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 potential V s x (r), (ii) the Kohn Sham exchange potential & x (r), and thus of the optimized potential method exchange potential & OPM x (r), and show these latter potentials to decay as 1 2 V s x (r). With the assumption that the Kohn Sham exchange-correlation potential & xc (r) decays asymptotically as the image potential, we thereby derive (iii) the analytical asymptotic structure of the Kohn Sham correlation potential & c (r). These potentials all decay as &x &1 with coefficients which depend upon the Fermi energy and surface barrier height. However, it is only & x (r) that is image-potential-like, the coefficient being 1 4 for 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 potential & x (r) decays as &x &2 and explain why this is the case. Finally, we show (v) analytically that two approximate Kohn Sham exchange potentials in the literature possess the correct 1 2 V s x (r) asymptotic structure in the vacuum. With exchange and correlation considered together, the assumption that the asymptotic image potential structure of & xc (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).
1997 Academic Press
I. INTRODUCTION
According to the theorems of Hohenberg and Kohn [1], the ground-state wavefunction 9 of a system of electrons in some local external potential &(r) is a functional of the electronic density (r). Thus, the ground-state energy and the expectations of all operators representing physical observables are unique functionals of the density. Furthermore, the energy is a minimum for the true groundstate density. In the Kohn Sham [2] version of density-functional theory [3], electron correlations due to the Pauli exclusion principle and Coulomb's law, as article no. PH975705 97 0003-4916Â97 25.00