Some tests of an approximate density functional for the ground-state kinetic energy of a fermion system

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Abstract
Any proposed approximation to the ground-state kinetic energy of a system of noninteracting fermions in terms of the particle density ρ(r) can be used to estimate the potential field v(r) that will give rise to a given ρ(r), or the ρ(r) that will result from placing a given number of particles in a given v(r). Comparison with exact quantum-mechanical calculations thus gives two possible types of tests for any proposed kinetic-energy functional. This paper reports such tests for a recently proposed nonlocal functional applicable to one-dimensional systems, comparing its predictions for several simple problems not only with the correct answers but also with the predictions of Thomas-Fermi and Thomas-Fermi-Weizsäcker functionals, to which the new functional proves considerably superior. The comparisons yield useful insights on the virtues and defects of the new functional, and on the directions in which improvements should be sought. Particular attention is devoted to Friedel oscillations and shell structure. It is shown that functionals of the Thomas-Fermi-Weizsäcker type can never predict multiple maxima in ρ if v has only a single minimum and no maxima; the new functional does not have this defect, though it yields density oscillations rather weaker than the exact quantum-mechanical calculations. A natural general inference from the present tests is that any satisfactory kinetic-energy functional must (as ours does) replace the entire Thomas-Fermi term by a nonlocal expression.