1. Electrons in a crystal lattice

Although the number of electrons in a solid is very large, and we must therefore work with N-electron antisymmetric wave functions, we can reduce the problem to one electron. We study the motion of each electron in the static mean potential, created by the nuclei and the other electrons; we then construct an antisymmetric total wave function, as required by Pauli's principle, and the total wave function takes on the appearance of a so-called Slater determinant. We could use all these determinants to treat the problem completely, but this is practically impossible. So we're generally content to represent the system by a single determinant. Even in this approximation, it is unrealistic to proceed rigorously, choosing a mean test potential, making one-electron wave functions in this potential, deducing a new potential and proceeding in this way until a self-coherent potential has been obtained....