Fractional spins and static correlation error in density functional theory

TL;DR

This paper demonstrates that the exact density functional energy for fractional-spin states remains constant, while all approximate functionals deviate significantly, causing static correlation errors in strongly correlated systems.

Contribution

It proves the constancy of the exact functional for fractional spins and highlights the need to incorporate this behavior into approximate functionals.

Findings

Exact functional energy for fractional spins is constant.

Approximate functionals show large deviations from constancy.

Numerical examples illustrate static correlation errors in molecular systems.

Abstract

Electronic states with fractional spins arise in systems with large static correlation (strongly correlated systems). Such fractional-spin states are shown to be ensembles of degenerate ground states with normal spins. It is proven here that the energy of the exact functional for fractional-spin states is a constant, equal to the energy of the comprising degenerate pure spin states. Dramatic deviations from this exact constancy condition exist with all approximate functionals, leading to large static correlation errors for strongly correlated systems, such as chemical bond dissociation and band structure of Mott insulators. This is demonstrated with numerical calculations for several molecular systems. Approximating the constancy behavior for fractional spins should be a major aim in functional constructions and should open the frontier for DFT to describe strongly correlated systems. The key results are also shown to apply in reduced density-matrix functional theory.