Differential virial theorem in density-functional theory in terms of the Pauli potential for spherically symmetric electron densities: Illustrative example for the family of Be-like atomic ions

The differential virial theorem relates the force -partial derivative V/partial derivative r associated with the one-body potential V(r) of density-functional theory to the Laplacian del(2)n of the ground-state density n(r) and to a quantity z(s)(r) involving the kinetic energy density tensor t(alpha beta)(r). Having the concept of the Pauli potential V-P(r), z(s) is derived for spherically symmetric ground-state densities n(r) in terms of the von Weizsacker kinetic energy density and the first derivative of V-P(r). z(s) is related solely to the gradient kinetic energy density t(G)(r) for Be-like atomic ions.