[QE-users] WFs confined within the unit-cell

[Nicola Marzari]

Dear Emanuel,


a few pointers here:

- in the calculations, you have a real space unit cell A (whatever it 
is; it can be the primitive one, or a larger one); its corresponding 
Brillouin Zone B is sampled with a k1xk2xk3 monkhorst-pack mesh, with 
k1/k2/k3 integers. In principle, k1/k2/k3 are integers that should be 
very large/go to infinity; in practice they are finite, but large enough 
to make sure your calculations are converged.

- the MLWFs live in a supercell S that is k1xk2xk3 times larger than the 
unit cell A of your calculation. The MLWFs are by definition periodic 
with the periodicity of this supercell S. If S is large enough (i.e. if 
the combination of your unit cell A and your k-sampling is fine enough - 
e.g. primitive with 8x8x8 sampling, for silicon, or - exactly 
equivalently - a unit cell A double the primitive one in each direction, 
and 4x4x4 sampling), the MLWFs will be able to decay to almost zero 
before they rise again (since, remember, they are in themselves periodic 
with the periodicity of S).

- the MLWFs will have the localization that the physics of the system 
tells them to have (I'm assuming here we are dealing with MLWFs in an 
insulator, obtained transforming the valence bands). They will be 
localized, but typically do not got to zero from their maximum after a 
distanca as short as a half primitive cell. Much longer. But they do 
decay exponentially.

- an interesting quantity to monitor is the matrix element of your 
operator of choice (e.g. the Hamiltonian) between the same MLWFs, but 
localized in different unit cells, or between different unit cells. Of 
course, once the unit cells get very far away, these matrix elements 
with rise again, because of the overall periodicity in S for the MLWFs

Some of these issues were discussed in here
https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.95.076804
but are explained better in Young-Su Lee PhD thesis - see e.g. Fig 4.11 
and 4.14 in https://dspace.mit.edu/handle/1721.1/37371

Hope this clarifies some of your doubts - important to make sure you 
master the points above if you want to work on this topic. The Rev Mod 
Phys from 2012 might be also a good starting point, with its 
introductory sections:
https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.84.1419


				nicola


On 01/08/2022 15:19, EMANUEL ALBERTO MARTINEZ wrote:
> Hello everyone,
> 
> I have a simple question regarding to the localization of WFs. It seems 
> to be intuitive that an individual spread for each MLWF less than the 
> size of unit-cell ensures that it is localized well-within the 
> unit-cell. I need to justify some models I'm creating and I recently 
> realized that It could exist rare systems in which the MLWFs can surpass 
> the size of the unit-cell or even have another periodicity. These cases 
> will break my approximations. Could you give me any references about the 
> spread values and the localization within the unit-cell? I know It could 
> sound senseless, but I need to make it as clear as possible.
> 
> Thanks in advance!
> 
> -- 
> 
> Emanuel A. Martínez
> 
> Departamento de Física de Materiales
> 
> Universidad Complutense de Madrid
> 
> 
> 
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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
Head, Laboratory for Materials Simulations, Paul Scherrer Institut
Contact info and websites at http://theossrv1.epfl.ch/Main/Contact