Angular Fock coefficients. Fixing the errors, and further development

TL;DR

This paper recalculates and corrects the angular Fock coefficients for two-electron atomic systems, introduces new techniques for solving related equations, and presents previously uncomputed components using Mathematica.

Contribution

It provides corrected and extended calculations of angular Fock coefficients, including new components, and develops methods for solving associated recurrence relations.

Findings

Corrected previous errors in $\psi_{2,0}$ representations.

Calculated all $j$-components of $\psi_{4,1}$ for the first time.

Presented most components of $\psi_{3,0}$ for the first time.

Abstract

The angular coefficients $\psi_{k,p}(\alpha,\theta)$ of the Fock expansion characterizing the S-state wave function of the two-electron atomic system, are calculated in hyperspherical angular coordinates $\alpha$ and $\theta$. To solve the problem the Fock recurrence relations separated into the independent individual equations associated with definite power $j$ of the nucleus charge $Z$, are applied. The "pure" $j$-components of the angular Fock coefficients, orthogonal to of the hyperspherical harmonics $Y_{kl}$, are found for even values of $k$. To this end, the specific coupling equation is proposed and applied. Effective techniques for solving the individual equations with simplest nonseparable and separable right-hand sides are proposed. Some mistakes/misprints made earlier in representations of $\psi_{2,0}$, were noted and corrected. All $j$-components of $\psi_{4,1}$ and the majority of components and subcomponents of $\psi_{3,0}$ are calculated and presented for the first time. All calculations were carried out with the help of the Wolfram \emph{Mathematica}.