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<p class="MsoNormal">Hello everyone,<o:p></o:p></p>
<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoNormal"><span lang="EN-US">In <a href="https://github.com/QEF/q-e/blob/de3035747f5d8f2ec9a67869827341ebb43f5b12/upflib/vloc_mod.f90">
/upflib/vloc_mod.f90</a> the Fourier transform of the local pseudopotential V_loc is calculated. My question is about how one can derive the
<a href="https://github.com/QEF/q-e/blob/de3035747f5d8f2ec9a67869827341ebb43f5b12/upflib/vloc_mod.f90#L160">
G=0 term</a>.<o:p></o:p></span></p>
<p class="MsoNormal"><span lang="EN-US"><o:p> </o:p></span></p>
<p class="MsoNormal"><span lang="EN-US">Now I will describe how I understand the G=0 term and how this differs from what is implemented.<o:p></o:p></span></p>
<p class="MsoNormal"><span lang="EN-US"><o:p> </o:p></span></p>
<p class="MsoNormal"><span lang="EN-US">Since the local potential is long-ranged, which results in problems when performing the Fourier transformation, the long-range part is subtracted in real-space and added back in reciprocal space.<o:p></o:p></span></p>
<p class="MsoNormal"><span lang="EN-US">We then calculate the Fourier transform of [V_loc(r) + erf(r)/r] – erf(r)/r. The Fourier transform of the term in []-parentheses is the integral over (r V_loc(r)+erf(r)) sin(Gr)/G where we integrate r from 0 to infinity.
The G=0 case for this integral is no problem since the function is continuous in the G -> 0 limit, where sin(Gr)/G becomes r. This is implemented in
<a href="https://github.com/QEF/q-e/blob/de3035747f5d8f2ec9a67869827341ebb43f5b12/upflib/vloc_mod.f90#L133">
this loop</a>.<o:p></o:p></span></p>
<p class="MsoNormal"><span lang="EN-US">The Fourier transform of the remaining –Ze^2 erf(r)/r is implemented in
<a href="https://github.com/QEF/q-e/blob/de3035747f5d8f2ec9a67869827341ebb43f5b12/upflib/vloc_mod.f90#L296">
this loop</a>, which is<o:p></o:p></span></p>
<p class="MsoNormal"><span lang="EN-US">4 pi/V 1/G^2 e^(-G^2/4).<o:p></o:p></span></p>
<p class="MsoNormal"><span lang="EN-US">There the G -> 0 limit is explicitly excluded and should, in my opinion, be the G = 0 term calculated
<a href="https://github.com/QEF/q-e/blob/de3035747f5d8f2ec9a67869827341ebb43f5b12/upflib/vloc_mod.f90#L157">
here</a> but is not. The limit G -> 0 of the above term is (using the series expansion of the exponential)<o:p></o:p></span></p>
<p class="MsoNormal"><span lang="EN-US">lim G -> 0 (4 pi/V 1/G^2 – pi/V) which is divergent<o:p></o:p></span></p>
<p class="MsoNormal"><span lang="EN-US"><o:p> </o:p></span></p>
<p class="MsoNormal"><span lang="EN-US">But, the G = 0 term implemented is the integral over r^2 (V_loc(r)+1/r), also called the "alpha Z" energy term in the code documentation, where I do not understand where the 1/r term comes from and, if added here, where
it is subtracted again to not change the local potential. This suggests that something like [V_loc(r) + 1/r] – 1/r is used for the G=0 term but the subtracted -1/r term is never calculated.<o:p></o:p></span></p>
<p class="MsoNormal"><span lang="EN-US"><o:p> </o:p></span></p>
<p class="MsoNormal"><span lang="EN-US">I thought that this can be explained by 4 pi/V 1/G^2 from the above limit which is the Fourier transform of 1/r, but then the V_loc(r) term is missing. As you see, I am confused.<o:p></o:p></span></p>
<p class="MsoNormal"><span lang="EN-US"><o:p> </o:p></span></p>
<p class="MsoNormal"><span lang="EN-US">Further, I could not find any literature about calculating the Fourier transform of the local pseudopotential. The only reference I found that also mentions this "alpha Z" energy term is
<a href="https://journals.aps.org/prb/abstract/10.1103/PhysRevB.69.075101">Phys. Rev. B 69, 075101</a> in equation (12). Since they do not provide a motivation of this term besides that it is “the non-Coulomb part of the pseudopotential at q=0”, I cannot understand
where this term comes from.<o:p></o:p></span></p>
<p class="MsoNormal"><span lang="EN-US"><o:p> </o:p></span></p>
<p class="MsoNormal"><span lang="EN-US">Can anyone help me understand the origin of the G=0 term implemented in QuantumEspresso?<o:p></o:p></span></p>
<p class="MsoNormal"><span lang="EN-US"><o:p> </o:p></span></p>
<p class="MsoNormal"><span lang="EN-US">Best regards<o:p></o:p></span></p>
<p class="MsoNormal">Erik Schultheis<o:p></o:p></p>
<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoNormal" style="line-height:14.0pt"><span lang="EN-US" style="font-size:10.0pt;font-family:"Arial",sans-serif;color:dimgray;mso-fareast-language:DE">#CallMeByMyFirstName<o:p></o:p></span></p>
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<p class="MsoNormal" style="punctuation-wrap:simple;text-autospace:none"><b><span lang="EN-US" style="font-size:10.0pt;font-family:"Arial",sans-serif;color:dimgray;mso-fareast-language:DE">German Aerospace Center</span></b><span lang="EN-US" style="font-size:10.0pt;font-family:"Arial",sans-serif;color:dimgray;mso-fareast-language:DE">
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<o:p></o:p></span></p>
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<span lang="EN-US" style="font-size:8.5pt;font-family:"Arial",sans-serif;color:dimgray;mso-fareast-language:DE">Linder Höhe | 51147 Cologne<o:p></o:p></span></p>
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<p class="MsoNormal" style="punctuation-wrap:simple;text-autospace:none"><b><span lang="EN-US" style="font-size:10.0pt;font-family:"Arial",sans-serif;color:dimgray;mso-fareast-language:DE">Erik Schultheis M. Sc.</span></b><span lang="EN-US" style="font-size:10.0pt;font-family:"Arial",sans-serif;color:dimgray;mso-fareast-language:DE"><o:p></o:p></span></p>
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<span lang="EN-US" style="font-size:8.5pt;font-family:"Arial",sans-serif;color:dimgray;mso-fareast-language:DE">Metallic and Hybrid Materials<o:p></o:p></span></p>
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