Nonrelativistic exchange-energy density and exchange potential in the lowest order of the1/Zexpansion for ten-electron atomic ions

Abstract

The Dirac exchange energy ${E}_{x}$ has a density ${\ensuremath{\varepsilon}}_{x}(r)$ in a spherically symmetric ten-electron atomic ion that is determined by the idempotent first-order density matrix (1 DM). In turn, this 1 DM has as its leading term in the $1/Z$ expansion, with Z the atomic number, the bare Coulomb result. This latter quantity is known analytically from the study of March and Santamaria [Phys. Rev. A 38, 5002 (1988)]. Here ${\ensuremath{\varepsilon}}_{x}(r)$ is calculated analytically, and presented graphically for $Z=92$ in this large-Z limit. The Slater approximation to the exchange potential ${V}_{x}(\mathbf{r}),$ namely, ${V}_{\mathrm{Sla}}(\mathbf{r})=2{\ensuremath{\varepsilon}}_{x}(r)/\ensuremath{\rho}(r)$ with $\ensuremath{\rho}(r)$ the ground-state electron density, is also plotted for $Z=92.$ In the large-Z limit, ${V}_{x}(\mathbf{r})$ can be obtained by functional differentiation of the resultant exchange energy, and is expressed in terms of electron and kinetic-energy densities. Numerical calculations, based on ${V}_{\mathrm{Sla}}(\mathbf{r})$ plus approximate corrections, are also presented.