Dear Erakulan E.S.,

I assume that you refer to k-points grid according to the "standard' Monkhorst-Pack generation procedure. Increasing the grid density simply means that you better sample your Brillouin zone. So a 2x2x2 grid is less dense (and your calculation less accurate) than when using 3x3x3, which is less dense than 4x4x4, and so on. There is then no particular significance about whether K-points are 'odd' or 'even''. You should find the compromise between accuracy (denser grid) and cost (denser grids mean more k-points,, then more states to compute). This is not a feature of VASP, but is general to all electronic structure calculations of periodic systems.

Being not a VASP user, I will let others answer the second point about ISYM.

Best regards

L.P.

Laurent Pizzagalli Thanks for your answer. However in a lecture it was mentioned that even/odd has something to do with shifting of the k-points that depends upon the type of material used and it has some significance.

Dear Erakulan E.S., so you are referring to whether the k-points grid is shifted relatively to the origin (not really the same thing as even or odd). To explain, in the Monkhorst-pack scheme, you can generate a grid centered on the origin, or shifted by a fractional distance in the BZ. While you can use any translation, people tends to use (1/2,1/2,1/2) or centered on gamma point (so translation is 0,0,0). This is what you called even or odd I guess.

Of course it could have some significance. For instance, in some case, you want to have the gamma point in your calculation, so you should use the gamma-centered grid (so even in your formulation). Also, early DFT calculations on semiconductors use only one k-point, the (1/2,1/2,1/2) point, because of the calculation cost and also because they found that they could got very good results with this k-point already (whereas if you use only the gamma point to model bulk semiconductors, it does not work well).

But note also that if you are doing well converged k-points calculations, so dense grid, the difference between odd- and even-k-points grid vanish.

Dear Erakulan E Siddharthan ,

K-point convergence test is a must in DFT calculations. If you do not use the crystal symmetries (ISYM=0), including more k-points will always take longer; but if you do, it will take almost the same time. David Sholl, in his book, "Density functional theory: a practical introduction", explains this very well:

"For example, for the 10x10x10 Monkhorst–Pack sampling of the BZ, only 35 distinct points in k space lie within the IBZ for our current example [fcc Cu](compared to the 1000 that would be used if no symmetry at all was used in the calculation)."

An even/odd k-points choice, together with the optional mesh shift that Laurent Pizzagalli mentions (5th line in KPOINTS file), results in including or excluding the Gamma point. What you heard in your lecture, was very likely the following, as D. Sholl continues:

"... pairs of calculations with odd and even values of M [the MxMxM grid] took the same time—they have the same number of distinct points to examine in the IBZ [Irreducible Brillouin Zone]. This occurs because in the Monkhorst–Pack approach using an odd value of M includes some k points that lie on the boundaries of the IBZ (e.g., at the Gamma point) while even values of M only give k points inside the IBZ. An implication of this observation is that when small numbers of k points are used, we can often expect slightly better convergence with the same amount of computational effort by using even values of M than with odd values of M**. Of course, it is always best to have demonstrated that your calculations are well converged in terms of k points. If you have done this, the difference between even and odd numbers of k points is of limited importance.

** In some situations such as examining electronic structure, it can be important to include a k point at the Gamma point."

I dimly recall that energy convergence is faster on even grids, e.g. if you plot the energy of Si for a n,n,n grid vs n you see it going down generally, but bouncing up for odd n. As to Gamma centering or not, I understand that centered is usually preferred in metals, and for some reason I don't quite know, in hexagonal symmetry; generally, I suppose, you want to sample well any region of the zone where some action occurs. As to ISYM as has been mentioned, ISYM=0 turn symmetry off, so you will get all the kpts in the grid (no symmetry reduction); also, perhaps more importantly, you won't get automatic force symmetrization, hence your accuracy in structural predictions will require more iterations and care. Finally, remember that kpts number is not a variational parameter (plane-wave cutoff, instead, is), i.e. there is no guarantee that your energy will not bounce up as n increases (this, presumably, may be the case if you have a Fermi surface: I recall the hex and fcc structure in noble metals to be still competing at extremely fine meshes).

Dear Erakulan E Siddharthan,

For grids centred on the Gamma-point, even grids sometimes have better convergence than odd grids. This is precisely because they avoid the gamma-point itself, which is a very high symmetry point and not a good representative sampling point.

For systems with hexagonal symmetry, however, you should basically *never* use an even-grid (gamma-centred) 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).

You may find this discussion useful:

https://www.researchgate.net/post/Is_there_an_optimum_number_of_k_points_in_the_irreducible_Brilloiun_zone_that_is_necessary_for_good_calculations_of_bulk_total_energy

Hope that helps,

Phil Hasnip

(CASTEP developer)

Article Special Points in the Brillouin Zone

D. J. Chadi and M. L. Cohen, Phys. Rev. B 8, 5747 (1973)