Introduction

Information regarding the electronic structure of a condensed matter system provides the basis for understanding the myriad of its physical properties: optical, mechanical, magnetic, electrical, etc. There are two main hurdles, however, that seriously hamper any attempt to derive the electronic states of condensed matter systems: The first arises from the gigantic difference in the time scales associated with the motions of electrons and nuclei, or ions, which can be in the order of 103−105. The second difficulty concerns the numbers of particles involved, which are at least of the order of Avogadro's number of ∼1024. In order to overcome the first hurdle, we invoke the adiabatic, or Born–Oppenheimer, approximation, which we will discuss in the following chapter. The only impact of this approximation here is that we treat the ions classically and fix all the ionic positions, {R}; we introduce their interactions with the electron system as an external potential, V (r, {R}). For simplicity, we drop {R} from the notation in the remainder of the chapter.

The one-electron approximations and self-consistent-field theories

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The many-body problem

Our objective here is to determine the ground-state properties of an interacting manyelectron system subject to an external potential V (r), representing the interaction with the frozen ions.