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

Stretched or noded orbital densities and self-interaction correction in density functional theory

Semilocal approximations to the density functional for the exchange-correlation energy of a many-electron system necessarily fail for lobed one-electron densities, including not only the familiar stretched densities but also the less familiar but closely related noded ones. The Perdew-Zunger (PZ) self-interaction correction (SIC) to a semilocal approximation makes that approximation exact for all one-electron ground- or excited-state densities and accurate for stretched bonds. When the minimization of the PZ total energy is made over real localized orbitals, the orbital densities can be noded, leading to energy errors in many-electron systems. Minimization over complex localized orbitals yields nodeless orbital densities, which reduce but typically do not eliminate the SIC errors of atomization energies. Other errors of PZ SIC remain, attributable to the loss of the exact constraints and appropriate norms that the semilocal approximations satisfy, suggesting the need for a generalized SIC. These conclusions are supported by calculations for one-electron densities and for many-electron molecules. While PZ SIC raises and improves the energy barriers of standard generalized gradient approximations (GGAs) and meta-GGAs, it reduces and often worsens the atomization energies of molecules. Thus, PZ SIC raises the energy more as the nodality of the valence localized orbitals increases from atoms to molecules to transition states. PZ SIC is applied here, in particular, to the strongly constrained and appropriately normed (SCAN) meta-GGA, for which the correlation part is already self-interaction-free. This property makes SCAN a natural first candidate for a generalized SIC. Published under license by AIP Publishing.Peer reviewe

Predicting cognitive dysfunction and regional hubs using Braak staging amyloid-beta biomarkers and machine learning

Abstract Mild cognitive impairment (MCI) is a transitional stage between normal aging and early Alzheimer’s disease (AD). The presence of extracellular amyloid-beta (Aβ) in Braak regions suggests a connection with cognitive dysfunction in MCI/AD. Investigating the multivariate predictive relationships between regional Aβ biomarkers and cognitive function can aid in the early detection and prevention of AD. We introduced machine learning approaches to estimate cognitive dysfunction from regional Aβ biomarkers and identify the Aβ-related dominant brain regions involved with cognitive impairment. We employed Aβ biomarkers and cognitive measurements from the same individuals to train support vector regression (SVR) and artificial neural network (ANN) models and predict cognitive performance solely based on Aβ biomarkers on the test set. To identify Aβ-related dominant brain regions involved in cognitive prediction, we built the local interpretable model-agnostic explanations (LIME) model. We found elevated Aβ in MCI compared to controls and a stronger correlation between Aβ and cognition, particularly in Braak stages III–IV and V–VII (p < 0.05) biomarkers. Both SVR and ANN, especially ANN, showed strong predictive relationships between regional Aβ biomarkers and cognitive impairment (p < 0.05). LIME integrated with ANN showed that the parahippocampal gyrus, inferior temporal gyrus, and hippocampus were the most decisive Braak regions for predicting cognitive decline. Consistent with previous findings, this new approach suggests relationships between Aβ biomarkers and cognitive impairment. The proposed analytical framework can estimate cognitive impairment from Braak staging Aβ biomarkers and delineate the dominant brain regions collectively involved in AD pathophysiology

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