GW band gap convergence vs k-mesh size

[sazadi]

Hi,

What's the best approach to check the convergence of GW band gap vs k-mesh size? In general, DFT gap converges faster than GW. So, having a converged DFT gap doesn't necessary guarantee the convergence of GW one. If one does GW calculations for few k-grid size, assuming that the number of k-points in each x,y,z direction increased by the same factor, does fitting approach and extrapolation to infinite k-gird size provide correct results ?

Best
Sam

[Daniele Varsano]

Dear Sam,
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As you say, convergence with k points can be very hard, mu suggestion is to check all the other convergences using a small k point grid as they are independent. Next increase the k point grid and extrapolate in case it is needed.

Please note that the RIM (Random Integration method), if you are dealing with a bulk, helps in accelerating convergences.

Best,
Daniele

[sazadi]

Dear Daniele,

Assuming that other parameter (cutoff, number of empty bands) are converged, does GW gap behave linearly as function of k-mesh size? There are two options for the parameters in a linear fitting and extrapolation; I. the number of k-point in one direction 1/nk_i (when the number of k-points in x,y,z direction equals) II. total number of k-point 1/((nk_i)^3). The quality of fit and systematic errors for extrapolation of direct and indirect gaps depend on choosing nk_i or nk_i^3. Is there any theory for this or just numerical issues? If I'm right, the quasiparticle energy of one state scales quadratically in the number of k-points.

Best
Sam
--
Sam Azadi
University College London

[Daniele Varsano]

Dear Sam,

I would say that there is no argument for that. Finally, this is an integration over a regular grid, and it is not even variational. Assuming you consider grids with the same ratio for each direction I would try to extrapolate as a function of the total number of kpoints as 1/nk in first approximation.

Best,

Daniele

[sazadi]

Dear Daniele,

Thanks for your reply. Do you know any reference where the behavior of GW (or G_0W_0) band gap as function of k-grid was systematically studied? Thanks.

Best
Sam

[Daniele Varsano]

Dear Sam,
many papers reports convergence study with respect the k point sampling, you should look to some work studying systems in the class of your system of interest (e.g. 2D/bulk semiconductor/metal...etc.)
For bulk systems you can have a look at this recent paper on assessment and reproducibility of GW calculation for solids.
Here is the arXive version.

Best,
Daniele

[sazadi]

Dear Daniele,

This is an interesting and helpful reference. Thanks indeed!

Your
Sam