Asymptotic Analysis of the Pauli Potential for Atoms

TL;DR

This paper investigates the asymptotic behavior of the Pauli potential in atoms under Lieb-Simon scaling, revealing complex regional behaviors that could inform the development of improved orbital-free density functional theory models.

Contribution

It provides a detailed mathematical analysis of the Pauli potential's behavior in large atoms, highlighting non-smooth convergence and distinct regional behaviors near the nucleus and in the outer core.

Findings

Near the nucleus, the potential approaches a constant related to eigenvalue differences.

The potential exhibits a transition region in the outer core with deviations from classical models.

The results suggest new constraints for improving orbital-free DFT functionals.

Abstract

Modeling the Pauli energy, the contribution to the kinetic energy caused by Pauli statistics, without using orbitals is the open problem of orbital-free density functional theory. An important aspect of this problem is correctly reproducing the Pauli potential, the response of the Pauli kinetic energy to a change in density. We analyze the behavior of the Pauli potential of non-relativistic neutral atoms under Lieb-Simon scaling -- the process of taking nuclear charge and particle number to infinity, in which the kinetic energy tends to the Thomas-Fermi limit. We do this by mathematical analysis of the near-nuclear region and by calculating the exact orbital-dependent Pauli potential using the approach of Ouyang and Levy for closed-shell atoms out to element Z=976. In rough analogy to Lieb and Simon's own findings for the charge density, we find that the potential does not converge smoothly to the Thomas-Fermi limit on a point-by-point basis but separates into several distinct regions of behavior. Near the nucleus, the potential approaches a constant given by the difference in energy between the lowest and highest occupied eigenvalues. We discover a transition region in the outer core where the potential deviates unexpectedly and predictably from both the Thomas-Fermi potential and the gradient expansion correction to it. These results may provide insight into semi-classical description of Pauli statistics, and new constraints to aid the improvement of orbital-free DFT functionals.