Nonuniform grids for Brillouin zone integration and interpolation*

We present two developments for the numerical integration over the Brillouin zone. First, we introduce a nonuniform grid, which we refer to as the Farey grid, that generalizes traditional regular grids. Second, we introduce symmetry-adapted Voronoi tessellation, a general technique to assign weights to the points in an arbitrary grid. Combining these two developments, we propose a strategy to perform Brillouin zone integration and interpolation that provides a significant computational advantage compared to the usual approach based on regular grids. We demonstrate our methodology in the context of first-principles calculations with the study of Kohn anomalies in the phonon dispersions of graphene and MgB₂, and in the evaluation of the electron-phonon driven renormalization of the band gaps of diamond and bismuthene. In both cases, we find large speedups, whether density functional perturbation theory or finite difference methods are used. Besides, our results of bismuthene reveal that it preserves a sizable topological band gap at room temperature. In summary, our method opens up an avenue for designing the most appropriate nonuniform grid for any given task, with the prospect of saving valuable computational time and allowing for new frontiers in computational condensed matter physics to be charted.