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 {ρ}(r)[1/{β}(r)]dr, where {β}(r) is the local temperature parameter of Ghosh, Berkowitz, and Parr [Proc. Natl. Acad. Sci. USA 81, 8028 (1984)]. The Gaussian ansatz gives K=({π}/2) F ρ²(r){β}(r)dr, while the uniform-gas ansatz gives K=(9{π}/20) F ρ²(r){β}(r)dr. Assuming further only the existence of a local equation of state of the form {β}={β}({ρ}), one then gets, by imposing the exact normalization condition for the first-order density matrix, from the Gaussian ansatz T=(3{π}/25/3) F ρ5/3(r)dr and K=2^-1/3 F ρ4/3(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.