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.
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