Theoretical Chemistry (SPR Theoretical Chemistry (RSC)) (Vol 4)

Theoretical Chemistry Volume 4

A Specialist Periodical Report

Theoretical Chemistry Volume 4

A Review of the Recent Literature

Senior Reporter C. Thomson, Department of Chemistry, University of St. Andrews Reporters J. Ladik, University of Erlangen-Niirnberg, West Germany N. H. March, University of Oxford S. Suhai, University of Erlangen-Nurnberg, West Germany S.WiIson, University of Oxford

The Royal Society of Chemistry Burlington House, London W1 V OBN

British Library Cataloguing in Publication Data Theoretical chemistry - (A Specialist periodical report) Vol. 4: A review of the recent literature 1. Chemistry, Physical and theoretical - periodicals I. Royal Society of Chemistry 541.2 QD453.2

ISBN 0-85186-784-7 ISSN 0305-9995

Copyright 0 1981 The Royal Society of Chemistry All Rights Reserved No part of this book may be reproduced or transmitted in any form or by any means - graphic electronic, including photocopying, recording, taping or information storage and retrieval systems - without written permission from The Royal Society of Chemistry

Printed in Great Britain by Adlard and Son Limited Bartholomew Press, Dorking

Foreword This fourth volume of the Specialist Periodical Reports on Theoretical Chemistry contains three articles dealing with topics which are perhaps less familiar to chemists but which reflect some of the contemporary interests in theoretical chemistry. Wilson reviews in detail many-body perturbation theory of molecules, which is one very useful technique for the inclusion of electron correlation in molecular calculations for small molecules. Ladik and Suhai at the other extreme describe the important advances which have recently been made in the study of the electronic structure of polymers, with emphasis on the use of ab initio methods, which have become practicable in recent years following the development of new computational schemes. Finally, March surveys the current status of the density functional approach, which gives an alternative approach to the description of atoms and molecules. A glossary of abbreviations used in the text is given on page xi. As in previous volumes, this Reporter has not attempted to restrict the authors to the use of SI Units, and conversion factors to SI units are given on page xii. I would like finally to thank my co-Reporter on Volumes 1-3, Professor R. N. Dixon, for his work in starting and maintaining this series of Specialist Reports. C. THOMSON

Contents Chapter 1 Many-body Perturbation Theory of Molecules By S. Wilson

1

1 Introduction

1

2 The Many-body Perturbation Theory General Remarks The Partitioning Technique Lennard-Jones Brillouin Wigner Perturbation Theory Rayleigh-Schrodinger Perturbation Theory The Many-body Perturbation Theory Diagrammatic Conventions Diagrammatic Perturbation Theory Generalizations

4 4 4 5 7 7 9 12 13

3 The Algebraic Approximation General Remarks The Algebraic Approximation Universal Basis Sets Basis Set Truncation

15 15 15 16 18

4 Truncation of the Many-body Perturbation Expansion General Remarks Pad6 Approximants and Perturbation Expansions Scaling of the Zero-order Hamiltonian Modified Potentials Upper Bounds to Total Energies Fourth-order and Higher-order Terms Quasi-degeneracy Effects Comparison with Other Methods

19 19 20 22 22 23 23 30 31

5 Computational Aspects General Remarks Third-order Many-body Perturbative Calculations Higher-order Terms Bubble Diagrams Vector Processing Computers

34 34 34 36 37 39

6 Some Applications General Remarks

40 40

Contents

viii Application to Liz, N, Potential Energy Curves Triple-excitations and Quadruple-excitations Molecular Properties 7 Concluding Remarks General Remarks Some Other Aspects Final Comments

Chapter 2 The Electronic Structure of Polymers By J. Ladik and S. Suhai

41 42 43 44 45

45 45 47

49

1 Introduction

49

2 Hartree-Fock LCAO Crystal Orbital Method A6 inifio Closed Shell Formalism

51 51 53

DODS Crystal Orbital Method Truncation of Infinite Lattice Sums Calculation of Wannier Functions

54 56

3 Excited States and Correlation Effects in Polymers Intermediate Exciton Theory of Excited States More General Treatments of Electron Correlation in Polymers

57 57 59

4 Semi-empirical Crystal Orbital Methods

61

5 Disorder Effects in the Electronic Structure of Polymers Application of Dean’s Negative Eigenvalue Theorem to Aperiodic Polymers Treatment of Point Defects in Polymers

63

6 Illustrative Examples Polyacetylenes (Polyenes) Infinite Stacks of TCNQ and TTF Molecules Periodic DNA Models Periodic Protein Models Impurity and Aperiodicity Effects in Polymers

65 65 77 80 83 84

Chapter 3 Electron Density Description of Atoms and Molecules By N. H. March

64 65

92

1 Introduction

92

2 Density-Potential Relation of Thomas-Fermi Statistical Theory Self-consistent Fields for Heavy Positive Atomic Ions

92 93

Contents

ix 3 Variation Principle and Chemical Potential of TF Theory Kinetic Energy Density of Electron Cloud Euler Equation for Density

95 96 96

4 Energy Relations for Heavy Positive Atomic Ions

Total Energy for Heavy Neutral Atoms Comparison with Bare Coulomb Field Scaling of Energies of Positive Ions

97 97 98 99

5 Relation of TF Theory to l/Z Expansion

100

6 Inhomogeneity and Exchange Corrections to TF Theory Origin of Corrections to TF Neutral Atom Energy Chemical Potential and Energy Relations

103 105

102

7 Ionic Binding Energies, Ionization Potentials, and Electron

-tY

105

8 Kinetic Energies Calculated from Density Gradient Expansion

108

Relation between Total Energy and Sum of One-electron Energies

110

9 Density and Potential Distribution in Molecules

Central Field Model of Tetrahedral and Octahedral Molecules

10 Energy Relations for Molecules at Equilibrium Adoption of Central Field Model at Equilibrium Test of Energy Relations on Small Molecules Regularities in Nuclear-Nuclear Potential Energy

111 112 114 114 115 116

11 Teller’s Theorem, Chemical Potential, and Molecular Binding

119

12 Form of Energy of Homonuclear Diatomic Molecules

120

Coulomb Field Scaling for Diatomic Molecules Proposed Scaling in Self-consistent Field Theory

120 121

13 Can the Total Energy of a Molecule be Represented as the Sum of Orbital Energies? Density Gradient Corrections Basis for the Derivation of Walsh’s Rules 14 Density Description of Molecular Vibrations

Localized Models of Electron Density in Molecules Point Charge Model of XY, Linear Symmetric Molecules

15 Inclusion of Correlation in Density Theory Gradient Correction to Local Exchange and Correlation Energy

123 124 124 127 127 129

131 132

Contents

X

133

16 Electronegativity and Chemical Potential Equivalence of Chemical Potent ial and Sanderson’s Electronegativity Electron Migration in a Model Heteronuclear Diatomic Molecule Electronegativity Equalization in Bond Charge Model of Diatomic Molecules Simple Charge Transfer Model for Electronegativity Neutralization Total Energy, Sum of Orbital Energies, and Electronegativity

134

17 Wave Function Calculations and Density Functional Theory First Row Diatomic Molecules Alkali Dimers Iron-series Dimers

142 143 148 151

18 Topology of Molecular Charge Distributions Theory of Topological Dynamics of Molecular Systems Topological Definition of Atoms, Bonds, and Structure

158 159 159

19 Summary and Future Directions

160

134

135 138 139

Appendix 1 Some Results on the Chemical Potential for Electrons Moving Independently in a Harmonic Well and in a Pure Coulomb Field Bare Coulomb Field

164 167

Appendix 2 Hohenberg-Kohn and Two Other Density Theorems

168

Appendix 3 One-body Potential in He and H a

169

Appendix 4 Electron Correlation, Including Spin Density Description Spin Density Description

171 172

Appendix 5 Exact Differential Equation for Particle Density for N Particles Moving in One-dimensional Harmonic Oscillator Potential 173

Author Index

175

Abbreviations

Atomic orbital Brillouin zone Configuration interaction CI Coherent potential approximation CPA Coupled electron pair approximation CEPA Complete neglect of differential overlap CNDO Crystal orbital co Charge transfer CT DEMBPT Double excitation many-body perturbation theory Different orbitals for different spins DODS Electron affinity EA Extended Hartree-Fock EHF Hartree-Fock HF Har tree-Fock-Slater HFS Ionization potential IP Linear combination of atomic orbitals LCAO LCMTO Linear combination of muffin tin orbitals MCSCF Multi-configuration self-consistent field MIND0 Modified intermediate neglect of differential overlap Modified neglect of differential overlap MNDO Neglect of diatomic differential overlap NDDO Polydiacetylenes PDA Pariser-Parr-Po ple PPP Sugar phosphate SP Thomas-Fermi TF Thomas-Fermi-Dirac TFD Unrestricted Hartree-Foc k UHF A0

BZ

Units A number of different sets of units are used throughout this volume. Conversions to SI units are as follows: Energy: 1 a.u. (hartree)=4.359 828 aJs2625.47 kJ mo1-1 1 eV=0.160210 aJ~96.4868kJ mo1-l 1 cm-l= 1.986 31 x J~11.9626J mol-l Length: 1 a.u. (bohr)=0.529 177 x 10-lo m 1 A (ingstrom)= 10-10 m Dipole moment : 1 D (debye) = 3.335 64 x Cm Magnetic moment: 1 ,UB (Bohr magneton)=9.2732 x J T-l

Many- body Perturbation Theory of Molecules BY S. WILSON

1 Introduction

Chemistry is primarily concerned not with the properties of single molecules but with periodic trends, homologous series and the like. It is, therefore, important that any method which we apply to the problem of molecular electronic structure depends linearly on the number of electrons in the system being studied. Meaningful comparisons of atoms and molecules of different sizes are then possible. This property has been termed size-consistency 2. Independent electron models, such as the widely used Hartree-Fock approximation, provide a size-consistent theory of atomic and molecular structure. Independent-electron models account for the major proportion, typically 99.5%, of the non-relativistic electronic energy of an atom or molecule. The Hartree-Fock model describes not only the Fermi interactionsof the electrons but also their averaged electrostatic interactions. It is unfortunate that the remaining energy is of the same order of magnitude as most energies of chemical interest. This remaining energy, the correlation energy, arises from the ‘instantaneous correlations’ of the individual electronic motions. Chemistry is primarily concerned with small energy differences, such as those between different nuclear geometries or different electronic states, and these differences may be seriously affected by the correlation energy. In the past twenty years, there has been increasing interest in the calculation of correlation energies and other properties of atomic and molecular systems by means of diagrammatic many-body perturbation theory techniques3--8 due to Brueckner l o and Goldstone.ll Diagrammatic many-body perturbation theory provides a simple pictorial representation of electron correlation effects in atoms 1g

J. A. Pople, J. S. Binkley, and R. Seeger, Int. J. Quantum Chem., 1976, 10, 1. E. R. Davidson and D. W. Silver, Chem. Phys. Lett., 1977, 52, 403 3 N. H. March, W. H. Young, and S . Sampanthar, ‘The Many-body Problem in Quantum Mechanics’, Cambridge University Press, 1967. 4 A. L. Fetter and J. D. Walecka, ‘Quantum Theory of Many-particle Systems’, McGrawHill, New York, 1971. 5 H. P. Kelly, Adu. Chem. Phys., 1969, 14, 129. * J. Paldus and J. Cizek, Adu. Quantum Chem., 1975, 9, 105. 7 I. HubaC and P. €Sirsky, Top. Curr. Chem., 1978, 75,97. 6 S. Wilson, in ‘Proceedings of Daresbury Study Weekend’, Dec. 1977, ed. V. R. Saunders, Science Research Council, London, 1978. 9 S. Wilson, in ‘Proceedings of Daresbury Study Weekend‘, November 1979, ed. M. F. Guest and S. Wilson, Science Research Council, London, 1980. 10 K. A. Brueckner, Phys. Rev., 1955, 100,36. 11 J. Goldstone, Proc. R. SOC.London, Ser. A , 1957, 239, 267. 1 2

1

2

Theoretical Chemistry

and molecules and aIso forms the basis of a tractable, non-iterative scheme for accurate calculation^.^^ 12-14 Perturbation theory provides perhaps the most systematic technique for the evaluation of corrections to independent electron models. The many-body perturbation theory is so called because it can be applied to arbitrarily large systems. In fact, the theory was originally devised to treat infinite fermion systems. It leads to expressions for correlation corrections to independent electron models which have a linear dependence on the number of electrons being considered. If the theory is applied to a system A, giving an energy E(A), and to a system B, giving an energy E(B), and then to the combined system AB, where A and B are an infinite distance apart, then the energy of thesupersystem, E(AB), is given by ,

The energy of any system may be written as a sum of the energies of its component parts no matter how these components are defined. This property is not shared by some of the other methods currently employed in the study of electron correlation, for example the widely used method of configuration mixing limited to single- and doubleexcitations, which, when a single determinantal reference function is used, leads to an expression for the correlation energy depending on the square root of the number of electrons under consideration.16s16Limited configuration mixing is not a size-consistent technique. The diagrammatic many-body perturbation theory may be derived from the Rayleigh-Schrodinger perturbation expansion. Bruecknerlo showed that certain terms arise in the Rayleigh-Schrodinger expansion which have a non-linear dependence on the number of electrons being studied. He showed that these unphysical terms cancel in each of the first few orders of the Rayleigh-Schrodinger perturbation series. Goldstone l1 generalized this result to all orders using the diagrammatic techniques of time-dependent perturbation theory. This leads to the linked diagram perturbation All terms corresponding to unlinked diagrams depend non-linearly on the number of electrons and thus mutually cancel in each order. This cancellation of unlinked diagrams not only eliminates unphysical terms but also leads to important computational simplicacations.12-14 The pioneering work on the application of the many-body perturbation theory to atomic and molecular systems was performed by Kelly.6117-21He applied the method to atoms using numerical solutions of the Hartree-Fock equations. Many other calculations on atomic systems were subsequently D. M. Silver, Comput. Phys. Commun., 1978, 14, 71. D. M. Silver, Comput. Phys. Commun., 1978, 14, 81. 1 4 S. Wilson, Comput. Phys. Commun., 1978, 14, 91. l5 A. Meunier, B. Levy, and G. Berthier, Znt. J. Quantum Chem., 1976,10,1061. I6 W. Kutzclnigg, A. Meunier, B. Levy, and G. Berthier, Int. J. Quantum Chem., 1977,12,77. l 7H. P. Kelly, Phys. Rev., 1963, 131, 684. l 8 H. P. Kelly, Phys. Rev., 1964, 136, 896. l o H. P. Kelly, Phys. Rev., 1966, 144, 39. 8 0 H. P. Kelly, Ado. Theor. Phys., 1968, 2, 75. z1 H. P. Kelly, Int. J. Quantum Chem. Symp., 1970, 3, 349. l2 l8

Many-body Peturbation Theory of Molecules

3

reported (e.g. refs. 22-26). The first molecular calculations using many-body perturbation theory used single-centre expansions and were limited to simple hydrides where it is possible to treat the hydrogen atoms as additional perturbat i o n ~ . ~ ~More - ~ ’ recently, the theory has been applied to arbitrary molecules by employing the algebraic which is fundamental to most molecular calculations. In this approximation, single-particlestate functions are parameterized in terms of a finite basis set. This is equivalent to replacing the true hamiltonian by a model hamiltonian whose domain is restricted to some subspace of the Hilbert space associated with the true hamiltonian. In this article, the results of atomic calculations will only be considered when they are relevant to the molecular situation. This is the case in a number of areas where the application to atoms is well established but remains to be extended to molecules. This article is concerned with the application of the many-body perturbation theory to arbitrary molecular systems. Recent work43-44has shown that this technique can be at least as if not more accurate than other techniques currently employed in the study of molecular electronic structure. The method is probably computationally more efficient than other schemes and certainly has a number of theoretical properties which make its use attractive. For example, in discussing the widely used method of configuration mixing, Shavitt states :46 ‘The fact that in a configuration interaction expansion unlinked cluster contributions can only be accounted for by including quadruple- (and higher-order) excitations is one of the principal drawbacks of the method. In contrast, such contributions are automatically accounted for without explicitly computing higher-order terms, in some cluster-based methods and in many-body perturbation theory. In this sense the CI expansion is much less compact and less efficient than these approaches and becomes progressively less efficient as the number of electrons increases.’ E. S. Chang, R. T. Pu, and T. P. Das, Phys. Rev., 1968,174, I . N. C. Dutta, C. Matsubara, R. T. Pu, and T. P. Das, Phys. Rev., 1969,177, 33. 24 R. T. Pu and E. S. Chang, Phys. Reu., 1966,151, 31. 25 T. Lee,N. C. Dutta, and T. P. Das, Phys. Reu. A., 1970, 1,995. 2 6 T.Lee, N. C. Dutta, and T. P. Das, Phys. Reu. A., 1971, 4, 1410. 27 H. P. Kelly, Phys. Rev. Lett., 1969, 23, 455. 2 8 J. H. Miller and H. P. Kelly, Phys. Rev. Lett., 1971, 26, 679. 2 9 T. Lee, N. C. Dutta, and T. P. Das, Phys. Reu. Lett., 1970, 25,204. 30 T. Lee and T. P. Das, Phys. Rev. A., 1972, 6,968. 31 C. M. Dutta, N. C. Dutta, and T. P. Das, Phys. Rev. Lett., 1970, 25, 1695. 32 S. Wilson and D. M. Silver, Phys. Rev. A., 1976, 14, 1949. 38 J. M. Schulman and D. N. Kaufman, J. Chem. Phys., 1970, 53,477. 84 J. M. Schulman and D. N. Kaufman, J. Chem. Phys., 1972,57,2328. 35 U. Kaldor, J. Chem. Phys., 1975, 62,4634. 3 6 U. Kaldor, J . Chem. Phys., 1975, 63,2199. 37 M. A. Robb, Chem. Phys. Lett., 1973, 20, 274; and in ‘Computational Techniques in Quantum Chemistry and Molecular Physics’, ed. G. H. F. Diercksen, B. T. Sutcliffe, and A. Veillard, D. Reidel, 1974, p. 435. 8 8 U. Kaldor, Phys. Rev. A., 1973, 7,427. 3 9 R. J. Bartlett and D. M. Silver, J. Chem. Phys., 1975, 62, 3258; erratum, 1976, 64,4578. 4 0 D. F. Freeman and M. Karplus, J. Chem. Phys., 1976, 64, 2641. 4 1 M. Urban, V. Kello, and I. HubaE, Chem. Phys. Lett., 1977, 51, 170. 42 S. Prime and M. A. Robb, Theor. Chim. A d a , 1976,42, 181. 43 S. Wilson and D. M. Silver, J. Chem. Phys., 1977, 66, 5400. 44 S. Wilson and D. M. Silver, J. Chem. Phys., 1977, 67, 1649. 45 I. Shavitt, in ‘Modern Theoretical Chemistry’, Vol 3, ‘Methods of Electronic Structure Theory’, ed. H. F. Schaefer 111, Plenum Press, New York, 1977. 22

23

4

Theoretical Chemistry

This article is divided into seven parts. The many-body perturbation theory is discussed in the next section. The algebraic approximation is discussed in some detail in section 3 since this approximation is fundamental to most molecular applications. In the fourth section, the truncation of the many-body perturbation series is discussed, and, since other approaches to the many-electron correlation problem may be regarded as different ways of truncating the many-body perturbation expansion, we briefly discuss the relation to other approaches. Computational aspects of many-body perturbative calculations are considered in section 5 . In section 6, some typical applications to molecules are given. In the final section, some other aspects of the many-body perturbation theory of molecules are briefly discussed and possible directions for future investigations are outlined. 2 The Many-body Perturbation Theory

General Remarks.-In this section a brief introduction to the many-body perturbation theory is given. In the second part the partitioning technique due to L O ~ d i and n ~ ~Feshbach4' is used to give a straightforward and general introduction to perturbation expansions. The perturbation series of Lennard-Jones,48 Brillo~in,*~ and Wigner50is then described. This series is not suitable for application to many-particle systems and we, therefore, indicate how the many-body perturbation theory can be derived from the Rayleigh-Schrodinger perturbation theory. Diagrammatic rules and conventions are then introduced enabling the diagrammatic formulation of the many-body perturbation theory to be given. Some generalizations of the theory are briefly considered in the final part of this section. The Partitioning Technique.-Let P denote the projector onto some zero-order model wave function I D o ) and Q its complement. The electronic Schrodinger equation

2 IY}

=

d

IY)

(2)

may then be written as a two by two block matrix equation P*P (Q*P

P#Q

(g;)

Q9Q)

=

'

(3)

where1Q0>=P(?Po) * Ql Yo) can now be eliminated to produce the effective Schrodinger equation

[ P S P + P S Q (80- Q 9 Q ) - ' Q S P ] I@o}

= bol@o>

(4)

or where =

[P*P

+ PJPQ (go- Q#Q)-l

Q*P]

P.0.Lowdin, J. Math. Phys., 1962,3,969 and references therein. H. Feshbach, Ann. Phys. (N.Y.), 1962,19, 287. 4 8 J. E. Lennard-Jones, Proc. R. SOC.London, Ser. A., 1930,129,598. 4 9 L. Brillouin, J. Physique, 1932, 7 , 373. 5 0 E. P. Wigner, Math. u. naturw. Anz. ungar. Akad. Wiss., 1935,53,475. 46

47

(6)

Many-body Perturbation Theory of Molecules

5

This effective hamiltonian has eigenfunctions in the model space but has the exact energy as an eigenvalue. Various forms of perturbation theory result from different expansions of the inverse in the effective hamiltonian using the identity n=co

(&B)-l

=

c

n=O

A-1 (&j-l)n

(7)

If &?'o denotes some zero-order hamiltonian and Eo its ground state eigenvalue then the perturbation series of Lennard-Jones,48Brillo~in,4~ and WignerS0is obtained by putting

a = 80-20

(8)

B

(9)

and =3P-20

The Rayleigh-Schrodinger perturbation expansion is obtained by putting

A

= Eo-A?o

(10)

and

Lennard-Jones Brillouin Wigner Perturbation Theory.-Let us write the total hamiltonian operator as a sum of a zero-order operator and a perturbation

9 =A%

+ 2 1

with ZoI@,t> =

EiI@i>

and * p i >

= &lYf> = (Et

+ A&)pJi>

We introduce the projection operators Po =

I@O>(@Ol

and Qo =

I-Po

and employ the intermediate normalization convention <@ol!Jfo> = 1 =

Now we can define the wave operator,

<@Ol@O>

D,with the following properties

JYO) =

61@0>

and POO = Po OPO = 0

OQo = 0

Theoretical Chemistry

6

and thus obtain an expression for the level shift, AEo, for the ground state i= co

AEo = C Eo") f= 1

=

<@Ol.@ll~O>

=
I @o>

- <@oIPl@o) where the reaction operator,

p,is given by

v=

al@.

In Lennard-Jones Brillouin Wigner perturbation theory the wave operator is written as n=co

a = [ I + n c= l

EO

+ AQoE o - 9 0 R,)'] Po

and the level shift has the form

These expressions for the wave operator and the reaction operator are formally equivalent to the integral equations O=Po+

Qo

E~

+ A E ~ - - 9~ 1 -

0

(26)

and

from which the corresponding perturbation expansions can be obtained by iteration. Explicitly, the first few terms in the Lennard-Jones Brillouin Wigner perturbation series take the form Eo = < @ o l 2 o l @ o > EL" = <@o 1 9 1 I @o>

EL21 = <@I)pP&.@l I @(I>

(284 (284

4 3 ) = <@o 191(9Jf1)2

I @o>

(2W

~ 6 4= '

I@o>

(2W

I @o>

(28f)

<@o

1.#1(&.@1>3

EL51 = (GO1#1(
)~

... where

d?is the resolvent

The Lennard-Jones Brillouin Wigner perturbation expansion is a simple geometric series. However, it contains the unknown exact energy within the denominators. This expansion is, therefore, not a simple power series in the perturbation.

Many-body Perturbation Theory of Molecules

7

The perturbation theory of Lennard-Jones, Brillouin, and Wigner is not size consistent. Rayleigh-Schriidinger Perturbation Theory.-In RayleighSchrodinger perturbation theory the unknown energy in the denominators of the Lennard-Jones Brillouin Wigner expansion is avoided. This enables a size-consistent theory to be derived. The wave operator, 0,may be written in an alternative form by replacing9, b y 9 , + QoAEoQoa n d 9 , b y 9 , - QoAEoQogiving O = P o + - Qo Eo-90

(91-~~o)~

Rearranging this expression in terms of powers of the perturbation, 91, we obtain

Clearly, the terms other than the first in the expressions depend on the number of electrons in a non-linear fashion. These terms exactly cancel components of the first terms in each of the expressionswhich also have a non-linear dependence on the number of electrons. The Many-body Perturbation Theory.-The Rayleigh-Schrodinger form of perturbation theory provides an expansion for expectation values which have a linear dependence on the number of electrons in the system, N. In each order, other than zero-, first-, and second-order, terms arise which have a non-linear dependence on N. BruecknerlO showed that for the first few orders the terms having a non-linear dependence on N mutually cancel in each order. Goldstonell showed, using time-dependent perturbation theory, that this result can be generalized to all orders. The terms having a non-linear dependence on N may be associated with unlinked diagrams while those having the desired linear

8

Theoretical Chemistry

dependence on N are associated with linked diagrams. This is the well known linked diagram theorem of many-body perturbation theory. It should perhaps be stated at this point that the use of diagrams in the manybody perturbation theory is not obligatory. The whole of the theoretical apparatus can be set up in entirely algebraic terms. However, the diagrams are both more physical and easier to handle than the algebraic expressions and it is well worth the effort required to familiarize oneself with the diagrammatic rules and convent ions. The linked diagram expansion has, indeed, been derived by many authors and we shall, therefore, content ourselves with a brief outline of the Goldstone derivation here referring the interested reader elsewhere for full detail^.^-^ Before outlining the Goldstone treatment, we shall briefly mention some other derivations of the linked diagram theorem. Of particular interest is the derivation given by Brandow61which is based on the expansion of the energy-dependent denominators in the Lennard-Jones Brillouin Wigner perturbation theory. Paldus and Cizeks have given a time-independent derivation using a generalization of Wick's theorem62for time-independent problems. This approach has also been followed by Hubac and Carsky.' The many-body perturbation theory is developed in terms of some set of single particle states, &, which are eigenfunctions of some single-particleoperator, {,

I$ = EP$P *

(35)

with eigenvalues cp. In the second-quantized formalism the zero-order hamiltonian has the form 90=

dri (L+(ri)t(ri) (L(ri)

(36)

and the perturbation operator may be written as

where y+(rl) and y(rl) are the usual creation and annihilation field operators, g(rl, r2)is the twoelectron potential, and V(r,) is the effective potential which is added to the bare-nucleus hamiltonian to give the one-electron operator $(rl). There is, of course, considerable freedom in the choice of the effective potential. Use of the interaction representation in time-dependent perturbation theory and an adiabatic switching, ( I a It ) , of the perturbation yields the evolution operator

Oa(t, - 0 0 ) = f

n=

03

+ n=c 1

&(t,

(38)

-00)

where f i s the identity operator and O$(t,--00) is proportional to the nth power of the perturbation:

0: (t, - 00)

= (- i)"

1' 1" dfl

-03

--oo

dtz

. . . Jrn-'

dtn*l

(t1)#1

(tz)

. . .91 (tn)

-03

(39) B. H. Brandow, Rev. Mod. Phys., 1967,39,771. 68 G . C. Wick, Phys. Reo., 1950,80, 268. I1

Many-bodyPerturbation Theory of Molecules

9

9 i ( t ) = exp(i2ot)9i exp(lorlt)exp(-i&ot)

(40)

The function

which obeys the intermediate normalization condition
+ AEo

(42)

where the level shift is

The denominator in this equation may be written as exp [<@o I 6. (0, - 00) I@O)IL where the subscript L indicates that only terms described by linked diagrams are to be included. From equations (41) and (44),we obtain after taking the limit, the Goldstone expression for the wave function IPo) = (O(0,

- 00) I@O>)L

(45)

and the level shift AEo = <@o I ~ I ( O ) O ( O ,

- 00) I@O>L

(46)

Analysis of the products of field operators in these equations leads to a representation of the wave function and of the level shift in terms of diagrams of the type first introduced by Feynman. These diagrams provide a simple pictorial description of electron correlation effects in terms of the particle-hole formalism. Diagrammatic Conventions-A diagram is a device through which a corresponding algebraic expression can be obtained. A number of different diagrammatic conventions are in use, We shall follow the work of BrandowS1below and then discuss briefly its relation with other commonly used conventions. The basic elements of the diagrams are shown in Figure 1. Figure 1 (a) shows the diagrammatic representation of a one-electron operator matrix element. Figure 1 (b) shows the representation of a two-electron matrix which in the Brandow scheme includes permutation of the two electrons involved. Upward (downward) directed lines represent particles (holes) created above (below) the Fermi level when an electron is excited. The rules for performing the translation from a given diagram of the Brandow form to the corresponding algebraic expression are as follows: (i) Label each downward directed line with a unique ‘hole’ index: i,], k, . . and label each upward directed line with a unique ‘particle’ index a, b, c, . . . (ii) There is a summation over each unique hole and particle index covering all permissible values of these indices.

.

fi*

M.Gell-mann and F.Low, P h p . Rev., 1951, 84, 350.

.

10

(Cl

Figure 1 Diagram elements: (a) one-electron operator, (b) two-electron operator, (c) particle line, (d) hole line

(iii) The numerator of the summand consists of a product of antisymmetrized twoelectron integrals

where 4 is a one-electron function and rla is the inter-electronic separation. For each interaction, i.e. horizontal dashed line, there is an integral of this type. The orbital indices can be read from the labelled diagram. Hence the indicesp, q, r, s, should correspond to hole or particle lines entering or leaving the interaction in the following order: ‘left-out, right-out, left-in, right-in’, respectively. (iv) The denominator of the summand consists of a product of factors. There are n- 1 factors corresponding to an nth order diagram such that there is a denominator factor arising between adjacent interactions in the diagram. A denominator factor consists of the terms

where the first summation is over all hole lines that extend between the adjacent interactions and the second summation is over all particle lines that extend between the interactions. The E are the oneelectron orbital energies. (v) There is a multiplicative factor of 1/2 for each pair of ‘equivalent’ lines. An equivalent pair of lines is defined to be two lines beginning at one interaction and ending at another and both going in the same direction. (vi) There is a multiplicative factor of (- l)p, where p is the sum of h and I. h is the number of unique hole lines in the diagram and I is the number of fermion loops. A fermion loop is determined by following the hole and particle lines in the direction of the arrows to form a continuous closed loop.

Many-body Perturbation Theory of Molecules

11

As an example of the application of these rules consider the diagram representing the third-order ‘holehole’ energy, E3 (h-h), shown in Figure 2(a). The lines in this diagram may be labelled to give Figure 2(b). The lowest interaction line gives rise to the integral / t j a b , the middle line implies the integral f k l t j , and the upper line the integral f a b k l . The lower half of the diagram yields the denomi, the upper half corresponds to a denominator nator factor E { + q- ca- ~ b while

Figure 2 Diagrammatic representation of the third-order ‘holt+particle’ energy

+

factor of ~k e l - ea- ~ b The . lines labelled i and j, k and I, and a and b are equivalent pairs giving a factor of (1/2)3=1/8. There are two fermion loops and four unique hole lines giving a factor of (- 1)6 = 1. E3 (h-h) thus corresponds to the algebraic expression

where dpq

...rs... - Ep + &q + . .-&a-

&b-.

..

(50)

Two other diagrammatic conventions are commonly used in perturbation theory. The Goldstone diagrams1’ are similar to those of BrandowS1except that the interaction lines do not include permutation of the two electrons involved. Thus there is a set of Goldstone diagrams, which are related by electron exchange, corresponding to each Brandow diagram. The diagrams of H u g e n h o l t ~are ~ ~in one-to-one correspondence with those of Brandow. The Hugenholtz diagrams can be obtained from the Brandow diagrams by replacing the interaction lines in the latter by a single dot. This correspondence is illustrated in Figure 3.

Figure 3 Correspondence between Brandow and Hugenholtz diagrams 64

N.H. Hugenholtz,Physica, 1957,23,481.

12

Theoretical Chemistry

Diagrammatic Perturbation Tbeory.-The zero-order and first-order terms in the diagrammatic perturbation theory for a closed-shell molecule, which is described by a single determinantal wave function in zero-order, sum to the expectation value of the total hamiltonian for that wave function. Thus for a Hartree-Fock reference function the sum of the zero-order and the first-order energies is the self-consistent-field energy. The correlation energy is given by the sum of the second-order and higher-order terms. The second-order and third-order diagrams, using the Brandow antisymmetrized vertices convention, are displayed in Figure 4. When the Hartree-Fock reference function is employed, there is one second-order diagram and three third-order diagrams, which are distinguished by the type of central interaction line : the particle-particle (p-p) diagram, the hole-particle (h-p) diagram, and the hole-hole (h-h) diagram. The second-order and third-order (p-p) diagrams have only two hole lines; the correlation effects which they describe are, therefore, exclusively two-body. The third-order (h-p) diagram contains three hole lines and can therefore describe three-body interactions; however, if two of the hole lines are associated with the same single-particle state, then it describes two-body correlations. Finally, the third-order (h-h) diagram has four hole lines and can describe two-body, three-body, and four-body interactions.

Figure 4 Second- and third-order diagrams which arise when the Hartree-Fock model is used to obtain a reference function

Explicit expressions corresponding to the four diagrams shown in Figure 4 can be written down by following the rules given in the previous section. The secondorder energy expression has the form

where the dijab will be discussed below but in the immediate discussion should be taken to be zero. The third-order (p-p) energy may be written

the third-order (h-p) energy written as (53)

and the third-order (h-h) energy written as (54)

Many-body Perturbation Theory of Molecules

13

Note that all of the above expressions are written in terms of singIe electron functions and no reference is made to many-electron functions. This is a fundamental characteristic of the many-body perturbation theoretic approach to the correlation problem. In the above expressions the N-electron Hartree-Fock model hamiltonian, so, was used as a zero-order operator. This leads to the perturbation series of the type first discussed through second-order by Msller and Plesset.66s66 However, it is clear that any operator 8 obeying the relation [20,21

=0

* B = xr, !.><.I Pl.X.1

where I x > is an eigenfunction of #o, series. The operator Sshiited

=

(55)

may be used to develop a perturbation

r, 1.><.1~1.><.1

(56)

x

gives rise to the shifted, or Ep~tein-Nesbet,~~-~~ perturbation series. The resulting perturbation expansion has the same diagrammatic representation as that discussed above for that based on the Hartree-Fock model hamiltonian. The corresponding algebraic expressions are as given above in equations (50)-(54) except that (i) in third-order terms we omit the diagonal scattering terms, i.e. the summations are modified as follows

-

xi j abcd z - - + xi j zabcd

(574

=

(57W

Qk abc

ab fcd

gzc i#k a # c

(ii) the denominators are ‘shifted’ by

The use of shifted denominators may also be interpreted as the inclusion of certain higher-order terms in the perturbation series based on the Hartree-Fock model hamiltonian. The sum of the perturbation expansion to infinite order is, of course, independent of the choice of the zero-order operator. Assuming that the perturbation series have converged, the model, or M0ller-Plesset, and shifted, or EpsteinNesbet, perturbation series will give identical results at sufficientlyhigh order. Generalizations-The expansion of Bruecknerlo and Goldstonell can only be applied when the unperturbed wave function can be described by a single Slater Chr. Merller and M. S. Plesset, Phys. Rev., 1934, 46, 618. J. S. Binkley and J. A. Pople, Znr. J. Quantum Chem., 1975,9,229. 57 P. S. Epstein, Phys. Rev., 1926, 28, 695. 5 8 R. K . Nesbet, Proc. R. SOC. London, Ser. A , 1955,230, 312, 322. 59 P. Claverie, S. Diner, and J. Malrieu, Znt. J. Quantum Chem., 1967, 1, 751. 65

66

Theoretical Chemistry

14

determinant. Brandow 61s 6o has derived a quasi-degenerate many-body perturbation theory which has been applied by Kaldor and his co-workers61,6ato some open-shell systems which cannot be described by a single determinant in zeroorder. The formulation is developed with respect to a multi-determinantreference function. In Brandow’s formalism,61~60 the occupied orbitals are divided into ‘core’ orbitals, which are occupied in all of the determinants included in the reference function, and ‘valence’ orbitals, which are only occupied in some of the determinants in the reference function. The problem of applying the manybody perturbation theory to open-shell and multiconfiguration reference functions has been considered by a number of Brandow’s derivation61~60 is time-independent. The derivation of Kuo et al.71is a generalization of the time-dependent treatment of Goldstone.ll Johnson and Baranger 7 0 give a detailed discussion of the so-called folded diagrams which arise in the generalized expansion. The work of Lindgre~~,’~ Levy,73and Kvasnicka 74 in this area should also be noted. The work of Hegarty and Robb 75 using quasi-degenerate RayleighSchrodinger perturbation theory is also of interest. In almost all derivations of quasi-degenerate many-body perturbation theory given to date, it is assumed that the model space, the space spanned by the determinants in the reference function, is complete; that is, all possible occupations of the valence orbitals are included. However, this definition of the model space is likely to cause the appearance of intruder states. Intruder states are determinants which have to be included in the reference function in order to complete the model space but which have an energy significantly above that of other determinants in the reference function. The presence of intruder states can impair or even destroy the convergence of the perturbation expansion. A recent theoretical development in this area is the work of Hose and Kaldor 76 which allows an incomplete model space to be used and thus enables intruder states to be omitted from the reference function. A second generalization of the many-body perturbation theory, which would certainly be useful in the calculation of potential energy curves and surfaces, would be to use a reference function constructed from non-orthogonal orbitals.

B. H. Brandow, Ado. Quantum Chem., 1977,10,187. U.Kaldor, J . Chem. Phys., 1975,62,4364; 1975,63,2199. 62 P. S. Stem and U. Kaldor, J. Chem. Phys., 1976,64, 2002. O3 C. Bloch, Nuclear Phys., 1958,6, 329. 64 J. Des Cloizeaux, Nuclear Phys., 1960,20, 321. 65 H.Primas, Heiu. Phys. Actu, 1961, 34, 1961. 6 6 H.Primas, Rev. Mod. Phys., 1963,35, 710. 67 T. Morita, Progr. Theor. Phys., 1963,29, 351. 6 8 H.P. Kelly, Phys. Rev., 1966,144, 39. 6 9 P. G.H. Sandars, Ado. Chem. Phys., 1969,14,365. 7 O M.B. Johnson and M. Baranger, Ann. Phys. (N.Y.), 1971, 62, 172. 7 1 T.T.S. Kuo, S. Y. Lee, and K. F. Ratcliffe, Nuclear Phys., 1971, A176,65. 72 I. Lindgren, J. Phys. B: Atom. Moi. Phys., 1974,7 , 2441 ; Znt. J. Qirantum Chem. Symp.,

60 61

1978, 12, 33. 73

€3. Levy, in ‘Proceedings of the Fourth Seminar on Computational Methods in Quantum

Chemistry, 1978’,ed. B. Roos and G. H. F. Diercksen, Max-Planck-Institut fur Physik und Astrophysik, Munich. 74 V. Kvasnicka, Czech. J. Phys., 1974,B24, 605; Adv. Chem. Phys., 1977,36, 345. 7 5 D.Hegarty and M. A. Robb, Moi. Phys., 1979, 37, 1455. 7 6 G.Hose and U. Kaldor, J. Phys. B: Atom. Mol. Phys., 1979, 12, 3827.

Many-body Perturbation Theory of Molecules

15

The work of G e ~ ~ a tGallup,78 t , ~ ~ and Goddard79suggests a possible reference function for use in such a scheme, e.g. ref. 80. Newman81n82has examined a many-body perturbative formalism which uses biorthogonal sets of orbitals. The use of non-orthogonal orbitals has also been discussed by K v a s n i ~ k a Moshin,~~ sky and Seligman,84G o ~ y e tCantu , ~ ~ et al.,86and Kirtman and C0le.8~ 3 The Algebraic Approximation General Remarks.-The algebraic approximiation is fundamental to most applications of the methods of quantum mechanics to molecules. When employing non-variational techniques, such as perturbation theory, in the study of correlation energies it is important to be very clear about the origin of truncation effects in a particular calculation. In molecular studies using diagrammatic perturbation theory there are, of course, two truncation effects, that arising from basis set restrictions (i.e. by invoking the algebraic approximation), and that arising from truncation of the perturbation series. In applications of perturbation theory to molecules, it is important that one does not obtain good results by a fortuitous cancellation of these two sources of error. In the first part of this section, the relationship between the solution of the Schrodinger equation and the hamiltonian in the space generated by a given basis set is discussed in some detail. Since basis set limitations appear to be one of the largest sources of error in most present day molecular calculations, the concept of a universal even-tempered basis set is discussed in the second part of this section. This concept represents an attempt to overcomethe incomplete basis set problem, at least for diatomic molecules. Further aspects of the basis set truncation problem are discussed in the fmal part of this section. The Algebraic Approximation.-The determination of the electronic structure of atoms and molecules containing N electrons involves the evaluation of an appropriate eigenvalue and associated eigenfunction of a semi-bounded selfadjoint hamiltonian operator, #, in Hilbert space &. A tractable scheme for solving such equations is the algebraic approximation in which eigenfunctions are parameterized by expansion in a finite set of functions. Integro-differential Hartree-Fock equations thus become algebraic equations for the expansion coefficents. The difficulties of the atomic correlation problem are compounded in molecular studies by the impossibility of factorizing the variables in a molecular field and use of the algebraic approximation becomes almost obligatory. J. Gerratt, Adv. Atom. Mol. Phys., 1971, 7 , 141; and in ‘Theoretical Chemistry’, ed. R. N. Dixon (Specialist Periodical Reports), The Chemical Society, London, 1974, Vol. 1, p. 60. 7 8 R. C. Morrison and G. A. Gallup, J. Chem. Phys., 1969, 50, 1214. 7 9 W. A. Goddard, Phys. Rev., 1967,157, 81. 8 0 S. Wilson and J. Gerratt, Mol. Phys., 1975, 30, 777; S. Wilson, Mol. Phys., 1978, 35, 1. 8 l D. J. Newman, J. Phys. Chem. Solids, 1969, 30, 1709. s2 D, J. Newman, J. Phys. Chem. Solids, 1970, 31, 1143. 8s V. Kvasnicka, Chem. Phys. Lett., 1977, 51, 165. 84 M. Moshinsky and T. H. Seligman, Ann. Phys. (N. Y.), 1971, 66, 311. 85 J. F. Gouyet, J. Chem. Phys., 1973, 59,4637. a6 A. A. Cantu, D. J. Klein, F. A. Matsen, and T. H. Seligman, Theor. Chim. Acta, 1975,38, 77

341. 87

B. Kirtman and S. Cole, J. Chem. Phys., 1978,69, 5055.

16

Theoretical Chemistry

The algebraic approximation results in the restriction of the domain of the operator to a finite dimensional subspace, 9, of the Hilbert space A. The algebraic approximation may be implemented by defining a suitable orthonormal basis set of M ( < N ) one-electron spin orbitals and constructing all unique N-electron determinants Ip ) using the M oneelectron functions. The number of unique determinants that can be formed is q =

(3

, and

7 is the

dimension of the subspace 9spanned by the set of determinants. The algebraic approximation restricts the domain of 9to this q-dimensional subspace. Within the algebraic approximation, the Schrodinger equation may, in principle, be solved by the method of configuration mixing. The wave function is then expressed as a superposition of configurations Ip } with linear coefficients Cp,the optimal expansion coefficients determined by the variation principle. In practice, difficulties arise in setting up and solving secular equations of high order. Thus, only a small subset of the I p } is usually employed in the expansion. Diagrammatic many-body perturbation theory may also be formulated within the algebraic approximation. The domain of the perturbation theory operators is restricted to the q-dimensional space spanned by the l p } and consequently the perturbation theory wave function is generated in terms of an 7-dimensional representation. The results of many-body perturbation theory calculations, when carried through infinite order, are identical to those of configuration mixing if the same basis set, i.e. subspace 9, is used in both calculations. However, in practice, the perturbation series must be truncated at some finite order and in this case the degree of agreement with the configuration interaction result is a measure of the convergence of the perturbation series at this order. Let us briefly discuss the relationship between approaches which use basis sets and thus have a discrete single-particle spectrum and those which employ the Hartree-Fock hamiltonian, which has a continuous spectrum, directly. Consider an atom enclosed in a box of radius R, much greater than the atomic dimension. This replaces the continuous spectrum by a set of closely spaced discrete levels. The relationship between the matrix Hartree-Fock problem, which arises when basis sets of discrete functions are utilized, and the Hartree-Fock problem can be seen by letting the dimensions of the box increase to infinity. Calculations which use discrete basis sets are thus capable, in principle, of yielding exact expectation values of the hamiltonian and other operators. In using a discrete basis set, we replace integrals over the continuum which arise in the evaluation of expectation values by summations. The use of a discrete basis set may thus be regarded as a quadrature scheme. Once the algebraic approximation has been invoked there is essentially no difference between the atomic problem and the molecular problem, except that the multicentre integrals which arise in the latter case are more difficult to evaluate. Universal Basis Sets.-Historically, it has been necessary to restrict the size of basis sets employed in molecular calculations to a reasonably small number of functions in order to keep the computation tractable. However, to achieve high accuracy, moderately large basis sets are ultimately required, especially if a significant fraction of the molecular correlation energy is to be recovered, Since

Many-body Perturbation Theory of Molecules

17

the flexibility of a basis set generally increases as the number of functions is increased, the need to optimize parameters becomes less important. This has led to the suggestion that a single moderately large basis set, with a reasonable choice of parameters, might be suitable for a variety of systems. Such a basis set has been termed a ‘universal basis set’.88-02 It has been shown that Slater type orbitals restricted to Is,2p, 3 4 . . . functions can lead to good results for total energies. Orbital exponents, C, may be chosen by the ‘even-tempered’formula n3-00 Ck: = a p - 1 k = 1,2,...,Ad It can be shown that the metric matrix, S, then has the property

&,5

=

sZ+l,j+l

(59)

(60

The parameters a, p, and M for the universal even-tempered basis set thus generated have been chosen quite arbitrarily using the following guidelines : (i) a must be small enough to ensure a wide range of orbital exponent values, (ii) p must be large enough to avoid possible near linear dependence, (iii) Mmust be large enough to generate a ‘near complete’ set. The following values of a, p, and M have been shown to be useful in studies of atoms and diatomic molecules:88-02

p = 1.55 M

IS:

CY

= 0.5

Zp:

CY

= 1.0

3d:

01

= 1.5 t9 = 1.65 M = 3

=9

= 1.60 M = 6

Thus the orbital exponents of the 1s functions range from 0.5 to 16.7 bohr-I, the 2p exponents range from 1.0 to 10.5 bohr-l, and the 3d exponents from 1.5 to 4.1 bohr-l. The energy is generally found to be more sensitive to the particular choice of p than the choice of a. Universal even-tempered basis sets have been employed in calculations on first-row and second-row atoms using the matrix Hartree-Fock 89 Atomic calculations including electron correlation have also been reported.Bo For molecules calculations using universal even-temperedbasis sets of s, p , and d functions have been reported at both the Hartree-Fock levelo1 and including correlation. 92 For the nitrogen, carbon monoxide, and boron fluoride molecules about 80 % of the empirical correlation energy is recovered by taking the perturbation series through third-order. D. M. Silver, S. Wilson, and W. C. Nieuwpoort, Znt. J. Quantum Chem., 1978,14,635. D. M. Silver and W. C. Nieuwpoort, Chem. Phys. Lett., 1978,57,421. 90 D. M. Silver and S. Wilson, J. Chem. Phys., 1978, 69, 3787. 91 S. Wilson and D. M. Silver, Chem. Phys. Lett., 1979, 63, 367. 92 S. Wilson and D. M. Silver, J. Chem. Phys., 1980,72,2159; unpublished work. 93 C. M. Reeves and J. Harrison, J. Chem. Phys., 1963, 39, 11. 94 K. Ruedenberg, R. C. Raffenetti, and R. D. Bardo, in ‘Energy Structure and Reactivity’, Proceedings of the 1972 Boulder Conference on Theoretical Chemistry, Wiley, New York 1973. 95 R. C. Raffenetti, J. Chem. Phys., 1973, 59, 5936. 9 6 R. D. Bardo and K. Ruedenberg, J. Chem. Phys., 1973,59, 5956, 5966. 9 7 R. C. Raffenetti and K. Ruedenberg, J. Chem. Phys., 1973,59, 5978. 9 8 R. D. Bardo and K. Ruedenberg, J. Chem. Phys., 1974, 60,918. 9 9 R. C. Raffenetti, Int. J. Quantum Chem. Symp., 1975,9, 289. 88

89

18

Theoretical Chemistry

Let us comment at this point on the use of the even-tempered method for generating orbital exponents. A universal basis set clearly need not be an eventempered one. However, even-tempered basis functions do span the one-electron space in a fairly uniform fashion. Unlike a set of exponents determined by energy optimization for a particular system, we expect that the even-tempered functions will be more suitable for transferring between systems. Even-tempered orbital exponents are defined through the recursion CkCl

=

clcg

(62)

A possibly useful generalization of this formula is the recursion Ck+l

=

Ck(B

+ kr)

(63)

where y is an additional parameter. Several advantages would accrue to the use of a universal basis set. Since most electronic structure studies of molecules begin with the valuation of one-electron and twoelectron integrals over the primitive basis functions, then, for a given set of nuclear positions, these integrals could be computed once and used for all subsequent studies without regard to the identity of the constituent atoms. This transferability extends, of course, to all integrals including multicentre twoelectron integrals. In order to be an acceptable concept, the universal basis set must be capable of providing a uniform description of a series of atoms. Since such a basis set is necessarily moderately large, it should yield reasonably high accuracy as well as uniformity. The concept of a universal gaussian basis set should be briefly discussed since gaussian functions are widely used in molecular studies, especially in studies of polyatomic systems. Clearly, gaussian basis sets could be transferred from molecule to molecule in precisely the manner we have demonstrated for Slater functions. However, it is generally accepted that Slater basis functions provide a more realistic description of the orbitals especially near nuclei. Integrals over universal basis sets are generated once and stored for future use and it would therefore seem more profitable to use Slater functions which would generate a smaller integral list, especially if a significant fraction of the correlation energy is to be recovered. Raffenetti B9 has investigated the use of even-tempered gaussian basis sets. His results demonstrate the inferior nature of gaussian functions in accurate work. However, it should be noted that multicentre integrals over gaussian functions can be evaluated with greater precision than integrals over Slater functions. It should be noted that the concept of a universal basis set could prove useful in almost all molecular studies in which finite basis sets are employed, e.g. electron-molecule scattering. Such basis sets lead to accurate results because they are moderately large. Furthermore, they afford a certain degree of uniformity in that they are not optimized for one particular property. Basis Set Truncation.-Basis set truncation does appear to be one of the main sources of error in the majority of calculations on small molecules. The efficiency of algorithms based on the diagrammatic perturbation expansion will allow increasingly large basis sets to be employed in molecular studies during the next few years.

Many-body Perturbation Theory of Molecules

19

Calculations for atomic systems have shown that almost all of the correlation energy may be recovered in second-order, if there are no near-degeneracy effects in the reference spectrum, when a sufficiently large basis set is employed, For the ground state of the neon atom, Eggarter and Eggarter 100-103 have shown that 98% of the empirical correlation energy can be recovered by including functions with Zquantum number up to 6 in the basis set. Since there is essentially no difference between the atomic and the molecular problems once a finite basis set is introduced (excepting the difficulty of evaluating the multicentre integrals, of course), it is expected that similar results will be obtained for molecules once sufficiently large basis sets are utilized. In order to discuss the convergence of atomic and molecular calculations with respect to basis set size, one must have an orderly procedure for extending the basis set. This is necessary if the relationship between various restricted basis sets and complete bases is to be understood. The recent work of Ruedenberg and his co-workers104~106 is of great interest in this repect. They have examined the effective convergence of orbital bases through systematic sequences of gaussian primitives. They develop various sets of even-tempered gaussian functions-such that OL

= a(M)

(64)

and where f f ( M )+. 0

P(M) +- 1

as M+W

(67)

Clearly, this approach can also be used in the case of Slater basis sets and, moreover, in the case of universal Slater basis sets. Ruedenberg and c o - w o r k e r ~ ~ ~ ~ J ~ ~ have shown that, within the molecular orbital approximation, this systematic approach gives a series of energy values which smoothly approach the HartreeFock limit. Similarly smooth convergence is to be expected in the calculation of correlated wave functions and expectation values, and will be the subject of future studies in this area.lo6 4 Truncation of the Many-body Perturbation Expansion

General Remarks.-Perturbation theory forms the basis of a unique approach to the calculation of accurate expectation values in that it provides a clearly defined order parameter indicating the relative importance of various terms. This order 100 E. Eggarter and T. 101 E. Eggarter and T.

P. Eggarter, J. Phys. B: Atom. Mol. Phys., 1978,11, 1157. P. Eggarter, J. Phys. B: Afom. Mof. Phys., 1978,11, 2069. 102 E. Eggarter and T. P. Eggarter, J. Phys. B: Arom. Mol. Phys., 1978,11,2969. 10s T . P. Eggarter and E. Eggarter, J. Phys. B: Atom. Mol. Phys., 1978, 11, 3635. 104 D. F. Feller and K. Ruedenberg, meor. Chim. Acta, 1979, 52, 23 1. 105 M. Schmidt and K. Ruedenberg, J. Chem. Phys., 1979,71, 3951. 106 S. Wilson, Theor. Chim. Acfa, 1980, 57, 53; ibid., 1980, 58, 31; ibid., 1981, in the press.

20

Theoretical Chemistry

parameter provides an unambiguous criterion for truncating expansions for expectation values. Brandow thus comments that perturbation theory ‘seems to be the “least biased’’ of all the many-body techniques’. In this section various aspects of the truncation of the perturbation expansion are discussed. The use of Pad6 approximants in perturbation theory is outlined and the special invariance properties of the [ N + l / N ] Pad6 approximants in Rayleigh-Schrodinger perturbation theory are emphasized. Scaling of the zeroorder hamiltonian is considered and the construction of upper bounds to the energy described. Fourth-order and higher-order terms are presented and discussed in some detail. Investigations of the effects of quasi-degeneracy on the convergence properties of the perturbation series are reviewed. Finally, since most other approaches to the electron correlation problem in molecules may be regarded as different ways of truncating the many-body perturbation series, a brief description of the relationship between many-body perturbation theory and some other methods is given. Pad4 Apptoximants and Perturbation Expansions.-The [a/Q] Pad6 approximant lo8,log to a Taylor series T ( x ) of order M is defined by CpIQl(-4 = W4/W); P + Q=M (68) where 9 is a polynomial of order P and 9 is a polynomial of order Q such that 9 ( x ) - 9 ( x ) [P/Q](x) = O(XM+l)

(69)

where O(xM+l)denotes terms of order M + 1 and higher. Pad6 approximants provide a useful representation of perturbation series. They often converge when the usual power series diverges. Unlike the power series, Pad6 approximants can handle a class of function with various types of singularities still providing correct uniform convergence.1o9In the LennardJones Brillouin Wigner perturbation theory, PadC approximants can be used to find bounds to the exact energy.l1° On the other hand, in Rayleigh-Schrodinger perturbation theory, the [ N + 1/N] Pad6 approximants are specia1111,112in that, when the expansion parameter is set equal to unity, the numerical value of this approximant is invariant to two modifications in the zero-order hamiltonian operator, namely a change of scale and a shift of origin. Let us consider these two modifications of a given zero-order operator, so; that is a uniform displacement of the zero-order energy spectrum and a uniform change of scale in the spacing of the zero-order energy levels. Consider the zeroorder operator

2Py = p&

4- v f

where f is the identity operator and p and perturbation operator is

2y

=9

1

v

are scalars. The corresponding

+ (l-p)9o-vP

(71)

B. H. Brandow, in ‘Effixtive Interactions and Operators in Nuclei’, ed. B. R. Barratt, Springer-Verlag, Berlin, 1975. 109 H.Padt, Ann. sci. Ecol. norm. sup. Paris (Suppl.), 1892, 9, 3. 109 G.A. Baker, ‘Essentials of Pad6 Approximants’, Academic Press, New York, 1975. 110 0. Goscinski, Int. J. Quantum Chem., 1967, 1, 769. ll1 S. Wilson, D.M. Silver, and R. A. Farrell, Proc. R. SOC.London, Set. A , 1977,356,363. 119 E. Feenberg, Ann. Phys. (N. Y.),1958, 3,292.

107

Many-body Perturbation Theory of Molecules

21 Note that when the perturbation parameter is put equal to unity the total hamiltonian operator is recovered, i.e. The energy coefficients, E/'sv, where i denotes the order of perturbation, may be related to the values of by the expressions EY**

+v

(73)

+ (l-p)u)E~*O-~

(74)

= pEOi0

Ef" = E?'' and

It can be shownlll that the [N+l/N] PadC approximants alone among all PadC approximants of order 2N+1 are invariant to an arbitrary choice of p and Y, when A is set to unity. The [N+ l/Nl Pad6 approximants to the energy series also have the advantage that they have a linear dependence on the perturbation parameter, A, as it becomes large. Furthermore, the Goldstone level shift formulall may be written = <@o

I&Ol

@oh

(76) where the evolution operator, 0, is unitary if all terms are included but not if the expansion is truncated. A unitary approximation to 0 can be obtained by forming the [ N / w operator PadC approximant.llSThis has the same form as a function of the perturbation parameter as the [N+ 1/N] scalar Pad6 approximant to the level shift. From third-order perturbation calculations, the [2/1]Pad6 approximants can be formed. Several additional reasons can be advanced for their use: (i) there is a large amount of evidence demonstrating that this leads to improved results,l14 (ii) this may be regarded as the choice of scaling parameter which makes the third-order energy v a n i ~ h , ~ l ~ - ~ ~ ~ (iii) it may also be regarded as the energy expression for which the energy is a minimum with respect to choice of scaling parameter.lla It should be noted that for one-electron properties, such as dipole moments, the [N/N] Pade approximants are invariant to changes of scale and shifts of origin in the reference spectrum8 whereas for second-order properties, such as polarizabilities, the [N/N+ 11 Pad6 approximants are to be preferred. Indeed, for polarizabilities the use of the form hE0

ffg

( 1 - y

(77)

has been advocated by a number of authors.s~lle,12* 113 J. L. Gammel and F. A. McDonald, Phys. Rev., 1966, 142, 114 S. Wilson, J. Phys. B: Atom. Mol. Phys., 1979,12, 1623. 115 A. T . Amos, J. Chem. Phys., 1970, 52, 603.

u0A. T. Amos, Int. J . Quantum Chem., 1972, 6, 125. 117A. T . Amos, J. Phys. B: Atom. Mol. Phys., 1978, 11,2053. 118 D. T. Tuan, Chem. Phys. Lett., 1970,7, 115.

2

1245.

22

Theoretical Chemistry

Pad6 approximants to perturbation series have been considered by numerous authors,12Lparticularly those interested in effective interactions in nuclei. They have been used in studies of the 'intruder state' problem which can arise when a multideterminantal function is used as a reference in a perturbative scheme.122 Scaling of the Zero-order Hamiltonian.-It has been d e m o n ~ t r a t e d l ~that ~-~~~ improved results can be obtained from second-order Rayleigh-Schrodinger perturbation studies of electron correlation energies using very large basis sets by introducing a scaling parameter, p, in the zero-order hamiltonian, so. The scaling parameter may be determined, from calculations taken to third-order in the energy and using a basis set of moderate size, by imposing the condition that the third-order energy coefficient corresponding to the zero-order operator, p s 0 , be zero. The correlation energy is then approximated by p-1E2, where E2 is the second-order energy resulting from the calculation using the large basis set. Second-order calculations with large basis sets are computationally tractable, One of the most time consuming parts of many calculationsis the transformation of the twoelectron integrals over atomic functions to integrals over molecular orbitals. For second-order calculations only a restricted list of two-electron integrals is required (although the restricted transformation is still an n5 process, where n is the number of basis functions). It is envisaged that the concept of a universal even-tempered basis set will prove useful in developing the large basis sets required for these calculations.88-92 The scaling of the zero-order hamiltonian is also useful in the calculation of molecular properties and has been demonstrafedLaofor the polarizability of the hydrogen molecule.126,12' Modified Potentials.-It is possible to modify the perturbation series by using alternative potentials in the zero-order hamiltonian. The modified potential

has been widely used in atomic studies.6 Recent calculation^^^^^^^^ on the hydrogen fluoride molecule have indicated that there is little difference between results obtained using the Hartree-Fock potential and the above modified potential when all terms are included through third-order. For the FH molecule using unshifted denominators, 80.2 % of the empirical correlation energy was l19 G. Howat, M. Trsic, and 0. Goscinski, Int. J . Quantum Chem., 1977,11,283. 1aoS. Wilson, Mol. Phys., 1980,39,5 2 5 ; S . Wilson and A. J. Sadley, to be published. 181 G. A. Baker, Ado. Theor. Phys., 1965,1,1;0.Goscinski andE. Brandas, Phys. Rev. A,

1970, 1,552. 1%' E. M. Krenciglowa and T. T. S. Kuo, Nuclear Phys., 1974, A235, 171 ; H.M.Hoffmann, S. Y. Lee,J. Richert, and H. A. Weidenmuller, Ann. Phys., 1974,85,410;J . M.Leinaas and T. T. S . Kuo, Ann. Phys., 1978,111, 19; T. H. Schucan and H. A. Weidenmuller, Ann. Phys., 1972,73, 108;J. Richert, T. H. Schucan, M. H. Scrubel, and H. A. Weidenmuller, Ann. Phys., 1976,96, 139; T. H. Schucan and H. A. Weidenmuller, Ann. Phys., 1973,76,483. la8 S. Wilson, J. Phys. B: Atom. Mol. Phys., 1979,12, L135. 124 F.W. Byron and C. J. Joachain, J. Phys. B: Arum. Mol. Phys., 1979,12,L597. l8ti S. Wilson, J. Phys. B: Atom. Mol. Phys., 1979,12,L599. 1'4T. Itagaki and A. Saika, Chem. Phys. Lett., 1977,52,530. 117T. Itagaki and A. Saika, J. Chem. Phys., 1979,70,2378. m D.M.Silver and R. J. Bartlett, Phys. Reu. A, 1976,13, 1. 119 D. M. Silver, S. Wilson, and R. J. Bartlett, Phys. Reu. A, 1977, 16,477.

Many-body Perturbation Theory of Molecules

23

recovered in second-order when the Hartree-Fock potential was employed, whereas 101.2 % of the correlation energy was recovered when the Y N - l modified potential was used. The [2/1] Pad6 approximant to the perturbation series using the Hartree-Fock potential gave 79.6% of the correlation, which is close to the 77.8% given by the [2/1] Pad6 approximant to the series based on the modified potential. The infinite-order result should, of course, be independent of the choice of zero-order hamiltonian. Upper Bounds to Total Energies.-In performing a calculation of the energy to third-order one also implicitly calculates the wave function to first-order. Let the wave function through first-order be written as 0 0

+ 101

(79)

where the parameter ;1 has been introduced. The truncated expansion for the energy is not bounded: however by substituting the first-order wave function in the Rayleigh quotient, an upper bound is obtained. This may be written asS* Ev~I(')= EO

+ + El 4- (21- 1Az) f12A1, E 2

AZE3

+

(80)

where the overlap integral, All, is given by

A may be regarded as a variational parameter. Its optimal value may be obtained by invoking the variation principle, giving LFlt =

&I[ 2 / 5 2 1 - 5 1

(82)

where 5 = 4 (1 -E3/E2)

(83)

Unfortunately, by forming an upper bound one introduces terms depending on the square of the number of electrons and thus the quality of this upper bound deteriorates as the number of electrons is increased. This is illustrated in Table 1 for an array of well-separated nitrogen molecules. Table 1 Size-consistency error in an array of well separated N2Molecules? rn

na

E

2

4

ec

1 14 0.00 -0.45 2 28 0.09 (10%) -0.90 3 42 0.30 (22 %) -1.35 4 56 0.59 (33%) -1.80 1 (44%) -2.25 5 70 ?Based on results given in ref. 92. a Number of electrons. Fourth-order size-consistency error; percentage of estimated correlation energy given in parentheses. C Estimated correlation energy.

.oo

Fourth-order and Higher-order Terms.-Most methods currently employed in the study of electron correlation in atoms and molecules, for example the coupledelectron pair theory, limited configuration mixing, etc., may be regarded as third-

24

Theoretical Chemistry

order theories in that they neglect or approximate fourth-order and higher-order terms in the perturbation analysis of the energy. Diagrammatic many-body perturbation theory offers a systematic scheme for extending such calculations. The perturbation series forms the basis of a balanced treatment of the correlation problem in that it provides a clearly defined order parameter which indicates the relative importance of various terms. There is considerable interest in the evaluation of the fourth-order terms in the many-body perturbative expansion since these terms represent, at least in part, the dominant corrections to most techniques currently being employed in calculating atomic and molecular correlation energies. Unlike the second-order and third-order energy diagrams, the fourth-order diagrams can involve intermediate states which are singly-excited, doublyexcited, triplyexcited, and quadruplyexcited with respect to the Hartree-Fock reference function.**130 There are four diagrams, using the Brandow convention61which involve an intermediate state which is singly-excited with respect to the Hartree-Fock function, these are shown in Figure 5. Diagrams AS and D S are related by time reversal, whereas BSand C Sare related by complex

8,

AS

CS

DS

Figure 5 Fourth-order diagrams which involve singly-excited intermediate states

The twelve diagrams of the fourth-order which involve only doubly-excited intermediate states are displayed in Figure 6. Diagrams (BD, CD), (ED, FD),and (GD, HD) are related by complex conjugation. Diagrams (AD, DD), (ED, HD), (FD,GD), and (JD, KD) are related by time reversal. The diagrams which involve a triply-excited intermediate state in fourth-order are shown in Figure 7. Explicitly, diagrams AT,BT,CT,and DT, for example, correspond to the expressions E4(A

T)

=

- -f

-+

E4(BT)

=

E4(cT)

= -

E~@T)=

-3

;c, ( B i l u b B a k c d & c b e k & e d i f ~ ( ~ ~ 5 u b ~ ~ ~ ~ b c ~ ~ t (84) l~e) ( & r ~ u b 8 a k e d 8 c d d & e ~ d k / ( ~ t 5 u b ~ ~ 5 ~ b ~ ~ ~(85) ~~be) (&ffab$PIfc$mbkZ

~ a , m J ) / ( ~ t j u b ~ ~ k Z u b c ~ ~ ~(86) u c )

( & i f u b ~ k l f c ~ m c k j & a b m l ) / ( ~ ~ f a b ~ ~ k Z u b c ~ e m u (87) b)

Time-reversal symmetry can be Seen to relate diagrams AT,BT, ET,and FT to diagrams DT, CT, HT, and GT,respectively. Diagrams (IT, LT)and (JT,KT) form complex conjugate pairs. Diagrams (MT, PT) and (NT, OT) are related by time reversal and (MT, NT) and (OT, PT) are complex conjugates. 130 S. Wilson and D. M.Silver, Int. J . Quantum Chem., 1979, 15, 683.

Many-body Perturbation Theory of Molecules

25

Q 0 CD

AD

Q ED

FD

I D

JD

GO

HD

LO

Figure 6 Fourth-order diagrams which involve doubly-excited intermediate states

Finally, there are seven fourth-order terms which involve an intermediate state which is quadruply-excited with respect to the Hartree-Fock reference function. These diagrams are shown in Figure 8. Time reversal relates diagrams (BQ, CQ), (DQ,GQ), and (EQ, FQ).Explicitly, diagrams AQ, BQ, and CQ, for example, correspond to the expressions

Gz

E4(AQ) = E@Q) =

*

ab&

( S i j a b&kZ cd& c biz &ad k j ) / ( g i j a b g i j k l a b cd g k j a d )

(&ijab&kZcd&cd~~&abkZ)/(azjabg~jkZabcd~~Zab)

ykl abcd

(8 8)

(89)

The fourth-order quadruple-excitation energy diagrams arise from the disconnected wave function diagrams shown in Figure 9. This observation leads to considerable simplification in the evaluation of this energy component. For example, using the identity

+

( a ~ ~ a b ~ ~ ~ k Z a b c o l ~ ( c(da t) j-elb g ~ j k Z a b c d a k Z a b ) - l = ( g i j a b g i j c d d k Z a b ) - l

(91)

the sum of the energy components corresponding to diagrams BQand CQmay be written as E4@Q

+ CQ) = -&

(%ijab&kZcd&cd~,%abkZ)/(~tjabg~jcd~kZab)

(92)

It should be noted that this simplification does not occur when the shifted, or

Theoretical Chemistry

26

B

CT

DT

AT

BT

ET

FT

GT

HT

I T

JT

KT

LT

MT

NT

OT

PT

----

Figure 7 Fourth-order diagrams which involve trbly-excited intermediate states

DQ

EQ

FQ

GQ

Figure 8 Fourth-order diagrams which involve quadruply-excited intermediate states

Many-body Perturbation Theory of Molecules

27

Figure 9 Connected and disconnected wave function diagram: (a) second-order connected triple-excitation diagram, (b) second-order disconnected quadruple-excitation diagram, (c) third-order disconnected triple-excitation diagram, (d) third-order connected quadruple-excitationdiagram

Epstein-Nesbet, perturbation scheme is used. All of the quadruple-excitation terms may be written in the form XfpQ

where

(93)

28

Theoretical Chemistry

and

M is a constant. p and v denote compound indices representing a maximum of four actual indices. The intermediates f, and g, differ only in that in the latter there is an additional denominator factor. A number of calculations of the quadrupleexcitation component of the fourth-order energy have been reported.131-140 The triple-excitation fourth-order energy, in contrast to the quadrupleexcitation component, arises from connected wave function diagrams. The algorithm required to evaluate this energy component is considerably less tractable than that for the quadruple-excitation energy, depending on n7,where n is the number of basis functions. The triple-excitation diagrams can be written in terms of the intermediates.

and gtjk;abc

=

cI &jlbcr$tkl gjlab

c

-__

(97)

Diagrams AT, BT,CT,and DTcorrespond to the expressions E~(AT)=

-*

~:jk;abcfijk;aeb/~t/kabc

(98)

E4(B T) =

-3 c f i j k ; a b c f f k j ; a b c / ~ ~ j k a b c

E~(CT) =

-3 c g i j k ; a b c g f 5 k ; a c b / ~ f / k a b c

(100)

E ~ @ T= )

-3

(101)

2 gijk;abcgtkj;abc/gtjkabc

(99)

The computation of these energy components has been rendered reasonably tractable by (i) using a spin-free formalism; (ii) recognizing certain permutational symmetry properties of the intermediatesfijk;abcand g i j k ; a b a ; 131

S. Wilson and D. M. Silver, in ‘Proceedings of the Fourth Seminar on Computational Problems in Quantum Chemistry’, ed. B. Roos and G. H. F. Diercksen, Max-PlanckInstitute fur Physik und Astrophysik, Munich, 1978.

S. Wilson and D. M.Silver, Mol. Phys., 1978, 36, 1539. S . Wilson and D. M. Silver, Comput. Phys. Comm.,1979, 17,47. 134 R. Krishnan and J. A. Pople, Int. J. Quantum Chem., 1978, 14, 91; R. Krishnan, M. J. Frisch, and J. A. Pople, J. Chem. Phys., 1980, 72, 4244; M. J. Frisch, R. Krishnan, and J. A. Pople, Chem. Phys. Lett., 1980, 75, 66. 135 L.T.Redmon, G. D. Purvis, and R. J. Bartlett, J. Chem. Phys., 1978,69,5386. lS6 R. J. Bartlett, I. Shavitt, and G. D. Purvis, J. Chem. Phys., 1979, 71, 281. 187 I. HubaE, M. Urban, and V. Kello, Chem. Phys. Lett., 1979,62,584; M. Urban, 1. HubaE, V. Kello, and J. Noga, J. Chem. Phys., 1980, 72,3378. l a *S. Wilson and V. R. Saunders, J. Phys. B: Atom. Mol. Phys., 1979, 12, L403; corrigenda, 1980, 13, 1505; M. F. Guest, and S . Wilson, Chem. Phys. Lett., 1980, 72, 49; S. Wilson and M. F. Guest, ibid., 1980, 73, 607. 1 3 9 V . KvasniEka, V. Laurinc, and S . BiskupiE, Chem. Phys. Lett., 1979, 67, 81. l40 D. M. Silver, S. Wilson, and C. F. Bunge, Phys. Rev., 1979, A19, 1375. la2 138

Many-body Perturbation Theory of Molecules

29

(iii) using a vector-processingcomputer. Again a number of calculations of the triple-excitation component of the correlation energy have been reported.13*s139~141 Disconnected triple-excitation wave function diagrams and connected quadruple-excitation diagrams first contribute to the energy in fifth-order. A full set of lifth-order diagrams, using the notation of Paldus and Wong148s143 for the sake of brevity, is given in Figure l0.O Calculations through fifth-order would enable [3/2] Pad6 approximants to the energy series to be constructed and variational upper bounds calculated from second-order wave functions. Full fifth-order calculations do not appear to be possible at present. However, the

Figure 10 Fqth-order diagrams, using the mod$ed Hugenholtz notation of Paldus and Wong 141 143

S. Wilson and V. R. Saunders, Comput. Phys. Comm., 1980, 19, 293. J. Paldus and H. C. Wong, Comput. Phys. Comm., 1973,6, 1. H. C. Wong and J. Paldus, Comput. Phys. Comm.,1973, 6,9.

2*

Theoretical Chemistry

30

recent work by KvasniEka et aI.*44on Wigner's (2n+ 1) rule in many-body perturbation theory will certainly simplify fifth-order studies. Quasi-degeneracy Effects.-The convergence properties of the non-degenerate formulation of the many-body perturbation theory deteriorate when quasidegeneracy is present in the reference spectrum. In view of its simplicity, however, there is considerable interest in exploring the range of applicability of the nondegenerate formalism. The low-lying 2p state in the Be atom ground state makes this a severe test of the non-degenerate perturbation series. It is an appropriate choice of system for investigating the limitations of a non-degenerate formulation of perturbation theory to systems involving near degeneracy. Full configuration mixing calculations and third-order many-body perturbation theory studies of the ground state of the beryllium atom have been performed140within the same basis sets. For a basis set containing only functions of s symmetry, there are no near degeneracy effects and the degree of agreement between the configuration mixing and perturbative studies is good. For a basis set of functions having s, p, and d symmetry, the perturbation series based on the Hartree-Fock model zero-order hamiltonian gave only 93 % of the correlation correction given by the configuration mixing calculation after forming the [2/1] Pad6 approximant to the energy series. On the other hand, by forming the [2/1] PadC approximant to the shifted perturbation series, 99.5 % of the configuration mixing correlation energy was recovered. Third-order non-degenerate many-body perturbation theory studies of the ground state of the CH+ ion have been ~ e p 0 r t e d . lCalculations ~~ were made using the same basis set of Slater functions employed by Green et a/.146in their configuration mixing calculations, which included all single- and double-excitations from a two-configuration reference function. For this system quasi-degeneracy effects become increasingly important as the nuclear separation is increased. It was again demonstrated that [2/1] Pad6 approximants to the shifted perturbation series are useful when near degeneracy is present. One interpretation of the significance of denominator shifts, which are used in the shifted perturbation series, is that they provide a method for summing certain higher-order diagonal scattering diagrams in the model perturbation scheme to infinite In this sense, the shifted third-order results include all third-order contributions in the model scheme plus a selection of additional diagrams summed to infinite order. The quasi-degeneracy problem can be overcome in the Be atom and the CH+ ion for internuclear distances < 4 bohr by including certain higher-order diagrammatic terms. Alternatively the shifted perturbation scheme can be interpreted as an expansion based on an alternative zero-order hamiltonian 14' whose spectrum does not contain near degeneracies. We prefer this latter interpretation since it represents a more balanced treatment

14(

V. KvasniEka, V. Laurinc, and S. BiskupiE Mol. Phjs., 1980, 39, 143, Wilson, J . Phys. B: Atom. Mol. Phys., 1979, 12, 1623. S. Green, P. S. Bagus, B. Liu, A. D. McLean, and M. Yoshmine, Phys. Rev. A , 1972, 5,

IQ5 S. 146

1614.

P. Claverie, S. Diner, and J. Malrieu, Inl. J . Chem., 1967, 1, 751.

lP7

Many-body Perturbation Theory of Molecules

31

in that all terms are included through third-order and no partial evaluation of higher-order terms is attempted. In order to eliminate the arbitrary use of the model and shifted perturbation series, depending on the degree of degeneracy in the corresponding reference spectra, the following zero-order hamiltonian has been considered:148 $o = (1 - p)$podel + p2,shifted (102) where p is an arbitrary scalar. Prototype calculations have been performed for the Be atom in which quasi-degeneracy effects arise, and for the N e atom in which no near degeneracy is present. For both Be and Ne, E[3/0]and E[2/1]were found to exhibit minima with respect to ,~i (although it should be noted that E[3/0]and E[2/1]are not bounded). For the Be atom the [2/1] Pad6 approximant to the energy series takes its minimum value at p 0.9 while for Ne E[2/1]is a minimum at ,u-0.2. For Be this minimum occurs closer to p = l , that is the shifted, or Epstein-Nesbet series, than ,u=0, the model or Msller-Plesset series. The reverse situation applies in the case of the Ne atom. The E[2/1]energy values for both Be and Ne are less sensitive to the value of p than the usual power series, E[3/0]. Clearly, given a set of zero-order functions which are eigenfunctions of some zero-order operator, So, it is possible to develop a perturbation with respect to any operator, A, provided that [9 , , A ] =0. The infinite-order perturbation expansion should of course be independent of the choice of reference hamiltonian and thus the degree of agreement between two perturbation series when truncated at some finite order gives a qualitative measure of convergence. Strictly, if two perturbation expansions agree exactly at some order this does not mean that they have converged, but only that they are converging at the same rate, The use of a combination of two operators as described above allows a set of zeroorder operators and the corresponding perturbation series to be investigated. Obviously, when severe near degeneracy is present in the reference spectrum, a full quasi-degenerate formulation of the perturbation series should be employed. However, there is considerable value in applying the much simpler non-degenerate formalism to as wide a range of problems as possible. Comparison with Other Methods.-Most methods for performing accurate calculations of the electronic properties of molecules involve some finite or infinite expansion for the wave function and the corresponding expectation values. This is the case in, for example, the method of configuration mixing, themany-body perturbation theory, cluster expansions, coupled-pair approximation, etc. Each method leads to the exact wave function if all terms in the expansion are evaluated. In practice, the expansions have to be truncated and methods differ only in the manner in which this truncation is effected. However, this truncation can significantly affect the theoretical properties, and to some extent the computational properties, of a method. The mere fact that one method includes more terms than another does not mean that it is superior. Which terms one leaves out is just as important as the terms one actually evaluates. Thus in the method of configuration mixing limited to single- and double-excitation with respect to a N

148

S. Wilson, Chem. Phys, Lett., 1979, 66,255.

32

Theoretical Chemistry

single determinantal reference function one includes many terms, corresponding to unlinked diagrams in the perturbation analysis, which cancel with terms involving higher-order excitations. Perturbation theory provides a clearly defined order parameter in the expansion for expectation values giving a ‘least-biased’ indication of the importance of various terms. To quote Brandow:lo7 ‘The structure of each formalism tends to suggest that certain types of approximations are most reasonable or “natural”, regardless of the actual quantitative characteristics of the physical system, The many-body literature is full of papers where people have tried to make the physics fit into their preconceived approximation schemes, instead of vice versa. perturbation theory . . seems to be the “least biased” of all the many-body techniques.’

...

.

The relationship between the coupled-electron pair approximation (c.e.p.a.) and the many-body perturbation theory has been discussed in detail by Ahlrichs.lq9All of the methods denoted by c.e.p.a. ( x ) (x=O, 1, 2, 3) may be related to the summation of certain classes of diagrams in the many-body perturbation theory to infinite order. For example, c.e.p.a. (0), which is Ciiek’s linear approximation or Hurley’s c.p.a. (0) ansatz150 is equivalent to the summation of all doubleexcitation linked diagrams in the perturbation series. This is also denoted d.e.m.b.p.t. (double excitation many-body perturbation theory) by some worker^.^^^^ 16* In the coupled pair approximation,160all diagrams which can be formed by considering products of disconnected double-excitations are summed to infinite order. Thus in fourth-order, the coupled pair approximation includes the linked double-excitation and linked quadruple-excitation diagrams shown in Figure 6 and Figure 8, respectively. The coupled pair approximation may be represented diagrammatically as follows:

The order of perturbation at which various levels of excitation first arise is illustrated in Figure 11 for three different reference functions. In Figure 1l(a), the Hartree-Fock orbitals are used to form the reference function, in Figure ll(b) the bare-nucleus model is used in zero-order, while in Figure ll(c) Brueckner orbitals are used to construct the reference function. The method of single- and double-excitation configuration mixing is not size-consistent. The method remains size-inconsistent even if a multi-configurational reference function is used. For a configuration mixing calculation, with

R. Ahlrichs, Comput. Phys. Comm., 1979, 17,31. A. C. Hurley, ‘Electron Correlation in Small Molecules’, Academic Press, 1976. 151 R. J. Bartlett and I. Shavitt, Znt. J . Quantum Chem. Symp., 1977, 11, 165. 152 M. R. A. Blomberg and P. E. M. Siegbahn, Znt. J . Quantum Chem., 1977,14,583.

149

150

Many-body Perturbation Theory of Molecules

L-1

33

34

Theoretical Chemistry

respect to a single determinant, a correction to fourth-order can be made by means of Davidson’s formula 2 v ls3*lS4 (1 - c:)ED (104) where ED is the double-excitation configuration mixing energy and Co is the coefficient of the reference function. The formula2,131~155~156

is correct to fifth-order. All of the methods discussed above, and some others which we have not discussed, represent a partial evaluation of fourth-order and higher-order terms in the many-body perturbation series. This must be done with great care since often there is a large degree of cancellation between terms in a given order. For example, it has been demonstrated that the two-body and many-body components of the energy are of opposite sign and for a system such as the hydrogen fluoride molecule the latter can have a magnitude which is z 25 % of the two-body In a fourth-order calculation on water, it was found that there was a significant degree of cancellation between the fourth-order double and quadruple- excitation components.156 5 Computational Aspects General Remarks.-In this section, the computational aspects of the application of the many-body perturbation theory to molecular correlation energies are discussed. Any molecular calculation starts with the evaluation of integrals over the basis functions. This is usually, but not necessarily, followed by a self-consistent field calculation and a transformation from integrals over atomic basis functions to integrals over molecular orbitals. Full details of these particular phases of calculation are well documented elsewhere and we do not consider them further here, Third-order calculations are considered in the next section. This is followed by a brief discussion of the computation of higher-order terms and of the evaluation of ‘bubble’ diagrams which are required when molecular properties are calculated or when a reference function other than the closed-shell Hartree-Fock function is employed. The impact of the new generation of computers, which have vector processing capabilities, on many-body perturbative calculations is discussed very briefly in the final section. Third-order Many-body Perturbative Calculations.-Starting from a list of oneelectron and two-electron integrals over molecular orbitals, the evaluation of the second-order and third-order energies12-14consists of two stages. In the first stage the integrals are sorted into various types12while in the second stage these integral lists are used to compute the required energy component^.'^,^^ R. Davidson, in ‘The World of Quantum Chemistry’ Proceedings of the First Intetnational Congress on Quantum Chemistry, ed. R. Daudel and B. Pullman, D. Reidel, 1974. 154 S. R. Langhoff and E. R. Davidson, Int. J . Quanfurn Chern., 1974, 8, 61. 155 P. E. M.Siegbahn, Chern. Phys. Lett., 1978, 55, 386. S. Wilson and D. M.Silver, Theor. Chim.Acra, 1979, 54, 83. 157 S. Wilson and D. M.Silver, J. Chem. Phys., 1977, 66, 5400.

15s E.

Many-body Perturbation Theory of Molecules

35

In the integral sorting phase of the calculation, the following lists of integrals are created:

u=- 1 1-12

where I,J,K, . . . are used to denote occupied orbitals and A,B,C, . . . are used to denote virtual orbitals. Integral types (3) and (4) are not required when a HartreeFock reference function is used in calculations through third-order. It is also convenient to produce a list of integrals of the form ( P e l ul PQ) and ( P e l vl QP> which are required for the denominator shift factors and, together with the oneelectron integrals, to evaluate the matrix Hartree-Fock energy. To illustrate the calculation of the energies corresponding to the four Brandow diagrams which arise through third-order we shall consider the third-order ‘hole particle’ diagram in some detail.14 The various spin types which can arise are shown in Figure 12. The first step is to process the integrals of the type (IJI vl AB). For an atom or molecule which is described by N doubly-occupied orbitals, N secondary lists are created. The Ith of these secondary lists contains the integrals K. Next the integrals (IJI ul A B ) and (JKIuIBC) are read into core from the Ith and Kth secondary lists, respectively. All integrals depending on a given I,J, and K are in core simultaneously.The integrals of the form (IJI vl AB) and <JKI ul BC) are read for all values of J in turn until the Ith and Kth secondary lists have been read through completely. A new set of integrals (IClvIAK) is then processed. For a given set of integrals depending on I, J, and K the following spin-free intermediates are formed :

The two remaining particle states can then be summed over to give g:yK = C

AC

F+j)KAC

((IC10 I A K > - ( I C 10 IKA})

7

8

9

10

I1

12

13

14

Figure 12 Spin types which arise for the third-order ring diagram gf)K

=

&

(Fij)KAC((lCIU

IAK)-
g$yK = g lZ 4 J) K

=

AC

la)) + F f J s $ A C < I c I o ( K A ) )

(113)

FI(J2kAC

(Ic10 IA K )

(1 14)

Fij)KAC


(1 15)

10

IAK>

Finally, the third-order 'hole-particle' energy is given by E301P) = -

IFK (g\':K

+ g$yK + g$$H + g i 2 f K )

(116)

Similar algorithms have been devised to evaluate the other third-order and the second-order energy components. Higherlorder Terms.-Diagrammatic perturbation theory provides a tractable scheme for calculating the dominant components of the correlation energy which

Many-body Perturbation Theory of Molecules

37

may be associated with triple-excitations and quadruple-excitations. As an example we consider the evaluation of the quadruple-excitation fourth-order linked diagram energy component. When the matrix Hartree-Fock operator is used as a reference hamiltonian, that is the scheme of Msller and Plesset, the evaluation of the quadrupleexcitation energy component is particularly easy. Only one type of integral is required, namely the (IJI vl A B ) integrals, For a system which is described by N doubly occupied orbitals, N secondary lists are created in the manner described in the previous part of this section for the third-order hole-particle energy. The computation of the correlation energy corresponding to the diagrams shown in Figure 8 requires that certain blocks of integrals be in computer core storage simultaneously. This is achieved by two file handling schemes; one for diagram AQ and the sum of diagrams EQand GQ,and one for the sum of diagrams BQ and CQ,and DQand EQ.In our method, diagram AQand the sum of EQand GQare evaluated by employing the same file handling scheme. Blocks ( I J ) and ( J K )are placed in core storage simultaneously. This is done by reading simultaneously through the Ith and Kth secondary lists for all values of J. The Ith and Kth secondary lists are read through completely for each value of I and K. When computing the energy component corresponding to the sum of diagrams BQand CQ, we require blocks (IJ) and (KL) in core storage at the same time. This is achieved by reading the block ( I J ) from the primary list of integrals, rewinding and reading all (KL) less than (IJ). The next value of (IJ) can then be treated, the file rewound and the whole process repeated. Spin orthogonalities considerably simplify the computations. For example, in the evaluation of diagram AQ six spin-free intermediates can be defined. Three of these take the form

while the remaining three merely include an extra denominator factor [see equation (95)]. The energy corresponding to diagram AQis then given by

We note that iterative schemes have been devisedlsl to evaluate higher-order diagrams involving double-excitations,These schemes can easily be generalized to handle triple-excitations and quadruple-excitations. Bubble Diagrams.-The bubble diagrams which are shown in Figures 13 and 14 are required when the reference function for a closed-shell system is not defined by the matrix Hartree-Fock model, when a restricted Hartree-Fock reference

38

Theoretical Chemistry

A

B

Figure 13 Second-order diagrams which arise when electron-electron interactions are completely neglected

A

Q-lo -3 D

B

(J:g --0. E

()--0.n, --

G

H

J

K

M

C

I

L

N

Figure 14 Third-order diagrams which arise when electron-electron interactions are completely neglected

Maity-body Perturbation Theory of Molecules

39

function is used for open-shell systems, or when molecular properties are to be evaluated. General programs have been written which allow any arbitrary oneelectron operator matrix elements to be used in the evaluation of the bubble diagrams.lss The expressions corresponding to each of the diagrams in Figure 13 are as follows

where A is the ‘bubble’ matrix. The expressions corresponding to the diagrams shown in Figure 14 are as follows E3(A) = 6 E3@)

=

- GkC

ab:d

r,

abc

(&i~ub&cdab&cdU)/(gijabgijcd)

(123)

(&cjab&kaic&bcjk)/(aijab~jkbc)

(124)

The energy associated with diagram G is the complex conjugate of that associated with diagram J. Diagrams H and K, I and L are similarly related. Vector Processing Computers.-We conclude this section by considering briefly the impact which vector-processingcomputers are likely to have on many-body perturbative calculations in the near future. As noted in the previous subsection, use of the CRAY 1 has already enabled the triple-excitation component of the correlation energy to be evaluated for a molecule using a fairly large basis set.’= The algorithms devised to perform many-body perturbation calculations are The scalar well suited to vector processing computers such as the CRAY 1?41*15e S. Wilson and D. M. Silver, unpublished work. R. W. Hockney, Contemp. Phys. 1979, 20, 149; M. F. Guest and S. Wilson, ‘Proceedings of the American Chemical Society Symposium on Supercomputers in Chemistry’, Las Vegas, 1980.

158

159

40

Theoretical Chemistry

operations on this machine are about twice as fast as those of the CDC 7600 or the IBM 370/195 whereas the measured time on the CRAY 1 for matrix multiplication, using the vector order repertoire is twenty times faster than the best hand-coded routines on the CDC 7600 or IBM 370/195. The innermost loop in the program written to evaluate the triple-excitation energy component described in the previous subsection has the form

DO 1 I D = l , N DD(1D)= Dl(ID)*D2(ID) DE(1D)= Dl(ID)*E2(ID) ED(1D)= El(ID)*D2(ID) EE(1D)= El(ID)*E2(ID) 1 CONTINUE where D1 and D2 contain direct integrals and El and E2 exchange integrals. We performed a timing test for this loop and the results are shown in Table 2.

Table 2 CPU times on the CRAY 1 and the IBM 360/195 computer required to execute the inner loop in the evaluation of the triple-excitation energy component Timeb IBM 360/195d 1c 41.8 16 82.9 32 124.4 48 165.6 64 a DO-loop range defined in text. b CPU time in microseconds. Using vector order repertoire. d FORTRAN H extended plus compiler with OPT=2. Na

CRAY 3.6 5.2 6.8 8.4

The efficiency of algorithms based on diagrammatic perturbation theory together with the capabilities of the new generation of vector-processing computers will allow increasingly large basis sets to be employed over the next few years, thus reducing one of the major sources of error in present-day molecular calculations, the basis set truncation 6 Some Applications General Remarks.-In this section we attempt to give a brief overview of the accuracy of electron correlation calculations which are possible at present using many-body perturbation theory. In the first part we describe some new calculations on the lithium dimer and some previously reported work O2 on the nitrogen molecule. In the second part we review the application of many-body perturbation theory to potential energy curves. In part three, the importance of tripleexcitations and quadruple-excitationsis discussed in the light of recent calculations. The calculation of molecular properties is discussed in the final part of this section.

Many-body Perturbation Theory of Molecules

41

Application to Li,, N,.-The results of calculations through third-order are shown in Table 3 and Table 4 for the lithium dimer and the nitrogen molecule, respectively. These calculations were performed by using the universal basis set containing functions with s, p , and d symmetry described in Section 3. For the lithium dimer the calculations were performed at an internuclear distance of 5.0507 bohr, the experimental equilibrium value, whereas for the nitrogen molecule the calculations were performed for a nuclear separation of 2.0 bohr. The experimental equilibrium nuclear separation in the nitrogen molecule is 2.068 bohr. The second-order and third-order energy components are given for both the model perturbation series and the perturbation series using shifted denominators. In addition to the simple perturbation results, [2/1] Pad6 approximants and variational upper bounds are presented. The model perturbation series is observed to converge more rapidly than the shifted expansion, if we use the ratio of the third-order energy to the secondorder energy as a measure of the rate of convergence. Note that for the total energies shown in Tables 3 and 4, the Pad6 approximants resulting from the two perturbation series are in closer agreement than the simple third-order energies. The difference between E[3/0] and E[2/1] is smaller for the model scheme than the shifted expansion: The E[2/1] values derived from the model scheme are thus of most interest although the optimized upper bound is also useful. The empirical correlation energy of the lithium dimer is estimated to be -0.126 hartree. For the model perturbation series the formation of [2/1] Pad6 approximants leads to Table 3 Components of the calculated correlation energy for the lithium dimer obtained using third-order perturbation theory and an s,p,d universal basis set a Escf

E[2/01 ~[3/01 EWI Evar (A= 1) Evar( Aopt) AOPt 0

Model series

Shifted series

-14.871 23 - 14.969 20 -14.982 19 - 14.984 17 - 14.978 31 - 14.979 20 1.102 08

-14.871 23 -14.993 76 - 14.983 74 - 14.984 49 -14.968 99 -14.97276 0.828 63

In hartrees.

Table 4 Components of the calculated correlation energy for the nitrogen molecule obtained using third-order perturbation theory and an s,p,d universal basis set@ Escf

E[2/01 ~[3/01 m/01 Eva*(A= 1) Evar(1opt) AOPt

In hartrees.

Model series

Shifted series

-108.992 10 -109.43908 -109.43676 - 109.436 77 - 109.392 51 -109.396 54 0.904 8

- 108.992 10

- 109.523 85 - 109.401 38 - 109.424 31

- 109.335 28 - 109.380 01 0.729 5

Theoretical Chemistry

42

a calculated correlation energy of -0.1 129 hartree which corresponds to 89.6 % of the empirical estimate. Formation of an upper bound leads to a correlation energy of -0.1080 hartree which is 85.7% of the estimated total correlation energy. For the nitrogen molecule calculations were performed for nuclear separations of 1.75, 2.00, 2.25, and 2.50 bohr and these values interpolated to obtain a correlation energy at the equilibrium nuclear geometry. The empirical Correlation energy for the nitrogen molecule is -0.538 hartree. The calculated correlation energy obtained by interpolation is - 0.4501 hartree which represents 83.7 % of the estimate. Both of the calculations described above recover a greater percentage of the correlation energy than previous studies. There are two deficiencies in the above calculations. The first, and probably the major effect, is that resulting from truncation of the basis set used in invoking the algebraic approximation. The concept of a universal even-tempered basis set should go a long way towards developing the large basis sets which are ultimately going to be required in order to achieve chemical accuracy (one millihartree). The basis sets used in the calculations described above do not include functions with 4 symmetry which will be required for an accurate description of angular correlation effects. The second deficiency results from the truncation of the perturbation series at third-order. Results of calculations for the water molecule will be discussed more fully in the fourth part of this section. We feel that it is always important to evaluate all terms, through whatever order the perturbation series is taken. Potential Energy Curves.-Potential energy curves have been calculated by manybody perturbation theory for a number of diatomic molecules, including FH,160 C0,929161 CSY 162 N B2*163 BF 92 F2, lo3Be2,164Mg2,165We take the results for the FH molecule as representative. The potential energy curves for FH are shown in Figure 15. Curves (d) and (e) in this Figure were determined by third-order many-body perturbation calculations. Full details of the calculations can be found elsewhere.1s0 Curve (a) is the potential energy curve obtained by the self-consistent-field approximation. Curves (b) and (c) were determined by Dunningls6 using the generalized valence bond method and by Meyer and Rosmusls7 using the coupled electron pair method, respectively. Curve (d) corresponds to the upper bound determined from third-order perturbation calculations using the model perturbation series and curve (e) to the [2/1] Pad6 approximant constructed from this series. Morse curves derived from experimental data are labelled (f) and (g). Curve (f) includes a correction for relativistic effects and represents an estimate of the nonrelativistic potential energy curve. 29

9

S. Wilson, Mol. Phys., 1978,35, 1. Int. J. Quantum Chem., 1977,12,604. 1 6 2 S. Wilson, J. Chem. Phys., 1977,67,4491. 183 M.Urban and V. Kello, Mol. Phys., 1979, 38, 1621. 1 6 4 R. J. Bartlett and G. D. Purvis, Znr. J . Quantum Chem. Symp., 1978, 14, 561; M.R. A. Blomberg and P. E. M. Siegbahn, ibid., p. 583; M. A. Robb and S. Wilson, Mol. Phys., 1980,40,1333. 165 R. J. Bartlett and G. D. Purvis,J. Chem. Phys., 1978,68, 21 14. 1 6 6 T.H.Dunning Jr., J. Chem. Phys., 1976,65, 3854. 1 6 7 W.Meyer and P. Rosmus,J. Chem. Phys., 1975,63,2356.

180

161

S. Wilson,

Many-body Perturbation Theory of Molecules

43

-99.8

- 99.9 -I 00.0

-100.1

?

2 -100.2

U

W

-100.3

-100.4

V

-100.5

-100.6

0

I.o

2.0

3.0

4.0

5.0

6.0

R ( a. u.1 Figure 15 Potential energy curve for the hydrogenfluoride molecule

The equilibrium nuclear distance determined from the self-consistent-field curve is 0.897 A which differs by 2.2% from the experimentally determined value of 0.917 A. Curve (e) yields an equilibrium distance of 0.908 A, which differs by 1.0 % from the experimentally determined value. The fundamental frequencies of vibration derived from curves (a), (e), and (g) are 4471, 4261, and 41 37 cm-l, respectively. Triple-excitations and Quadruple-excitations.-Only double excitations contribute to the correlation energy and other properties through third-order ; triple-, quadruple and higher-order excitations are usually neglected. Triple- and quadruple-excitations first arise in the fourth-order of the perturbation series. To illustrate the importance of triple and quadruple excitations we present in 13* 166 This Table 5 a full fourth-order calculation for the water m01ecule.~~~~ calculation corresponds to the equilibrium nuclear geometry and the basis set of

44

Theoretical Chemistry

Table 5 Contribution of tr@le-excitations and quadruple-excitations to the correlation energy of the water molecule at its equilibrium nuclear geometry

Diagram

A B C D E F G H I&)

Triplea excitation energy

Quadruple excitation

-2.29

-7.14

- 1.33 -1.35 -2.39 -9.42 -7.11

energy

-1.02 +3.51

- 5.94

+7.86

Total linked diagram energy Unlinked diagram energy

-7.77 +2.36 +9.73 + 1.53 +1.15 -7.86 -

3-3.21 -17.11

12=[2/1I b

-0.285 1

JW) M(N) OW

E 4 . 5

E~D In millihartrees.

b

-0.0020 -0.004 3

In hartrees.

39 Slater functions first presented by Rosenberg and Shavitt.lss The quadrupleexcitation component can be divided into a part which corresponds to unlinked diagrams and a part which corresponds to linked diagrams, The unlinked diagram is only of interest since it represents the largest contribution to the size-consistency error in the widely used method of configuration interaction limited to single and double excitations with respect to a single-determinantal reference function. It can be seen that the linked triple excitation energy is in fact quite large and certainly chemically significant. There is clearly a danger in any technique which attempts to make a partial evaluation of higher order terms in the perturbation sries.1s1,156 showed that the many-body perturbation theory Molecular Properties.-Kelly provides a powerful method for the calculation of atomic polarizabilities. Typical diagrams which arise in such calculations are shown in Figure 16. Itagaki and Saika126slS7 have extended Kelly's work to molecules. The present authorlZ0has demonstrated how the use of a scaling parameter can lead to improved resuIts in such calculations. Itagaki and Saika have also very recently reported applications to nuclear spin-spin coupling constants.16B Many-body perturbation theory can also form the basis of a useful technique for the calculation of photoionization effects in atom and molecules. Kelly has recently reviewed this area of research.l7O 168

B. J. Rosenberg and 1. Shavitt, J. Chem. Phys., 1975, 63, 2163. itagaki and A. Saika, J. Chem. Phys., 1979,70,2378, H. P. Kelly, Comput. Phys. Comrn., 1979, 17, 99.

16s T. 1'0

Many-body Perturbatation Theory of Molecules

Figure 16 Diagrams which arise in the calculation of polarizabilities. The heavy dot represents a dipole moment matrix element

7 Concluding Remarks General Remarks.-In this final section, some other aspects of the many-body perturbation theory of molecules are briefly reviewed. The final comments address the general properties which we believe methods for treating electron correlation in molecules must possess. Some Other Aspects.-In 1972, Coulson (quoted in ref. 6) suggested that diagrammatic techniques would become increasingly important in theoretical chemistry. The impact of diagram methods is perhaps only just being fully realized. In this section very recent work and possible directions for future research are discussed. In particular an extension of the work described in the second part of section 4 on invariants is described, the possibility of employing group-theoretical developments in many-body perturbation theory is indicated, recent work on electron-molecule scattering is mentioned, and the calculation of energy derivatives is briefly discussed. Even-order Invariants for Rayleigh-Schrodinger Perturbation Theory. Full fourthorder correlation energy calculations are now possible and it is, therefore, important to have fourth-order forms which share the invariance properties of the [2/1] Pad6 approximants in third-order. The expression

where E[2/1] = E2/(l-E3/E2)

has recently, been shown to be useful in this 171

S.Wilson, Int. J . Quantum Chem., in press.

46

Theoretical Chemistry

Graphical Methods of Spin Algebras. In most problems of chemical interest, the molecular hamiltonian can be taken to be spin-free and can be ~ r i t t e n l ~ * , l ~ ~ in terms of the generators of the unitary group, U(n), where n is the number of basis functions

2=

c + 3abcd < a b I u I c d ) ( b a c ~ b d - 5 b c ~ a d )

ab

(136)

in which 8ab

=

abu U

(1 37)

and a+ and a are the usual creation and annihilation operators. A spin-adapted perturbation expansion can be obtained by defining

and

The matrix elements of the generators, bab, can be evaluated using the graphical methods of spin a l g e b ~ a . l ~The ~ - l particle-hole ~~ formalism introduced by Flores and M ~ s h i n s k yand l ~ ~recently discussed by Paldus and Boyle17*is most suitable for such developments. Electron-Molecule Scattering. Kaldor and c o - ~ o r k e r shave ~ ~ ~recently shown that diagrammatic perturbation theory can be used to expand the optical potential in electron-molecule scattering calculations. The second-order diagrams, including those resulting from electron exchange are shown in Figure 17. Pade‘ Approximants to Energy Derivatives. The evaluation of derivatives of the energy, with respect to the nuclear geometry, forms the basis of an efficient approach to the calculation of potential energy curves and surfaces. It is easily shown that

where

J. Paldus, Theor. Chem., Advances and Perspectiv es, 1976, 2, 131. I. Shavitt, Int. J. Quantum Chem., Symp., 1978, 12, 5 ; and in ‘Proceedings of Daresbury Study Weekend’, November 1979, ed. M. F. Guest and S. Wilson, Science Research Council, 1980. 1 7 4 A. P. Yutsis, I. Levinson, and V. Vanagas, ‘Mathematical Approaches of the Theory of Angular Momentum’ Israel Program for Scientific Translation, 1962. 175 E. El Baz and B. Castel, ‘Graphical Methods of Spin Algebras’, Dekker, New York, 1972. 176 S. Wilson, in ‘Proceedings of Daresbury Study Weekend’, November 1979, ed. M. F. Guest and S. Wilson, Science Research Council, 1980. J. Flores and M. Moshinsky, Nircl. Phys., 1967, A93, 81. l7* J. Paldus ar.d M. Boyle, Physica Scripta, 1980,21, 295; M.Boyle, Dissertation, University of Waterloo, 1979; J. Paldus, in ‘Proceedings of Daresbury Study Weekend’, November 1979, ed. M. F. Guest and S. Wilson, Science Research Council, 1980. 17@ A. Klonover and U. Kaldor, J. Phys. B: Atom. Mol. Phys., 1978, 11, 1623; ibid., 1979,12,

172 173

323.

Many-body Perturbation Tfieory of Molecules

47

li

Figure 17 Second-order diagrams for electron molecule scattering

is the kth order energy corresponding to the zero-order hamiltonian 2oand & is the kth order energy corresponding to ~ 9 It therefore ~ .follows that the Ek

[N+1/NJ Pad6 approximants to the energy derivatives share the invariance properties of the [ N + 1/N] Pad6 approximants to the energy itself. Final Comments.-Many-body perturbation theory has several theoretical properties that make its use in molecular studies attractive. If a system is composed of non-interacting subsystems, then this fact should be evident at any level of the theory. It is possible to formulate any quantum mechanical problem in such a way that, if the whole system is separated into subsytems, all quantities appearing in this formalism are additive. There is a close connection between this question of additivity of properties and linked diagram expansions and Lie algebraic formulations of quantum mechanics. Many-body perturbation theory provides a formulation of the correlation problem which satisfies these requirements. The energy of any system is always written as a sum of the energies of its component parts. The energy is written in terms of single-particle state functions rather than N-electron functions. These theoretical properties lead to tractable computational schemes for treating correlation effects with little more effort than a conventional self-consistent field calculation.lsO Coulson181thought that it would take 15 years for the impact of diagrammatic techniques to be fully realized in theoretical chemistry. It is therefore not surprising that the first half of this period has been devoted mainly to the development of new methods and algorithms. Although this development will undoubtedly continue, it is clear that, armed with these new techniques, theoretical chemists will be able to attack problems with an accuracy which was not previously attainable. For example, it will be possible to calculate rotation constants for small molecules more precisely. Although the accuracy of calculated rotation M. F. Guest and S. Wilson, ‘Proceedings of the American Chemical Society Symposium on Supercomputers in Chemistry’, Las Vegas, 1980. 181 C. A. Coulson, 1973, Progress Report, Department of Theoretical Chemistry, Oxford, 180

1972-73.

48

Theoretical Chemistry

constants will still not match that of microwave experiments, calculations are of great value when experiments are difficult or impossible. In the past few years, theoretical studies have been of great value in the detection of certain interstellar radicals and ionsIs2that are very difficult to study in terrestrial experiments.

181

S. Wilson, Chem. Rev., 1980, 80, 263.

2 The Electronic Structure of Polymers BY J. LADIK AND S. SUHAI

1 Introduction During the past decade there has been very intense experimental and theoretical research on the physics and chemistry of polymeric materials. This work has led to the discovery of a great number of organic and inorganic crystals built from polymers or molecular stacks which have a quasi-one-dimensional electronic system and exhibit fascinating electric, magnetic, etc. properties. Other polymers have proved to play a fundamental role in the biosciences. Owing to recent developments in theoretical and computational methods, the quantum mechanical approach to the polymer electronic structure problem has begun to associate very fruitfully with experimental research in this field. Combination of the methods of molecular quantum theory with the ideas of theoretical solid-state physics has provided a really efficient tool, not only for the interpretation of experimental results, but also for investigation of fine details in the electronic structure which would be only barely accessible in experiments. The present Report does not aim to review comprehensively this rapidly expanding field. Instead we shall give only a brief outline of the theoretical methods and restrict the applications to a few examples to provide the reader with an impression of the present capabilities of this approach. Emphasis will be laid on the a priuri or ab initiu realization of the various computational schemes proposed for polymers. We have to recall at this point that the great advantage of the ab initiu techniques in molecular as well as in solid-state physics is that (within the framework of a given atomic basis set) all one- and two-electronic integrals are calculated exactly without recourse to any empirical information apart from the structural data. The above definition has to be made more precise in the case of solids. Namely, as it will be later shown, to reach a meaningful precision for all physical properties in question (i) integrals whose absolute value is smaller than a given threshold (usually 10-6-10-7 hartree) can be neglected and (ii) certain core-electron and electron4ectron integrals describing interactions between oery far elementary cells can also be neglected pairwise (even if the members of the pairs are above the prescribed threshold value). The fulfilment of the requirements contained implicitly in the second point has proved to be important in actual polymer ab initio calculations. Calculation of the infinite lattice sums remains the most difficult step in a6 initiu polymer calculations. They can be evaluated in configuration space as well as in momentum space (or the two procedures can be also combined). There is not enough experience accumulated in the literature to decide which approach

49

50

Theoretical Chemistry

is more favourable from the computational point of view. Theoretically, the first one would be preferable for covalently bonded semiconducting polymers while the second one seems to be more appropriate for metallic ones. In the works reviewed in this article the configuration space approach has been used with appropriately truncated lattice sums. It has been found that different manyelectron and one-electron properties are affected to differing extents by the truncation procedure. The most sensitive are the total energies per elementary cell and the relative conformational energies between different isomeric structures which require calculation of all interactions within a radius of 70-80 A in the polymer. To predict energy differences between similar isomers or to calculate potential energy surfaces for the same isomer on the other hand, the interaction radius can be reduced to 18-20 A. Computations with such a radius usually also provide satisfactory charge distributions. If one is interested only in one-electron properties they can be obtained even less expensively; ionization potentials, electron affinities, bandwidths, and gaps usually converge after a few neighbours. It is stressed here, however, that the above conclusions hold only for the case of a correct (electrostatically balanced) truncation. If this requirement is not fulfilled, the calculations may completely lose their ab initio nature. The use of a one-particle picture (Hartree-Fock theory) is a reasonable starting point in polymer conformational studies though the Hartree-Fock absolute energies are, of course, in considerable error (correlation error). For conformational problems one is concerned, however, with energy differences and it is well established that the correlation error in the Hartree-Fock energy remains relatively constant in the neighbourhood of the extremal points of the potential energy surface thus allowing acceptable predictions for the geometry. Other quantities (like band gaps, for instance) may be much more sensitive to correlation effects; therefore, only predictions concerning the relative values of these quantities for different structures seem to be significant at Hartree-Fock level and further calculations with explicit allowance for electron correlation are needed to obtain them more accurately. In conclusion we can state that careful handling of lattice sums and efficient use of symmetry makes it now possible to perform a priori studies even for polymers with larger elementary cells. These computations are not ‘horrendous’ as claimed earlier and this approach is thus able economically to supplement experimental work on these systems. I n Section 2 we briefly summarize the basic mathematical expressions of the LCAO Hartree-Fock crystal orbital method both in its closed-shell and DODS (different orbitals for different spin) forms and describe the difficulties encountered in evaluating lattice sums in configuration space. Various possibilities for calculating optimally localised Wannier functions are also presented. They can be efficiently used in the calculation of excited states and correlation effects discussed in Section 3. Through a short description of the semi-empirical MNDO crystal orbital method we demonstrate in Section 4 how the computational difficulties of the a priori approach can be reduced. While the procedures described in the above Sections all refer to strictly periodic polymers, we present in Section 5 three methods which also can be efficientlyapplied for the treatment of deviations

The Electronic Structure of Polymers

51

from periodicity. A number of applications of the described theoretical procedures are collected in Section 6. These include highly conducting polymers (polyacetylenes, polydiacetylenes, infinite stacks of TCNQ and TTF moIecules) as well as biopolymers (DNA and protein molecules). A short discussion of the results of impurity and aperiodicity calculations closes the Report. 2 Hartree-Fock LCAO Crystal Orbital Method The introduction of the concept of one-electron crystal orbitals (CO’s) considerably reduces difficulties associated with the many-electron nature of the crystal electronic structure problem. The Hartree-Fock (HF) solution represents the best possible description of a many-electron system with a one-determinantal wavefunction built from symmetry-adapted one-electron CO’s (Bloch functions). The H F approach is, of course, only a first approximation to the many-particle problem, but it has many advantages both from practical and theoretical points of view:

(i) The H F total energy per elementary cell in the crystal is variationally determined and admits quite realistic geometry optimizations, potential energy surface calculations (phonon spectra), etc. (ii) The electronic charge distributions obtained by the H F method are very accurate and suitable not only for qualitativebut also for quantitative discussions and interpretations of properties related to them. (iii) The one-determinantal wavefunction of the method is precisely defined; therefore, it can serve as a reasonable starting point for the investigation of effects lying beyond the capabilities of the one-particle approach (correlation corrections, optical properties, etc.).

The HF CO method is especially efficient if the Bloch orbitals are calculated in the form of a linear combination of atomic orbitals (LCAO)’, since in this case the large amount of experience collected in the field of molecular quantum mechanics can be used in crystal HF studies. The atomic basis orbitals applied for the above mentioned expansion are usually optimized in atoms and molecules. They can be Slater-type exponential functions if the integrals are evaluated in momentum space3 or Gaussian orbitals if one prefers to work in configuration space. The specific computational problems arising from the infinite periodic crystal potential will be discussed later. Ab initio Closed Shell Formalism.-The method for calculation of HF CO’s in polymers and crystals at the ab initio level has been discussed many times;1-5 therefore, we repeat here only the basic expressions to allow a self-contained discussion of the procedures applied. We will work entirely in configuration space, writing down the many-electron crystal wavefunction as a Slater-deter1 2

3

G. Del Re, J. Ladik, and G. Bicz6, Phys. Rev., 1967,155,967. J.-M. Andre, L. Gouverneur, and G. Leroy, Int. J. Quantum Chem., 1967,1,427,451. F. E. Harris and H. J. Monkhorst, Phys. Rev. Lett., 1969,23, 1026; Phys. Rev. B, 1970,2, 4400.

4 5

And& in ‘Electronic Structure of Polymers and Molecular Crystals’, ed. J.-M. And& and J. Ladik, Plenum Press, New York, 1975, p. 1 ; (6) J. Ladik, ibid, p. 23. J. Ladik and S. Suhai, in ‘Molecular Interactions’, ed. H. Ratajczak and W. J. OrvilleThomas, J. Wiley and Sons, New York, 1980, p. 151. (a) J.-M.

52

Theoretical Chemistry

minant built from doubly filled Bloch functions y i ( r ) representing a oneelectron state with quasi-momentum k in band n. These functions are expressed as linear combinations of Bloch basis orbitals in the form

where Y is the number of basis orbitals per site and the Bloch orbitals y i ( r ) are symmetry adapted combinations of the AO’s: +:(r) = N-Il2X exp {ikR,}x!(r) J

(2)

N stands here for the number of elementary cells and ~ i ( r=)xa(r- R j - R a) is an A 0 centred in the cell j at Rj+R,. Applying the non-relativistic electronic Hamiltonian (in atomic units)

(ZAbeing the core charge of the atom A at position RA) and using the orthonormality condition ( yE(r) I yk(r))= d,, we can perform a Ritz-variation and

arrive at the complex pseudo-eigenvalue problem1#

where

Here the atomic overlap integrals between the reference cell and cell j are defined (xi@) [ &)> and the corresponding Fock-matrix elements have the form

by s s =

The electronic density matrix elements ptfi connecting the two orbitals c and d i n cells h and f, respectively, are calculated by summation over all occupied states (k,n)in the first Brillouin zone (BZ):

while the two electron integrals in equation (6) are defined by

The Electronic Structure of Polymers

53

It should be noted here that the condition of charge neutrality in the crystal, whose satisfaction plays an important role in actual calculations, requires that

Finally, the total energy per elementary cell of the crystal is obtained from

where from the summation over B the B = A term is excluded if j refers to the reference cell. It should be remarked here that the Bloch-form of the one-electron orbitals [equation (2)] automatically implies translational symmetry. In the case of onedimensional polymers this symmetry operation can be combined with a simultaneous rotation around the polymer axis (helix operation). It can be shown that if the AO's &r) are properly transformed in the translated and rotated elementhe above described formalism can still be applied. tary DODS Crystal Orbital Method.-The method of different orbitals for different spins (DODS) is one possible way to take into account part of the electron correlation within the framework of the one-particle model. Using, however, one single Slater determinant built up from different spatial orbitals for the electrons with spin o! and p, respectively (unrestricted Hartree-Fock, UHF method), a difficulty arises because the many-electron wavefunctions will not be an eigenfunction of the operator s2of the total spin. To overcome this difficulty one has to project out with the aid of a projection operator from the Slater determinant the component with the desired multiplicity 2S+ 1, annihilating all other 'contaminating' components. This can be done either after an already performed calculation (spin-projection after variation, UHF with annihilation), or, as Lowdin has pointed out,Bone would expect a more negative total energy, if the variation is performed with an already spin-projected Slater determinant (spinprojection before variation, spin-projected extended Hartree-Fock method). If one performs the variation of the expectation value of the hamiltonian formed with a spin-projected Slater determinant, one obtains the extended HartreeFock (EHF) equations.OBlo The investigation of these equations has shown that if the number of electrons M+co and M % S , the EHF equations go over into the unprojected UHF equations.ll Therefore, the expectation values of all one- and two-electron operators and all one-electron energies and wavefunctions will be the same in the projected and unprojected cases. This means that if we wish to apply the DODS method to a polymer we can use its simple unprojected form. The unprojected UHF equations can be obtained again4bby calculating the expectation value of hamiltonian (3) with the DODS many-electron function and performing the variation of the coefficients in the Bloch functions 7

@

10 11

I. 1. Ukrainski, Theor. Chim. Acta (Berlin), 1975, 38, 139. C. Merkel, Elektronische Eigenschaften von Molekiilkristallen, Thesis, Munich, 1977. P.-0. Lowdin, Phys. Rev., 1955,97, 1474, 1490, 1509. F. Martino and J. Ladik, J. Chern. Phys., 1970, 52, 2262. I. Mayer, J. Ladik, and G. Bicz6, Int. J. Quantum Chem., 1973,7,583. F. Martino and J. Ladik, Phys. Rev. A , 1971, 3, 862.

3

54

Theoretical Chemistry

We note that the basis orbitals &r) are common for both a and /3 spin electrons, therefore, the evaluation of one- and two-electron integrals has to be performed only once (or they can be taken over from the corresponding closed-shell calculations). The procedure analogous to the one applied for the closed-shell case leads to a set of coupled complex pseudo-eigenvalueequations of the form

F k #s'C

=

Sk c,".fl

(12b)

The matrices FkJ' (y = a,8) are again Fourier transforms of the Fock matrices f0i.y ( y = a,B) as in the previous case, the only difference being that the part containing the electron-electron interactions becomes spin dependent :4b

The definition of the density matrix elements ph;$ (y=a,@) can be obtained from equation (7) substituting the spin-dependent eigenvectors resulting from equations (12). One has to pay attention, however, to the possibility that the number of filled bands and for certain bands the occupied parts of the BZ may be different for a and B spin electrons, respectively. We can see further from equation (13) that to calculate the average potential felt by an electron with a certain spin also the charge distribution of the electrons with opposite spin is needed. Owing to this fact a coupled self-consistent-field procedure has to be used to solve equations (12a) and (12b) simultaneously. We remark finally, that the total energy expression in the DODS case is a simple generalization of equation (lO).O Truncation of Infinite Lattice Sums.-The crucial part of all ab initio HF CO studies in configuration space is the calculation and handling of the two-electron contribution to the Fock matrix. In principle, as we can see from equations (5a) and (6), the number of these integrals is proportional to N S x v4. This relation holds in practice, however, only for a few nearest neighbouring cells in the crystal and goes over very quickly into an Nx v 2 relation (the transition depends, of course, on the lattice parameters and basis set).12 The feasibility of correct computations depends now basically on whether the non-necessary twoelectron integrals lying at an absolute value below a certain threshold (usually to lo-' hartree) can be picked out on the basis of a guess before they are actually calculated. Such a procedure has been developed in our Group on the 1*

S. Suhai, J. P h p . Chem. Solids, submitted.

Zke Electronic Structure of Polymers

55

basis of Mulliken's approximation to four-centre integrals13 and built into the HF CO program system applied for the computations reported below. This procedure, as well as the efficient use of symmetry in both the upper and lower indices of the two-electron integrals, makes it possible that extended systemswith many atoms in the elementary cell can also be investigated with the above method using reasonable computer time (in the case of an STO-3G atomic basis set l4 the computation of all significant two-electron integrals for an alternating transpolyacetylene (C,H,)* up to the twenty-eight neighbouring cells [ 70 A] takes, for instance, 8 min on a CDC 176 The interaction radii beyond which the various intercell quantities appearing in equations ( 5 ) and (6)can be cut off are very different :l2, l6

-

N

159

(i) The simplest are the overlap and kinetic energy integrals, they fall off exponentially and usually even the largest of them is smaller in absolute value than lo-' a.u. after 9-10 A. (ii) More problematic are the elements of the charge density-bond order matrix (which are needed for the calculation of the first-order density matrix). From previous analytic studies on simple model systems1' we know that for large distances they show an r-l dependence. This decay is rather slow and, in fact, for most polymers one has to calculate bond orders for a distance of 3 8 4 A using a reasonable threshold of 5 x for this quantity. (iii) It is usual to anticipatels that the exchange part of the Fock matrix is as well behaved as the kinetic energy and that it disappears after a few neighbours. This statement is, however, not valid and this is why the use of the correct nunlocal exchange is inevitable in polymers. The leading exchange contributions are obtained in equation (6), namely from the terms

and since the two-electron integral is proportional to r-l the exchange decays with r2. On the other hand, the corresponding Coulomb part

contains two (exponentiallydiminishing) overlap-distributions for both electrons. It is evident that the sphere of elementary cells within which Fock-matrix elements have to be calculated is determined only by the exchange integrals. It was found for various polymers that the radius of this sphere is again 40 A.13 (iv) The undoubtedly most difficult problem is, however, to truncate the two 'electrostatic' interactions in equation (6), namely the core attraction and the Coulomb repulsion terms with small differencesin the (0,j ) and (h, r ) index pairs, respectively. It is clear that, though the sums of the attractive and repulsive interactions diverge individually due to the r-l dependence, the charge neutrality N

13 14

15

16 17

R. S. Mulliken, J. Chim. Phys., 1949,46,497. W. Hehre, R. F. Stewart, and J. A. Pople, J. Chem. Phys., 1969, 51, 2657. S. Suhai, J. Chem. Phys., 1980, 73, 3843. L. Piela and J. Delhalle, Int. J. Quantum Chem., 1978, 13, 605. J. CiPek, G. Bicz6, and J. Ladik, Theor. Chirn. Acta (Berlin), 1967, 8, 175.

56

Theoretical Chemistry

requirement [equation (9)] still ensures that they accurately cancel each other for large distances, i.e. the electrostatic interaction between very distant elementary cells will be determined by interactions between higher electric moments (for polymers with inversion centre in the elementary cell like polyacetylene, polydiacetylenes, polyethylenes, etc. the leading term is the quadrupole-quadrupole interaction falling off as r5). It is, however, very difficult to foresee for any system how far one has explicitly to calculate these interactions to reach, in spite of the necessary truncation, the prescribed threshold in all quantities required. Analysing the results obtained for various polymers we found12 that actually different physical properties are sensitive to this truncation to a different degree. We can still get a h n t for the lower bound of the corresponding interaction radius by the following reasoning: we know from the decay of the overlap matrix elements that the electronic charge distributions p z ( r l )= x:(rl) &rl) and pt$(rz)= xt(rz)1cfi(r2)in equation (6) extend over approximately 20 8,. If now we look at the electrostatic interaction of an electronic charge distributed around the reference cell [in the sense of equation (9)] with another charge distribution centred in cell h, we can say that the distance between the two cells has to be at least 3 0 - 4 0 8, to avoid monopole-type interactions. It is evident, however, that the total energy per elementary cell will not be converged for this distance, and thus we can expect that the two charge spheres have to be separated for about 60-80 8, to be able correctly to calculate quantities like cohesion, conformational energy differences, erc. It is to be stressed here that besides choosing a large enough ‘cutoff’radius one must pay attention to two further points (for any finite radius): (a) The two-electron interactions have to be truncated in such a way that the charge distributions p$ experience the whole electronic charge belonging to cell h. This is not possible if one works with a general cutoff radius for all twoelectron integrals since then for the maximum value of lR,,l, for instance, a considerable part of the electronic repulsion will be left out due to IRI > [I?,,I in p::. We shall analyse in some detail the consequences of this error on concrete examples later and emphasize here only that there is less electronic repulsion than nuclear attraction in many ab initio calculations containing this methodological inconsistency.18-21 (b) The truncation of both the one- and two-electron integrals has to be performed at the level of the lower indices in equation (6) preserving in this way all additional symmetries within the elementary cell. Calculation of Wannier Functions.-In most further applications of the wavefunctions obtained in HF CO studies (calculation of excitonic effects, CDW’s, impurity and vacancy levels, etc.) the use of Wannier functionsz2instead of the original Bloch functions seems to be very promising.23The connection between the two basis sets is given by the transformation

I

18 19

20 21 22

23

J.-M. Andrt and G. Leroy, Inr. J . Qitantirm C/retn., 1971,5,557. M . Kertesz, J. Koller, and A. Aiman, J . Chem. Phys., 1977, 67, 1180; M. KertBsz, J. Koller, and A. Aiman, J . Chem. Sac., Chem. Commun., 1978, 575. A, Karpfen and J. Petkov, Theor. Cliim. Acta (Berlin), 1979, 53, 65. M. Kerttsz, J. Koller, and A. Aiman, Chem. Phys. Lett., 1978, 56, 18. G. H. Wannier, Phys. Rev., 1937, 52, 191. S. Suhai, to be published.

The Electronic Structure of Polymers

51

+; (r) = N-’lZX y: (r) exp (- ikR,) BZ

(141

k

where the Wannier function +i(r)=+,(r-Rj) is centred around the cell at R j . From the point of view of the accuracy and economy of the above mentioned calculations the extension of $i in direct space is of great importance. As it is well known,24on the other hand, there is still a residual degree of freedom in the Bloch functions represented by the renormalization

where R,(k) can be any analytic function of k possessing the symmetry of the BZ. The phase factor exp{iA,(k)} can be used to predetermine certain properties of the Wannier functions obtained by equation (14). These properties are partly in conflict; therefore, one has to consider which of them is more important from the point of view of further calculations. If the transformed phase is written in the form A,@)= A$g- (k)+ A&), where ATg. (k)stands for the phase of the Bloch function obtained during the band structure calculations, the following statements hold :

(i) The choice Xn(k)= - An( - k) makes the Wannier functions real. (ii) The choice A(, - k)= An@) preserves spatial symmetries present in the Bloch functions in addition to the translational symmetry. (iii) One can apply variational procedures to determine the functional form of which will minimize the spatial extension of the Wannier Different criteria can be used for the latter purpose. We found that maximization of the expectation values

1#;

(r)+z2#;

(4 d3r

a

for a small region Q around the reference cell ( j = 0, z is the co-ordinateparallel to the polymer axis) also provides efficientlocalization with reasonable computational efforts at a6 initio level. Even better localization can be achieved for semi-empirically determined wavefunctions by generalization of the method of Edmiston and R ~ e d e n b e r g but , ~ ~ the application of this procedure to ab jnitjo Wannier functions seems to be too ineffi~ient.~~ It is evident that making choices (i) and (ii) there remains no room for a further optimization of A&). On the other hand, one of these properties can always be combined with (iii). Actually, we found that the use of (i) and (iii) provides the best starting point for calculation of optical properties and localized impurity levels.23 3 Excited States and Correlation Effects in Polymers Intermediate Exciton Theory of Excited States.-It is well known that the HF picture does not permit a reasonable calculation of excited states and optical 24

For a review see: E. I. Blount, Solid State Phys., 1963, 13, 305.

25

C.Edmiston and K. Ruedenberg, Rev. Mod. Phys., 1963, 35, 457,

58

Theoretical Chemistry

properties in The first step towards a correct description of such phenomena is provided by the exciton theory of solids.27The exciton model takes into account the simplest correlation effect in the crystal, namely the interaction between an excited electron in the conduction band and a hole left behind in the valence band. The calculation of such states in polymers is complicated by the fact that neither of the two usually applied models of exciton theory can be used with these systems. Both of these models in fact assume a limiting case: the Frenkel picture is valid only for nearly localized excitations (within the same elementary cell), while the Wannier model applies to strongly delocalized ones. From inspection of the electronic indices of most polymers it is clear, however, that excitations between neighbouring elementary cells should play an important role but it is also evident that a simple effective mass theory (continuum model) would not work for them. This conceptual difficulty was removed by Takeuti28 who devised the so called 'intermediate exciton' scheme based on the mathematical procedure proposed by Slater and Koster 2g-31 to treat localized impurity levels in semiconductors. We describe here briefly his scheme for the case when the ground state is a completely filled valence band represented by a Slaterdeterminant containing doubly filled Wannier functions and there is only one empty conduction band (the generalization of his expressions to the case of more valence and conduction bands is obvious and would complicate only the notations). The many-particle wavefunction of Takeuti's method is constructed in the form

where the function WDEC(Rj)itself is a symmetry-adapted linear combination of singly excited configurations with appropriate multiplicity (M):

The Slater determinant "./$:+j can be obtained if we substitute the Wannier function +t(r) in the ground-state determinant by 45+j(r)(the index M takes care of the possible spin change during excitation). Each function M@&(Rj)thus represents in the exciton state WfKan excitation wave corresponding to an electron-hole separation by a lattice vector Rj and moving with a wave vector K. This 'exciton representation', proposed by Wannier,22combines the expected explicit dependence of the matrix elements on the electron-hole separation in direct space with the fact that the total momentum of the electron-hole pair has to remain a good quantum number (K-dependence). Treating the electron-hole interaction as perturbation and using expansion (16) for the perturbed wavefunction, the Schrodinger equation of this simple configuration interaction problem l6 l7

l6

so s1

For a review see: J. Ladik, in 'Excited States in Quantum Chemistry', ed. C. A. Nicolaides and D. R. Beck, D. Reidel, Dordrecht, 1978, p. 495. R. S. Knox,Theory of Excitons, Solid State Phys. Suppl., 1963,5, 1, Y. Takeuti, Prog. Theor. Phys. (Kyoto), 1957, 18,421. J. C. Slater, Technical Report, No. 5, 1953. G. F. Koster and J. C. Slater, Phys. Reo., 1954, 95, 1167. G. F. Koster, Phys. Rev., 1954, 95, 1436.

59

The Electronic Structure of Polymers H M y K = MEKM~IK

(18)

can be rewritten (applying the Slater-Koster idea) in the form28 MUK(R,)=

h

I:GK(Rj,R,, MER)*MVK(Rl, RJ-"U'(R&

(19)

The Green's function for the electron-hole pair is defined here by

) the energy dispersions in the conduction and valence where E@) and ~ " ( kare bands, respectively. Finally, the matrix elements of the electron-hole interaction are given by

We have used the notation of equation (8) but the basis functions are here, of course, Wannier functions with the appropriate band index (the constant 8 M in the exchange part is equal to one in the case of singlet excitons and to zero for triplet ones). The most tiine-consumingstep in exciton calculations (as well as in CI ones) is the transformation of the two-electron interaction integrals (which have been evaluated during the band structure calculations in the atomic basis) to the Wannier basis. The matrix elements in equation (21) have to be calculated then for each value of K, and the zero values of the determinant corresponding to the system of homogeneous linear equations (19) as functions of MEK provide solutions to the Schrodinger equation (18). More details of such calculations on highly conducting polymers and biopolymers as well as the extension of the above formalism to doubly excited configurations will be presented el~ewhere.~a More General Treatments of Electron Correlation in Polymers.-The introduction of excitonic states was just a simple example to show how one can go beyond the HF approximation to obtain correlated electron-hole pairs, whose energy level(s) may fall into the forbidden gaps in H F theory, and form the basis for interpretation of optical phenomena in semiconducting polymers. The schemes described until now for investigation of certain types of correlation effects (the DODS method for ground-state properties and the exciton-picture for excited states) are relatively simple from both the conceptual and computational points of view and they have been actually used at the ab initiu level. It is evident, on the other hand, that further efforts are needed in polymer electronic structure calculations if we want to reach the level of sophistication in correlation studies on polymers which is quite general nowadays in molecular quantum mechanics. If the valence band is completely filled in a polymer (or in a molecular crystal) we can subdivide the ground-state correlation into a long range and a short range part. For the long range correlation the electronic polaron can be used. It has been applied already to the periodic DNA model polycytosine (polyC),S4 32 33 34

S. Suhai, to be published. Y . Toyozawa, Prog. Theor. Phys. (Kyoto), 1954, 12, 421; A. B. Kum, Phys. Rev., 1972, B6,606; J. T. Devreese, A. B. Kunz, and T. C. Collins, Solidstate Commun.,1972,11,673. J. Ladik, S. Suhai, P. Otto, and T. C. Collins, Int. J. Quantum. Chem., 1977, QBS4, 4.

60

Theoretical Chemistry

to the periodic protein models polyglycine and p ~ l y a l a n i n e ,and ~ ~ to 1-D TCNQ These calculations resulted in about 10% decrease in the widths of conduction and valence bands and in the gap between them. On the other hand, in the case of simple metals and ionic 3D crystals the long range correlation has a much larger effect.37 To treat the short range correlation in the ground state of a polymer one has to Fourier-transform the delocalized Hartree-Fock Bloch orbitals into localized Wannier functions. Though this localization procedure does not eliminate ‘the tails’ of the Wannier functions,23it does localize the overwhelming major part of the crystal orbitals around a site (molecule). Using these Wannier functions as a basis instead of the MO’s of the free molecules one can apply the usual quantum chemical methods like CT, CEPA,38or the coupled cluster expansion method of Ciiek and P a l d ~ to s ~obtain ~ the correlation energy per unit cell due to the short range correlation. In this procedure one must also use charge-transfer excitations to the neighbouring cells, apart from excitations within that cell to which the Wannier function is localized, The problem of ground-state correlation becomes much more difficult if (i) the subunits are strongly coupled [as in a (CH)z chain], where to achieve the desirable accuracy charge-transfer excitations to more distant neighbours cannot be neglected and (ii) if the valence band is partially filled [as in (SN)J, because in this latter case the localized Wannier functions could be formed only if one takes into account the unfilled part of the band in the Fourier transformation (i.e. the single ground-state HF Slater determinant had to be mixed with excited Bloch functions). Furthermore, in this case the ground-state Correlation energy cannot be subdivided into long and short range contributions. In such cases in solid-state physics electron gas methods are usually Instead of starting from HF Bloch orbitals, the density ( p ) functional formalism is applied to describe all contributions to the electronic energy. Since, however, the Hohenberg-Kohn theorem41 is only an existence theorem and the explicit form of the exact E [ p ] functional is unknown, there seems to be no systematic way to improve the results obtained with this formalism. On the other hand, starting with the Hartree-Fock method (which requires much harder numerical work), one can systematically improve the results, if a comparatively simple and accurate method is found to treat the ground-state correlation. Such a method could also be provided for polymers by the further development of an approximate CI technique4z which uses, instead of excitations from single levels to single levels, 35 36 37

38 39

40 41

42

S. Suhai, T. C. Collins, and J. Ladik, Biopolymers, 1978, 18, 899. S. Suhai, Phys. Lett., 1977, 62A, 185. For a review see: T. C. Collins, in ‘Electronic Structure of Polymers and Molecular Crystals’, ed. J.-M. Andre and J. Ladik, Plenum Press, New York, 1975, p. 405. R . Ahlrichs and W. Kutzelnigg, J. Chcm. Phys., 1968,48, 1819; W. Meyer, J. Chem. Phys., 1973,58, 1017. J. Ciiek, J . Chem. Phys., 1966, 45, 4256; J. Ciiek and J. Paldus, Int. J. Quantum Chem., 1975,5, 359; J . Paldus and J . Ciiek, A h . Quantum Chem., 1975, 9 , 105. For a review see: N. H. March, in ‘Quantum Theory of Polymers’, ed. J.-M. And& J. Delhalle, and J . Ladik, D. Reidel, Dordrecht, 1978, p. 48. P. Hohenberg and W. Kohn, Ph-vs. Reu., 1964, 136, 3864. J. Ladik, in ‘Recent Advances in the Quantum Theory of Polymers’, ed. J.-M. Andre, J.-L. Bredas, J. Delhalle, J. Ladik, G. Leroy, and C. Moser, Springer Verlag, Berlin, 1979, p. 155.

The Electronic Structure of Polymers

61

excitations from a region of a given band to other regions of the same band or to regions of other bands. To achieve this, one has to subdivide each band into specific regions by analysing the density of states curves of the bands in question and taking into account the dependence of the partial charge density distribution of these particular bands on k.42In this connection, the problem of size consistency is important and the question of by how many units the orbitals representing the excited regions should be extended seems to require further invest igation. 4 Semi-empirical Crystal Orbital Methods The most important motivations for the application of ab initio procedures in polymer electronic structure investigations are, in our opinion, the following:

(i) the fact that certain quantitatively well defined features of the calculation (basis sets applied, amount of correlation taken into account, etc.) provide a solid basis for judging the accuracy to be expected; (ii) the precisely defined theoreiical framework makes it possible to improve the results systematically; (iii) after having reached a certain level of sophistication they should be able to predict all properties of all kinds of polymers. The quantum theory of polymers has not yet reached the level required to meet the third requirement using reasonable computational efforts. Therefore, it is obvious to turn to molecular physics and apply those semi-empirical calculation schemes which proved to be successful in solving certain molecular problems, and problems in polymer physics. The common feature of these methods is that they avoid the computational bottle-neck of the ab initio work by neglecting the overwhelming majority of two-electron integrals in equation (8). Since the number of semi-empirical molecular methods is confusingly large one must keep in mind before making a choice for polymer purposes that even carefully parametrized methods work correctly only for those properties which have been taken into account explicitly during the determination of the parameter set. From the point of view of the consideration of economy uersus flexibility, we think that the level of sophistication represented by the NDD043and MND044 methods is a reasonable compromise. The other semi-empirical methods are basically very similar to these only they go further in neglecting integrals. To demonstrate their common methodology we describe here briefly the MNDO CO procedure. As a first step all integrals involving core orbitals or diatomic differential overlap are neglected, i.e. from equation (8) we obtain

or zero if one of the lower indices stands for a core-orbital (here orbital a belongs to atom A , efc.) This assumption removes the computational difficulties connected with the multicentre integrals. For Fock matrix elements of the polymer defined in equation (6), for instance, the following MNDO expressions are 43 44

J. A. Pople, D. L. Beveridge, and P. A. Dobash, J. Chem. Phys., 1967,47,2026. M. J. S. Dewar, and W. Thiel, J. Am. Chem. SOC.,1977,99,4899,4907.

3*

Theoretical Chemistry

62

obtained,45 where p, Y, 1 and stand for valence orbitals, SL is an s-type valence orbital belonging to atom L and the prime on the summation means that the termL=Mhas tobeomittedinh=o:

The various integrals appearing in these quations are not evaluated analytically. Some of them are determined from semi-empirical expressions, which contain adjustable parameters to fit certain experimental data, others are taken directly from experiment. To be able to perform thus an MNDO calculation the following quantities have to be defined:44

+

core attraction of (i) One-centre one-electron integrals Up,(kinetic energy atom M) and onecentre twoelectron repulsion and exchange integrals

and

They are all determined by fitting the theoretical energies of several valence states of an atom and its ions to the corresponding spectroscopic values. (ii) Twocentre one-electron core resonance integrals [first term in equation (23c)l: these are taken proportional to the interatomic overlap integrals which are calculated analytically with Slater-type orbitals (whose exponents, on the other hand, are adjusted empirically). The atomic parameters #IM are again adjusted to experiments. (iii) The two-centre one-electron attractions between an electron distribution x p xp on atom A4 and the core of atom L are calculated by representing the core with the help of an s-type valence electron distribution [second term in the first line of equation (23a)J. (iv) The two-centre two-electron repulsion integrals

):;I;( 45

M.J. S. Dewar, Y. Yamaguchi, and S. H. Suck, Chern. Phys., 1979,43, 145.

The Electronic Structure of Polymers

63

are again evaluated semi-empirically using a multipole-multipole interaction scheme and including a certain amount of correlation (which is necessary since some allowance for correlation effects was also made in the case of the one-centre repulsion integrals). It should be mentioned finally that the core-core repulsions appearing in equation (10) are calculated in a complicated way in the MNDO method to be consistent with other approximation^.^^ To determine the adjustable interatomic parameters, the experimental values of four properties (heat of formation, gradient of the total energy with respect to geometrical variables, fist ionization potential, and dipole moment) have been fitted for a set of more than thirty We can expect, therefore, that similar properties of polymers can be reasonably described by the MNDO CO method. Its fist application for the investigation of vibrational spectra of polyethylene proved promising.45To calculate other properties of interest some changes in the parameter set seem to be desirable. Work along these lines is in progress in our laboratory. 5 Disorder Effects in the Electronic Structure of Polymers

The electronic properties of polymers have been discussed until now using the assumption of strict periodicity (translational symmetry). In reality, however, in many polymers of great practical importance (polyethylenes, polyacetylenes, polydiacetylenes, etc.) the presence of imperfections (vacancies, impurities, structural disorder, etc.) is assumed to play an important role in their physical and chemical properties. Other polymers, polypeptides for instance, are by their very nature aperiodic owing to the presence of different side groups bound to the !-carbon atoms of the polypeptide backbone. To investigate the influence of these groups on the electronic structure of the otherwise regular polypeptide, models with more complicated translational elementary cells, composed of a dior tri-peptide with different amino-acid residues, e.g. Gly-Gly-Ala, Gly-Ala-Ser, etc. have been in~estigated.~~ The generally accepted opinion (in agreement with the earlier intuitive feeling of the present authors) was that since, both possible channels of electron flow" (the backbone and the hydrogen bonded peptide units, respectively) contain regularly repeated elements, this periodicity should be the primary feature of the system and the potential fluctuation due to the different side chains should play a minor role in electrical conduction. This opinion is apparently also supported by the fact that the widths of the energy bands and forbidden gaps are very similar, for instance, in poly(GIy), poly(Ala), and poly(Ser). The differences in the potential of the three side groups (-H, -CH3, and -CH,--OH, respectively) change the positions of the valence and conduction bands of the corresponding homopolymers only by some tenths of an electron volt but not their form. The same apparently small differences produce, however, strong modifications in the band structures of the composite systems, as we shall see later. The calculations performed on models with such complicated elementary cells have the advantage that the effect of the perturbing potentials 48

S. Suhai, J. Kaspar, and J. Ladik, Znt. J. Quantunz Chem., 1980,17,995.

47

J. Ladik, Int. J. Quantum Chem., 1974,QBSl, 651;S. Suhai, Biopolymers, 1974,13, 1701.

Theoretical Chemistry

64

can be investigated in the framework of the same self-consistent-fieldprocedure that was used in the case of the host systems. The size of the enlarged unit limits, however, the applicability of this procedure. Application of Dean’s Negative Eigenvalue Theorem to Aperiodic Polymers.-The basic idea of this method has been proposed by Dean4ato interpret vibrational spectra in disordered solids. It can be most easily understood through the example of a simple linear chain (one orbital per site) consisting of N units. It is described in the framework of a simple tightly-binding scheme with the &st nearest neighbours’ interactions (in the absence of periodic boundary conditions) by the secular equation

=O

(24)

..

Here ai (i= 1,2, . . ., N) and bi (i= 2,3, . ,N) are the diagonal and off-diagonal matrix elements, respectively, of an effective one-electron hamiltonian and il is its eigenvalue. The above secular determinant can be factorized as

Assuming that some other convenient factorization is found in the form

Dean’s negative eigenvalue states that the number of eigenvalues less than a particular 3. value is equal to the number of negative factors &((A). If we transform equation (24) with the aid of successive Gaussian eliminations into an upper triangular form, the q ( A ) values are given by the simple recurrence relation && = qI-) A, &i(A)

=

ai-A-b;/&i-l(A),

(i = 2, 3 , .

,,a)

The eigenvalue distribution of the polymer can be calculated thus simply by counting the numbers of negative factors q ( A ) . By giving 3, different values throughout the range of the energy spectrum of interest and then taking the differencein the number of negative factors belonging to consecutive values of A, the distribution of eigenvalues of H (density of states) can be determined to any desired accuracy. 48

P. Dean, Proc. R. SOC.London, Ser. A , 1960, 254, 507; 1961, 260, 263; Rev. Mod. Phys., 1972, 44,127,

The Electronic Structure of Polymers

65

The representation of a polymer unit by one orbital is, of course, a rather crude approximation but calculations of this type49*50 may still throw light on the qualitative effects of aperiodicity. There seems to be, however, no difficulty in applying Dean’s method in a more sophisticated way, namely substituting the matrix elements a$ and bi by Fock matrix blocks which could be taken from appropriate cluster calculations. Treatment of Point Defects in Polymers.-Both previously described methods (periodic clusters and negative eigenvalue counting) for the calculation of disorder effects in polymers have disadvantages (cluster size and non-self-consistent nature, respectively) which make them less suitable for the quantitative description of impurities, vacancies, and other point defects in polymers. In the case of these properties, the correct handling of the bulk-impurity interaction plays a predominant role. Perturbative methods based again on the Slater-Koster idea 29-31 seem to be very promising for such calculation^.^^ The total crystal potential is taken in these methods as U = Uo+ V and it is assumed that the problem of the host crystal (bulk) with the periodic potential Uohas been solved previously. As in the case of excitons in Section 4 one has to calculate again the matrix elements of the perturbing potential V with the periodic functions (preferably in Wannier representation). With the help of Green’s function (20) the analogue of equation (19) has to be solved to obtain perturbed energy levels and wavefunctions. Since one expects an extensive redistribution of the electrons around the defect it seems to be very important that the potential I/ be reconstructed using the perturbed wavefunctions and the whole procedure be repeated until self-consistency is reached. Further improvement can be achieved if not only the changes in the electron distribution but also in the ionic positions around the defect are taken into account. 6 Illustrative Examples Polyacetylenes (Polyenes).-The electronic structure of polyacetylenes [called also polyenes, (CH)z, (C2H2)z]has been a subject of interest for several decades in theoreiical chemistry due to the central role of the polyene backbone in various organic compounds. These investigations have been considerably stimulated in the past years by the fascinating solid-state physical properties of different doped semiconducting polyacetylene crystals. It has been shown that by doping with electron acceptors (bromine, iodine, As&,) or donors (sodium) the electrical properties can be varied over a wide range under The specific conduc49 50

51 52

53

M. Seel, Chem. Phys., 1979, 43, 103. M. Seel, in ‘Recent Advances in the Quantum Theory of Polymers’, ed. J.-M. Andrk, J.-L. Bredas, J. Delhalle, J. Ladik, G. Leroy, and C. Moser, Springer-Verlag, Berlin, 1979, p. 271. R. Day and F. Martino, Phys. Rev. B., submitted. J. Callaway, J. Math. Phys., 1964, 5, 783; J. Ladik and M. Seel, Phys. Rev. B., 1976, 13, 5338; G. A. Baraff and M. Schluter, Phys. Rev. Lett., 1978, 41, 892; J. Berholc, N. D. Lipari, and S. T. Pantelides, Phys. Rev. Lett., 1978, 41, 895; G. Del Re and J. Ladik, Chem. Phys., in press. H. Shirakawa, E. J. Louis, A. G . MacDiarmid, C. K. Chiang, and A. J. Heeger, J . Chem. Sue., Chern. Commun., 1977, 578; C. K. Chiang, M. A. Drug, S. C. Gau, A. J. Heeger, H. Shirakawa, E. J. Louis, A. G. MacDiarmid, and Y.W. Park, J. Am. Chem. SOC.,1978, 100, 1013.

66

Theoretical Chemistry

tivity of films of (CH)z varies, for instance, over twelve orders of magnitude from insulator ( u w 10-gfl-l cm-l) to highly conducting metallic polymer (aw 103 R-l cm-1).54-56Furthermore, compensation and junction formation have been demonstrated on various n- and p-type samples6' Partial orientation of the polymer fibres results in highly anisotropic electrical55 and optical p r ~ p e r t i e s . ~ ~ Optical absorption studies suggest for most cases a direct band-gap semiconductor with very anisotropic band structure. For certain dopant concentrations in AsF5-doped polyacetylene, however, some qualitative changes in the electrical and optical properties 6 o indicate a semiconductor-to-metaltransition with conductivities in excess of 2000 Sz-l cm-l. There are two possible isomers of pure polyacetylene (the cis- and trans-forms) which show characteristic differences in their physical and chemical properties. The cis-isomer is thermodynamically unstable in pure form and has thus attracted little interest until now. Its doped form shows, however, electrical conductivities The two isomers can which exceed in some cases those of the trans-m~dification.~~ also appear as a mixture and their ratio can be well controlled.61Early experimental observations suggested a bond-alternate molecular structure for the transpolyeneeZand this model was also used as a possible explanation of the observed energy gap in long p01yenes.~~ In fact, for the cis-polyene there are two further possible structures with bond alternation, the cis-transoid and trans-cisoid forms. Since highly oriented polymer samples of pure polyacetylenes are still not available, it has not been possible yet directly to determine the structural parameters from X-ray diffraction alone.64Although Raman spectral investigations suggest65 a cis-transoid structure for the cis-isomer, the relative stability of the various polyene structures is still an open problem (especiallyin their doped form). The overwhelming majority of theoretical works on polyenes has been concentrated on the gap-problem. The majority of authors based their investigations on a one-electron picture (band theory), while a smaller group stressed the importance of many-electron (correlation) effects. It has been shown that, in fact, the two mechanisms can be active simultaneously (for a recent review of this field see ref. 15). We briefly summarize here the main results of ab initiu investigations of five 599

58

C. K. Chiang, C. R. Fincher, Jr., Y. W. Park, A. J. Heeger, H. Shirakawa, E. J. Louis, S. C. Gau, and A. G. MacDiarmid, Pliys. Rev. Lett., 1977, 39, 1098. Y. W. Park, M.A. Drug, C. K. Chiang, A. J. Heeger, A. G. MacDiarmid, H. Shirakawa, and S. Ikeda, J. Polym. Sci., Polym. Chem. Ed., 1979, 17, 195. C. K.Chiang, Y. W. Park, A. J. Heeger H. Shirakawa, E. J. Louis, and A. G. MacDiarmid, J. Chem. Phys., 1978, 69, 5098. C. K. Chiang, S. C. Gau, C. R. Fincher, Jr., Y. W. Park, A. G. MacDiarmid, and A. J. Heeger, Appl. Phys. Lett., 1978, 33, 181. C. R. Fincher, jun., D. L. Peebles, A. J. Heeger, M. A. Drug, Y.Jatsumara, and A. G. MacDiarmid, Solid State Commun., 1978,27,489. C. R. Fincher, jun., M. Ozaki, A. J. Heeger, and A. G. MacDiarmid,Phys. Rev. B., 1979,

6o

Y. W. Park, A. Denenstein, C. K. Chiang, A. J. Heeger, and A. G. MacDiarmid, Solid

54

55

56 57

58

19,4140. State Commun., 1979, 29, 747. 62

63 64 65

H. Shirakawa, T. Ito, and S. Ikeda, Makromol. Chem., 1978, 179, 1565. L. G. S. Brooker, J. Am. Chem. SOC.,1951,73, 1087, 5332. H. Kuhn, Helv. Chim. Acta, 1948, 31, 1441. R. H. Baughman, S. L. Hsu, G. P. Pez, and A. J. Signorelli, J. Chem. Phys., 1978,68, 5405. T. Ito, H. Shirakawa, and S. Ikeda, J . Polym. Sci., Polym. Chem. Ed., 1975, 13, 1943.

The Electronic Structure of Polymers

67

different polyacetylene chains.lSModels I and I1 are the two trans-isomers with regular and alternating backbone structures, respectively (Figure 1). The three proposed cis-isomers,namely the regular (Model 111), the trans-cisoid (Model IV), and the cis-transoid structures are shown in Figure 2. The bond length and bond

Figure 1 Regular (Model I) and alternating (Model 11) trans-polyacetylene structures. The unit cells are surrounded by broken lines

H

\C I

I

I

I

Model V. -------------

1

I

Figure 2 cis-Polyacetylene structures: regular (Model 111)' trans-cisoid (Model IV) and cis-transoid (Model V). The unit cells are surrounded by broken lines

68

Theoretical Chemistry

angle values of these models have been proposed by Baughman et al. on the basis of X-ray diffraction studies combined with a crystal packing analysis.g4Throughout these calculations a minimal atomic basis set has been applied and each Slater-typeorbital has been combined from three Cartesian Gaussians (STO-3G).14 Total Energyper Elementary Cell. On the basis of our previous discussion concerning the role of long range effects in performing lattice sums in configuration space we expect that the total energy will very strongly depend on the truncation of these infinite sums. The experiences in the case of many different polymers prove that this expectation is correct. As a demonstration of this fact the total energy per C2H2 unit (Etot.)is shown in Figure 3 for the cis-transoid polyacetylene

Etot. {U.U.)

-75.c

-75.:

@ 10

20

N

Figure3 Total energy per C2Hz unit as function of the number of interacting CXH2 neighbows ( N ) in the cis-transoid polyacetylene model. The encircled values are those obtained by the truncation method of refs. 19 and 20

structure (Model V in Figure 2) as a function of the number of interacting C2H, neighbours N (the Nth neighbour interaction means in our terminology that, starting from the reference cell in both directions along the polymer until the Nth C2H2unit, all significant interactions are explicitly calculated, including two-

The Electronic Structure of Polymers

69

electron integrals for which one of the functions lies outside of this region, if that function is a member of an electron distribution centred inside the above region). We can see that Etot.is a very sensitive function of N for the first 8-10 neighbours and it saturates only very slowly for larger values of N . Besides the number of included neighbours it is important how the truncation of the lattice sums is performed. In previous ab initio calculations on polyacetylenes18-21a general cut-off radius was applied neglecting all one- and twoelectron integrals if the distance for any function-pair was larger than a prescribed value. It was shown l5 that this procedure leads to non-negligible errors: owing to the missing part of the electron-electron repulsion integrals the core-electron attractions become over-represented. Violation of the electrostatic balance in the crystal has serious consequences. Comparable errors are also contained in calculations in which the interactions between distant cells are cut off too early since in this case the partial summation of the two long range interactions (which cancel each other accurately only for infinite distance) makes the results inaccurate. The relative total energies for different conformations are, of course, physically much more interesting than the absolute values. Though the functional form of Etot.(N)is very similar for all five models studied it is noteworthy that the total energy differences are very strongly dependent on N. In Figure 4 the total energy of the cis-transoid structure (Model V) has been chosen as reference for each value of N . The relative conformational energies (EE!:)are given in kcal mol-l in Table 1. In summary we can state on the basis of these calculations that : (i) The regular structures are unstable for both the trans- and cis-isomer (by 8.4 and 9.5 kcal mol-l, respectively) as compared with the alternating models.

(ii) From the alternating cis-structures, the cis-transoid model is more stable than the trans-cisoid one by 2.5 kcal mol-l. It is interesting to note here that this has been correctly predicted from the semi-empirical calculations of Yamabe et a1.66though the CNDO method exaggerates the energy difference (to 4.8 kcal mol-l). Our calculations thus support the Raman spectral results of Ito et a1.66 who also found that the cis-transoid is the stable pure cis-isomer. (iii) The energy difference between the alternating-trans and trans-cisoid structures is too small to be significant (0.1 kcal mol-l); further interchain calculations are needed, in our opinion, to decide which is the more stable in the crystalline environment. Finally, it must be mentioned that the above discussion refers only to pure polyacetylene samples. According to the results of calculations on halogen-doped polyacetylenes6 7 the presence of impurity atoms not only significantly changes the band structures but may completely reverse the conformations. One-electron Properties. Unlike the total energy which is calculated from the many-electron wavefunction, these quantities depend much less sensitively on N and in the case of a correct truncation they can be obtained with reasonable accuracy after a few neighbours. This is demonstrated by the example of the 66 67

T. Yamabe, K. Tanaka, H. Termata,-e K. Fukui, A. Imamura, H. Shirakawa, and S. Ikeda, Solid State Commun.,1979,29,329. S. Suhai, Solid State Commun.,submitted.

Theoretical Chemistry

70 .

..

(au.1 0.10

-

0.05 -

0.00-

10

20

N

Figure4 Relative total energies per elementary cell of four polyacetylene models as function of the number of interacting cells ( N ) . As reference for each value of N the corresponding energy of the cis-transoid model has been chosen. : regular trans, 0:alternating-trans, A :regular&, 0: trans-cisoid, :cis-transoid

Table 1 Total energy values per CJ3, unit for the five diferent polyacetylene models obtained by correct truncation of the two-electron lattice sumsb Polyacetylene modela Regular-trans (I) Alternating-trans (II) Regular-cis (111) trans-Cisoid (IV) cis-Transoid (V)

(Etot./CsHz)/ hartree -75.851 01 - 75.864 45

-75.853 45 -75.864 61 - 75.868 57

(q:;:/C2H2)/ kcal mol-1 c 11 .o 2.6 9.5 2.5 0

For various polyacetylene structures see Figures 1 and 2. * From ref. 15 : calculations up to the 28th neighbouring C ~ H unit, Z electrostatically balanced cutoff. C E!::: is the relative total energy per C2H2 unit measured from the corresponding value obtained for model V (1 hartree = 627.49 kcal mol-1). (1

ionization potentials, which are shown in Figure 5 as functions of the number of interacting neighbours (N). If the electrostatic balance is violated, however, the above statement does not hold. The ionization potentials (IP), electron affinities (EA), fundamental gaps (AEg), and valence- and conduction-band widths (SE, and 6Ec, respectively) of

The Electronic Structure of Polymers

71

the investigated polyacetylenes are given in Table 2 corresponding to the case of N=28.Study of the values in Table 2 shows that the one-electron properties of

IP (a.u.

0.

0.:

0.1 10

20

N

Figure 5 Ionization potential of polyacetylenes as function of the number of interacting cells (N).0 :alternating-trans, A : regular-cis, 0: trans-cisoid, A : cis-transoid

Table 2 Some one-electron properties of the investigated polyacetylene models; IP (ionization potential), E A (electron affinity),fundamental gap (AE,), valence band width (SE,), and conduction band width (SE,), respectively, obtainedfrom calculations up to the 28th neighbouring CzHa unit (all values in eV)

Polyacetylene modela Regular-trans ( I ) Alternating-trans (11) Regular-cis (111) Trans-cisoid (IV) Cis-transoid (V)

IP 0.21 5.27 2.81 4.54 4.56

EA -0.21b 4.47 2.40 3.29 3.63

AEk! 0

9.74 5.21 7.83 8.19

0 For the corresponding polyacetylene structures see Figures 1 and 2. level. c Half filled band.

6EY 25.56c 6.39 9.52 7.52 7.58

6EC

8.24 11.23 10.15 9.86

Position of the Fermi

72

Theoretical Chemistry

various polyacetylene models exhibit significant differences, which will be the subject of our further investigations concerning the optical and transport properties of these systems. Wavefunctions and Charge Distributions. Though the quality of the wavefunction obtained in a crystal orbital study cannot be assessed by direct comparison with experiment it is of decisive importance from the point of view of prospective transport calculations on conducting polymers (calculation of electron-phonon interaction matrix elements, optical properties, etc.). Of course, the wavefunction also plays a fundamental role when properties related to the many-electron energy are calculated, and therefore the quality of these quantities partially characterizes that of the wavefunction. Some features (e.g. symmetry properties) of the wavefunctions, on the other hand, may be very helpful in analysing certain trends in the structural properties of polymers independently of the actual quality of the wavefunctions. A nice example of this has been given by Yamabe et a1.66who have shown that the orbital patterns of the Bloch functions belonging to the highest filled band of the regular cis-polyacetylene provide a qualitative explanation for the relative stability of the cis-transoid structure. The same observations can also be made for the corresponding ab initio wavefunction. In spite of this qualitative agreement, however, the CND0/2 wavefunction fails to predict the correct atomic polarization in the polymer. We can see from the second column in Table 3 that the carbon atoms should have a net positive charge according to this method, contradicting not only the results of ab initio calculations cited in Table 3 but also general quantum-chemical experiences. We have to conclude that the semiempirical wavefunction used for calculation of these charge distributions is not accurate. Table 3 Net charges in the carbon atoms in various polyacetylene models obtained by diferent methods using a Mullikerr-type population analysis (in millielectrons) CNDOIZ

Polyacetylene model" Regular-trans (I) A1ternating-trans (11) Regular-cis 011) trans-Cisoid (IV) cis-Transoid (V) 0

calculation b -

-1.2 -4.5 -5.7

Ab initio calcirlation Ab initio calculation with balanced lattice with unbalanced lattice .wmsC sumsd 58.2 64 64 54 157 106

For the corresponding polyacetylene structures see Figures 1 and 2. Ref. 15.

56.7 61.5 57.7 59.4

* Ref. 66. c Ref.

20.

Polydiacetylenes. The polydiacetylenes (PDA's) are unique among highly conducting polymers discovered in the past years in that they can be obtained as highly perfect macroscopic single Upon solid-state polymerization of 68

G. Wegner, 2.Naturforschung, 1969, 24b, 824; G. Wegner, Makromol. CIiem., 1971,145, 8 5 ; 1972, 154,35; R. H . Baughman, 3. Polym. Sci., Polym. Phys. Ed., 1974, 12, 1511; E. P. Goodings, Chem. SOC.Reo., 1976,5,95.

The Electronic Structure of Polymers

73

single crystals of various substituted diacetylenes (RC-C-C-CR), nearly defect-free polymers with a fully conjugated backbone are formed. The planar backbones have a trans conformation and their electronic structure strongly depends on the nature of the side group R. The two extreme representations of the backbone bonding sequence are the acetylene structure [=RC-C=CCR=], and the butatriene structure [-CR=C=C=CR-In, respectively. The actually observed bond distances in PDA crystals fall between these two ideal structures. The experimentally most intensively investigated systems are PTS (R = CH2S03CsH4CH,)and TCDU [R= (CH2),0CONHC6H5],the former with an acetylene-like backbones0 while the latter is best represented by the butatriene The experiments performed on these and other PDA crystals involved photoconductivity measurements,71 Raman spectroscopic studies,72 investigation of the core- and valence-electron spectra by UPS and XPS,73 as well as by absorption and reflectivity rnea~urements.~~ While the experimental results on PDA's are fascinating the theoretical picture is rather confusing. The band gaps obtained for the same structure range from 0.5 75 to 11 eV,76even the signs of the net atomic charges are different (without taking into account the magnitudes), the band positions and widths do not even resemble each other, e t ~In. some ~ ~ cases a lucky parametrization of a semi-empirical method may reproduce reasonably a certain experimentally observed quantity (like the band gap in X , calculations) 'O but at the same time the predictions for other quantities (relative conformational energies, wavefunction properties, etc.) are considerably in error. Therefore, we would also like to keep level with present day sophistication in experiments in the case of PDA's. A priori computational methods are needed which do not use any empirical parameter. The first step in this direction has been made by KertCsz, Koller, and A ~ m a n Their . ~ ~ ab initio crystal orbital calculations still contain, however, the methodological inaccuracies mentioned in Section 2 under the heading 'TruncaN

N

789

69 70

71

72

73

74

75 76

77 78 70

D. Kobelt and E. F. Paulus, Acta Cryst., 1974, B38, 232. A. Enkelmann and J. Lando, Acta Cryst., 1978, B34,2352. B. Reimer and H. Bassler, Phys. Stat. Sol., 1975, A32, 435; B. Reimer and H. BPssler, Chem. Phys. Lett., 1976,43, 81 ; R. R. Chance and R. H. Baughman, J. Chern. Phys., 1976, 64,3889; R. R. Chance, R. H. Baughman, R. J. Reucroft, and K. Takahashi, Chem. Phys., 1976, 13, 181; H. Muller, C. J. Eckhardt, R. R. Chance, and R. H. Baughman, Chern. Phys. Lett., 1977, 50, 22. R. H. Baughman, J. D. Witt, and K. C. Yee, J. Chem. Phys., 1976, 60, 4755; J. Iqbal, R. R. Chance, and R. H. Baughman, J. Chem. Phys., 1977,66, 5520. D. Bloor, G . C. Stevens, P. J. Page, and P. M. Williams, Chem. Phys. Lett., 1975, 33, 61; J. Knecht, B. Reimer, and H. Bassler, Chem. Phys. Lett., 1977, 49, 327, G. C. Stevens, D. Bloor, and P. M. Williams, Chem. Phys., 1978, 28, 399. D. Bloor, D. J. Ando, F. H. Preston, and G . C. Stevens, Chem. Phys. Lett., 1974, 24,407; D. Bloor, F. H. Preston, and D. J. Ando, Chem. Phys. Lett., 1976, 38, 33; D. Bloor, Chem. Phys. Lett., 1976, 42, 174; C. J. Eckhardt, H. Miiller, J. Tylickis, and R. R. Chance, J. Chem. Phys., 1976, 65, 4311; R. R. Chance, R. H. Baughman, H. Muller, and C. J. Eckhardt, J. Chem. Phys., 1977, 67, 3616. D. E. Parry, Chem. Phys. Lett., 1977, 46, 605. M. Kerttsz, J. Koller, and A. Aiman, Chem. Phys. Lett., 1978, 56, 18; M. Kerttsz, J. Koller, and A. Aiman, Chem. Phys., 1978,27,273. For a recent review see: S. Suhai, Chem. Phys., 1980, 54, 91. D. S. Boudreaux, Chem. Phys. Lett., 1976, 38, 341. D. S. Boudreaux and R. R. Chance, Chem. Phys. Lett., 1977,51,273.

74

meoretical Chemistry

tion of Infinite Lattice Sums’ and this fact makes it difficult to analyse the physical properties of PDA’s on the basis of their wavefunctions. To elucidate the differences in the electronic structures of various PDA backbones, four models have been investigated recently at the minimal basis ab initio CO level using properly converged lattice sums.77In these calculations, which will be reviewed briefly here, the PDA side groups have been substituted by H atoms. Models I and IV were the two previously mentioned ideal structures with acetylene- and butatriene-like bonding sequences, respectively. The building unit of these polymers and the corresponding atomic distances used for the calculations are shown in Figure 6 and Table 4, respectively. The other two backbone

’ *c

Figure 6 Segment of a typicalpolydiacetylene (PDA) backbone. The side chain groups are substituted by hydrogen atoms and the translationally invariant unit cell is surrounded by broken lines. The carbon-carbon bond distances vary for diferent models as defined in Table 4

Table 4 Atomic distances and chain repeat lengths in the four investkated P D A backbones” PDA model Ideal acetylene (I) FTS backbone (11) TCDU-backbone (III) Ideal butatriene (IV) a

Rc1,ca

Rcz. c3

Rc4.c1

Chain repeat

1.45 1.43 1.38 1.35

1.20 1.21 1.24 1.26

1.34 1.36 1.42 1.48

4.91 4.89 4.87 4.87

See ref. 79; the numbering of atoms is defined in Figure 6; all distances are given in A.

structures (Models I1 and 111) were taken from PTS and TCDU crystals, respectively. We can see that the changes in bond lengths between the four structures are continuous; thus they cover nicely all the regions of interest. These variations in the bond lengths are, of course, consequences of the differences between the side chain groups, and therefore the influence of these groups on the backbone electronic structure is partly reflected by them. We disagree, however, with theoretical efforts which try to explain h e differences in the physical properties of different PDA’s (e.g. the observed blue shift in the optical spectra of ETCD by 0.3 eV on going from the acetylene backbone to the butatriene one)74on the basis of these. bond length variations alone. In our opinion the changes in

The Electronic Structure of Poiymers

75

chemical structure and also in the geometrical configuration of the side groups have to be explicitly taken into account in any such calculation. Conformational Properties. The total energy per elementary cell also changes dramatically as a function of the number of interacting cells N in PDA’s in the region of the first two to three neighbouring units (over which the electron distribution of the reference cell is delocalized as well) and it saturates only very slowly in the region of N = 10-16 [the function Etot.(N) has the same shape as shown for the case of polyacetylenes in Figure 31. The energy differences between the various conformers are again more interesting. In Figure 7 we present the values of the functions E z . (i) =E& (i) - E& (IV), (i = I, 11,111) for N = 1 ,2, . 16. As we can see these relative energies converge faster with respect to N than the absolute ones but the values for N = 8-10 are still quite unreliable and those for N = 1-2 even predict the energetic order of the different structures incorrectly (see also Table 5).

..

( kcaVmol1

10 -

0-

-10

-

-20

-

-30-

-40 *

-501

,

. . . , 5

. . . .

, 10

,

,

.

, .

,

15 N

Figure 7 Relative total energy per C4Ha unit as function of the number of interacting units ( N ) . As reference level for each value of N,the corresponding total energy of model IV has been chosen. : model I, 0 : model 11, A : model III, A : model IV

Theoretical Chemistry

76

Table 5 Total energies per diacetylene unit in four diferent P D A backbones (model I to IV as defined in Table 4) obtained with balanced lattice sum truncation using one and sixteen interacting neighbours ( N = 1, N = 16), respectively (values from reJ 77) Model I

I1 I11 IV

EE,'I

E 3 kcal mol-l

hartree - 149.033 03 - 149.030 64 -149.02021

EN I e=l .l 6 /

kcal mol-1

- 150.576 22

6.1 7.6 14.2 0.

- 149.042 78

E Nt o=t 1. 6 /

hartree - 150.573 54 -150.55526 150.543 44

-20.6 -18.9 7.4

0

In summary, we can state that predictions concerning stability problems, energy changes due to phase transitions, etc. in PDA's (assuming a level of significance of 1 kcal mol-l) can be made on the basis of apriori HF CO studies if 8-10 neighbouring PDA units are correctly treated in the crystal. The relative stability of the PDA backbone increases by -20 kcal mol-l per diacetylene unit as the bonding sequence goes over from the ideal butatriene structure to the acetylene one. Prediction of the corresponding energy difference in the case of a TCDU to PTS transition is 11 kcal mol-l if only the structural changes in the backbone are taken into One-particle energies. These quantities can be calculated again quite reliably with a few interacting neighbours. The ionization potentials, electron affinities, and forbidden gaps are shown in Table 6 for an electrostatically balanced calculation (N= 16). As we can see all these quantities have converged practically to the third unit.

-

Table 6 Ionization potentials (IP), electron affinities (EA), and fundamental gaps (AE,) of four PDA backbones (model I and IV as dejned in Table 4) calculated with the correct lattice sum truncation using sixteen interacting neighbours ( N = 16, ref. 77). All quantities in eV Model I I1 I11 IV

IP 5.63 5.22 3.94 3.63

EA 3.84 3.44 2.11 1.76

A& 9.47 8.66

6.05 5.39

The ionization energies obtained can be reasonably compared with experiment in the following way. From PDA cluster calculations we have learned 8o that the valence levels obtained in the STO-3G basis have to be shifted downwards by 1.5 eV to mimic the effect of extension of the atomic basis beyond the minimal level. Applying the same correction to the valence bands obtained in CO calculations, the upper edges of the n-type valence bands for the PTS-backbone (Model 11) and for the TCDU-backbone (Model 111) will lie at -6.7 and -5.5 eV, respectively, while the corresponding experimental values are - 7.3 k 1 and - 6.6 k 1 eV, respectively.

-

S. Suhai and J. Kaspar, unpublished results.

77

The Electronic Structure of Polymers

No such comparison is meaningful in our opinion for the forbidden gaps at the HF level since they are extremely sensitive to correlation effects. Another problem is that to reproduce theoretically the observed gap differences between various substituted PDA chains (ranging from 0.3 to 1 eV) it is necessary also to take explicitly into account the side chains since the gaps seem to be very sensitive to structural (i.e. potential changes). We can see from the last column in Table 6 that the minor bond length changes between Models I and I1 as well as between 111and IV produce gap changes of -0.6-4.8eV, respectively (at the HF level). The energy band structures are qualitatively very similar for all four PDA backbones. The four core bands of practically zero width are situated around -299 eV. The other nine doubly occupied bands lie in the region of - 4 to - 3 0 eV. Both the highest filled and lowest unfilled (valence and conduction) bands have TC symmetry and both are crossed by the nearest o-bands. These band dispersions are also relatively stable against various approximations and some quantitative differences between them may play an important role in transport calculations on these polymers : (i) With correct truncation it has been found77that the valence band width increases monotonically on going from Model I to IV: 6Ev= 3.33, 5.12, 6.18, and 6.20 eV. (ii) A similar trend is obtained for the conduction bands:776Ec=5.11, 5.42, 6.56, and 6.68 eV.

- -

Charge Distribution. In Table 7 the net atomic charges on the two carbon atoms (not related by symmetry) and on the hydrogen are shown for all four PDA structures studied. These charges have been calculated again with a Mullikentype population analysis of HF Bloch functions obtained with an electrostatically balanced (N= 16) 7 7 truncation procedure. We found that the charge distributions are also very sensitive to the method of truncation as well as to a proper convergence with respect to N.

Table7 Net atomic charges on the carbon and hydrogen atoms in four PDA backbones (model I to IV as defined in Table 4 and Figure 6 ) calculated with correct lattice sum truncation using sixteen interacting neighbours ( N = 16, ref. 77) (in millielectrons) Model I I1 I11

IV

C 1

cz

39.9 41.8 46.1 54.8

46.5 44.9 39.2 27.4

H -86.4 -86.7 -85.3 -82.2

Infinite Stacks of TCNQ and TTF Molecules.-The quasi one-dimensional charge-transfer molecular crystal TCNQ (7,7',8,8'-tetracyanoquin0dimethane)TTF (tetrathiofulvalene) has received considerable attention in the past decade because of its interesting solid-state physical properties. In recent publications 81 81

J. Ladik, A. Karpfen, G. Stollhoff, and P. Fulde, Chem. Phys., 1975, 7, 267; A. Karpfen, J. Ladik, G. Stollhoff, and P. Fulde, Chem. Phys., 1975, 8, 215; S. Suhai, J. Phys. C, Solid State Phys., 1976,9, 3073; R. D. Singh and J. Ladik, Phys. Lett., 1978,65A, 264; S. Suhai, J. Phys. C, Solid State Phys., submitted.

78

Theoretical Chemistry

we have given references to review papers on the experimental work done on highly conducting TCNQ-TTF systems, on the semi-empirical and ab initio SCF LCAO MO calculations on the cons,ituents of these stacked chains (monomers, dimers, and a TCNQ-TTF pair), and have presented different all-valence-electron semi-empirical band structures of the neutral stacks. Here we summarize briefly the main results of an ab initio HF CO calculation on infinite neutral TCNQ and TTF stacks.82These ab initio Hartree-Fock band structures serve on the one hand as starting points for the calculation of some transport properties of neutral TCNQ and TTF columns, which are of great interest themselve~.~~ On the other hand, with the help of the wavefunctions and band structures obtained for these systems, further calculations become feasible which will take into explicit account the amount of transferred charge between the TCNQ and TTF columns. The calculations have been performed on the second neighbours’ interactions approximation using the STO-3G basis set,l*and for stacked chains the geometry found in the mixed TCNQ-TTF crystala4 (3.18 8, interplane distance in the TCNQ and 3.47 A in the TTF stack, respectively) has been applied. In Table 8 we show the positions and widths of the valence and conduction bands of poly(TCNQ) and of poly(TTF). The most interesting result obtained in this study was that the valence band of poly(TTF) [from which charge transfer (CT) occurs] is comparatively broad (-0.3 eV) and the conduction band of poly(TCNQ) (to which the charge is transferred) is broad (- 1.2 eV), while both the valence band of poly(TCNQ) and the conduction band of poly(TTF) (which do not take part in the CT process) have widths less than 0.1 eV. It is worthwhile to note that the positions of the conduction band of poly(TCNQ) and of the valence band of poly(TTF), respectively, favour the CT process more than LEMO and HOMO, respectively, in the corresponding single molecule (E + E $ ~ =4.212 eV, &cqnd. m,n,TmQ - Eval. max,nF=3.011 eV). This may throw some light on the fact that

~s~p~

neither theoretical (ab initio SCF LCAO MO calculations on a TCNQ-TTF molecular pair with realistic structural data modelling in the mixed crystals),86 nor actual measurements on aqueous solutions containing both TCNQ and mF molecules as indicated CT, which seems to be a purely solid-state physical effect in the mixed crystal (-0.6 eV per TCNQ-TTF pair).*’ We are aware of the fact, of course, that the mechanism of CT is very complicated in these systems and no simple orbital energy consideration could explain it. Further studies on interacting chains are desirable, therefore, to elucidate this problem. The most important conclusions will hold, however, if interchain interactions are taken into account, namely: (1) the conduction and valence bands of both systems are site dispersions, 82

83 84 85

86

87

7c

bands with oppo-

S. Suhai and J. Ladik, Phys. Lett., in press. S. Suhai, to be published. T. J. Kistenmacher,T. E. Phillips, and D. 0. Cowan, Acta Cryst., 1974, 33, 76. F. Cavallone and E. Clementi, J. Chem. Phys., 1975, 63, 4304. J. J. Andre, in ‘Recent Advances in the Quantum Theory of Polymers’, ed. J.-M. Andrk, J.-L. Bredas, J. Delhalle, J. Ladik, G. Leroy, and C. Moser, Springer-Verlag, Berlin, 1980, p. 35. F. Denoyer, R. Comes, D. F. Garito, and A. J. Heeger, Phys. Reo. Lett., 1975,35,445.

Q"

Table 8 Valence and conduction bands of poly(TCNQ) and poly(TTF) and their widths (in ev). For comparison the Table contains also Q the locations of the corresponding MOs 2 2,

Valence band EM0

poly(TCNQ) POlY(TTF)

-6.838 -3.774

e l %

- 7.252(n)a -3.801(~)

In parentheses the corresponding k.5 values.

e

x

-7.157(0) -3.498(0)

BE

0.095 0.303

,=* 0.438 8.466

Conduction band

G?Ii

-0.487(0) 8 . 5 1 l(n)

el%

0.687(n) 8.594(0)

BE

1.174 0.083

80

Theoretical Chemistry

(2) bands which participate in the CT process are relatively broad and that the valence or conduction band, respectively, which plays no role in CT is very narrow, and finally (3) the position of the conduction band in poly(TCNQ) and of the valence band in poly(TTF) still favours more CT than the corresponding LEMO and HOMO levels. Periodic DNA Models.--Owing to the central role of DNA in biochemistry and biophysics the electronic structures of various periodic polymers built from nucleotide bases and base-pairs have excited much theoretical interest. Most calculations have been performed at the n-electron level using the PariserParr-Pople (PPP) CO method,88 which has proved to be very useful in the description of those properties of planar conjugated molecules which are related mostly to the n-electron system. The main results obtained with this approximate CO method are still serving, however, as a guide in more sophisticated calculations. In the investigation of periodic DNA models the structural data of DNA B have been The energy band structures obtained for these systems with the aid of the PPP CO method can be divided into two groups. The five homopolynucleotides (also including polyU) and the two poly(base pairs) poly(A-T) and poly(G-C) have rather broad bands (the widths of the valence bands are 0 . 2 4 . 3 eV, those of the conduction bands are 0.1 eV, and the widths of the lowest filled bands are in most cases - 1 eV). On the other hand for the more complicated periodic DNA models, where there are two different base pairs in the unit cell, like poly&E) etc. the bands are very narrow (the widths of the valence bands are usually 0.01-0.03 eV, those of the conduction bands are 0.01 eV, and the widths of the lowest filled bands are -0.1 eV).88To take into account the effect of the other valence electrons as well, CND0/2 CO calculations have also been performed for the homopolynucleotidesgOand for the sugar-phosphate (SP) chain of DNA.g1These resulted in valence band widths of 0.15-0.50 eV and in conduction band widths of 0.1-0.25 eV for both kinds of system. As the next step in the systematic investigation of the electronic structure of DNA, minimal basis ab itzitio band structure calculations were performed for the four nucleotide base stacks, for the sugar-phosphate (polySP) chain, and for a whole cytosine-sugar-phosphate (polyCSP) chain. In a preliminary calculation on polycytosinea2each orbital of the heavy atoms was expanded into two Gaussians, while in later studies, g3 which we will review here, an STO-3G basis set l4 was applied. In the case of polySP and polyCSP, instead of a Kf ion, a proton was attached to the PO- group, thus keeping these chains neutral. In all calculations the second neighbours' interactions have been included with a correct (electrostatistically balanced) cutoff. Table 9 contains the characteristics of the valence band and conduction band of the four nucleotide base stacks, while 88

go g1 92 88

For a review see: J. Ladik, Ado. Quantum Chem., 1973, 7 , 397. M. Spencer, Acta Cryst., 1959, 12, 66; S. Arnott, S. D. Dover, and A. J. Wonacott, Acta Cryst., 1969, BE, 2192. S. Suhai and J. Ladik, Int. J. Quantum Chem., 1979, 7 , 547. S. Suhai, Biopolymers, 1974, 13, 1799. S. Suhai, C. Merkel, and J. Ladik, Phys. Lett., 1977, 61A, 487. J. Ladik and S. Suhai, Int. J, Quantum Chem., in press.

@2 a

Table 9 Limits and widths of the valence and conduction bands of the four nucleotide base stacks (in eV). For comparison the table 3 5 also contains the locations of the corresponding MOs 2 Chain PolyC PolyT PolyA PolyG

EM0

-5.61 -6.66 -6.10

-5.08

Valence band

egyn(kming

- 5.51 - 6.48(0) -6.04(0)

(j~)

-5.16(0)

Ez:x(kmsxa)

-4.65(0)+

8.9

- 5.88(n) -5.57(n)

0.86 0.60 0.47

-4.34(n)

0.82

&*O

6.00 5.97 6.46 6.45

Conduction egyn(krninZ) 6.07(0) 6.02(n) 6.56(n) 6.41(n)

0,

band

6.91(n) 6.33(0) 6.86(0) 7.15(0)

5

6~

>x(kmaxg

%

0.84 0.31 0.29 0.74

ki

.1"

$

x

Table 10 Limits and widths of the valence and conduction bands of polyC and polySP and of the two highest filled and lowest unfilled bands of the polycytidine chain (in ev) PoIyC &g?n(kmina) &g:x(kmaxa)

Conduction band Valence band

6.07(0) -5.51(~)

6.91(n) -4.65(0)

Poly CSP

Polj6P 6e

0.84 0.86

ez:n(kming

7.43(0) -6.44(~)

E:;Jkmsx@

S.OO(n)

-6.28(0)

E ~ ~ n ( k m i n& ~ )~ ~ x ( k m a x ~ )

0.57 0.16

7.40(0) 6.55(0) -5.19(n) -6.79(n)

7.96(n) 7.38(n) -4.36(0) -6.71(0)

d&

0.56 0.83

0.83 0.08

Theoretical Chemistry

82

Table 10 gives the same information for the polySP and polyCSP chains. In Table 9 all the valence and conduction bands of the four nucleotide base stacks originate from the 7c HOMO and LEMO levels of the constituent molecules. Since the SP and CSP units are not planar, such a classification in the case of polySP and polyCSP chains is not possible. Looking at Tables 9 and 10 one can see that the valence and conduction bands of the stacked bases and of the polySP chain are several tenths of an eV in width (values between 0.16 and 0.86 eV) indicating that there is the possibility of a Bloch-type conduction in these systems if free charge carriers are generated in them. On the other hand, the gap in all cases is more than 10 eV. Although one knows that the Hartree-Fock calculation gives too large a gap for conduction, this rules out the possibility of intrinsic semiconduction in DNA. Furthermore, our calculation on the polyCSP superchain has resulted in a CT of 0.187e per molecule pair from the sugar-phosphate unit to cytosine. Although in a restricted Hartree-Fock superchain calculation because of the method one only obtains completely filled bands, if there is considerable CT from one chain to the other, the calculated CT indicates the possibility of creation of free charge carriers in the system. The valence band of polySP is, according to this calculation, rather narrow (0.16 eV) but one should bear in mind that, as previous n electron band structure calculations have indicated,a4the presence of positive ions may increase the band widths to a significant degree (in some cases by a factor of 2 or 3). Therefore, if the polySP and polyCSP band structure calculations were repeated (work is in progress) assuming the presence of Kfions around the PO; groups and not putting a proton chemically bound to them, one would expect a large broadening of the valence band of polySP. It should be further noted that if a superchain calculation is performed, as we did in the case of polyCSP, the conceptual difficulty is encountered that independent of how large is the CT from one chain to the other (from one molecule to the other in the supermolecule) in the restricted Hartree-Fock framework two bands are always obtained (the valence band and the band below it) which are completely filled. In this way one would always conclude that the system is an insulator, though as we know, for instance, from the case of the mixed TCNQTTF crystal, the individual molecular stacks may become metallic conductors. One possibility of avoiding this conceptual difficulty would be to perform a different orbitals for different spins calculation for the superchain, which would probably result in the correct conduction properties for these interacting chains with CT (the Fermi level would be inside a band). The simpler approach is to take the amount of transferred charge per pair of units in the interacting chains from a molecular calculation (possibly not a simple restricted Hartree-Fock calculation of the corresponding supermolecule, but from a MCSCF calculation which allows for interactions of the AD and A-Df configurations, where A stands for the acceptor and D for the donor) and populate the energy bands of the single chains according to this amount of CT.In the case of the TCNQ-TTF system this approach would not work, because the CT in this case seems to be a purely solid-statephysical effect,but for the CT between the sugar-phosphate and

- -

94

B. F. Rozsnyai and J. Ladik, J. Chenr. Phys., 1970,52,5711; 1970,53,4325.

The Electronic Structure of Polymers

83

base pair regions of DNA it probably would work. Calculations of this type are in progress. Finally, we point out that CT from the sugar-phosphate (SP)unit to the cytosine (C) molecule cannot be explained by the naive HOMO (D)-LEMO (A) picture, because the valence band of polySP (see Table 10) lies below the valence band of polyC by about -0.8 eV. The same is true if one compares the upper limits of the valence bands of the three other nucleotide base stacks (see Table 9) with the upper limits of the valence band of polySP. Periodic Protein Models.-There is increasing evidence that the electrical properties of proteins may play an essential role in their biological functions.95According to Szent-Gyorgyi's theory 96 the possibility of electron transport through these macromolecules may also be closely related to the problem of cancer. Since the early suggestions of Szent-Gy6rgyiQ7and of Lakig8 concerning the possibility of semiconduction in proteins, a number of theoretical investigations have been devoted to determination of the energy band structuresgs and of possible pathways for electron delo~alization~~ in these systems. The basic common feature of these calculations was that the proteins were represented by simple periodic models. The elementary cell of these models consisted either of the four atoms of the planar peptide unit (taking into account only the nelectrons of the resulting hydrogen-bonded chain or all the valence electrons forming a polyformamide chain) or of a glycine residue forming a polyglycine chain. Usually semi-empirical methods were used in the latter case, with the exception of one ab initio all-electron calculation.99 Since proteins build two-dimensional networks with simultaneous interactions between the peptide units along the protein backbone and forming hydrogen bonds perpendicular to it, the strictly periodic two-dimensional polyformamide network has been investigated as the simplest fairly realistic model for a protein in which formamide molecules are chemically bonded in one direction (main chain) and are hydrogen-bonded in the perpendicular direction.loO The calculations have been performed with both the CND0/2 and MIND0/2 CO methods. When the interactions were taken along the hydrogen bonds the widths of the valence and conduction bands were 0.2 and 0.3 eV, respectively, in the CND0/2 case, whereas the MIND0/2 CO calculations provided smaller widths (-0.1 and -0.05 eV, respectively). On the other hand, if the interactions were taken along the main chain, the corresponding band widths were much larger [width of the valence band (66,) -2.6 eV, width of the conduction band (dec) -0.6 eV in the CNDO/2 CO case, whereas the MIND0/2 CO results were de, -0.9 eV, d~~ -2.5 eV]. Finally, taking into account both types of interactions, the CND0/2 CO results for the two-dimensional polyformamide

-

N

A. Szent-Gyorgyi, Bioenergetics, 1973, 4, 535 ; A. Szent-Gyorgyi,Acta Biochim. Biophys. Acad. Sci. Hung., 1973, 8, 177; R. Pethig and A. Szent-Gyorgyi, Proc. Natl. Acad. Sci. USA, 1977, 74, 226; S. Bone, T. F. Lewis, R. Pethig, and A. Szent-Gyorgyi, Proc. Natl. Acad. Sci. USA, 1978,75,315. 0 6 A. Szent-Gyorgyi, Int. J. Quantum Chem., 1976, QBS3,45. 9 7 A. Szent-Gyorgyi, Nature, 1941, 148, 157. 9 8 K. Laki, Studies from the Inst. Med. Chem. University Szeged, 1942, 2,43. 99 For a recent review see: S. Suhai, T. C. Collins, and J. Ladik, Biopolymers, 1979,18,899. 100 S. Suhai and J. Ladik, Theor. Chim. Acta (Berlin), 1972,28,27. 95

84

Theoretical Chemistry

-

network were dcv -2.6 eV, dcc -0.8 eV and the MIND0/2 CO ones asv -2.0 eV, 6eC 1.6 eV. Later the MIND0/2 CO calculation was repeated for the two-dimensional parallel-chain pleated sheet B-polyglycine network (two glycine molecules in the unit cell) using second neighbours’ interactions.lol These resulted in values of 6 E~ 1.2 eV, 6 E~ 1.7 eV and a forbidden band width of 4.8 eV. It should be mentioned that while in the case of the polyformamide we had planar structures and could therefore define n- and a-electron bands (in both cases the valence band was a n- and the conduction band a a-band), this classification becomes impossible in the case of the parallelchain pleated sheet conformation of ppolyglycine because it is not planar. Recently ab initiu SCF LCAO CO band structure computations have also been performed on polygly~ine.~~ For the geometry of the polypeptide chain the same conformation was used, as the chains are in the parallel-chain p-pleated sheet conformation and the basis set was a Gaussian lobe expansion of the minimal basis proposed by Mely and Pullman.1o2The Hartree-Fock band structures obtained were corrected for long-range correlation effects using the formalism of the electron polaron Table 11 contains the results obtained for the main polypeptide and the hydrogen-bonded chains of polyglycine. As we can see from Table 11 in the case of the main chain the bands are rather broad (the width of the valence band is 2.1 eV and that of the conduction band is 1.4 eV), while the bandwidths of the hydrogen-bonded chain are one order of magnitude smaller. These results agree qualitatively with those of the semi-empirical all-valence-electroncalculations discussed above. The long-range correlation correction decreases the bandwidths by about 10%. The Hartree-Fock gap for the main chain is 12.4 eV (which decreases to 11.6 eV due to the long-range correlation), while for the hydrogen-bonded chain its value is (due to the smaller bandwidths) still larger ( 14.8 eV). Owing to this large gap intrinsic semiconduction in proteins seems to be negligible and the role of various impurities becomes i m p ~ r t a n t . ~ ~ Impurity and Aperiodicity Effects in Polymers.-The presence of various impurity centres (cations and water in DNA, halogens in polyacetylenes, etc.) contributes basically to the physics of polymeric materials. Many polymers (like proteins or DNA) are, however, by their very nature aperiodic. The inclusion of these effects considerably complicates the electronic structure investigations both from the conceptual and computational points of view. We briefly mentioned earlier the theoretical possibilities of accounting for such effects. Apart from the simplest ones, periodic cluster calculations, virtual crystal approximation, and Dean’s method in its simplest form, the application of these theoretical methods [the coherent potential approximation (CPA),lo3Dean’s method in its SCF form,51 the Hartree-Fock Green’s matrix (resolvent) method, etc.] is a tedious work, usually necessitating more computational effort than the periodic calculations

-

-

-

-

N

-

N

-

Suhai, Theor. Chim. Acru (Berlin), 1974, 34, 157. B. Mely and A. Pullman, Theor. Chim. Acta (Berlin), 1969, 13, 278. loS P. Soven, Phys. Rev., 1967, 156, 809; R. J. Elliot, J. A. Krumhansl, and P. L. Leath, Reo. Mod. Phys., 1974, 46,465; M. Seel, T. C. Collins, F. Martino, D. K. Rai, and J. Ladik, Phys. Rev. B., 1978, 18, 6460. lol S. lo2

Table 11 Valence and conduction bands of polyglycine calculated with the ab initio SCF LCAO CO method (in ev). The bands belonging to the main chain are corrected for long-range correlation effects (numbers in parentheses) Polyglycine main chain ~~~n(kmin~)

Conduction band Valence band

3.817(3~/8) (3.102) - 11 .252(0) (-10.373)

(kmaxa)

5.195(n) (4.357) -9.154(~/2) (- 8.465)

6.5

1.378 (1.255) 2.098 (1.908)

Polyglycine H-bondedperpendicular chain ~~~x(kmax~) 88 3.755(0) 3.896( n) 0.141

csn(kmjn$

-11.382(~)

-11.091(0)

0.291

86

Theoretical Chemistry

themselves (on the other hand they depend heavily on the quality of the description of the periodic or host crystal itself). Periodic Cluster Calculations. In this method the elementary celI(s) of the host crystal and the impurity atom(s) are treated together as a cluster. This cluster is periodically repeated and if it is large enough compared with the size of the impurity itself the method provides a fairly realistic description of impurity levels, changes in the electron density around the impurities, etc. We present here two applications of this procedure to DNA and protein chains. Eflect of Water and Divalent Metal Ions on the Electronic Structure of DNA. As a first step in investigating the effect of impurities on the electronic structure of DNA, the band structure of poly(G-C) was computed in the PPP CO approximation assuming that one or two water molecules are bound by hydrogen bonds to the NH2 group of each C molecule orland to the C=O group of each G molecule.Q4According to the results obtained for these systems the additional n-orbital in the H 2 0 molecules produced an extra n-band between the lowest filled bands, while the other bands remained practically unchanged. In the next step the band structures of poly(G-C) and poly(A-T) were recalculated in the presence of Mg2+ions again using the PPP CO method modified suitably to account for the presence of charged (The ability of divalent metal ions, especially Mg2+ions, to react with a variety of electron-donor sites in polynucleotide chains was demonstrated experimentally.lW) Calculations have been performed for the poly(G-C) and poly(A-T) systems with all possible types of Mg2+ attachment to the heterocyclic bases (including both in the plane of the base pairs and the out of plane positions), taking one Mg2+per unit cell. According to the results obtained for these model calculations the presence of Mg2+ions drastically changes the band structures: the bands become generally broader by a factor 2-3 and their positions also change considerably. This causes great changes in the band gap: for instance if the Mgz+ion is attached to the NH2 nitrogen of C in the G-C base pair it decreases from 6.0 to 2.0 eV. This means that the exciton band (which always lies below the conduction band) will probably overlap with the valence band in this case, which may give rise to the possibility of a phase transition to the excitonic insulator state. Efect of Di‘erent Side Groups on the Electronic Structure of Proteins. For these studies a polypeptide backbone from the chain of the /%pleatedsheet configuration has again been used.46Figure 8 shows the structure of this system. The ‘host polymer’ is polyglycine [poly(Gly]) in which all three side groups are hydrogens (R1 = R2= R3= H). In real proteins, however, these groups have different chemical structures. In the periodic cluster calculations of ref. 46 they were chosen according to the amino-acids Gly (R1=H), Ala (R1=CH3), and Ser (R1= CH,OH). From the combination of these side groups large elementary cells (clusters) were built containing three amino acid residues and these were repeated periodically. For the resulting polymers MIND0/3 CO calculations were performed. As an example we show in Table 12 the positions of the six most important bands in the composite polymer poly(G1y-Ala-Ser). For comparison, in Table 13 the valence and conduction bands of the three ‘pure’ polylo‘ G . L. Eichhorn and Y. A. Shin, J. Am. Chem. SOC.,1970,90, 7323.

m e Electronic Stucture of Polymers 1 I I

R'

0

I

I

I'1

Ll

RiaiirP FI Plomontnrv unit ----- .Qriirtiiro n f tho .-.'..'-".-', -I.--.-.-

v,

. . I

R3

--

of

-..I. "J

",._.,.

tho y"-,Y.y'.-I nnlvnontido rhnin with rid0

I .

,...I"

I

R2 and R3

urnimr * '"..F"

""-1

R1,

. A

Table 12 Energy parameters of six bands in the poly(G1y-Ala-Ser) mixed polymer lying in the energy region of the conduction and valence bands of the pure systems (all quantities in ev) Type of banda n*+3 n*+2 n*+l n* n*-1 n*-2

Band minimum 0.9933 0.6071 0.3347 -9.2669 -9.5311 - 9.7833

Band maximum 1.1887 0.9462 0.4696 -9.2650 -9.3600 -9,6648

Bandwidth 0.1954 0.3391 0.1349 0.0019 0.1711 0.1185

a n * and n* -C 1 ctanrl fnr the valence and cnndiictinn hands resnectivelv

poly(Ala), and poly(Ser) (all quantities in eV) System POlY(G1Y)

Poly(A1a) Poly(Ser)

Type of banda n*+ 1

n* n*+ 1 n* n*+l n*

Band minimum Band maximum 1 .lo40 0.2755 - 9,9196 -9.4242 0.4723 1.2454 -9.6414 -9.1552 0.1970 1.1754 -9.7440 -9,2480

Bandwidth 0.8285 0.4954 0.7731 0.4862 0.9784 0.4960

an* and n* -e 1 stand for the valence and conduction bands. resDectiveIv.

peptides are also presented. We can see that the effect of the side group potentials is quite moderate: the bands are shifted but their overall properties are not substantially different. However, small deviations are enough to produce drastic effects in the electronic structure of the composite system. It can be seen that owing to the presence of three different side groups in the elementary cell both the valence and conduction bands (as well as other bands) are split into new bands separated by gaps. This 'disorder' effect of the side groups is shown graphically in Figures 9 and 10, where the density of electronic states defined by

Theoretical Chemistry

88

r-

- 9.8

I -

I

I

- 9.6

- 9.4

.Elev 1

Figure 9 Density of electronic states plotted in the valence band region of the poly(G1yAh-Ser) polymer (in relative units)

.4

.6

.8

1.

Figure 10 Density of electronic statesplotted in the conduction band region of thepoly(GIyAh-Ser) polymer (in relative units)

The Electronic StructureIof Polymers

89

can be seen in the valence and conduction band regions of the host polymer. In both regions two new energy gaps of considerable width develop and the new valence and conduction band widths are substantially smaller than the original values. The very important biological implications of these phenomena have been discussed in more detail in ref. 46. It should be mentioned that the appearance of these new forbidden gaps in the electronic spectrum of proteins is independent of the method of the CO calculation. Similar gaps have also been obtained, namely in ab initio calculations of composite polypeptide mode1s.lm Calculation of the Electronic Spectra of Aperiodic Protein Models Using the Negative Eigenvalue Theorem. To illustrate how Dean’s method4scan be applied to aperiodic polymers we present here the results of a calculation performed for two-component Gly(A)-Ala(B) model chains.4BThe Hiickel parameters a{ and bt appearing in equation (24) have been chosen to match the locations and widths of the valence bands of the corresponding periodic poly(G1y) and poly(A1a) chains obtained in ref. 46. The values aA= -9.6719 eV, bAA= -0.1239 eV, aB= -9.3983 eV, bBB= -0.1215 eV have been obtained. When in the aperiodic chains a B unit followed an A unit (or vice versa) ~ A =B 1/2 (bAA+ bBB) has been used. The random two-component chains have been generated employing a random number generator but keeping the composition prespecified.The sequence of numbers [ ~ ~ ( j l )has ] been computed for n=200 values of il in the energy region between -9.920 and -8.925 eV (step width 0.005 eV). The eigenvaIue spectra have been plotted by noting the numbers of negative E( values. Thus histograms give the number of energy eigenvalues (states) between Ed and Eg-l, the histogram interval being 0.005 eV. The computer time for calculation of a spectrum of a chain of length lo00 units is - 6 s on a Cyber 172, including the random chain generator (without the Monte Carlo routine - 5 s), and for the chain length 10 OOO units it is 42 s. Figures 11 and 12 show the eigenvaluespectra for randomly generated chains of 1OOO units length. The percentage of alanine residues varies from 1 % in Figure 1 l(a) to 50% in Figure 12(b) as indicated on the diagrams. The histograms exhibit a complex structure, a complicated system of peaks and valleys at the upper energy end of the spectrum starts to develop, the whole energy range spanned by periodic (A) and periodic (B) becomes covered with increasing concentrations of B, and the eigenvalue spectrum for the case of 50% Ala residues [Figure 12(b)] bears no resemblance whatever to the ordered chain. The explanation for the existence of the well-defined peaks at the upper energy end of the spectrum, some of which are denoted by the letters A to D, is the same as in the case of vibrational ~ p e c t r (for a ~ more ~ ~ ~ details ~ we refer to ref. 49). At low concentration of Ala units (Figure 11) the spectrum consists mainly of the well-known spectrum of the poly(G1y) valence band with some of the structure in the Ala region dominated by peak A due to isolated Ala residues. At 5 % Ala concentration, peak A increases in intensity. Also, peaks such as B and D which are due to clusters of BB and BAB appear. At 10%Ala concentration peak A reaches its maximum intensity in the computed spectra and then J. Ladik, in ‘Submolecular Biology and Cancer’, Ciba Foundation Symposium 67, Excerpta Media, Amsterdam, 1979, p. 58. l o 8J. Hori, ‘Spectral Properties of Disordered Chains and Lattices’, Pergamon Press, Oxford, 1968, p. 34. 105

Theoretical Chemistry

90 tJ 70

50

A

30 1

L

10

( b ) cg=O.05

70

B

D

A

D

B

50 n

30

10

Figure 11 Eigenvalue spectra for t wo-component (Gly-Ala) disordered chains containing loo0 units. The parameter C B refers to the fraction of Ala residues in the chain. The spectral lines Iabelled by the letters A to D are associated with particular local chain sequences. N is the number of eignevalues in an histogram interval of 0.005 eV

declines at the expense of secondary peaks: the probability of clusters of the form BB, BAB, and BBB (peak denoted by C) increases. At 30% [Figure 12(a)] the tertiary peaks due to three-Ala clusters are already quite pronounced and at 50 % [Figure 12(b)] they are as high as the primary and secondary peaks. At this concentration the spectrum becomes very complicated indeed, although it is clear that the identity of the individual peaks still holds. AcknowZe&mnt: The authors are indebted to Prof. T. C. Collins, G. Del Re, J. Ciiek, F. Martino and to Drs P. Otto and M. See1for many helpful discussions during the development and application of the methods described in this Report.

The Electronic Structure of Polymers

91

N

( a ) Cg"0.30

70 C

SO -

B,

D A D B

C

30-

10-

MJ)$Jl!,*":

Figure 12 Eigenvalue spectra for two-component (Gly-Ala) disordered chains of length lo00 units. 27ie parameter C B refers to the fraction of Ala residues in the chain. N is the number of eigenvalues in an histogram interval of 0.005 eV

J

Electron Density Description of Atoms and Molecules BY N. H. MARCH

1 Introduction The problem of calculating accurate ground-state wave functions for atoms, and to a lesser degree for molecules, is tractable with modern computers for relatively small numbers of electrons N. However, for large numbers of electrons, a different approach is clearly required. This is afforded by the density description. Instead of a wave function with 3N spatial co-ordinates, one works with the ground-state electron density p(r), where this is explicitly the number of electrons per unit volume at position r. This is evidently a three-dimensional quantity, independent of the number of electrons, and is therefore a favourable tool for really large molecules. That one could describe the ground-state of an electronic assembly by its electron density p ( r ) was known to the pioneers of the density description, Thomas' and FermL2 These workers described the ground-state of an atom, with nuclear charge 2, by treating the electrons as a completely degenerate Fermi gas. They recognized, of course, the essential inhomogeneity of the charge distribution, but they pointed out that for a sufficientlylarge number of electrons, N, it would become increasingly accurate to describe the electron cloud by applying free Fermi gas relations locally. It is therefore appropriate to begin this review by setting up the basic equations of the Thomas-Fermi statistical theory. 2 Density-Potential Relation of Thomas-Fermi Statistical Theory As already remarked, the idea underlying the Thomas-Fermi (TF) statistical theory is to treat the electrons around a point r in the electron cloud as though they were a completely degenerate electron gas. Then the lowest states in momentum space are all doubly occupied by electrons with opposed spins, out to the Fermi sphere radius corresponding to a maximum or Fermi momentum p f ( r ) at this position r. Therefore if we consider a volume dr of configuration space around I, the volume of occupied phase space is simply the product dr 4npf(r)/3.However, we know that two electrons can occupy each cell of phase space of volume hS and hence we may write for the number of electrons per unit volume at r,

1

L. H. Thomas, Proc. Camb. Philos. SOC.,1926, 23, 542. E. Fermi, Z.Physik, 1928,48, 73.

92

Electron Density Description of Atoms and Molecules

93

This is the first basic relation of the TF theory. It amounts to taking the result for a uniform Fermi gas and applying it locally at r. Next we write down the classical energy equation for the fastest electron at r, namely that the Fermi energy ,u is given by

This equation defines the Fermi energy in terms of the kinetic energy of the fastest electron, m being the electronic mass, and the potential energy V(r) in which the electrons move. It is very important to note that whereas in equation (2) the two terms on the right-hand side both depend on position r, the sum p must be independent of r. This is easy to understand, for if neighbouring regions of the electronic cloud p ( r ) had different Fermi energies, then electron redistribution could occur to lower the energy. When we discuss the variational principle for the TF theory later, we shall see that the Fermi energy can be identified with the chemical potential, as already anticipated by the notation ,u used in equation (2). Therefore, equation (2) expresses the constancy of the chemical potential throughout the atomic or molecular charge cloud. One can now eliminate the Fermi momentum p f ( r ) between equations (1) and (2) to obtain the density-potential relation of the T F statistical theory

This equation (3) is to be used provided ,u- V(r)2 0, while p (r)is zero if p - V(r) GO. This is readily recognized to be a condition stemming from the semiclassical nature of the TF theory. Electrons are not allowed to occupy regions of negative kinetic energy, i.e. there are no electrons in classically forbidden regions. Self-consistent Fields for Heavy Positive Atomic Ions.--Let us immediately turn to the use of equation (3) to establish the self-consistent field in a heavy atomic ion with nuclear charge Ze and total number of electrons N. We merely combine the form (3) with the Poisson equation V2V = -4ne2p(r)

(4)

to obtain the self-consistent TF equation

= 0 otherwise

Near the nucleus the self-consistent potential energy V(r)must obey the boundary condition V ( r )+ -z"a, r 4-0

and at infinity V(r) must tend to zero. In the spherically symmetric cloud of an 4*

Theoretical Chemistry

94

atomic ion it is useful to work with dimensionless variables 4(x) and x defined through p- Y ( r ) =

Ze2

----#

and r = bx

where b is a length chosen to simplify the resulting differential equation for $(x). With the choice 9n2

113

b = + ( z )

a

o

0.8853 ao

=

T

(9)

where a. is the Bohr radius h2/me2,one obtains the dimensionless TF equation for atomic ions

to be solved subject to the boundary condition

!w) = 1

(1 1)

from equations (6) and (7). The second boundary condition needed to specify uniquely the solution to equation (10) is best examined after discussing the nature of the solutions to equation (10) subject to boundary condition (11). For small x , an expansion due to Baker exists of the form #(x) = 1

+ azx + u9x3/2+ . . .

(12)

of $ ( x ) at the origin, aS=4/3 etc. For different choices of initial slope a, the types of solution shown in Figure 1 are generated. Solution I

a , being the slope f(0)

tends to zero at infinity and is readily interpreted as the solution representing neutral atoms, with N = Z . From the Gauss theorem, the positive ion solutions are of type I1 in Figure 1, the construction shown allowing the ionicity to be obtained from

N

-xo+yxo) = 1 - 2

(1 3)

which is essentially a statement of the Gauss theorem. The radius R o of the positive ion is given by

Numerical solutions of equation (10) subject to the boundary conditions (11) and (13) are available (see, for example ref. 4) and hence the self-consistent field V(r) in heavy positive ions is established. It should be stressed that there are no solutions in this statistical limit corresponding to negative ions, the solutions of type 111 in Figure 1 having a quite 3 4

E. B. Baker, Phys. Rev., 1930,36, 630. P. Gombds, ‘Die StatisticheTheorie des Atoms und Ihre Anwendungen’, Springer-Verlag Vienna, 1949, p. 360.

Electron Density Description of Atoms and Molecules

95

f

1

I

i

I

N

9 7

I I I

J.

*

I

X

XO

X

I Figure 1 Types of solution of the dimensionless Z%omas-Fermi equation (10). Function

4 expresses the potential distribution in the atomic ion as a function of distance from the nucleus. Type I solution: neutral atom, potential and electron density have infinite extent. Type 11 solution: corresponds to positive ions, these have a finite radius. If N is the number of electrons, and Z the atomic number, the constructionshown determines N/Z (< 1) for the given solution

different interpretation, which need not concern us here as we deal with free atomic and molecular ions. Later on, we shall see that there is a simple explanation why negative atomic ions are not bound in such a statistical theory. 3 Variation Principle and Chemical Potential of TF Theory It is of considerable importance to note that the density-potential relationship (3) of the TF theory follows from a variational principle for the total energy. To see this, we note first that the classical electrostatic potential energy U consists of the sum of two terms in an atomic ion, the electron-nuclear potential energy Venand the electron-electron potential energy Vee.We can write

u=

Ven

+ Vee

where V N ( ~the ) nuclear potential energy equals -Ze2/4ns,r, and for convenience we shall work with units such that 4ns0=1 throughout. V&) is the potential energy of the electronic cloud and so the self-consistent potential energy V(r) is given by

+

V(r) = V N ( ~ ) Ve(r)

(16)

Theoretical Chemistry

96

To obtain the total energy E(2,N) of a positive atomic ion we must simply add the kinetic energy T to U in equation (15). Kinetic Energy Density of Electron Cloud.-Again we apply Fermi gas relations locally. The probability of finding an electron at r with momentum of magnitude between p and p + dp is

where p
Using equation (1) one obtains immediately

and eliminatingpr(r) in favour of the density p(r) using equation (1) yields tr

=

c k (p(f)}5/3

Equation (18) is again simply the kinetic energy density of a uniform Fermi gas, but now applied Iocally to the electron cloud at position r. The total kinetic energy Tis given by

T=

j

tr dr

=

ck

I

{ ~ ( r ) } dr ~/~

(19)

and hence the total energy E of an atomic ion is given by

Euler Equation for Density.-A pillar of the TF theory is that the density-potential relationship (3) can be obtained as the Euler equation of the variation problem posed by minimizing E in equation (20) with respect to p ( r ) subject to the normalization requirement

1

p(r)dt = N

(21)

Thus we introduce a Lagrange multiplier p and minimize according to (22)

d(E-pN) = 0

This can be expressed in the equivalent form p =dE/W

(23)

which is recognizable as the defining equation of the chemical potential. Performing the minimization (22) with E given by equation (20) leads, after a short calculation, to the Euler equation F =

8 c k (P(r)}2’3

f VN ( r )

+ v e (r)

(24)

Electron Density Descr@tion of Atoms and Molecules

97

and using equations (16) and (18) we readily regain equation (3). Evidently the Euler equation (24) is, in semiclassical theory, identical to the energy equation (2) of the fastest electron, plus relation (1). The principle underlying the whole of the density theory of atoms and molecules has essentially been exposed by the argument leading to equation (24). We shall see below, however, that there is a need to refine the elementary approximation (18) for the kinetic energy density, in order to transcend TF statistical theory. Nevertheless, even without refinement, certain useful relations follow from this simplest form of density theory and we will discuss these now for positive atomic ions. 4 Energy Relations for Heavy Positive Atomic Ions

Equation (20), combined with the positive ion solutions of type I1 in Figure 1, provides a route for determining the total energy E ( Z , N ) for heavy positive ions. This is already of considerable interest, but it will be convenient to discuss it in the context of the 1/Z expansion in Section 5 below. Therefore, we will next consider the results from Figure 1, and the Euler equation (24) of TF theory. Multiplying equation (24) by p ( r ) and integrating throughout the space occupied by the electron cloud p ( r ) of the positive ion yields the result Np = 8 T

+ Ven + 2Vee

(25)

using equations (21), (19), and (15). As FockS was the first to demonstrate, the virial theorem is valid for the TF theory, and for an atomic ion takes the usual form 2T

or equivalently E=

+ Yen + Vee = 0

(26)

- T.

Total Energy for Heavy Neutral Atoms.-The fact that the neutral atom solution has the form I in Figure 1 implies that d(x) -+ 0 as x --f 03, in fact as 144/x8 which is readily verified to be an exact solution of the dimensionless TF equation (lo), not however satisfying the atomic boundary condition (11). Since V ( r )--+ 0 at infinity, it follows from equation (7) that, for this neutral case with N = Z , we must have p=O. The condition that, in the simplest density description of neutral atoms, the chemical potential is zero is important for the arguments which follow. We shall see below that one of the objectives of more sophisticated density descriptions must be to find p. Equations (25) and (26) can be rewritten in the form, using E= - T and p=O, g E = Yen

+ 2Vee

2E = Yen

+ Vee

(27)

and It follows that, by eliminating Vee between equations (27) and (28),

+

E ( Z , Z ) = Ven V. Fock, PhysZs. d. Sowjetunion, 1932, 1, 747.

(29)

Theoretical Chemistry

98

But from equation (15), Yen, which is the interaction energy between the nuclear , equally well be considered as potential h ( r ) and the electronic cloud ~ ( r )can the potential energy of the nuclear charge Ze sitting at an electrostatic potential Ve(0)/(-e) in the electron cloud at the position of the nucleus. Hence we can rewrite equation (29) as E=

-qzve(o)

(30)

But combining equations (6)--(9) and (12) it is easy to show that

and hence we have from equation (30) and (31) the important result for heavy neutral atoms that, within a non-relativistic framework, E ( Z , Z ) = -0.768727/3e2/u~ (32) The number in equation (32) has come from the numerical solution of type I in Figure 1,which has a slope at the origin of a2= - 1 S88. Comparison with Bare Coulomb Field.-The origin of the behaviour of the total energy as Z713 is easy to understand. Consider a pure Coulomb field with an energy level spectrum

(33) n being the principal quantum number. Since there are 2n2 electrons in a closed shell of principal quantum number n, we have for the energy per shell the result -Z2e2/a0,independent of n for a bare Coulomb field. Now suppose there are JY” closed shells. Then, for a neutral atom we must have en = -(Z2/2n2)e21ao

In the spirit of the TF statistical theory, not only 2 but N must be taken to be large i.e. JV%1, when equation (34) yields, in units of e2ao

and hence the energy per shell timesM yields E ~ o ” ~ o m b ( Z= , Z -(;)1’3z7/3 )

=

-1.127/3

(36)

It is clear that equation (36) will have too large a binding of outer shells; the self-consistent field reduces equation (36) to the form of equation (32). It is not simple to demonstrate that 7/3 is the correct power of Z for real atoms when 2 100. In fact Foldy6 showed that up to this value of 2,Hartree and Hartree-Fock data are slightly better fitted with Z l Z i s .Nevertheless, for a non-relativistic theory there can be no doubt that for really large values of 2 the total binding energy must vary as Z 7 I S .We shall see below, when we deal with L. L.Foldy, Phys. Rev.,

1951, 83, 397.

Electron Density Description of Atoms and Molecules

99

the relationship between the TF statistical theory and the 1/Z expansion, that equation (32) should be viewed as the leading term in a power series in Z-113, significant corrections for atoms involving terms of O(Z2) and O(Z6/8).The origin of these terms will be discussed below. Scaling of Energies of Positive Ions.-For positive ions, with finite radius Ro=bxo it is evident that, just outside r = R o , say at R: e2 V(R$)= -(Z-N)

Ro

(37)

since the electron cloud is spherical and outside the charge cloud, Gauss theorem tells us that the potential is just as though all the charge were lumped at the nucleus. But y5 (xo)= 0 from Figure 1 and hence it follows from equation (7) that

Evidently, from equations (25), (26), and (38) we have for N f Z IE=

en + 2 ~ e e+ ; ~ N ( z - N )

(39)

and 2E= Ye,+ Vee. [If N > Z , equation (38) gives p > 0, and negative ions are unstable.] Eliminating Vee we now find the generalization of equation (29) when N f Z a s

Equation (31) now has p on the I.h.s., and a 2 = a 2( N / Z )as seen from Figure 1. Therefore, using equation (9) for the quantity b in equation (M),

Equation (41) is the generalization for positive ions when N < Z of equation (32) for neutral atoms. Clearly we can write equation (41) in the form

We shall refer to equation (42) as the scaling property of the binding energy of heavy positive ions. Whereas in general we expect E to be a function of two variables 2 and N, equation (42) shows that in the limit of applicability of statistical theory, the energy is a function which is a product of an explicit form Z7i3times a function of the ratio N/Z. This is directly traceable to the properties shown in Figure 1, where a given solution of the dimensionless TF equation (10) is characterized only by this ratio. Thus, fo(N/Z) in equation (42) can be calculated from the known solutions of equation (lo), the form of fo(N/Z), taken from the work of March and White,’ being plotted in Figure 2. Since the density description focuses so directly on E ( Z , N), as in equation (42), it is natural that we should bring the result (42) of this simplest (TF) density theory into contact with the 1/Z expansion of E ( 2 , N) for atomic ions. That these two treatments are very intimately related

Theoretical Chemistry

100 0.8

0.70.60.5-

r-l

&-

-

2'nc 0.4 U

Id? 0.3-

0.1

0

0.1

0.2

0.3

04

0.5 N/ Z

0.6

0.7

0.8

0.9

1.0

Figure 2 Form offunctionf o ( N / Z )in equation (42). Quantity acfuallyplottedis - (E/Z7j3)( N / Z ) 2 / 3against N/Z, with E in Hartree units

was demonstrated by March and White;7 see also the later work of Dmietrieva and Plindov.* 5 Relation of TF Theory to 1/Z Expansion A landmark in atomic theory was provided by the work of L a y ~ e rwho , ~ pointed out that regularities in the properties of atomic ions,1° which were hard to relate via numerical Hartree-Fock (HF) studies, could be understood via the so-called 1/Z expansion. Layzerg showed that the total non-relativistic energy of an atomic ion could be expanded as

+

1

E ( 2 , N ) = Z2[&o(N) ,EI(N)

+ 1221 e ( N ) + . . .]

(43)

where the job of the theory was then reduced to calculating E n ( N ) . The quantity E,,(N)is determined entirely by the bare Coulomb field, but the higher coefficients depend on the self-consistent field, supplemented by electron correlation effects. The work of Katoll guarantees convergence of equation (43) for sufficiently large 2. It is clear therefore that equation (43) must be approximately summed to all orders in 1/Z, in the limit of a large N value, by the TF energy of an atomic N. H. March and R. J. White, J. Phys., 1972, B5,466. I. K. Dmitrieva and G. I. Plindov, Phys. Left. A., 1975, 55, 3. 9 D. Layzer, Ann. Phys., 1959, 8, 271. lo N. H.March, 'Self-consistent Fields in Atoms', Pergamon, Oxford, 1975. l1 T.Kato, Commun.Pure Appl. Math.. 1957.10,151; J. Fac. Sci. Univ. Tokyo, 16,145. 7

8

Electron Density Description of Atoms and Molecules

101

ion given by equation (42). March and White7 pointed out that a connection could be established and that equation (42) implied asymptotic behaviour, ~n ( N ) K

Nn+'I3, N + HI

(44) of the coefficients in the 1/Z expansion (43). e , ( N ) was shown explicitly to have this property, and approximate forms for e l ( N ) and eZ(N)were proposed. These forms, based on equation (44),were made semi-quantitative by a least squares fit to e 1 ( N )and e 2 ( N )which are known for small values of N. Refinements of the coefficients obtained from the least squares fit have been given by Dmietrieva and Plindov * who propose

+ ... + .. . .

~1 (N) 0.485 N4/3-0.354 N2/3 E Z ( N2 ) -0.1MN7/3 0.130N513

+

(45) (46)

These join on quite well to the known small N values as shown in Table 1. Table 1 Coeficients in 1/Z expansion of equation (43) N 2 6 8 10 15 24 28 60 110

( N ): exact 0.625 3.2589 5.6619 8.7708 13.950 28.952 37.970 108.49

EI

~l(N)from equation (45) 0.660 4.12 6.34 8.80 15.8 30.6 38.0 108.47 247.44

- 4 N ):exact 0.1577 3.2880 8.1319 16.273

-EZ ( N )from equation (46) 0.11 4.2 9.1 16.29 45.65 146.3 213.2 1342 5684

March and Parr12 have recently pointed out that, if we assume En ( N )

-

-

a: Nn+'/3 a)

+ a: N n + a! Nn-113 f . . . -

0%Nn+1/3 m / 3

(47)

m=O

then insertion into equation (43) allows us to write 03

E(Z,N)=Z2 X O3 -

.;Nn+1/3-m/3

n=o Z n m = 0 0

0

1

This result, which follows by combining the Layzer 1/Z expansion with the asymptotic expansion (47) for large values of N, represents the formal generall a N. H. March and

R. G . Parr, Proc. Natl. Acad. Sci. USA, 1980,77, 6285.

102

Xheoretical Chemistry

ization of equation (42) to include significant corrections of O ( Z 2 )and of order

Z5I3. fo(N/Z) is known from the TF theory discussed fully above. 6 Inhomogeneity and Exchange Corrections to TF Theory Before discussing the consequences of equation (48) for the total energy of positive ions, it is clearly of importance to understand how the density description has to be generalized beyond the TF approximation to account for the terms 0(Z2) and 0(Z5i3) in equation (48). This takes us back to the total TF energy in equation (20). When we consider again the basis of this, we note first that the kinetic energy density t, has been approximated by equation (18), which is a local free electron relation. Formally, it is straightforward to take the variation of T = J t, d t with respect to p, and then one can write the Euler equation (24) in the generalized form

Clearly, equation (49) reduces to equation (24) if t, is replaced by the approximation (18); formally it now takes full account of the (rapid) variation of electron density in the atom, in contrast to the semiclassical TF Euler equation. Unfortunately, t, is only presently known in two special cases: (i) to low order in gradient expansion corrections to equation (18) as in equation (76) below and (ii) in a perturbative development about the uniform electron assembly.l3 Form (i) will be referred to again below. However, as Scott14 was first to argue for the neutral atom, the origin of the Z 2 term in equation (48) resides in the inhomogeneity correction to the TF theory, which is formally contained in equation (49). Fortunately, an approximation based on the Coulomb field treatment of Section 4 suffices to gain a useful estimate of the order of the Z 2 term in the neutral atom. Before discussing this, let us consider, in relation to the O ( Z 5 / 3 )term in equation (48), the introduction of exchange into the TF model, following Dirac.I5 Essentially, Dirac argued that, just as for kinetic energy density, one should calculate the exchange energy for a uniform electron assembly and then apply it locally. The exchange energy is calculable for a Slater determinant of plane waves l o and the result for the exchange energy density E X for density p ( r ) is EX

=

Ce

= 4 (3/n)'13

- ~ e( ~ ( r ) } ~ / 3

where 3e

Using this result, one can add on to the total energy a term for the exchange energy A = E X d t and then minimizing with respect to the density p one finds p = 13 l4 l5

'? 'P

+ VN (f) + ~ e ( r -) i:

Ce { p ( f ) } 1 / 3 .

J. C. Stoddart and N. H. March, Proc. R. SOC.London, Ser. A., 1967,299,279. J. M. C.Scott, Philos. Mag., 1952, 43,859. P.A. M.Dirac, Proc. Camb. Philos. SOC.,1930, 26, 376.

Electron Density Description of Atoms and Molecules

103

This then is the Euler equation of the density method, generalized (formally) to include inhomogeneity in the kinetic energy density and to incorporate electron exchange in what is now often referred to as the Dirac-Slater exchange approximation. Thus, from the form of this Euler equation (51), one can regard -4/3 ce{p(r)}'13 as an exchange potential VX adding to the Hartree potential VHartree = VN Ve. VX is the Dirac-Slater exchange potential. Historicallyl6 it is worthy of note that if one resorts in equation (51) to the TF approximation (18) for t,, then the Euler equation of the Thomas-FermiDirac method results. We shall not go into the solutions of the Thomas-FermiDirac equation in this review, though there has been recent interest in this area. Suffice it to say that in the full form of the Euler equation (51), we are working at the customary Hartree-Fock-Slater level. However, we shall content ourselves, until we come to Section 17 below, with understanding in a more intuitive, but inevitably less detailed, way how the corrections to the TF energy in equation (48) arise. We want to emphasize that in writing equation (51) we are still working at the level of a single Slater determinant, no electron correlation therefore being embodied as yet in the density description. The formal relaxation of this final restriction will be carried out in Section 15 below. Origin of Corrections to TF Neutral Atom Energy.-Since presently we do not know how to write the kinetic energy density t, in any precise way, let us use the Coulomb example of Section 4 to estimate the error in the TF energy for this case. It is then easy to verify for particles moving in this bare r-l potential that, using equation (3) with V(r)= -Ze2/r, yields, for Z electrons,

+

Coulomb =

~ T F

- 24/3 !f 181J3ao

and

Therefore, since the virial theorem holds,

This is the result of the kinetic energy density being approximated by equation (18), and agrees with the exact Coulomb energy in the limit of very large 2 values, given in Section 4. In fact, developing the exact energy

f o r N closed shells, making use of equation (34), we readily find, l 7with e2/ao=1, ECoulomb = -Z7/3(3/2)1/3

+ +Z2 + O(Z5/3)

(56)

The inhomogeneity correction in this case is seen to be + Z z .Scott, to whom this result is due,14 examined the origin of this correction, and found it came 16 17

N. H. March, Adv. Phys., 1957, 6, 1. R. A. Ballinger and N. H. March, Philos. Mag., 1955, 46, 246.

104

Theoretical Chemistry

dominantly from the K shell. He argued therefore that it should not be sensitive to the self-consistent field and that it should be taken over as it stands into the TF energy formula for self-consistent fields, which leads to equation (32) being transcended to read, again in Hartree units, E ( 2 , Z ) = -0.7687Z7/3

+ +Z2 + O(Z5/3)

(57)

Evidently, the kinetic energy T is reduced from the T F value in equation (54) by + Z 2because of the virial theorem; this then is the inhomogeneity correction for the Coulomb field, in the sense of the Z-1/3expansion. It now remains to estimate the exchange energy. To do so, we return to formula (50) and note that, since in the Thomas-Fermi-Dirac method the energy is stationary with respect to small changes in density, we can usefully approximate the Thomas-Fermi-Dirac total energy by retaining the T F density, characterized for the neutral atom by the function #(x) which tends to zero, as l M / x 3 , as x tends to infinity. In terms of this function we have for the total exchange energy of the TF(D) neutral atom

Scott1* evaluated the integral involving the neutral atom solution $ ( x ) of the TF equation (10) and thus obtained

March and Plaskett l a investigated corrections to O(Z5iS)other than exchange and concluded that formula (56) should be extended to read E(Z, Z) = -0.76872713

+ +22-0.26625/3 + O(Z413)

(60)

Thus, in relation to equation (48) proposed by March and Parr for N # Z , one has the approximate estimates that f l ( l ) = + and f2(l)= -0.266. Some work has been done on the exchange energy for N < Z from statistical theory,8’1Bbut we shall not go into further details here. Rather, we conclude the discussion by showing in Table 2 a few values of atomic binding energies, the ‘observed’ values having been obtained, following Scott, in such a way that relativistic corrections are removed. The agreement is already satisfactory, and points strongly to the fact that the terms of O(Z4j3)must really contribute little to atomic binding energies, in spite of the fact that the O ( Z 5 / 3term ) makes an essential contribution over the entire range of Z in Table 2. It is clearly of interest to attempt, both theoretically and by analysis of empirical data, to represent the functionsfl(N/Z) and f i ( N / Z ) in equation (48).

Table 2 Atomic binding energies in Hartree units (non-relativistic) Z

6 37.8 Equation (60) 37.6

Observed

18

lo

9 99.7 99.4

13 242.4 240.1

18 528 524

23 945 941

N. H. March and J. S. Plaskett, Proc. R . SOC.London, Ser. A., 1956, 235, 419 N. H. March, J. Phys. By 1976,9, L73.

Electron Density Description of Atoms and Molecules

105

A further property of these functions which should then be useful will emerge below when we examine briefly other consequences of equation (48). In summary, equation (48) represents both a natural generalization of the simplest density description (TF approximation) and a valuable rearrangement of the Layzer l/Z expansion. In principle, equation (43) and its rearrangement and partial sum (48) include many-electron correlation effects. However, the numbers fi(l)=+ and f2(l)=-0.266 do not have any correlation included; it may be that this enters only in the higher-order terms in equation (48) but this has not presently been established. We shall later discuss the inclusion of correlation in the density description, Chemical Potential and Energy Relations.-As followed from the scaling property (42) characteristic of the TF theory, so formula (48) implies certain relations which we shall now examine. First, the chemical potential p=(aE/aN)lzcan at least be determined formally. Thus we have

In the TF limit when 2 and N become very large, we know that p + 0 for the neutral atom case N=Z and hence fi(l)=O. March and Parr12 argue that this property must also imply fi (1) =fi (1) =fL( 1)= 0 and possibly also fi (1) = 0, leaving p(2, Z ) W Z - ~but / ~again , this has not been presently proved. One can also obtain V,, from the Hellman-Feynman theorem

From equations (48) and (62) we then find, for the neutral atom case

The virial theorem yields, with exchange and correlation now in Vee,

and hence Ye, =

- $ Z 7 / 3 f ~ (+ 1 )$ Z 5 l 3 f 2 ( 1 )+ 0(24/3)

(65)

It is worth emphasizing that Vee has no term in Z2,as expected from the origin of this term discussed above. 7 Ionic Binding Energies, Ionization Potentials, and Electron Afhity The density description focused attention on the total ionic energy E(2,N ) and led to the Z-ll3 expansion (48), when combined with the 1/Z series (43). Two further developments of E(2,N)will be recorded here, following the work

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106

of Lawes et aLaOThe first of these uses the 1/Z expansion to relate the energy of positive ions of atomic number Z to that of the neutral atom E ( 2 - 1,Z- 1). The second relation is concerned with ionization potentials. It uses a Taylor expansion to second order to relate E(Z, N) to the neutral atom energy E(Z, 2 ) when N is near to 2. Its relation to the work of Pyper and Granta1 will be emphasized below. Lawes et al. rearranged the 1/Z expansion, in order to relate the energy of an ion E ( 2 , Z - n) to that of the neutral atom E(Z- n, 2-n). We motivate rearrangement of the 1/ Z expansion by considering first the case of a singly charged positive ion, n=1, starting from the neutral atom energy E(2-1,Z-1) and then adding a proton to the nucleus. The merit from the standpoint of the 1/Z expansion is obvious; N is kept constant at the value 2- 1 and by varying Z the 1/Z expansion makes the dependence explicit. But now it has been shown [cf.equation (30) above] that the binding energies of atoms are closely related to the electrostatic potential created at the nucleus by the electronic charge cloud. If we assume the unperturbed charge cloud to be that of a neutral atom (2- 1,Z- l), then its interaction with the nucleus is increased by a factor Z / Z - 1 on adding a proton and this strongly suggests that one should form the quantity E(Z, 2- 1)- ( Z / Z - 1) E ( Z - 1,Z- 1). One obtains from the 1/Z expansion E(Z,z- 1)- (Z/Z- 1 ) E(Z- 1,z-1 ) = 2 EO (2-1) - €2 ( Z - l)/(Z- 1)

+ higher order terms.

(66)

In Table 3, accurate energies of light atoms and ions taken from the compilation by Weiss22are recorded. Using the neutral atom values E ( Z - 1,Z- 1) available in Table 3, Lawes et al. constructed Table 4 to show the convergence of the terms in equation (66). Because of the satisfactory nature of the convergence demonstrated there, it is worth recording the generalization for n= 1, E(Z, 2-n)-(Zl.2-n) E(2-n, 2 - n ) = nZ EO (2-n) -n €2 (2-n)/(Z- n)

+ higher-order terms

(67)

Table 3 Non-relativistic total energies E(Z, N ) of light atoms and ions in Hartree units Z 3 4 5 6 7 8 '0

1'

N=2 -7.280 -13.656 -22.031 -32.406 -44.781 -59.157

E(Z, N) N= 3 -7.478 -14.325 - 23.425 -34.776 - 48.377 -64.229

N=4 -7.496 -14.667 - 24.349 -36.535 -51 -224 -68.413

G . P. Lawes, N. H. March, and M.S. Yusaf, Phys. Lett. A, 1978, 67,342, N. C. Pyper and I. P. Grant, Proc. R. SOC.London, Ser. A , 1978, 359, 525. A. W.Weiss, Phys. Rev., 1961, 122, 1826.

Electron Density Description of Atoms and Molecules

107

Table 4 Contributions to singly chargedpositive ion energy E ( Z ,Z - 1) from 1/Z expansion result [equation (66)].In this Table, E ( Z - 1, Z - 1) has been taken from Table 3, Coefficients c0 and E~ are collected by March;10see also Table 1 above Predicted

Z 3 4 5 6 7

E(Z,2- 1) -7.28 -14.32 -24.35

( Z / Z - 1) x E(Z- 1,Z- 1) -4.36 -9.97 -18.33 (- 29.44)* (-43.90)*

E2 (2- 1) E ( 2 ,. Z - 1) from Z&o(Z-1) -3 -4.5 -6.25 -8.25 -10.5

2- 1 0.08 0.14

equation (66)

-7.28 -14.33 -24.36 (- 37.32) (-53.85)

0.22 0.37 0.55 * Since these are not available from Table 3, the Hartree-Fock of Clementiashave been used to illustrate convergence.

However, the movement of n protons, for n > 1, will usually be a very substantial perturbation and the convergence must be expected to become poorer. A modification of the above formulae (66) and (67) has been Lawes et al. also focused on the variation of E(2,N ) with N at constant Z , in contrast to the above constant N argument. For this variation with N, the 1/Z expansion is not immediately appropriate. The further development of E(2, N ) given by Lawes et al. is closely related to, and motivated by, the relativistic Hartree-Fock studies of Pyper and Grant though the considerations below are restricted purely to non-relativistic theory. The motivation for the work of Pyper and Grant was the regularity that is found empirically between successive ionization potentials over a wide area of the Periodic Table, as discussed for example by Phillips and Williams.25In the course of their analysis of light ions, Lawes et al. noticed regularities when the energies of the ions, for a fixed atomic number 2,were considered in relation to the neutral atom energy E(Z, 2).These regularities prompted an investigation of the Taylor expansion of E(2,N ) around the neutral atom value E(2,Z ) , namely

The analysis of Lawes et al. of the results of Table 3 above led to the result that the first- and second-order terms explicitly displayed above could give an accurate representation of Table 3, but now at constant atomic number 2. Furthermore, they noticed that there was a close relation between the derivatives(aE/aN) IN - z and ( a 2 E / a N 2 ) I ~The = ~ .obvious interest from the studies of Pyper and Grant prompted Lawes et al. to study further the relation of the above treatment of E(2,N ) to the Hartree-Fock studies. An essential point in the explanation by Pyper and Grant of the empirical result (74) below relating successive ionization potentials was that the electron affinity, that is the energy of binding of an electron added to a neutral atom to 93

E. Clementi, J. Chem. Phys., 1963, 38, 996.

a4

T. Shibuya, Phys. Lett. A , 1979, 71, 39.

45

C. S. G. Phillips and R. J. P. Williams, ‘Inorganic Chemistry’, Oxford University Press, 1965, Vol. 1

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108

form a negative ion, was small compared with, say, the first ionization potential II.To impose, then, this requirement on equation (68) it must next be noted that I1 = E ( Z , Z - 1 ) - E ( Z , Z )

and that the electron affinity A is similarly A = E(Z,Z

+ 1)-E(Z,Z)

N-z

+

1 a2E 2 aN2

I

N-z

For A to be small, the two terms displayed must obviously be of the same order of magnitude and then it follows that

We shall write (aE/aN)IN = Z = ,u(Z,Z)=p , because of the definition (23) of the chemical potential. Then equations (69) and (71) lead immediately to the result I1

+ -2p.

(72)

The nth ionization potential, defined by In

= E(Z,Z-n)-E(2,Z-n

+ 1)

(73)

then becomes In 9

+n(n

+ l)Z1-+(n-l)nI1

= nZ1

(74)

which was the result explained by Pyper and Grant within their relativistic Hartree-Fock framework. The most elementary approximate form therefore of the relation between E ( 2 , N), E(Z, Z ) , and the chemical potential, which has a magnitude approximately equal to half the first ionization potential from equation (72), is E(Z,N) = E(Z,Z)

+ + ( Z - N ) ( Z - N + l)Ii(Z)

(75)

Of course this equation then implies zero electron affinity A and the correlation between successive ionization potentials follows. Evidently there is some further information to be gained by comparing the Taylor expansion form (75) with the Taylor expansion of equation (48) around the point N/Z=1. From the property of the chemical potential, terms in (Z- N ) 2 arise from fl, fi,and fs,whereas terms proportional to (Z- N)arise from higher terms in the series (48). Equation (75) shows that approximate relations must obtain between the coefficients of (Z- N) and (Z- N ) 2 and that further work is required. 8 Kinetic Energies Calculated from Density Gradient Expansion

We shall conclude this discussion of atoms and ions by returning to the inhomogeneity correction treated formally in Section 6. There it was noted that the TF approximation ckp’’’ to the single-particle kinetic energy density is rigorously valid only for a constant density. When p varies by but a small fraction of itself over a characteristic electron de Broglie wavelength, one can

Electron Density Description of Atoms and Molecules

109

contemplate correcting this approximation by adding terms dependent on the density gradient and higher-order derivatives.The lowest-ordergradient correction leads to the corrected kinetic energy density as

Hence for the total kinetic energy T we may write

T = To + T2 + T4 +

...

(77)

where To is the Thomas-Fermi term, T2comes from the ( V P ) ~term, and the fourth-order term, which we shall not display here, has been given by Hodges.26 Historically it was KirznitsY2'following pioneering work by von Weizsacker 28 who first gave the correct coefficient for T2as in the above equations. As noted by Lawes and one can demonstrate the inequality

where the equality is readily verified to hold when a single level only is occupied. Thus from equations (77) and (78) one can write T 2 9T2,

(79)

T2 being the lowest-order gradient correction to the Thomas-Fermi value. Hence we have from equations (77) and (79) that To

+ T4 + ... 2 8T2

(80)

and if T4is genuinely small, then we see from this result that the lowest-order gradient correction is < 1/8 of the Thomas-Fermi term To,which is encouraging for the usefulness of the expansion. Wang et aL30 have calculated To, T2,and T4using good wave-mechanical densities for closed-shell atoms and a selection of their results is recorded in Table 5. The inequality (80) is seen to be fulfilled. Furthermore, since the ThomasFermi statistical theory becomes correct for sufficiently large numbers of electrons, it follows that the importance of T2diminishes continually for heavier atoms. Table 5 Total energies of closed-shell atoms built up from gradient expansion of equation (77). Energies are in Hartree units and are non-relativistic He Ne Ar Kr 26

To 2.56 117.8 490.6 2594

TZ

T4

0.32 10.1 34.3 142

0.08 1.9 6.2 24

C. H. Hodges, Can. J. Phys., 1973, 51, 1428. A. Kirznits, Sou. Phys. JETP, 1957, 5, 64. C. F. Von Weizsacker, Z . Physik, 1935, 96, 431. G . P . Lawes and N. H. March, Phys. Lett. A , 1978, 66, 285. W. P. Wang, R. G . Parr, D. R. Murphy, and G . A. Henderson, Chem. Phys. Lett., 1976, 43, 409.

z7 D.

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110

We shall comment later on the analogue of the above gradient expansion of the kinetic energy for the exchange plus correlation energy.

Relation between Total Energy and Sum of One-electron Energies.-March and Plaskettle demonstrated for the TF neutral atom that there was a simple numerical relation between the total energy and the sum of the one-electron energies Ef for the potential energy V= VN+ Ve. This is readily seen as follows. The sum of the one-electron energies, Es say, is given by En=

Zet = occupied levels

T + F/Y:Widr I

= T + SP(VN = T + Ven

+ Ve)dr

(81)

+ ZVee,

where the !Pis are the eigenfunctions for motion in the one-body potential V and we have used the fact that P(r) =

y f y l ,

(82)

occupied levels

Between equations (25) and (81), Ven+2Vee can be eliminated and applying once again the virial theorem in the form T = - E yields31 (Es-Np) = Q E

(83)

Since for the neutral TF atom we have shown that p=O, we obtain March and Plaskett’s l8result that E ( Z , Z ) = 9 Es

(84)

Equation (83) generalizes their result for positive ions; the chemical potential then being given by equation (38). For some numerical estimates based on equations (38) and (83) the reader is referred to ref. 31. We shall return to relations (83) and (84) when we deal with molecules at equilibrium later. In this later connection, it will be of interest to examine the inhomogeneity correction to the relation (83). To understand the nature of this, we consider the general Euler equation (49). Multiplying this equation by the density p and integrating over the whole of space, yields

where we have added and subtracted (5/3) T= (5/3)J trdr on the right-hand side. One can estimate the correction term

in equation (85) to the lowest order in the density gradient by means of equation (76), and one obtains

31

N. H. March, J. Chem. Phys., 1980, 72, 1994.

Electron Density Description of Atoms and Molecules

111

For the model of a pure Coulomb field, the correction on the left-hand side of equation (86) can, in fact be estimated exactly, the result being given in Appendix 1 together with results for a harmonic well. We shall return to this discussion when we treat molecular energies below.

9 Density and Potential Distribution in Molecules Having discussed the basic equations of the density description and their application to atomic ions we turn now to the much more difficult problem of molecules. Even the simplest density description afforded by the TF theory presents severe computational problems for multicentre problems, as well as some conceptual difficulties on which we shall attempt to throw light in the ensuing discussion. From the practical standpoint, the first attempt to solve the self-consistent T F equation for a diatomic molecule was made by Hund.S2Following this, the density method was applied to the benzene molecule and compared with both the molecular orbital prediction for the density and with relevant experiment^.^^ Various other early molecular calculations are discussed in ref. 16; we refer here to the recent studies of Dreizler and his c o - ~ o r k e r sThe . ~ ~importance of such self-consistent calculations will be emphasized below, even though we shall not use them in any detail in the ensuing discussion. Indeed, we shall focus on generalizing much of the above discussion of atoms to molecules, especially in their equilibrium configuration. With regard to atoms, the important questions which arise are then: (i) What can be learnt about the magnitude of the chemical potential, a central quantity in the density description? (ii) Are there simple energy relations for molecules at equilibrium, such as a generalization of, for example, equation (29), for neutral atoms? (iii) Can the total molecular energy at equilibrium be related simply to the sum of one-electron (orbital) energies, as in equation (84) for atoms? (iv) As in the approximate equation (72) for atoms, the chemical potential is related to the ionization potential; are there simple and useful relations for ,u for molecules? All these questions, as we shall see, can be discussed fruitfully from the density description of molecules. But because, as we have already emphasized, the multicentre problem is difficult to tackle even in the simplest TF density description, we shall attempt to tackle questions (i)-(iii) above by turning immediately to a central field model which was solved by MarchSSin the TF density description. The model was set up with tetrahedral and octahedral molecules in mind, for example GeH,, UF, etc. It has been used recently by Mucci and MarchS6in a discussion of energy relations for molecules at equilibrium. We shall summarize their main results below,after discussing the solution of the central field model.a5 33 s5

F. Hund, 2.Physik, 1932, 77, 12. N. H. March, Acta Crystallogr., 1952, 5, 187. R,M.Dreizler, E. K. U.Gross, and A. Toepfer, Phys. Lett. A, 1979, 71, 49. N,H. March, Proc. Camb. Philos. SOC.,1952, 48, 665. J. F. Mucci and N. H. March, J. Chem. Phys., 1979, 71, 5270.

112

Theoretical Chemistry

Central Field Model of Tetrahedral and Octahedral Molecules.-The idea is very simple, and has long been exploited in the sense of one-centre expansions of molecular wave functions in a molecule like CH,. However, to exemplify the way the density description can afford answers to questions (i)-(iii) above, we take the model literally in which, in methane for example, we smear the protons uniformly over the surface of a sphere of radius R, equal, in the methane example to the C-H bond length. Thus, we have in this model a nuclear potential energy VN(Y)given by

Here the central nucleus carries charge ze (for methane, z = 6 ) and the total surface charge spread uniformly on the sphere of radius R is ne (n=4 for methane). We stress the model character of this example of the density description. While it might be hoped that for the series CH,, SiH,, GeH, it may have some relevance to experiment, it is obvious on chemical grounds that it could not be expected to have much to say quantitatively about a molecule like CCl,, the approximation of smearing the C1 nuclei being, of course, unrealistic. Nevertheless, the model has considerable merit in exemplifying some of the salient features of the density description applied to molecules. Since the nuclear framework is spherical according to equation (87), we can immediately use the TF equation (10) to describe the self-consistent field, provided that : (a) we redefine the unit of length b in equation (9) by scaling with the charge on the central nucleus, namely

and (b) we redefine the boundary conditions, to take account of the discontinuity in the electric field across a surface charge distribution, as already reflected quantitatively in the nuclear framework described by equation (87). Then we need merely to return to Figure 1 and recognize that the type of solution we require for the neutral molecule is obtained by taking a solution of type IV for r = bx less than R = bX, and type V for x > X . This solution of type V again behaves as 144/x3 for large x, as for the TF neutral atom. The discontinuity in the slopes of solutions of type IV and V when they meet at X tells us the value of n/z for this particular solution. Without the need to go into more quantitative detail, which is given fully elsewhere,%there are two immediate consequences of the TF solution for this central field model of molecules: namely that the chemical potential for the neutral molecules under discussion is identically zero, and secondly there is an equilibrium bond length R= Re say, which is specified by

Electron Density Description of Atoms and Molecules

113

Though solutions are available away from equilibrium, we shall below restrict ourselves solely to a discussion of the molecular energy terms at equilibrium. Indeed, it is easy, from an argument about the net force acting on the shell of surface charge, to write down an expression for dE/dR in terms of the solutions of the dimensionless TF equation (lo), but this is not given in detail here. Suffice it to say that it then follows that the equilibrium bond length Re takes the form Re = bXe = 2-113 f ( n / z )

(90) where X e is shown in Figure 3 taken from ref. 35. The reason there are two curves is that the total energy E in equation (89) obviously involves the nuclearnuclear potential energy Vnn, which one calculates from the true tetrahedral and octahedral nuclear frameworks as

V,

ne2 R

= -(z

+ cn)

N Z

i.

0.4

L

Figure 3 Equilibrium bond length in density description of central fietd model of tetrahedral molecules. Curve I: tetrahedral molecules. Curve 11: octahedral molecules. There are two curves because the nuclear-nuclear potential energyfor the tetrahedral and octahedral nuclear fvameworks difer by diferent values of constant c in equation (91)

114

Theoretical Chemistry

where c =

3 43

32 for tetrahedral molecules

+2t"for octahedral molecules -

--

Bowers3' represented curve 1 of Figure 2 by the approximate analytic form Re = z-lI3 3.12(n/~)-o.6A

(92)

the range of validity of equation (92) in the fitting being 0.8 < n/z< 3.0. Though, as emphasized, we must not expect equation (92) to be realistic for tetrahedral and octahedral molecules with heavy atoms in the outer positions, its form is of some interest. However, we will enquire, within this density description, what the answer to question (i), namely that the chemical potential is zero in this treatment, has to say about questions (ii) and (iii). We tackle the question of the energy relations in the next section. 10 Energy Relations for Molecules at Equilibrium The above treatment of the central field model was based on the TF Euler equation (24), applied self-consistently with vN(r) which was given by equation (87). But as for atoms, we have generally, by multiplying equation (24) by the density and integrating over all space N / . = if T

+ Ven + 2Vee

(93)

where naturally yenis to be calculated for the appropriate nuclear framework. T is the total kinetic energy, and the total molecular energy E, including the nuclear-nuclear potential energy, is defined as E =T

+ Ven + Yee + Vnn

(94)

Equations (93) and (94) are true for all molecules in the TI: density description. Adoption of Central Field Model at Equilibrium.-To make progress, we now adopt the central field model, for which, for neutral molecules, we saw that the chemical potential p was zero. Secondly, we established the existence of an equilibrium bond length at Re, given by equation (89), where the virial theorem for equilibrium under purely Coulomb forces takes the usual form 2T

+ Ven + Vee + Vnn = 0

(95)

or equivalently E= - T. Eliminating Ven between equations (93) and (94), putting p=O and using the virial theorem then yields

This simple relation will be confronted with the results of accurate self-consistent calculations for a variety of small molecules in the next section, where we shall see that, in spite of its derivation from the central field model above, it is valid in a much wider context, to a useful approximation. 07

W.A. Bowers, J . Chem. Phys., 1953, 21, 1117.

Electron Density Description of Atoms and Molecules

115

If, instead of eliminating V e n to obtain equation (96), we use equations (93) and (94) to relate V e e and Ynn, we find

a relation first proposed by P ~ l i t z e r . ~ ~ If V n n is eliminated between equations (96) and (97),one obtains

Evidently any two of the relations (96)-(98) imply the third. Test of Energy Relations on Small Molecdes.-Mucci and MarchSshave tested the result (96) from self-consistent field calculations on light molecules. Using the data given by Snyder and B a s ~ h they , ~ ~ plotted Vnn- V e e against T, the result being shown in Figure 4. The linearity shown confirms the result (96) when it is noted that the slope of the straight line in Figure 4 is near to 1/3. Similarly, their results for 2Vee+ V e n against Tfor the same series of molecules are plotted in Figure 5. Again there is an excellent linear relationship, and the slope of the straight line drawn is near to 5/3, in accord with the relation (98).

Figure 4 Diference Vflf l - Vee between nuclear-nuclear and electron-electron potential energy against total kinetic energy T for light molecules. Energies are in Hartree units 98

a9

P.Politzer, J. Chem. Phys., 1976,64, 4239. L. C. Snyder, and H. Basch, ‘Molecular Wave Functions and Properties’, Wiley, New York 1972.

116

Theoretical Chemistry

-250

1 / 3 BH3

50

100

150

T

200

Figure 5 Quantity 2Vee+ Vcn with Vcn the electron-nuclear potential energy against total kinetic energy T for light molecules. Energies are in Hartree units

Thus, although equations (96) and (98) were derived above by working out the density description for the central field model of tetrahedral and octahedral molecules, Figures 4 and 5 confirm the validity of these relations for a wide variety of molecules, using self-consistent wave function calculations. This is the more remarkable because the simplest TF density description is ensured, as a statistical theory, to become asymptotically valid for large numbers of electrons N, whereas the results of Figures 4 and 5 are for molecules with N G 2 4 (cf. Appendix 1). But in view of this last point, it is obviously important to study molecules with a larger number of electrons. This leads us back to the tetrahedral and octahedral molecules. Regularities in Nuclear-Nuclear Potential Energy.-Mucci and March first noted, in their study of regularities in the nuclear-nuclear potential energy Vnn for tetrahedral and octahedral molecules, that for the central field model of Section 9 one could combine equations (91) and (92) to obtain for F‘nn at equilibrium Vnn = 0.170 nl-6z-O.27 (z + cn) e2/ao

+

= 0.170 ( N - Z ) ~ . ~ Z -[cN * . ~ ~z(l-c)] eZ/ao

(99)

Here, in the second line of equation (99) we have substituted for n in favour of the total number of electrons N, which for the neutral molecules under consideration is given by N = n z. The result (99) was calculated for the selected tetrahedral and octahedral molecules shown in Figure 6, taken from ref. 36. The straight lines drawn in

+

Electron Density Description of Atoms and Molecules

l5

117

t

10

5

100

N

150

Figure 6 Nuclear potential energy V A for tetrahedral and octahedral molecules as given bu equation (99)

Figure 6 do not represent the function (99) but merely join the calculated points. But the trend with the total number of electrons N shown in Figure 6 is important for comparison with the empirical results for this same group of tetrahedral and octahedral molecules, to which we now turn. Evidently equation (91) can be used to calculate Vnn at equilibrium, provided we use as input data the equilibrium bond lengths Re. Taking these from ref. 40, Vnn is plotted against the number of electrons N in Figure 7, taken from ref. 36. This plot is rather striking and in contrast to the dependence of the model results in Figure 6 on the charge ze on the central nucleus, the empirical Vnnis much less sensitive to variation of z, as Figure 7 clearly demonstrates. This can be understood in general terms in that smearing heavy nuclei like C1 or Br over the surface of a sphere reduces their attractive influence for electrons, and therefore the role of the central atom is proportionately overemphasized in the central field model. Of course, while Figure 7 shows that Vnn = f ( N ) to a useful degree of accuracy, if one wanted to work back from such an average curve to obtain bond lengths to chemical accuracy, as opposed to gross trends within a few tenths of an Bngstrom, it would be necessary to discuss carefully the fluctuations about the average curve in Figure 7.But the fact that, empirically, V n n depends in a simple way, on average, on the total number of electrons is encouraging for the density description. 40

A. D. Mitchell and L. C. Cross, ‘Tables of Interatomic Distances and Configurations in Molecules and Ions’, (Special Publication No. 1 l), The Chemical Society, London, 1958.

5

Theoretical Chemistry

118

2500 -

ZOO0

-

SiBrr,

C

'1500 -

1000 -

150

100

200

N

Figure 7 Nuclear-nuclear potential energy Vnn against total number of electrons N for tetrahedral and octahedral molecules. In contrast to Figure 6, empirical data for bond lengths are used to construct this figure

In summary then, the main conclusions that follow from the work of Mucci and Marchs6 are that, even for the relatively light molecules shown in Figures 4 and 5, the simple energy relations (96) and (98) are obeyed. Secondly, there are some remarkable regularities in the nuclear-nuclear potential energy for molecules at equilibrium. Though this regular behaviour is studied for other than tetrahedral and octahedral molecules also in ref. 36, in general, knowledge of J'nn at equilibrium will only suffice to establish one relation between bond lengths, so that the tetrahedral and octahedral molecules discussed here represent a favourable case. But it seems worthwhile, in quantum chemical problems, to pay more attention to regularities in the nuclear-nuclear potential energy than has been done hitherto. The relations (96H98) were established on the basis of the simplest density description, the TF theory. We mentioned earlier a conceptual difficulty that

Electron Density Description of Atoms and Molecules

119

arises in this theory; this is the point at which we must now confront this, and suggest a way to resolve it. 11 Teller’s Theorem, Chemical Potential, and Molecular Binding The conceptual difficulty to which we have referred is connected with the result of Teller 41 that there is no molecular binding in the TF theory. This will be called Teller’s theorem below and it means, of course, that there is no nuclear configuration in which the energy of the molecule is lower than that of the separated atoms. The first difficultythat we then need to resolve is why this theorem is apparently violated by the central field model of Sections 9 and 10 where it was proved that an equilibrium bond length Re exists for which (dE/dR)R,=O. The situation here is that Teller’s theorem is not applicable in the sense that one cannot regain atoms by letting R tend to infinity in the central field model. Thus, what is established is an energy minimum, not that the molecular energy is lower than the isolated atoms’ energy by equation (89). It must be pointed out secondly that the Teller theorem tells us that inhomogeneity corrections of the type discussed in Section 6 are essential for molecular binding. But replacing point nuclei by a surface charge distribution leads to smaller density gradients and therefore the T F approximation is more accurate for the surface charge model than for point nuclei. But March and Parrla have gone further, and interpreted the Teller theorem as showing that, in the limit of a large number N of electrons, the nuclear-nuclear potential energy is a smaller term in the number of electrons than the electronic energy terms. This proposal is supported by the fact that the curve of Mucci and March shown in Figure 7 can be approximately represented by

YA = constant N4/3 Wm where N is the total number of electrons. This is to be contrasted with, say, the kinetic energy of the neutral atom which is known from equation (54) to be proportional to N ‘I3. The fact that in the tetrahedral and octahedral molecules the bond length at equilibrium, in the TF density description of the central field model, goes as R N Z - ~in/ ~equation (92) is reflected in the fact that the model gives Vnntoo large a value, as a comparison of the magnitudes of Vnnin Figures 5 and 6 shows. In tetrahedral and octahedral molecules, equations (91) and (100) show that the relation is more like ReccN1I3. March and Parr l2 also consider the chemical potential in the same limit. They argue that the meaning of ,u=O in the Euler equation of the density description is that N p in this equation is a smaller-order term in the number of electrons than the other energy components. Thus gross features, of the kind exhibited in the energy relations (96)--(98), can be treated but the chemical potential p, and the nuclear-nuclear potential energy, require special care. Notwithstanding this, the above considerations suggest that a more refined theory, motivated by the density description, may be possible for some mole41

E. Teller, Rev. Mod. Phys., 1962, 34, 627.

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Theoretical Chemistry

cules, and in particular for homonuclear diatomics. These arguments use, following the atomic approach of Section 4, the scaling properties of a bare Coulomb field, that is of a hydrogen-like molecule ion, and therefore we must next consider this system. 12 Form of Energy of Homonuclear Diatomic Molecules Coulomb Field Scaling for Diatomic Molecules-Dreizler and March 4 3 have derived asymptotic scaling laws for both the chemical potential and the total energy for a hydrogen-like molecular ion. This is relevant to scaling laws for homonuclear diatomic molecules considered in the following section and therefore we first outline this bare Coulomb two-centre treatment. As stressed in Section 5 , the first term of the 1/Z expansion of the total energy is the eigenvalue sum E for the Coulomb potential. In generalizing the 1/Z expansion to homonuclear diatomic molecules, one therefore needs, as the first term, the sum of the one-electron energies for the potential V(r) = -Ze2

(X + jb) -

-

where the nuclei a and 6 are separated by 2R. The energy levels for this potential energy can be obtained 4 4 but the results cannot be readily presented in simple analytical form. Therefore Dreizler and March have reported results for a large number of electrons (say 2N), moving independently in the pure Coulomb potential (101), and present in the molecular ion. The asymptotic eigenvalue sum presented below, we must stress, will only be rigorously valid in the limit when 2 and N tend to infinity, in a non-relativistic framework. Their main results can be summarized as follows: 431

(i) The relation between N, 2,and R and &, = R Ip I/Ze2is given by

N =

c(to); = 38n h2

(ze2~)3/2

(102)

(24312

where the function G has been evaluated numerically and is plotted in Figure 8, taken from ref. 42. (ii) As follows from (i) immediately above, Ze2 = F[N/(z~~R)~/~I

a useful scaling relation for the chemical potential. It reduces

(103) p,

in units of

Ze2/R,to a function of merely one variable N/(Ze2R)a/2. (iii) The total eigenvalue sum Es for the pure Coulomb field (lol), in this asymptotic limit of large 2 and N, can either be obtained from

42 48

44

R. Dreizler and N. H. March, 2. Physik, 1980, A294,203. E. Teller and H. L. Shalin, ‘Physical Chemistry’,Academic Press, New York, 1970, Vol. 5. D. R. Bates, K. Ledsham, and A. L. Stewart, Philos. Trans. R. SOC.London, Ser. A, 1953, 246, 215.

Electron Density Description of Atoms and Molecules

121

Figure 8 Function G determining chemical potential for asymptotic limit of hydrogen-like molecular ion from equation (I 02)

or by evaluating the Thomas-Fermi sum

ES = ck

p5j3 dt

+

1

pYdt

One obtains in either case EsmDptotic (23, R) = 2* z

R H(N/(Ze2R)3/2)

(106)

These then are the asymptotic scaling laws in a pure Coulomb field for two identical centres. We apply these results to scaling laws in molecules, including the interactions self-consistently below. Proposed Scaling in Self-consistent Field Theory.-March and Parr have proposed an extension of the 1/Z expansion for atoms, analogous to rearrangement (48), for the case of diatomic molecules. Introducing a scaled length X = R Z , they note first that from the above discussion of the bare Coulomb field one -has E O (N, X) = N l / 3 G (N/X3l2) (107)

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Theoretical Chemistry

Then they effectthe generalization of equation (47) to homonuclear diatomics as En(N,

X) = N n + 1 / 3 A n ( N / X 3 / 2 ) + N n Bn(N/X3l2)+ Nn-1I3 Cn(N/X'/') +

.-

(1 08)

Hence, the total energy E (2,N, R )

We can sum this formally to read

This then is the general scaling proposed l 2 for homonuclear diatomic molecules with nuclear charge Z and N electrons. If one takes the limit N -+ 2 of neutral molecules, then one obtains

+

E ( Z , Z , R) = Z 7 / 3 j ~ ( Z 1 / 3+ R )Z2j2(Z1/3R)+ Z5/3j3(Z1/3R) O(Z4/3) (111)

That the first, TF, term will scale as shown is already clear from the work of Hund32and of Townsend and Handler.45 As in the case of atoms, one can determine the individual components of the energy from this expression. Thus, using Feynman's theorem one can obtain the electron-nuclear potential energy as

+

z10/3

dZ

+ Z3 d12+ z 8 / 3 d j 3

dZ

dZ

At this stage we invoke the virial theorem in the form

where Etotal includes Ynn. But it is now clear that because of the scaling property of equation ( l l l ) , derivatives of E with respect to R are related to dj,JdZ. Since at equilibrium dEtotal/dR=0, the derivative terms can be eliminated, so that we have, specifically at equilibrium, Ve, =

8 Z 7 / 3 j l ( Z 1 / 3 R+) 2Z2j2(Zf/3R)+ !j Z5l3j3(Z1l3R)

+

+ ;$ v,,

o(z413)

(114)

One can also calculate Vee from the virial theorem at equilibrium, since T= -Etotal. The result is Vee =

- $ Z 7 / 3 j ~ ( Z 1 / 3+R$Z5/3j3(Z1/3R) ) + 0(Z4/3)+ Q

Vnn

(115)

March and Parr have argued as discussed above that the proper interpretation of Teller's result that there is no molecular binding in the Thomas-Fermi limit 2-t 00 is that we must then find that Re tends to infinity and this means that 45

J. R.Townsend and G.S. Handler, J. Chem. Phys. 1962, 36,3325.

Electron Density Description of Atoms and Molecules

Vnn is a smaller term in an expansion in

123

Z-lI3 than V e e etc. Thus the term

O ( Z 7 I 3 )in equation (115) arises entirely from Vee. There is no term 0 ( Z 2 )in V e e , just

as for atoms. March and Parr also point out as already mentioned above that the interpretation of the result p=O for the neutral Thomas-Fermi atom (cf. the result for the central field model for molecules in Section 9, where the chemical potential is also zero for neutral molecules) is that the gross trend with 2 is p c ~ Z - l /at ~ large 2. Of course, such a discussion would have to be refined considerably to reproduce the chemically important periodic effects in p, which will be focused upon below. 13 Can the Total Energy of a Molecule be Represented as the Sum of Orbital Energies? In Section 8, it was shown that the simplest density description of a heavy neutral atom led to relation (84), namely that the sum of the one-electron energies over occupied levels, times 3/2, gives the total energy. For molecules, early work using Hiickel theory, and the discussion of molecular shapes by W a l ~ h following ,~~ the pioneering work of M ~ l l i k e n repre,~~ sented the total energy by the eigenvalue sum. A number of groups pointed out that this could not be correct, for in Hartree self-consistent field theory the electrostatic energy of the electronic charge cloud is counted twice over in the eigenvalue sum, as is explicitly shown in equation (81). Interest in this question was revived by R ~ e d e n b e r g ,who ~ ~ demonstrated from available self-consistent field calculations that relation (84) was obeyed well for molecules at equilibrium. He suggested a coefficient 1.55 from his semiempirical studies, which involved relation (97) of Politzer. The author drew attention to the fact that the simplest density description, combined with the equilibrium form of the virial theorem, would lead back to equation (84) for molecules. Because of the problem associated with Teller's theorem, discussed in Section 11, let us again examine the predictions of the central field model of molecules of Sections 9 and 10. From this model stemmed the energy relations (96)--(98). Equation (81) is again the complete expression for the sum of the eigenvalues in this simplest density description. Using equation (93), with the chemical potential equal to zero, as was demonstrated to be so for neutral molecules in the central field model, one can eliminate Ven 2 Veeby subtracting equations (81) and (93), to obtain *@

+

-#T=

Z E ~ =Es

occupied levels

where, as usual, the er denote the orbital energies. But since T= - E from the virial theorem at equilibrium, equation (84) is regained from equation (1 16). 46

41 48

49

A. D. Walsh, J. Chem. SOC.,1953, 2260. R. S. Mulliken, Rev. Mod. Phys., 1942, 14, 204. K. Ruedenberg, J. Chem. Phys., 1977,66, 375. N. H. March, J. Chem. Pliys., 1977, 67, 4618.

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Theoretical Chemistry

Density Gradient Corrections.-Mucci and March 5 0 have discussed the corrections expected to the result (84) of the simplest density description. As they emphasize, three steps are involving in deriving equations (116) and (84) for molecules at equilibrium: (a) The chemical potential ,u is put equal to zero. (b) The density gradient terms are neglected in equation (76) for the kinetic energy density. (c) Electron exchange and correlation are not included. Their argument for the corrections due to (a) and (b) is briefly summarized below. Correction (c) will be considered in Section 15 and Appendix 4. The inhomogeneity correction (b) above, and the non-zero chemical potential, are both incorporated in the generalized Euler equation (49). Multiplying this by the electron density p and integrating through space yields

Combining this with equation (81) for the eigenvalue sum and noting that T= J' t, d t leads immediately to the result

Using result (86) to lowest order in a density gradient expansion gives

Finally, using T= - E at equilibrium yields the generalization of equation (84) as

In this result, a misprint in the sign of the term in (VP)~in St,/Sp given in ref. 51 has been corrected; this misprint was also transmitted to ref. 50. The author is indebted to Professor Parr and his colleagues for drawing his attention to this, Since one expects, for bound molecules, that the chemical potential ,u will be negative, the fact that $(Vp)2/pdt is a positive quantity shows from equation(l20) that deviations from relation (84) of the simplest density treatment can, in principle, be of either sign, depending on the relative magnitudes of the chemical potential and density gradient corrections. We shall discuss numerical values for the deviation from equation (84) in Section 16 below, in the light of equation (120). But before doing this it is of interest to re-examine the theoretical basis of Walsh's rules. Basis for the Derivation of Walsh's Rules.-As already remarked (see also ref. 52), Walsh's rules governing molecular shapes were based, at least implicitly, on the assumption that the total energy was the sum of orbital energies. Though this 5O 51 52

J. F. Mucci and N. H. March, J . Chem. Phys., 1979, 71, 1495. N. H. March, J. Chem. Phys., 1979, 71, 1004. 'Codson's Valence', ed. R. McWeeny, Oxford University Press, 1979, p. 263.

Electron Density DescrQtion of Atoms and Molecules

125

is not correct, the above discussion shows that in the simplest density theory they differ at equilibrium by a scale factor 3/2. To be specific, we show in Figure 9, following Walsh, the correlation diagram as we go from a linear HAH molecule to the case when the HAH angle is 90". In the linear molecule, the classification of the lowest states is quite clear, into two CT states, even and odd, and into a doubly degenerate nu non-bonding state,

90

180

Angle HAH

Figure 9 Variation of orbital energies in H A H molecule on going from 90"bent molecule to linear molecule. The classificationof states, built from s and p atomic orbitals, is discussed in the main text. The steep rise in the curve joining a1 and n,&favours the bent molecular form for HzO, whereas with four valence electrons, as in BeH2 or HgHz, the linear configuration is favoured. This argument is based on an intimate relation, which Walsh assumed, between the sum of orbital energies and total energy. Density theory in its simplest form supplies such a relation, namely equation (84). The figure is a schematic version of that of Walsh,46 who noted that the line 180" must be either a maximum or a minimum

assuming we are building from s and p atomic orbitals. The degeneracy is due to the possibility of rotating a p orbital about the HAH axis by 90 which would not change the energy. When one considers the 90" bent molecule, symmetry classification C 22), the lowest energy states can be thought of as follows : (i) Two orbitals binding the H atoms to the central atom. These can be considered as formed from the overlap of a purep atomic orbital on A with the O,

126

Theoretical Chemistry

1s on H. Equivalently, one can form in-phase and out-of-phase orbitals which are delocalized, denoted by a1 and b , respectively, in Figure 8. (ii) A p orbital on atom A pointing perpendicular to the plane of the molecule; classification b,. (iii) An s orbital on A. This is non-bonding and of type a,. It is not our purpose here to make a detailed explanation of Figure 9. But the variation in the one-electron energies is schematically as shown for qualitative reasons. Because of the steep rise in the eigenvalue curve joining al and xu, it is argued by Walsh to be energetically unfavourable to have a linear molecule when that state is filled. Since four electrons fill the lowest two levels, it is predicted for five, six, seven, and eight valence electrons that one will have bent molecules, whereas with BeH, and HgH2, for example, one will have linear geometry. The density description throws light on the equilibrium configuration, through equation (84), since the total energy can be determined directly from the eigenvalue sum, But a difficulty arises as one changes, in for example AH2, the angle from its equilibrium value. Fortunately, at the level of relation (84), one can now complete the argument as follows. Relations (81) and (93) are true for all molecular configurations, neglecting density gradient corrections. For the case where p = 0, therefore, equation (1 16) is regained for any molecular geometry. The only step that remains is to relate T to the total energy E. This seems, at first sight, to present a severe difficulty away from equilibrium. Fortunately, however, the difficulty is only concerned with bond distances away from equilibrium, not with angles. For instance, in the case of a diatomic molecule, we should have, writing U for the total potential energy, and E= T+ U 2T+

u=

dE -RdR

The term on the right-hand side is only zero at equilibrium; otherwise there would be a contribution to the virial from the forces required to hold fast the nuclei at separation R. Let us consider that, in the case of AH,, we have drawn Figure 9 such that, at each angle, the energy has been minimized with respect to R. Then as, for example, Nelander 5 3 has shown, for every set of angles for which the energy is minimized with respect to bond distances, the virial theorem holds in its simple form T= -E. This then enables us to complete the first principles argument to provide a basis for Walsh’s rules. Of course, the argument is approximate; it neglects corrections of the type exhibited in equation (120) from density gradient and chemical potential terms. Both of these corrections will vary somewhat with configuration, but one can expect that only in borderline cases one will have to study either carefully. However, this is a direction in which to examine exceptions in the context of Walsh’s rules. In connection with changes in molecular configuration from equilibrium, this is the natural point at which to say something, briefly, about the use of the density description for molecular vibrations.

‘*

B. Nelander, J.

Chem. Phys., 1969, 51, 469.

Electron Density Description of Atoms and Molecules

127

14 Density Description of Molecular Vibrations

Briefly, we record here that the problem of molecular vibrations can be expressed, at least formally, in terms of the ground-state density at equilibrium, as discussed by Handler and Everything is subsumed then into an equation expressing the density change, Ap say, from its equilibrium value, when the nuclei are subjected to small displacements from equilibrium. The equation determining the density change has the form Ap ( r ) =

1

F(r, r’) A V ( r ’ )dr’

(122)

where F is a one-body linear response function which, as Handler and March argue, is a function of the equilibrium density. Because the total number of electrons in the molecule is fixed, it is clear that JApdt=O, and hence from equation (122) the response function F must satisfy

1

F(r, r’) dr = 0

:all r‘

(123)

The change in potential energy A V is first order in the nuclear displacements, and of course, Ap is obtained correctly to the same order from equation (122). Handler and March show that the Thomas-Fermi approximation to the linear response function F has the form FTF(ry

” = dr dr’ S ( r r 9 j d s S ( r , s ) / d t S ( t , r ’ ) - S ( r r ’ )

(124)

which satisfies condition (123) for any choice of S for which the integrals exist. They then obtain S from the response function of a uniform gas, replacing the Fermi wave number kr, related to the density through equation (2), or its equivalent (kj/3n2)=p,by the local density at the point (r+r’)/2. The uniform gas response function is determined by the first-order spherical Bessel function jl(x)= (sin x--x cos x)/x2, and in terms of this the function S in equation (124) is

To transcend this approximation involves accurate knowledge of the Green function for bound 56 While the above affords a fundamentalroute for the future, a more immediately practical approach is provided by modelling the charge density according to bond charge models. One example of these will be given below, from the work of Pam and his colleagues. Localized Models of Electron Density in Molecules.-Based on the linear response equation (122), applied however to periodic monatomic crystals, Jones and March 57 have argued that in discussion of vibrational properties the correct tool 54

55 56

57

G. S. Handler and N. H. March, J. Chem. Phys., 1975, 63, 438. H. Hameka, J. Chem. Phys., 1963, 39, 2085. G. S. Handler and N. H. March, in press. W. Jones and N. H. March, Proc. R. SOC.London, Ser. A , 1970, 317, 359.

Theoretical Chemistry

128

to use is the gradient of the electron density Vp. Whereas older views of vibrational motions in crystals considered an atom to carry with it rigidly its ‘own’ charge cloud as it vibrated, they stressed that the fundamental decomposition into localized distributions must be carried out on the gradient of the density Vp. Thus one must write vp = Z R ( r - l ) I

(126)

where I represents the atomic sites on the regular lattice. They then showed that one could construct the density for small displacements from this vector ‘rigid atom’ R(r). It seems clear that in molecules with one type of atom only, say ozone, one should have a similar sort of decomposition. In independent work and from a different direction, Bader and his co-workers have likewise insisted on the importance of Vp in dividing a molecule into fragments which are localized and transferable. As one example, the work of Bader and Fkddal15*effects such a partitioning into fragments based on the virial theorem, and the interested reader is referred to this and various other papers which throw very considerable light on electron densities in specific molecules and on partitioning the electron clo~d.~~-~~ Turning from these fundamental arguments, which are still difficult to implement, we shall discuss models embodying, independently, some related ideas. We refer first to the work of Anderson and Parr,66who write for a diatomic molecule with nuclei at R, and Rg

The third term in equation (127) is the deviation from a rigid ‘atoms in molecules’ model, that is, it represents a ‘non-perfectly-following’ part of the density as the nuclei move. Then, in the Born-Oppenheimer approximation, they show that the total potential energy E is such that V i E is the sum of the orbital densities from the other atoms B, multiplied by 4nZ,, 2, being the charge on the E nucleus. This is an approximate result, in which PNPF is assumed small in equation (127). One of the successes of their method is that they can evaluate higher-order derivatives of E and can relate these together to agree with experiment. It has to be stressed that equation (127) has not, so far, been given a first principles basis. One could envisage, by comparing model (127) with X-ray scattering experiments, getting a useful test on a homonuclear diatomic molecule but as far as we are aware that has not been done to date. In concluding this 68

59

6O

61 63 15‘ *6 66

R. F. W. Bader and P. M. Beddall, J. Chem. Phys., 1972,56, 3320. R. F. W. Bader and P. M. Beddall, J. Am. Chem. SOC.,1973,95, 305. R. F. W. Bader, A. J. Duke, and R. R. Messer, J. Am. Chem. SOC.,1973, 95, 7715. R. F. W. Bader and R. R. Messer, Can. J. Chem., 1974,52,2268. S . Srebenik and R. F. W. Bader, J. Chem, Phys., 1974, 61, 2536, R. F. W. Bader and G. Runtz, Mol. Phys., 1975,30, I 17. G. Runtz and R. F. W. Bader, Mol. Phys., 1975,30, 129. R. F. W. Bader, Acc. Chem. Res., 1975, 8, 34. A. B. Anderson and R. G . Parr, J. Chem. Phys., 1970,53, 3375.

Electron Density Descrbtion of Atoms and Molecules

129

section on localized models, the reader is referred to a more extensive discussion of this topic in refs. 67 and 68. Point Charge Model of XY Linear Symmetric Molecules.-It is possible to model the potential energy in a molecule usefully by more primitive models than those just discussed. We consider here some consequences of a point charge model employed by Ray and Parr.B9So that we can be more specific than in the previous section, let us consider briefly the ideas of their approach with reference to linear triatomic molecules of the symmetric type XY 2. If R1and R 2are the YX and XY distances in a molecule YXY and 8 is the YXY angle, then the total Born-Oppenheimer energy E= T+ V has a potential energy Vgiven by53

Then it follows that

and

For the equilibrium configuration R 1 = R 2 = R e and O=x, the first term on the r.h.s. of equation (129) vanishes. The vibrational problem of a symmetric molecule XY, is most frequently described in terms of a quadratic valence force field. This is given in terms of a bond stretching force constant K 11, the bond-bond interaction constant K 1 and the bending force constant Ko0:

In the conventional simple valence force field approximation, mixed derivatives of the type ( P E / a R l a 0 2 ) ~and 2 (a3E/aR2a02)~1 are neglected. In fact they are usually relatively small. Using these approximations, it follows from the above equations that

and

67 68

69

R. F. W. Bader, in ‘Localization and Delocalization in Quantum Chemistry’, ed. 0. Chalvet, R. Daudel, S. Diner, and J. P. Malrieu, Reidel, Dordrecht, 1975, p. 15. N. H. March, in ref. 67, p. 115. N. K. Ray and R. G. Part, J . Chem. Phys., 1973,59, 3934.

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Theoretical Chemistry

Following Simons,'" the potential energy Y is approximated by placing charges + 2/3 qe at each atom and -qe at the middle of each bond. The potential energy V is then given by qZe2 22qzeZ y = -20 q2e __ -20 - + (9Rz) [9(Rf R g - 2 R i R z c o ~8)1/2] (9RI) 2q2e 2q2e2 [3 (R: + Rg4- Ri Rz cos 8)li2] - [3(-:Ri Rz COS 8

+

(136)

I _ _ _ _ _ -

d

Hence, in this model

and

Thus KO0

+ O.O26(Krl + K12)

( 1 39)

This formula related the bending force constant to the stretching force constants. Now the stretch-stretch interaction force constant K 1 is usually smaller by a factor of 10 or so than K l l and hence often can be neglected. Then from equation (137) K11 = iiq2e2/R8,

(140)

and K00 = 0.026 K11

For diatomic molecules, Borkman et al. found (see also ref. 69) k = 2 qi ez/Rd,

(142)

k being the stretching force constant,

R d the equilibrium distance, and qd the bond charge. Table 6 shows a comparison of triatomic and diatomic bond charges in halides. One can also model the potential energy Vin a slightly different, but natural manner, by placing a charge + +qe on each end atom Y ,qe on the central atom X, and -qe at the centre of the bond. Then

K11 = 109 ---q2e21R3, 48

and

which are seen to be very similar to equations (140) and (141). 70

G . Simons, J. Chem. Phys., 1972, 56, 4310.

(143)

Electron Density Description of Atoms and Molecules

131

Table 6 Triatomic and diatomic chargesfor halides (from Ray and Parr's results in ref. 69) MoIecuIe ZnFz ZnClz ZnBrz ZnIz CdFz CdClz CdBrz CdI2 HgFz HgClz HgBrz HgI2

q (diatomic)

1.85 1.93 2.00 2.18 1.9 2.0 2.0 2.2 1.9 2.0 2.1 2.3

q (triatomic)

2.25 2.15 2.24 2.3 2.4 2.3 2.4 2.4 2.7 2.5 2.6 2.6

The triatomic and diatomic bond changes are seen to be rather similar, consistent with spatially localized, and independent chemical bonds.

We have given this primitive modelling of the density to stress the importance of finding localized descriptions of the electron density in a molecule (or solid, especially amorphous materials, e.g. Si, where the periodicity that is so helpful in a crystalline solid no longer is present). The considerations leading to equation (126), plus the work of Bader and his colleague^,^^-^^^^^ point the way to a break up into blobs or localized entities using the gradient of the charge density. At very least a localized picture ought not only to reproduce the correct equilibrium ground-state density but also enable one to construct the density for small displacements of the nuclei from equilibrium, without the need to go through a new solution of the equations for the density. This is an area in which further progress is to be expected, going beyond the point charge modelling discussed above. Density theory should be particularly helpful in this programme. At this point we must return to the basis of the electron density description. So far, we have corrected the simplest theory for inhomogeneity and exchange. In the next section, we shall consider the genuinely many-body effects, that is electron4ectron correlation.

15 Inclusion of Correlation in Density Theory The way to include correlation in the density description is readily set down.4 The argument given below is justified by the fundamental theorem of Hohenberg and Kohn, proved in Appendix 2, where two other density theorems are also discussed briefly. The theorem establishes, for a non-degenerate ground state, that the total energy of a many-electron assembly is a unique functional of the electron density p(r). This was, of course, assumed in the pioneering work of Thomas, Fermi, and Dirac and therefore the Hohenberg-Kohn theorem formally completes the Thomas-Fermi-Dirac theory. Thus, recognizing that from this theorem the exchange and correlation energy density, E X C say, is a unique functional of the density, one now writes for the

Theoretical Chemistry

132

total energy E of a molecule with nuclear-nuclear potential energy

Vnn

The Euler equation of the variation problem now involves the functional derivatives of the single-particle kinetic energy density tr corresponding to the exact many-body density p ( r ) , and the exchange and correlation energy density EXC. Such a minimization of equation (145) yields immediately

Because of the definition of t, which was adopted, this has the form of singleparticle theory, but with potential energy7'$7 2

In other words, V ( r ) is a sum of the Hartree potential energy, but calculated with the exact many-body density, and apart from the exchange and correlation interactions. The same philosophy used for exchange in Section 6 can now be used to approximate EXC in the one-body potential V(r) of equation (147). Thus one takes the best calculation available for the exchange and correlation energy for a homogeneous electron assembly, say E ~ C ,and uses it locally to obtain the total exchange and correlation energy as Exc =

1

&-[p(r)l

dt

(148)

Evidently then equation (147) takes the approximate, but explicit, form V ( r ) = VHartree

SEOXdP) + __--dP

(149)

and we regain Dirac-Slater exchange potential if we approximate E$&) by equation (50). We can include correlation through approximate formulae for the homogeneous interacting electron assembly such as those of W i g r ~ e r , ~ ~ Nozitires and Pines,74or Gordon and Kim.75 Gradient Correction to Local Exchange and Correlation Energy.-Just as we discussed gradient corrections to the single-particle kinetic energy tr in Section 8 , so one can contemplate adding such corrections to the local density approximation to the exchange and correlation written explicitly in equation (149). Herman et af.76pointed out that the leading term must take the form e2(Vp)2/p4/3 on dimensional grounds. Thus ( V P ) ~ (length)-8 and p4/lS (length)-4 yielding correctly EXC(P) e2p4/3-e2 (length)-4. However, care is needed, for if N

-

N

71

'2 73 74

75 76

W. Kohn and L. J. Sham, Pliys. Rev. A , 1965,140, 1133. N. H. March, in 'Orbital Theories of Molecules and Solids', Clarendon Press, Oxford, 1974. D. 95. E. P: Wigner. Trans. Faraday SOC.,1938, 34, 678. P. Nozitres and D . Pines, Phys. Rev., 1958, 111, 442. R. G. Gordon and Y. S. Kim, J. Chem. Phys., 1972,56, 3122; ibid., 1974,60, 1842. F. Herman, J. P. van Dyke, and I. B. Ortenburger, Phys. Reo. Letr., 1969, 22, 807.

Electron Density Description of Atoms and Molecules

133

we treat exchange and correlation energies separately, neither will have a gradient e x p a n ~ i o n . ~ ~ - ~ ~ Actually Geldart and Ras01t'~write

and they determined CXCas a weak function of the density p. It is important to record here that pioneering work in this area is that of Ma and Brueckner,*O who essentially combined the density functional philosophy with the use of diagrammatic techniques in many-body perturbation theory. We cannot give the details here, but by such arguments they arrived at an energy density of the form ( V P ) ~ / P *proposed /~ by Herman et al.76 and discussed above, with a calculated value of the coefficient. A discussion of how partial summation of a gradient series can be carried out may be found in ref. 81. In concluding this section on exchange and correlation, which hinges so much in its implementation on the homogeneous interacting electron assembly, it should be emphasized that much of the pioneering work was done by Lundqvist and his co-workers.82We shall refer to this again when we discuss wave function calculations for molecules in the framework of density theory. We will also discuss briefly in Appendices 3 and 4, respectively, the form of V(r) in equation (147) for simple two-electron systems and the modifications to the relation between total energy and sum of orbital energies due to exchange and correlation interactions. 16 Electronegativity and Chemical Potential This is the point at which we must return to discuss fully the significance of the chemical potential p. From the elementary derivation of the TF theory in Section 1, it is clear that p must be a constant, independent of position, and characteristic of the electronic distribution in the atom or molecule in question. This is, so to speak, the condition that no redistribution of charge is possible. For though p is built up as the sum of spatially varying components, for example in the most general Euler equation (146) from a sum of kinetic, Hartree, and exchange plus correlation energy contributions, the sum must be independent of r for otherwise electrons could spill over from one point to another in the molecule to lower the total energy. Should it prove possible, in the future, to solve the Euler equation (146) to full chemical accuracy, the values of p for isolated atoms, and the single value of p for the molecule built up from these atoms, will be of great interest. It is clear from the way that p was introduced in the density theory [see especially equation (23)] that one could contemplate estimating it in an atomic ion from 77 78 79 80

81

82

A. M. Beattie, J. C. Stoddart, andN. H. March, Proc. R. SOC.London, Ser. A. 1971,326,97. A. Sjolander, G. Niklasson, and K. S. Singwi, Phys. Rev. B, 1975, 11, 113. D. J. W. Geldart and M. Rasolt, Phys. Rev. B, 1976, 13, 1477. S. K. Ma and K. A. Brueckner, Phys. Rev., 1968, 165, 18. N. H. March, in 'Electrons in Finite and Infinite Structures', ed. P. Phariseau and L. Scheire, Plenum, New York, 1977, p. 236. L. Hedin and S. Lundqvist, in 'Solid State Physics', ed. F. Seitz, D. Turnbull, and H. Ehrenreich, Academic Press, New York, 1969, Vol. 23, p. 1.

134

Theoretical Chemistry

E ( 2 , N ) - E(2,N- l), that is from an ionization potential. Indeed it has already been seen in Section 7 that, for a neutral atom, p is in magnitude approximately half the ionization potential, for the case when the electron affinity is sufficiently small to be neglected. Mulliken’s definition of electronegativity, namely

Z being the ionization potential and A the electron affinity, is then to be expected

to be an approximation to I p I, as we shall discuss below. Equivalence of Chemical Potential and Sanderson’s E1ectronegativity.-In an important paper, Parr and his co-workers83have drawn attention to the work of Sanderson on electronegativity84-88 and they have proposed that the negative of the chemical potential p is identified with his definition of electronegativity. What is quite clear, as already stressed, is that the readjustment of the electron distribution as chemical bonds form on assembling a molecule from its separate atoms is usefully quantified by comparing ,LL for the molecule with the various chemical potentials of the separate atoms. Rather than go into detail of approximate numerical calculations of the chemical potential from various approximate forms of the density theory, for instance the Hartree-Fock-Slater equation (51), we shall focus some attention on models that have been built as a result of this recognition of the very intimate relation between the chemical potential and electronegativity. Electron Migration in a Model Heteronuclear Diatomic Molecule.-One model introduced by Politzer and W e i n ~ t e i nfollowing ~~ the work of Parr et aLa3 is illuminating, and will be referred to here in some detail. They specifically consider a diatomic molecule, separation R, and regard the total energy E(NA,NB, ZA,ZB,R)as varying due to variation in the number of electrons of atom A, charge ZA,say dNA and atom B, dNB, and finally R + R dR. Then the energy change d E can be written

+

Now we consider the molecule in its ground state, with R fixed at its equilibrium value Re. Since the system is at equilibrium, dE- 0 for an infinitesimal movement of charge dN= -dNA=dNB. Then the above equation becomes

or

83 84

85 87

R. G. Parr, R. A. Donnelly, M. Levy, and W. E. Palke, J. Chem. Phys., 1978,68,3801. R. T. Sanderson, Science, 1955, 121, 207. R. T. Sanderson, J. Am. Chem. SOC.,1952, 74, 272. R. T.Sanderson, ‘Chemical Bonds and Bond Energy’, Academic Press, New York, 1971. P. Politzer and H. Weinstein, J. Chem. Phys., 1979, 71, 4218.

Electron Density DescrlIption of Atoms and Molecules

135

Following the lead of various ~ o r k e r s , ~ Politzer * - ~ ~ and Weinstein take as and , Nthus ~ the above definition of the electronegativities X A = - ( ~ E / ~ N A ) R equation (154) is a statement that the electronegativities of the two atoms are equal at equilibrium. Politzer and Weinstein take this same model a little further by considering some general, fixed, separation R’say. They assume the nuclei have been brought to this internuclear distance from another separation so quickly that the electrons have had no time to redistribute themselves. For this redistribution dE< 0, and if it involves a charge transfer dN= - dNA= d h , then

=

k-4-x~)~’ dN < 0.

Thus, since dN is positive, XA < XB and electron migration must be to the more electronegative atom. A second model, related to the ‘non-perfectly following’ charge density of equation (127), due to Anderson and Parr,66 is also developed by Politzer and Weinstein, with similar conclusions. But from these models, let us return to the density description. Since Parr et al. have defined electronegativity from density functional theory as the negative of the chemical potential p, the value of the electronegativity x is then given by equation (146).

Unfortunately, d/Sp[tr+ E X C ] is not yet known to great accuracy. So direct quantititive study based on equation (156) has not yet proved possible to chemical have presented results of two kinds of calaccuracy. Therefore, Ray et dD1 culation to illustrate the idea of electronegativity neutralization. First, they discuss the idea in the context of the simple bond charge model for diatomic molecules developed by Parr and his co-workers, one example of which was discussed in Section 13. Then they show how the idea can be developed from two alternative primitive hypotheses on the effects of charge transfer on electronegativity. Electronegativity Equalization in Bond Charge Model of Diatomic Molecules.PasternakD2has considered electronegativity in the simple bond charge model of diatomic molecules. While his definition is not based on equation (156) it is not at variance with it, and Parr et al. base their first treatment of electronegativity neutralization on it. Then one can obtain a reasonable estimate of the electronegativity of AB from the electronegativity of separate atoms A and B and one can also describe the effect of heteropolarity on force constants and bond lengths. 88

90 91

92

J. Hinze, M. A. Whitehead, and H. H. Jaffk, J. Am. Chem. Soc., 1963, 85, 148. G. Klopman, J. Am. Chem. SOC.,1964, 86, 1463. N. C. Baird, J. M. Sichel, and M. A. Whitehead, Theor. Chim. Acta, 1968, 11, 38. N. K. Ray, L. Samuels, and R. G. Parr, J. Chem. Phys., 1979,70, 3680. A. Pasternak, Chem. Phys., 1977, 26, 101.

136

Theoretical Chemistry

Consider a diatomic molecule AB with equilibrium bond length RAB. In the simple bond charge model, we let ZA 7 and ZB- q be the charges on nuclei A and B, where ZAis the charge on A in the diatomic molecule AA, ZBthe charge on B in BB, and 7 measures the charge transferred. A point charge carrying -(ZA+ZB)electrons is located at a distance r l from A and r 2 from Bywith r l + r g = R ~ B it ; is assumed that the bond charge in AB is the average of the bond charges in AA and BB. Assuming the bond length in AA is 2rA and in BB is 2rB, Pasternak took as a definition of the electronegativity of atom A a formula

+

xa

= c(zA/rA)

(157)

With C=9/16 for single bonds, this gave values in accord with Mulliken’s definition (151) and other values, and the form of this relation is in excellent agreement with the early considerations of G ~ r d y Ray . ~ ~ et al. extend this definition to multiple bonds by letting the constant depend on bond type, and fitting each constant to obtain the best agreement with Mulliken’s values. For single (s), double (d), and triple (t) bonds, respectively, Ray et aLa1 find f A d s t = Cs,d, t(ZA/rA);

ca = 0.484, c d

= 0.354, Ct = 0.273

(158)

Table 7 indicates the good agreement between these values and values of Mulliken’s electronegativity, given according to equation (151) by %A, Mullfken

= 3(IA f

(1 59)

AA)

Table 7 Binding energy of a bond electron to the atom and Mulliken’s electronegativity x* Molecule Single bond Ha Liz Naz K2

Fa c12 Brz I2

%lev C= Cs=O. 484 7.2 2.8 2.5 2.2 8.8 8.9 8.2 7.4

Double bond

C= cdo. 354

0 2

9.5 7.7 7.1 6.2

S2 Se2 Te2 Triple bond N2 P2 As2

Sbz

Biz

9.6 7.4 7.0 6.5

C= Ct =0.273 9.8 6.3 5.7 5.0 4.3

After Ray et al. in ref. 9 1. 98

7.1 3 .O 2.8 2.4 10.4 8.3 7.6 6.8

W.J. Gordy, J. Chem. Phys., 1951, 19, 792.

8 .O 7.2 7 .O

6.0 4.2

Electron Density Description of Atoms and Molecules

137

where IA and A A denote ionization potential and electron affinity, respectively, of atom A. In the simple bond charge model for the diatomic AB, the natural definition of the electronegativity of atom A, in the final molecule after charge transfer, has the form

and similarly

where C depends on bond type. These must be equal to each other and to the molecular electronegativity XAB. Assuming all molecules have the same bond type, this leads to the formulae

The AB electronegativityis a weighted arithmetic mean of the electronegativities of atoms A and B. Tables 8 and 9 give some electronegativity values obtained by Ray et al.91from the above formula. The values are in good general agreement with those estimated or determined by other methods, for example from the geometric mean XAB

= (xAXB)1'2

(163)

Equation (163) is not clearly superior to the formula (162) which is a direct

Table 8 Electronegatiuities x for some AB systems* Molecule HF HCl HBr BrF CIF

ICl BRCl LiH NaH KH

so

SeO TeO PN AsN SbN SbBi * After Ray et al. in ref. 91.

Equation (162) 8.8 7.9 7.5 8.7 9.2 7.5 8.0 4.2 4.1 3.6 6.8 7.2 6.3 6.5 6.3 5.7 4.4

Equation (163) 11.o 8.5 8 .O

9.2 9.7 7.4 6.9 4.2 3.7 3.3 8.6 8.3 7.8 7.5 7.3 6.6 5.1

Other estimates 8.6 7.7 7.4 8.9 9.3 7.5 7.9 4.6 4.5 4.2 8.5 8.2 7.9 7.6 7.5 6.9 5 .O

138

Theoretical Chemistry

Table 9 Electronegativities of some polyatomic molecules* Molecule NHa CFz NFz c02

NO2 Ha0 CSa SO2

CH3 BF3 SF3 Pc13

cos SF5

Value of Ray et al. 7.3 8.7 9.5 7.1 7.6 7.3 6.3 7.0 6.9 8.2 9.0 7.8 6.6 9.4

(IP+ EA)/2 6.1 7.2 7.5 6.6 6.5 7.8 5.6 6.7 5.5 9.1

7.8 5.4 5.8 9.3

* After Ray et al. in ref. 91. consequence of the simple bond charge model. It should be noted that the result in equation (162) is independent of the location of the bond charge in AB, i.e. of the relative magnitudes of rl and r z . Simple Charge Transfer Model for Electronegativity Neutralization.-As discussed by Pam et al.,83a molecule may be regarded as a superposition of atoms, each in an appropriate state with not necessarily integral charge. A molecule after charge transfer may be regarded as arising from charge transfer between neutral atoms in valence states, the driving forces for the charge transfer being the electronegativity difference between these atoms. In the final state, which is a state of minimum energy, the electronegativities have equalized. It is possible to compute the final electronegativity if the electronegativities of atoms are known as functions of the number of electrons they contain. as well as other workers, Ray et dS1 Following Iczkowski and assume that an equation of the form E ( N ) = CiR

+ C2R2

(1 64)

adequately represents the energy of an atom in its various states of ionization, relative to the neutral atoms; IV is the number of electrons minus the nuclear charge. The constants C, and C2 are characteristic of the nucleus under consideration and can be determined from available ionization potential and electron affinity data (cf. discussion in Section 7). For atoms A and B we write € A ( N )= arm

+ a2R2

and

(165)

€B(N)= b i N + bzR2 (166) The electronegativity equalization principle then demands that for the final molecule

'4

R. P. Iczkowski and J. L. Margrave, J. Am. Chem. Soc., 1961,83, 3547.

Electron Density Description of Atoms and Molecules

139

from which it follows that

Electronegativity values obtained by Ray et al. from this equation are recorded in Tables 8 and 9; good agreement with other estimates is obtained. An interesting connection between the above formula and equation (162) can be demonstrated. In the quadratic approximation of equation (164), the constants a,, b,, a2,and 6 2 are given by and Hence one can write the above equation for XAB as

where the quantities

are characteristic properties of atoms A and B, respectively. If equations (162) and (172) are both valid, there would be an approximate proportionality between 01 and the covalent radius; if one simplified equation (162) further by replacing RABwith +(RAA+RBB)it would be the same as equation (171). Figure 10 shows that such a proportionality exists: approximately RAA

IA-AA- 3.6 2

(173)

Extension of these results to polyatomic molecules is straightforward. Relations for ABm and ABC systems, as given by Ray et al., are listed in Table 10 and some numerical predictions are given in Table 9. Again there is good agreement with other estimates. Total Energy, Sum of Orbital Energies, and E1ectronegativity.-To conclude this discussion on electronegativity and chemical potential, let us return briefly to the relation (84) between the total energy of a molecule at equilibrium and the sum of orbital energies. As discussed in Section 13, Mucci and March50 have made corrections to this relation due to the non-zero value of the chemical potential and to density gradients. We reproduce their results in Table 11 for the deviation A from equation (84), defined by A = E-3Es

(174)

Table 11 has been constructed from available self-consistent field calculations for a variety of molecules. As the argument of Mucci and March implied, both signs of A are found in Table 11.

140

Theoretical Chemistry

15

10

5

RAA 2

Figure 10 Shows approximate proportionality between OL defined in equation (172) and covalent radius, following Ray et al.gl Formula (173) follows from this plot, which confirms the approximate equivalence of the electronegarivity formulae (162) and (171)

Table 10 Formulae for A B , and ABC molecules*

ABm

Charge conservation N'+mN"=O

ABC

N1+N1l+N1ll=O

Molecule

Electronegativity equalization a1 2mN1 =br 2b2N al+2azN1

+

+

X

albz+ mazbl - ( m a + &-) a1bzc2 azblcz a z b m

+

+

= c1+ 2czN'll

* After Ray et al. in ref. 91. Mucci and March emphasized that the large deviations in negative sign in the selection of results in Table 12 are associated with the electronegative elements 0 and F and confirm that the deviations A depend on the chemical potential, if we use the equivalence of electronegativity and p proposed by Pam et aZ.,ss and displayed in equation (156). We note that, in principle, one can contemplate making a gradient expansion of the chemical potential,g6but in practice this does not seem a very useful way of calculating p for molecules. It does emphasize though that the separation between density gradient v p and p in equation (120) may not be the most useful way to discuss quantitatively the deviation A defined in equation (174). Nevertheless the above study of deviation A from equation (84) strongly supports the proposal of Parr et ~ l . linking * ~ electronegativity and chemical potential. It may 96

N.H. March, J. Chem. Phys., 1979,71, 1004.

141

Electron Density Description of Atoms and Molecules

Table 11 Values of sum of orbital energies E8 and total energy E from sevconsistentfield calculations at equilibrium configuration* Molecule H202 F2

CHsOH c02

C3H4 CHzCO CH2N2 N2O NHzCN BH3CO CH3CN CHsNC CHaN2 0 3

CF2 BH3 CHI NH3 H2O HF C2H2 HCN N2

co

BF B2H6 C2H4 N2H2 HzCO C2He N2H4

(3P) Es -141.1 - 180 -111 - 180 -118 - 149 - 147 - 178 - 146 - 138 -132 - 132 - 147 -208 - 219 -28 -41 -55 -71 -90 -78 -92 - 107 - 109 -116 -57 -79 - 108 -110 -81 - 109

E - 151 - 199 -115 - 187 - 116 - 152 - 148 - 183 - 148 - 139 - 132 - 132 - 148 -224 - 236 - 26 -40 - 56 -76 - 100 - 77 -93 - 109 -113 -124 -53 - 78 -110 - 114 - 79

-111

* Energies are in Hartree units (e2/ao= 1). Table 12 Deviation A as percentage of I El Molecule H202 F2

CH30H CH3F c02

N2O 0 3

CF2 H2O HF BF BH3 B2H6 C2H4 C2H6 C8H4

(A! IE I)I % -6 - 10 -4 -7 -4 -3 -8 -7 -6 - 10 -7 +6 +7 +2 +2 +1

Diference A=E-(3/2)Ee - 10 - 19 -4 -7 +2 -3 -1 -5 -2 -1 0 0 -1 - 16 - 17 3-2 +1 -1 -5 - 10 +1 -1 -2 -4 -8 +4 +1 -2 -4 +2 -2

142

Theoretical Chemistry

be some time yet, however, before direct calculations of chemical accuracy of p, or the Sanderson electronegativity x from equation (156), can be made from the full Euler equation of the density theory. For the time being, wave function calculations of the type to be discussed in the following section offer a practical way of obtaining numerical values for the density and energies, 17 Wave Function Calculations and Density Functional Theory The general Euler equation of the density description, as written in equation (156), shows that the sum t, + E X C is required as a functional of the density. The reason we stress this is that, in density theory based on this equation, we can effect separation of kinetic and electron exchange and correlation contributions either in a fundamental manner, in which we include kinetic energy in t and leave only potential energy in E X C , or we can choose t, to be single-particle kinetic energy, corresponding to the correct many-electron density p(r), and include correlation kinetic energy in EXC. The latter approach, already adopted in writing equation (147), is the one that has been used historically in introducing correlation into the density theory (see, for example, ref. 4). For present purposes, what is important is the observation of Kohn and Sham7' that with this separation into single-particle kinetic energy and electron exchange and correlation, equation (146) has then the form of a single-electron problem, the potential energy in which the electrons move being equation (147). This, as we have already stressed, shows that this oneelectron potential energy is the sum of two parts: a Hartree term, but using the many-electron density, plus the contribution E X C from exchange and correlation. In Dirac-Slater theory, one simply uses the p l l s form in equation (51), or with a variable multiplying coefficient in the so-called Xa method; this being thought of as a way of approximately simulating correlation. A more basic method, as stressed in Section 15, is to use for E X C a local density assumption in which one takes over locally the results of the uniform interacting electron gas correlation and exchange energy, numerous approximate forms of this being available and all giving rather similar values. Kohn and Sham71 emphasized that, to avoid approximating the kinetic energy density within this single-particle treatment of t,, one could go back to the one-electron Schrodinger equation, with potential V(r) now including electron exchange and correlation through equation (147), and solve this by already well established methods. This solves one major problem of the density description; it avoids making any approximations in the single-particle kinetic energy. However, in work on atoms and molecules this means that one is resorting, almost immediately, to numerical procedures, and one does not retain the simple, approximate underlying relations which we have exposed in the earlier part of this article. Nevertheless, this step, while losing some of the advantages of the density description, can, as we shall see in some examples below, lead to results of chemical accuracy. We can say then that Kohn and Sham's'' observation builds a bridge between the density description and earlier work in which one proceeded entirely by wave function calculations. In the following, the wave equation has been solved for

Electron Density Description of Atoms and Molecules

143

single electron equations in which the potential energy V is given by equation (147), with Y(r) approximated using the approximate solution of the homogeneous intelacting electron assembly. This is oversimplified in one respect which we shall state here and which is set out briefly in Appendix 4. There are obviously cases, in molecules and solids, where it is important to discuss spin density. One then has to generalize the density description to include the density pt of the upward spin electrons and pi of the downward spin. In Appendix 4 we show how the density description has to be generalized to write the total energy of the system as a function(a1)of these spin densities. The total density is the sum

P(4

=

Pf(4

+ fw

(175)

and the spin or magnetization density m is the difference This treatment, based on a local approximation to the many-electron terms of the energy will be referred to, for example, when we come to discuss the work of Harris and JonesDson the iron-series dimers in Section 17 below. First Row Diatomic Molecules.-The single-particle potential version of the density description based on equation (146) has been applied to first row diatomics by a number of workers, the exchange and correlation contribution to V(r) being approximated by a local density form. Extensive calculations were reported by Heijser et aLg7They used the discrete variational method to solve the Hartree-Fock-Slater version of the local density method. We shall not go into the technical details here. The agreement with experiment was fairly good. In later work Gunnarsson, Harris, and Jones 99 following related studies on H2100have employed the density functional method to calculate spectroscopic constants for B2,N,, 02,F,, CO, and BF and for the four lowest lying states of C,. They likewise find that their results are in good agreement with experiment. They are consistently better than Hartree-Fock accuracy, and comparable with the results of configuration interaction calculations. The method they used to solve the density functional equations, the so-called linear combination of m u f k tin orbitals (LCMTO), due originally to Andersen and Woo11ey,101,102is described by them in detail and for these molecules we shall simply discuss the results. The first row homonuclear molecules B2,C , , N,, 02,and F, have been frequently used to test calculational procedures, owing to the small number of orbitals involved in the bonding and the relative simplicity of the ground states. Changes in the nature of the bonding with position in the Periodic Table have long been recognized and used as bases for descriptions of the chemical bond, J. Harris and R. 0 .Jones, J. Chem. Phys., 1979,70, 830. Heijser, A. T. Van Kessel, and E. J. Baerends, Chem. Phys., 1976, 16, 371. 08 0. Gunnarsson, J. Harris, and R. 0. Jones, Phys. Rev. B, 1977, 15, 3027. 99 0. Gunnarsson, J. Harris, and R. 0 .Jones, J. Chem. Phys., 1977,67, 3970. loo 0. Gunnarsson and P. Johansson, Int. J. Quantum Chem., 1976, 10, 307. 101 0. K. Andersen and R. G. Woolley, Mol. Phys., 1973, 26, 905. lo2 0. K. Andersen, Phys. Rev. B, 1975, 12, 3060. 86

@7 W.

Theoretical Chemistry

144

as in one of Slater's books loS for example. Of the possible heteronuclear mole cules in the first row, CO and BF have been the most widely studied. Together with N 2, they form an isoelectronic series characterized by a triple bond. Results obtained for these molecules will be discussed with other cases below. In Figure 11, the calculated binding energies EB, equilibrium internuclear separations re, and vibrational frequencies we are compared with experiment.lo4 As can be seen immediately, all the trends are correctly reproduced, and discrepancies between theory and experiment are systematic across the series.

x---x

EXPERIMENT

0

-2 m 5 w

I

10 X

';"-/-.. O"

I

X

1500-

-5

\

0

ClJlooO-

3

i 02 C2 N2 02 F2 CO BF 31g

;ll

I q 3rg ;ll

l1+ 11+

Figure 11 Calculated binding energies EB, equilibrium internuclear separation re and vibrational frequencies we obtained by Gunnarsson et al.,gQcompared with experiment. The trencis are correctly reproduced by the density functional calculations across the whole series of first row diatomic molecules

As for the charge density plots, it is found that for the homonuclear molecules features are similar to those found in HF calculations. In each case, charge is transferred from the region close to the nuclei, but the redistribution varies C. Slater, 'Quantum Theory of Molecules and Solids', The Self-consistent Field for Molecules and Solids, McGraw-Hill, New York, 1974, Vol. 4. K. P. Huber, 'Constants of Diatomic Molecules' (in American Inst. Phys. Handbook), McGraw-Hill, New York, 1972, Sec. 7g.

loS J. Io4

Electron Density Description of Atoms and Molecules

145

greatly across the series. Some brief comments on the individual molecules are made below. C, Multiplets. The ground state of the C, molecule was a matter of controversy for many years. The four most tightly bound states have energies within 1 eV. It is now established that the l Z ; state is 0.08 eV lower in energy than the 311u state. Subsequently, Fougere and Nesbet lo5showed that a configuration interaction calculation could give the correct trends for the low-lying states, though the sensitivity of their results to basis set and choice of configurations is quite pronounced. The energetic ordering of the states obtained by Gunnarsson et aZ.08*O D is not correct, with the 3rIustate lying lowest in energy and lC$ lying slightly above both lIIU and 3C;. Ground State of N,. In Table 13, the spectroscopic constants of the N, ground state (lZ$) obtained by Gunnarsson et al. are compared with experimentlo*and with the Hartree-Fock results of Cade et aZ.lOd Table 13 Spectroscopic constantsfor N, ('C:)

E~/ev re(ad oe/cm-l a

HartreeFocka

Gunnarsson

5.27 2.0134 2729.6

7.8 2.16 2170

et

a1.b

ExperimentC

9.90 2.0742 2358.1

Ref. 106; ref. 99; ref. 104.

The charge density difference maps from the Gunnarsson et aI.O9 calculations are compared with experimentlo7 and with Hartree-Fock results l o 8 in Figure 12. There are differences between theory and experiment which are not presently resolved to the author's knowledge. Ground State of CO. The CO molecule has been extensively studied, both experimentally and theoretically. Table 14 compares ground-state ( l X + ) spectroscopic constants calculated by the Hartree-Fock methodlogand by the density functional approach with experiment.lo4 In addition to these spectroscopic constants, the polar nature of the molecule provides a further measurable quantity, the dipole moment. Since the intensities of infrared vibrationrotation bands allow the dipole moment to be determined as a function of C-0 separation this provides a useful comparison with the results of ab initio calculations. For example, the positive sign obtained from the equilibrium dipole moment by Hartree-Fock calculations was viewed as a reason to question the negative value found experimentally, whereas the current view is that the positive sign is a defect of the Hartree-Fock method. Charge density contours for CO are shown in Figure 1 3 a - c for three internuclear separations. The transfer of charge which takes place during the 105 108

107 108

100

P. F. Fougere and R. K. Nesbet, J. Chem. Phys., 1966,44, 285. P. E. Cade, K. D. Sales, and A. C. Wahl, J. Chem. Phys., 1966,44, 1973. M. Fink, D. Gregory, and P. G. Moore, Phys. Rev. Lett., 1976, 37, 15. D. A. Kohl and L. S. Bartell, J. Chem. Phys., 1969, 51, 2891, 2896. W . M . Huo, J. Chem. Phys., 1965,43, 624.

Theoretical Chemistry

146

1 a. HARTREE-FOCK

c. PRESENT WORK

DIFFERENCE DENSITY CONTOURS

Figure 12 Charge density diflerence maps of Nz from reJ 99, compared with HF results of ref. 108

Table 14 Spectroscopic constants for CO(lC+)from work of Gunnarsson et al.Oo HartreeFock

EdeV re(ad w,/cm-l

7.89 2.08 2431

Gunnarsson et al.QQ 9.6 2.22 2100

Experiment 11.2 2.13 2170

formation of the CO molecule was discussed a long time ago by Mulliken. For large internuclear separations, C+O- dominates, whereas the reverse polarization occurs for small distances. Near the equilibrium separation the dipole moment passes through zero with a steep slope. Detailed measurements have subsequently confirmed this general behaviour. The picture of dipole moment versus internuclear distance is as shown in Figure 13d, the Hartree-Fock results of Huo logand the experimental finding of Chakerian 110 also being plotted. Summary on First Row Diatomic Molecules. It is satisfyingthat the density description gives consistently accurate results for binding energy curves and dipole 110

C.Chakerian, J. Chem. Phys., 1976,65, 4228.

Electron Density Description of Atoms and Molecules

147

t c1

W

> m W

Q

Y

Figure 13 (a)-(c) Charge density contoursfor CO for three internuclearseparations,from the work of Gunnarsson et al.Q9For large separations, the density m p s show that C+O- dominates. For small separations, the opposite direction of polarization obtains. The consequences of these density maps for the d&ole moment are shown in Figure 13(d) of dipole moment of CO against internuclear separation. This picture is a consequence of the contours of constant charge density shown in Figure 13(a)--(c). The results are taken from the work of Gunnarsson

et al.QQ moments of first-row diatomic molecules. We stress that Gunnarsson et al. achieved this accuracy with a local density approximation for the exchange and correlation energy. In order to examine the sensitivity of their numerical results to the choice of EXC in equation (147), they repeated some of the calculations with the X a approximation to EXC, namely, in atomic units,

which is a constant (3a/2)times the exchange energy density of a homogeneous electron gas of density p given in equation (50). A comparison of their results F2,and CO. This table shows and the Xa scheme is shown in Table 15 for NZ, that the two schemes give remarkably similar results for r e and me. If spinpolarization effects are excluded, binding energies for X a and the local density functional are also very similar. Since the molecular and separated atomic energies are significantly different, this is further evidence for the cancellation of errors which results from the consistent use of a single functional. However, the spin-polarized local density functional energies are consistently lower. Since

148

Theoretical Chemistry

Table 15 Comparison of results of Gunnarsson et al. with X a results (u=O.7) Xa

Gunnarsson et al.g9

Experiment 104

N2

re (ao) W€./CNl-'

2.16

2060 5.6

2.16 2070 7.8

2.074 2359 9.76

2.91 790 0.6

2.679 891.9 1.60

F2 re

me

EB

2.91

840 0.3

co re We

EB

2.22

2Ooo 8.3

2.22 2090 9.6

2.13

2170 11.22

molecules N,,F,, and CO have no spin polarization in the ground-state, this arises because the Xa functional overestimates the energy lowering due to spin polarization in the constituent atoms. This effect is also observed in solids, where the Xa approximation overestimates the tendency towards the ferromagnetic state. Dunlap et a2.l1l have considered the first row diatomics by a local density treatment of exchange and correlation, and this work supports the conclusion of Gunnarsson et aZ.,@@ and of the independent work of Baerends and Roos,l12 that, by comparison with experiment, local density methods give excellent answers for most properties of the first row diatomics. Alkali Dimem.-Hams and Jones1lShave also calculated binding energy curves for the lZiground state of the alkali dimers Li ,-Fr ,, using the density functional method. They obtain a satisfactorydescription of this series, based on the approximations of (i) a frozen core and (ii) a local exchange and correlation energy function. We shall briefly summarize the approach, and their results below. It must be stated that other ab initiu methods have been applied successfully to Lis and Na,. What Harris and Jones (HJ) show is that the resulting spectroscopic constants they calculate agree with multiconfiguration self-consistent field (MCSCF)work for Li, and Na,, and agree well with experiment for the heavier dimers. Energy Functional. The energy functional adopted, assuming a local density approximation for the exchange and correlation energy, takes the form, with &PXC denoting now the exchange and correlation energy per particle,

111 112 11s

B. I. Dunlap, J. W. Connolly, and J. R. Sabin, J. Chem. Phys., 1979, 71, 4993. E. J. Baerends and P. ROOS, Int. J. Quantum Chern., Symp., 1978, 12, 169. J. Harris and R. 0. Jones, J. Chern. Phys., 1978,68, 1190.

Electron Density Description of Atoms and Molecules

149

Here the electron density

is that corresponding to the electronic ground state for nuclei with charge Zf and positions Rt. The yn(r) terms are the self-consistent solutions of the one-electron equations of density functional theory. The exchange and correlation energy per particle sgc(p), following Section 15, is taken as that of a homogeneous interacting electron assembly of density p. Finally the fn terms are occupation numbers determined by the symmetry of the state under consideration. In the approximation in which the core eigenfunctions are independent of internuclear separation and have zero overlap, the above energy expression separates as E = Ec

+ Ev

(180)

where Ec, which involves only the cores, is independent of the internuclear separation, and a term

In this equation, en are the self-consistent one-electron energies for the valence orbitals, dv is the Coulomb potential of the valence electrons, Zvt is the net charge of the i t h core and the electron density has been written d r ) = Pc(4

+

P V W

(182)

The importance of separating E as in equation (180) is clear from the case of Cs,, where Ec 3 x lo4 Ry, while the binding energy is only 0.03 Ry. Since Ec is independent of internuclear distance Rnn it need not be calculated and the energy curve is given by N

EB(r) = Ev(r)-Ev(00)

(183)

where each of the terms is, for the alkali dimers, <0.5 Hartree. As HJ point out, in practice the above scheme is complicated by the slight overlap of the core densities on different atoms. In their work, the ground-state energy EA, its components EZ and E & and core and valence densities p,"(r) and p $ (r), respectively, are determined self-consistently for each constituent atom. The core density is then renormalized to a suitable radius Re by the addition of a term ApS(r) = ar3-br5, r < Rc

where a and b are chosen so that

and 6

(184)

150

Theoretical Chemistry

This core renormalization is such that the core density is changed predominantly in the region near Rc. The renormalized core then has the correct total charge and the core density vanishes for r 3 Rc. With the new core density fixed, the iterations were continued by HJ until the valence density was again self-consistent, with a corresponding value of E$ denoted by E $ ( R c ) .The change in E t due to the core renormalization, which we denote by AEA,(Rc),is always positive, due to the increase in the potential felt by the valence electrons close to the nucleus. The valence-valence Coulomb energy is reduced owing to the spreading of the valence charge and the eigenvalues move upwards. It should be noted that the remaining terms in equation (181) generally give a small contribution. The binding energy curve E i c ( r ) in equation (183) was then found by HJ by evaluating Ev(r) using the frozen core density determined above. The energy curve depends on Rc and can be calculated only for rb2Rc. The usefulness of the procedure lies in the large cancellation of the effects of core renormalization in the molecule and atoms, so that E:C(r) is much less dependent on Rc than its two components separately. In some cases, such as Cu,, HJ find that this error cancellation is almost complete, and the cancellation is substantial even in the heavier alkali dimers, which have very extended cores. Results. We refer the interested reader to the series of papers by HJ for full details of the calculations. We note that spin polarization is included by HJ for the atoms and for the molecular "I;'u states. Turning to the ground state lZ;, the results are summarized and compared with experiment in Figure 14. The agreement is seen to be pretty satisfactory, the energetics of the dimer formation being described quite accurately even for the heaviest observed member Cs,. Regarding the charge density, the alkali dimers behave very differently from the s-p bonds in the first row molecules B2-F2. In particular, in these cases the molecular density at the nuclei is less than in the isolated atoms. HJ point out that in the detailed work on H , by Kol0s and Wolniewicz,l14 the first excited state "c", was found to have a very weak minimum at a large separation. This binding presumably arises from a van der Waals force which is not included in the density functional theory when a local approximation to exchange and correlation is employed. Nevertheless, as HJ point out, their study of the corresponding state of the dimers Li,-Cs, revealed a weak, but definite maximum in each case. Rough estimates of binding energy and equilibrium separation are shown in Table 16. It is, of course, possible that these results are a consequence of the local spin-density approximation, so that further work will Table 16 Estimates of binding energy arid equilibrium separation of alkali dimers, for first excited state ":, from the work of Harris and Jones l I 3 re(ao)

114

Liz 6.8

Na2 9.2

0.10

0.04

K2

Rb2

10.4

12.0

W.Koles and L. Wolniewicz, J. Chem. Phys.,

0.06

0.03

1965, 43, 2429.

cs2 N

14 0.03

Electron Density Description of Atoms and Molecules

151

1.d-

Liz

1

I

I

I

I

No2

K2

Rb2

Cs2

Fr2

Figure 14 Results of Harris and Jonesll3 for the ground state 1ct of the alkali dimers. Binding energies, equilibrium separations, and weighted vibrational frequencies are shown (,u1J2w, where p is reduced mass).Experimentalvalues(crosses)from ref. 114. Circles from Harris and Jones’ calculations

be needed to clarify this point. However, in the latter context, HJ also note that earlier work by Kutzelnigg et al.l15 on the 3 Z i state of Liz showed a weak minimum at re- 8a, with a binding energy EB=0.03 eV. Iron-series Dimers.-Harris and Jonesgs have in addition calculated binding energy curves for low-lying states of the 3d-dimers K,-Cu2 using the density functional formalism with a local spin-density approximation for the exchange and correlation energy. Although band spectroscopic data are available only for K,, Ca,, and Cu,, dissociation energies have been estimated from high-temperature mass-spectrometric data and recent advances in matrix isolation spectroscopy promise to yield more information concerning the low-lying states. Because of the large number of possible configurations resulting from the degeneracy of the atomic d orbitals, only limited configuration interaction calculations are currently avail115

W. Kutzelnigg, V. Staemmler, and M Gtlus, Chem. Phys. Lett., 1972, 13, 496.

152

Theoretical Chemistry

able for Cu, and Ni, and most of the theoretical work has employed the extended Huckel method. The energy of a given dimer configuration can be usefully discussed in terms of three components. If bonding orbitals predominate, there is a bonding energy which acts to pull the nuclei together. This contribution is a maximum at the centre of the series, for the state where all bonding orbitals are occupied and all antibonding orbitals are empty (Cr ,, 'Xi). Predominant occupation of bonding orbitals generally means, however, a lowering of the spin compared with infinitely separated atoms, which tend to have ground states of maximum spin. A large bonding energy therefore implies a large spin energy and this competition is responsible for differences between binding energy trends in the first and third rows. An extreme example is the lX; state of the Cr, dimer referred to above, which has a bonding energy calculated to be -9 eV. The energy cost of the six spin flips required to create this configuration, however, is 5.3 eV per atom, and the calculations of Harris and Jones predict this state to be unbound. If spins are now flipped in the dimer to lower the spin energy, the bonding energy is reduced and the result is a family of energy curves lying within 1-2 eV of the energy of two isolated atoms. In this dimer, therefore, the spin energy dominates, although it will be argued that one spin flip, leading to a llZ+ ground state, is energetically favourable. The third quantity determining the binding energy of a given configuration is the extent of s-d transfer. Most of the 3d atoms have 4s2 ground states and since formation of a 1og 4s-like bond is an essential feature of many low-lying configurations, the process of dimer formation involves the promotion of s electrons to the d shell. The local spin density approximation does not adequately describe this contribution to the binding energy. It is likely that a considerable part of the discrepancy between the calculations of Harris and Jones and experimental trends resides in this contribution. Because spin and s--d transfer energies are equally important in the 3d series, Harris and Jones have carried out numerous calculations for excited states of the 3d atoms, for which detailed spectroscopic data are available, to try to establish the level of accuracy of local spin density calculations of these quantities. These calculations are summarized below and will be used later in interpreting energy curves of the dimers. Spin Density Functional Results for Transition Metal Atoms. Extensive experimental data exist on 3d and 4d atoms. Th- provide a valuable check on the accuracy of the spin-promotion energy, which plays an important part in determining which dimer state lies lowest. The interconfigurational energy A E = E(dn-lsl)- E(dn-,s2) calculated by Harris and JonesQeis shown in Figures 15a and b for the 4d and 5d series, respectively. The squares show the non spin-polarized results, the full circles the spin-polarized calculations and the triangles show the experimental values. The trends in the density calculations are seen to agree well with experimental values. Earlier work of Slater et al. considered mixed configurations 3dn-Z4sz of Co and Ni.

Electron Density Description of Atoms and Molecules

Ca Sc

Ti

V

Cr

Mn Fe Co Ni

153

Cu

(b) Figure 15 Interconfigurationalenergy AE= E(dn-W) -E(dn-2s2).(a) For the 4d series, (b) For the 5d series, taken from the work of Harris and JoneseQ60, Nonspin-polarizedresults ; 0, spin-polarizedcalculations; A,experimental values

Theoretical Chemistry

154

Full details are given by Harris and JonesBs but we may summarize their conclusions from these atomic calculations as follows : (i) The energy cost of flipping an s spin in a high spin configuration is overestimated by 0.2-0.5 eV. The energies needed to reduce the spin in such a configuration are not unphysically large, however, since the subsequent d-spin flip energies are underestimated by the functional. (ii) The local spin density approximation favours d configurations quite consistently, so that the binding energy will be overestimated or underestimated if there is a nett transfer to or from the d shell, respectively. Although systematic, these errors are of the same order as the binding energy itself, particularly at the centre of the series. Therefore below, attention will be focused on trends in the dimer binding energies: precise numerical values cannot be expected because of the limitations of the local spin density approximation. Trends in Bonding of 3d Dimers. Following Harris and Jones, we first discuss the behaviour of the eigenvalues en. In their treatment, HJ include spin effects perturbatively and spin-up and spin-down eigenvalues are not split. For atoms, this procedure gives values of En near to the average of EL, ef and this is expected to be true generally. The form of the eigenvalue spectrum for a given dimer depends on the relative spreading of the 3d and 4s atomic orbitals, which varies markedly across the series. The consequences for the eigenvalue behaviour are shown in Figures 16 and 17 for ground-state configurations of Ti and Cu ,,respectively. The atomic 4s eigenvalue of Cu is 1 eV above the 3d value, for which the corresponding orbital is very compact. At large separations, therefore, levels of predominantly d character remain at their atomic energy while the overlap of the 4s tails generates 2uy, 20: bonding and antibonding orbitals with a small splitting. As this splitting approaches 1 eV, hybridization of 4s and 3d2 atomic functions becomes strong and the four hybrid a orbitals form bonding-antibonding combinations which appear to converge on the atomic 3d and 4s levels, respectively. This can be seen at the largest separation shown in Figure 17. In contrast, the d-like n and 6 orbitals tend to move together, reflecting potential changes close to the nuclei, due to the contraction of the 4s charge towards the molecular axis. At considerably smaller re values, the d overlap becomes substantial and the nu level moves down rapidly. Since the 2aZ orbital lies so high, it is clear that the ground state is 'C; and is substantially lower than the band of triplets formed by promoting one spin from either lo:, n; or 6;. At all separations, the atomic-like 6 orbitals have energies close to the atomic c 3 d , indicating that the state of Cu, corresponds to the 4s atom configuration, i.e. there is no s-d transfer on going from atom to dimer. As the atomic number Z decreases the main effect on the eigenvalues is due to the upward movement of with respect to E~~ and the changed s-d,~hybridization which results. In Ni, and Co,, the 20: orbital is still separated from the remainder, but below Fe, it becomes part of a dense band of antibonding levels. Concurrently, the la; orbital separates from the 3d-like orbitals and, in Sc,, la, and 10: lie lowest at intermediate separations. Eigenvalues for the '2; ground state of Ti, shown in Figure 16 illustrate the situation at the lower end of the series. It should be noted that the splittings of the n and 6 orbitals are

-

,

Electron Density Description of Atoms and Molecules

155

d31 -0.

1

-x

E.-.

w -0.

s31

522 322

-0.

Figure 16 Eigenvalue behaviour for ground-state conJguration of dimer Ti 2 taken from work of Harris and Jonesgs

greater relative to the a splittings, reflecting the increased importance of d overlap. In addition, the 4s-like 1 ug tends to flatten out at small re as the outermost antinode of the 4s function approaches the bond centre and 2a, shows strong bonding due to the substantial d overlap. The 6 levels do not approach the atomic d levels corresponding to d3s1or d2s2,indicating that the strong hybridization in Ti, ('Xu+) leads to a non-integral number of 4s electrons (-2.5). In this case, therefore, there is an s-d transfer on forming the dimer from two atoms. Although it might appear from Figure 16 that 1 a; 2ai +u ('Xi) could lie spin splittings comparable lower than the ground state la; 26; n: 8; l a z l ('Z;), with those one finds in the atom would raise 2ah, nh above d;, "1. Since eigenvalue orderings change substantially with configuration and with re, such arguments are generally insufficient to predict the dimer ground states and many calculations are necessary to establish which state lies lowest. Bonding Energy and Eigenualues. Eigenvalue behaviour suggests the following picture of the bonding energy.es At large separations bonding 4s-like orbitals provide an attractive force which, neglecting the effects of spin and d electrons, would give a contribution varying from -0.5 eV (K,) to -2 eV (Cu,), the increase resulting from the contraction of the 4s tails. As d overlap increases an additional force draws the nuclei together and the binding energy is raised. The equilibrium separation is determined by the point where this force, together with

Theoretical Chemistry

156

01-

0 5-

3

'%I I

I

5

6

Figure 17 Eigenvahe behauiour for ground-state configuration of dimer Cu2 taken from work of Harris and Jonesg6

the 4s force which becomes repulsive at small re due to overlap with neighbouring cores, balances the electrostatic repulsion due to incomplete screening of the nuclei. The form of the binding energy curve depends on the relative strengths of the s and d bonding. For example, a closed s-shell configuration with several d bonds (e.g. azt) will show a weak variation at large separations but a strong minimum with a large vibration frequency. In contrast a closed d-shell configuration will have a shallower curve with a minimum at a larger value of re. Ground State Binding Energies. Assuming that the 3d metals have singlet ground states, GriffithllO was able to explain trends in the cohesive energy on the basis of a competition between bonding, spin, and s-d promotion energies. In the dimers, the same competition occurs, but the situation is complicated by the fact that the ground-state configurations are unknown. The calculations of HJ show that, except for Ca,, Cu2, and possibly Sc,, the bonding energy in low-spin ground states is less than the spin energy. The approximate density functional theory therefore predicts high-spin ground states for most of the 3d dimers. The binding energies of the iron-series dimers are shown in Figure 18. The trend is similar to that found in density calculations of the cohesive energies,l" two peaks and a minimum at Cr,. Binding energies are generally considerably less than the cohesive energies, as can be seen from the following argument. 116 117

J. S. Griffith, J . Znorg. Nucl. Chem., 1956, 3, 15. J. Friedel, in 'Physics of Metals 1, Electrons', ed. J. M. Ziman, Cambridge University Press, 1969.

Electron Density Description of Atoms and Molecules

" K

Ca sc Ti

v

Cr

Mn

Fe

157

co

Ni

cu

Figure 18 Binding energies of iron series dimers, taken from the work of Harris and Jones 96

In a cluster of N atoms, the bonding energy increases faster than N, because each atom can bond to more than one neighbour, while the spin energy remains linear. Neglecting magnetic ordering energies, larger clusters should tend increasingly to have low-spin strong-bonding ground states. The pronounced minimum at Cr, is due to the dominance of the spin energy. As noted above, this is an extreme case where, according to the functional, even one spin flip over the non-bonding state of maximum spin is energetically unfavourable. Experimental values for the binding energies EB are also shown, together with estimated errors and while agreement is good at each end of the series, serious discrepancies occur for the dimers Ti ,-Co ,. Thus, the local spin density method fails to reproduce the experimental trend. But as will be argued below, most of the trouble can be put down to the defects in the local spin density approximation revealed in the atomic calculations described above. In particular, this approximate functional tends to favour configurations involving d electrons. It is striking that minima in dimer binding energies and cohesive energies occur at Cr according to theory, and at Mn experimentally. The reason is that the Mn atomic ground state (3d6 4s2) is stable by 2.1 eV against s + d transfer. The low-lying dimer states, on the other hand, have a configuration close to 4s1J so that a net s-d transfer of 1 electron occurs on dimer formation. Summary and Conclusions on Iron-series Dimers. In spite of reservations about the local spin density functional, calculations based on this approach yield much interesting information about these dimers. In particular, the spin energy is such a dominating influence that dimer ground states tend to have a net spin only one less than the separated atoms. Calculated binding energiesagree with experiN

N

6*

158

Theoretical Chemistry

ment at the ends of the series, and the discrepancies in Ti,-Co, can be explained plausibly. Although the dimers exist in high spin states, bulk metals are spinless (if we neglect magnetic ordering energies)owing to the extra freedom ford and s orbitals to bond with more than one neighbour. For each element, therefore, a minimum cluster size should exist beyond which the ground state is a singlet. As we stressed earlier, this section has been devoted to somewhat arbitrarily chosen applications of the use of the one-body potential Y ( r ) of the density description, combined with solution of the resulting one-electron Schrodinger equation. As already emphasized, this one-body potential is the bridge between the density description and the customary quantum chemical route uia the one-electron Schrodinger equation. We have, of course, throughout this section been attempting to account for exchange and correlation interactions through the choice of the one-body potential. In the penultimate section of this article, we return to the electron density itself, as a conceptual framework within which some basic aims of quantum chemistry can be discussed. But the shift in focus is away from detailed description of the density and towards its topological characteristics.

18 Topology of Molecular Charge Distributions Bader and his colleagues have developed topological arguments for molecular charge distributions. In particular Bader, Nguyen Dang, and Tal 118 have emphasized that the concepts which are essential to the description of a chemical system are: (i) the existence of atoms, or functional groupings of atoms in molecules as evidenced by characteristic sets of properties; (ii) the concept of bonding, i.e. that the stability of a molecule may be understood by assuming the existence of particularly strong interactions between certain pairs of atoms within a molecule; and (iii) the associated concepts of molecular structure and molecular shape. Bader and his co-workers have presented evidence that these concepts have evolved because they are consequences of fundamental, topological properties of the distribution of charge (electronic and nuclear) in a molecular system. Both for molecules and for crystals, attention is usefully focused on the gradient of the charge density V p ( r ) . The properties of this quantity, according to Bader et al., are determined by the number and nature of its ‘critical points’ defined as points at which V p ( r ) = O . This point of view, which evolved from extensive studies of the properties of molecular charge distributions, was recently formalized by Collard and Halllls who demonstrated the utility of orthogonal trajectories, i.e. the paths traced by the gradient vectors of a scalar field, in the analysis of scalar functions of several variables. The universal properties of p ( r ) for molecules in their equilibrium configuration have been identified through such analysis by Bader et al.llSIt is shown by these workers that the topological properties of p ( r ) yield definitions of an atom in a molecule and of a chemical I*#

R. F. W. Bader, T. T. Nguyen Dang, and Y. Tal, J. Chem. Phys., 1979,70,4316.

Electron Density Description of Atoms and Molecules

159

bond, and provide a basis for the definition of molecular structure and molecular shape. We here summarize the results of Bader et aZ.ll*which are concerned with the definition of molecular structure and with the extension of this concept, together with the associated concept of a bond, to the dynamic case. A precise description and physical interpretation of the making and breaking of chemical bonds is presented by these workers in a quantitative analysis of the evolution of molecular structure. The topological analysis of the dynamic system, as pointed out by Collard and Hall,119 falls naturally into the realm of an existing and elegant mathematical theory, the catastrophe theory of Thom.120 Theory of Topological Dynamics of Molecular Systems.-The description of a molecular system of interest within the Born-Oppenheimer approximation is realized by associating to each point X of the nuclear configuration space, a scalar field which is the electronic charge distribution p(r, X). The topological properties of p ( r , X) are characterized by its gradient field V,p(r,X). For a given nuclear configuration X, we can write the defining equation of the gradient path of p ( r , X)as

The topological analysis of p ( r , X) then proceeds through the search for and identification of its critical points. In the neighbourhood of a critical point, the field p ( r , X) is expanded by Taylor’s theorem, the first non-trivial terms being those quadratic in the variables r. The collection of the nine second derivatives of p(r, X)constitute the so-called Hessian matrix A ofp(r, X) at the critical point. Topological Definition of Atoms, Bonds, and Structure.-The definitions of an isolated atom, of an atom in a molecule, of a chemical bond, and of molecular structure derive from the properties of critical points of the charge density. In the neighbourhood of a critical point rc, defined by

equation (187) becomes

where A is the Hessian matrix of p at r c :

Let Al, A,, A3, be the eigenvalues and vl, v,, and v 3 the associated eigenvectors of A. A general solution of equation (189) is then r (s) = 119 l20

a u1 e h l s

+

~

u eA2s 2

+y

u3 eA3s

(191)

K. Collard and G. G. Hall, Inr. J. Quantum Chem., 1977, 12,623. R. Thorn, ‘Structural Stability and Morphogenesis’, Benjamin, Reading, Massachusetts, 1975.

160

Theoretical Chemistry

The nature of the critical point is determined by the sign of the real eigenvalues Ai. The collection of gradient paths which terminate at a given nucleus defines the volume of space associated with that nucleus; they define an atom. The common boundary between two neighbouring atoms contains a particular critical point which generates a pair of gradient paths linking the two neighbouring nuclei. The union of this pair of gradient paths and their end points is called a bond path. The network of bond paths defines a molecular graph of the system. Having defined a unique molecular graph for any molecular geometry, the total nuclear configuration space is partitioned into a finite number of regions. Each region is associated with a particular structure defined as an equivalence class of molecular graphs. A chemical reaction in which bonds are broken and/or formed is therefore a trajectory in configuration space which must cross one of the boundaries between two neighbouring structural regions. These boundaries form the catastrophe set of the system. The partitioning of configuration space by the catastrophe set into different structural regions defines a structure diagram. A structure diagram defines all structures and all possible mechanisms of structural change for a given system. The properties of the topologically defined atoms and their temporal changes are identified within a general formulation of subspace quantum mechanics. It is shown that the quantum mechanical partitioning of a system into subsystems coincides with the topological partitioning: both are defined by the set of zero flux surfaces in V p ( r ) . Consequently the total energy and any other property of a molecular system are partitioned into additive atomic contributions. For full development of all this, the reader must refer to Bader et aZ.118 Figure 19, as an illustration, shows the properties of the charge density of the ground state of H 2 0 at its equilibrium geometry. Figures 1 9 b - e show how the density varies as the geometry changes. In Figure 19c, a singularity in p ( r ) exists between the protons. It is at this geometry that a bond is formed between hydrogen atoms in the dissociation of water. Bader et al. argue that the primary concepts of chemistry find precise definition in terms of the topological properties of a molecular charge distribution. One is led to view molecular structure and its stability as the result of competition between the various nuclei within the molecule for the electronic charges of the system related to the discussion of electronegativities for different geometry. Returning to the water molecule for an illustration, the equilibrium charge distribution is dominated by the oxygen nucleus. The transfer of electronic charge from the H atoms to oxygen which occurs during the formation of H 2 0 is clear from Figure 19. The accumulated charge initially present between the protons as a result of their combined attractive powers, the H-H bond, is transferred eventually to the oxygen. The H-H bond is broken and each proton is directly bonded to the oxygen. 19 Summary and Future Directions We have emphasized in this article the development of the density description of atoms and molecules from the theory of its pioneers, Thomas' and Fermi.*

Electron Density Description of Atoms and Molecules

B

I

1

b

161

I

C

8

1

Figure 19 Properties of the charge density of the ground state of the water molecule at its equilibriumgeometry. (b)-(e) show density variation with geometrical changes. A singularity in p ( r )occurs between the protons in (c). This, according to Bader and his co-workers, to whom this Figure is due, is the geometry at which a bond is formed between the hydrogens in the dissociation of water

This simplest, statistical form of the density description already has much to say about the energies of atoms and molecules. But, as we have stressed, it rests on two major assumptions: (i) Use of a local density p5I3form of the kinetic energy density t,. (ii) Neglect of exchange and correlation interactions between electrons. Even neglecting (i) and (ii) above, the description’s statistical foundations ensure that it becomes a correct, non-relativistic theory in the limit when the number of electrons becomes very large. For atoms, this can be expressed quantitatively for the neutral case by the result that the total energy varies as Z7j3for large Z . For atoms, which have 2 5 100, the corrections due to (i) and (ii)

1 62

Theoretical Chemistry

contribute terms that are important quantitatively, O ( Z 2, and O(Z513),respectively. The Z513term comes from the Thomas-Fermi-Dirac theory, which as we have emphasized is the forerunner of the Dirac-Slater p1i3exchange potential.1° The Thomas-Fermi-Dirac theory was formally completed in 1964 by the theorem of Hohenberg and Kohn that the ground-state energy of a manyelectron system is a unique function(a1) of the electron density (see Appendix 2). But it is quite clear that exact knowledge of the functional is equivalent to exact solution of the many-body problem, which may be impossible in a closed analytic form. Therefore, the emphasis of recent developments has been to approximate to the exchange and correlation energy density by a local theory based on the solution of the homogeneous electron assembly. We strongly emphasize that this is in the spirit of the Thomas-Fermi method, as refined by Dirac, as is clear from the book by G ~ m b a sand , ~ the early review of the author.ls The present form of the local density approximation to the exchange and correlation energy density has been largely due to the work of Lundqvist and his 121* 122 There is a body of evidence in favour of the approach, which, however, does not preclude future progress in including gradient series, no doubt partially summed, into a non-local exchange and correlation energy density.*’$123 Of course, the formal simplicity of the density description, set out in the form including exchange and correlation in Section 15 of this review, is an attractive aspect of it. As Parr and his colleagues have emphasized, one return for a really accurate treatment of the Euler equation (146) of the density description would be if the chemical potential ,u reflected Sanderson’s electronegativity. The fact that in the Thomas-Fermi theory the chemical potential of a neutral atom is identically zero warns us that sophisticated approximations to the functionals, or a return to wave function calculations using the one-body potential (147) of the density description, may well be necessary for accurate work. However, the work of Parr has shown that the conceptual content of the density description in relation to electronegativity motivates simple chemical models, bond charges and charge transfer for instance, and allows thereby further insight into chemical bonding. The status of Walsh’s rules for molecular shapes still needs further fundamental clarification, though their overall usefulness can hardly be in doubt. However, the simplest density description, as shown by March and Plaskett l 8 for neutral atoms, and by R ~ e d e n b e r gfor ~ ~ neutral molecules, does indeed relate the total energy to the sum of orbital energies, with a multiplying factor of 3/2, both for atoms and for molecules at equilibrium. Corrections to this relation undoubtedly involve the chemical p ~ t e n t i a l . ~ ~ As remarked above, considerable progress has resulted from use of the onebody potential of the density description in a one-electron Schrodinger equation approach. In the language of the density description, this is tantamount to treating the single-particle kinetic energy density exactly, as suggested by Kohn 121 122

123

0.Gunnarsson, B. I. Lundqvist, and J. W. Wilkins, Phys. Rev., 1974, B10, 1319. 0. Gunnarsson and B. I. Lundqvist, Phys. Reu. B, 1976, 13, 4274. N. H. March, in ‘Quantum Theory of Polymers’, ed. J. M. AndrC, J. Dclhalle, and J. Ladik, Reidel, Dordrecht, 1978.

Electron Density Descrbtion of Atoms and Molecules

163

and Sham.71The bridge, built by Slater,124 between the density description and the Xa method based on equation (177) and its r e f i n e m e n f ~ ~is~ thereby ~v~~~ formally extended to cover fully electron-electron correlation. The important ideas in the topological considerations of Bader and his colleagues on the molecular electron density have also been emphasized. The connection made with catastrophe theory l l S ,119 may well prove important in future work on bond breaking, potential energy surfaces, and chemical reactions. Also combination of topological arguments with a detailed study of the gradient of the electron density in molecules, and its use in defining localized frag6 s may well be a fruitful way forward. ments At least two areas that have been omitted from the present article require comment. These are (a) study of interatomic forces using the density description and (b) study of the dynamics of electron density distributions in atoms and molecules. The historical background to (a) is covered to some extent in GombM book,* and for recent work reference may be made to the work of Gordon and Kim.75We have regarded (a) as properly dealing with a province different from that covered in the present article. Here, we have viewed the molecule as a multi-centre problem to be solved for the electron density. In the description of interatomic forces by the density method,75the two-centre density is a superposition of atomic (or ionic) densities. In the present article, the redistribution of electrons, required to equalize the chemical potentials as atoms are brought together to form a molecule, has been a central theme. As regards (b), the pioneering work of Lundqvist and Brandt is already summarized in ref. 10 (p. 121). This has led to exciting developments relating to collective manifestations of electron interactions in some specific shells of heavy atoms. But so far this has not been posed at any quantitative level in terms of the electron density and would take us too far outside the scope of this article. While we anticipate that the many-electron effects, subsumed here into E X C of equation (1 56), are going to necessitate approximate treatments, such as local density methods, plus non-local gradient series corrections, for some time to come, it is possible that a point is being reached where a route will soon exist to a direct calculation of the electron density, with the full shell structure absent from the Thomas-Fermi theory, in atoms and molecules. The ultimate aim here is to set up and solve a partial differential equation for the electron density, given a suitable approximate form of the exchange and correlation energy density E X C . Because of the attractive nature of this, we have felt it worthwhile to record one, admittedly very elementary, example, the linear harmonic oscillator in Appendix 5. There we demonstrate, following the work of Lawes and March,lZ7that an exact third-order differential equation exists for the particle density, for an arbitrary number of particles filling the oscillator levels. For large numbers of particles, this equation contains the Thomas-Fermi density as a limit, as it of course must do. It is too naive to expect that a simple generalization 589

124 125

126

127

67y

J. C. Slater, Phys. Rev., 1951, 81, 385. K. H. Johnson, Adv. Quantum Chem., 1973, 7, 147. N. Rosch, in ‘Electrons in Finite and Infinite Structures’, ed. P. Phariseau and L. Scheire, Plenum, New York, 1977, p. 1. G. P. Lawes and N. H. March, J . Chem. Phys., 1979,71, 1007.

164

Theoretical Chemistry

of this will exist for a general three-dimensional molecular potential energy V ( r ) . But Lawes and March128have discussed a route whereby approximate calculation of the electron density should be possible, without appeal to wave function calculations, and we are currently embarking on a pilot calculation on the heavy molecule UF,, though by a non-relativistic approach, which will eventually need relativistic refinement. If this can be handled, the way should be clear to make density calculations on large molecules of biological interest.lZ9 To explore molecules with a really large number of electrons is, of course, the ideal context for the density description, as its historical background makes plain. The concluding remark is that it can only enrich our fundamental understanding of the electron density description if further studies can be made on low-order density matrices, along lines laid down in the pioneering studies of L 6 ~ d i n . Indeed, l~~ using the natural orbitals that he introduced, Gilbert 131and Donnelly and Parr 132 have established that the chemical potential is the same for each orbital, a result fundamentally different from Hartree-Fock theory. Further fundamental progress is to be expected both in this direction and with regard to the topological properties of the first-order density

Appendix 1 Some Results on the Chemical Potential for Electrons Moving Independently in a Harmonic Well and in a Pure Coulomb Field.-To test some features of the approximate density description presented in the main text, we here present some results for independent electrons moving in a harmonic well. We shall discuss one-, two-, and three-dimensional wells and demonstrate the important dependence of the results on the dimensionality d.I3* As we saw in the main text, the chemical potential is defined as

where E is the total energy and N the number of particles. Using the level spacing as the unit of energy, the energy levels en are given by (n 4 2 ) . The total energy, which for independent particles is the sum of the eigenvalues, is for the nondegenetate one-dimensional case given by

+

N- 1 n=O

where the electrons singly occupy the lowest N levels. Hence from equations (Al) and (A2) we have p = N

as was noted by Lawes and March.127 G . P. Lawes and N. H. March, Physica Scripta, 1980, 121, 402. Ladik and S. Suhai, this volume, p. 49. l30 P. 0. Lowdin, Phvs. Rev., 1955, 97, 1474. 131 T. L. Gilbert, Phys. Reu. B, 1975, 12, 2111. 132 R. A. Donnelly and R. G. Parr, J. Chem. Phys., 1978, 69, 4431. I33 R. F. W. Bader, personal communication. la4 N. H. March, 1981, to be published. 12*

l Z 9J.

(A31

Electron Density Description of Atoms and Molecules

165

Turning to the two-dimensional oscillator, with isotropic force constants, one has then for JV levels singly occupied and for filled 'shells' E=

M- 1

c

n=O

N3

dv 6

Jv-2

(n+l)2=-+--+3 2

while the total number of particles N occupyingM closed shells is N =

M- 1

x

n=O

Jv-2

(n+l)=-+2

dv 2

(A51

+

since the degeneracy is (n 1). Thus in two dimensionsthe chemical potential is

which evidently tends to Jfr for large Jfr. In three dimensions, the levels are (n+ 3/2) and the degeneracy is +(n+ 1). (n+ 2) and hence

while, again for closed shells, Jcr-1

N = nC = O +(n

N 2 JV + 1)(n + 2) = N3 6 + __ 2 +3

Hence it follows that L

3

2 P =

+3

q +

;Jf+ 1

4

1 "y+dv-+5

which tends to M as Jfr becomes large. Having derived p exactly for 1-3 dimensions for a harmonic well, we focus on the Euler equation of the density description

where t as usual is the kinetic energy density as a functional of p. Integrating this after multiplication by p, as in the main text, yields

where dt indicates integration over all co-ordinates in d dimensions while U is the potential energy. But for a harmonic oscillator we have

166

Theoretical Chemistry

from the virial theorem. Thus it follows from equations ( A l l ) and (A12) that in one dimension

But E is known from equation (A2) and hence from (A12) T= N 2 / 4 for the onedimensional harmonic well. Thus it follows that $ p &6td r =

3T= 3 S t d a

(A 14)

We have demonstrated here that for the one-dimensional harmonic oscillator the integral required in the Euler equation, involving the functional derivative Gt/Gp, can be exactly expressed in terms of the total kinetic energy. Indeed, the relation, involving a factor of 3, is exactly that given by the T F statistical theory. This latter theory gives for the density in d dimensions p =

constant ( p - V ) d / 2

(A 15)

and for the kinetic energy density the form kinetic energy density-

pxp =

constant p l + 2 / d

(A16)

Thus, in this case, the functional derivative is such that

and hence the integral required from the Euler equation argument is related to the total kinetic energy by

We have demonstrated by the above argument that, for the one-dimensional harmonic well, the statistical relation (A18) becomes exact for all N, and not just in the limit of very large N. When we turn to the two- and three-dimensional cases, this is no longer true. However, it is worth exhibiting the ‘correction’ term to the T F relation (A18), which was central, for three dimensions, in the derivation of the ‘scaling relations’ for molecular energies in Section 10. Thus, in two dimensions we form the difference between the left and right hand sides of equation (Al8), when we obtain, using equations (A1 I), (A12), (A4), and (A5),

If we contemplate using equation (A19) to estimate fp(dt/dp) dt, we find that, even for M = 1, we obtain Jp(dt/dp) dt accurately to 5 % of the total kinetic energy, f o r N = 2, 2% and so on. Thus, for relatively small numbers of particles, the approximation (A18) appears already to be very useful, without correction terms. As remarked, for the one-dimensional oscillator it is exact for all N

Electron Density Descretion of Atoms and Molecules

167

Similarly in three dimensions we obtain the relation analogous to equation (A19) as

The accuracy is again for N =1, 8 % and for N =2, 3 %. It remains only to consider a potential with features more appropriate to atoms and molecules, namely the Coulomb potential. Bare Coulomb Field.-As a final example, we consider here the result for the chemical potential for a bare Coulomb potential; the example considered in another connection in Section 4. If we generalize that discussion to a nucleus with charge 2, using atomic units, and a number of electrons N, then for J)r closed shells we can write E = -Z2N

(A211

where Nand .Nare related by the generalization of equation (34) for N f 2 N = N(N

+ 1) ( 2 N + 1)/3

(A221

From equation (Al), the chemical potential is given by

Using the virial theorem for the Coulomb field, T = -E, we can again obtain both terms in equation (A18) for three dimensions, when we find

For .N= 1 the integral accuracy is given to 8 % by the approximation (5/3)T and for .N=2 to 5 % accuracy. Here again is a demonstration of the usefulness of this way of approximating to the integral Jp(dt/dp)d t appearing in the Euler equation method. We want to stress that one is dealing here with an approximate relation for the left hand side of equation (A18), given an excellent approximation to the total kinetic energy T. Naturally, if one used the TF statistical value of T for small numbers of electrons, this would introduce serious quantitative error. The burden of this Appendix has been to establish the usefulness of the approximation (A18) beyond the TF statistical limit in which it becomes asymptotically exact. We have seen that it works well for small numbers of particles in both the harmonic well and in the Coulomb field. It is therefore reassuring that the use made of it, especially in derivation of approximate energy relations for molecules in Section 10 of the main text, leads to rather better results than, a priori, one might have expected for relatively small numbers of electrons. In this connection, it is worth noting that all the models in this Appendix, plus the low-order gradient expansion result (86) for three dimensions, are consistent with

I[ $-- + i) p

(1

t ] dz

< 0,

Theoretical Chemistry

168

the equality applying only to the one-dimensional harmonic oscillator, for the reasons set out in Appendix 5. It seems therefore that inequality (A25) may represent a general result. Therefore, the main conclusion of this Appendix is that often the statistical relation (A18) is very useful even for small numbers of electrons. Appendix 2 Hohenberg-Kohn and Two Other Density Theorems.-Below a proof is given that a unique charge density exists for each external potential, following Hohenberg and K ~ h n . ' ~ ~ Let us suppose that corresponding to two external potentials Yand Vl there are corresponding many-body ground-state wave functions Y and !PI. If the hamiltonians are H and H 1 then one can write for the ground-state energy E of H, assuming a non-degenerate ground state

Now let us suppose further that the electron densities associated with the wave functions Yl and Y a r e the same. One can then write

I

Y*[V-

V13

Ydt =

Y:[ V - V I ]Y1 dz =

[V - V l ]p ( r ) d t

(A27)

in this case. Also one has

and hence from the inequality in equation (A26) above

Using the variational principle for H I , with energy El,

By addition these two inequalities yield E + E l < E + E l and one must conclude to escape this absurdity that two different external potentials cannot generate the same electron density. But E is uniquely determined by the external potential and hence one deduces that the ground-state energy is a unique functional of the electron density p ( r ) . This result, known as the Hohenberg-Kohn theorem, was assumed in all the early work on the density description (see refs. 4 and 16). Of course, the problem of finding the energy functional remains; to date it is a matter of judicious approximation for the problem under consideration. It is relevant in this context to refer to the earlier result of WilsonlSBthat, for a molecule with nuclear charges Z,, and internuclear distances R,,, the l35

136

P. Hohenberg and W. Kohn, Phys. Rev. B, 1964, 136, 864. E. B. Wilson, J . Chem. Phys.. 1962, 36, 2232.

Electron Density Description of Atoms and Molecules

1 69

total energy E may be written

This tells us that knowledge of the way p ( r ) varies as the nuclear charges are varied from zero to their true value in the molecule under discussion is sufficient information to determine the energy of the system. Of course, this by-passes the search for energy functionals, but leaves the problem of the molecule with variable nuclear charge to be solved. Frost 137 has done some related work and, remembering Wilson’s result, the scaling arguments presented in Section 12 may afford an approximate route to the calculation of the ground-state energy, and perhaps eventually the energy functional itself. A better known relation, also involving the density, is the theorem of Kato.ll For a spherically symmetric atom or ion, it relates the electron density at the nucleus to the derivative of p also taken at the nucleus through

2 being the nuclear charge. The generalizations of Steiner, lS8of Bingel, lSQ and

of Pack and Byers-Brown140are of interest in this connection. A good application of equation (A32) has been made in the work of Goscinski and Linder,141 in determining the form and amplitude of the asymptotic decay of the atomic scattering factor.

Appendix 3 One-body Potential in He and H,.-To make certain of the nature of the one-body potential V ( r )which incorporates electron correlation effects [cf:equation (147)], we summarize here the results of March and Stoddart for He and H,. These workers do not strictly discuss He, but a two-electron ion with large atomic number in a non-relativistic framework. The electron density for this system has been given by S ~ h w a r t zand , ~ ~by ~ Hall, Jones, and Rees,14*who obtained p (r) =

exp (- 2Zr) G (r)

(A331

with G(r)= 1

(2Zr) + Rz-

(e-”- 1) 137 138 139 140

141 142

143 144

+3

so x

dt

e-t-1

(A33

A. A. Frost, J. Chem. Phys., 1962, 37, 1147. E. Steiner, J. Chem. Phys., 1963, 39, 2365. W. A. Bingel, 2. Naturforsch. Teil A , 1963, 18, 1249. R. T. Pack and W. Byers-Brown, J. Chem. Phys., 1966, 45, 556. 0. Goscinski and P. Linder, J. Chem. Phys., 1970, 52, 2539. N. H. March and J. C. Stoddart, in ‘Computational Solid State Physics’, ed. F. Herman, N. W. Dalton, and T. R. Koehler, Plenum, New York, 1972. C. Schwartz, Ann. Phys., 1959, 6, 156. G. G. Hall, L. L. Jones, and D. Rees, Proc. R. London, Ser. A , 1965, 283, 1393.

170

Theoretical Chemistry

As both electrons are in the same orbital y ( r ) then we may write

and we can obtain immediately from the Schrodinger equation the one-body potential V(r) which will reproduce the exact density. Thus we have the Euler equation

Hence we obtain, using equations (A33) and (A36)

which, for the case 2Zr < 1 yields

The full form is plotted against r for two examples, Z=5 and 10 in ref. 142 to which the reader is referred for further details. For this particular twoelectron example, March and Stoddart show that, in addition to the potential V(r) discussed above, which reproduces the exact Schwartz electron density ~ ( r ) it, is possible to define a one-body potential which reproduces exactly the momentum density in the ground-state of the two-electron ion treated above. Naturally, because of electron correlation, this is different from V ( r ) above, as can be Seen from its expansion for 2 Z r < 1. This is given by AVmomentum density

(r) =

7

(1

(

)

l I 2 3Z2r T) 7I 0.67

0.67

-112

(A40)

where we have indicated explicitly that this is the potential which exactly yields the momentum density. In general, for more than two electrons, it has not been established that such a potential can be defined uniquely for the momentum density, and we have mentioned the above result because a comparison of these two one-body potentials highlights the effect of electron correlation (see figure in ref. 142). Related to the momentum density, the Compton profile of heavier atoms has been studied by the density method in refs. 145 and 146. Briefly, we turn to the hydrogen molecule in the Heitler-London limit which was also discussed in ref. 142. Again, from the Heitler-London function for the spatial part of the ground-state wave function we construct the charge density. Considering the potential Vonly along the internuclear axis, and with r, the distance from nucleus A, being such that r G R , the internuclear separation, the form is V(r) = 1 aor 145 146

+s (---) 1 1 exp (-5 +") ao r

R

J. R. Sabin and S. B. Trickey, J. Phys., 1975, 38, 2593. B. Y.Tong and L. Lam, Phys. Rev. A, 1978,18, 552.

ao

ao

Electron Density Description of Atoms and Molecules

171

where the overlap integral S is

s = LRJa0 [I + R

1R2

+ Jq]

For large R, the corrections to the ionic potential are small and repulsive. We mention here, finally, that for larger systems the work of Sharp and Horton 14' and later work by Talman and co-workers 148, lQ9 takes a single Slater determinant for the wave function and minimizes with respect to the one-body potential. This yields results for closed-shell atoms, with a one-body potential thus obtained, which are practically indistinguishable from the Hartree-Fock results. Of course, unlike the two-electron systems treated above, the resulting one-body potential does not incorporate correlation, but only exchange. The full implementation of the local density scheme for atoms by Tong and Sham 150 led to distinctly good results also. Appendix 4 Electron Correlation, Including Spin Density Description.-In this Appendix we discuss: (i) The relation between total energy and eigenvalue sum, in the presence of electron correlation and (ii) The idea behind the generalization of density theory to treat non-zero spin density, introduced via equations (175) and (176). On the first point, let us return to the Euler equation (156). By the now familiar procedure of multiplying by the density p and integrating through the whole of space we obtain Np =

Yen

+ 2 V e e + 1p sp6 + (tr

EXC)

dt

(A431

Using equation (147) for the one-body potential V(r), which generates oneelectron energies E Z , we find for the eigenvalue sum Es

By subtraction we find immediately

If we add and subtract 2/3 of the single-particle kinetic energy from the righthand-side of this equation, we find Np-Es =

/(

p

2-i

t f ) dt

+

51

tr dt

As remarked in the main text, since Ts= J t, d t is the single-particle kinetic energy, we cannot relate it directly to the total energy E a t equilibrium. Rather, 147 148 149

150

R. T. Sharp and G. K. Horton, Phys. Rev., 1953, 90, 317. J. D. Talman and W. F. Shadwick, Phys. Rev. A , 1976,14, 1136. J. D. Talman and M. M. Pant, Phys. Rev. A , 1979, 19, 52. B. Y . Tong and L. J. Sham, Phys. Rev., 1966, 144, 1.

172

Theoretical Chemistry

if T is the exact total kinetic energy at equilibrium, let AT be the difference between the single-particle kinetic energy at equilibrium and T, defined precisely by

Ts + A T = T

(A471

Hence, using E= - T from the virial theorem at equilibrium we obtain the final result relating total energy E, and the sum of orbital energies Es, in the presence of electron correlation as E = -&--Np 3 3 2

+ i/(p$i-;fr)dr--bT

2

This relation generalizes the discussion of Section 13 and Appendix 1. The deviations from relation (84) are qualitatively of the form that the chemical potential correction term in equation (A48) is positive, since p is negative for bound atoms and molecules, while both the kinetic energy contributions are expected to be negative, the term [p(dt/dp - 5 / 3 t ) ] d t because of equation (86) and the inequality obtained from the models of Appendix 1, and the correlation kinetic energy AT is expected to be positive. This is because, in forming a wave function from, say, the single-particle basis generated by the one-body potential V ( r ) of equation (147), one will expect the correlated wave function to involve creation of particles and holes and hence to increase the kinetic energy. In general, we can expect the correlation kinetic energy to be a small fraction of the total kinetic energy. But one must warn that it may well be of the same order as correlation contributions to N p in equation (A48). Spin Density Description.-To conclude this Appendix, we shall briefly summarize the idea behind the generalization of the density description to allow for non-zero spin density m ( r ) defined in equation (176). This is discussed in the work of Stoddart and March 151and by a number of other workers later.152~16s One can now write the ground-state energy as

where this equation represents formally the fact that the total energy E is now a functional of both the electron density p of equation (175) and of the spin density m in equation (176). In equation (A49), the total ground-state energy is split into three parts, a piece G which is thought of as a generalization of the energy J (t,+ E X C ) d t for the case of zero spin density, the self-electrostatic energy of the total electron density, and the energy of interaction with an external potential or field. It is obvious, as in the case of zero spin density, that calculation of G as a functional of p and rn is tantamount to a complete solution of the many-electron problem, which is at present impossible. Therefore, one must again resort to local approximations based on the uniform interacting electron assembly, but now with density p and magnetization m. J. C. Stoddart and N. H. March, Ann. Phys., 1971, 64, 174. U. von Barth and L. Hedin, J. Phys. C, 1972, 5, 1629. lb8 A. K. Rajagopal and J. Callaway, Phys. Rev. B, 1973, 7, 1912. 15*

Electron Density Description of Atoms and Molecules

173

Some examples to illustrate this approach may be found in articles in ref. 81 and 154. The local approximations to the exchange and correlation part of G introduced above are discussed in further detail in refs. 121 and 122, and are encouraging for local density approximations to the many-electron part of G above. The kinetic part of G can again be treated by solving the single-particle Schrodinger equation, by a generalization of the approach described in Section 17, but now with different potentials for the two different spin directions. Appendix 5 Exact Differential Equation for Particle Density for N Particles Moving in Onedimensional Harmonic Oscillator Potential.-One of the major aims of density theory must remain direct calculation of the particle density p from one-body potential of the form of equation (147). So far, this has not proved possible exactly for any three-dimensional potential. There is one, admittedly elementary, example where an exact differential equation has been derived by Lawes and This is for N particles moving in a one-dimensional harmonic oscillator potential. The motivation of their argument was to study the functional derivative Stz/Sp appearing in the Euler equation (49). Adapted to the linear harmonic oscillator, this reads, with suitable choice of units

A local density form for ts will clearly lead to

and from the work of March and Young155one has for a general one-dimensional potential V ( x )

Combining equations (A5 1) and (A52) leads immediately, for the harmonic oscillator case, to the differential equation for the density

But from Appendix 1, the chemical potential for this case is simply p= N. As Lawes and March127 demonstrate directly, this is the differential equation satisfied for the exact density given in terms of the Hermite polynomials by Husimi.156 Thus equation (A53) contains the density of the harmonic oscillator for an arbitrary number of particles N. Of course, it is only for this very elementary potential field that the local density assumption (A51) becomes exact. For a 154

155

156

N. H. March, in ‘Charge and Spin Density’, ed. P. Becker, Plenum, New York, 1980. N. H. March and W. H. Young, Nuclear Phys., 1959, 12, 237. K. Husimi, Proc. Phys. Math. SOC. Japan, 1940, 22, 264.

174

Theoretical Chemistry

single-particle N = 1, the function(a1) tz has been given by Lawes and March as tz = + p

+ +p(Inp + +Inn)

(A54)

This simple example illustrates the philosophy underlying the density description; equation (A53) contains the TF limit if one neglects the third derivative of p. It is worth pointing out again that, for the TF limit, tzccp3 and in this TF theory it follows that

In fact this result is exact for the linear harmonic oscillator for an arbitrary number of particles N,because of the result lS7

+

Multiplying both sides by p and integrating from - m to 00 yields equation (A53 as r.h.s. vanishes because dp/dx is zero at f m. Related earlier work is that of Light.16**15@ It is a pleasure to thank Dr G. P. L a w s and Professors Bader, Mucci, and Pam, for both valuable discussions and helpful correspondence. I thank the Science Research Council for financial support.

15' 158

15'

N. H. March, unpublished work. J. C. Light, J. Chem. Phys., 1973,sS. 660. J. C. Light, J. Chern. Ph-vs., 1974, 61, 3417

Author Index Ahlrichs, R.,32, 60 Amos, A. T., 21 Andersen, 0.K.,143 Anderson, A. B., 128 Ando, D. J., 73 Andre, J. J., 78 Andrb, J.-M., 51, 56 Arnott, S., 80 Afman, A., 56, 73 Bader, R. F. W., 128, 129, 158. 164 ~ - 2

-

Baerends, E. J., 143, 148 BPssler, H., 73 Bagus, P. S., 30 Baird, N.C., 135 Baker, E. B., 94 Baker, G. A., 20,22 Ballinger, R. A., 103 Baraff, G. A., 65 Baranger, M., 14 Bardo, R. D., 17 Bartell, L. S., 145 Bartlett, R. J., 3, 22, 28, 32, 43

Basch, H., 115 Bates, D. R.,120 Baughman, R.H., 66,72,73 Beattie, A. M.,133 Beddall, P. M.,128 Berholc. J.. 65 Berthier, G., 2 Beveridge D. L., 61 Bicz6, G.,’51 53, 55 Bingel, W.A:, 169 Binkley, J. S., 1, 13 BiskupiE, S.,28, 30 Bloch, C., 14 Blomberg, M. R.A., 32 Bloor, D., 73 Blount, E. I., 57 Bone, S.,83 Boudreaux, D.S., 73 Bowers, W.A., 114 Boyle, M., 46 Brandas, E.,22 Brandow, B. H.,8, 14, 20 Brillouin, L.,4 Brooker, L. G. S., 66 Brueckner, K.A., 1, 133 Bunge C. F., 28 Byers-Brown, W., 169 Byron, F. W.,22 Cade, P. E., 145 Callaway, J., 65, 172 Cantu, A. A., 15 Chrsky, P., 1 Castel, B., 46 Cavallone F., 78 Chakeriai, C., 146 Chance, R. R.,73

Chang, E. S., 3 Chiang, C. K., 65, 66 CiEek, J., 1, 55, 60 Claverie, P., 13, 30 Clementi, E., 78, 107 Cole, S., 15 Collard, K.,159 Collins, T.C.,59,60,83,84 Comes, R.,78 Connolly, J. W.D., 148 Coulson, C. A., 47 Cowan, D. O.,78 Cross, L. C., 117 Das, T. P., 3 Davidson, E. R.,1, 34 Day, R.,65 Dean, P.,64 Delhalle, J., 55 Del Re, G., 51, 65 Denenstein, A., 66 Denoyer, F., 78 Des Cloizeaux, J., 14 Devreese, J. T.,59 Dewar, M.J. S., 61,62 Diner, S., 13, 30 Dirac, P. A. M., 102 Dmitrieva, I. K.,100 Dobosh, P. A., 61 Donnelly, R. A., 134, 164 Dover, S. D., 80 Dreizler, R. M.,1 1 1, 120 Drug, M. A., 65,66 Duke, A. J., 128 Dunlap, B. I., 148 Dunning, T.H., jun., 42 Dutta, C. M., 3 Dutta, N. C., 3 Eckhardt, C. J., 73 Edmiston, C., 57 Eggarter, E., 19 Eggarter, T. P.,19 Eichhorn, G.L.,86 El Baz E., 46 Elliot, k. J., 84 Enkelmann, A., 73 Epstein, P. S., 13 Farrell, R. A., 20 Feenberg, E., 20 Feller, D. F., 19 Fermi E., 92 Feshbkch, H., 4 Fetter, A. L., 1 Fincher, C. R.,jun., 66 Fink, M., 145 Flores, J., 46 Fock V 97 Fold; L: L.,98 Fougire, P. F., 145 Freeman, D. F., 3

175

Friedel, J., 156 Frisch, M. J., 28 Frost, A. A., 169 Fukui, K., 69 Fulde, P.,77 Gallup, G. A., 15 Gammel, J. L.,21 Garito, D. F., 78 Gau, S. C., 65, 66 Geldart, D. J. W., 133 Gell-mann, M.,9 Gelus, M.,151 Gerratt J 15 Gilbert,’ T.’L.,164 Goddard, W.A., 15 Goldstone, J., 1 Gombtis P.,94 Goodings, E. P.,72 Gordon, R. G., 132 Gordy, W. J., 136 Goscinski, O.,20,22, 169 Gouverneur, L., 51 Gouyet, J. F., 15 Grant, I. P..106

47 162 H[all, G. G., 159, 169 H[ameka, H., 127 H[andler, G. S., 122, 127 Hjarris, F. E.,51 H.amis. J.. 143. 148 H arrison,’J., 17 H‘edin, L.,133, 172 H eeger, A. J., 65, 66,78 H egarty, D., 14 H ehre, W.,55 H eijser, W., 143 Henderson, G.A., 109 H erman, F., 132 H inze, J., 135 H ockney, R.W.,39 H odges, C. H., 109 H offmann, H. M.,22 H ohenberg, P., 60, 168 H ori. J.. 89 Horton,’ G. K., 171 Hose, G., 14 Howat G., 22 Hsu, S: L 66 HubaE, I.;’l,3, 28 Huber, K. P.,144 Hugenholtz, N.H.,1 1 Hund, F.,1 1 1 Huo, W.M.,145 Hurley A. C.,32 Husim;, K., 173

Author Index

176 Iczkowski, R. P., 138 Ikeda, S., 66, 69 Imamura, A., 69 Iqbal, J., 73 Itagaki, T.,22, 44 Ito, T., 66 Jaffe, H. H., 135 Jatsumara, Y., 66 Joachain, C. J., 22 Johanssson, P., 143 Johnson, K. H., 163 Johnson, M. B, 14 Jones, L. L., 169 Jones, R. O., 143, 148 Jones, W., 127 Kaldor, U., 3, 14, 46 Karpfen, A., 56, 77 Karplus, M., 3 Kaspar, J., 63, 76 Kato, T., 100 Kaufman, D. N., 3 Kello, V., 3, 28, 42 Kelly, H. P., 1, 2, 3, 14, 44 KertCsz, M., 56, 73 Kim, Y.S., 132 Kirtman, B., 15 Kirznits, D. A., 109 Kistenmacher, T. J., 78 Klein, D. J., 15 Klonover, A., 46 Klopman, G., 135 Knecht, J., 73 Knox, R. S., 58 Kobelt, D., 73 Kohl, D. A., 145 Kohn, W., 60, 132, 168 Koller, J., 56,+73 Kobs, W., 150 Koster, G. F., 58 Krenciglowa, E. M.,22 Krishnan, R., 28 Krumhansl, J. A., 84 Kuhn, H., 66 Kunz, A. B., 59 KUO,T. T. S., 14, 22 Kutzelnigg, W., 2, 60,151 KvasniEka, V., 14, 15, 28, 30

Ladik, J., 51, 53, 55, 58, 59,

60, 63, 65, 77, 78, 80, 82, 83, 84, 89, 164

Laki, K., 83 Lam, L., 170 Lando, J., 73 Langhoff, S. R., 34 Laurinc, V.,28, 30 Lawes, G. P., 106, 109, 163, 164

Layzer, D., 100 Leath, P. L., 84 Ledsham, K., 120 Lee, S. Y.,14, 22 Lee, T., 3 Leinaas, J. M.,22 Lennard-Jones, J. E., 4 Leroy, G., 51, 56 Levinson, I., 46 Levy, B., 2, 14 Levy, M., 134 Lewis, T. F., 83

Light, J. C., 174 Linder, P., 169 Lindaren. I.. 14 Lipa;, N’. D., 65 Liu, B., 30 Lowdin, P. O., 4, 53, 164 Louis, E, J., 65, 66 Low. F.. 9 Lundqvist, B. I., 162 Lundqvist, S., 133 Ma, S. K., 133 MacDiarmid, A. G., 65, McDonald, F. A., 21 McLean, A. D., 30 Malrieu, J., 13, 30 March, N. H., I, 60,100, 101, 103, 104, 106, 109, 110, 111,120, 123,124,127, 129, 132. 133. 140. 162. 163. 164; 173, 174 Margrave, J. L., 138 Martino, F., 53, 65, 84 Matsen, F. A., 15 Matsubara, C., 3 Mayer, I., 53 Mely, B., 84 Merkel, C.,53, 80 Messer, R. R., 128 Meunier, A., 2 Meyer, W., 42, 60 Miller, J. H., 3 Mitchell, A. D., 117 Msller, C., 13 Monkhorst, H. J., 51 Moore, P. G., 145 Morita, T., 14 Morrison, R. C., 15 Moshinski, M., 46 Moshinsky, M., 15 Mucci, J. F., 111, 124 Miiller, H., 73 Mulliken, R. S., 55, 123 Murphy, D. R., 109

Nelander, B., 126 Nesbet, R. K., 13,145 Newman, D. J., 15 Nguyen-Dang, T. T., 158 Nieuwpoort, W. C., 17 Niklasson, G., 133 Noga, J., 28 Nozieres, P., 132 Ortenburger, I. B., 132 Otto, P., 59 Ozaki, M., 66 Pack, R. T., 169 Padt, H., 20 Page, P. J., 73 Paldus, J., 1, 29, 46, 60,73 Pant, M. M., 171 Pantelides, S, T., 65 Park, Y. W.,65, 66 Parr, R. G., 101, 109, 128, 129, 134, 135, 164

Parry, D. E., 73 Pasternak, A., 135 Paulus, E. F., 73 Peebles, D. L., 66 Pethig, R., 83 Petkov, J., 56

Pez, G. P., 66 Phillips, C. S. G., 107 Phillips, T, E., 78 Piela, L., 55 Pines, D., 132 Plake, W. E., 134 Plaskett, J. S., 104 Plesset, M. S., 13 Plindov, G. I., 100 Politzer, P., 115, 134 Pople, J. A., 1, 13, 28, 55,61 Preston, F. H., 73 Primas, H., 14 Prime, S., 3 Pu, R. T., 3 Pullman, A., 84 Purvis, G. D., 28, 42 Pyper, N. C., 106 Raffenetti, R. C., 17 Rai, D. K., 84 Rajagopal, A. K., 172 Rasolt, M., 133 Ratcliffe, K. F., 14 Ray, N. K., 129, 135 Redmon, L. T., 28 Rees, D., 169 Reeves, C. M., 17 Reimer, B., 73 Reucroft, R. J., 73 Richert, J., 22 Robb, M. A., 3, 14 Rosch, N., 163 Roos, P., 148 Rosenberg, B. J., 44 Rosmus, P., 42 Rozsnyai, B. F., 82 Ruedenberg, K., 17, 19, 57, 123

Runtz, G., 128 Sabin, J. R., 148, 170 Saika, A., 22, 44 Sales, K. D., 145 Sampanthar, S., 1 Samuels, L., 135 Sandars, P. G. H., 14 Sanderson, R. T., 134 Saunders, V. R., 28, 29 Schluter, M., 65 Schmidt, M., 19 Schucan, I. H., 22 Schulman, J. M., 3 Schwartz, C., 169 Scott, J. M. C., 102 Scrubel, M. H., 22 Seeger, R., 1 Seel, M., 65, 84 Seligman, T. H., 15 Shadwick, W. F., 171 Shalin, H. L., 120 Sham, L. J., 132, 171 Sharp, R. T., 171 Shavitt, I., 3, 28, 32, 44, 46 Shibuya, T., 107 Shin, Y.A., 86 Shirakawa, H., 65, 66, 69 Sichel. J. M.. 135 Siegbahn, P.’E. M., 32, 34 Signorelli, A. J., 66 Silver, D. M., 1, 2, 3, 17, 22, 24, 28, 34, 39

Author Index Silver, D. W., 1 Simons, G., 130 Singh, R. D., 77 Singwi, K. S., 133 Sjolander, A., 133 Slater, J. C., 58, 144, 163 Snyder, L. C., 115 Soven, P., 84 Spencer, M., 80 Srebenik, S., 128 Staemmler, V., 151 Steiner, E., 169 Stern, P. S., 14 Stevens, G. C., 73 Stewart, A. L., 120 Stewart, R. F., 55 Stoddart, J. C., 102, 133, 169, 172 Stollhoff, G., 77 Suck, S. H., 62 Suhai, S., 51, 54, 55, 56, 59, 60, 63, 69, 73, 76, 78, 80, 83, 84, 164 Szent-Gyorgyi, A., 83 Takahashi, K., 73 Takeuti, Y., 58 Tal, T., 158 Talman, J. D., 171

177 Tanaka, K., 69 Teller, E., 119, 120 Termata, H., 69 Thiel, W., 61 Thom, R., 159 Thomas, L. H., 92 Toepfer, A., 111 Tong, B. Y., 170, 171 Townsend, J. R., 122 Toyozawa, Y., 59 Trickey, S. B., 170 Trsic, M., 22 Tuan, D. T., 21 Tylickis, J., 73 Ukrainski, I. I., 53 Urban, M., 3, 28, 42 Vanagas, V., 46 van Dyke, J. P., 132 Van Kessel, A. T., 143 von Barth, U., 172 Von Weizsacker, C. F., 109 Wahl, A. C., 145 Walecka, J. D., 1 Walsh, A. D., 123 Wang, W. D., 109 Wannier, G. H., 56

Wegner, G., 72 Weidenmuller, H. A., 22 Weinstein, H., 134 Weiss, A. W., 106 White, R. J., 100 Whitehead, M. A., 135 Wick, G. C., 8 Wigner, E. P., 4, 132 Wilkins, J. W., 162 Williams, P. M., 73 Williams, R. J. P., 107 Wilson, E. B., 168 Wilson, S., 1,2, 3, 15, 17, 19, 20, 21, 22, 24, 28, 29, 30, 31, 34, 39, 42, 45, 46, 47, 48 Witt, J. D., 73 Wolniewicz, L., 150 Wonacott, A. J., 80 Wong, H. C., 29 Woolley, R. G., 143 Yamabe, T., 69 Yamaguchi, Y.,62 Yee, K. C., 73 Yoshmine, M., 30 Young, W. H., 1,73 Yusaf, M. S., 106 Yutsis, A. P., 46