Exchange energy density and some approximate exchange potentials obtained from Hartree–Fock theory of the ground state of the Be atom

Hartree–Fock (HF) theory of the ground state of the Be atom is used to calculate first the exchange energy density ϵx(r) from the Dirac density matrix. Beyond r=2a0, with a0=ℏ2/me2, ϵx(r) rapidly approaches the general asymptotic form −12e2ρ(r)/r, with ρ(r) the HF electronic density. The nuclear cusp condition1ϵx(r)∂ϵx∂rr→0=−2Za0with atomic number Z=4, is also accurately satisfied by the present numerical data. Since a quantum Monte Carlo (QMC) exchange-correlation potential exists for the Be atom, we have compared this with (a) the Slater potential VSL(r)=2ϵx(r)/ρ(r) and (b) the Harbola–Sahni form. Both have the main features of the QMC exchange-correlation potential, though the magnitude of VSL(r) at r=0 is too large by some 16%. We have also studied how well these two approximate HF exchange potentials fare when inserted into the Levy–Perdew relation between the total exchange energy and the `virial-likeʹ form involving the gradient of the exchange potential.