5 Density Functional Theories and Self-energy Approaches Rex W. Godby∗ and Pablo Garc´ıa-Gonz´alez† ∗ Department of Physics, University of York, Heslington, York YO105DD, United Kingdom
[email protected] † Departamento de F´ısica Fundamental, Universidad Nacional de Educaci´on a Distancia, Apto. 60141, 28080 Madrid, Spain
[email protected] Rex Godby 5.1 Introduction One of the fundamental problems in condensed-matter physics and quantum chemistry is the theoretical study of electronic properties. This is essential to understand the behaviour of systems ranging from atoms, molecules, and nanostructures to complex materials. Since electrons are governed by the laws of quantum mechanics, the many-electron problem is, in principle, fully described by a Schr¨odinger equation (supposing the nuclei to be fixed). How- ever, the electrostatic repulsion between the electrons makes its numerical resolution an impossible task in practice, even for a relatively small number of particles. Fortunately, we seldom need the full solution of the Schr¨odinger equation. When one is interested in structural properties, the ground-state total energy of the system is sufficient. In other cases, we want to study how the system responds to some external probe, and then knowledge of a few excited-state properties must be added. For instance, in a direct photoemission experiment a photon impinges on the system and an electron is removed. In an inverse photoemission process, an electron is absorbed and a photon is ejected. In both cases we have to deal with the gain or loss of energy of the N electron system when a single electron is added or removed, i.e.