An optimal approach to computing phonons and their interactions via finite difference

Phonons and their interactions are critical to predicting a wide range of materials properties. Therefore, efficiently extracting a high resolution Taylor series expansion of the Born-Oppenheimer surface from an arbitrary first-principles approach is of great importance. Here we present an optimal formalism to compute phonons and their interactions at arbitrary order on a uniform grid using finite difference. Our approach ensures that a given derivative is always obtained from the smallest possible supercell dictated by the translation group, in addition to the smallest number of runs dictated by space group symmetry. We demonstrate that our approach is superior to any single-supercell finite difference approach, which is commonplace in the literature for phonons and cubic interactions. Applications are presented for graphene and PbTe, providing phonons and interactions up to 5th order at an unprecedented q-space resolution with minimal discretization errors. Additionally, we present the phonons of the rare-earth nickelates, demonstrating the utility of our approach for large complex unit cells.