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\title{Maximally localized Wannier functions: Theory and applications}

\author{Nicola Marzari}
\affiliation{Theory and Simulation of Materials (THEOS),
\'Ecole Polytechnique F\'ed\'erale de Lausanne, Station 12, 1015 Lausanne, Switzerland}
\author{Arash A. Mostofi}
\affiliation{Departments of Materials and Physics, and the 
Thomas Young Centre for Theory and Simulation of Materials,
Imperial College London, London SW7 2AZ, UK}
\author{Jonathan R. Yates}
\affiliation{Department of Materials,
  University of Oxford, Parks Road, Oxford OX1 3PH, UK}
\author{Ivo Souza}
\affiliation{Centro de F\'{\i}sica de Materiales (CSIC) and DIPC, Universidad
del Pa\'\i s Vasco, 20018 San Sebasti\'an, Spain}
\affiliation{Ikerbasque, Basque Foundation for Science, 48011
Bilbao, Spain}
\author{David Vanderbilt}
\affiliation{Department of Physics and Astronomy, Rutgers University,
        Piscataway, NJ 08854-8019, USA}

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\begin{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 {\it 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.
\end{abstract}                                                                 

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\tableofcontents
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% Section 1
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\section{INTRODUCTION}
\label{sec:intro}
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\section{REVIEW OF BASIC THEORY}
\label{sec:theory}
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\section{RELATION TO OTHER LOCALIZED ORBITALS}
\label{sec:locorb}
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\section{ANALYSIS OF CHEMICAL BONDING}
\label{sec:bonding}
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\section{ELECTRIC POLARIZATION AND ORBITAL MAGNETIZATION}
\label{sec:pol}
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\section{WANNIER INTERPOLATION}
\label{sec:winterp}
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\section{WANNIER FUNCTIONS AS BASIS FUNCTIONS}
\label{sec:basis}
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\section{WANNIER FUNCTIONS IN OTHER CONTEXTS}
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\section{SUMMARY AND PROSPECTS}
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