Analysis of the Large Z Exchange Expansion of Neutral Atoms Beyond the Leading Order

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. The exact large Z expansion for the exchange energy has never been explicitly calculated beyond the leading order Z^5/3 term. It is typically considered that the next term is of order Z, however in general terms of order Z^4/3 and ZlogZ are possible. We test these considerations via calculations using the Optimized Effective Potential (OEP) method for atoms up to Z=120, and the LDA and popular GGAs up to Z=978. This data shows that the LDA exchange analyzed down columns of the periodic table has a leading order of Z^4/3. From analysis of the beyond-LDA contributions to the OEP exchange energy we conclude that there is no additional Z^4/3 contribution from exact exchange, but rather, that there is a distinct dependence on ZlogZ. Current GGAs repoduce the limiting trend in Z qualitatively but not quantitatively. We do find reasonably good agreement with theoretical estimates based on analyzing the inner shells of very large Z atoms.