  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.
