ABSTRACT
We know that all materials are made of atoms that are in turn made of electrons and nuclei. In other words, a material is made of electrons and nuclei that interact with each other in accordance with Coulomb’s law. Classical mechanics fails to describe the behavior of electrons; therefore, only quantum mechanics can give us a correct description of materials. Thus, to understand it, we have to solve a quantum mechanical many-body problem, that is, Schrödinger’s equation for interacting electrons and nuclei. Although not necessary, but for convenience and simplicity, we shall divide the electrons into valence and core electrons. For example, in sodium, the 3s electron, which is the outermost electron, will be called a valence electron and the remaining will be called core electrons. When a solid is formed, the valence electrons get unbound from atoms and can move in the solid. Since core electrons do not play much of a role in solid-state properties, we can combine core electrons with their nucleus and call the combined entity an ion. Thus, a condensed matter system can be described as a many-body system of interacting electrons and ions and we are interested in finding a solution of Schrödinger’s equation for this system. In Section 2.2, we shall discuss how to solve this problem approximately by separating the electronic problem from the ionic problem using the Born-Oppenheimer approximation. In the remaining sections, we focus on the electronic problem. In Section 2.3, we discuss the simplest solution using the Hartree approximation and in Section 2.4, we discuss the Hartree-Fock (H-F) approximation. In Section 2.5, we briefly discuss the configuration interaction (CI) method. In Sections 2.6 and 2.7, we discuss the application of the Hartree and H-F approximations to homogeneous electron gas (HEG).
