Exact-exchange energy density in the gauge of a semilocal density functional approximation

TL;DR

This paper develops a method to transform the conventional exact-exchange energy density into a form compatible with semilocal density functionals, improving the modeling of static correlation in density functional theory.

Contribution

It introduces a gauge transformation for the exact-exchange energy density that aligns it with semilocal functionals, ensuring better integration in local hybrid functionals.

Findings

The transformed energy density satisfies exact constraints and is most accurate where a single orbital dominates.

The difference between semilocal and exact-exchange energy densities increases under bond stretching.

The method requires uncontracted basis functions for the resolution-of-the-identity calculation.

Abstract

Exact-exchange energy density and energy density of a semilocal density functional approximation are two key ingredients for modeling the static correlation, a strongly nonlocal functional of the density, through a local hybrid functional. Because energy densities are not uniquely defined, the conventional (Slater) exact-exchange energy density $e_\mathrm{x}^\mathrm{ex(conv)}$ is not necessarily well-suited for local mixing with a given semilocal approximation. We show how to transform $e_\mathrm{x}^\mathrm{ex(conv)}$ in order to make it compatible with an arbitrary semilocal density functional, taking the nonempirical meta-generalized gradient approximation of Tao, Perdew, Staroverov, and Scuseria (TPSS) as an example. Our additive gauge transformation function integrates to zero, satisfies exact constraints, and is most important where the density is dominated by a single orbital shape. We show that, as expected, the difference between semilocal and exact-exchange energy densities becomes more negative under bond stretching in He$_2^{+}$ and related systems. Our construction of $e_\mathrm{x}^\mathrm{ex(conv)}$ by a resolution-of-the-identity method requires uncontracted basis functions.