Visualizing the large-$Z$ scaling of the kinetic energy density of atoms

TL;DR

This paper visualizes how the kinetic energy density of atoms scales with atomic number, revealing asymptotic behaviors and proposing orbital-free models to improve density functional theory calculations.

Contribution

It provides new insights into the scaling of kinetic energy density in atoms and introduces orbital-free meta-GGA models to better describe these features.

Findings

KED within atomic cores fits a gradient expansion involving gradient and Laplacian.

Gradient expansion differs from that of a slowly-varying electron gas.

Quantum oscillations in inner shells complicate complete parametrization.

Abstract

The scaling of neutral atoms to large $Z$, combining periodicity with a gradual trend to homogeneity, is a fundamental probe of density functional theory, one that has driven recent advances in understanding both the kinetic and exchange-correlation energies. Although research focus is normally upon the scaling of energies, insights can also be gained from energy densities. We visualize the scaling of the positive-definite kinetic energy density (KED) in closed-shell atoms, in comparison to invariant quantities based upon the gradient and Laplacian of the density. We notice a striking fit of the KED within the core of any atom to a gradient expansion using both the gradient and the Laplacian, appearing as an asymptotic limit around which the KED oscillates. The gradient expansion is qualitatively different from that derived from first principles for a slowly-varying electron gas and is correlated with a nonzero Pauli contribution to the KED near the nucleus. We propose and explore orbital-free meta-GGA models for the kinetic energy to describe these features, with some success, but the effects of quantum oscillations in the inner shells of atoms makes a complete parametrization difficult. We discuss implications for improved orbital-free description of molecular properties.