Study of the exchange energy of an inhomogeneous electron gas at a surface

In this work we have studied the exchange energy property of an inhomogeneous electron gas at a jellium surface. The nonlocal surface exchange energy is determined exactly for the accurate set of single-particle wave functions generated within the linear-potential approximation to the effective potential at a surface, and written in terms of a universal function of the field strength. It is shown, that over the wide range of density profiles considered, this universal function increases by a factor of 6 as the profile is changed from rapidly to slowly varying. For a Hartree-Fock gas with bulk density such that 5>=r_s>=1, the corresponding increase in the surface exchange energy is three orders of magnitude. The local density approximation (LDA) functional of the surface exchange energy is also determined for the same set of wave functions, and for the first time its convergence to the exact results as the density profile is made slowly varying demonstrated. It is shown that this LDA functional is a poor approximation for metallic densities, overstimating the exact results by 17-55%. The results of a previous calculation within the step-model approximation are also reinterpreted, and similar conclusions with regard to the inaccuracy of the LDA functional arrived at. The applicability of the nonlocal calculation of this work to the evaluation of other approximate nonlocal exchange energy functionals, and to the study of the inhomogeneous Hartree-Fock and fully correlated systems is also indicated.