Perturbation Theory of Molecules

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-consistency1-2. Independent electron models, such as the widely used Hartree-Fock approximation, provide a size-consistent theory of atomic and molecular structure. 

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 techniques3-9 due to Brueckner10 and Goldstone.11 Diagrammatic many-body perturbation theory provides a simple pictorial representation of electron correlation effects in atoms 

Davidson and D. W. Silver, Chem. Phys. Lett., 1977, 52, 403 8 N. H. March, W. H. Young, and S. Sampanthar, The Many-body Problem in Quantum Mechanics , Cambridge University Press, 1967. 

Fetter and J. D. Walecka, Quantum Theory of Many-particle Systems , McGraw-Hill, New York, 1971. 

Wilson, in Proceedings of Daresbury Study Weekend , Dec. 1977, ed. V. R. Saunders, Science Research Council, London, 1978. 

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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. 

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. 

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. 

Stephen Wilson contributed a chapter to Volume 4 of Theoretical Chemistry entitled Many-body Perturbation Theory of Molecules , in which he described the beginnings of diagram techniques. Again, it seemed appropriate to ask Stephen to tell us how things now stand. He has done just this, with key references through May 1999. 

The expansion coefficients are therefore not determined uniquely by the function which is to be expanded, and we can show that (15) defines the most general expansion of this sort which is possible. Especially, the constants af can be chosen in such a way that the series expansion starts only for an arbitrary high constant t. This multiplicity would be inconsequential if the result would only depend on the function which is to be expanded. However, in the peculiar case of the perturbation theory of molecule formation the perturbation energy in second order is strongly dependent on the form of the expansion of the first order function. It is not the case that quantities from the higher orders are cancelled, but quite new quantities are introduced, which vanish only later. 

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