Analysis of Smooth and Oscillatory terms in the Large Z Exchange Expansion of Atoms

Lieb-Simon zeta-scaling, which describes the scaling of neutral closed-shell atoms as Z approaches infinity, has been used with marked success to provide theoretical constraints on density functionals and to help generate perturbative expansions with Z. It has long been shown explicitly that the LDA recovers the dominant term, of order Z^5/3, for the atomic exchange energy, but little else is known formally. To remedy this lack, we analyze OEP calculations for exchange for neutral atoms up to Z=978. Prior work shows consistent numerical and analytical evidence for an anomalous ZlnZ term as the leading order correction to LDA. [1] We report here an analysis of the LDA contribution which is complicated by a complex dependence on shell-structure. We find that it is characterized by a smooth Z^4/3 term and an oscillatory pattern with a two-row period. We propose an accurate model of the alkali earth column and qualitative model for all closed shell atoms as functions of the partial occupation of the two-row period. These results yield a stringent test for orbital free density functionals, and may also be applicable to characterize the oscillations that occur in the post-LDA data and thus be of help testing and developing functionals.

[1] Nathan Argaman, Jeremy Redd, Antonio C. Cancio, and Kieron Burke Phys. Rev. Lett. 129, 153001 (2022)