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

The March-Suhai (MS) partial differential equation for the Dirac density matrix γs(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 ts(r). Here, we examine the relation between the equation of motion for γs(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 ts(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 γs(x,x') to satisfy for arbitrary level occupancy, in the form of a MS-type differential equation.