Abstract
Phonons and their interactions are necessary for determining a wide range of materials properties. Here we present four independent advances which facilitate the computation of phonons and their interactions from first principles. First, we implement a group-theoretical approach to construct the order Taylor series of a -dimensional crystal purely in terms of space group irreducible derivatives (ID), which guarantees symmetry by construction and allows for a practical means of communicating and storing phonons and their interactions. Second, we prove that the smallest possible supercell which accommodates given wave vectors in a -dimensional crystal is determined using the Smith normal form of the matrix formed from the corresponding wave vectors; resulting in negligible computational cost to find said supercell, in addition to providing the maximum required multiplicity for uniform supercells at arbitrary and . Third, we develop a series of finite displacement methodologies to compute phonons and their interactions which exploit the first two developments: lone and bundled irreducible derivative (LID and BID) approaches. LID computes a single ID, or as few as possible, at a time in the smallest supercell possible, while BID exploits perturbative derivatives for some order less than (e.g., Hellman-Feynman forces) in order to extract all ID in the smallest possible supercells using the fewest possible computations. Finally, we derive an equation for the order volume derivatives of the phonons in terms of the order ID. Given that the former are easily computed, they can be used as a stringent, infinite ranged test of the ID. Our general framework is illustrated on graphene, yielding irreducible phonon interactions to fifth order. Additionally, we provide a cost analysis for the rocksalt structure at , demonstrating a massive speedup compared to popular finite displacement methods in the literature.
3 More- Received 16 April 2019
- Revised 4 June 2019
DOI:https://doi.org/10.1103/PhysRevB.100.014303
©2019 American Physical Society