Maximally localized Wannier functions: Theory and applications

TL;DR

This paper reviews the theory and diverse applications of maximally localized Wannier functions, highlighting their role in analyzing chemical bonding, polarization, magnetization, and their use in interpolation and other physical systems.

Contribution

It provides a comprehensive overview of the methods to construct maximally localized Wannier functions and explores their wide-ranging applications beyond electronic structure.

Findings

Methods for constructing unique maximally localized Wannier functions are reviewed.

Applications include analyzing chemical bonding, electric polarization, and orbital magnetization.

Wannier functions are used in interpolation schemes and in systems like phonons, photonic crystals, and cold atoms.

Abstract

The electronic ground state of a periodic system is usually described in terms of extended Bloch orbitals, but an alternative representation in terms of localized "Wannier functions" was introduced by Gregory Wannier in 1937. The connection between the Bloch and Wannier representations is realized by families of transformations in a continuous space of unitary matrices, carrying a large degree of arbitrariness. Since 1997, methods have been developed that allow one to iteratively transform the extended Bloch orbitals of a first-principles calculation into a unique set of maximally localized Wannier functions, accomplishing the solid-state equivalent of constructing localized molecular orbitals, or "Boys orbitals" as previously known from the chemistry literature. These developments are reviewed here, and a survey of the applications of these methods is presented. This latter includes a description of their use in analyzing the nature of chemical bonding, or as a local probe of phenomena related to electric polarization and orbital magnetization. Wannier interpolation schemes are also reviewed, by which quantities computed on a coarse reciprocal-space mesh can be used to interpolate onto much finer meshes at low cost, and applications in which Wannier functions are used as efficient basis functions are discussed. Finally the construction and use of Wannier functions outside the context of electronic-structure theory is presented, for cases that include phonon excitations, photonic crystals, and cold-atom optical lattices.