, 510, 773 Pekeris, C.L., 495, 693t Pekeris, C.L., 692, 849. See Frankowski, K. Pelloni, S., 727t. See Lazzeretti, P. Perico, A., 566. See Magnasco, V. Persico, M., 863 Petelin, A.N., 872 Peterson, K.A., 640. See Xie, D. Peverati, R., 617, 622, 634, 635. See Magnasco, V. Phillips, E.G., 220, 221, 223, 225 Pipin, J., 760, 762t. See Bishop, D.M. Piquemal, J.-P., 52, 857. See Reinhardt, P. Pitzer, R.M., 558, 559, 563, 564, 564t Plesset, M.S., 696. See Møller, C. Polder, D., 744, 755. See Casimir, H.B.G. Polyansky, O.L., 857. See Császár, A.G. Polyansky, O.L., 856, 857. See Pyykkö, P. Polymeropoulos, E.E., 49. See Adams, W.H. Pople, J.A., 548, 580, 584, 768, 873, 874 Porter, R.N., 449, 568. See Karplus, M. Potts, A.W., 564t, 872 Power, J.D., 121. See Byers Brown, W. Pratolongo, R., 775, 800. See Figari, G. Price, W.C., 564t, 872. See Potts, A.W. Proctor, T.R., 775. See Koide, A. Prosser, F.P., 185 Pryce, M.H.L., 872. See Longuet-Higgins, H.C. Pryce, M.H.L., 872. See Öpik, U. Purvis III, G.D., 698, 698t. See Bartlett, R.J. Pyper, N.C., 689 Pyykkö, P., 832, 855, 856, 857 Pyykkö, P., 856. See Labzowsky, L. Pyykkö, P., 564, 607, 726. See Sundholm, D.

R Racah, G., 502 Raffenetti, R.C., 562. See Ruedenberg, K. Raimondi, M., 640, 641, 663, 664, 691 Raimondi, M., 689. See Clarke, N.J. Raimondi, M., 600, 623, 654, 688, 689, 691. See Cooper, D.L. Raimondi, M., 689. See Gerratt, J. Raimondi, M., 640, 641. See Tantardini, G.F. Ralston, A., 181, 184

Ramsay, D.A., 874. See Dressler, K. Ransil, B.J., 21, 522, 567 Rapallo, A., 813 Rapallo, A., 181, 763, 771, 812, 813, 825, 829. See Magnasco, V. Reinecke, M., 53. See Ruder, H. Reinhardt, P., 52, 857 Remiddi, E., 699 Renner, E., 872 Rice, B., 877. See Clinton, W.L. Rice, J.E., 777. See Amos, R.D. Rijks, W., 777. See Wormer, P.E.S. Riley, M.E., 751 Ritz, W., 21, 496 Robinson, P.D., 121 Robinson, P.D., 77, 121. See Coulson, C.A. Roetti, C., 563, 564. See Clementi, E. Roos, B., 21, 687 Roothaan, C.C.J., 182, 473, 534, 550, 553, 555, 556, 792, 799, 803, 812, 828, 876 Roothaan, C.C.J., 729. See Cohen, H.D. Roothaan, C.C.J., 812. See Wahl, A.C. Rose, M.E., 431, 432, 740, 748, 773, 774, 775 Rosen, N., 183, 615t, 792, 809 Rosenberg, B.J., 564, 564t, 698t Rosenthal, J.E., 298 Rossetti, C., 100, 188, 189, 195, 196, 197, 230, 235, 237 Ruder, H., 53 Ruedenberg, K., 183, 562, 809, 813 Ruedenberg, K., 567. See Edmiston, C. Rui, M., 731, 770, 771t, 777. See Bendazzoli, G.L. Rui, M., 745, 747, 782. See Magnasco, V. Rutherford, D.E., 12, 349 Rychlewski, J., 773. See Ko1os, W. Ryzhik, I.M., 142, 518, 790, 819. See Gradshteyn, I.S.

S Sack, R.A., 740 Sack, R.A., 872. See Longuet-Higgins, H.C. Saika, A., 813. See Yasui, J. Salahub, D.R., 759. See Briggs, M.P. Salem, L., 568, 877 Salpeter, E.E., 848, 855, 856. See Bethe, H. Salvetti, O., 876. See Arrighini, G.P. Santry, D.P., 582 Santry, D.P., 580. See Pople, J.A. Satchler, G.R., 115, 190, 361, 397, 418, 431, 432, 433, 468, 469, 756, 775, 800, 813. See Brink, D.M. Saunders, V.R., 475, 817 Schiff, L.I., 55, 147, 832, 835, 839 Schlegel, H.B., 562. See Frisch, M.J. Schnuelle, G.W., 664. See Maclagan, R.G.A.R.

Author Index

Schrader, D.M., 41. See Lebeda, C.F. Schroedinger, E., 25, 76, 96, 98, 124, 125, 127, 161, 459, 832 Schwenke, D.W., 857. See Partridge, H. Schwerdtfeger, P., 855. See Thierfelder, C. Schwinger, J., 383 Scrocco, E., 736 Scrocco, E., 800. See Guidotti, C. Segal, G.A., 580. See Pople, J.A. Segal, G.A., 580, 582. See Santry, D.P. Serber, R., 688 Sham, L.J., 713, 715. See Kohn, W. Shavitt, I., 177, 475, 817, 819, 822, 824 Shavitt, I., 698, 698t. See Bartlett, R.J. Shavitt, I., 563, 564, 564t, 698t. See Rosenberg, B.J. Shaw, G., 45, 48. See Murrell, J.N. Sherman, J., 655, 657, 657t Shilov, G.E., 213. See Gel’fand, I.M. Shortley, G.H., 76, 96, 128, 130, 148, 417, 439, 469. See Condon, E.U. Shull, H., 7, 739, 740 Shull, H., 664. See Ellison, F.O. Shull, H., 7, 739, 740. See Löwdin, P-O. Siciliano, A., 42, 46, 734t Siciliano, A., 7, 42, 202, 531-532, 694t. See Magnasco, V. Silver, D.M., 707. See Bartlett, R.J. Silver, D.M., 197, 707, 708, 708f, 709f, 710f, 711. See Wilson, S. Silverstone, H.J., 77, 131 Silverstone, H.J., 849, 849t. See Gázquez, J.L. Silverstone, H.J., 185. See Todd, H.D. Simonetta, M., 640 Simonetta, M., 800. See Gianinetti, E. Simonetta, M., 640, 641, 664. See Raimondi, M. Simonetta, M., 640, 641. See Tantardini, G.F. Singh, T.R., 202, 738 Slater, J.C., 254, 347, 431, 473, 534, 535, 536, 546, 618, 620, 682 Smeyers, Y.G., 353 Smirnov, V.I., 88, 175 Sneddon, I.N., 102, 103, 104, 114, 127, 152, 153, 154, 157, 158, 159, 161, 163, 165, 167, 170, 171, 178, 180, 185, 207 Sneddon, I.N., 832, 833. See Mott, N.F. Sommerfeld, A., 845 Soni, R.P., 177. See Magnus, W. Spelsberg, D., 433, 757 Spiegelman, F., 878 Stam, P., 777. See Visser, F. Stegun, I.A., 108, 114, 127, 129, 152, 154, 156, 168, 172, 174, 178, 180, 181, 189, 190, 195, 287, 432, 604, 808, 809, 814. See Abramowitz, M. Steinborn, E.O., 185. See Filter, E. Steinborn, E.O., 178, 185, 213. See Weniger, E.J.

Steiner, E., 77, 93, 121, 123. See Byers Brown, W. Stewart, A.L., 93, 121, 123. See Bates, D.R. Stone, A.J., 117, 190, 468 Stone, A.J., 777. See Amos, R.D. Stone, A.J., 691. See Hayes, I.C. Strauss, H.I., 876–877. See Coulson, C.A. Sugiura, Y., 180, 512, 809 Sundholm, D., 564, 607, 726, 727t, 846 Szabo, A., 696 Szalewicz, K., 700. See Bukowski, R. Szalewicz, K., 48. See Cha1asinski, G. Szalewicz, K., 52. See Jeziorska, M. Szalewicz, K., 52. See Jeziorski, B. Szalewicz, K., 694t, 695. See Ko1os, W. Szalewicz, K., 52. See Patkowski, K.

T Tang, K.T., 244 Tantardini, G.F., 641 Tantardini, G.F., 640, 641, 664. See Raimondi, M. Tarczay, G., 856, 857. See Pyykkö, P. Tauber, G.E., 806, 807 Taylor, B.N., 14, 831. See Mohr, P.J. Taylor, H.S., 53. See Grozdanov, T.P. Tedder, J.M., 568. See Murrell, J.N. Teller, E., 878 Teller, E., 872. See Jahn, H.A. Tennyson, J., 857. See Császár, A.G. Tennyson, J., 856, 857. See Pyykkö, P. Termath, V., 699 Thierfelder, C., 855 Thomson, G.P., 458 Thouless, D.J., 200 Tinkham, M., 299, 300t, 320, 334, 351 Tipping, R.H., 53. See Fernández, F.M. Todd, H.D., 185 Tokman, M., 856. See Labzowsky, L. Tomasi, J., 736 Tomasi, J., 736. See Scrocco, E. Torello, F., 634, 767. See Feltgen, R. Torkington, P., 666 Tough, R.J.A., 117. See Stone, A.J. Troup, G., 465 Trucks, G.W., 562. See Frisch, M.J. Tunega, D., 700, 726, 730, 731, 770, 771t Tunega, D., 699, 700. See Noga, J. Turbiner, A.V., 53

U Uhlenbeck, G.E., 381 Ujiie, M., 745, 782. See Koga, T.

917

918

Author Index

Unsöld, A., 32 Urdaneta, C., 695 Urdaneta, C., 695. See Largo-Cabrerizo, A.

V Valiron, P., 701, 713. See Noga, J. Vallini, G., 666. See Musso, G.F. Vance, R.L., 688. See Gallup, G.A. Van der Avoird, Ad, 45 Vandoni, I., 640. See Simonetta, M. Van Duijneveldt, F.B., 489, 490t Van Duijneveldt, F.B., 562. See Van Duijneveldt-Van de Rijdt, J.G.C.M. Van Duijneveldt-Van de Rijdt, J.G.C.M., 562 Van Lenthe, J.H, 689 Van Vleck, J.H., 425 Visser, F., 777 Von Neumann, J., 878 Vosko, S.H., 197, 716

W Wahl, A.C., 695, 812 Walker, R.L., 200. See Mathews, J. Waller, I., 59, 124, 130 Walter, J., 107, 108, 110, 127, 334, 351, 353, 383, 417, 472, 796. See Eyring, H. Wang, S.C., 615t, 694t Wannier, G.H., 618 Watson, G.N., 177, 738, 821 Weinbaum, S., 615t Weniger, E.J., 178, 185 Wentzel, G., 55, 57, 130 Werner, H.-J., 729, 730 Weyl, H., 298, 626 Wharton, L., 726 Wheatley, R.J., 490t Wheland, G.W., 637 Wheland, G.W., 655, 657t. See Pauling, L. Wigner, E.P., 37, 40, 190, 254, 298, 338, 350, 378, 398

Wigner, E.P., 878. See Von Neumann, J. Wilk, L., 197, 716. See Vosko, S.H. Wilson, S., 197, 689, 707, 708, 708f, 709f, 710f, 711 Wilson, S., 40, 202. See Hubac, I. Wilson, S., 40. See Papp, P. Wilson Jr., E.B., 863, 865, 866 Wilson Jr., E.B., 21, 91, 140, 496, 642, 863. See Pauling, L. Wolniewicz, L., 694t, 695, 852 Wolniewicz, L., 531, 532, 606, 610, 611t, 614, 615t, 616t, 617, 694, 694t, 739, 760, 773, 783, 852. See Ko1os, W. Woon, D.E., 565 Wormer, P.E.S., 433, 757, 777 Wormer, P.E.S., 349. See Paldus, J. Wormer, P.E.S., 777. See Visser, F. Wright, J.S., 615, 616t Wunner, G., 53 Wunner, G., 53. See Ruder, H.

X Xie, D., 640

Y Yan, Z.C., 730, 745, 751, 752, 762t Yang, W., 714, 716. See Lee, C. Yang, W., 714, 716. See Parr, R.G. Yassen, R.F., 813. See Guseinov, I.I. Yasui, J., 813 Yokoyama, H., 641 Yoshimine, M., 724, 756. See McLean, A.D.

Z Zanasi, R., 562. See Lazzeretti, P. Zauli, C., 814. See Magnusson, E.A. Zener, C., 473, 495, 509t, 883 Zener, C., 615t. See Guillemin Jr., V. Zerner, M.C., 584 Zobov, N.F., 857. See Császár, A.G.

Subject Index Note: Page numbers followed by “f” denote figures; “t” tables.

A abstract group theory, 323–380 Abelian group, 324 axial groups, 356–358 axioms of, 324 characters of, 333–338 conjugation and classes, 331–332 continuous groups, 353–356 direct-product groups, 332–333 double groups, 357, 364 examples of, 325–326 irreducible representations of, 338–340 isomorphism, 328–331 molecular point groups, 351–353 multiplication table, 324, 326–327 problems and solutions, 364–379 projectors, 338, 340–345 representations of, 333–338 rotation groups, 356–364 spherical group, 358–361 transformation properties of, 362–364 subgroups, 327–328 symmetric group, 345–351 symmetry-adapted functions of, 333–334, 336, 340–345 active representations, of symmetry operations, 302–303 adjoint matrix, 256 allyl radical (N ¼ 3), 627–629 Pauling’s formula for, 646–647 alternant hydrocarbons, 572–577, 576f, 578f analytic continuation of complex functions, 230–232 angular momentum, 359–361, 417–445 Clebsch–Gordan coefficients, 431–433 commutation properties of, 13, 60, 65–66, 359 coupling of, 418–420 rules, 433–434 Gaunt coefficients, 433–434 Landé g-factor, 424, 434, 437 matrix method, 425–431 multiplet structure of, 421–425 operators, 13 problems and solutions, 434–445 projection operator method, 431, 436, 443–445 Russell–Saunders (LS) coupling for light atoms, 421–425, 424f, 425f, 501f spherical coordinates, expression in, 78–79, 81–84 vector model, 418–425

Wigner 3-j and 9-j symbols, 431–433 annihilation operators, 702–703 anti-commutative relations, 9, 702–703 anti-Fourier transform. See inverse Fourier transform anti-Hermitian operators, 12 approximation methods DFT, 550, 713–716 perturbative, 25–54 Ritz, 21–24 variational, 19–24, 483–532 WKB, 55–59 Argand diagram, 216 associated Laguerre polynomials, 159–161 basic integrals over, 161–163 associated Legendre functions, 155 associative relations, 9 asymptotic expansions, 56–57 atomic interactions diatomic systems, accurate theoretical results for, 772–773 H–H expanded interaction up to second order, 742–745 H–H long-range dispersion interaction, higher-order terms in, 746–747 H–H non-expanded interaction up to second order, 735–737 H–H+ non-expanded interaction up to second order, 731–735 H–H non-expanded second-order dispersion energy, multipole analysis of, 740–742 H–H non-expanded second-order induction energy, multipole analysis of, 737–739 many-electron atoms, expanded dispersion interaction for, 748–752 problems and solutions, 779–788 atomicity of matter, 458–459 atomic one-electron system hydrogen-like atomic orbitals, 115–117 F equation, solution of, 109 2p excited state of, 493 2s excited state of, 490–492 1s ground state of, 487–490 radial equation, solution of, 105–108 Schroedinger eigenvalue equation for, 91–93 Q equation, solution of, 109–115, 132–135 atomic orbitals (AOs), 449, 467–481 Gaussian-type orbitals (GTOs), 473–477 hydrogen-like orbitals (HAOs), 115–117, 468–470, 470f, 502

919

920

Subject Index

atomic orbitals (AOs) (Continued ) problems and solutions, 477–481 Slater-type orbitals (STOs), 473–475 atomic two-electron system 1s2p excited state of, 499–502, 501f first 1s2s excited state of, 497–498, 498f, 501f 1s2 ground state of, 493–496, 493f atomic units, 14 axial groups, 356–358 axioms of group theory, 324

B band theory of solids, 577–579 basic integrals, 790–793 definite integral, 791–793 indefinite integral, 790–791 basis set, 9 benzene, Pauling’s formula for, 647–655 Bessel functions of first kind, 176–177 of half-integral order, 171–172 of integral order, 170, 204–209 modified, 176–178 reduced, 177–178 of second kind, 176–177 spherical, 172–176 Bessel’s inequality, 15 bond chemical, 601–614, 623–641 hydrogen, 761–765, 764f, 768–769, 769f Keesom, 770–771, 787–788 Pauli repulsion, 765–767, 785–786 Van der Waals (VdW), 767–769, 768f, 769t Borel summation, 237 Born interpretation, 460 Born–Oppenheimer approximation, 89, 546, 601–602, 850, 853, 857, 867–872, 883, 889–890 Brillouin’s theorem, 697–698, 721–722 Brillouin–Wigner (BW) perturbation theory, 37–40, 710–711 butadiene, Pauling’s formula for, 645–646

C calculus of residues, 232–241 Jordan lemma, 234–235 real variable functions integrals, evaluation of, 238–241 residue theorem, 232–233 sum of non-convergent series, 235–238 Cartesian coordinates, 74 Cartesian Gaussian orbitals, 476–477, 562, 695 Casimir operator, 359

Casimir–Polder integral, 744, 755, 760 Cauchy combinatorial formula, 348 Cauchy expansion of determinants, 251–254, 251f Cauchy–Hadamard test, 220 Cauchy integral representation, 227 Cauchy–Riemann’s differential equations, 218 Cauchy theorem, 225 Cayley–Hamilton theorem, 270, 296 CC-R12-MBPT, 712–713, 730 CCSDT-R12, 712–713, 726, 730–731 CH+4 , Jahn–Teller effect in, 874–877 circular functions, of complex variable, 221–222 classical mechanics, normal coordinates in, 864–866 Clebsch–Gordan coefficients, 431–433, 800 closed chain, general solution for, 570–572 closure property, 32 coefficients of fractional parentage, 502 commutative relations, 9, 13, 248, 359, 387 complete neglect of differential overlap (CNDO) method, 580–584 completeness relation, 15–16 complex conjugate matrix, 256 complex integral calculus, 223–232 analytic continuation, 230–232 Cauchy’s integral representation, 227 Cauchy theorem, 225 complex plane, integrals in, 224–225 integration over not simply connected domain, 225–227 Laurent’s expansion, 228–230 line integrals, 223–224 Taylor’s expansion, 228 zeros of regular functions, 230 complex numbers, 215–217 defined, 215–216 division of, 217 imaginary part of, 215–216 modulus of, 216 complex plane, integrals in, 224–225 complex variable, functions of, 215–245 circular functions, 221–222 continuous, 217 elementary operations, 219 exponential functions, 221–222 hyperbolic functions, 221–222 logarithmic functions, 222 many-valued functions, 223 power series of elementary functions, 219–222 problems and solutions, 241–245 rational functions, 220 regular functions, 218 Compton effect, 457–458

Subject Index

Condon–Shortley phase, 115, 362, 433, 758, 813 configurational interaction (CI), 40, 686–695 cusp-corrected, 691–692 generalized valence bond (GVB), 688–691 Kolos–Wolniewicz wavefunctions, 692–695 large-scale, 687–688 confluent hypergeometric functions, 168–170 confocal spheroidal coordinates, 70, 75–76 conical intersections in polyatomic molecules, 878–883 conjugation, 331–332 continuous groups, 353–356 convergence, 219 radius of, 219–220 coordinate space, 302–310 active representation, 302–303 passive representation, 302–303 symmetry transformation in, 303–305 coordinate systems, 69–86 Cartesian coordinates, 74 generalized coordinates, 73–74 orthogonal coordinates, systems of, 71–72 parabolic coordinates, 70–71, 76–78 problems, 78–86 spherical coordinates, 69, 74–75 spheroidal coordinates, 70, 75–76 correlation energy, 549, 696 coupled-cluster many-body perturbation theory (CC-MBPT), 700–701, 711–713 CC-R12-MBPT, 712–713, 730 CCSDT-R12, 712–713, 726, 730–731 Cramer’s rule, 257 creation operators, 702–703 crystallographic notation, 300, 300t cusp-corrected configurational interaction, 691–692 cyclobutadiene (N ¼ 4), 630–631 Pauling’s formula for, 644–645, 659–660, 676–677

D Dalgarno interchange theorem, 28 Darboux inequality, 225 definite integral, 791–793 auxiliary functions of, 791–793 density electron, 540–543 Fock–Dirac, 550–551 matrices, 540, 543 spin, 541 transition, 737, 751, 754

921

density functional theory (DFT), 713–716 determinants bordered, 253–254 Cauchy expansion of, 251–254, 251f defined, 248, 250 Laplace expansion of, 254–255 ordinary expansion, 250–251 parent, 623 properties of, 250–255 Slater, 347, 535–539, 682–685, 697–698 diagonalization of matrices, 258, 266–267 diagonal matrix, 255 diagrammatic theory, 706–713 CC-R12-MBPT, 712–713 coupled-cluster MBPT, 711–712 fourth-order, 708–718, 708f, 709f, 710f linked-cluster theorem, 710, 710f Padé approximants and perturbation expansions, 710–711 second-order, 707, 708f third-order, 707, 708f differential equations, in quantum mechanics, 87–150 atomic one-electron system, 104–117 hydrogen-like atomic orbitals, 115–117 F equation, solution of, 109 radial equation, solution of, 105–108 Q equation, solution of, 109–115 free particle in one-dimension, 139–140 hydrogen atom in electric field, 117–120 hydrogen atom in parabolic coordinates x-equation of, 124–128, 135 hydrogen atom in spherical coordinates F equation, solution of, 109 radial equation of, solution of, 105–108, 131–132 Q equation, solution of, 109–115, 132–135 hydrogen molecular ion, 120–124 near singular points, 100–101 one-dimensional harmonic oscillator, 101–104 partial, 88–89 particle in one-dimensional box, 140–142, 143–147 problems and solutions, 135–150 separation of variables, 89–98 atomic one-electron system, 91–93 hydrogen atom in uniform electric field, 96–98 molecular one-electron system, 93–96 particle in three-dimensional box, 89–91 three-dimensional harmonic oscillator, 91 solution by series expansion, 98–100 Stark effect, in atomic hydrogen, 124–131 x-equation solution, in zero-field case, 124–128 first-order, 128–131 higher-orders, 130–131 dipole polarizability, of hydrogen atom, 510–512

922

Subject Index

dipole pseudospectra, 745t, 752t, 760t, 761t Dirac delta function, 16, 184–185 formula, for many-electron spin, 395 notation, 4, 10, 15, 28, 37, 461 relativistic equation for central field, 838–846 hydrogen-like atom, 843–846 radial equation separation, 841–842 relativistic equation for the electron, 834–835 wave equation, 834 direct-product groups, 332–333 dispersion coefficients, 745–747, 751–752, 758–761, 762 dispersion constant, 744–745, 757 dual spin functions, 350–351 dynamic propagator, 755–756, 777

E eigenvalue equation of matrices, 257–258 eigenvalue problem of matrices, 256–261 Eisenschitz–London–Hirschfelder–Van der Avoird (EL–HAV) perturbation theory, 50–51 electric properties of molecules, 724–731 molecular moments of, 724–727 polarizability of, 724–731 electron density, 540–543 electron distribution functions, of many-electron wavefunctions, 539–544 electron density, 540–543 one-electron distribution functions, 539–540 spin density, 540–543 two-electron distribution functions, 543–544 electronic energy, average value of, 546 electronic ground state, 869–870 (1s2j1s2) electron repulsion integral, 798–799 different orbital exponents, 799 same orbital exponent, 798–799 electron spin, 381–416 Dirac formula, for many-electron spin, 395 Kotani’ synthetic method, 397–398, 408–410 Löwdin spin projection operators, 398–399, 405–408 many-electron spin theory, 394–397 one-electron spin theory, 386–389 operators, matrix representation of, 389–391 Pauli’s postulate, 383 problems and solutions, 400–416 two-electron spin theory, 392–394 Wigner’s formula, 396 Zeeman effect, 382–385 electron spin resonance (ESR), 385 electrostatic potential evaluation, 795–798 spherical coordinates, 795–797

spheroidal coordinates, 797–798 Epstein–Nesbet (EN) perturbation theory, 42–45 equivalent line pair, 707 Euler’s formulae, for imaginary exponentials, 116, 221–222, 362–363 Euler–Lagrange equation, 34, 42 exchange-overlap energies, 605, 607t, 609, 610, 611t exchange perturbation theories, 45–52 Eisenschitz–London–Hirschfelder–Van der Avoird (EL–HAV) perturbation theory, 50–51 Jeziorski-Ko1os (JK) theory, 49–50 many-electron systems theory, 50–51 Murrell–Shaw–Musher–Amos (MS-MA) theory, 48–49 paradox, 45–49 polarization (P) theory, 47–48 SAPT2006 program, 52 excitation energies, 34, 36, 751 excited electronic state, 870–872 excited pseudostates, 36, 735–736 exclusion principle, 535, 537 expanded molecular energy corrections up to second order, 756–761 expansion theorem, 14–17, 461 exponential functions, of complex variable, 221–222 exponential integral function, 179–181 generalized, 181 extended Hückel’s theory (EHT), 580

F Fermi correlation, 685–686 fermion loop, 707 first-order Rayleigh–Schroedinger perturbation theory, 30–31 first order, variational principles in, 484–485 Fock–Dirac density matrix, 551–552 Fock space, 706 four-centre two-electron integral, over 1s Slater-type orbitals, 815–817 four-electron integrals, 699, 790 Fourier transform (FT), 185–188, 211–213 inverse, 185–188 fourth-order diagrammatic theory, 708–710, 708f, 709f, 710f functional derivatives, 34, 42, 714–715 Hylleraas, 33–34, 36 Rayleigh, 19–20, 484 relations, 73 functions basis set, 9 Bessel. See Bessel functions circular functions, of complex variable. See complex variable, functions of

Subject Index

complex functions, analytic continuation of, 230–232 confluent hypergeometric, 168–170 Dirac delta, 16, 184–185 electron distribution, 539–544 exponential, 221–222 exponential integral, 179–181 gamma, 178–180, 208–211 Gaussian, 817–822 Green’s, 199–202 Hermite, 163–166 hyperbolic, 221–222 hypergeometric, 166–170 Kummer’s, 168–169, 178–179 Laguerre, 7–8, 158–163 Legendre, 152–158 logarithmic, 222 many-valued, 222 orthonormal, 8–9 rational, 220 regular, 4–5, 218, 230 space, 305–310 state function, 18–19, 454–456

G gamma function, 178–180, 208–211 incomplete, 178–179 gamma ray properties, 458t Gaunt coefficients, 433–434 Gaussian functions integrals of, 819–820 properties of, 817–819 Gaussian-type orbitals (GTOs), 475–477, 817–824 molecular integrals over, 817–824 Gaussian functions, integrals of, 819–820 Gaussian functions, properties of, 817–819 integral transforms, 820–822 Gauss–Legendre method, 815 Gegenbauer polynomials, 197 generalized coordinates, 73–74 generalized exponential integral function, 181 generalized valence bond (GVB) methods, 688–691 gradient vector operator in Cartesian coordinates, 12–13, 74 in generalized coordinates, 73 in parabolic coordinates, 77 product of, 12–13 in spherical coordinates, 74 in spheroidal coordinates, 76 Gram–Schmidt orthogonalization. See Schmidt orthogonalization Green’s functions, 199–202 ground-state H2

923

chemical bond in, 601–617 Coulson–Fischer AOs, 616–617, 617f covalent valence bond theory, orthogonality catastrophe in, 618–623 dipole polarizability of, 760 Heitler–London (HL) theory, 608–609 molecular orbital-configurational interaction and full valence bond, equivalence between, 611–617 molecular orbital (MO) theory, 602–608, 611–613 Pauling’s formula for, 657–659 relativistic theory, 850–853 valence bond (VB) theory, 611–616

H H2O molecular orbital description of, 563–564, 564t Mulliken population analysis of, 559–561 sp2 hybridization in, 661–662 valence bond description of, 662–664, 663f Hamiltonian operator, 10 Hamiltonian partitioning, perturbation methods without, 40–45 harmonic oscillator, 91, 101–104, 165–166 Hartree–Fock (HF) theory for closed shells, 550–567 basic theory of, 550–551 fundamental invariant r, properties of, 550–551, 586, 593 HF equations, Roothaan’s variational derivation of, 553–555 HF wavefunction, electronic energy for, 552–553 HOMO-LUMO, 556 LCAO-MO-SCF equations, Hall–Roothaan’s formulation of, 555–558 molecular orbitals localization, 565–567 Mulliken population analysis, 558–561 Pyykkö HF/2D, 564 quantum chemical calculations, atomic bases in, 561–565 He atom, 493–496, 497–502, 501f, 692, 693t, 751–752, 752t, 847–850 HeH+ molecule, 850–852 Heisenberg’s uncertainty principle, 458–459, 463–465 Heitler–London (HL) correlation in HL wavefunction, 686, 719 theory for H2, 608–611 Hellmann–Feynman theorem, 728, 871–872 Hermite functions, 163–166 integrals over, 165–166 Hermite polynomials, 163–164, 197 Hermitian matrix, 256, 259, 261–265, 273–276, 277–288, 290–295 analytic functions of, 261 canonical form of, 262 examples of, 262–265

924

Subject Index

Hermitian matrix (Continued ) projectors of, 262 Hermitian operators, 10–14, 16, 59–60, 62–65 (HF)2 first-order electrostatic energy in, multipole expansion of, 761–765 homodimer, dispersion damping in, 778 H–H dispersion coefficients, 512–513, 512t, 513t, 745, 745t, 747t, 761t, 762t expanded interaction up to second order, 742–745 long-range dispersion interaction, higher-order terms in, 746–747 non-expanded interaction up to second order, 735–737 non-expanded second-order dispersion energy, multipole analysis of, 740–742 non-expanded second-order induction energy, multipole analysis of, 737–739 HHe+2 molecule, 846–847 H–H+ non-expanded interaction up to second order, 731–735 Hückel’s theory (HT), 567–579 alternant hydrocarbons, 572–577, 576f, 578f band theory of solids, 577–579 of benzene molecule p electrons, eigenvalue problem in, 270–273 diagonalizing matrix, by unitary transformation, 272–273 general considerations, 270–272 closed chain, general solution for, 570–572 of chain hydrocarbons, 30–31 extended Hückel theory (EHT), 580 linear chain general solution for, 569–570 recurrence relation for, 568–569 hybridization, 560–561 properties of, 664–667 hydrocarbons, alternant, 572–577, 576f, 578f hydrogen atom (H) dipole polarizability of, 510–512 in parabolic coordinates, 96–98, 124–128 in spherical coordinates angular eigenfunctions, 115–116 energy levels, 108 F equation, solution of, 109 quantum numbers, 107–108, 113, 115 radial eigenfunctions, 108, 470f radial equation, solution of, 105–108, 131–132 Q equation, solution of, 109–115, 132–135 relativistic equation, 843–846 Schroedinger eigenvalue equation for, 69–70, 104–115, 136–139 Stark effect in, 124–131

two ground-state hydrogen atoms, London attraction between, 512–513 in uniform electric field, 96–98, 117–120 hydrogen-like atomic orbitals (HAOs), 115–117, 468–473 hydrogen molecular ion (H+2 ), 120–124, 503–510, 503f, 507t, 508t, 509f, 509t, 772–773 ground and first excited state of, 503–505 interaction energy, 505–510 hydrogen molecule (H2) chemical bond in, 601–617 Coulson–Fischer AOs, 616–617, 617f covalent valence bond theory, orthogonality catastrophe in, 618–623 dipole polarizability, 760 excited triplet state, 609–611 Heitler-London (HL) theory, 608–611 molecular orbital-configurational interaction and full valence bond, equivalence between, 611–617 molecular orbital (MO) theory, 602–608, 611–613 molecular orbital theory failure for, 602–608 Pauling’s formula for, 657–659 relativistic theory, 850–853 valence bond (VB) theory, 611–616 Hylleraas method, 33–34 hyperbolic functions, of complex variable, 221–222 hypergeometric differential equation, 166–168 hypergeometric functions, 166–170 confluent, 168–170 hypergeometric series, 166–168

I identity matrix, 256 incomplete gamma function, 178–179 indefinite integral, 790–791 independent particle model (IPM), 534, 549–550 inequalities Bessel’s, 15 Darboux, 225 Schwarz, 5 infinitesimal operator, 355, 359 inner product. See scalar product integral operator, 9 integration by parts, 64, 224 integration over a not simply connected domain, 225–227 interatomic potentials, 731–752 H–H expanded interaction up to second order, 742–745 H–H long-range dispersion interaction, higher-order terms in, 746–747 H–H non-expanded interaction up to second order, 735–737 H–H non-expanded second-order dispersion energy, multipole analysis of, 740–742

Subject Index

H–H non-expanded second-order induction energy, multipole analysis of, 737–739 H–H+ non-expanded interaction up to second order, 731–735 many-electron atoms, expanded dispersion interaction for, 748–752 intermediate neglect of differential overlap (INDO) method, 584 intersection of vibronic states, 925 inverse Fourier transform, 185 inverse matrix, 256, 270, 275, 276, 283, 287–288 irreducible representations of groups, 338–340 irregular solid harmonics. See spherical tensors isomorphism, 328–331

J Jacobi polynomials, 196 Jahn–Teller effect, in CH+4 , 874–877 James–Coolidge generalized wavefunctions, 694–695, 694t Jeziorski and Ko1os (JK) perturbation theory, 49–50 Jordan lemma, 234–235

K Kirkwood method, 34–35 Klein–Gordon relativistic equation, 833 Kolos–Wolniewicz wavefunctions, 692–695 Koopmans’ theorem, 563 Kotani branching diagram, 396–397, 396f synthetic method, 397–398 Kummer’s function, 168–169, 178–179

L ladder operators, 13, 359–360 Lagrange interpolation formula, 268–269, 276, 287–288 multipliers, 553, 586–587, 594 Laguerre functions, 7–8, 158–163 associated, 159–161 integrals over, 161–163 recurrence relations for, 159 Landau–Zener rule, 883 Landé g-factor, 424, 434, 437 Laplace equation, 155 expansion of determinants, 254–255 transform, 188–189, 189t, 213–214 Laplacian Cartesian, 11–12, 73–74, 76–77, 89, 93–94, 118, 451–453 radial, 13, 69–70, 74–75, 91–92, 452–453 Laurent’s expansion, 228–230 LCAO-MO-SCF equations, Hall–Roothaan’s formulation of, 555–558

925

Lebeda–Schrader (LS) perturbation theory, 41–42 Legendre functions, 152–158 associated, 153–155 of first kind, 155–157 Neumann’s formula for, 157–158 of second kind, 155–157 Legendre operator, 452–453 Legendre polynomials, 152–154, 197 recurrence relations for, 154 series of, 154–155 Leibnitz’s rule, 175 Lennard-Jones density matrix, 254, 550–551, 586, 591–593 Lie groups, 354–356 linear chain general solution for, 569–570 recurrence relation for, 568–569 linear operator, 9 linear parameters, 21–24, 496–510 linear pseudostates, 35–36 line integrals, 223–224 linked-cluster theorem, 710, 710f logarithmic functions, of complex variable, 222 Löwdin orthogonalization, 8, 267–268 spin projection operators, 398–399, 405–408 lowering operators. See ladder operators

M many-body perturbation theories (MBPTs), 696 CC-R12-MBPT, 712–713 coupled-cluster (CC-MBPT), 711–712 many-electron atoms, expanded dispersion interaction for, 748–752 many-electron relativistic atomic systems, 854–857 many-electron relativistic molecular systems, 854–857 many-electron spin theory, 394–397 many-electron wavefunctions antisymmetry of, 534–539 electron distribution functions, 539–544 electron density, 540–543 one-electron distribution functions, 539–540 spin density, 540–543 two-electron distribution functions, 543–544 electronic energy, average value of, 546 Hartree–Fock theory for closed shells, 550–567 basic theory of, 550–551 fundamental invariant r, properties of, 550–551 HF equations, Roothaan’s variational derivation of, 553–555 HF wavefunction, electronic energy for, 552–553 HOMO-LUMO, 556

926

Subject Index

many-electron wavefunctions (Continued ) LCAO-MO-SCF equations, Hall–Roothaan’s formulation of, 555–558 molecular orbitals localization, 565–567 Mulliken population analysis, 558–561 Pyykkö HF/2D, 564 quantum chemical calculations, atomic bases in, 561–565 Hückel’s theory (HT), 567–579 alternant hydrocarbons, 572–577, 576f, 578f band theory of solids, 577–579 of chain hydrocarbons, 30–31 closed chain, general solution for, 570–572 linear chain, general solution for, 569–570 linear chain, recurrence relation for, 568–569 one-electron operators, symmetrical sums of, 544–545 Pople’s two-dimensional chart of Quantum Chemistry, 548–550, 548f problems and solutions, 585–598 semiempirical molecular orbital methods, 579–584 CNDO method, 580–584 EHT method, 580 INDO method, 584 ZINDO method, 584 Slater determinants for, 347, 535–539 Slater’s rules for, 546–548, 682–683 two-electron operators, symmetrical sums of, 545 many-valued functions, 223 matrices, 247–296 adjoint, 256 canonical form of, 262 Cayley–Hamilton theorem, 270, 277, 296 complex conjugate, 256 Cramer’s rule, 257 defined, 247–248 definite angular momentum, 425–431 density, 540, 543 diagonal, 255 eigenvalue problem of, 256–261 Hermitian, 256 identity, 256 inverse, 256, 269, 270, 276, 283, 287–288 Lagrange interpolation formula, 268–269 multiplication by a complex number, 248 null, 255 orthogonal, 256 partitioning of, 248–249 problems and solutions, 273–296 product rows by columns, 248 projectors of, 262–265 properties of, 247–248

pseudoeigenvalue problem of, 266–268 real, 256 rectangular, 248 representation of spin operators, 389–391 representatives of Hermitian operators, 16–17 representatives of symmetry operators, 308–309 scalar, 256 secular equation, 257 special, 255–256 square, 248 symmetric, 256 trace of, 248 transpose, 256 unitary, 256 unsymmetrical, 276, 288–290 maximum overlap principle, in valence bond theory, 667–669 MBPT(4)-R12, 712, 726 mean square error, 15 molecular integrals, evaluation of, 789–830 basic integrals, 790–793 definite integral, 791–793 (1s2j1s2) electron repulsion integral, 798–799 four-centre two-electron integral, over 1s STOs, 815–817 four-centre two-electron integral, over 1s STOs, by GTO transform technique, 822–824 indefinite integral, 790–791 molecular integrals, over Gaussian-type orbitals, 817–824 multicentre integrals, 814–817, 822–824 one-centre one-electron integrals, 793–795 one-centre two-electron integrals, 799–800 problems and solutions of, 824–830 1s Slater-type orbitals, two-centre integrals over, 800–811 three- and four-electron integrals, 699, 790 three-centre one-electron integral, over 1s STOs, 814–815 two-centre integrals, 812–814 over 1s Slater-type orbitals, limiting values of, 809–811 spherical coordinates, 813–814 spheroidal coordinates, 812–813 two-electron integrals, 795 molecular integrals, over Gaussian-type orbitals, 817–824 four-centre two-electron integral, over 1s STOs, by GTO transform technique, 822–824 Gaussian functions, integrals of, 819–820 Gaussian functions, properties of, 817–819 integral transforms, 820–822 molecular interactions, 752–765 expanded molecular energy corrections up to second order, 756–761 generalized multipole expansion for, 773–778 (HF)2 homodimer, dispersion damping in, 778 intermolecular potential, expansion of, 773–776

Subject Index

molecular interaction energies, 777–778 molecular moments and polarizabilities, 776–777 (HF)2 homodimer, first-order electrostatic energy in, multipole expansion of, 761–765, 764f non-expanded molecular energy corrections up to second order, 753–756 problems and solutions, 779–788 molecular moments, of molecules, 724–727 generalized, 776–777 molecular one-electron system, Schroedinger eigenvalue equation for, 93–96 molecular orbitals (MOs) HOMO-LUMO, 556 localization, 565–567 semiempirical methods, 579–584 CNDO method, 580–584 EHT method, 580 INDO method, 584 ZINDO method, 584 molecular point groups, 351–353 molecular symmetry, 297–322, 300t, 301t coordinate space, 303–305 active representations, 302–303 passive representations, 302–303 transformations in, 303–305 function space, 305–308 function reflection, 307–308 function rotation, 306–307 similarity transformations, 310 symmetry operators, matrix representatives of, 308–309 fundamental theorem of, 310–311 polyatomic molecules, ground state electron configuration of, 312–313 problems and solutions, 313–322 selection rules, 311 molecular vibrations, 863–893 Born–Oppenheimer approximation, 867–872, 889–890 CH+4 , Jahn–Teller effect in, 874–877, 876t classical/quantum mechanics, normal coordinates in, 864–866 GF method, 865, 883–884 Landau–Zener rule, 883 polyatomic molecules, conical intersections in, 878–883 problems and solutions of, 883–893 Renner’s effect in NH2, electronically degenerate states and, 872–874, 884, 891–892 translational–rotational motion separation, 863–864 valence force field (VFF) for HCN, 883, 884–889 vibronic interactions, 868–872, 871f Von Neumann and Wigner non-crossing rule, 878 Møller–Plesset (MP) perturbation theory, 696–701 moment method, 53–55

927

Mulliken population analysis, 558–561 multicentre integrals, 814–817 four-centre two-electron integral over 1s STOs, 815–817 four-centre two-electron integral, over 1s STOs, by GTO transform technique, 822–824 three-centre one-electron integral over 1s STOs, 814–815 multiconfigurational self-consistent-field (MC-SCF) method, 695, 695t multiple bonds, structural valence bond construction for, 626–627 multiplet structure of light atoms, 421–425, 501t multiplication table, 324, 326–327 Murrell–Shaw–Musher–Amos (MS-MA) perturbation theory, 48–49

N naphthalene, Pauling’s formula for, 655–657 near singular points, differential equations solution in, 100–101 Neumann’s expansions, 806, 813 Neumann’s formula, for Legendre functions, 157–158 NH2 Renner’s effect in, electronically degenerate states and, 872–874, 884, 891–892 non-convergent series, sum of, 235–238 non-expanded molecular energy corrections up to second order, 753–756 non-linear parameters, 21, 487–496, 515, 522–525 non-orthogonality between VB structures, 625, 641, 647 intermolecular, 765–767, 780, 785–786 Löwdin’s density matrices for, 683–685 orbital, 5, 490–492, 492f, 691 normalization factor, 5 no-virtual pair-approximation, 854 nuclear magnetic resonance (NMR), 385 null matrix, 255

O observables average values of, 454–455 measure of, 461–463 and operators, correspondence between, 450–454 state function of, 454–455 occupation number representation, 706 one-centre atomic problems, 181–182 one-centre integrals, 793–795 one-centre two-electron integrals, 799–800 one-dimensional harmonic oscillator, Schroedinger eigenvalue equation for, 101–104 one-electron distribution functions, 539–540

928

Subject Index

one-electron integrals, 793–795 over 1s Slater-type orbitals, 801–803 one-electron operators, 544–545, 703–704 symmetrical sums of, 544–545 in second quantization formalism, 703–704 one-electron relativistic molecular systems, 846–847 one-electron spin theory, 386–389 operators angular momentum, 13 annihilation, 702 anti-commutator, 9 anti-Hermitian, 12 antisymmetrizer, 347 Casimir, 359 commutator, 9 creation, 702 definition, 9 Fock, 553 gradient vector. See gradient vector operator Hamiltonian, 10 Hermitian (or self-adjoint), 10–14, 16, 59–60, 62–65 identity, 14–16 integral, 9 Kohn-Sham, 715 ladder (or shift), 13, 359 Laplacian in Cartesian coordinates, 11, 60–61, 66, 74 in generalized coordinates, 73 in parabolic coordinates, 71, 77 radial, 13, 69–70, 74–75 in spherical coordinates, 13, 69–70, 74 in spheroidal coordinates, 76 Legendrian, 452 linear, 9 projection, 262, 340–345, 349, 398–399, 400–401, 405–408, 431, 436, 443–445 vector, 12 orbital model, 449–450 orthogonal atomic orbitals (OAOs), 618–623 orthogonal coordinates, systems of, 71–72 orthogonalization Löwdin (symmetrical), 8, 267–268 Schmidt (unsymmetrical), 5–8, 59, 62 orthogonal polynomials, 195–197 orthonormal functions, 8–9 orthonormal set, 461

P Padé approximants, 197–199 and perturbation expansions, 710–711 parabolic coordinates, 70–71, 76–78

hydrogen atom in, x-equation of, 135 partial differential equations, 88–89 particle in three-dimensional box, Schroedinger eigenvalue equation for, 89–91 partitioning of matrices, 248–249 passive representation, of symmetry operations, 302–303 Pauli antisymmetry principle, 535 equations, for one-electron spin, 386 exclusion principle, 537 postulate, 383 repulsion between closed shells, 633–634, 765–767 spin matrices, 389 Pauling’s valence bond theory for conjugated and aromatic hydrocarbons, 641–660 allyl radical, 646–647 benzene, 647–655 butadiene, 645–646 cyclobutadiene, 644–645, 659–660 H2, 657–659 naphthalene, 655–657 singlet covalent valence bond structure matrix elements, formula for, 642–643 perfect-pairing approximation, 623 permutation group. See symmetric group perturbative methods, for stationary states, 25–55 Brillouin–Wigner (BW) perturbation theory, 37–40 Epstein–Nesbet (EN) perturbation theory, 42–45 exchange perturbation theories, 45–52 Lebeda–Schrader (LS) perturbation theory, 41–42 Møller–Plesset (MP) perturbation theory, 696–701 moment method, 53–55 Rayleigh–Schroedinger (RS) perturbation theory, 25–36 second-order approximation methods in, 32–36 symmetry-adapted perturbation theories (SAPTs). See exchange perturbation theories F equation of H-atom in spherical coordinates, 93 solution, for atomic one-electron system, 109 photon mass, 457t physical observables average value of, 18, 455–456 and Hermitian operators, correspondence between, 17–18, 450–454 physical principles, of quantum mechanics, 456–465 atomicity of matter, 458–459 Born interpretation, 460 Heisenberg’s uncertainty principle, 458–459, 463–465 measure of observables, 461–463 problems and solutions, 463–465

Subject Index

Schroedinger’s wave equation, 459–460 state function, 454–456 wave-particle dualism, 456–458 Planck’s law, 385 polarizability, of molecules, 724–731 dynamic (FDP), 726 generalized, 776–777 static, 725–726 polarization (P) perturbation theory, 47–48 polyatomic molecules conical intersections in, 878–883 ground state electron configuration of, 312–313 point groups of, 301t Pople’s two-dimensional chart of Quantum Chemistry, 548–550, 548f population analysis, 541–543, 558–561 post-Hartree–Fock methods, 681–722 density functional theory (DFT), 713, 716 diagrammatic theory, 706–713 CC-R12-MBPT, 712–713 coupled-cluster many-body perturbation theory, 711–712 fourth-order diagrammatic theory, 708–710, 708f, 709f, 710f linked-cluster theorem, 710, 710f Padé approximants and perturbation expansions, 710–711 second-order diagrammatic theory, 707, 708f third-order diagrammatic theory, 707, 708f problems and solutions of, 716–722 second quantization methods, 702–706 annihilation operators, 702–703 creation operators, 702–703 energy expressions, 705–706 Fock space, 706 one-electron operators, 703–704 two-electron operators, 704–705 Slater determinants, matrix elements between, 682–685 non-orthogonal determinants, Löwdin’s density matrices for, 683–685 orthonormal determinants, Slater’s rules for, 546–548, 682–683 spinless pair functions and the correlation problem, 685–686 power series, of elementary functions, 219–222 projection operator method, 431, 436, 443–445 projectors, 262, 338, 340–345, 349, 351–352, 352f, 398–399, 400–401, 405–408, 431, 436, 443–445 pseudoeigenvalue problem of matrices, 266–268, 277, 290–295 pseudospectra, 744 dipole, 512t, 531t, 745t octupole, 747t quadrupole, 747t reduced, 751–752, 752t, 760t, 761t pseudostates, 36, 511–512, 530t

929

Q quantum chemical calculations accuracy scale of, 549, 549f atomic bases in, 561–565 Quantum Chemistry, two-dimensional chart of, 548–550, 548f quantum mechanics basic principles of, 17–19 differential equations, 87–150 fundamental postulates, 450–456 normal coordinates in, 864–866 physical principles of, 449–465 symmetry and, 298–299

R Racah’s formula, 434 Racah’ spherical harmonic, 758 radial equation separation, in Dirac’s relativistic equation for a central field, 841–842 radial equation solution, for atomic one-electron system, 105–108 raising operators. See ladder operators Ransil optimization, 522–523 rational functions, of complex variable, 220 ratio test, 220 Rayleigh expansion, 773 functional, 19 ratio, 19–22 variational approximations, 20–21 variational principles, 19 Rayleigh–Schroedinger (RS) perturbation theory, 25–36, 710–711 first-order theory for degenerate eigenvalues, 30–31 Hylleraas method, 33–34 Kirkwood method, 34–35 Ritz method for second-order energies, 35–36 second-order approximation methods in, 32–36 Unsöld approximation, 32–33 real matrix, 256 real variable functions integrals, evaluation of, 238–241 recurrence relations, for Hermite polynomials, 164 recurrence relations, for Laguerre polynomials, 159 recurrence relations, for Legendre polynomials, 154 reduced pseudospectra, 751–752, 752t, 760t, 761t regular functions, 4–5 of complex variable, 218 regular solid harmonics. See spherical tensors

930

Subject Index

relativistic molecular quantum mechanics, 831–862 Dirac’s equation for a central field, 838–846 hydrogen-like atom, 843–846 radial equation separation, 841–842 Dirac’s relativistic equation for the electron, 834–835 Klein–Gordon relativistic equation, 833 many-electron atomic systems, 854–857 many-electron molecular systems, 854–857 one-electron molecular systems, 846–847 problems and solution, 857–862 Schroedinger’s relativistic equation, 832–833 spinors, 835–838 two-electron atomic systems, 847–850 two-electron molecular systems, 850–853 Renner’s effect in NH2, electronically degenerate states and, 872–874, 891–892 renormalization terms, 30 residue theorem, 232–233 Ritz–Hylleraas method, 35–36, 745, 761 Ritz method, 21–24, 487, 496–510, 555, 611, 687, 732–733 atomic applications of, 497–502 molecular applications of, 503–510 for second-order energies, 35–36 variational approximation due to, linear combination of, 496 Rodrigues’ formula, for Legendre polynomials, 153 Rodriguez formula, for quadratic polynomials, 196 rotation groups, 356–364 axial groups, 356–358 spherical group, 358–361 Russell–Saunders (LS) coupling, for light atoms, 421–425, 424f, 425f, 501f

S scalar matrix, 256 scalar product, 4–5 Schmidt orthogonalization, 5–8, 59, 62 Schoenflies notation, 299, 300t, 301, 351–353 Schroedinger equation eigenvalue, 25, 28 relativistic, 832–833 time-dependent, 18–19, 455–456 wave, 459–460 Schwarz inequality, 5 second order, variational principles in, 510–513 second-order approximation methods, in RS perturbation theory, 31–36 Hylleraas method, 33–34 Kirkwood method, 34–35 Ritz method for second-order energies, 35–36 Unsöld approximation, 32–33 second order diagrammatic theory, 707, 708f

second quantization methods, 702–706 annihilation operators, 702–703 creation operators, 702–703 energy expressions, 705–706 Fock space, 706 one-electron operators, 703–704 two-electron operators, 704–705 selection rules, 311, 340 semiempirical molecular orbital methods, 579–584 CNDO method, 580–584 EHT method, 580 INDO method, 584 ZINDO method, 584 separation of variables (differential equations), 89–98 atomic one-electron system, 91–93 hydrogen atom in a uniform electric field, 96–98 molecular one-electron system, 93–96 particle in three-dimensional box, 89–91 three-dimensional harmonic oscillator, 91 series expansion, solution of differential equations by, 98–100 shift operators. See ladder operators similarity transformations in function space, 305–308 singlet covalent valence bond structure matrix elements, formula for, 642–643 Slater determinants for many-electron wavefunctions, 535–539 matrix elements between, 546–548, 682–685 non-orthogonal determinants, Löwdin’s density matrices for, 683–685 orthonormal determinants, Slater’s rules for, 546–548, 682–683, 697–698 Slater’s rules, for orthonormal determinants, 546–548, 682–683, 697–698 Slater-type orbitals (STOs), 7, 473–475 1s Slater-type orbitals four-centre two-electron integral, 815–817 one-electron integrals, 801–803 three-centre one-electron integral, 814–815 two-centre integrals, limiting values of, 809–811 two-electron integrals, 803–809 special matrices, 255–256 spherical Bessel functions, 171–176 spherical coordinates, 69, 71f, 74–75 electrostatic potential in, 795–797 hydrogen atom in F equation, solution of, 109 radial equation of, 105–108, 131–132 Q equation, solution of, 109–115, 132–135 two-centre integrals in, 813–814 spherical Gaussian orbitals, 475–476 spherical harmonics, 115–116, 361

Subject Index

irregular, 468–469 modified, 190 Racah’s, 758, 813 regular, 468–469, 758 transformation properties of, 362–364 spherical tensors, 190–195 in complex form, 190 generalized, 195 in real form, 191–194, 192f, 193f, 194f Wigner–Eckart theorem, 190 spheroidal coordinates, 70, 72f, 75–76 confocal, 70, 72f electrostatic potential in, 797–798 two-centre integrals in, 812–813 spin density, 540–543 spin-free quantum chemistry, 351 spinless pair functions and the correlation problem, 685–686, 717–719 spinors, 835–838 square matrix, 256 standardization factor, 196 Stark effect in atomic hydrogen x-equation solution, in zero-field case, 124–128, 135 first-order, 128–130 higher-orders, 130–131 state function, 18 of observables, 454–455 time evolution of, 18–19, 455–456 stationary states, perturbative methods for, 25–55 Brillouin–Wigner (BW) perturbation theory, 37–40 Epstein–Nesbet (EN) perturbation theory, 42–45 exchange perturbation theories, 45–52 without Hamiltonian partitioning, 40–45 Lebeda–Schrader (LS) perturbation theory, 41–42 Møller–Plesset (MP) perturbation theory, 696–701 moment method, 53–55 Rayleigh–Schroedinger (RS) perturbation theory, 25–36 second-order approximation methods in, 32–36 symmetry-adapted perturbation theories (SAPTs). See exchange perturbation theories subgroups, 327–328 submatrices, 248 sum-over-pseudostates expression, 36 symmetric group, 345–351 symmetric matrix, 256 symmetry, 298–299 -adapted functions, 334, 336, 340–345 fundamental theorem of, 339 molecular. See molecular symmetry transformations, in coordinate space, 303–305 transformations, in function space, 305–309 symmetry-adapted perturbation theories (SAPTs), 45–52

931

Eisenschitz–London–Hirschfelder–Van der Avoird (EL–HAV) theory, 50–51 Jeziorski-Ko1os (JK) theory, 49–50 many-electron systems theory, 51–52 Murrell–Shaw–Musher–Amos (MS-MA) theory, 48–49 paradox, 45–46 polarization (P) theory, 47–48 SAPT2006 program, 51–52

T Taylor approximants, 198 Taylor’s expansion, 198, 228 Tchebichef polynomials, 197 tesseral harmonics (real), 116, 191, 470, 774 Q-equation solution, for atomic one-electron system, 109– 115, 132–135 third-order diagrammatic theory, 707, 708f three-centre one-electron integral, over 1s Slater-type orbitals, 814–815 three-dimensional harmonic oscillator, Schroedinger eigenvalue equation for, 91 three-electron integrals, 699, 790 time-dependent Schroedinger equation, 18–19, 455 translational–rotational motion separation, 863–864 transpose matrix, 256 truncation error, 15 turn-over rule, 10 two-centre integrals, 799–814 over 1s Slater-type orbitals, 800–809 limiting values of, 809–811 spherical coordinates, 813–814 spheroidal coordinates, 812–813 two-electron distribution functions, 543–544 two-electron integrals, 795 over 1s Slater-type orbitals, 798–799, 803–809 two-electron operators, 545, 704–705 symmetrical sums of, 545 in second quantization problems, 704–705 two-electron relativistic atomic systems, 847–850 two-electron relativistic molecular systems, 850–853 two-electron spin theory, 392–394 two-electron wavefunctions, antisymmetry of, 534–535

U uniform electric field, hydrogen atom in, 117–120 Schroedinger eigenvalue equation for, 96–98 unitary matrix, 256 Unsöld approximation, 32–33 unsymmetrical matrix, 276, 288–290

932

Subject Index

upper bounds second-order energy, 33 total energy, 20, 428

V valence bond (VB) theory, 599–680 allyl radical (N ¼ 3), 627–629 cyclobutadiene (N ¼ 4), 630–631 general formulation of, 623–626 ground-state H2 Heitler–London theory for, 608–609 Pauling’s formula for, 657–659 VB theory for, 611–616 H2, chemical bond in, 601–623 H2O sp2 hybridization in, 661–662 valence bond description of, 662–664 hybridization, properties of, 664–667 maximum overlap principle in, 667–669 multiple bonds, structural construction for, 626–627 Pauling’s valence bond theory for conjugated and aromatic hydrocarbons, 641–660 allyl radical, 646–647 benzene, 647–655 butadiene, 645–646 cyclobutadiene, 644–645, 659–660 H2, 657–659 naphthalene, 655–657 singlet covalent valence bond structure matrix elements, formula for, 642–643 problems and solutions of, 669–680 simple molecules, valence bond description of, 631–641 Van der Waals (VdW) bond, 767–771 variational method, 19–24, 483–532 atomic applications of, 497–502

basis functions and parameters, 486–487 in first order, 21–24, 484–487 linear parameters (Ritz method), 21–24, 496 molecular applications of, 503–510 non-linear parameters, 21, 487–496 problems and solutions of, 514–532 in second order, 510–513 variational approximations, 485–486 vector coupling coefficients. See Clebsch–Gordan coefficients vector model, 418–425 coupling of angular momenta, 418–420 Russell–Saunders (LS) coupling, 421–425, 424f, 425f, 501f virial theorem, 481, 613–614 Von Neumann and Wigner non-crossing rule, 878

W wave-particle dualism, 456–458 Wentzel–Kramers–Brillouin (WKB) method, 55–59, 67–68, 130–131 Weyl’s formula, 626 Wigner–Eckart theorem, 190 Wigner’s formula, 396, 625 Wigner 3-j and 9-j symbols, 431–433

Y Young diagrams and tableaux, 349 Young–Yamanouchi–Wigner projector, 349

Z Zeeman effect, 382–385 Zerner intermediate neglect of differential overlap (ZINDO) method, 584 zeros of regular functions, 230