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 calc ulate first the exchange energy density e(x)(r) from the Dirac density matr ix. Beyond r = 2a(0), with a(0) = h(2)/me(2), epsilon (x)(r) rapidly approa ches the general asymptotic form -1/2e(2)rho (r)/r, with p(r) the HF electr onic density. The nuclear cusp condition 1/epsilon (x)(r) partial derivative epsilon (x)/partial derivativer /(r --> 0) = -2Z/a(0) with atomic number Z = 4, is also accurately satisfied by the present numer ical data. Since a quantum Monte Carlo (QMC) exchange-correlation potential exists for the Be atom, we have compared this with (a) the Slater potentia l V-SL(r) = 2 epsilon (x)(r)/rho (r) and (b) the Harbola-Sahni form. Both h ave the main features of the QMC exchange-correlation potential, though the magnitude of V-SL(r) at r = 0 is too large by some 16%. We have also studi ed how well these two approximate HF exchange potentials fare when inserted into the Levy-Perdew relation between the total exchange energy and the 'v irial-like' form involving the gradient of the exchange potential. (C) 2001 Elsevier Science B.V. All rights reserved.