Coulomb truncation and convergence in a CNT

Please post here any issue related to convergence parameters in linear response GW and BSE calculations.

Moderators: Davide Sangalli, Daniele Varsano

Post Reply
User avatar
wachr
Posts: 32
Joined: Wed Sep 24, 2014 4:43 pm

Coulomb truncation and convergence in a CNT

Post by wachr » Mon Oct 17, 2016 4:06 pm

Dear yambo developers,
dear yambo community,

I often see that information on converging calculations of low-dimensional systems using yambo is scattered in this forum. As this meant a lot of trouble for me, I would like to share about my personal experience. This were strange convergence issues in low-dimensional systems in yambo 3.4.1 and 3.4.2 (latest release).

In this post, I would like to adress the convergence for the self-energy corrections in the GW scheme.

We converged the band gap of the (8,0)-CNT using a (1x1xN)-kgrid. The litarature value is 1.75 eV (Appl. Phys. A 78, 1129). As the system is quasi one-dimensional, Coulomb truncation has to be applied in order to converge the supercell. The random integration method (RIM) has to be activated in order to obtain meaningful results; the k-grid is (1x1x40). We note that we will never achieve supercell convergence:
GW_conv_cell_std.png
If one does not activate RIM, the picture looks a bit strange:
GW_conv_cell_std_awkward.png
The reason is that the GW self energy does not converge without RIM for an increasingly dense k-mesh and the resulting value of the gap has no meaning.

Now, we need to get rid of the convergence issue for the unit cell. We apply a Coulomb truncation using a cylinder of half the cell size. And the results that we find are somewhat discouraging:
GW_conv_cell_trunc_awkward.png
[to be continued]
You do not have the required permissions to view the files attached to this post.
Christian Wagner
Institute of Physics
Chemnitz University of Technology, Germany

User avatar
wachr
Posts: 32
Joined: Wed Sep 24, 2014 4:43 pm

Re: Coulomb truncation and convergence in a CNT

Post by wachr » Mon Oct 17, 2016 4:50 pm

[continued post]

In the reference above, it was mentioned that maybe some more k-points are required for convergence (64 kpts were used). Therefore, we converge the k-points:
GW_conv_k_awkward.png
We see several issues: In the case of "GW" (RIM and truncation deactivated), we see no convergence wrt. #kpts due to the self energy issue. In the case of RIM activated, the gap converges - but due to the insufficient supercell, the gap is by far too low. And for the truncated case, we see that the gap does not converge entirley but wiggles somwhere in the range between 1.65 and 1.75 eV. Something else seems to be underconverged.

We plotted the head element of the dielectric matrix vs. q (not shown here) and found that the only difference are sampling artefacts. Therefore, there was no problem with the Coulomb truncated, dielectric function. The reason must lie somewhere in the sum of GW. We therefore took a look at the number of G-vectors in the exchange sum (20'000), which were converged in the non-truncated case. However, the truncation seems to have an effect on this convergence:
GW_conv_NG.png
The 20'000 case does not look too bad, but also qualitatively different than the case with more G-vectors. Therefore, in the following, we use 100'000 number of G-vectors in the exchange sum to be sure (as this is computationally relatively cheap):
GW_conv_k_trunc.png
This finally leads to a convergence with respect to the number of k-points. It is expected (but not yet calculated) that the unit cell convergence makes much more sense, now. And the convergence with respect to the number of bands (not shown here) must be a lot smoother.

I hope that sharing this experience with other yambo users helps to resolve convergence issues in calculations for low-dimenstional systems.
You do not have the required permissions to view the files attached to this post.
Christian Wagner
Institute of Physics
Chemnitz University of Technology, Germany

User avatar
wachr
Posts: 32
Joined: Wed Sep 24, 2014 4:43 pm

Re: Coulomb truncation and convergence in a CNT

Post by wachr » Mon Oct 17, 2016 4:58 pm

Therefore, I propose a strategy to insure convergence in these difficult cases:

Especially for new users, I recommend the following convergence scheme for the calculation of low-dimensional systems - this is an extension to the useful guide by Claudio Attacelite.

GENERALLY: If you plot convergence plots, always display over 1/X, where X is the quantity to be converged. With this, you quickly find out if the calculation converges.
  1. Use a unit cell, where 99% of the electron density is within the truncation range ("99% rule", [pptx]).
    • First, throughly converge the DFT calculations (especially the cell size in the non-periodic directions)
    • Use ypp -e d to print the electron density for the above unit cell
    • Look at the output, e.g. with VESTA, and plot the extension of the electron density at the 99% limit. Take the unit cell twice as large.
    • Recalulate the DFT in the new unit cell (with a sufficient number of bands)
  2. Converge the untruncated calculation as proposed in here for this unit cell [May require some DFT calculations]
  3. Apply a sherical / cylindrical / box cutoff for Coulomb integral at half the cell size. [Use the converged DFT calculation from 1)]
  4. Now, re-converge the resulting gap with different numbers of EXXRLvcs (and maybe NGsBLkXd for the screening; preferably converge this at very last) [Use the converged DFT calculation from 1)]
  5. Converge the cell size, again: Try some smaller values for the cell extension in order to see if the 99%-rule is already converged or even overconverged. A fully converged kpoint-set is not yet required. [A few more DFT calculations of different cells are required]
    • NEW TIP! Instead of converging the cell size, one could also reduce the size of the truncation to mimic a smaller unit cell. This makes it possible to use the converged DFT calculation from 1) and prevents more DFT calculations.
  6. Now, converge the k-points again (!). It is known that the truncation requires a larger number of k-points (about 1.5x-2x the untruncated number of kpts) [A few more DFT calculations are required]
  7. Now, converge the number of bands [hopefully, the DFT calculation from 1) contained sufficient amounts of bands. If so, just one DFT calculation is required. If not, a nscf-calculation for more bands is mandatory.]. The convergence can be improved by activating the flag GTermKind= "BRS" in the input file.
    • NEW TIP! Leave the number of bands for the calculation of the dielectric function constant and use the same database (yambo -F input_{bands} -J same_database). Then, just the GW part is recalculated each time and the expensive screnning is just calculated once.
  8. For those who want to run GW self-consistently:
    • The calculation of GW0 eigenvalues (iteration over G, only) will come much closer to the correct result than full self-consistent GW for low-dimensional systems (due to large excitonic effects!).
    • Theoretically, one could take the "exact" screening for reliable full-GW calculations into account by running G0W0 -> G1 -> BSE (with full frequency- and q-dependence (hardly implemented in any code; also the static limit for BSE would have to be dropped)) -> W1 -> G2 -> ...
      This would be the most accurate way to iteratively solve the Hedin's equations more exactly than by GW, only. However, such calculations have not yet been reported in literature and are still out of reach. Therefore, stick to a) - it's the best one can do at the moment.
  9. FINISH!
If one now calculates a "derived setup" (such as applying strain, consider an absorped molecule, ...), the convergence from above should be good. Still: think about possibly larger extensions of "derived setups" before convergence. E.g. if you add another molecule, think about its electron density extension. This affects the very first step to get a converged cell (!). The benefit will be: the same set of parameters is used for all the calculations of the (derived) setup- and differences between the setups do not suffer further numerical issues.
Last edited by wachr on Tue Oct 18, 2016 11:24 am, edited 1 time in total.
Christian Wagner
Institute of Physics
Chemnitz University of Technology, Germany

User avatar
Daniele Varsano
Posts: 2992
Joined: Tue Mar 17, 2009 2:23 pm
Contact:

Re: Coulomb truncation and convergence in a CNT

Post by Daniele Varsano » Mon Oct 17, 2016 4:59 pm

Dear Christian,

Many thanks for sharing your exciting experience in converging low dimensional system!!
Seriously, Thanks a lot for your contribution.

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/

User avatar
wachr
Posts: 32
Joined: Wed Sep 24, 2014 4:43 pm

Re: Coulomb truncation and convergence in a CNT

Post by wachr » Mon Oct 17, 2016 5:16 pm

Thanks a lot, Daniele! It was a lot of work and I hope that someone else has less work and headache on converging / understanding the results in a low-D- system!

I think that the community (and also I) am happy to receive feedback, reports on similar experiences or even different experience.

I will soon report on the BSE for the same system - as we faced some surprises, too. But not too soon.
Christian Wagner
Institute of Physics
Chemnitz University of Technology, Germany

luca.montana
Posts: 27
Joined: Fri Jun 13, 2014 6:52 pm

Re: Coulomb truncation and convergence in a CNT

Post by luca.montana » Mon Oct 17, 2016 5:28 pm

Dear Christian,
"The calculation of GW0 eigenvalues (iteration over G, only) will come much closer to the correct result than full self-consistent GW for low-dimensional systems (due to large excitonic effects!). "
In my opinion this is related to the fact that W in full self consistent GW gets underscreened upon self consistency (due to artificial transfer of spectral weight to the incoherent part of the spectral function),
leading to overestimated band gaps. However, inclusion of static vertex corrections (second order diagrams) in the polarization (P and not in the self energy) counteracts the usual overestimation and restores
the gap of the partial self consistent GW (GW0). Therefore, GW0 is quite accurate because of fortunate effect cancellations.

Further it is interesting to see the convergence wrt the NGsBlkXd and number of bands, as they correlate with each other and should be increased upon increase of the cell size.

By the way i do not understand why you used GW with RIM when using cylinder ?
" Apply a sherical / cylindrical / box cutoff for Coulomb integral at half the cell size. "
I thought that the truncated cell should be just a bit smaller than the original one.

Bests
LUCA
Luca Montana
PhD student
University of York, UK

User avatar
wachr
Posts: 32
Joined: Wed Sep 24, 2014 4:43 pm

Re: Coulomb truncation and convergence in a CNT

Post by wachr » Tue Oct 18, 2016 9:35 am

Dear Luca,
In my opinion this is related to the fact that W in full self consistent GW gets underscreened upon self consistency (due to artificial transfer of spectral weight to the incoherent part of the spectral function),
leading to overestimated band gaps. However, inclusion of static vertex corrections (second order diagrams) in the polarization (P and not in the self energy) counteracts the usual overestimation and restores
the gap of the partial self consistent GW (GW0). Therefore, GW0 is quite accurate because of fortunate effect cancellations.
I completely agree. That's another way to express what I meant to say. Fully self-consistent GW leads to band gap opening -> reduced screening -> even more band gap opening -> even more reduced screeing -> ... -> maybe, one converges towards Hartree Fock ;) (no screening).

BSE (vertex corections for P) reduces the eigenvalues of the optical transitions and leads to an increased screening (especially at omega=0, where the effect is the largest). And GW+BSE screening is not "too" far from the DFT screening - at least the eigenvalues for the dominant transitions are somewhat close (for that particular CNT: GW+BSE transition is at about 1.55 eV; the DFT transition at about 1.65 eV; the G0W0 transition is at about 2.8 eV). The exact shape of the spectrum, however, strongly differs from GW+BSE vs. DFT due to the excitonic nature (e.g. see Appl. Phys. A 78, 1129).
By the way i do not understand why you used GW with RIM when using cylinder ?
As I read in the forum, the RIM gets ineffective and only the cylindrical truncation plays a role for GW. However, in order to activate Coulomb truncation, RIM is automatically activated in the input files. Even if the number of RIM-point is set to 0.

But I am not very much in code development rather than application of the results. Therefore, I guess, Daniele (or one of the other developers) can explain this a bit more precise and more in detail.
Further it is interesting to see the convergence wrt the NGsBlkXd and number of bands, as they correlate with each other and should be increased upon increase of the cell size.
Good point. One should stick to the recommendation to set the value for NGsBlkXd in mHa as described in the blog of Claudio Attacelite. And the EXXRLvcs should be overconverged, if one wishes increase the unit cell further (increasing this parameter is not very costly).

NGsBlkXd can lead to quite expensive calculations [memory and cpu] for the screening - therefore, I think, this is one of the parameters to converge towards the end of the calculations. This parameter should not depend too much on k-point or band convergence rather than geometrical effects (please correct if I am wrong on this).
I thought that the truncated cell should be just a bit smaller than the original one.
In principle, right. Especially for the box cutoff, as I read, one should rather go to 40% (1/(1+sqrt(2)) ) of the unit cell size. But the difference is relatively small. Here, I have a plot for the CNT varying the cylinder truncation radius:
GW_rcut.png
The cylinder should be larger (or at least, close) to the diameter of the 99% charge density; which is half the unit cell in my case (green line). If the diameter of the cylinder is smaller than the CNT (red line), the results make no sense. If it becomes larger, the gap increases as the "virtual cell" defined by the cylinder includes more space with less electron density. If one comes close to the 99%, the gap only slightly changes.

I hope this gives a bit a better feeling for the impact of the truncation radius. In the "yambo truncation publication" (PRB 73, 205119 (2006)), it is recommended that for practical application, the cylinder can be about half the cell size without loss of too much accuracy. In case of doubt: one can check for the differences in for reasonable truncation values and include incertainties in the gap, e.g. 1.80 +/- 0.02 eV for this case.

Kind regards,
Christian

P.S. A minor edit improving clarity.
You do not have the required permissions to view the files attached to this post.
Christian Wagner
Institute of Physics
Chemnitz University of Technology, Germany

Post Reply