Alternative expressions for the Fermi hole curvature

Abstract

The Fermi hole curvature C(r,s) is defined as the Laplacian of the parallel-spin pair distribution, evaluated at zero separation r′=r for a pair of Fermions in a many-Fermion system. It has been used by a number of authors to discuss electron localization, properties of the exchange and correlation hole, and exchange and correlation energies of inhomogeneous electron gases. Here, the discussion of this quantity is extended in two directions. First, for the special case of a single-determinant many-electron state, it is shown that a previously derived macroscopic expression for C can be generalized in a simple fashion to apply to current-carrying states. Second, it is shown that a recently given interpretation of C(r,s), in terms of relative kinetic energy of pairs, is valid for a general many-Fermion state and is not limited to the single-determinant case investigated previously.