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

[Paolo Giannozzi]

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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-- 
Paolo Giannozzi, Dip. Scienze Matematiche Informatiche e Fisiche,
Univ. Udine, via delle Scienze 206, 33100 Udine Italy, +39-0432-558216
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