  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.
