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

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

Figure 1

${\epsilon }_x^{\text{Slab}}\left(z\right)$ for $r_s=2.07$ and slab width $d=4.3{\lambda }_F$. The curves-denoted “slab” have been evaluated from Eq. (12), by using exchange-only LDA orbitals (solid line) and exact-exchange (OEP) orbitals (dotted line). The curves denoted “LDA” have been evaluated from the well-known LDA formula ${\epsilon }_{x,\text{LDA}}\left(z\right)=-\left(3/4\pi \right){\left[3{\pi }^{2}n\left(z\right)\right]}^{1/3}$ (hartrees) with $n\left(z\right)$ being the exchange-only self-consistent electron density obtained with the use of exchange-only LDA orbitals (wide solid line) and exact-exchange (OEP) orbitals (wide dotted line). Inset: enlarged view of the bulk region near the surface. The bulk value of ${\epsilon }_x$ for this electron density is $-0.22134\left(e^2/a_0\right)$.

Figure 2

Position-dependent exchange energy per particle for a thin jellium slab with $r_s=2.07$ and $d=0.3{\lambda }_F\approx 2a_0$. Solid line: full numerical calculation of Eq. (12). Dashed-dotted lines: asymptote of Eq. (18); the dashed (dotted) lines represent the asymptote of Eq. (18) with the last term (last two terms) neglected. Inset: enlarged view of the asymptotic region.

Figure 3

Same as Fig. 2 but for a slab with $d=4{\lambda }_F$, with nine SDL occupied. Full thick line, ${\epsilon }_x^{\text{Slab}}\left(z\right)$ from Eq. (12); dotted, dashed, and dashed-dotted lines, asymptotic expansions from Eq. (22). The contribution from each pair $\left(i,j\right)$ of occupied SDL to the total ${\epsilon }_x^{\text{Slab}}\left(z\right)$ are represented with thin full lines, except for the last contribution $\left(i=j=9\right)$. Inset: enlarged view of the asymptotic region.

Figure 4

Parameter $z_x^{\text{Slab}}\left(d,r_s=2.07,z\right)$ from Eq. (38), as a function of $d$, in several approximations. Full line, $z_x^{\text{Slab}}\left(d,r_s=2.07,z\to \infty \right)+d/2=-2/\pi k_F^m\left(d,r_s=2.07\right)$; squares, Eq. (40); circles, fit to Eq. (38). Upper panel, $k_F^m\left(d\right)$ as a function of $d$.

Figure 5

${\epsilon }_x^{\text{Slab}}\left(z\right)$ for $r_s=2.07$, slabs with $d=4$, 6, 8, and 12 ${\lambda }_F$. Dotted line, ${\epsilon }_x^{\text{SI}}\left(z\right)$ for the semi-infinite case, from Eq. (32). Upper inset: enlarged view of the bulk region. Lower inset: asymptotic region.

Figure 6

Asymptotic behavior of ${\epsilon }_x^{\text{SI}}\left(z\right)$ for the semi-infinite case, and comparison with Eq. (36) and the asymptote form of Eq. (22).

Figure 7

$\alpha \left(r_s\right)=\sqrt{{\epsilon }_F\left(r_s\right)/W\left(r_s\right)}$ versus $r_s$, for the semi-infinite case. Inset, the coefficient $-\left[\pi +2\alpha \text{ln}\left(\alpha \right)\right]/\left[2\pi \left(1+{\alpha }^2\right)\right]$ of Eq. (36) versus $r_s$.