Simple hydrogenic estimates for the exchange and correlation energies of\n atoms and atomic ions, with implications for density functional theory

Exact density functionals for the exchange and correlation energies are\napproximated in practical calculations for the ground-state electronic\nstructure of a many-electron system. An important exact constraint for the\nconstruction of approximations is to recover the correct non-relativistic\nlarge-$Z$ expansions for the corresponding energies of neutral atoms with\natomic number $Z$ and electron number $N=Z$, which are correct to leading order\n($-0.221 Z^{5/3}$ and $-0.021 Z \\ln Z$ respectively) even in the lowest-rung or\nlocal density approximation. We find that hydrogenic densities lead to\n$E_x(N,Z) \\approx -0.354 N^{2/3} Z$ (as known before only for $Z \\gg N \\gg 1$)\nand $E_c \\approx -0.02 N \\ln N$. These asymptotic estimates are most correct\nfor atomic ions with large $N$ and $Z \\gg N$, but we find that they are\nqualitatively and semi-quantitatively correct even for small $N$ and for $N\n\\approx Z$. The large-$N$ asymptotic behavior of the energy is pre-figured in\nsmall-$N$ atoms and atomic ions, supporting the argument that widely-predictive\napproximate density functionals should be designed to recover the correct\nasymptotics. It is shown that the exact Kohn-Sham correlation energy, when\ncalculated from the pure ground-state wavefunction, should have no contribution\nproportional to $Z$ in the $Z\\to \\infty$ limit for any fixed $N$.\n