  The large-$Z$ asymptotic expansion of atomic energies has been useful in
determining exact conditions for corrections to the local density approximation
in density functional theory. The correction for exchange is fit well with a
leading $Z \ln Z$ term, and we find its coefficient numerically. The gradient
expansion approximation also has such a term, but with a smaller coefficient.
Analytic results in the limit of vanishing interaction with hydrogenic orbitals
(a Bohr atom) lead to the conjecture that the coefficients are precisely 2.7
times larger than their gradient expansion counterparts, yielding an analytic
expression for the exchange-energy correction which is accurate to $\sim 5\%$
for all $Z$.
