On possible gradient approximations to the one-dimensional kinetic energy density functional compatible with the differential virial theorem

Abstract

When the kinetic energy density in k (defined positive definite) of a system of one-dimensional, non-interacting fermions is approximated by an ordinary function of the density rho , and of the lowest n derivatives of rho , in k>or=0 can satisfy the differential virial theorem for arbitrary density distributions only if in k actually depends on rho and rho ' only. Thus the case n>1 is ruled out, and one is left with the result in k= kappa rho 3+ rho '2/(8 rho ) ( kappa >or=0) already obtained in previous work.