Various functionals for the kinetic energy density of an atom or molecule

Various approximate density functionals for the kinetic energy density of atoms and molecules are analyzed. These include the results of a gradient expansion to first and second orders and a form recently derived from a new Green's function approximation [W. Yang, preceding paper, Phys. Rev. A 34, 4575 (1986)]. All the approximate functionals studied diverge to minus infinity at a nucleus, due to the ${\ensuremath{\nabla}}^{2}$\ensuremath{\rho} term that is in them, while the exact functional is positive and finite everywhere. Away from nuclei, however, the Hartree-Fock results are well reproduced, including the atomic shell structure. New functionals are proposed to correct the divergent behavior, and accurate total kinetic energy values are obtained from a new formula for kinetic energy density ${t}_{\mathrm{MP}(\mathrm{r})={C}_{k}\mathrm{\ensuremath{\rho}}{(\mathrm{r})}^{5/3}}$ +(1/72)\ensuremath{\Vert}\ensuremath{\nabla}\ensuremath{\rho}(r)${\ensuremath{\Vert}}^{2}$/\ensuremath{\rho}(r)+( 1/12)${\ensuremath{\nabla}}^{2}$\ensuremath{\rho}(r), with a divergence correction.