Variations of the Hartree-Fock fractional-spin error for one electron

TL;DR

This paper investigates the variation of Hartree-Fock fractional-spin errors in one-electron systems, revealing how different schemes minimize these errors and their impact on various electronic properties.

Contribution

It provides a detailed analysis of fractional-spin errors in Hartree-Fock theory using ensemble densities and explores their effects on electronic structure calculations.

Findings

Unrestricted and generalized Hartree-Fock schemes reduce fractional-spin errors.

Fractional-spin errors influence the Coulomb hole and density depletion.

Errors affect Møller-Plesset adiabatic connection and excited states.

Abstract

Fractional-spin errors are inherent in all current approximate density functionals, including Hartree-Fock theory, and their origin has been related to strong static correlation effects. The conventional way to encode fractional-spin calculations is to construct an ensemble density that scales between the high-spin and low-spin densities. In this article, we explore the variation of the Hartree-Fock fractional-spin (or ghost-interaction) error in one-electron systems using restricted and unrestricted ensemble densities, and the exact generalized Hartree-Fock representation. By considering the hydrogen atom and H$_2^+$ cation, we analyze how the unrestricted and generalized Hartree-Fock schemes minimize this error by localizing the electrons or rotating the spin coordinates. We also reveal a clear similarity between the Coulomb hole of He-like ions and the density depletion near the nucleus induced by the fractional-spin error in the unpolarized hydrogen atom. Finally, we analyze the effect of the fractional-spin error on the M{\o}ller-Plesset adiabatic connection, excited states, and functional- and density-driven errors.