In this chapter, we begin our study of crystalline solids by concentrating on multielectron systems. The first topic we consider is called the Born–Oppenheimer or adiabatic approximation. This approximation enables us to treat the electronic and the ionic components of a solid as two separate, distinct subsystems coupled only through the electron–lattice scatterings. The next topic is that of multielectron systems. The nature of the exchange interaction is examined, and the energies or multielectron systems are studied. Finally, we examine crystalline symmetries and how these properties facilitate our understanding of crystalline solids.

The Born-Oppenheimer Approximation

The basic assumption implicit to the theory of crystalline solids is called the Born–Oppenheimer or the adiabatic approximation. Simply put, the Born–Oppenheimer approximation assumes that a solid can be treated as being composed of two separate subsystems, the electronic and the lattice systems. The elections are considered as moving in a stationary lattice, and the lattice system is treated as being embedded within a uniform electron gas. The dynamics of each subsystem can then be treated independently of the dynamics of the other system, which holds to zeroth order. Coupling between the two subsystems occurs through electron–lattice scatterings. Therefore the Born–Oppenheimer approximation enables us to treat a solid symbolically in the same way as the two separate subsystems shown in Figure 5.3.1 when energy exchange is allowed between them.

The use of the adiabatic approximation is justified in the following way. The masses of the electrons and the ions differ by several orders of magnitude.