Exact electronic properties in the classically forbidden region of a metal surface

We derive exact properties of the inhomogeneous electron gas in the asymptotic classically forbidden region at a metal-vacuum interface within the framework of local effective potential energy theory. We derive a new expression for the asymptotic structure of the Kohn-Sham density functional theory (KS-DFT) exchange-correlation potential energy v x c (r) in terms of the irreducible electron self-energy. We also derive the exact asymptotic structure of the orbitals, density, the Dirac density matrix, the kinetic energy density, and KS exchange energy density. We further obtain the exact expression for the Fermi hole and demonstrate its structure in this asymptotic limit. The exchange-correlation potential energy is derived to be V x c (z → ∞) = -α K S , x c /z, and its exchange and correlation components to be v x (z → ∞) = -a K S , x /z and v c (z → ∞) = -α K S , c /z, respectively. The analytical expressions for the coefficients α K S , x c and a K S , x show them to be dependent on the bulk-metal Wigner-Seitz radius and the barrier height at the surface. The coefficient a K S , c = 1/4 is determined in the plasmon-pole approximation and is independent of these metal parameters. Thus, the asymptotic structure of V x c (z) in the vacuum region is image-potential-like but not the commonly accepted one of -1/4z. Furthermore, this structure depends on the properties of the metal. Additionally, an analysis of these results via quantal density functional theory (Q-DFT) shows that both the Pauli W x (z → ∞) and lowest-order correlation-kinetic W ( 1 ) t c (z → ∞) components of the exchange potential energy v x (z → ∞), and the Coulomb W c (z → ∞) and higher-order correlation-kinetic components of the correlation potential energy v c (z → ∞), all contribute terms of O(1/z) to the structure. Hence correlations attributable to the Pauli exclusion principle, Coulomb repulsion, and correlation-kinetic effects all contribute to the asymptotic structure of the effective potential energy at a metal surface. The relevance of the results derived to the theory of image states and to KS-DFT is also discussed.