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 e²/r times the electron density {ρ}. Thus two lengths enter the asymptotic form in contrast to ρ^-1/3 alone of local-density theory.