[Wannier] Band structure interpolation with "cut" mode
Deyu Lu
deyu.lu at gmail.com
Wed Jan 14 04:39:22 CET 2015
Dear wannier90 users and developers:
I'm relatively new to this code, and recently I've been using the
band structure interpolation function. My motivation is to understand how
the Hamiltonian in the Wannier basis ( H(R) ) decays as a function of
distance. For practical purpose, I run my
tests on bulk silicon in example 11 using occupied bands only and a 4x4x4
k-mesh. As bands_plot_mode = cut computes distance directly, I was testing
the "cut" mode, and using "sk" as references. As bands_plot_dim = 3 was not
allowed in version 1.2, but is enabled in version 2.0, all my calculations
were performed with a 2.0 linux intel ifort build.
I noticed several inconsistencies between "sk" and "cut", which I
describe below. Perhaps someone can help me to
clarify these points.
1. Degeneracy
Wigner-Seize (WS) grid points in "sk" are properly renormalized, satisfying
the sum rule: \sum_R 1/g(R)=64.
Grid points in the "cut" method do not satisfy the sum rule. This can be
seen for points with degeneracy in
"sk" greater than one. The renormalization factor for the points in the
"cut" mode is always one.
As a result, in "cut", a 729-point real space mesh was transformed to a
64-point k mesh, without proper weight.
2. WS grid points
Let us look at H11 (wannier centers i=j=1) to avoid the complexity from
translation_centre_frac
and shift_vec for different wannier centers.
In "sk", there are 93 WS grid points. The first 87 grid points have the same
coordinates and distances as in the "cut" mode. However, the last six are
different.
In "sk":
distance H11 R
10.795215 0.003258 -2 -2 2
10.795215 0.003258 -2 2 2
10.795215 0.003258 -2 2 -2
10.795215 0.003258 2 -2 2
10.795215 0.003258 2 -2 -2
10.795215 0.003258 2 2 -2
In "cut", the last six are:
distance H11 R
10.0980 0.003509 -1 -2 0
10.0980 0.003509 1 2 0
10.0980 0.003509 -1 3 0
10.0980 0.003509 1 -3 0
10.0980 0.003509 -2 -1 0
10.0980 0.003509 2 1 0
The discrepancy comes from the fact that these points in the "cut" mode are
double-counted. Because, e.g., the WS grid point of (-1, -2, 0) should be
(-1, 2, 0) with a smaller distance of 6.610692 angstrom, after being
translated by (0, 4, 0).
Finally, I wonder whether it is a better to stick to H(R) calculated for 93
WS points,
and apply distance cutoff directly to this matrix instead of creating the
9x9x9 H(R) matrix in
the "cut" mode. I did my own implementation, and the resulting interpolated
band structure
looks reasonable.
Best regards,
Deyu Lu
Associate Physicist, Theory & Computation Group
the Center for Functional Nanomaterials
Rm 1002, Building 735, Brookhaven National Lab
Upton, NY, 11973
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