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

The position-dependent exact-exchange energy per particle ε_x(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 ε_x^Slab(z→∞)→−e²/2z, independent of the bulk electron density, which is exactly half the corresponding exact-exchange potential V_x(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 ε_x^Slab(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 ε_x^SI(z) is somehow different. As in the case of jellium slabs ε_x^SI(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 ε_x^Slab(z) and ε_x^SI(z) only coincide in the low-density limit (r_s→∞), where the density-dependent coefficient of the semi-infinite jellium approaches the slab-universal coefficient 1/2.