Meta-generalized gradient approximation for the exchange-correlation hole with an application to the jellium surface energy

We propose a model for the angle- and system-averaged exchange-correlation hole of a many-electron system. This hole analyzes the exchange-correlation energy into contributions of various distances $u$ from an electron. The model is ``reverse-engineered'' (derived from and not used to derive a density functional). It satisfies known exact hole constraints, including ones that can only be satisfied by a meta-generalized gradient approximation or meta-GGA. It incorporates the exchange-correlation energy density of the Tao-Perdew-Staroverov-Scuseria (TPSS) nonempirical meta-GGA. The hole model is tested for atoms and applied to jellium surfaces. The Fourier transform $(u\ensuremath{\rightarrow}k)$ of the hole is needed for wave-vector interpolation of the jellium surface energy from an exact small-$k$ or large-$u$ asymptote. We find essentially the same surface energies (close to the uncorrected TPSS values) whether we apply the wave-vector interpolation correction to the local spin density approximation, the GGA or the meta-GGA. These and other considerations suggest that these surface energies are accurate. Moreover, we find that the uncorrected TPSS surface energies have a realistic wave-vector analysis. Our TPSS hole model can be used to build the hole model for a TPSS-based global hybrid functional, or for a hyper-GGA that uses full exact exchange.