Maximally localized Wannier functions: Theory and applications

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2. Self-interaction and DFT + Hubbard U
In the approaches just described, the results of a DFT
calculation are used to parametrize the model Hamiltonian
of a strongly correlated electron system. In contrast,
in a DFT+U formulation (Anisimov et al., 1993, 1991)
the energy functional is explicitly augmented with a Hubbard
term U (Hubbard, 1963) aimed at improving the description
of strong interactions, such as those associated
with localized d and f electronic states, and at repairing
the mean-field underestimation of on-site Coulomb
repulsions.
In DFT+U the Hubbard manifold is defined by a set of
projectors that are typically atomic-like orbitals of d or f
character. Localization of this manifold plays a key role,
since DFT+U effectively corrects for the lack of piecewise
linearity in approximate energy functionals (Cococcioni
and de Gironcoli, 2005; Perdew et al., 1982) and thus
greatly reduces self-interaction errors (Kulik et al., 2006;
Mori-Sánchez et al., 2008; Perdew and Zunger, 1981).
Since strongly localized orbitals are those that suffer most
from self-interaction, MLWFs can become an appealing
choice to define Hubbard manifolds adapted to the chemistry
of the local environment. In fact, MLWFs have
been successfully used as Hubbard projectors (Anisimov
et al., 2007; Fabris et al., 2005; Miyake and Aryasetiawan,
2008), and it has been argued that their shape can
constitute an additional degree of freedom in the calculations
(O’Regan et al., 2010), provided their localized,
atomic character is maintained. It should also be pointed
out that the value of U entering the calculations should
not be considered universal, as it depends strongly on
the manifold chosen (e.g. for pseudopotential calculations
on the oxidation state of the reference atomic calculation
(Kulik and Marzari, 2008)), or on the structure
or electronic-structure of the problem studied.
Last, it should be pointed out that functionals that
aim to correct directly for some effects of self-interaction
– such as the Perdew-Zunger correction (Perdew and
Zunger, 1981) or Koopmans-compliant functionals (Dabo
et al., 2010) – can lead naturally in a periodic system to
Wannier-like localized orbitals that minimize the total
energy (Park et al., 2011; Stengel and Spaldin, 2008),
while the canonical orbitals that diagonalize the Hamiltonian
still preserve Bloch periodicity.
VIII. WANNIER FUNCTIONS IN OTHER CONTEXTS
As described in Sec. II.A, Wannier functions provide an
alternative, localized, description of a manifold of states
spanned by the eigenstates (energy bands) of a singleparticle
Hamiltonian that describes electrons in a periodic
potential. The equivalence of the Wannier representation
and the eigenstate representation may be expressed
in terms of the band projection operator ˆP , see
Eq. (15). This operator satisfies the idempotency condition
P 2 = P , which embodies simultaneously the requirements
of orthogonality and Pauli exclusion.
From their conception, and until relatively recently,
Wannier functions have been used almost exclusively in
this context, namely to represent a manifold of singleparticle
orbitals for electrons. Furthermore, as discussed
in Sec. II, we need not restrict ourselves to an isolated
group of states, such as the occupied manifold: the disentanglement
procedure enables a subspace of a larger
manifold, e.g., of occupied and unoccupied states, to be
selected which may then be wannierized. This has, for
example, opened up areas of application in which Wannier
functions are used as tight-binding basis functions
for electronic structure and transport calculations, as described
in Sec. VI and Sec. VII.
From a general mathematical point of view, however,
the set of orthogonal eigenfunctions of any self-adjoint
(Hermitian) operator may be “rotated” by unitary transformation
to another orthogonal basis that spans the
same space. As we have seen, the unitary transformation
is arbitrary and may be chosen to render the new
basis set maximally-localized, which has computational
advantages when it comes to representing physical quantities
that are short-ranged. When the operator in question
has translational symmetry, the maximally-localized
functions thus obtained are reminiscent of the Wannier
functions familiar from electronic structure theory. Often,
such a basis is also preferable to using another localized
basis because information regarding the symmetries
of the self-adjoint operator from which the basis is derived
is encoded within it.
These ideas have led to the appropriation of the MLWF
formalism described in Sec. II for contexts other than
the description of electrons: the single-particle electronic
Hamiltonian is replaced by another suitable periodic selfadjoint
operator, and the Bloch eigenstates are replaced
by the eigenfunctions of that operator, which may then
be transformed to give a MLWF-like representation that
may be used as an optimal and compact basis for the desired
calculation, for example, to analyze the eigenmodes
of the static dielectric matrix in ice and liquid water (Lu
et al., 2008).
Below we review the three most prominent of these
applications, namely to the study of phonons, photonic
crystals, and cold atom lattices.
A. Phonons
Lattice vibrations in periodic crystals are usually described
in terms of normal modes, which constitute a
delocalized orthonormal basis for the space of vibrations
of the lattice such that an arbitrary displacement
of the atoms in the crystal may be expressed
in terms of a linear combination of its normal modes.