Cutoff for metals or metallic materials?

[RKBK]

Is there any literature or rules of thumb concerning how to choose the cutoff length for metals or metallic materials?

[ajjackson]

This is a good question. In my experience when studying metals the total energy and geometry are usually well-converged for cutoff lengths around 25Å and above. I would always recommend to check the convergence with respect to the property of interest! For example, if interested in the bandgap it may be that the conduction band continues to move when the total energy is well-converged. That being said, it is good to have some kind of starting point; I usually start at 20Å and try a few calculations with higher cutoffs up to about 32Å. I will leave this Issue open for further discussion, with a view to improving the README.

[RKBK]

I've done a little bit of testing for a system I am interested in. The property of interest is the adsorption energy of O on a 4-layer slab of Rh(111), calculated with RPBE in quantum espresso, and I've tested cutoff lengths from 10 to 32 Å. This is how the adsorption energy (vs 1/2 * the total energy of molecular O2 in vacuum) converges with higher cutoff:

The red dashed lines mark the adsorption energy + or - 0.05 eV at the most dense k-point sampling.

The different cutoffs result in the following k-point samplings (only one k-point is used in the z axis in all cases):
10: (4, 4, 1)
14: (6, 6, 1)
18: (7, 7, 1)
22: (9, 9, 1)
26: (10, 10, 1)
30: (12, 12, 1)
34: (13, 13, 1).

The adsorption energy is converged already at cutoffs of about 15 Å.

[ajjackson]

That is interesting. It's great that convergence of the order 0.02 eV is achieved so quickly. However, the system still seems far from higher levels of convergence, say of the order 1 meV. The nice thing about convergence testing is that it lets you report approximate "error bars". In this case you're converged within 1-2% of the value you're computing, good enough for most purposes :-)

[RKBK]

Since I'm still interested in using cutoff lengths rather than just "common sense" when choosing k-point sampling, I've now done convergence calculations for O adsorption on a 4 Å thick jellium slab (with the Wigner-Seitz radius of 2.81 a.u., similar to that of Ru). The results are below:

The different cutoffs correspond to the following k-point samplings:
10: (4, 2, 1)
14: (5, 2, 1)
18: (6, 3, 1)
22: (8, 4, 1)
28 (yes, I now realize 22+4=26 rather than 28 :-( ): (10, 4, 1)
32: (11, 5, 1)

Again, for my purposes, a cutoff length of 14 seems to be enough.

[ajjackson]

Intriguing that the convergence appears to be so much smoother for the jellium slab. Was the mesh gamma-centered in both cases? Is the occupation broadening the same in both cases?

[RKBK]

I am not quite sure about whether the mesh is gamma-centered. The first calculation was from QE and the second calculation was done with GPAW, where apparently you can force the gamma point to be included if you so choose. Is is required that the gamma point is included for the cutoff length method to work?

The occupation broadening should be the same in both cases.

[ajjackson]

In the Moreno-Soler paper, they note that displacing the origin of the grid has no bearing on the cutoff length and may allow additional symmetry reduction. In the Monkhorst-Pack paper there doesn't seem to be any mention of origin shifts. The conventional wisdom seems to be that Gamma-centering is an obstacle to convergence. I've heard anecdotal complaints about odd-even effects when not using a Gamma-centered grid, but haven't seen any papers on it. FWIW I always include the Gamma point just because it's usually the most interesting part of the band structure.

In practice it does seem to be hard to converge k-points to meV levels without a lot of broadening, which in turn can change the final value. There are some scary plots in this presentation.

[RKBK]

Thanks for your comments! My calculations are most likely not including the k point (unless QE and GPAW do that by default sometimes). Yeah, that looks worrying.

By the way, I encountered something of a bugbear when using k cutoffs today. I was (still) converging jellium parameters, this time the adsorption energy as the width of the slab (in one direction) was changed. This is using a cutoff length of 22 Ang for k, but the red lines now delimit +-0.01 eV. The results are below:

It turns out that what happens here is that at 12 Ang, the width of the slab changes so that the k point sampling becomes (4, 4, 1) rather than (5, 4, 1), and this is why two of the lines (blue and green lines, representing calculations with and without dipole correction, with circles and down-pointing triangles) show a significant change in energy. I guess one has to be somewhat conservative in how the k cutoff is chosen to avoid effects such as these.

[RKBK]

@ajjackson Regarding "origin shifts" and whether it is positive to include the gamma point, I recently found some discussion on this topic in the 2013 paper by Sun and Ceder (10.1016/j.susc.2013.05.016, page 57, end of section 3 just before start of 3.1): "One must also be mindful that the k-point sampling accurately reflects the reciprocal Bravais lattice of the SBZ — for example, the (111) surfaces of FCC and BCC crystals are hexagonal and require a gamma-centered odd k-point grid."