Analytical Asymptotic Structure of the Slater Exchange and Kohn–Sham Exchange and Correlation Potentials at a Metal Surface

Abstract

In this paper we consider the semi-infinite jellium and structureless pseudopotential models of a metal surface and derive the exact analytical asymptotic structure in the classically forbidden region of (i) the Slater exchange potentialVsx(r), (ii) the Kohn–Sham exchange potentialνx(r), and thus of the optimized potential method exchange potentialνOPMx(r), and show these latter potentials to decay as 12Vsx(r). With the assumption that the Kohn–Sham exchange-correlation potentialνxc(r) decays asymptotically as the image potential, we thereby derive (iii) the analytical asymptotic structure of the Kohn–Sham correlation potentialνc(r). These potentials all decay as −x−1with coefficients which depend upon the Fermi energy and surface barrier height. However, it is onlyνx(r) that is image-potential-like, the coefficient being 14for stable jellium. Thus, with exchange and correlation effects considered separately, the principal contribution to the asymptotic image potential structure arises due to Pauli correlations, the Coulomb correlation contribution being weak. We also show (iv) analytically that for a slab-metal geometry, the potentialνx(r) decays as −x−2and explain why this is the case. Finally, we show (v) analytically that two approximate Kohn–Sham exchange potentials in the literature possess the correct 12Vsx(r) asymptotic structure in the vacuum. With exchange and correlation considered together, the assumption that the asymptotic image potential structure ofνxc(r) is independent of metal parameters provides an alternate “classical” interpretation of the physical origin of this structure. It is due to those Coulomb correlations which contribute to that part of the Coulomb hole (of charge −e) localized to the surface region, the component (of charge +e) delocalized in the metal bulk screening out the corresponding delocalized Fermi hole distribution (of charge −e).