Position-dependent exact-exchange energy for slabs and semi-infinite jellium

The position-dependent exact-exchange energy per particle εx(z) (defined as the interaction between a given electron at z and its exact-exchange hole) at metal surfaces is investigated, by using either jellium slabs or the semi-infinite (SI) jellium model. For jellium slabs, we prove analytically and numerically that in the vacuum region far away from the surface εSlabx(z→∞)→−e2/2z, independent of the bulk electron density, which is exactly half the corresponding exact-exchange potential Vx(z→∞)→−e2/z [Horowitz et al., Phys. Rev. Lett. 97, 026802 (2006)] of density-functional theory, as occurs in the case of finite systems. The fitting of εSlabx(z) to a physically motivated imagelike expression is feasible but the resulting location of the image plane shows strong finite-size oscillations every time a slab discrete energy level becomes occupied. For a semi-infinite jellium, the asymptotic behavior of εSIx(z) is somehow different. As in the case of jellium slabs εSIx(z→∞) has an imagelike behavior of the form ∝−e2/z but now with a density-dependent coefficient that, in general, differs from the slab-universal coefficient 1/2. Our numerical estimates for this coefficient agree with two previous analytical estimates for the same. For an arbitrary finite thickness of a jellium slab, we find that the asymptotic limits of εSlabx(z) and εSIx(z) only coincide in the low-density limit (rs→∞), where the density-dependent coefficient of the semi-infinite jellium approaches the slab-universal coefficient 1/2.C.M.H wishes to acknowledge the financial support received from CONICET of Argentina. C.R.P. was supported by the European Community through a Marie Curie IIF -Grant No. MIF1-CT-2006-040222 and CONICET of Argentina under Grant No. PIP 5254. J.M.P. acknowledges partial support by the University of the Basque Country, the Basque Unibertsitate eta Ikerketa Saila, and the Spanish Ministerio de Educación y Ciencia -Grants No. FIS2006- 01343 and No. CSD2006-53.Peer reviewe