Introduction to Maximally Localized Wannier Functions

The maximally localized Wannier functions (MLWFs) provide a practical basis set for the analysis and computation of the electronic structure of periodic systems, which can be applied in all those cases where real space localization represents a significant advantage. In close analogy with MLWFs, localized molecular orbitals (LMOs) have a long tradition in computational chemistry. To provide a simple and practical introduction to MLWFs, this chapter reviews the basic ideas underlying periodic lattices and Bloch functions. Among the potentially infinite applications of MLWFs, a number of selected examples are presented in the chapter, providing a practical and intuitive solution to common problems in electronic structure calculations. These encompass the classical cases of charge distribution and bonding analysis, followed by the closely related study of defected and amorphous structures, where information on bonding and electronic charge distribution complements the standard geometrical analysis of the system.