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

## TL;DR

This paper introduces a simplified Green's function method for calculating angular Fock coefficients in helium-like atoms, enabling both analytical and numerical analysis, especially for collinear particle arrangements.

## Contribution

The paper develops a generalized Green's function approach for angular Fock coefficients, including cases previously intractable, and applies it to collinear particle configurations in helium-like atoms.

## Key findings

- Formulas simplified for analytical and numerical use
- Green's function method applied to collinear arrangements
- Solutions provided for cases where Green's function approach is not applicable

## 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.