Simple hydrogenic estimates for the exchange and correlation energies of atoms and atomic ions, with implications for density functional theory
Abstract
Exact density functionals for the exchange and correlation energies are approximated in practical calculations for the groundstate electronic structure of a manyelectron system. An important exact constraint for the construction of approximations is to recover the correct nonrelativistic 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 lowestrung 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 semiquantitatively correct even for small $N$ and for $N \approx Z$. The large$N$ asymptotic behavior of the energy is prefigured in small$N$ atoms and atomic ions, supporting the argument that widelypredictive approximate density functionals should be designed to recover the correct asymptotics. It is shown that the exact KohnSham correlation energy, when calculated from the pure groundstate wavefunction, should have no contribution proportional to $Z$ in the $Z\to \infty$ limit for any fixed $N$.
 Publication:

arXiv eprints
 Pub Date:
 July 2020
 DOI:
 10.48550/arXiv.2007.01917
 arXiv:
 arXiv:2007.01917
 Bibcode:
 2020arXiv200701917K
 Keywords:

 Physics  Chemical Physics
 EPrint:
 This work has been accepted for publication at the Journal of Chemical Physics. Revisions: new Appendix A (former Appendix A is now Appendix B) discussing exact KohnSham perturbation series for Ec. Added material discussing the Becke 1988 functional. More discussion of nonempirical functionals' recovery of the asymptotic series, and their accuracy in predicting atomic/molecular energies