Convergence with respect to the size of the super-cell

Now we test convergence w.r.t the size of the super-cell. To do this, go back to Sec.  and repeat the steps to generate a super-cell, for example a $4 \times 4 \times 4$. In order to make comparable calculations, modify the runphon script by halving the size of the displacement (in order to still have a displacement with a length of 0.04 Å). To start with, modify also the k-point sampling by setting 1 1 1 in the KPOINTS file. This k-point sampling is equivalent to a 2 2 2 grid on the $2 \times 2 \times 2$ super-cell, so the present results can be compared with the very first calculations we performed. Now execute runphon and phon and plot the new FREQ.cm together with the old one. Here it is even more evident that the k-point sampling is insufficient, but at least we are directly verifying that the phonon frequencies predicted with the two super-cells are the same at zone boundary. In the previous section we found that a 4 4 4 k-point grid on the $2 \times 2 \times 2$ super-cell produced good results, so let's try now to use this density of points in the BZ also for the calculation also with the $4 \times 4 \times 4$ super-cell. This means using a 2 2 2 k-point grid in the file KPOINTS. Now run the calculations and see how the phonon dispersions change. Finally, to obtain a fully converged phonon spectrum on a $2 \times 2 \times 2$ super-cell we observed that we required a 8 8 8 k-point grid, which corresponds to a 4 4 4 grid on the $4 \times 4 \times 4$ super-cell.