July 10, 1997 | Nicola Marzari and David Vanderbilt
This paper presents a method for determining optimally-localized generalized Wannier functions for composite energy bands in crystalline solids. The method minimizes a functional representing the total spread of Wannier functions in real space, using unitary matrices to describe rotations among Bloch bands at each k-point. It is suitable for use with conventional electronic-structure codes and returns the total electric polarization and Wannier center locations. The approach is motivated by recent developments in linear-scaling methods and the theory of electric polarization. The method is demonstrated for Si, GaAs, molecular C₂H₄, and LiCl. The paper discusses the arbitrariness in defining Wannier functions, the spread functional in real and k-space, and special cases such as one-dimensional systems, isolated bands, and inversion symmetry. It also describes a numerical algorithm for computing optimally localized Wannier functions on a k-space mesh and discusses the implications of the results for electronic polarization in disordered or distorted insulating materials. The work also explores generalizations of the problem, including non-orthonormal Wannier functions and larger sets of functions spanning a space containing the desired bands. The paper concludes with a discussion of the significance of the work and its potential applications.This paper presents a method for determining optimally-localized generalized Wannier functions for composite energy bands in crystalline solids. The method minimizes a functional representing the total spread of Wannier functions in real space, using unitary matrices to describe rotations among Bloch bands at each k-point. It is suitable for use with conventional electronic-structure codes and returns the total electric polarization and Wannier center locations. The approach is motivated by recent developments in linear-scaling methods and the theory of electric polarization. The method is demonstrated for Si, GaAs, molecular C₂H₄, and LiCl. The paper discusses the arbitrariness in defining Wannier functions, the spread functional in real and k-space, and special cases such as one-dimensional systems, isolated bands, and inversion symmetry. It also describes a numerical algorithm for computing optimally localized Wannier functions on a k-space mesh and discusses the implications of the results for electronic polarization in disordered or distorted insulating materials. The work also explores generalizations of the problem, including non-orthonormal Wannier functions and larger sets of functions spanning a space containing the desired bands. The paper concludes with a discussion of the significance of the work and its potential applications.