[Pw_forum] k point convergence of dielectric properties

[Stefano de Gironcoli]

Dear Dario,

this is an issue related to the fact that a shifted grid in the fcc  
Bravais lattice has  lower symmetry than cubic... just think of the  
simplest 
1 1 1 1 1 1 grid that should contains one single point  L=(1/2 1/2 1/2) 
but actually represents  the 4 (rotationally but not translationally)
equivalent points (1/2 1/2 1/2), (-1/2,-1/2,1/2), 
(-1/2,1/2,-1/2),(1/2,-1/2,-1/2) . The argument can be easily extended to 
denser grids and shows that the shifted grid represents 4*N^3 points in 
the fcc BZ.
This explains why the shifted grid appears more accurate (it is actually 
denser than you think) and why the number of points it contains is 
larger than you may have thought ... In fact avoiding high symmetry 
points like the Gamma point,  contained in the unshifted grid, should 
help reducing the number of inequivalent points while instead the number 
of  shifted points for a given nominal density is larger than the number 
of unshifted ones... true, but the corresponding parent grid is actually 
4 times denser than you may have thought...and the "symmetry reductions"
factor is indeed larger for the shifted grid
for example:
shifted     4*4*4 -> 256 points in the full BZ -> 10 points in the IW 
=>  sym. red. fact. = 25.6
unshifted 4*4*4 ->  64 points in the full BZ ->   8 points in the IW =>  
sym. red. fact. = 8
 
Some time (especially if your system does not have much symmetry to 
start with) you don't really care about sampling the BZ according to a 
fully cubic symmetry but a regular grid in the 3 directions is enough.. 
in this case the use of shifted grids may results in a large amount of  
"un-needed" k-points which you may want NOT to include in the 
calculation... this annoying fact is the origin of the infamous 
nosym=.true. options that much confusion generates in most users until 
they realize what its meaning and purpose really is ... there are many 
threads on that in the archive....

As to the fact that total energy are less sensitive than dielectric 
properties to k-points sampling this is a well known fact
[see for instance  S.Baroni and R. Resta, PRB 33, 7017 (1986)]

Hope this helps,

stefano
--

Stefano de Gironcoli - SISSA and DEMOCRITOS


dario rocca wrote:
> Dear Users
> I have an issue related to the convergence of the static dielectric  
> matrix using the PH code. I have performed calculations
> on bulk silicon using different k point meshes and I have obtained the 
> following results:
>
> k grid              diagonal component of      number of k points in
>                          the dielectric tensor           the 
> irreducible Brillouin zone
>
> 4*4*4                   23.668350065                        8
>  6 *6*6                16.297485614                       16
> 8 *8*8                  14.044830694                       29
> 10 *10*10             13.288531964                     47
> 12*12*12              13.029602882                     72
> 16 *16*16             12.908820645                    145
> 20 *20*20             12.894538380                    256
>
> k grid+ 1 1 1 shift              diagonal component of      number of 
> k points in
>                                             the dielectric 
> tensor           the irreducible Brillouin zone
>
> 4*4*4                                      13.840844632 
>                           10
> 6*6*6                                     
> 12.997009732                           28
> 8*8*8                                     
> 12.903849607                           60
> 10*10*10                               
> 12.893711596                         110
> 12*12*12                               
> 12.892685597                         182
> 16*16*16                               
> 12.892482798                         408
> 20*20*20                               
> 12.892537346                         770
>  
> I was surprised of the improvement in the convergence due to the shift 
> of the grid. I don't think this is related to the number of k points
> in the IBZ (at least not exclusively).
> I have observed a similar behavior in diamond.
> The ground state energy convergence also benefits from the shift, but 
> the improvement is not so striking.
> Does someone has any hint on why the shift of the grid improves the 
> calculation of the dielectric properties of silicon?
> Thanks a lot
> Dario Rocca, dept. of chemistry, UC Davis
>
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