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 ${\gamma }_s(\stackrel{\rightharpoonup }{r},{\stackrel{\rightharpoonup }{r}}^{\prime })$, proved for one- and two-level occupancies, involves both the ground-state density $n(\stackrel{\rightharpoonup }{r})$, with its low-order derivatives, and the positive definite kinetic energy density $t_s(\stackrel{\rightharpoonup }{r})$. Here, we examine the relation between the equation of motion for ${\gamma }_s(\stackrel{\rightharpoonup }{r},{\stackrel{\rightharpoonup }{r}}^{\prime })$, 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 $t_s(\stackrel{\rightharpoonup }{r})$ 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 ${\gamma }_s(x,x^{\prime })$ to satisfy for arbitrary level occupancy, in the form of a MS-type differential equation.