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 $e_{\text{x}}^{\text{ex}\left(\text{conv}\right)}$ is not necessarily well suited for local mixing with a given semilocal approximation. We show how to transform $e_{\text{x}}^{\text{ex}\left(\text{conv}\right)}$ 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 $e_{\text{x}}^{\text{ex}\left(\text{conv}\right)}$ by a resolution-of-the-identity method requires uncontracted basis functions.

Figure 1

Conventional exact-exchange energy per electron in the CO molecule along the internuclear axis evaluated at the experimental geometry using the approximate resolution of the identity in four basis sets. The curves for the last three basis sets are close together everywhere.

Figure 2

Exact-exchange energy per electron in the TPSS gauge in the NaCl molecule evaluated at the experimental geometry using the approximate resolution of the identity in four basis sets. The curves for the last three basis sets are almost indistinguishable.

Figure 3

Radial exchange energy densities of the H atom computed at the exact ground-state density: exact conventional [ex(conv)], exact in the TPSS gauge [ex(TPSS)], exact from a transformed exchange hole [$\text{ex}\left(\omega \right)$, $\omega =0.92$], and semilocal (TPSS).

Figure 4

Exchange energies per electron in the H atom computed at the exact ground-state density. For the explanation of the legend, refer to Fig. 3.

Figure 5

Radial exchange energy densities of the Ne atom computed at the converged Hartree-Fock orbitals in the UGBS basis set: exact conventional [ex(conv)], exact in the TPSS gauge [ex(TPSS)], and semilocal TPSS approximation.

Figure 6

Same as in Fig. 5 for the Kr atom.

Figure 7

Difference ${\epsilon }_{\text{x}}^{\text{TPSS}}-{\epsilon }_{\text{x}}^{\text{ex}}$, where ${\epsilon }_{\text{x}}^{\text{ex}}$ is in the conventional and TPSS gauges, in a free N atom and along the internuclear axis of the ${\text{N}}_2$ molecule at the experimental geometry. (b) shows only the right half of the molecule with the N nucleus placed at $z=0$. All quantities were computed at the converged TPSS orbitals using the UGBS1P basis set.

Figure 8

Difference ${\epsilon }_{\text{x}}^{\text{TPSS}}-{\epsilon }_{\text{x}}^{\text{ex}\left(\text{TPSS}\right)}$ in the $\text{He}_2{}^{+}$ molecule along the internuclear axis and along a parallel axis offset by 0.5 bohr. Each panel shows only the region near the right nucleus which is always placed at $z=0$. The static correlation in the stretched molecule ${\text{He}}^{0.5+}\cdots {\text{He}}^{0.5+}$ (dashed line) is more negative than at the equilibrium geometry (solid line). All quantities were computed at the converged TPSS orbitals using the uncontracted cc-pVQZ basis set.

Figure 9

Radial exchange energy densities for a spherical jellium cluster of $N=2$ electrons computed at the exchange-only OEP orbitals: exact conventional [ex(conv)], exact in the TPSS gauge [ex(TPSS)], and semilocal (TPSS).

Figure 10

Same as in Fig. 9 for a cluster of $N=58$ electrons.