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Locality of correlation in density functional theory

The Hohenberg-Kohn density functional was long ago shown to reduce to the Thomas-Fermi approximation in the non-relativistic semiclassical (or large-$Z$) limit for all matter, i.e, the kinetic energy becomes local. Exchange also becomes local in this limit. Numerical data on the correlation energy of atoms supports the conjecture that this is also true for correlation, but much less relevant to atoms. We illustrate how expansions around large particle number are equivalent to local density approximations and their strong relevance to density functional approximations. Analyzing highly accurate atomic correlation energies, we show that the correlation energy tends to $-A_c Z ln Z + B_c Z$ as $Z$ tends to infinity, where $Z$ is the atomic number, $A_c$ is known, and we estimate $B_c$ to be about 37 millihartrees. The local density approximation yields $A_c$ exactly, but a very incorrect value for $B_c$, showing that the local approximation is less relevant for correlation alone. This limit is a benchmark for the non-empirical construction of density functional approximations. We conjecture that, beyond atoms, the leading correction to the local density approximation in the large-$Z$ limit generally takes this form, but with $B_c$ a functional of the TF density for the system. The implications for construction of approximate density functionals are discussed.