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

 

Aaron D. Kaplan^1,a), Biswajit Santra¹, Puskar Bhattarai¹, Kamal Wagle¹, Shah Tanvir ur Rahman Chowdhury¹, Pradeep Bhetwal¹, Jie Yu¹, Hong Tang¹, Kieron Burke², Mel Levy³, and John P. Perdew^1,4

• ¹Department of Physics, Temple University, Philadelphia, Pennsylvania 19122, USA
• ²Departments of Chemistry and Physics, University of California, Irvine, California 92697, USA
• ³Department of Chemistry and Quantum Theory Group, Tulane University, New Orleans, Louisiana 70118, USA
• ⁴Department of Chemistry, Temple University, Philadelphia, Pennsylvania 19122, USA
• ^a)Author to whom correspondence should be addressed: kaplan@temple.edu

 

 

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 (−0.221Z^5/3 and −0.021Z ln Z, respectively) even in the lowest-rung or local density approximation. We find that hydrogenic densities lead to E_x(N, Z) ≈ −0.354N^2/3Z (as known before only for Z ≫ N ≫ 1) and E_c ≈ −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.

 

https://sci-hub.se/https://aip.scitation.org/doi/abs/10.1063/5.0017805?journalCode=jcp