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

Abstract

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