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

The 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 electron density, through a local hybrid functional. Because energy densities are not uniquely defined, the conventional (Slater) exact-exchange energy density ${e}_{\text{x}}^{\text{ex}(\text{conv})}$ is not necessarily well suited for local mixing with a given semilocal approximation. We show how to transform ${e}_{\text{x}}^{\text{ex}(\text{conv})}$ 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 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 $\text{He}_{2}{}^{+}$ and related systems. Our construction of ${e}_{\text{x}}^{\text{ex}(\text{conv})}$ by a resolution-of-the-identity method requires uncontracted basis functions.