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

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

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 Ex(N, Z) ≈ $−0.354N^{2/3}Z$ (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.

Reference:

Simple hydrogenic estimates for the exchange and correlation energies of atoms and atomic ions, with implications for density functional theory Kaplan,Aaron D., Santra,Biswajit, Bhattarai,Puskar, Wagle,Kamal, Chowdhury,Shah Tanvir ur Rahman, Bhetwal,Pradeep, Yu,Jie, Tang,Hong, Burke,Kieron, Levy,Mel and Perdew,John P., The Journal of Chemical Physics 153, 074114 (2020).

Bibtex Entry:

@article{KSBW20, Pub-num = {200}, Title = {Simple hydrogenic estimates for the exchange and correlation energies of atoms and atomic ions, with implications for density functional theory}, Author = {Kaplan,Aaron D. and Santra,Biswajit and Bhattarai,Puskar and Wagle,Kamal and Chowdhury,Shah Tanvir ur Rahman and Bhetwal,Pradeep and Yu,Jie and Tang,Hong and Burke,Kieron and Levy,Mel and Perdew,John P.}, 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 Ex(N, Z) ≈ $−0.354N^{2/3}Z$ (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.}, Doi = {10.1063/5.0017805}, %% Issn = {}, Year = {2020}, Month = {August}, Journal = {The Journal of Chemical Physics}, Volume = {153}, %% Issue = {}, Number = {7}, Pages = {074114}, %% Publisher = {}, Url = {https://doi.org/10.1063/5.0017805}, arXiv = {arXiv:2007.01917}, %% keywords = {} }

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