Improving the calculations

Let us now address the two issues that determine the accuracy of the calculations, beginning with the k-points for the electronic problem. We started with a 2 2 2 k-point grid, which on a $2 \times 2 \times 2$ supercell corresponds to a 4 4 4 grid on the primitive cell, which is typically not a dense enough grid for a system like Al, which is a metal (usually metals require dense k-points sampling). This suggests that a first improvement of the phonon dispersions might be obtained by increasing the number of k-points.