Relativistic variational methods and the Virial Theorem

TL;DR

This paper extends the Virial Theorem to many-electron atoms, analyzing how electron interactions and Slater integrals influence the convergence of self-consistent field calculations.

Contribution

It provides a detailed study of the Virial Theorem for subshells in many-electron atoms and examines the impact of Slater integrals on convergence.

Findings

Some Slater integrals impose conditions on individual subshells.

Other Slater integrals impose conditions between different subshells.

Interactions between subshells slow the convergence of the self-consistent field process.

Abstract

In the case of the one-electron Dirac equation with a point nucleus the Virial Theorem (VT) states that the ratio of the kinetic energy to potential energy is exactly $-1$, a ratio that can be an independent test of the accuracy of a computed solution. This paper studies the virial theorem for subshells of equivalent electrons and their interactions in many-electron atoms. It shows that some Slater integrals impose conditions on a single subshell but others impose conditions between subshells. The latter slow the rate of convergence of the self-consistent field process in which radial functions are updated one at a time. Several cases are considered.