# k-points: Odd vs. Even

As mysterious as k-points are, why is there a cut-off for an odd and even k-mesh preference?

On the VASP wiki page titled, "Number of k points and method for smearing", they say results when using odd k-meshes shouldn't be compared to those when using even k-meshes (section: Comparing different k-point meshes). In the section "Some other considerations", they go on to say that even meshes are better up to 8 k-points in a direction while odd meshes are better beyond 8. What is the physics behind it or is it a code-level preference?

As an aside: I recently asked a question about odd/even q-points for phonon DoS calculations where I received a comment suggesting that the problem could be similar to that of k-points in electronic calculations.

• Convergence is generally quicker with an even k-mesh because you avoid sampling the high-symmetry points. That is why even-grids are preferred. I am not sure if there is a concrete reason for why odd-centered grids are preferred for more than 8 points. In general, it's just easier to use a grid like 10x10x10 instead of 9x9x9 because even though the even grid is denser, it will typically converge quicker than the latter because it avoids sampling high-symmetry points, which are atypical points. – Xivi76 Oct 7 at 17:43
• @Xivi76: I also don't understand the preferrence of odd-centered grids for grids with many k points. In the documentation for the Fleur code we have an example for k-point set convergence for fcc Cu. The even-odd behavior is very nicely visible there, but it seems like the even k-point sets feature the more friendly convergence behavior: flapw.de/MaX-4.0/documentation/parameterConvergence/… – Gregor Michalicek Oct 8 at 11:07
• While you may get slower convergence w.r.t. mesh size with odd grids, you also can greatly reduce the number of points you need to calculate due to symmetry. I suppose there's a trade-off that begins to favor odd grids at high k-point densities. You can also shift even grids to include high-symmetry points... The symmetry of your system is probably important in making this decision, as the VASP manual mentions (including $\Gamma$ with fcc, etc.) – Kevin J. M. Oct 13 at 22:03

• Thank you for your answer. Isn't the data in the table a bit counterintuitive to what the VASP wiki says? Up to $8$, odd values of $M$ have a shorter execution time, whereas, for $11,12$ and $13,14$, even values of $M$ have a shorter execution time. – Hitanshu Sachania Oct 14 at 11:44