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 tau of an electronic ground state of density n. This meta-GGA is designed to recover the fourth-order gradient expansion tau(GE4) in the appropriate slowly varying limit and the von Weizsacker expression tau(W)=parallel to del n parallel to(2)/(8n) in the rapidly varying limit. It is constrained to satisfy the rigorous lower bound tau(W)(r)<=tau(r). 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 del(2)n carry about the same information beyond that carried by n and del 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.