Differential equation for the Dirac single-particle first-order density matrix in terms of the ground-state electron density

Abstract

The March-Suhai (MS) partial differential equation for the Dirac density matrix {gamma}{sub s}(r,r'), proved for one- and two-level occupancies, involves both the ground-state density n(r), with its low-order derivatives, and the positive definite kinetic energy density t{sub s}(r). Here, we examine the relation between the equation of motion for {gamma}{sub s}(r,r'), with input now being the one-body potential of density-functional theory, and the MS equation. The important link is the differential virial theorem, which can be used to remove t{sub s}(r) from the MS differential equation. For multiple occupancy, the Pauli potential enters in an important manner. In one dimension, however, the appearance of the Pauli potential can be avoided, obtaining a necessary condition for {gamma}{sub s}(x,x{sup '}) to satisfy for arbitrary level occupancy, in the form of a MS-type differential equation.