Helium-like atoms. The Green's function approach to the Fock expansion calculations

Abstract

The renewed Green's function approach to calculating the angular Fock coefficients, $\psi_{k,p}(\alpha,\theta)$ 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 $\theta=0,\pi$ (arbitrary $\alpha$) or $\alpha=0,\pi$ (arbitrary $\theta$) are indicated as corresponding to the angular Fock coefficients possessing physical meaning. The most interesting case of $\theta=0$ 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 ($\alpha=0$) and electron-electron ($\alpha=\pi/2$) coalescences. It is shown that the Green's function method for $\theta=0$ enables us to calculate any component/subcomponent of the angular Fock coefficient in the form of a single series representation with arbitrary angle $\theta$. Those cases, where the Green's function approach can not be applied, are thoroughly studied, and the corresponding solutions are found.