r- and p-space electron densities and related kinetic and exchange energies in terms of s states alone for the leading term in the 1/Z expansion for nonrelativistic closed-shell atomic ions

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 (-Ze²/r), and for an arbitrary number of closed shells it is known that ∂ρ(r)/∂r=-(2Z/a₀)ρ_s(r), where ρ(r) is the ground-state electron density while ρ_s(r) is the s-state (l=0) contribution to ρ(r). Here, the kinetic-energy density t(r) is also derived as a double integral in terms of ρ_s(r) and Z. Although the exchange energy density ɛ_x(r) is more complex than t(r), a proof is given that, in the Coulomb limit system, ɛ_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.