Explicit estimation of ground-state kinetic energies from electron densities

This paper discusses some initial steps toward the goal of finding explicit procedures for calculating, to a good approximation, the minimum kinetic energy consistent with a given particle density ρ(r) for a system of fermions. The strategy proposed begins by separating the desired kinetic energy into a sum $T_W$+$T_{\mathrm{theta}}$, where $T_W$≥0 is the Weizsaaumlcker energy F $d^3$r ‖∇ρ${\Vert }^2$/8ρ , and $T_{\mathrm{theta}}$≥0. Approximations are applied to $T_{\mathrm{theta}}$ alone, and are sought in the form of interpolations that will be nearly correct in two limits: small departures from uniform density, for which exact results are known from linear-response theory, and cases where a region containing no more than one particle of a given spin becomes isolated from the rest of the distribution by regions of nearly vanishing density. Only highly nonlocal functionals can behave properly in either of these limits. A few other conditions for satisfactory approximations to $T_{\mathrm{theta}}$ are noted. Some explicit interpolation formulas are offered for one-dimensional problems, and are tested on a variety of examples; one such is found to give kinetic energies to within a few percent in nearly all cases examined. More detailed tests are possible by comparing correct and approximate potentials yielding a given density, or correct and approximate densities for the ground state of a given potential; tests on the position dependence of kinetic energy density, however, are physically meaningless. A few remarks are offered on the additional problems that beset extension of the scheme to three dimensions; foremost among these is that of computational tractability.