Maximally localized Wannier functions: Theory and applications

17
Problems associated with reaching local minima of the
spread functional, and with obtaining Wannier functions
that are not real-valued, are more pronounced in the case
of entangled bands. They are usually alleviated by careful
reconsideration of the energy windows used, in order
to include higher energy states of the appropriate
symmetry, and/or by using a better initial guess for the
projections. We infer, therefore, that such problems are
associated not with the wannierization part of the procedure,
but rather with the initial selection of the smooth
subspace from the full manifold of entangled bands.
It is worth noting that the Γ-point formulation
(Sec. II.F.2) appears to be less affected by these problems.
In cases where it is not intuitive or obvious what
the MLWFs should be, therefore, it can often be a fruitful
strategy to use the Γ-point formulation to obtain approximate
MLWFs that may then be used to inform the initial
guess for a subsequent calculation with a full k-point
mesh.
J. Many-body generalizations
The concept of WFs is closely tied to the framework
of single-particle Hamiltonians. Only in this case can we
define J occupied single-particle Bloch functions at each
wavevector k and treat all J of them on an equal footing,
allowing for invariance with respect to unitary mixing
among them. Once the two-particle electron-electron
interaction is formally included in the Hamiltonian, the
many-body wavefunction cannot be reduced to any simple
form allowing for the construction of WFs in the usual
sense.
One approach is to consider the reduced one-particle
density matrix
∫
n(r, r ′ ) = Ψ ∗ (r, r 2 ...) Ψ(r ′ , r 2 , ...) dr 2 dr 3 ... (54)
for a many-body insulator. Since n(r, r ′ ) is periodic in
the sense of n(r + R, r ′ + R) = n(r, r ′ ), its eigenvectors –
the so-called “natural orbitals” – have the form of Bloch
functions carrying a label n, k. If the insulator is essentially
a correlated version of a band insulator having J
bands, then at each k there will typically be J occupation
eigenvalues ν nk that are close to unity, as well as some
small ones that correspond to the quantum fluctuations
into conduction-band states. If one focuses just on the
subspace of one-particle states spanned by the J valencelike
natural orbitals, one can use them to construct oneparticle
WFs following the methods described earlier,
as suggested by Koch and Goedecker (2001). However,
while such an approach may provide useful qualitative
information, it cannot provide the basis for any exact
theory. For example, the charge density, or expectation
value of any other one-particle operator, obtained
by tracing over these WFs will not match their exact
many-body counterparts.
A somewhat related approach, adopted by Hamann
and Vanderbilt (2009), is to construct WFs out of the
quasiparticle states that appear in the GW approximation
(Aryasetiawan and Gunnarsson, 1998). Such an approach
will be described more fully in Sec. VI.A.3. Here
again, this approach may give useful physical and chemical
intuition, but the one-electron quasiparticle wavefunctions
do not have the physical interpretation of occupied
states, and charge densities and other ground-state
properties cannot be computed quantitatively from them.
Finally, a more fundamentally exact framework for a
many-body generalization of the WF concept, introduced
in Souza et al. (2000), is to consider a many-body system
with twisted boundary conditions applied to the manybody
wavefunction in a supercell. For example, consider
M electrons in a supercell consisting of M 1 × M 2 × M 3
primitive cells, and impose the periodic boundary condition
Ψ q (..., r j + R, ...) = e iq·R Ψ q (..., r j , ...) (55)
for j = 1, ..., M, where R is a lattice vector of the superlattice.
One can then construct a single “many-body
WF” in a manner similar to Eq. (3), but with k → q and
|ψ nk ⟩ → |Ψ q ⟩ on the right side. The resulting many-body
WF is a complex function of 3M electron coordinates,
and as such it is unwieldy to use in practice. However, it
is closely related to Kohn’s theory of the insulating state
(Kohn, 1964), and in principle it can be used to formulate
many-body versions of the theory of electric polarization
and related quantities, as shall be mentioned in
Sec. V.A.4.
III. RELATION TO OTHER LOCALIZED ORBITALS
A. Alternative localization criteria
As we have seen, WFs are inherently non-unique and,
in practice, some strategy is needed to determine the
gauge. Two possible approaches were already discussed
in Sec. II, namely, projection and maximal localization.
The latter approach is conceptually more satisfactory, as
it does not depend on a particular choice of trial orbitals.
However, it still does not uniquely solve the problem of
choosing a gauge, as different localization criteria are possible
and there is, a priori, no reason to choose one over
another.
While MLWFs for extended systems have been generated
for the most part by minimizing the sum of
quadratic spreads, Eq. (17), a variety of other localization
criteria have been used over the years for molecular
systems. We will briefly survey and compare some of
the best known schemes below. What they all have in
common is that the resulting localized molecular orbitals