Symmetry of the Atomic Electron Density in Hartree, Hartree-Fock, and Density Functional Theory

TL;DR

This paper investigates the symmetry properties of atomic electron densities across various quantum mechanical models, demonstrating that approximate methods slightly violate expected symmetries, which can be corrected without significant energy changes.

Contribution

It introduces a perturbation theory analysis of symmetry violations in approximate density calculations and proposes a constrained-search method to restore correct symmetry.

Findings

Approximate schemes slightly violate the expected spherical harmonic content.

Correct symmetry can be restored with minimal impact on energies.

Exact density functional theory has the correct symmetry, but its potentials and wavefunctions are more complex.

Abstract

The density of an atom in a state of well-defined angular momentum has a specific finite spherical harmonic content, without and with interactions. Approximate single-particle schemes, such as the Hartree, Hartree-Fock, and Local Density Approximations, generally violate this feature. We analyze, by means of perturbation theory, the degree of this violation and show that it is small. The correct symmetry of the density can be assured by a constrained-search formulation without significantly altering the calculated energies. We compare our procedure to the (different) common practice of spherically averaging the self-consistent potential. Kohn-Sham density functional theory with the exact exchange-correlation potential has the correct finite spherical harmonic content in its density; but the corresponding exact single particle potential and wavefunctions contain an infinite number of spherical harmonics.