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

Abstract

The position-dependent exact-exchange energy per particle ${\epsilon }_x(z)$ (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 ${\epsilon }_x^{\text{Slab}}(z\to \infty )\to -e^2/2z$, {\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 ${\epsilon }_x^{\text{Slab}}(z)$ 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 ${\epsilon }_x^{\text{SI}}(z)$ is somehow different. As in the case of jellium slabs ${\epsilon }_x^{\text{SI}}(z\to \infty )$ 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 ${\epsilon }_x^{\text{Slab}}(z)$ and ${\epsilon }_x^{\text{SI}}(z)$ 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.

26 pages, 7 figures, to appear in Phys. Rev. B

References in corpus (4)

• Collapse of the Electron Gas to Two Dimensions in Density Functional Theory
• Kohn-Sham Exchange Potential for a Metallic Surface
• Dimensional crossover of the exchange-correlation energy at the semilocal level
• Exact-exchange Kohn-Sham potential, surface energy, and work function of jellium slabs

Cited by in corpus (8)

• Semilocal density functional theory with correct surface asymptotics
• Adiabatic-Connection-Fluctuation-Dissipation approach to the long-range behavior of the exchange-correlation energy at metal surfaces: A numerical study for jellium slabs
• Global hybrids from the semiclassical atom theory satisfying the local density linear response
• Localized versus extended systems in density-functional theory: some lessons from the Kohn-Sham exact exchange potential
• Improving the applicability of the Pauli kinetic energy density based semilocal functional for solids
• Asymptotic behavior of the Kohn-Sham exchange potential at a metal surface
• Asymptotics of the metal-surface Kohn-Sham exact exchange potential revisited
• Semilocal approximations to the Kohn-Sham exchange potential as applied to a metal surface