Asymptotic formula far from nucleus for exchange energy density in Hartree-Fock theory of closed-shell atoms

In Hartree-Fock theory, the exchange energy density can be expressed solely in terms of the first-order density matrix. Far from the nucleus of a closed-shell atom, idempotency of the density matrix yields the exchange energy density as the magnitude of the Coulomb energy $e^2$/r times the electron density ρ. Thus two lengths enter the asymptotic form in contrast to ${\rho }^{\mathrm{-}1/3}$ alone of local-density theory.