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

[Erik.Schultheis at dlr.de]

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/de3035747f5d8f2ec9a67869827341ebb43f5b12/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/de3035747f5d8f2ec9a67869827341ebb43f5b12/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/>
> 
> 
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