Random q-points.

[sitangshu]

Dear Sir,
While doing Electron-Phonon calculation in Yambo-py we go for random q-point generation unlike the phonon calculation done in yambo where we do sampling using Monkhorst-Pack mesh. I want to understand that why we need to take random q-points? Is it possible to do electron-phonon calculation with Monkhorst-Pack grid sampling??

Regards,
Himani Mishra

[miranda.henrique]

Dear Himani,

Both methods are ok to sample the Brillouin zone and you can choose the one that you think its better.
The calculations of the electron self-energy due to electron-phonon interaction require a good sampling of the Brillouin zone and careful convergence checks.
Using random q-grids has the advantage that you can increase the number of q-points included in your calculation by just calculating more q-points and giving them a new weight in the integral as 1/Nq (Nq is the number of q-points).
Also in random grids, you can give more importance to regions on the Brillouin zone that have a larger contribution to the integrals (this requires some kind of adaptative sampling).
With regular meshes normally you increase the number of k and q points until you are converged. This has a drawback that if you are not converged with for example 12x12x12 you will have to do for example a 16x16x16 which means you will be calculating repeated points.

The general consensus in the literature is to use random q-grids for the self-energy:
https://journals.aps.org/prl/abstract/1 ... 107.255501
https://journals.aps.org/prl/abstract/1 ... 105.265501
https://journals.aps.org/prb/abstract/1 ... .93.155435
https://www.sciencedirect.com/science/a ... via%3Dihub

Now for the convergence of the two cases, you can find some useful discussions here:
In the case of a random Q-grid the error should go as 1/sqrt(Nq)
(see https://en.wikipedia.org/wiki/Monte_Carlo_integration)
In the case of regular grids, the error should decrease as 1/[Nq^(2/d)] where d is the number of dimensions
(see https://math.stackexchange.com/question ... -integrals)
This would seem to indicate that regular grids are better for integrals in less than 4 dimensions.

But don't take my word for it.
If you reach some conclusion about this topic please share.

Cheers,
Henrique

[sdwang]

Using random q-grids has the advantage that you can increase the number of q-points included in your calculation by just calculating more q-points and giving them a new weight in the integral as 1/Nq (Nq is the number of q-points).
Henrique

Dear Henrique,
How do we understand this? Do you mean for the increased q-points, we can add some new points just behind the previous ones and give a another weight?
Let's take a example: in my calculation I used 80 random q-points and all the weights are 1.0, then I can add anther q-points after this list of q , like
another 30 points after the 80 ones, and give the weights as 1/110? If so, the ph. x calculation should use 'recover=true?

Thanks!

Shudong