General and efficient algorithms for obtaining maximally localized Wannier functions

Recent advances in the theory of polarization and the development of linear-scaling methods have revitalized interest in the use of Wannier functions for obtaining a localized orbital picture within a periodic supercell. To examine complex chemical systems it is imperative for the localization procedure to be efficient; on the other hand, the method should also be simple and general. Motivated to meet these requirements we derive in this paper a spread functional to be minimized as a starting point for obtaining maximally localized Wannier functions through a unitary transformation. The functional turns out to be equivalent to others discussed in the literature with the difference of being valid in simulation supercells of arbitrary symmetry in the $\mathit{\Gamma }$-point approximation. To minimize the spread an iterative scheme is developed and very efficient optimization methods, such as conjugate gradient, direct inversion in the iterative subspace, and preconditioning are applied to accelerate the convergence. The iterative scheme is quite general and is shown to work also for methods first developed for finite systems (e.g., Pipek-Mezey, Boys-Foster). The applications discussed range from crystal structures such as Si to isolated complex molecules and are compared to previous investigations on this subject.