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 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 (−0.221Z 5/3 and −0.021Z ln Z respectively) even in the lowest-rung or local density approximation. We find that hydrogenic densities lead to Ex(N, Z) ≈ −0.354N 2/3 Z (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 for 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 .

Gradient expansions for the large-coupling strength limit of the Møller-Plesset adiabatic connection

“…As we will discuss later, while A HF is universal, the coefficient B HF seems to depend on how the slowly varying limit is approached, similarly to what happens with the DFT exchange functional. 23,27 3.1 LDA coefficient A HF…”

“…• p = − 2 3 : the Thomas-Fermi scaling of the Bohr atoms; 27,28,38,39 • p = 0 : the scaling used in For any density functional G[ρ] that under the uniform coordinate scaling of eq (22) behaves as…”

“…with ρTFna (r) the TF profile (integrating to 1) for neutral atoms, [25][26][27][28]37 led to the conclusion that their exchange energy as a function of N = Z should have, to leading orders, the large-N expansion a x N 5/3 +b x N . Extracting b x from exchange energies of neutral atoms allowed to fix the GEA2 coefficient for exchange in ref.…”

“…A case for which it is even simpler to make a detailed numerical analysis of I GEA2 is the Bohr atoms, 27,28,38 which have densities constructed by occupying hydrogenic orbitals…”

“…18 The purpose of this work is to derive the missing GEA2 for the strong-coupling functionals of the MP AC, in order to reduce empiricism and hopefully increase the accuracy of the interpolations along the MP AC. To this purpose, we use the ideas derived from the semiclassical limit of neutral atoms, which have been used in recent years in DFT for the analysis of the exchange and correlation functionals, [23][24][25][26][27][28] yielding new approximations such as PBEsol. 29 As we shall see, the functionals we need to approximate allow us to probe more extensively these ideas, revealing several interesting features that could be used more generally to build DFT approximations.…”