Fourth-order gradient contributions in extended Thomas-Fermi theory for noninteracting fermions at finite temperature

Abstract

Nonlocal corrections to the free energy of a system of noninteracting fermions are considered within the framework of extended Thomas-Fermi theory. The density and temperature dependence of gradient corrections is given through fourth order in powers of the density gradient. The calculation is based on long-wavelength expansions of linear and nonlinear response functions of a nearly uniform system. Results are given in terms of Fermi-Dirac integrals and convenient forms are given which span the full density and temperature range from the degenerate limit to the classical limit