Abstract
The position-dependent exact-exchange energy per particle $\varepsilon_x(z)$ (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 $\varepsilon_{x}^{\text{Slab}}(z \to \infty) \to - e^{2}/2z$, {\it independent} of the bulk electron density, which is exactly half the corresponding exact-exchange potential $V_{x}(z \to \infty) \to - e^2/z$ [Phys. Rev. Lett. {\bf 97}, 026802 (2006)] of density-functional theory, as occurs in the case of finite systems. The fitting of $\varepsilon_{x}^{\text{Slab}}(z)$ to a physically motivated image-like 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 $\varepsilon_{x}^{\text{SI}}(z)$ is somehow different. As in the case of jellium slabs $\varepsilon_{x}^{\text{SI}}(z \to \infty)$ has an image-like behavior of the form $\propto - e^2/z$, 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 $\varepsilon_{x}^{\text{Slab}}(z)$ and $\varepsilon_{x}^{\text{SI}}(z)$ only coincide in the low-density limit ($r_s \to \infty$), where the density-dependent coefficient of the semi-infinite jellium approaches the slab {\it universal} coefficient 1/2.
Highlights
The jellium model of a metal surface, introduced by Bardeen in 1936,1 is the simplest model which reproduces qualitatively, and sometimes quantitatively, the physical properties of real-metal surfaces.[2]
For an arbitrary finite thickness of a jellium slab, we find that the asymptotic limits of xSlabzand xSIzonly coincide in the low-density limitrs → ρ, where the density-dependent coefficient of the semi-infinite jellium approaches the slab-universal coefficient 1/2
We analyze the asymptotic behavior of the exact xrin the vacuum region far away from the surface, and we find that there is a qualitative difference between xSlabz → ρand xSIz → ρ: both exhibit an imagelike behavior of the form −ae2 / za Ͼ 0͒ but with a coefficient a that while in the case of jellium slabs is universal and equal to 1/2 in the case of a semi-infinite jellium depends on the density of the bulk material and only approaches 1/2 in the low-density limit
Summary
The jellium model of a metal surface, introduced by Bardeen in 1936,1 is the simplest model which reproduces qualitatively, and sometimes quantitatively, the physical properties of real-metal surfaces.[2] While in his work Bardeen applied an approximated Hartree-FockHFtheory for the study of the electronic structure since the seminal work of Lang and Kohn[3] the standard theoretical tool applied to the study of the electronic structure of metal surfaces has been density-functional theoryDFT.[4] As in the original work of Lang and Kohn, most of the subsequent investigations have applied the local-density approximationLDAof DFT or some of its semilocal variantsgeneralized gradient approximationGGA, meta-GGA, etc.͔. This qualitative failure of the LDA xc potential translates to a similar failure of the position-dependent xc energy per particle, xcr, which is defined through[7]
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