Many-body theories of electron correlation

Unfortunately, the determination of exact solutions of the SchrOdinger equation is intractable for almost all systems of practical interest. On the other hand, independent particle models are not sufficiently accurate for most studies of molecular structure. In particular, the Hartree-Fock model, which is the best independent particle model in the variational sense, does not support sufficient accuracy for many applications. Some account of electron correlation effects has to be included in the theoretical apparatus which underpins practical computational methods. Although the energy associated with electron correlation is a small fraction of the total energy of an atom or molecule, it is of the same order as most energies of chemical interest. However, such theories may not be true many-body theories. They may contain terms which scale non-linearly with electron number and are therefore unphysical and should be discarded. Any theory which contains such unphysical terms is not acceptable as a true many-body method. Either the theory is abandoned or corrections, such as that of Davidson [7] which is used in limited configuration interaction studies, are made in an attempt to restore linear scaling. 

Lindgren and Morrison in their treatise on Atomic many-body theory [5] recognize that  

One of the present authors [1] makes a similar point in a text on Electron correlation in molecules published in 1984 (p. 65) 

The perturbation expansion of Lennard-Jones, Brillouin and Wigner does not lead to expressions which are directly proportional to the number of electrons in the system being studied. 

Likewise, Harris, Monkhorst and Freeman in their monograph Algebraic and Diagrammatic Methods in Many-Fermion Theory [6] wrote (p. 224) 

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Under certain conditions the Brillouin-Wigner pertiubation theory forms the basis for a many-body theory. Whether it can provide a robust multireference many-body theory of electron correlation effects is the subject of current research. 

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