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I am currently traveling internationally and may have limited access to e-mail. I will return your message when I return to the office on August 9.<br>
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On Aug 1, 2022, at 11:03 AM, Nicola Marzari via users <users@lists.quantum-espresso.org> wrote:<br>
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<div class="EmailQuote"><font color="black">Dear Emanuel,<br>
<br>
<br>
a few pointers here:<br>
<br>
- in the calculations, you have a real space unit cell A (whatever it <br>
is; it can be the primitive one, or a larger one); its corresponding <br>
Brillouin Zone B is sampled with a k1xk2xk3 monkhorst-pack mesh, with <br>
k1/k2/k3 integers. In principle, k1/k2/k3 are integers that should be <br>
very large/go to infinity; in practice they are finite, but large enough <br>
to make sure your calculations are converged.<br>
<br>
- the MLWFs live in a supercell S that is k1xk2xk3 times larger than the <br>
unit cell A of your calculation. The MLWFs are by definition periodic <br>
with the periodicity of this supercell S. If S is large enough (i.e. if <br>
the combination of your unit cell A and your k-sampling is fine enough - <br>
e.g. primitive with 8x8x8 sampling, for silicon, or - exactly <br>
equivalently - a unit cell A double the primitive one in each direction, <br>
and 4x4x4 sampling), the MLWFs will be able to decay to almost zero <br>
before they rise again (since, remember, they are in themselves periodic <br>
with the periodicity of S).<br>
<br>
- the MLWFs will have the localization that the physics of the system <br>
tells them to have (I'm assuming here we are dealing with MLWFs in an <br>
insulator, obtained transforming the valence bands). They will be <br>
localized, but typically do not got to zero from their maximum after a <br>
distanca as short as a half primitive cell. Much longer. But they do <br>
decay exponentially.<br>
<br>
- an interesting quantity to monitor is the matrix element of your <br>
operator of choice (e.g. the Hamiltonian) between the same MLWFs, but <br>
localized in different unit cells, or between different unit cells. Of <br>
course, once the unit cells get very far away, these matrix elements <br>
with rise again, because of the overall periodicity in S for the MLWFs<br>
<br>
Some of these issues were discussed in here<br>
<a href="https://urldefense.com/v3/__https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.95.076804__;!!PirnAJVEmx4jpg!_N-MxLl4U-I1ei0Q5lQ-jG654R9HiVG1S72bbZub81w4Zu4ewpNhN1ZaeLzGUCd8u18kslNYooIebJyzRzsNAAoYjXj88g$">https://urldefense.com/v3/__https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.95.076804__;!!PirnAJVEmx4jpg!_N-MxLl4U-I1ei0Q5lQ-jG654R9HiVG1S72bbZub81w4Zu4ewpNhN1ZaeLzGUCd8u18kslNYooIebJyzRzsNAAoYjXj88g$</a>
<br>
but are explained better in Young-Su Lee PhD thesis - see e.g. Fig 4.11 <br>
and 4.14 in <a href="https://urldefense.com/v3/__https://dspace.mit.edu/handle/1721.1/37371__;!!PirnAJVEmx4jpg!_N-MxLl4U-I1ei0Q5lQ-jG654R9HiVG1S72bbZub81w4Zu4ewpNhN1ZaeLzGUCd8u18kslNYooIebJyzRzsNAApy-jXIKw$">
https://urldefense.com/v3/__https://dspace.mit.edu/handle/1721.1/37371__;!!PirnAJVEmx4jpg!_N-MxLl4U-I1ei0Q5lQ-jG654R9HiVG1S72bbZub81w4Zu4ewpNhN1ZaeLzGUCd8u18kslNYooIebJyzRzsNAApy-jXIKw$</a>
<br>
<br>
Hope this clarifies some of your doubts - important to make sure you <br>
master the points above if you want to work on this topic. The Rev Mod <br>
Phys from 2012 might be also a good starting point, with its <br>
introductory sections:<br>
<a href="https://urldefense.com/v3/__https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.84.1419__;!!PirnAJVEmx4jpg!_N-MxLl4U-I1ei0Q5lQ-jG654R9HiVG1S72bbZub81w4Zu4ewpNhN1ZaeLzGUCd8u18kslNYooIebJyzRzsNAAqkNxgzUQ$">https://urldefense.com/v3/__https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.84.1419__;!!PirnAJVEmx4jpg!_N-MxLl4U-I1ei0Q5lQ-jG654R9HiVG1S72bbZub81w4Zu4ewpNhN1ZaeLzGUCd8u18kslNYooIebJyzRzsNAAqkNxgzUQ$</a>
<br>
<br>
<br>
nicola<br>
<br>
<br>
On 01/08/2022 15:19, EMANUEL ALBERTO MARTINEZ wrote:<br>
> Hello everyone,<br>
> <br>
> I have a simple question regarding to the localization of WFs. It seems <br>
> to be intuitive that an individual spread for each MLWF less than the <br>
> size of unit-cell ensures that it is localized well-within the <br>
> unit-cell. I need to justify some models I'm creating and I recently <br>
> realized that It could exist rare systems in which the MLWFs can surpass <br>
> the size of the unit-cell or even have another periodicity. These cases <br>
> will break my approximations. Could you give me any references about the <br>
> spread values and the localization within the unit-cell? I know It could <br>
> sound senseless, but I need to make it as clear as possible.<br>
> <br>
> Thanks in advance!<br>
> <br>
> -- <br>
> <br>
> Emanuel A. Martínez<br>
> <br>
> Departamento de Física de Materiales<br>
> <br>
> Universidad Complutense de Madrid<br>
> <br>
> <br>
> <br>
> _______________________________________________<br>
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<br>
-- <br>
----------------------------------------------------------------------<br>
Prof Nicola Marzari, Chair of Theory and Simulation of Materials, EPFL<br>
Director, National Centre for Competence in Research NCCR MARVEL, SNSF<br>
Head, Laboratory for Materials Simulations, Paul Scherrer Institut<br>
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The Quantum ESPRESSO community stands by the Ukrainian<br>
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country and on the free and peaceful scientific, cultural,<br>
and economic cooperation amongst peoples<br>
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