In the previous chapter we saw that except for the simplest solids, like those formed by noble elements or by purely ionic combinations which can be described essentially in classical terms, in all other cases we need to consider the behavior of the valence electrons. The following chapters deal with these valence electrons (we will also refer to them as simply “the electrons” in the solid); we will study how their behavior is influenced by, and in turn influences, the ions.

Our goal in this chapter is to establish the basis for the single-particle description of the valence electrons. We will do this by starting with the exact hamiltonian for the solid and introducing approximations in its solution, which lead to sets of single-particle equations for the electronic degrees of freedom in the external potential created by the presence of the ions. Each electron also experiences the presence of other electrons through an effective potential in the single-particle equations; this effective potential encapsulates the many-body nature of the true system in an approximate way. In the last section of this chapter we will provide a formal way for eliminating the core electrons from the picture, while keeping the important effect they have on valence electrons.