Maximally localized Wannier functions: Theory and applications

11
To proceed further, we write
dΩ = 2 tr [X ′ dX + Y ′ dY + Z ′ dZ] , (40)
exploiting the fact that tr [X ′ X D ] = 0, and similarly
for Y and Z. We then consider an infinitesimal unitary
transformation |w i ⟩ → |w i ⟩ + ∑ j dW ji|w j ⟩ (where
dW is antihermitian), from which dX = [X, dW ], and
similarly for Y , Z. Inserting in Eq. (40) and using
tr [A[B, C]] = tr [C[A, B]] and [X ′ , X] = [X ′ , X D ], we
obtain dΩ = tr [dW G] where the gradient of the spread
functional G = dΩ/dW is given by
{
}
G = 2 [X ′ , X D ] + [Y ′ , Y D ] + [Z ′ , Z D ] dW . (41)
The minimization can then be carried out using a procedure
similar to that outlined above for periodic boundary
conditions. Last, we note that minimizing ˜Ω is equivalent
to maximizing tr [X 2 D + Y 2 D + Z2 D ], since tr [X′ X D ] =
tr [Y ′ Y D ] = tr [Y ′ Y D ] = 0.
2. Γ-point formulation for large supercells
A similar formulation applies in reciprocal space when
dealing with isolated or very large systems in periodic
boundary conditions, i.e., whenever it becomes appropriate
to sample the wavefunctions only at the Γ-point
of the Brillouin zone. For simplicity, we start with the
case of a simple cubic lattice of spacing L, and define the
matrices X , Y, and Z as
X mn = ⟨w m |e −2πix/L |w n ⟩ (42)
and its periodic permutations (the extension to supercells
of arbitrary symmetry has been derived by Silvestrelli
(1999)). 6 The coordinate x n of the n-th WF center
(WFC) can then be obtained from
x n = − L 2π Im ln⟨w n|e −i 2π L x |w n ⟩ = − L 2π Im ln X nn ,
(43)
and similarly for y n and z n . Equation (43) has been
shown by Resta (1998) to be the correct definition of
the expectation value of the position operator for a system
with periodic boundary conditions, and had been
introduced several years ago to deal with the problem
of determining the average position of a single electronic
orbital in a periodic supercell (Selloni et al., 1987) (the
6 We point out that the definition of the X , Y, Z matrices for extended
systems, now common in the literature, is different from
the one used in the previous subsection for the X, Y, Z matrices
for isolated system. We preserved these two notations for
the sake of consistency with published work, at the cost of making
less evident the close similarities that exist between the two
minimization algorithms.
above definition becomes self-evident in the limit where
w n tends to a Dirac delta function).
Following the derivation of the previous subsection,
or of Silvestrelli et al. (1998), it can be shown that the
maximum-localization criterion is equivalent to maximizing
the functional
Ξ =
J∑ (
|Xnn | 2 + |Y nn | 2 + |Z nn | 2) . (44)
n=1
The first term of the gradient dΞ/dA mn is given by
[X nm (Xnn ∗ − Xmm) ∗ − Xmn(X ∗ mm − X nn )], and similarly
for the second and third terms. Again, once the gradient
is determined, minimization can be performed using
a steepest-descent or conjugate-gradients algorithm; as
always, the computational cost of the localization procedure
is small, given that it involves only small matrices
of dimension J × J, once the scalar products needed to
construct the initial X (0) , Y (0) and Z (0) have been calculated,
which takes an effort of order J 2 ∗ N basis . We
note that in the limit of a single k point the distinction
between Bloch orbitals and WFs becomes irrelevant,
since no Fourier transform from k to R is involved in the
transformation Eq. (10); rather, we want to find the optimal
unitary matrix that rotates the ground-state selfconsistent
orbitals into their maximally localized representation,
making this problem exactly equivalent to the
one of isolated systems. Interestingly, it should also be
mentioned that the local minima alluded to in the previous
subsection are typically not found when using Γ
sampling in large supercells.
Before concluding, we note that care should be taken
when comparing the spreads of MLWFs calculated in supercells
of different sizes. The Wannier centers and the
general shape of the MLWFs often converge rapidly as
the cell size is increased, but even for the ideal case of an
isolated molecule, the total spread Ω often displays much
slower convergence. This behavior derives from the finitedifference
representation of the invariant part Ω I of the
localization functional (essentially, a second derivative);
while Ω I does not enter into the localization procedure, it
does contribute to the spread, and in fact usually represents
the largest term. This slow convergence was noted
by Marzari and Vanderbilt (1997) when commenting on
the convergence properties of Ω with respect to the spacing
of the Monkhorst-Pack mesh, and has been studied in
detail by others (Stengel and Spaldin, 2006a; Umari and
Pasquarello, 2003). For isolated systems in a supercell,
this problem can be avoided simply by using a very large
L in Eq. (43), since in practice the integrals only need to
be calculated in the small region where the orbitals are
non-zero (Wu, 2004). For extended bulk systems, this
convergence problem can be ameliorated significantly by
calculating the position operator using real-space integrals
(Lee, 2006; Lee et al., 2005; Stengel and Spaldin,
2006a).