  We derive a two-term asymptotic expansion for the exchange energy of the free
electron gas on strictly tessellating polytopes and fundamental domains of
lattices in the thermodynamic limit. This expansion comprises a bulk
(volume-dependent) term, the celebrated Dirac exchange, and a novel surface
correction stemming from a boundary layer and finite-size effects. Furthermore,
we derive analogous two-term asymptotic expansions for semi-local density
functionals. By matching the coefficients of these asymptotic expansions, we
obtain an integral constraint for semi-local approximations of the exchange
energy used in density functional theory.
