Generalization to higher dimensions of one-dimensional differential equation for Slater sum of an inhomogeneous electron liquid for a given local potential

Abstract

The Slater sum (r, β) is the diagonal element of the canonical or Bloch density matrix, and by spatial integration yields the partition function. In one dimension, for independent electrons moving in a common potential v(x), the work of March and Murray (Phys. Rev., 120, 831 (1960)) already yielded a third-order partial differential equation for S(x, β). But to date, a generalization to higher dimensions for independent electrons in a given v(r) has not been effected. Here, using the differential virial equation (Holas and March, Phys. Rev. A, 51, 2040 (1995)) such generalization is derived. As a special case, the known differential equation for S(r, β) for three-dimensional spherically symmetric harmonic confinement is recovered. This equation is shown to be valid also for a wider, specific class of three-dimensional spherical systems.