Nonrelativistic exchange-energy density and exchange potential in the lowest order of the $1/\mathit{Z}$ expansion for ten-electron atomic ions

The Dirac exchange energy $E_x$ has a density ${\varepsilon }_x\mathrm{}\left(r\right)\mathrm{}$ 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 ${\varepsilon }_x\mathrm{}\left(r\right)\mathrm{}$ is calculated analytically, and presented graphically for $Z=92\mathrm{}$ in this large-Z limit. The Slater approximation to the exchange potential $V_x\left(\mathbf{r}\right),$ namely, $V_{\mathrm{Sla}}\left(\mathbf{r}\right)=2\mathrm{}{\varepsilon }_x\mathrm{}\left(r\right)/\rho \mathrm{}\left(r\right)\mathrm{}$ with $\rho \mathrm{}\left(r\right)\mathrm{}$ the ground-state electron density, is also plotted for $Z=92.\mathrm{}$ In the large-Z limit, $V_x\left(\mathbf{r}\right)$ 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 $V_{\mathrm{Sla}}\left(\mathbf{r}\right)$ plus approximate corrections, are also presented.