We construct a Laplacian-level meta-generalized-gradient-approximation (meta-GGA) for the noninteracting (Kohn-Sham orbital) positive kinetic energy density $\tau$ of an electronic ground state of density $n$. This meta-GGA is designed to recover the fourth-order gradient expansion ${\tau }^{\mathit{GE}4}$ in the appropriate slowly varying limit and the von Weizsäcker expression ${\tau }^W={\mid \nabla n\mid }^2∕\left(8n\right)$ in the rapidly varying limit. It is constrained to satisfy the rigorous lower bound ${\tau }^W\left(\mathbf{r}\right)⩽\tau \left(\mathbf{r}\right)$. Our meta-GGA is typically a strong improvement over the gradient expansion of $\tau$ for atoms, spherical jellium clusters, jellium surfaces, the Airy gas, Hooke’s atom, one-electron Gaussian density, quasi-two-dimensional electron gas, and nonuniformly scaled hydrogen atom. We also construct a Laplacian-level meta-GGA for exchange and correlation by employing our approximate $\tau$ in the Tao-Perdew-Staroverov-Scuseria (TPSS) meta-GGA density functional. The Laplacian-level TPSS gives almost the same exchange-correlation enhancement factors and energies as the full TPSS, suggesting that $\tau$ and ${\nabla }^{2}n$ carry about the same information beyond that carried by $n$ and $\nabla n$. Our kinetic energy density integrates to an orbital-free kinetic energy functional that is about as accurate as the fourth-order gradient expansion for many real densities (with noticeable improvement in molecular atomization energies), but considerably more accurate for rapidly varying ones.

• Received 16 December 2006

DOI:https://doi.org/10.1103/PhysRevB.75.155109