[Wannier] Question on spread values and localization in Wannierization

[Nicola Marzari]

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