[QE-users] [SPAM] AW: AW: Fourier Transform of Local Pseudopotential and G=0 limit

Mon Sep 9 14:49:26 CEST 2024

Hi Paolo,
Hi everyone,

I have a follow-up question about the G=0 limit of the local pseudopotential.
You wrote:
> 2. The local potential V(r) behaves as -Ze^2/r for large r, making direct computation of V(G) problematic. One removes the long-range behavior by adding to V(r) a function f(r)=Ze^2 erf(r)/r in real space; performs the Fourier transform; subtracts out f(G)=4\pi Z e^2 exp(-G^2)/\Omega G^2 or something like that from V(G). All this applies to G!=0.

In the Quantum ESPRESSO source code in vloc_mod.f90 (line 140: https://github.com/QEF/q-e/blob/de3035747f5d8f2ec9a67869827341ebb43f5b12/upflib/vloc_mod.f90#L140) the G=0 term of V(r) + Z erf(r)/r. But, Paolo, you wrote that the addition of Z erf(r)/r only applies to G!=0 since the G=0 term is treated separately as we already discussed. This makes sense to me and also results in the fact that the G=0 limit of the Fourier transform of - Z erf(r)/r (4\pi Z e^2 exp(-G^2)/\Omega G^2 or something like that) is never treated/calculated. But, similarly, why would one calculate the G=0 limit of V(r) + Z erf(r)/r?

This comment in the source code

!! q = 0 case, continuous for q -> 0
!! NOT THE SAME AS THE G=0 TERM, computed below

Also explicitly differentiates between q and G but both are momentum vectors and q=0 should implicate G=0.

Best regards
Erik

-----Ursprüngliche Nachricht-----
Von: Paolo Giannozzi <paolo.giannozzi at uniud.it> 
Gesendet: Freitag, 23. August 2024 12:36
An: Schultheis, Erik <Erik.Schultheis at dlr.de>
Cc: users at lists.quantum-espresso.org
Betreff: Re: AW: [QE-users] [SPAM] Fourier Transform of Local Pseudopotential and G=0 limit

Yes, exactly.

Paolo

On 23/08/2024 11:47, Erik.Schultheis at dlr.de wrote:
> Hi Paolo,
> 
> Thanks for your quick answer.
> 
> Point 2 is clear to me since it only applies to G!=0.
> After thinking what you wrote in Point 1 and finding further information in Warren E. Picketts " Pseudopotential Methods in Condensed Matter Applications" (https://www.sciencedirect.com/science/article/pii/0167797789900026), especially around equation (5.20), I now have an idea what might happen here.
> 
> You are saying that the local potential V_loc plus the Hartree potential yield the "alpha Z" term for G=0. In the work of Pickett he writes that the formally divergent terms in the ion-ion interaction, the coulomb interaction of the electrons, and the local pseudopotential combine to give the finite Ewald energy (which is clear to me) and the "alpha Z" term. From this, I guess that the electronic background, i.e. the coulomb interaction of the electrons, cancels the -Z/r behavior of the local pseudopotential V_loc. But since V_loc is not exactly -Z/r only its -Z/r part is canceled and the remaining (non-Coulomb) part is still to be calculated in the G=0 limit. Let's write V_loc(r) = V_loc(r) - (-Z/r) + (-Z/r) = (V_loc(r) + Z/r) - Z/r. Then the first term is the non-Coulomb part of V_loc and the second term (-Z/r) is the Coulomb part of V_loc. The second term is cancelled by the electronic background (Hartree potential) while the first term is not. So the G=0 limit of (V_loc(r) + Z/r), i.e. the "alpha Z" term, has to be calculated to get the correct G=0 limit of V_loc.
> 
> Is this the correct reasoning behind the "alpha Z" term?
> 
> Best regards
> Erik
> 
> -----Ursprüngliche Nachricht-----
> Von: Paolo Giannozzi <paolo.giannozzi at uniud.it>
> Gesendet: Donnerstag, 22. August 2024 18:35
> An: Schultheis, Erik <Erik.Schultheis at dlr.de>
> Cc: Quantum ESPRESSO users Forum <users at lists.quantum-espresso.org>
> Betreff: Re: [QE-users] [SPAM] Fourier Transform of Local 
> Pseudopotential and G=0 limit
> 
> You are mixing up two different aspects:
> 1. V(G=0) for the local+Hartree potential is not divergent and yields the "alpha Z" term. Of course, one sets V_H(G=0)=0.
> 2. The local potential V(r) behaves as -Ze^2/r for large r, making direct computation of V(G) problematic. One removes the long-range behavior by adding to V(r) a function f(r)=Ze^2 erf(r)/r in real space; performs the Fourier transform; subtracts out f(G)=4\pi Z e^2 exp(-G^2)/\Omega G^2 or something like that from V(G). All this applies to G!=0.
> 
> Paolo
> 
> On 20/08/2024 11:54, Erik Schultheis via users wrote:
>> Hello everyone,
>>
>> In /upflib/vloc_mod.f90
>> <https://github.com/QEF/q-e/blob/de3035747f5d8f2ec9a67869827341ebb43f5b12/upflib/vloc_mod.f90> the Fourier transform of the local pseudopotential V_loc is calculated. My question is about how one can derive the G=0 term <https://github.com/QEF/q-e/blob/de3035747f5d8f2ec9a67869827341ebb43f5b12/upflib/vloc_mod.f90#L160>.
>>
>> Now I will describe how I understand the G=0 term and how this 
>> differs from what is implemented.
>>
>> 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.
>>
>> 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 this loop
>> <https://github.com/QEF/q-e/blob/de3035747f5d8f2ec9a67869827341ebb43f5b12/upflib/vloc_mod.f90#L133>.
>>
>> The Fourier transform of the remaining –Ze^2 erf(r)/r is implemented 
>> in this loop 
>> <https://github.com/QEF/q-e/blob/de3035747f5d8f2ec9a67869827341ebb43f
>> 5b12/upflib/vloc_mod.f90#L296>, which is
>>
>> 4 pi/V 1/G^2 e^(-G^2/4).
>>
>> There the G -> 0 limit is explicitly excluded and should, in my 
>> opinion, be the G = 0 term calculated here 
>> <https://github.com/QEF/q-e/blob/de3035747f5d8f2ec9a67869827341ebb43f
>> 5b12/upflib/vloc_mod.f90#L157> but is not. The limit G -> 0 of the 
>> above term is (using the series expansion of the exponential)
>>
>> lim G -> 0 (4 pi/V 1/G^2 – pi/V) which is divergent
>>
>> 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.
>>
>> 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.
>>
>> 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 Phys. Rev. B 
>> 69, 075101 
>> <https://journals.aps.org/prb/abstract/10.1103/PhysRevB.69.075101> 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.
>>
>> Can anyone help me understand the origin of the G=0 term implemented 
>> in QuantumEspresso?
>>
>> Best regards
>>
>> Erik Schultheis
>>
>> #CallMeByMyFirstName
>>
>> **
>>
>> *German Aerospace Center*(DLR)
>>
>> Institute of Materials Research
>>
>> Linder Höhe | 51147 Cologne
>>
>> *Erik Schultheis M. Sc.*
>>
>> Metallic and Hybrid Materials
>>
>> Telephone: +49 (0) 2203 601 1311
>>
>> erik.schultheis at dlr.de <mailto:erik.schultheis at dlr.de> | LinkedIn 
>> <https://www.linkedin.com/in/erik-schultheis-930549243/>
>>
>>
>> _______________________________________________
>> The Quantum ESPRESSO community stands by the Ukrainian people and 
>> expresses its concerns about the devastating effects that the Russian 
>> military offensive has on their country and on the free and peaceful 
>> scientific, cultural, and economic cooperation amongst peoples 
>> _______________________________________________
>> Quantum ESPRESSO is supported by MaX (www.max-centre.eu) users 
>> mailing list users at lists.quantum-espresso.org 
>> https://lists.quantum-espresso.org/mailman/listinfo/users
> 

--
Paolo Giannozzi, Dip. Scienze Matematiche Informatiche e Fisiche, Univ. Udine, via delle Scienze 206, 33100 Udine Italy, +39-0432-558216 X SCUOLA ENERGIE RINNOVABILI 16-21/9/2024 - PRE-ISCRIZIONE https://agenda.infn.it/event/39573/