Maximally localized Wannier functions: Theory and applications

29
straightforward. The electric dipole is
d = −e ∑ j
⟨ψ j |r|ψ j ⟩ (84)
and the orbital moment is
m = − e ∑
⟨ψ j |r × v|ψ j ⟩ , (85)
2c
j
where the sum is over occupied Hamiltonian eigenstates
|ψ j ⟩, r is the position operator, v = (i/ħ)[H, r] is the
velocity operator, and Gaussian units are used. However,
these formulas cannot easily be generalized to the case of
crystalline systems, because the Hamiltonian eigenstates
take the form of Bloch functions |ψ nk ⟩ that extend over
all space. The problem is that matrix elements such as
⟨ψ nk |r|ψ nk ⟩ and ⟨ψ nk |r × v|ψ nk ⟩ are ill-defined for such
extended states (Nenciu, 1991).
To deal with this problem, the so-called “modern theory
of polarization” (King-Smith and Vanderbilt, 1993;
Resta, 1992, 1994; Vanderbilt and King-Smith, 1993) was
developed in the 1990’s, and a corresponding “modern
theory of magnetization” in the 2000’s (Ceresoli et al.,
2006; Shi et al., 2007; Souza and Vanderbilt, 2008; Thonhauser
et al., 2005; Xiao et al., 2005). Useful reviews of
these topics have appeared (Resta, 2000, 2010; Resta and
Vanderbilt, 2007; Vanderbilt and Resta, 2006).
These theories can be formulated either in terms of
Berry phases and curvatures, or equivalently, by working
in the Wannier representation. The basic idea of the
latter is to consider a large but finite sample surrounded
by vacuum and carry out a unitary transformation from
the set of delocalized Hamiltonian eigenstates ψ j to a
set of Wannier-like localized molecular orbitals ϕ j . Then
one can use Eq. (84) or Eq. (85), with the ψ j replaced by
the ϕ j , to evaluate the electric or orbital magnetic dipole
moment per unit volume in the thermodynamic limit. In
doing so, care must be taken to consider whether any
surface contributions survive in this limit.
In this section, we briefly review the modern theories
of electric polarization and orbital magnetization and related
topics. The results given in this section are valid
for any set of localized WFs; maximally localized ones
do not play any special role. Nevertheless, the close connection
to the theory of polarization has been one of the
major factors behind the resurgence of interest in WFs.
Furthermore, we shall see that the use of MLWFs can
provide a very useful, if heuristic, local decomposition
of polar properties in a an extended system. For these
reasons, it is appropriate to review the subject here.
A. Wannier functions, electric polarization, and localization
1. Relation to Berry-phase theory of polarization
Here we briefly review the connection between the
Wannier representation and the Berry-phase theory of
polarization (King-Smith and Vanderbilt, 1993; Resta,
1994; Vanderbilt and King-Smith, 1993). Suppose that
we have constructed via Eq. (8) a set of Bloch-like functions
| ˜ψ nk ⟩ that are smooth functions of k. Inserting
these in place of |ψ nk ⟩ on the right side of Eq. (3), the
WFs in the home unit cell R=0 are simply
|0n⟩ =
V
(2π) 3 ∫BZ
To find their centers of charge, we note that
r |0n⟩ =
V
(2π) 3 ∫BZ
dk | ˜ψ nk ⟩ . (86)
dk (−i∇ k e ik·r ) |ũ nk ⟩ . (87)
Performing an integration by parts and applying ⟨0n| on
the left, the center of charge is given by
r n = ⟨0n|r|0n⟩ =
V
(2π) 3 ∫BZ
dk ⟨ũ nk |i∇ k |ũ nk ⟩ , (88)
which is a special case of Eq. (23). Then, in the home
unit cell, in addition to the ionic charges +eZ τ located
at positions r τ , we can imagine electronic charges −e
located at positions r n . 13 Taking the dipole moment of
this imaginary cell and dividing by the cell volume, we
obtain, heuristically
P = e ( ∑
Z τ r τ − ∑ )
r n (89)
V
τ
n
for the polarization.
This argument can be put on somewhat firmer ground
by imagining a large but finite crystallite cut from the
insulator of interest, surrounded by vacuum. The crystallite
is divided into an “interior” bulk-like region and
a “skin” whose volume fraction vanishes in the thermodynamic
limit. The dipole moment is computed from
Eq. (84), but using LMOs ϕ j in place of the Hamiltonian
eigenfunctions ψ j on the right-hand side. Arguing
that the contribution of the skin to d is negligible in the
thermodynamic limit and that the interior LMOs become
bulk WFs, one can construct an argument that arrives
again at Eq. (89).
If these arguments still seem sketchy, Eq. (89) can be
rigorously justified by noting that its second term
P el = −
e ∑
∫
(2π) 3 dk ⟨ũ nk |i∇ k |ũ nk ⟩ , (90)
n
BZ
is precisely the expression for the electronic contribution
to the polarization in the Berry-phase theory (King-
Smith and Vanderbilt, 1993; Resta, 1994; Vanderbilt and
13 In these formulas, the sum over n includes a sum over spin. Alternatively
a factor of 2 can be inserted to account explicitly for
spin.