Maximally localized Wannier functions: Theory and applications

8
1. Real-space representation
An interesting consequence stemming from the choice
of (18) as the localization functional is that it allows
a natural decomposition of the functional into gaugeinvariant
and gauge-dependent parts. That is, we can
write
where
and
Ω I = ∑ n
[
Ω = Ω I + ˜Ω (19)
⟨ 0n | r 2 | 0n ⟩ − ∑ Rm
˜Ω = ∑ n
∑
Rm≠0n
∣
∣⟨Rm|r|0n⟩ ∣ ]
2
(20)
∣
∣⟨Rm|r|0n⟩ ∣ ∣ 2 . (21)
It can be shown that not only ˜Ω but also Ω I is positive
definite, and moreover that Ω I is gauge-invariant,
i.e., invariant under any arbitrary unitary transformation
(10) of the Bloch orbitals (Marzari and Vanderbilt, 1997).
This follows straightforwardly from recasting Eq. (20) in
terms of the band-group projection operator P , as defined
in Eq. (15), and its complement Q = 1 − P :
Ω I = ∑ ⟨0n|r α Qr α |0n⟩
nα
= ∑ α
Tr c [P r α Qr α ] . (22)
The subscript ‘c’ indicates trace per unit cell. Clearly
Ω I is gauge invariant, since it is expressed in terms of
projection operators that are unaffected by any gauge
transformation. It can also be seen to be positive definite
by
∑
using the idempotency of P and Q to write Ω I =
α Tr c [(P r α Q)(P r α Q) † ] = ∑ α ||P r αQ|| 2 c.
The minimization procedure of Ω thus actually corresponds
to the minimization of the non-invariant part
˜Ω only. At the minimum, the off-diagonal elements
|⟨Rm|r|0n⟩| 2 are as small as possible, realizing the best
compromise in the simultaneous diagonalization, within
the subspace of the Bloch bands considered, of the three
position operators x, y and z, which do not in general
commute when projected onto this space.
2. Reciprocal-space representation
As shown by Blount (1962), matrix elements of the
position operator between WFs take the form
∫
V
⟨Rn|r|0m⟩ = i
(2π) 3 dk e ik·R ⟨u nk |∇ k |u mk ⟩ (23)
and
⟨Rn|r 2 |0m⟩ = −
V ∫
(2π) 3 dk e ik·R ⟨u nk |∇ 2 k|u mk ⟩ . (24)
These expressions provide the needed connection with
our underlying Bloch formalism, since they allow to express
the localization functional Ω in terms of the matrix
elements of ∇ k and ∇ 2 k
. In addition, they allow to
calculate the effects on the localization of any unitary
transformation of the |u nk ⟩ without having to recalculate
expensive (especially when plane-wave basis sets are
used) scalar products. We thus determine the Bloch orbitals
|u mk ⟩ on a regular mesh of k-points, and use finite
differences to evaluate the above derivatives. In particular,
we make the assumption that the BZ has been discretized
into a uniform Monkhorst-Pack mesh, and the
Bloch orbitals have been determined on that mesh. 4
For any f(k) that is a smooth function of k, it can
be shown that its gradient can be expressed by finite
differences as
∇f(k) = ∑ b
w b b [f(k + b) − f(k)] + O(b 2 ) (25)
calculated on stars (“shells”) of near-neighbor k-points;
here b is a vector connecting a k-point to one of its neighbors,
w b is an appropriate geometric factor that depends
on the number of points in the star and its geometry
(see Appendix B in Marzari and Vanderbilt (1997) and
Mostofi et al. (2008) for a detailed description). In a
similar way,
|∇f(k)| 2 = ∑ b
w b [f(k + b) − f(k)] 2 + O(b 3 ) . (26)
It now becomes straightforward to calculate the matrix
elements in Eqs. (23) and (24). All the information
needed for the reciprocal-space derivatives is encoded in
the overlaps between Bloch orbitals at neighboring k-
points
M (k,b)
mn = ⟨u mk |u n,k+b ⟩ . (27)
These overlaps play a central role in the formalism, since
all contributions to the localization functional can be
expressed in terms of them. Thus, once the M mn
(k,b)
have been calculated, no further interaction with the
electronic-structure code that calculated the ground state
wavefunctions is necessary - making the entire Wannierization
procedure a code-independent post-processing
step 5 . There is no unique form for the localization functional
in terms of the overlap elements, as it is possible
4 Even the case of Γ-sampling – where the Brillouin zone is sampled
with a single k-point – is encompassed by the above formulation.
In this case the neighboring k-points are given by reciprocal lattice
vectors G and the Bloch orbitals differ only by phase factors
exp(iG · r) from their counterparts at Γ. The algebra does become
simpler, though, as will be discussed in Sec. II.F.2.
5 In particular, see Ferretti et al. (2007) for the extension to
ultrasoft pseudopotentials and the projector-augmented wave
method, and Freimuth et al. (2008); Kuneš et al. (2010); and
Posternak et al. (2002) for the full-potential linearized augmented
planewave method.