I'm using a genetic algorithm based global optimisation package, USPEX, to search stable structures of non-stoichiometric transition metal compounds. The code has a limitation - we aren't able to specify the number of k points explicitly, instead only a value called 'kresolution' can be entered. Higher is this value, lower is the density of the k grid. The thing is that it uses the same value of kresolution for all of the structures it searches. I get somewhere around 600 k points in the irreducible Brillouin zone for triclinic structures, which for most part are just slightly distorted versions of other crystal systems (cubic, hexagonal, monoclinic, etc), whereas I get anywhere around 30 to 70 k points for structures other than triclinic, all of these for the same value of kresolution. How reliable are these calculations? Can you please shed some light on what we may call an optimum number of k points in the irreducible Brillouin zone.
Dear Hitanshu,
As is well know the number of k-points depend on the symmetry of the crystal. The best way to use the right number ok k-points or grid is to perform calculations with different number of k-points. Then perform a curve k-points vs total energy and choose the precision, which usually is suggested to be equal or less than 1 men/atom.
Regards
Dear Sinhue,
Thank you for such a quick response. I'm familiar with the concept of k points sampling and I do that for every ab initio calculation on transition metal compounds I run with VASP. The thing about USPEX is that it won't allow you to do such a sampling. It only lets you enter a value, 'kresolution', which decides how dense your k points grid shall be. I just wanted some intuitive description of the optimum number of k points required for all different sorts of systems, viz. metals, semi-conductors, insulators, cubic crystals, hexagonal crystals, triclinic crystals, intermetallics, etc.
Dear Hitanshu,
I am not familiar with USPEX software/ code but I can comment on K-point sampling based on my experience with VASP and CP2K code. The K-point sampling does not depend on what kind of system are you using, such as metals, semi-conductors, insulators, cubic crystals, hexagonal crystals, triclinic crystals etc but it depends on system size. I would say you can increase your lattice vector lengths and use less number of K-points/ gamma point (or in your case increase Kresolution value) because large system size requires less k-points because of reciprocal space sampling. In my view, you can't really define good guess for K-points unless you don't perform k-point convergence calculations.
Hope it will help you little.
just remember that k-pt number is not a variational parameter. also, a good idea is using even monkhorst-pack grids (4,4,4, say, not 3,3,3). a further distinction is between gamma centered and shifted: the choice here will depend on the specific band structure and the type of problem.
Dear Vincenzo Fiorentini, can you elaborate a bit more. Why even grids and also your 2nd statement?
I have no precise idea why even is better. Empirically it seems to be so, at least for energy convergence. Gamma-centered: I suppose you would want the Gamma point included (it won't if you shift and reduce) if your band structure has features near Gamma relevant to your problem (say, defects, or thermoelectric coefficients, in direct gap materials; and I bet one can think of many others).
Dear Hitanshu Sachania,
I think people already have given you most of the insight you need here. I just would like to add that checking the k-point convergence is an essential step on ab initio calculations. I would say you have to try a resolution, and then redo the calculation with a better kresolution, and check whether your calculations have converged. Furthermore, with time, you will get a feeling on how much kpoints (k resolution) you need for each system. Probably you have to learn this yourself. Cheers.
Dear Flaviano Santos, that's absolutely true. It should become an intuitive process once I'm well acquainted with the code (hopefully).
Dear Hitanshu,
The reason gamma-centred even grids sometimes have better convergence is because they avoid the gamma point. The gamma point is an extremely high symmetry point, meaning that almost all states will have an energy maximum or minimum there -- in other words, it is highly atypical and hence a poor sampling point.
However if you have hexagonal symmetry then you *must* use the gamma-point, i.e. odd grids if you're using a gamma-centred grid. This is because the hexagonal symmetry operations applied to an even grid generate points outside the Brillouin zone (see the papers of Chadi and Cohen for details; link below).
In practice, for most simulations it is usually sufficient to plot the k-point convergence for all the odd grids separately from all the even grids, and you'll see that both converge to the same value but at different rates.
The k-point convergence depends crucially on two things:
1) the size of the space being sampled (the Brillouin zone)
The larger the real-space cell, the smaller the Brillouin zone and so the fewer k-points you need. This is why a k-point spacing (or "resolution") is usually a better guide to the sampling quality than just the number of points. You need to be careful to check the units: some codes use units of 1/A and some use 2pi/A (and some use inverse Bohrs);
2) the smoothness of the Fermi surface
The Fermi-surface is smooth and well-behaved for an insulator, but for a metal it is discontinuous at 0 K. This means metals typically need much higher k-point sampling than insulators, not to get the band-energies accurately, but to determine the band occupancies. Most codes allow some kind of smoothing function to be applied to the Fermi level, usually a Gaussian, Fermi-Dirac distribution, or a Methfessel-Paxton scheme. This is essentially like heating the electrons up, so the Fermi-surface is smoothed out and isn't so ill-behaved, allowing calculations with fewer k-points.
EDIT: for more details, see:
Article Special Points in the Brillouin Zone
D. J. Chadi and M. L. Cohen, Phys. Rev. B 8, 5747 (1973)
Article On Special Points for Brillouin Zone Integrations
H. J. Monkhorst and J. D. Pack, Phys. Rev. B 13, 5188 (1976)
Article Mean-value point and dielectric properties of semiconductors...
A. Baldereschi and E. Tosatti, Phys. Rev. B 17, 4710 (1978)Hope that helps,
Phil Hasnip
(CASTEP developer)
The short answer is one should optimize k-points for each individual structure.
More details and implementation here:
Preprint Convergence and machine learning predictions of Monkhorst-Pa...
The optimum value of k-points is the number of unit-cells in the lattice.
Dr Philip James Hasnip, thank you for your reply, it was helpful. Sorry for the late acknowledgement.
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