Maximally localized Wannier functions: Theory and applications

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operators that are represented in a localized basis may
be exploited in order to achieve computational efficiencies
and improved scaling of numerical algorithms with
respect to system size. Recently, therefore, MLWFs have
been used for this very purpose in a number of contexts
and we mention, in brief, some of them here.
In quantum Monte Carlo (QMC) calculations (Foulkes
et al., 2001), a significant computational effort is expended
in evaluating the Slater determinant for a given
electronic configuration of N electrons. These Slater
determinants are usually constructed from a set of extended
single-particle states, obtained from, e.g., a DFT
or Hartree-Fock calculation, represented in a basis set of,
e.g., extended plane-waves. This gives rise to O(N 3 ) scaling
of the computational cost of evaluating such a determinant.
Williamson et al. (2001) suggested, instead, to
use MLWFs that were smoothly truncated to zero beyond
a certain cut-off radius that is independent of system size.
This ensures that each electron falls only within the localization
region of a fixed number of MLWFs, thus reducing
the asymptotic scaling by one factor of N. Furthermore,
by representing the MLWFs in a basis of localized
spline functions, rather than plane-waves or even Gaussian
functions, the evaluation of each orbital is rendered
independent of system size, thereby reducing the overall
cost of computing the determinant of the given configuration
to O(N). More recently, rather than truncated
MLWFs, the use of non-orthogonal orbitals obtained by
projection (Reboredo and Williamson, 2005) or other localization
criteria (Alfè and Gillan, 2004) has also been
suggested.
In another development, Wu et al. (2009) use ML-
WFs in order to compute efficiently Hartree-Fock
exact-exchange integrals in extended systems. Hybrid
exchange-and-correlation functionals (Becke, 1993) for
DFT calculations, in which some proportion of Hartree-
Fock exchange is included in order to alleviate the wellknown
problem of self-interaction that exists in local and
semi-local functionals such as the local-density approximation
and its generalized gradient-dependent variants,
have been used relatively little in extended systems. This
is predominantly due to the computational cost associated
with evaluating the exchange integrals between extended
eigenstates that are represented in a plane-wave
basis set. Wu et al. (2009) show that by performing a
unitary transformation of the eigenstates to a basis of
MLWFs, and working in real-space in order to exploit
the fact that spatially distant MLWFs have exponentially
vanishing overlap, the number of such overlaps that
needs to be calculated scales linearly, in the limit of large
system-size, with the number of orbitals (as opposed to
quadratically), which is a sufficient improvement to enable
Car-Parrinello molecular dynamics simulations with
hybrid functionals.
Similar ideas that exploit the locality of MLWFs have
been applied to many-body perturbation theory approaches
for going beyond DFT and Hartree-Fock calculations,
for example, in the contexts of the GW approximation
(Umari et al., 2009), the evaluation of local
correlation in extended systems (Buth et al., 2005; Pisani
et al., 2005), and the Bethe-Salpeter equation (Sasioglu
et al., 2010). The improved scaling and efficiency of these
approaches open the possibility of such calculations on
larger systems than previously accessible.
Finally, we note that MLWFs have been used recently
to compute van der Waals (vdW) interactions in an approximate
but efficient manner (Andrinopoulos et al.,
2011; Silvestrelli, 2008, 2009b). The method is based
on an expression due to Andersson et al. (1996) for the
vdW energy in terms of pairwise interactions between
fragments of charge density. MLWFs provide a localized
decomposition of the electronic charge density of a system
and can be used as the basis for computing the vdW
contribution to the total energy in a post-processing (i.e.,
non-self-consistent) fashion. In order to render tractable
the necessary multi-dimensional integrals, the true ML-
WFs of the system are substituted by analytic hydrogenic
orbitals that have the same centers and spreads as the
true MLWFs. The approach has been applied to a variety
of systems, including molecular dimers, molecules physisorbed
on surfaces, water clusters and weakly-bound
solids (Andrinopoulos et al., 2011; Espejo et al., 2012;
Silvestrelli, 2008, 2009a,b; Silvestrelli et al., 2009). Recently,
Ambrosetti and Silvestrelli (2012) have suggested
an alternative, simpler formulation that is based on London’s
expression for the van der Waals energy of two
interacting atoms (Eisenschitz and London, 1930).
B. WFs as a basis for strongly-correlated systems
For many strongly-correlated electron problems, the
essential physics of the system can be explained by considering
only a subset of the electronic states. A recent
example is understanding the behavior of unconventional
(high-T c ) superconductors, in which a great deal
of insight can be gained by considering only the ML-
WFs of p and d character on Cu and O, respectively,
for cuprates (Sakakibara et al., 2010), and those on As
and Fe, respectively, for the iron pnictides (Cao et al.,
2008; Hu and Hu, 2010; Kuroki et al., 2008; Suzuki et al.,
2011). Other strongly-correlated materials for which ML-
WFs have been used to construct minimal models to
help understand the physics include manganites (Kovacik
and Ederer, 2010), topological insulators (Zhang et al.,
2009a,b) (see also Sec. VI.A.4), and polyphenylene vinylene
(PPV), in particular relating to electron-hole excitations
(Karabunarliev and Bittner, 2003a,b).
Below we outline some of the general principles behind
the construction and solution of such minimal models.