The Green's function approach to the Fock expansion calculations of two-electron atoms

The renewed Green's function approach to calculating the angular Fock coefficients, φk,p(α,⊖) is presented. The final formulas are simplified and specified to be applicable for analytical, as well as numerical calculations. The Green's function formulas with the hyperspherical angles ⊖ = 0, π (arbitrary α) or α = 0,π (arbitrary α) are indicated as corresponding to the angular Fock coefficients possessing physical meaning. The most interesting case of ⊖ = 0 corresponding to a collinear arrangement of the particles is studied in detail. It is emphasized that this case represents the generalization of the specific cases of the electron-nucleus (α = 0) and electron-electron (α = π/2) coalescences. It is shown that the Green's function method for ⊖ = 0 enables us to calculate any component/subcomponent of the angular Fock coefficient in the form of a single series representation with arbitrary angle ⊖. Those cases where the Green's function approach cannot be applied, are thoroughly studied, and the corresponding solutions are found.