Extensions of the diagrammatic approach

Diagrammatic methods were first developed for the study of continuous media, such as nuclear matter and solids, where the properties of interest are extensive. If, for example, the size of the system is doubled, keeping the particle density constant, then the total energy must also be doubled—remaining proportional to the number of particles N. Thus, for a free-electron gas, the total energy and the correlation energy 

It is instructive to consider a very simple system for which, using a limited basis, a direct comparison of perturbation and full-CI calculations is possible. The Cl approach is finite and leads at once to the basis-set limit, but the perturbation theory rests upon the expansions (9.4.7) and (9.4.8), which are infinite series in powers of a perturbation parameter and may or may not converge to the same limit. 

Let us consider the molecule H2 and start from the IPM wavefunction in which the two electrons occupy the lOg bonding MO, A say, which may be assumed to be an exact eigenfunction of the HF Hamiltonian. The virtual MOs B, C, D. may then be added to provide an orbital basis for the Cl/perturbation calculations and to make matters more transparent we truncate this basis after the first two orbitals. There are then two occupied spin-orbitals Aa, Afi and two virtual Ba, Bfi, and the calculations run as follows. 

From the Brillouin theorem, the Cl involves only Pq and one doubly excited function These are respectively 

From (9.2.22) and (9.2.23), the unperturbed Hamiltonian Hq gives a zero-order energy hf, namely, 

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