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.schulth...@dlr.de <mailto:erik.schulth...@dlr.de> | LinkedIn
<https://www.linkedin.com/in/erik-schultheis-930549243/>
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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
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Quantum ESPRESSO is supported by MaX (www.max-centre.eu)
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