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

Phonons and their interactions are critical to predicting a wide range of materials properties. Therefore, efficiently extracting a high resulotion Taylor series expansion of the Born-Oppeheimer 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 and crystal dimension on a regular grid using finite displacements; yielding a Taylor series purely in terms of group theoretically irreducible derivatives. Building on a key theorem we derive, our approach ensures that a given derivative can always obtained from the smallest possible supercell dictated by the translation group. Our approach maximally exploits any derivatives the first-principles approach can efficiently deliver perturbatively (e.g. Hellman-Feynman forces, etc) to obtain higher order derivatives. We prove that our approach is superior to any single-supercell finite displacement approach. Applications are presented for graphene, computing and tabulating the irreducible derivatives up to 5th order. A number of critical tests are performed to demonstrate the fidelity of our results.