9. Electron-Electron Interactions 9.1 Introduction Bloch's theory for periodic solids brilliantly solves the problem of a single electron in a periodic potential. While engaged in the process of solving this problem, it is easy to forget the severity of the approximations needed to reduce a real periodic solid to a single-electron problem. The problem has two different sides. First, how conceptually can interacting electrons be treated within a one-electron framework at all? An answer to this question will be provided in Section 17.5 by Fermi liquid theory. Yet Fermi liquid theory provides little practical guidance in constructing the effective one-electron potential it shows may exist. The construction is instead provided by a sequence of approximations, whose validity is in principle not quite clear but in practice lies behind all attempts at realistic calculations. There is no internally consistent test for the validity of these calculations, and sometimes they fail rather badly. Even more often, however, they achieve detailed comparison with experiment that is much better than might have been expected. The goal of this chapter is to describe only those treatments of electron-electron interactions leading to practical band structure calculations. The Hamiltonian that one really should solve is Eq. (6.1), in which electrons and nuclei all appear on an equal quantum-mechanical footing. A first simplifica- tion is to remove the nuclei from the quantum mechanics problem. Because nuclei are thousands of times more massive than electrons, they move that much more slowly. Born and Oppenheimer (1927) suggested an approximation scheme that is employed quite universally throughout condensed matter physics. So far as the electrons are concerned, take the nuclei to be static, classical potentials, and solve the electronic problem without worrying about the nuclei further. So far as the nu- clei are concerned, the electrons are a rapidly moving shroud of charge that follows them wherever they go. Because the motion of nuclei is accompanied by charge redistribution, the energies involved in moving nuclei about depend upon the so- lution of the electron problem, and the nuclei interact with complicated effective potentials. All the ideas of different types of interatomic bonding arise from this viewpoint and will be discussed further in Chapter 11. The Born-Oppenheimer approximation may have to be abandoned whenever the electrons and nuclei cannot be disentangled so neatly. The world is full of phenomena where the approximation fails. For example, in striking a flint to create a spark, mechanical motion excites electrons into a plasma that then emits light. 233 Condensed Matter Physics, Second Edition by Michael P. Marder Copyright © 2010 John Wiley & Sons, Inc.