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

Abstract

The position-dependent exact-exchange energy per particle ${\ensuremath{\epsilon}}_{x}(z)$ (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 ${\ensuremath{\epsilon}}_{x}^{\text{Slab}}(z\ensuremath{\rightarrow}\ensuremath{\infty})\ensuremath{\rightarrow}\ensuremath{-}{e}^{2}/2z$, independent of the bulk electron density, which is exactly half the corresponding exact-exchange potential ${V}_{x}(z\ensuremath{\rightarrow}\ensuremath{\infty})\ensuremath{\rightarrow}\ensuremath{-}{e}^{2}/z$ [Horowitz et al., Phys. Rev. Lett. 97, 026802 (2006)] of density-functional theory, as occurs in the case of finite systems. The fitting of ${\ensuremath{\epsilon}}_{x}^{\text{Slab}}(z)$ 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 ${\ensuremath{\epsilon}}_{x}^{\text{SI}}(z)$ is somehow different. As in the case of jellium slabs ${\ensuremath{\epsilon}}_{x}^{\text{SI}}(z\ensuremath{\rightarrow}\ensuremath{\infty})$ has an imagelike behavior of the form $\ensuremath{\propto}\ensuremath{-}{e}^{2}/z$ 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 ${\ensuremath{\epsilon}}_{x}^{\text{Slab}}(z)$ and ${\ensuremath{\epsilon}}_{x}^{\text{SI}}(z)$ only coincide in the low-density limit $({r}_{s}\ensuremath{\rightarrow}\ensuremath{\infty})$, where the density-dependent coefficient of the semi-infinite jellium approaches the slab-universal coefficient 1/2.