Divergence for W(G=0,G'=0)

[juya]

Dear deverlopers,

I am currently using Yambo and I am writing to ask what approximations Yambo uses to treat the divergence of the coupling term H_cvk_c'v'k' when W(G=0,G'=0) and c=c' (or v=v') and k=k' ?
On the documentation webpage http://www.yambo-code.org/doc/docs/doc_BSS.php, it is written

It should be noted that some terms in the sums diverge if G, G prime and eventually q are equal to zero, this means that some tricks have to be used in order to cure these divergencies, altough it is beyond the scope of this introductory description of the Bethe-Salpeter to present these details, the interested reader can refer to Albrecth.

.
Can you explain how Yambo get around this divergence a bit more in detail ? Thanks in advance.

Jun Yan
CAMD, DTU
http://dcwww.camd.dtu.dk/~juya/

[andrea marini]

Dear Jun Yan,

as the interaction W appears always inside as summation on the transferred momenta Yambo integrates the 1/|q|^2 in on a small sphere centered in the origin. The volume of the sphere is equal to the volume of the RL unit cell divided by the number of transferred momenta. More elaborated integration tools are available but, more or less, they rely on the same philosophy.

Hope it helps

Cheers

Andrea

[juya]

Dear Andrea,

Thanks for the reply ! This integration method is very useful when you have a 3D periodic system. However, for surfaces or 1D systems, the volume of the RL unit cell will depend on the size of the vacuum, which can be chosen almost arbitrarily. Another problem is that the, fx, for surfaces, the surface Broullione zone is 2D unless you sample the non-periodic direction with kpoints. Is it still reasonable to perform an intergration in a small sphere around Gamma point ? Thanks in advance for replying.

Jun Yan
CAMD, DTU

[Daniele Varsano]

Dear Ju Yan,
in order to perform integrals in the Bz, in 1D or 2D systems (i.e. 1D or 2D k-point sampling),
it is implemented in Yambo the so-called Random INtegration Method, RIM, based on Monte Carlo
integrals. You can find details in the Yambo paper in Computer Physcs Communicataion.

Cheers,

Daniele

[juya]

Dear Daniele,

Thanks for the reply. I have another question: Yambo uses Coulomb cutoff for 0D-2D periodic systems. For example, for 0D case, the Coulomb kernel becomes : (4*pi/G**2)(1-cos(GR)) for G!=0 and 2*pi*R**2 for G=0. In the case of G=0 and optical limit (q-> 0), although the original Coulomb kernel diverges with 4*pi/q**2, the dipole transition matrix |<nk|e*{iq.r}|mk-q>|**2 scales with q**2, so the product of two will not diverge. However, if I apply a Coulomb cutoff, the G=0 component will become 2*pi*R**2, after its multiplied by the dipole transition matrix (scales with q**2), the product of two will become too small. so what does Yambo do with the Coulomb cutoff when G=0 and q=0 ? Thanks in advance for replying.

All the best
Jun Yan
CAMD, DTU

[Daniele Varsano]

Dear Jun Yuan,
as you state the spherical cut coulomb potential does not diverge for G=0 and q->>0.
Anyway note that in 0D system, you want to calculate polarizability and not macroscopic
dielctrical matrix. Then, the absorption will be proportional to
\alpha(omega) = lim_q->>0 -1/q**2\chi(\omega,q), here q is the small transferred momentum
and not the coulomb potential (of course in 3D periodic systems they do coincides),
and the limit to q-->0 is performed numerically, i.e. using a small q ~10^-5.
Again the product in the limit of q-->0 does not diverges as \chi ~ q^2.
The cutoff potential enters, when you calculate \chi including local field effects:
\chi = \chi^0+\chi^0 V \chi, here V cut is used.

Cheers,

Daniele

[juya]

Dear Daniele,

Ah, do you mean that, when calculate dielectric function with \epsilon**-1 = 1 + V \chi, you use the origianl V=1/q**2, but when calculate \chi = \chi_0 + \chi0 V \chi, you use the cutoff V ?

All the best,
Jun Yan

[Daniele Varsano]

Dear Jun,
in some sense is right, anyway the macroscopic epsilon does not make much sense for an isolated molecule.
The bigger the volume of the super cell, or if you want, the more the system is isolated, the epsilon tend to one, i.e.
the dielectric constant of the vacuum.
What it is well defined is the polarizability, which is calculated as -\chi/q^2, here the q is a small q,
which is equivalent in the long wavelength limit to the real space integral: \int z \chi(r,r') z' drdr (this is the diagonal z,z').

Cheers,

Daniele

'

[juya]

Dear Daniele,

You are right for the molecules. What I am concerning is also 1D and 2D periodic systems. Then it means that you use cutoff when calculate \chi, but dont use cutoff when calculate \epsilon**-1_GG = \delta_GG'+V_G \chi_GG'. I dont know whether this is reasonable ?

All the best,
Jun

[Daniele Varsano]

Dear Jun,
let me remind you to write your affiliation in the posts, you can do it easily
filling the signature in your profile. Yes, this is essentially what the code does,
I do not know either if this is the best way to treat 1D and 2D systems. Note that
1D and 2D cutoff are still diverging in the periodic direction (even if, not as 1/q^2).
Anyway, I think that it is done only for the G=G'=0 components, for the optics (long wavelength limit),
while in the construction of 'eps_G,G' (for the calculation of the screening in GW approx and
BSE Kernel) the cutoff potential is used.

Cheers,

Daniele