Gaussian and other approximations to the first-order density matrix of electronic systems, and the derivation of various local-density-functional theories

Various approximations and models for Hartree-Fock kinetic energy T and Hartree-Fock exchange energy K are systematically derived with a minimum of assumptions and comparatively studied by testing on atoms. A Gaussian ansatz for the spherical average of the first-order density matrix and the uniform-gas ansatz are shown both to lead to the same formulas for T and K except for the values of certain numerical coefficients. Without any assumption one gets in all cases T=(3/2) F \ensuremath{\rho}(r)[1/\ensuremath{\beta}(r)]dr, where \ensuremath{\beta}(r) is the local temperature parameter of Ghosh, Berkowitz, and Parr [Proc. Natl. Acad. Sci. USA 81, 8028 (1984)]. The Gaussian ansatz gives K=(\ensuremath{\pi}/2) F ${\ensuremath{\rho}}^{2}$(r)\ensuremath{\beta}(r)dr, while the uniform-gas ansatz gives K=(9\ensuremath{\pi}/20) F ${\ensuremath{\rho}}^{2}$(r)\ensuremath{\beta}(r)dr. Assuming further only the existence of a local equation of state of the form \ensuremath{\beta}=\ensuremath{\beta}(\ensuremath{\rho}), one then gets, by imposing the exact normalization condition for the first-order density matrix, from the Gaussian ansatz T=(3\ensuremath{\pi}${/2}^{5/3}$) F ${\ensuremath{\rho}}^{5/3}$(r)dr and K${=2}^{\mathrm{\ensuremath{-}}1/3}$ F ${\ensuremath{\rho}}^{4/3}$(r)dr, respectively, 3.4% and 7.5% larger than the Thomas-Fermi-Dirac T and K which result from the uniform-gas ansatz. Numerical evidence is presented that shows preference for the Gaussian ansatz. A modified Gaussian ansatz also is examined.