On Hartree-Fock integro-differential equations for atoms

Hartree-Fock integro-differential equations have been classified to avoid confusion arising from use of the same name for approximations of different natures. The proper orthonormality conditions for the perturbation expansions of these equations have been established. For the totally unrestricted Hartree-Fock equations for the ground state of helium-like ions it has been shown that corresponding to the expansion of the one-electron functions u and v as power series in 1/Z, u ≡ v is the only solution. The status of the perturbation expansion of these one-electron functions as power series in 1/Z½ has been considered in some detail and it has been shown that the half-order functions satisfy well-defined nonlinear integro-differential equations. Various misunderstandings in the literature have been corrected.