Atomic Many-Body Problem. II. The Matrix Components of the Hamiltonian with Respect to Correlated Wave Functions

The wave function for an atom with N electrons in arbitrary configuration will be written in the form $${\mathrm{\Psi =\Psi }}_0+{\displaystyle \sum _{\text{i=1}}^N}{\displaystyle \sum _{\text{j=i+1}}^N}\mathrm{f(ij)}$$ , where ψ₀ is a Slater determinant and f(ij) is the antisymmetrized product of (N − 2) one‐electron spin‐orbitals and one 2‐electron function Φ(ij/12). The correlation between the two spin‐orbitals φ_i and φ_i can be taken into account by introducing r₁₂ (the interelectronic distance) explicitly into the 2‐electron function Φ. The purpose of the paper is to analyze the structure of the matrix components of the Hamiltonian with respect to the wave function given above. Starting from exact, general formulas for the matrix components it will be shown that, all integrals which occur in the diagonal, as well as in the nondiagonal matrix components can be reduced to six basic integrals which are 2‐ , 3‐ , and 4‐electron integrals, containing interelectronic distances. It will be indicated that, five of the six basic integrals can be calculated in closed form whereas one of them, (an exchange integral) can be given only in the form of an infinite series.