Best practice(s) for q-points convergence in phonon density of states calculations

Question

The answer can be a complete description of all the things we must be attentive to while converging q-points.

A peculiar thing that I feel contradicted about: Generally, we don't take an equal # of k-points in an electronic DFT calculation when the cell has different lengths in different directions. VASP even prints a warning - "Reciprocal lattice and k-lattice belong to different class of lattices", whereas phonopy doesn't give any error/warning if I take an equal # of q-points for such cells.

Answer 1

In a calculation of the phonon density of states, $\mathbf{q}$-points feature in two ways:

1. Explicitly calculated $\mathbf{q}$-points. These are the $\mathbf{q}$-points for which you explicitly calculate the dynamical matrix, and are typically referred to as forming the "coarse $\mathbf{q}$-point grid". If you are using finite differences to calculate phonons, then these will correspond to the $\mathbf{q}$-points commensurate with the supercells that you are using in the calculations. If you are using DFPT to calculate phonons, then these will correspond to the $\mathbf{q}$-points for which you explicitly calculate the response. You should converge the density of states with respect to the coarse $\mathbf{q}$-point grid.
2. Interpolated $\mathbf{q}$-points. A coarse $\mathbf{q}$-point grid that is converged (in the sense that the corresponding matrix of force constants decays to zero) is typically not enough to obtain a smooth phonon density of states. Instead, what you typically do is to perform a new phonon calculation on a much denser grid, typically called the "fine $\mathbf{q}$-point grid". For this second calculation, the dynamical matrices on the fine grid are not explicitly calculated but instead are constructed by Fourier interpolation over those of the coarse grid. You should converge the density of states with respect to the fine $\mathbf{q}$-point grid.

In practice, the computational bottleneck is in converging the coarse grid because you are explicitly doing the calculations in that case. My advice would be to pick a fixed size for the fine grid that is relatively large (larger than typical converged coarse grid sizes). Then perform a series of calculations for that fixed fine grid by increasing the size of the coarse grid until the DOS profile converges. This allows you to converge the coarse grid. Then fix the coarse grid to the converged value and now increase the size of the fine grid until the DOS profile converges again. The final combination of coarse and fine grids should be a good set of converged parameters.

As to how many $\mathbf{q}$-points to pick along each crystallographic direction, there is no fundamental reason why you should not be able to use the same number in all directions. However, convergence is typically faster along a direction that is longer, so a general rule of thumb for practical calculations is to use a grid of uniform density.