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

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 ($-0.221 Z^{5/3}$ and $-0.021 Z \ln Z$ respectively) even in the lowest-rung 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 semi-quantitatively correct even for small $N$ and for $N \approx Z$. 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 $Z\to \infty$ limit for any fixed $N$.Comment: 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 Kohn-Sham perturbation series for Ec. Added material discussing the Becke 1988 functional. More discussion of non-empirical functionals' recovery of the asymptotic series, and their accuracy in predicting atomic/molecular energie

How Good is the Density-Corrected SCAN Functional for Neutral and Ionic Aqueous Systems, and What is so Right about the Hartree-Fock Density?

Density functional theory (DFT) is the most widely used electronic structure method, due to its simplicity and cost effectiveness. The accuracy of a DFT calculation de- pends not only on the choice of the density functional approximation (DFA) adopted but also on the electron density produced by the DFA. SCAN is a modern functional that satisfies all known constraints for meta-GGA functionals. The density-driven errors, defined as energy errors arising from errors of the self-consistent DFA electron density, can hinder SCAN from achieving chemical accuracy in some systems, including water. Density-corrected DFT (DC-DFT) can alleviate this shortcoming by adopting a more accurate electron density which, in most applications, is the electron density obtained at the Hartree-Fock level of theory due to its relatively low computational cost. In this work, we present extensive calculations aimed at determining the accuracy of the DC-SCAN functional for various aqueous systems. DC-SCAN (SCAN@HF) shows remarkable consistency in reproducing reference data obtained at the coupled cluster level of theory, with minimal loss of accuracy. Density-driven errors in the description of ionic aqueous clusters are thoroughly investigated. By comparison with the orbital-optimized CCSD density in the water dimer, we find that the self-consistent SCAN density transfers a spurious fraction of an electron across the hydrogen bond to the hydrogen atom (H⋆, covalently bound to the donor oxygen atom) from the acceptor (OA) and donor (OD) oxygen atoms, while HF makes a much smaller spurious transfer in the opposite direction, consistent with DC-SCAN (SCAN@HF) reduction of SCAN over-binding due to delocalization error. While LDA seems to be the conventional extreme of density delocalization error, and HF the conventional extreme of (usually much smaller) density localization error, these two densities do not quite yield the conventional range of density-driven error in energy differences. Finally, comparisons of the DC-SCAN results with those obtained with the Fermi-Lowdin orbital self-interaction correction (FLOSIC) method show that DC-SCAN represents a more accurate approach to reducing density-driven errors in SCAN calculations of ionic aqueous clusters

How does HF-DFT achieve chemical accuracy for water clusters?

Bolstered by recent calculations of exact functional-driven errors (FEs) and density-driven errors (DEs) of semi-local density functionals in the water dimer binding energy [Kanungo et al., J. Phys. Chem. Lett. 2023, 15, 323], we investigate approximate FEs and DEs in neutral water clusters containing up to 20 monomers, charged water clusters, and alkali- and halide-water clusters. Our proxy for the exact density is r2SCAN50, a 50% global hybrid of exact exchange with r2SCAN, which may be less correct than r2SCAN for the compact water monomer but importantly more correct for long-range electron transfers in the non-compact water clusters. We show that SCAN makes substantially larger FEs for neutral water clusters than r2SCAN, while both make essentially the same DEs. Unlike the case for barrier heights, these FEs are small in a relative sense, and become large in an absolute sense only due to an increase in cluster size. SCAN@HF produces a cancellation of errors that makes it chemically accurate for predicting the absolute binding energies of water clusters. Likewise, adding a long-range dispersion correction to r2SCAN@HF, as in the composite method HF-r2SCAN-DC4, makes its FE more negative than in r2SCAN@HF, permitting a near-perfect cancellation of FE and DE. r2SCAN by itself (and even more so, r2SCAN evaluated on the r2SCAN50 density), is almost perfect for the energy differences between water hexamers, and thus probably also for liquid water away from the boiling point. Thus the accuracy of composite methods like SCAN@HF and HF-r2SCAN-DC4 is not due to the HF density being closer to the exact density, but to a compensation of errors from its greater degree of localization. We also give an argument for the approximate reliability of this unconventional error cancellation for diverse molecular properties. Finally, we confirm this unconventional error cancellation for the SCAN description of the water trimer via Kohn-Sham inversion of the CCSD(T) density

Understanding Density-Driven Errors for Reaction Barrier Heights

Delocalization errors, such as charge-transfer and some self-interaction errors, plague computationally efficient and otherwise accurate density functional approximations (DFAs). Evaluating a semilocal DFA non-self-consistently on the Hartree–Fock (HF) density is often recommended as a computationally inexpensive remedy for delocalization errors. For sophisticated meta-GGAs like SCAN, this approach can achieve remarkable accuracy. This HF-DFT (also known as DFA@HF) is often presumed to work, when it significantly improves over the DFA, because the HF density is more accurate than the self-consistent DFA density in those cases. By applying the metrics of density-corrected density functional theory (DFT), we show that HF-DFT works for barrier heights by making a localizing charge-transfer error or density overcorrection, thereby producing a somewhat reliable cancellation of density- and functional-driven errors for the energy. A quantitative analysis of the charge-transfer errors in a few randomly selected transition states confirms this trend. We do not have the exact functional and electron densities that would be needed to evaluate the exact density- and functional-driven errors for the large BH76 database of barrier heights. Instead, we have identified and employed three fully nonlocal proxy functionals (SCAN 50% global hybrid, range-separated hybrid LC-ωPBE, and SCAN-FLOSIC) and their self-consistent proxy densities. These functionals are chosen because they yield reasonably accurate self-consistent barrier heights and because their self-consistent total energies are nearly piecewise linear in fractional electron numbertwo important points of similarity to the exact functional. We argue that density-driven errors of the energy in a self-consistent density functional calculation are second order in the density error and that large density-driven errors arise primarily from incorrect electron transfers over length scales larger than the diameter of an atom

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

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