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Dear Hung,
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<div class="">You need to think of the projection function g_{n}(r) as being localised in all space, and the integral A_{mn}^{k} = < \psi_{mk} | g_{n} > as being over all space:</div>
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<div class="">A_{mn}^{k} = \int_{all space} dr \psi_{mk}(r) g_{n}(r)</div>
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<div class="">Then use Bloch’s theorem to write \psi_{mk}(r) = u_{mk}(r) e^{ik.r}, where u_{mk}(r) is a function that has the periodicity of the unit cell and, therefore, can be expressed as a Fourier series u_{mk}(r) = \sum_{G} c_{mk,G} e^{iG.r}, where G is
a reciprocal lattice vector of the unit cell. Then you’ll see that you can write A_{mn}^{k} as a sum in G-space involving the Fourier coefficients of u_{mk}(r) (essentially the eigenvectors that come out of an electronic structure calculation) and the Fourier
transform of g_{n}(r) at specific k+G vectors. So A_{mn}^{k} can be evaluated quite easily on the reciprocal-space Fourier mesh. This is, I believe, the way it’s done in pw2wannier90 — Stefano de Gironcoli did the original implementation of that. </div>
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<div class="">I’m not sure about implementations in the interfaces to other electronic structure codes but, in principle, you could evaluate A_{mn}^{k} in real space too. You would need to use a supercell and your domain of integration needs to be large enough
for g_n(r) to have decayed sufficiently to converge the truncated real-space integral.</div>
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<div class="">Best wishes,</div>
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<div class="">Arash</div>
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<div class="">On 28 Sep 2018, at 18:55, Hung Pham <<a href="mailto:phamx494@umn.edu" class="">phamx494@umn.edu</a>> wrote:</div>
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<div dir="ltr" class="">Dear Wannier90 developers,
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<div class="">Firstly, I do apologize for my long email. Thank you for your patience.</div>
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<div class="">I have implemented the interface between Wannier90 and PySCF.</div>
<div class="">I have a question over Amn(k) matrix. I just recall the formula of Amn(k)</div>
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<div class="">* In my 1st implementation, | psi_m^k(r)> is represented by a uniform grid of the
<b class="">unit cell</b>. The gn(r) is similarly described in a similar way (only computed at the grid points in the unit cell). And <span class="gmail-Grammar gmail-gr_gramm gmail-gr_646 gmail-gr-alert gmail-gr_run_anim gmail-gr_inline_cards gmail-replaceWithoutSep gmail-gr_ gmail-only-ins" id="gmail-646" style="display:inline;border-bottom:2px solid transparent;background-repeat:no-repeat">Amn</span>(k)
is calculated in a numerical manner. Here, I assume that <span class="gmail-ins-del gmail-gr_283 gmail-gr-alert gmail-gr_spell gmail-gr_run_anim gmail-ContextualSpelling gmail-gr_inline_cards gmail-gr_" id="gmail-283" style="display:inline;border-bottom:2px solid transparent;background-repeat:no-repeat">gn</span>(r)
is not k-dependent.</div>
<div class="">There is a concern over this. If we put a p orbital, for example, as a guess at (0.0,0.0,0.0), then we will lose the partially information of the <span class="gmail-ins-del gmail-gr_283 gmail-gr-alert gmail-gr_spell gmail-gr_run_anim gmail-ContextualSpelling gmail-gr_inline_cards gmail-gr_" id="gmail-283" style="display:inline;border-bottom:2px solid transparent;background-repeat:no-repeat"><span class="gmail-gr_918 gmail-ins-del gmail-gr-progress gmail-gr-alert gmail-gr_spell gmail-gr_run_anim gmail-ContextualSpelling gmail-gr_inline_cards gmail-multiReplace gmail-gr_" id="gmail-918" style="display:inline;border-bottom:2px solid transparent;background-repeat:no-repeat;color:inherit;font-size:inherit">gn</span></span>(r),
i.e. the part outside the unit cell (only grid points in the unit cell is considered).</div>
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<div class="">* In my 2nd implementation, t<span class="gmail-ins-del gmail-gr_283 gmail-gr-alert gmail-gr_spell gmail-gr_run_anim gmail-ContextualSpelling gmail-gr_inline_cards gmail-gr_" id="gmail-283" style="display:inline;border-bottom:2px solid transparent;background-repeat:no-repeat">he
gn</span>(r) is represented by a uniform grid for the <b class="">supercell </b>that is consistent with the k-mesh used. Then <span class="gmail-ins-del gmail-gr_283 gmail-gr-alert gmail-gr_spell gmail-gr_run_anim gmail-ContextualSpelling gmail-gr_inline_cards gmail-gr_" id="gmail-283" style="display:inline;border-bottom:2px solid transparent;background-repeat:no-repeat"><span class="gmail-ins-del gmail-gr-progress gmail-gr_disable_anim_appear gmail-gr-alert gmail-gr_spell gmail-ContextualSpelling gmail-gr_inline_cards gmail-gr_ gmail-gr_2080" id="gmail-2080" style="display:inline;border-bottom:2px solid transparent;background-repeat:no-repeat;color:inherit;font-size:inherit">gn</span></span>(r)
is Fourier-transformed to the k-space representation <span class="gmail-ins-del gmail-gr_283 gmail-gr-alert gmail-gr_spell gmail-gr_run_anim gmail-ContextualSpelling gmail-gr_inline_cards gmail-gr_" id="gmail-283" style="display:inline;border-bottom:2px solid transparent;background-repeat:no-repeat"><span class="gmail-ins-del gmail-gr-progress gmail-gr_disable_anim_appear gmail-gr-alert gmail-gr_spell gmail-ContextualSpelling gmail-gr_inline_cards gmail-gr_ gmail-gr_2080" id="gmail-2080" style="display:inline;border-bottom:2px solid transparent;background-repeat:no-repeat;color:inherit;font-size:inherit">gn<b class="">^k</b></span></span>(r).
This implementation is backed up by the fact that one wants <span class="gmail-ins-del gmail-gr_283 gmail-gr-alert gmail-gr_spell gmail-gr_run_anim gmail-ContextualSpelling gmail-gr_inline_cards gmail-gr_" id="gmail-283" style="display:inline;border-bottom:2px solid transparent;background-repeat:no-repeat"><span class="gmail-ins-del gmail-gr-progress gmail-gr_disable_anim_appear gmail-gr-alert gmail-gr_spell gmail-ContextualSpelling gmail-gr_inline_cards gmail-gr_ gmail-gr_2080" id="gmail-2080" style="display:inline;border-bottom:2px solid transparent;background-repeat:no-repeat;color:inherit;font-size:inherit"><span class="gmail-ins-del gmail-gr-alert gmail-gr_spell gmail-gr_run_anim gmail-ContextualSpelling gmail-gr_inline_cards gmail-gr_2179 gmail-gr_ gmail-multiReplace" id="gmail-2179" style="display:inline;border-bottom:2px solid transparent;background-repeat:no-repeat;color:inherit;font-size:inherit">gn</span></span></span>(r),
as a guessed function, is close to the WFs (the Fourier-transformed functions of Block states).</div>
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<div class="">Personally, I think the 2nd scheme should be more reasonable.</div>
<div class="">However, in fact, the 1st scheme often results in more localized WFs (the img/real ratios are smaller).</div>
<div class="">Is there anything wrong with these both schemes?</div>
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<div class="">I would appreciate any comments, especially developer for QE_wannier90.</div>
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<div class="">Thank you,</div>
<div class="">Hung Pham</div>
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<div class="">Hung Q. Pham</div>
<div class="">Gagliardi Group</div>
<div class="">Office: Smith 101</div>
<div class="">Email: <a href="mailto:phamx494@umn.edu" target="_blank" class="">phamx494@umn.edu</a></div>
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<div dir="ltr" class=""><span style="font-size:12.8px" class="">Department of Chemistry</span><br class="">
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<div dir="ltr" class=""><span style="font-size:12.8000001907349px" class="">University of Minnesota - Twin Cities, </span><span style="font-size:12.8px" class="">Minneapolis, MN 55455</span></div>
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