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

Aaron D Kaplan,John P Perdew,Pradeep Bhetwal,Mel Levy,Kieron Burke,Jie Yu,Puskar Bhattarai,Hong Tang,Kamal Wagle,Shah Tanvir Ur Rahman Chowdhury,Biswajit Santra

doi:10.1063/5.0017805

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 the leading order (-0.221Z5/3 and -0.021Z ln Z, respectively) even in the lowest-rung or local density approximation. We find that hydrogenic densities lead to Ex(N, Z) ≈ -0.354N2/3Z (as known before only for Z ≫ N ≫ 1) and Ec ≈ -0.02N ln N. These asymptotic estimates are most correct for atomic ions with large N and Z ≫ N, but we find that they are qualitatively and semi-quantitatively correct even for small N and N ≈ 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 → ∞ limit for any fixed N.

Highlights

In this work, we will find closed-form formulas for the exchange energy and correlation energy of an atom or atomic ion with electron number N and proton number Z
The primary difference is in the coefficient of Z5/3, in part because the density functional for the exchange energy depends on the detailed shape of the electron density but not so for correlation in leading order
In the limit 1 ≪ N ≪ Z, the Thomas-Fermi hydrogenic density becomes slowly-varying on the scale of the exchange energy, but not on the scale of the correlation energy

Summary

Introduction

We will find closed-form formulas for the exchange energy and correlation energy of an atom or atomic ion with electron number N and proton number Z. In Kohn-Sham theory, the exchange and correlation energies are functionals of the electron density. The primary difference is in the coefficient of Z5/3, in part because the density functional for the exchange energy depends on the detailed shape of the electron density (e.g. hydrogenic vs self-consistent neutral Thomas-Fermi) but not so for correlation in leading order.

Results

Conclusion