Theoretical Chemistry Volume 4

A Review of the Recent Literature

By C. Thomson

The Royal Society of Chemistry

Copyright © 1981 The Royal Society of Chemistry
All rights reserved.
ISBN: 978-0-85186-784-7

Contents

Chapter 1 Many-body Perturbation Theory of Molecules By S. Wilson, 1,
Chapter 2 The Electronic Structure of Polymers By J. Ladik and S. Suhai, 49,
Chapter 3 Electron Density Description of Atoms and Molecules By N. H. March, 92,
Author Index, 175,

CHAPTER 1

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. 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 interactions of 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 techniques due to Brueckner and Goldstone. Diagrammatic many-body perturbation theory provides a simple pictorial representation of electron correlation effects in atoms and mo1ecu1es and also forms the basis of a tractable, non-iterative scheme for accurate calculations. 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 the super-system, E(AB), is given by

E(AB) = E(A) + E(B) (1)

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 double-excitations, 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. Limited configuration mixing is not a size-consistent technique.

The diagrammatic many-body perturbation theory may be derived from the Rayleigh–Schrödinger perturbation expansion. Brueckner showed that certain terms arise in the Rayleigh–Schrödinger 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–Schrödinger perturbation series. Goldstone generalized this result to all orders using the diagrammatic techniques of time-dependent perturbation theory. This leads to the linked diagram perturbation expansion. 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.

The pioneering work on the application of the many-body perturbation theory to atomic and molecular systems was performed by Kelly. He applied the method to atoms using numerical solutions of the Hartree–Fock equations. Many other calculations on atomic systems were subsequently 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 perturbations. More recently, the theory has been applied to arbitrary molecules by employing the algebraic approximation which is fundamental to most molecular calculations. In this approximation, single-particle state 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 work has 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: ‘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 Jess compact and less efficient than these approaches and becomes progressively less efficient as the number of electrons increases.’

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 Löwdin and Feshbach is used to give a straightforward and general introduction to perturbation expansions. The perturbation series of Lennard-Jones, Brillouin, and Wigner is 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– Schrödinger 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 |Φ0> and Q its complement. The electronic Schrödinger equation

H|Ψ> = E|Ψ > (2)

may then be written as a two by two block matrix equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

where |Φ0 > = P |Ψ0 > · Q|Ψ0 > can now be eliminated to produce the effective Schrödinger equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

or

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

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

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

If H0 denotes some zero-order hamiltonian and E0 its ground state eigenvalue then the perturbation series of Lennard-Jones, Brillouin, and Wigner is obtained by putting

A = E0 – H0 (8)

and

B = H – H0 (9)

The Rayleigh–Schrödinger perturbation expansion is obtained by putting

A = E0 – H0 (10)

and

B = H – H0 – E + E0 (11)

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

H = H0 + H1 (12)

with

H0|Φi> = Ei|Φi (13)

and

H|Ψi > = Ei|Ψi = (Ei + ΔEi) |Ψi > (14)

We introduce the projection operators

P0 = |Φ0 >

and

Q0 = I – P0 (16)

and employ the intermediate normalization convention

= 1 = (17)

Now we can define the wave operator, Ω, with the following properties

|Ψ0 = Ω|Φ0 (18)

and

P0Ω = P0 (19)

ΩP0 = Ω (20)

ΩQ0 = 0 (21)

and thus obtain an expression for the level shift, ΔE0, for the ground state

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (22)

where the reaction operator, [??], is given by

[??] = H1Ω (23)

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

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (24)

and the level shift has the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (25)

These expressions for the wave operator and the reaction operator are formally equivalent to the integral equations

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

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

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (28a)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (28b)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (28c)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (28d)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (28e)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (28f)

where R is the resolvent

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (29)

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. The perturbation theory of Lennard-Jones, Brillouin, and Wigner is not size consistent.

Rayleigh–Schrödinger Perturbation Theory. — In Rayleigh –Schrödinger 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, Ω, may be written in an alternative form by replacing H0and giving by H0 + Q0ΔE0Q0 and H1 by H1 – Q0ΔE0Q0 giving

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (30)

Rearranging this expression in terms of powers of the perturbation, H1, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (31)

The j-th order energy is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (32)

Explicitly, the first few terms in the Rayleigh–Schrödinger perturbation expansion may be written in the form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (33a)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (33b)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (33c)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (33d)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (33e)

where R0 is the reduced resolvent

R0 = Q0/E0 – H0 (34)

Clearly, the terms other than the first in the expressions for E(3)0, E(4)0 E(5)0etc. depend on the number of electrons in a non-linear fashion. These terms exactly cancel components of the first terms in each of the expressions which also have a non-linear dependence on the number of electrons.

The Many-body Perturbation Theory. — The Rayleigh–Schrödinger 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. Brueckner showed that for the first few orders the terms having a non-linear dependence on N mutually cancel in each order. Goldstone 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 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 many-body 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 conventions.

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 details.

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 Brandow which is based on the expansion of the energy-dependent denominators in the Lennard-Jones Brillouin Wigner perturbation theory. Paldus and Cizek have given a time-independent derivation using a generalization of Wick’s theorem for 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, φp, which are eigenfunctions of some single-particle operator, [??],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (35)

with eigenvalues εp. In the second-quantized formalism the zero-order hamiltonian has the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (36)

and the perturbation operator may be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (37)

where Ψ+(r1) and Ψ(r1) are the usual creation and annihilation field operators, g(r1, r2) is the two-electron potential, and V(r1) is the effective potential which is added to the bare-nucleus hamiltonian to give the one-electron operator [??](r1). 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, (|α|t), of the perturbation yields the evolution operator

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (38)

where I is the identity operator and Uαn (t, – ∞) is proportional to the nth power of the perturbation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (39)

(Continues…)Excerpted from Theoretical Chemistry Volume 4 by C. Thomson. Copyright © 1981 The Royal Society of Chemistry. Excerpted by permission of The Royal Society of Chemistry.
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