Laplacian-level density functionals for the kinetic energy density and exchange-correlation energy

We construct a Laplacian-level meta-generalized-gradient-approximation (meta-GGA) for the noninteracting (Kohn-Sham orbital) positive kinetic energy density $\ensuremath{\tau}$ of an electronic ground state of density $n$. This meta-GGA is designed to recover the fourth-order gradient expansion ${\ensuremath{\tau}}^{\mathit{GE}4}$ in the appropriate slowly varying limit and the von Weizs\"acker expression ${\ensuremath{\tau}}^{W}={\ensuremath{\mid}\ensuremath{\nabla}n\ensuremath{\mid}}^{2}∕(8n)$ in the rapidly varying limit. It is constrained to satisfy the rigorous lower bound ${\ensuremath{\tau}}^{W}(\mathbf{r})\ensuremath{\leqslant}\ensuremath{\tau}(\mathbf{r})$. Our meta-GGA is typically a strong improvement over the gradient expansion of $\ensuremath{\tau}$ 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 $\ensuremath{\tau}$ 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 $\ensuremath{\tau}$ and ${\ensuremath{\nabla}}^{2}n$ carry about the same information beyond that carried by $n$ and $\ensuremath{\nabla}n$. 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.