2.2 Electrons

The electronic structure of a solid, characterized by its energy band structure, is the fundamental quantity that determines the ground state of the solid and a series of excitations involving electronic states. In the first part of this chapter, several basic concepts are summarized in order to establish the notation used and to repeat essential theorems from group theory and solid-state physics that provide the definitions that are needed in this context (Brillouin zones, symmetry operators, Bloch theorem, space-group symmetry). Next the quantum-mechanical treatment, especially density functional theory, is described and the commonly used methods of band theory are outlined (the linear combination of atomic orbitals, tight binding, pseudo-potential schemes, the augmented plane wave method, the linear augmented plane wave method, the Korringa–Kohn–Rostocker method, the linear combination of muffin-tin orbitals, the Car–Parinello method etc.). The linear augmented plane wave scheme is presented explicitly so that concepts in connection with energy bands can be explained. The electric field gradient is discussed to illustrate a tensorial quantity. In the last section, a few examples illustrate the topics of the chapter. Keywords: Bloch function; Bloch states; Bloch theorem; Bravais lattices; Brillouin zone; Car–Parrinello method; Korringa–Kohn–Rostocker method; Seitz operator; Sommerfeld model; atomic orbitals; augmented plane wave method; band structure; chemical bonding; core electrons; crystal harmonics; density functional theory; density of states; electric field gradient; energy bands; exchange–correlation; free-electron model; full-potential methods; itinerant electrons; linear combination of atomic orbitals; linear combination of muffin-tin orbitals; linearized augmented plane wave method; local coordinate system; localized electrons; muffin-tin approximation; nuclear quadrupole moment; partial charges; periodic boundary conditions; pseudo-potential; quantum-mechanical treatment; reciprocal lattice; relativistic effects; representations; semi-core states; small representations; tight binding