Maximally localized Wannier functions: Theory and applications

Nicola Marzari, Arash A. Mostofi, Jonathan R. Yates, Ivo Souza, and David Vanderbilt
Rev. Mod. Phys. 84, 1419 – Published 10 October 2012

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.

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  • Received 9 December 2011

DOI:https://doi.org/10.1103/RevModPhys.84.1419

© 2012 American Physical Society

Authors & Affiliations

Nicola Marzari

  • Theory and Simulation of Materials (THEOS), École Polytechnique Fédérale de Lausanne, Station 12, 1015 Lausanne, Switzerland

Arash A. Mostofi

  • Departments of Materials and Physics, and the Thomas Young Centre for Theory and Simulation of Materials, Imperial College London, London SW7 2AZ, United Kingdom

Jonathan R. Yates

  • Department of Materials, University of Oxford, Parks Road, Oxford OX1 3PH, United Kingdom

Ivo Souza

  • Centro de Física de Materiales (CSIC) and DIPC, Universidad del País Vasco, 20018 San Sebastián, Spain and Ikerbasque Foundation, 48011 Bilbao, Spain

David Vanderbilt

  • Department of Physics and Astronomy, Rutgers University, Piscataway, New Jersey 08854-8019, USA

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Vol. 84, Iss. 4 — October - December 2012

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