This chapter is devoted to representative examples that bring out the range of phenomena observed in materials with strong local atomic-like interactions, and the types of properties that can be calculated with various methods. The foremost examples are elements and compounds of the series of transition elements with d and f states that are localized as depicted in Fig. 19.1. As emphasized in the previous chapter, the theory must deal with the complexities of many competing interactions: direct Coulomb, exchange, spin–orbit, crystal field splitting, hybridization with other orbitals, and other effects, all of which may be essential for understanding any particular material.

Many-body perturbation theory, especially the GW approximation and beyond, is a firstprinciples method that can be applied directly to ordered states at low temperature. Several examples of applications to transition metal systems are given in Ch. 13 and referred to here to provide a unified picture. Density functional theory and approximate static mean-field approximations also provide useful results and insights. However, many of the most striking phenomena can be understood only by taking into account strong correlation, including large renormalization of electronic states, satellites in excitation spectra, magnetic phase transitions, metal–insulator transitions, and other phenomena. Many phenomena can be understood only if temperature is taken into account. Dynamical mean-field theory brings this capability, but at a price because it is feasible to treat explicitly only a small subset of the degrees of freedom of the electrons. All the results presented here are done in the single-site approximation (single-site DMFA), except the two-site cluster for VO2 in Sec. 20.6.