Dear Ayush,
there might be two issues at play here.
The first one is, say, more trivial; the spread of the Wannier functions
is estimated with a finite-differences representation of the Laplacian
(eq 32 in the 1997 paper), and this finite-difference representation
converges slowly with the k-point sampling. So, as you increase the
mesh, your estimate of the spread becomes (slowly) better and better,
but the Wannier fucntions per se had already been converged well (they
are obtained by minimizing Omega_tilde, that contains only matrix
elements of the position operator, not its square, and the matrix
elements of the position operator converge much faster with k-point
sampling.
Re the decay of the real-space Hamiltonian, note first that the valence
Wannier functions (in principle exponentially localized in a gapped
system) remain periodic if your k-point sampling is finite - in your
case, if you move by 12 (or 24) primitive cells, you find the Wannier
function repeated. So, a small sampling would make the WF plateau and
then increase again sooner.
Not sure what your case is - if you were to plot them along an axis, I
would expect the 12-sampled and the 24-sampled to be almost identical
around the center, and the 12-sampled to plateau earlier. But are these
Wannier functions for the valence only? Or disentangled? If the latter,
the denser sampling might capture better the fact that your manifold is
not isolated, and would explain possibly your results.
nicola
On 11/06/2026 10:57, Ayush Gaurav wrote:
Dear Wannier90 Team,
I am performing Wannierization for a 2D semiconductor using two
different k-point meshes. For a 12×12×1 mesh, the total spread converges
to about 6 Ų, whereas for a denser 24×24×1 mesh, it increases to about
10–12 Ų. In both cases, the Wannier-interpolated bands reproduce the
DFT bands very accurately.
I also observe that the real-space Hamiltonian decays much faster for
the 12×12×1 mesh (about 4 orders of magnitude by ~30 Å), while for the
24×24×1 mesh significant terms persist up to ~80 Å.
Since a denser k-mesh is generally expected to provide a more accurate
representation, I am puzzled by the increase in spread and the slower
Hamiltonian decay. *Does a larger spread necessarily imply less
localized Wannier functions, or are there better metrics to assess
localization and Wannierization quality in this situation?*
Thank you for your guidance.
Best regards,
Ayush Gaurav
Master's Student, Materials Engineering
Indian Institute of Technology Kharagpur
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Prof Nicola Marzari, Chair of Theory and Simulation of Materials, EPFL
Director, National Centre for Competence in Research NCCR MARVEL, SNSF
Laboratory Head, PSI Center for Scientific Computing, Theory, and Data
Contact info and websites: https://theos-wiki.epfl.ch/en/Main/Contact
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