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

TL;DR

This paper derives simple hydrogenic estimates for exchange and correlation energies in atoms and ions, highlighting their asymptotic behavior and implications for constructing accurate density functionals in quantum chemistry.

Contribution

It introduces hydrogenic estimates for exchange and correlation energies, emphasizing their accuracy for large atomic numbers and their importance for density functional theory development.

Findings

Hydrogenic densities approximate exchange energy as -0.354 N^{2/3} Z.

Correlation energy approximates -0.02 N ln N.

Asymptotic behaviors are valid even for small N and N≈Z.

Abstract

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