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 $(-{\mathrm{Ze}}^2/r),$ and for an arbitrary number of closed shells it is known that $\partial \rho \mathrm{}\left(r\right)/\partial r=-{(2Z/a}_0\left){\rho }_s\mathrm{}\right(r),$ where $\rho \mathrm{}\left(r\right)\mathrm{}$ is the ground-state electron density while ${\rho }_s\mathrm{}\left(r\right)\mathrm{}$ is the s-state $\mathrm{}(l=0\mathrm{})\mathrm{}$ contribution to $\rho \mathrm{}\left(r\right).$ Here, the kinetic-energy density $t\left(r\right)\mathrm{}$ is also derived as a double integral in terms of ${\rho }_s\mathrm{}\left(r\right)\mathrm{}$ and Z. Although the exchange energy density ${\epsilon }_x\mathrm{}\left(r\right)\mathrm{}$ is more complex than $t\left(r\right),$ a proof is given that, in the Coulomb limit system, ${\epsilon }_x$ is indeed also determined by s-state properties alone. The same is shown to be true for the momentum density $n\left(p\right),$ 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.

• Received 26 December 2000

DOI:https://doi.org/10.1103/PhysRevA.63.062501