Thank you very much Behnam Farid for the resources and explanation.

the shift away from Gamma was originally meant to reduce the grid size. one reason for using Gamma-centered sampling sometimes is that shifted grids may overlook "important" band character may near Gamma (often there is a gap there in semiconductors, e.g.) if your sampling is coarse. so the choice depends on your application. notice that in metals Gamma-centered sampling (NOT sampling Gamma only!) is often a better choice (a faster-converging choice, let us say) than the shifted grid.

also, let me point you to the original paper (by my own supervisor ;) on special points: http://journals.aps.org/prb/abstract/10.1103/PhysRevB.7.5212

Thank you Vincenzo Fiorentini.

btw this is not to imply that k-points number or grid sizes are a variational parameter: they are not, unlike plane waves cutoff.

We strongly recommend to use only Gamma centered grids for hexagonal lattices. Many tests we have performed indicate that the energy converges significantly faster with Gama centered grids than with standard Monkhorst Pack grids. Grids generated with the M setting in the third line, in fact do not have full hexagonal symmerty.

Nisar Muhammad ,

"Many tests we have performed indicate that the energy converges significantly faster with Gama centered grids than with standard Monkhorst Pack grids"

could you please cite few studies in this regard?

Many tests we have performed indicate that the energy converges significantly faster with Gama centered grids than with standard Monkhorst Pack grids.

https://books.google.az/books?id=3kvCDwAAQBAJ&pg=PA22&lpg=PA22&dq=What+is+the+difference+between+Monkhorst+-Pack+and+Gamma+centered+k-sampling?&source=bl&ots=RGRxzyYDPP&sig=ACfU3U3qCrqRpd_m6o2pPP9UwPxy0RR0Qw&hl=ru&sa=X&ved=2ahUKEwiSgeTawZPqAhUBuaQKHfhDAEYQ6AEwAHoECAoQAQ#v=onepage&q=What%20is%20the%20difference%20between%20Monkhorst%20-Pack%20and%20Gamma%20centered%20k-sampling%3F&f=false

In principle, so mathematically both ways of generating the grid would converge at the same pace with the increasing number of k points if the function to be integrated was arbitrary: They both are equi-distance samplings of a random function with the same distance between the sampling points. The difference and possible preference of one over the other are:

- As stated in the communication between Monkhorst and Pack and Chadi and Cohen on the other hand, the original sets of Monkhorst-Pack grids did not include the Gamma point, and they had a lower symmetry in some cells - in particular the hexagonal one - than the maximum symmetry of the cell (after the Comment by Chadi and Cohen on their original article Monkhorst and Pack devised the present Monkhorst-Pack algorithm where the whole grid of points can be translated - for example so that one of the points hits the Gamma point); this leads to either asymmetric sets of k points and thus a breaking of the symmetries, or, like probably in most codes, to a generation of many more k points that would be in the original set in order to preserve the symmetries. So if there are symmetries, like C_3v in many hexagonal systems, it is more economic and symmetric to use the Gamma-centred sets of Monkhorst-Pack grids

- The underlying function to be integrated is not fully random, rather it contains regularity - what we call bands. In the band structure at the Gamma point is always (well, except maybe with spin-orbit coupling, but in general) the bands have their extrema, either a local minimum or maximum (the bands arrive at the Gamma point with zero derivative). So when people could only afford few k points in their calculations they first had the special points - like the Baldereschi point - that all avoided the Gamma point, as the band structure was thought to be more accurately averaged/integrated over if one did not take the Gamma point but a point that was away from the Gamma, kind of averaging over bands better than at Gamma, where one has those local minima and maxima. When one has enough k points it naturally does no longer matter whether one includes the Gamma point or not, as the weight of the Gamma (and possibly other points at the boundaries of the first Brillouin zone) diminishes

So in general, if one has a hexagonal cell and symmetries in the atomistic structure, it is better to take the Gamma-centred Monkhorst-Pack sets, as one gets all symmetries included with fewer k points that the translated ones. Overall the convergence depends on the underlying band structure, in exotic cases (Kohn anomalies, Dirac point like in graphene, ...) one might need to adjust case by case. But, in "normal" cells with "normal" band structure the difference between the two kinds of choice in the speed of convergence with the number of k points and thus computational speed is probably small.