The flexible nature of exchange, correlation and Hartree physics: resolving "delocalization" errors in a 'correlation free' density functional

TL;DR

This paper introduces the Linear EXX (LEXX) theory to improve the treatment of fractional occupations in density functional theory, addressing delocalization errors and explaining the success and limitations of existing methods.

Contribution

The paper develops the LEXX approach, providing explicit piecewise linear expressions for energies and pair distributions, enhancing the understanding of exchange-correlation in ensemble systems.

Findings

LEXX energies are bounded by EEXX and EXX energies.

LEXX explains the success of standard OEP methods for diatoms at large spacing.

LEXX accurately reproduces correlation-free energies for various fractional ions.

Abstract

By exploiting freedoms in the definitions of 'correlation', 'exchange' and 'Hartree' physics in ensemble systems we better generalise the notion of 'exact exchange' (EXX) to systems with fractional occupations functions of the frontier orbitals, arising in the dissociation limit of some molecules. We introduce the Linear EXX ("LEXX") theory whose pair distribution and energy are explicitly \emph{piecewise linear} in the occupations $f^{\sigma}_{i}$. {\hi}We provide explicit expressions for these functions for frontier $s$ and $p$ shells. Used in an optimised effective potential (OEP) approach it yields energies bounded by the piecewise linear 'ensemble EXX' (EEXX) energy and standard fractional optimised EXX energy: $E^{EEXX}\leq E^{LEXX} \leq E^{EXX}$. Analysis of the LEXX explains the success of standard OEP methods for diatoms at large spacing, and why they can fail when both spins are allowed to be non-integer so that "ghost" Hartree interactions appear between \emph{opposite} spin electrons in the usual formula. The energy $E^{LEXX}$ contains a cancellation term for the spin ghost case. It is evaluated for H, Li and Na fractional ions with clear derivative discontinuities for all cases. The $p$-shell form reproduces accurate correlation-free energies of B-F and Al-Cl. We further test LEXX plus correlation energy calculations on fractional ions of C and F and again shows both derivative discontinuities and good agreement with exact results.