Hartree-Fock theory of the inhomogeneous electron gas at metallic surfaces

Abstract

In this paper we derive fundamental properties of the interacting inhomogeneous electron gas at jellium metal surfaces within the Hartree-Fock approximation. We also discuss the question of what constitutes quantum-mechanically the image charge at a metal surface, and provide a physical interpretation for the quantum-mechanical origin of the image potential. The self-consistent solution of the Hartree-Fock equations is in general formidable due to the integral exchange operator. We show that from a more fundamental physical viewpoint, the difficulty for the metal surface physics problem manifests itself in the complex structure of the orbital-dependent exchange charge densities that give rise to the orbital-dependent potentials. However, it is possible to solve the problem of the Pauli-correlated electron gas within the exchange-only formalism of density-functional theory. By employing the variational principle for the energy with the exchange energy component treated in its non-local form, and the variationally accurate “displaced-profile-change-in-self-consistent-field” expression for the work function, we derive rigorous upper bounds to the surface energy and accurate work functions. In order to understand what constitutes quantum-mechanically the image charge at a metal surface we study the structure of the average exchange charge density or Fermi hole as an electron is removed from within a metal to infinity outside. The study shows that the Fermi hole is localized to the surface region and is part of the image charge only for electron positions close to the surface. As the electron is removed further into the vacuum region, the width of the hole increases. In the asymptotic limit when the electron is removed to infinity, the hole is completely delocalized and spread throughout the crystal, its center of mass being singular. As a consequence it appears that it is the Coulomb hole charge distribution localized at the surface that is the image charge, but this has yet to be shown. Finally, we provide insights into the quantum-mechanical origin of the image potential. These ideas are based on our interpretation of the local exchange-correlation potential of density-functional theory as being the work done to remove an electron against the electric field of its Fermi-Coulomb hole charge density. Since the Coulomb hole charge is zero, the image potential in the asymptotic region far from the surface is the work done against the electric field of its Fermi hole. The results of preliminary calculations confirm this conclusion. Whether the Coulomb hole contributes to making the total effective potential the image potential for electron positions closer to the surface is yet unanswered. The above physical interpretation is based on the fact that the Fermi-Coulomb hole charge distribution is dynamic as a function of electron position. This explains why the exchange potential as calculated by the Slater method is incorrect. As such we also present the complete structure of the Slater potential at a metal surface and show that it leads to an erroneous value in the interior of the metal and that its asymptotic structure though image-potential-like has a coefficient approximately twice as large as that of the image potential.