R- andp-space electron densities and related kinetic and exchange energies in terms ofsstates alone for the leading term in the1/Zexpansion for nonrelativistic closed-shell atomic ions

Abstract

As a step towards constructing nonlocal energy density functionals, the leading term in the so-called $1/Z$ expansion for closed-shell atomic ions is the focus here. This term is characterized by the properties of the bare Coulomb potential $(\ensuremath{-}{\mathrm{Ze}}^{2}/r),$ and for an arbitrary number of closed shells it is known that $\ensuremath{\partial}\ensuremath{\rho}(r)/\ensuremath{\partial}r=\ensuremath{-}{(2Z/a}_{0}){\ensuremath{\rho}}_{s}(r),$ where $\ensuremath{\rho}(r)$ is the ground-state electron density while ${\ensuremath{\rho}}_{s}(r)$ is the s-state $(l=0)$ contribution to $\ensuremath{\rho}(r).$ Here, the kinetic-energy density $t(r)$ is also derived as a double integral in terms of ${\ensuremath{\rho}}_{s}(r)$ and Z. Although the exchange energy density ${\ensuremath{\epsilon}}_{x}(r)$ is more complex than $t(r),$ a proof is given that, in the Coulomb limit system, ${\ensuremath{\epsilon}}_{x}$ is indeed also determined by s-state properties alone. The same is shown to be true for the momentum density $n(p),$ which here is obtained explicitly for an arbitrary number of closed shells. Finally, numerical results are presented that include (a) ten-electron atomic ions $(K+L$ shells), (b) the limit as the number of closed shells tends to infinity, where an appeal is made to the analytical r-space study of Heilmann and Lieb [Phys. Rev. A 52, 3628 (1995)], and (c) momentum density and Compton line shape for an arbitrary number of closed shells.