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 integrated 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.

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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 Levy and Ouyang 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 the semi-classical description of Pauli statistics and new constraints to aid the improvement of orbital-free density functional theory functionals.

The large-Z asymptotic expansion of atomic exchange energies has been useful in determining exact conditions for corrections to the local density approximation in density functional theory. We find that the necessary correction is fit well with a leading ZlnZ term, and find its coefficient numerically. The gradient expansion approximation also displays such a term, but with a substantially smaller coefficient. Analytic results in the limit of vanishing interaction with hydrogenic orbitals (a Bohr atom) are given, leading to the conjecture that the true coefficients for all atoms are precisely 2.7 times larger than their gradient expansion counterpart. Combined with the hydrogen atom result, this yields an analytic expression for the exchange-energy correction which is accurate to ∼ 5% for all Z.