Analyzing the Large Z Exchange Expansion

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. Because the electronic core of an atom as the nuclear charge approaches infinity relatively approaches a homogeneous electron gas (HEG), energy data for large Z atoms can be used to probe perturbative corrections to density functionals. The exact large Z expansion about a HEG for the exchange energy has never been explicitly calculated beyond the leading order Z^5/3 term. It is typically assumed that the next order term is of order Z. We calculate exact exchange energies using the Optimized Effective Potential (OEP) method for closed-shell atoms up to Z=120 and other standard DFT models up to 976. Regression analysis shows that LDA exchange has a second-order term of order Z^4/3, and support a similar behavior for GGA’s. Exchange may also have a small contribution of order ZlogZ. Due to the high agreement between OEP and Becke 1988 exchange, we suspect this will be true of exact exchange as well. This may lead to improved constraint information to develop more robust exchange functionals.