## meaning of LongDrXd and macroscopic limit of epsilon

Deals with issues related to computation of optical spectra, in RPA (-o c) or by solving the Bethe-Salpeter equation (-o b). Includes local field effects, excitons, etc.
Christian Koenig
Posts: 30
Joined: Mon Nov 18, 2019 10:48 am

### meaning of LongDrXd and macroscopic limit of epsilon

Dear all,

For some reason I'm getting confused with what the q -> 0 limit means when calculating the macroscopic dielectric function.
Basically the question is if LongDrXd is the direction of propagation of the electric field (q/k) or its polarization.

The way I understood this problem was that the optical limit is q -> 0 (G = G' = 0) because the wave length is large and in that
case a direction of the electric field has to be specified somehow. However, the polarization would have to be specified for
any other G+q anyway as the wave vector alone doesn't define any polarization direction (being perpendicular to E).
This discussion:
www.yambo-code.org/forum/viewtopic.php? ... imit#p8370
would tell me LongDrXd is the direction of the electric field.

Now, on the other hand LongDrXd gets rescaled (absolute value << 1) which wouldn't be necessary if it was a direction only.
According to this:
http://www.yambo-code.org/wiki/index.ph ... cal_fields
LongDrXd is actually the q/k vector. What's the issue with q = 0 in this case and why would we not have to specify the polarization?

Best,

Christian
Christian Koenig
Tyndall National Institute
Cork, Ireland

Daniele Varsano
Posts: 2605
Joined: Tue Mar 17, 2009 2:23 pm
Contact:

### Re: meaning of LongDrXd and macroscopic limit of epsilon

Dear Christian,
I'm not sure I've understood your doubt. The polarization of an electromagnetic wave refers to the direction of the electric field, to not be confused with the propagation of the wave.

Now, on the other hand LongDrXd gets rescaled (absolute value << 1) which wouldn't be necessary if it was a direction only.
In the input file you specify a direction, then in the code a very small value of q is considered to perform the limit->0
According to this:
http://www.yambo-code.org/wiki/index.ph ... cal_fields
LongDrXd is actually the q/k vector. What's the issue with q = 0 in this case and why would we not have to specify the polarization?
Why do you say that here LongDrXd is the wave vector? it is the polarization.

I hope that now it is more clear,
Best,

Daniele
Dr. Daniele Varsano
S3-CNR Institute of Nanoscience and MaX Center, Italy
MaX - Materials design at the Exascale
http://www.nano.cnr.it
http://www.max-centre.eu/

Christian Koenig
Posts: 30
Joined: Mon Nov 18, 2019 10:48 am

### Re: meaning of LongDrXd and macroscopic limit of epsilon

Dear Daniele,

Thanks for your explanation. I was thinking along the lines of the dielectric response to an electromagnetic wave with some wave vector and a perpendicular electric field. From your answer I take it that q is in fact just a small Fourier component of the polarization/response function due to a perturbation and not the wave vector of an incoming wave. Therefore, q is parallel to the perturbation and it is clear that it has to be > 0 because the displacement of the charges has to be finite.
Including more iterations of Hedin's equation (Bethe Salpeter) I would have to use a similar parameter for the absorption spectrum which is BLongDir for the polarization of the applied electric field.
Does this make sense?

Best,

Christian
Christian Koenig
Tyndall National Institute
Cork, Ireland

Daniele Varsano
Posts: 2605
Joined: Tue Mar 17, 2009 2:23 pm
Contact:

### Re: meaning of LongDrXd and macroscopic limit of epsilon

Dear Christian,
yes, it is exactly as you say.
Once you have calculated the screening you can calculate the spectrum looking at your direction of interest. In case the system absorbs in anisotropic way you can calculate the spectrum for different polarizations, e.g. (100), (010), (001) or any combination of them by using the same screening as it should not change much provided that you calculated it for a relevant direction. For example of a 2D system in xy plane, you can calculate the screening along (1,1,0) and the absorption along (100) and (001).

Best,
Daniele
Dr. Daniele Varsano
S3-CNR Institute of Nanoscience and MaX Center, Italy
MaX - Materials design at the Exascale
http://www.nano.cnr.it
http://www.max-centre.eu/

haseebphysics1
Posts: 145
Joined: Sat Aug 17, 2019 2:48 pm

### Re: meaning of LongDrXd and macroscopic limit of epsilon

Dear Daniele,

1: I want to ask that when we compute the "Static screening" for the BSE calculations, then which component does it includes as far as dielectric function goes?
I mean, will only the electronic contribution (ε_∞) be included or the ionic contribution (ε_vib) only taken into account? I guess, the later case if phonons/non-clamped nuclei are allowed? Right?

2: And in total, the macroscopic static dielectric constant tensor should in principle include the sum of the electronic and vibrational part in the low field limit (photon energy =0). Right?

3: I found that in literature, the static dielectric constant and static electronic dielectric constant is somewhat confusing! I could not figure out when I take the static limit of the real part of the response function, should it be called static dielectric or static electronic dielectric?

4: And finally, what if the material is polar? In that case, lattice screening does indeed play its role, so can we include those effects in Yambo?

Thanks,
MS - Physics,
LUMS - Pakistan

Daniele Varsano
Posts: 2605
Joined: Tue Mar 17, 2009 2:23 pm
Contact:

### Re: meaning of LongDrXd and macroscopic limit of epsilon

Dear Haseeb,

Only the electronic contribution is taken into account. These electronic structure calculations are meant in the limit of zero kelvin.
Best,
Daniele
Dr. Daniele Varsano
S3-CNR Institute of Nanoscience and MaX Center, Italy
MaX - Materials design at the Exascale
http://www.nano.cnr.it
http://www.max-centre.eu/

haseebphysics1
Posts: 145
Joined: Sat Aug 17, 2019 2:48 pm

### Re: meaning of LongDrXd and macroscopic limit of epsilon

Dear Daniele,

I want to use the same model dielectric function for the lattice screening as we do for the electronic screening (https://journals.aps.org/prb/abstract/1 ... vB.47.9892)

I think the same above-mentioned function has been used in Yambo!

So, instead of using the dielectric constant at high frequency (eps_infinity), if we use the static dielectric constant (eps_zero), obtained by DFPT, in the same dielectric model formula, I can introduce the lattice screening effects...!

Can you suggest if this can be done easily with Yambo?

Thanks,
MS - Physics,
LUMS - Pakistan

Daniele Varsano
Posts: 2605
Joined: Tue Mar 17, 2009 2:23 pm
Contact:

### Re: meaning of LongDrXd and macroscopic limit of epsilon

Dear Haseeb,
I think the same above-mentioned function has been used in Yambo!
Unfortunately, I'm not aware of the implementation of this model, maybe others can give you more insight.

Best,
Daniele
Dr. Daniele Varsano
S3-CNR Institute of Nanoscience and MaX Center, Italy
MaX - Materials design at the Exascale
http://www.nano.cnr.it
http://www.max-centre.eu/

haseebphysics1
Posts: 145
Joined: Sat Aug 17, 2019 2:48 pm

### Re: meaning of LongDrXd and macroscopic limit of epsilon

Dear Daniele,

Alright, can you please refer to me how the Yambo computes the electronic screening? That might be helpful as well.

Regards,
MS - Physics,
LUMS - Pakistan

Daniele Varsano
Posts: 2605
Joined: Tue Mar 17, 2009 2:23 pm
Contact:

### Re: meaning of LongDrXd and macroscopic limit of epsilon

Dear Haseeb,
you can follow the 2009 yambo paper
Equations 6-9.

The steps are:
1) calculate non interacting response function Xo
2) calculate reducible response function X solving a Dyson equation X=X0+X0vX
3) evaluate the screening: eps^-1(q)_gg'=delta_gg'+v(q+g)X_(q)gg'

In the plasmon pole approximation, the frequency dependence can be modeled by a single pole function (which requires the calculation at two frequencies). The macroscopic dielectric function is calculated as 1/eps^-1_00

Best,
Daniele
Dr. Daniele Varsano
S3-CNR Institute of Nanoscience and MaX Center, Italy
MaX - Materials design at the Exascale
http://www.nano.cnr.it
http://www.max-centre.eu/