Maximally localized Wannier functions: Theory and applications
Maximally localized Wannier functions: Theory and applications
Maximally localized Wannier functions: Theory and applications
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13<br />
of the states in this larger manifold. Once a suitable J-<br />
dimensional Bloch manifold has been identified at each<br />
k, the same procedure described earlier for an isolated<br />
group of b<strong>and</strong>s can be used to generate <strong>localized</strong> WFs<br />
spanning that manifold.<br />
The problem of computing well <strong>localized</strong> WFs starting<br />
from entangled b<strong>and</strong>s is thus broken down into two distinct<br />
steps, subspace selection <strong>and</strong> gauge selection. As we<br />
will see, the same guiding principle can be used for both<br />
steps, namely, to achieve “smoothness” in k-space. In the<br />
subspace selection step a J-dimensional Bloch manifold<br />
which varies smoothly as function of k is constructed.<br />
In the gauge-selection step that subspace is represented<br />
using a set of J Bloch <strong>functions</strong> which are themselves<br />
smooth <strong>functions</strong> of k, such that the corresponding WFs<br />
are well <strong>localized</strong>.<br />
1. Subspace selection via projection<br />
The projection technique discussed in Section II.B<br />
can be easily adapted to produce J smoothly-varying<br />
Bloch-like states starting from a larger set of Bloch<br />
b<strong>and</strong>s (Souza et al., 2001). The latter can be chosen, for<br />
example, as the b<strong>and</strong>s lying within a given energy window,<br />
or within a specified range of b<strong>and</strong> indices. Their<br />
number J k ≥ J is not required to be constant throughout<br />
the BZ.<br />
We start from a set of J <strong>localized</strong> trial orbitals g n (r)<br />
<strong>and</strong> project each of them onto the space spanned by the<br />
chosen eigenstates at each k,<br />
|ϕ nk ⟩ =<br />
∑J k<br />
m=1<br />
|ψ mk ⟩ ⟨ψ mk |g n ⟩ . (46)<br />
This is identical to Eq. (16), except for the fact that the<br />
overlap matrix (A k ) mn = ⟨ψ mk |g n ⟩ has become rectangular<br />
with dimensions J k × J. We then orthonormalize<br />
the resulting J orbitals using Eq. (17), to produce a set<br />
of J smoothly-varying Bloch-like states across the BZ,<br />
| ˜ψ nk ⟩ =<br />
J∑<br />
m=1<br />
|ϕ mk ⟩(S −1/2<br />
k<br />
) mn . (47)<br />
As in Eq. (17), (S k ) mn = ⟨ϕ mk |ϕ nk ⟩ V = (A † k A k ) mn, but<br />
with rectangular A k matrices.<br />
The above procedure achieves simultaneously the two<br />
goals of subspace selection <strong>and</strong> gauge selection, although<br />
neither of them is performed optimally. The gauge selection<br />
can be further refined by minimizing ˜Ω within<br />
the projected subspace. It is also possible to refine iteratively<br />
the subspace selection itself, as will be described<br />
in the next section. However, for many <strong>applications</strong> this<br />
one-shot procedure is perfectly adequate, <strong>and</strong> in some<br />
cases it may even be preferable to more sophisticated iterative<br />
approaches (see also Sec. III.C.) For example, it<br />
Energy (eV)<br />
15<br />
10<br />
5<br />
0<br />
-5<br />
-10<br />
L Γ X K Γ<br />
FIG. 5 (Color online) B<strong>and</strong> structure of bulk crystalline Si.<br />
Solid lines: original b<strong>and</strong>s generated directly from a densityfunctional<br />
theory calculation. (Blue) triangles: <strong>Wannier</strong>interpolated<br />
b<strong>and</strong>s obtained from the subspace selected by<br />
an unconstrained projection onto atomic sp 3 orbitals. (Red)<br />
circles: <strong>Wannier</strong>-interpolated b<strong>and</strong>s obtained with the same<br />
procedure <strong>and</strong> the additional constraint of reproducing exactly<br />
the original valence manifold <strong>and</strong> parts of the conduction<br />
manifold, using a frozen energy window (see text).<br />
often results in “symmetry-adapted” WFs which inherit<br />
the symmetry properties of the trial orbitals (Ku et al.,<br />
2002).<br />
As an example, we plot in Fig. 5 the eight disentangled<br />
b<strong>and</strong>s obtained by projecting the b<strong>and</strong> structure of silicon,<br />
taken within an energy window that coincides with<br />
the entire energy axis shown, onto eight atomic-like sp 3<br />
hybrids. The disentangled b<strong>and</strong>s, generated using <strong>Wannier</strong><br />
interpolation (Sec. VI.A), are shown as blue triangles,<br />
along with the original first-principles b<strong>and</strong>s (solid<br />
lines). While the overall agreement is quite good, significant<br />
deviations can be seen wherever higher unoccupied<br />
<strong>and</strong> unwanted states possessing some significant sp 3<br />
character are admixed into the projected manifold. This<br />
behavior can be avoided by forcing certain Bloch states<br />
to be preserved identically in the projected manifold - we<br />
refer to those as belonging to a frozen “inner” window,<br />
since this is often the simplest procedure for selecting<br />
them. The placement <strong>and</strong> range of this frozen window<br />
will depend on the problem at h<strong>and</strong>. For example, in order<br />
to describe the low-energy physics for, e.g., transport<br />
calculations, the frozen window would typically include<br />
all states in a desired range around the Fermi level.<br />
We show as red circles in Fig. 5 the results obtained by<br />
forcing the entire valence manifold to be preserved, leading<br />
to a set of eight projected b<strong>and</strong>s that reproduce exactly<br />
the four valence b<strong>and</strong>s, <strong>and</strong> follow quite closely the<br />
four low-lying conduction b<strong>and</strong>s. For the modifications<br />
to the projection algorithm required to enforce a frozen<br />
window, we refer to Sec. III.G of Souza et al. (2001).<br />
Projection techniques can work very well, <strong>and</strong> an ap-<br />
Frozen Window