Gradient correction in Thomas-Fermi theory

Abstract

A new derivation of the Weizsacker-type gradient corrections to Thomas-Fermi (TF) kinetic energy functional is presented. The development is based on the first-order reduced density matrix as obtained from the one-body Green’s function in the mean-path approximation devised for the purpose, using the Feynman path-integral approach; the mean-path approximation turns out to be essentially equivalent to the eikonal approximation used in quantum collision theory for high-energy collisions. This derivation agrees with the conventional gradient expansion truncated at second order, in that it gives the kinetic energy functional of the TF-(1/9)W model, that is, the sum of the original TF kinetic energy and (1/9) of the Weizsacker gradient correction. However, in the present derivation, TF-(1/9)W results from a reduced density matrix of closed form; the original TF local relation between particle, density, and one-body potential is preserved; and the kinetic energy density contains a Laplacian of particle density with a factor half of that from the gradient expansion. Most significantly, the TF-(1/9)W kinetic energy functional is the consequence of representing both the diagonal and off-diagonal elements of the density matrix correctly to zero order through the mean-path approximation to the one-body Green’s function, whereas in the conventional TF approximation, the zero order of the gradient expansion, off-diagonal elements are not correct to the same order. Other results of the present approach include a nonlocal exchange energy functional of density, a one-body effective potential that contains a contribution from the kinetic energy functional derivative, and the construction of closed-form density matrices that give various kinetic energy functionals of TF-λW form (justifying various existing empirical λ values). Also presented are the results of numerical calculation for rare-gas atoms of TFD-λW models (TFD denotes Thomas-Fermi-Dirac) with λ=(1/3), 0.186, (1/6), and (1/9). .AE