A group theoretical approach to computing phonons and their interactions

TL;DR

This paper introduces a comprehensive group-theoretical framework for calculating phonons and their interactions from first principles, improving efficiency and symmetry handling in computational materials science.

Contribution

It presents a novel approach combining group theory, Smith Normal Form, and finite displacement methods to compute phonons and interactions more efficiently and accurately.

Findings

Implemented a symmetry-guaranteed Taylor series construction

Developed a supercell size determination method using Smith Normal Form

Achieved significant computational speedup in phonon calculations

Abstract

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 N Taylor series of a d-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 N given wavevectors in a d-dimensional crystal is determined using the Smith Normal Form of the matrix formed from the corresponding wavevectors; resulting in negligible computational cost to find said supercell, in addition to providing the maximum required multiplicity for uniform supercells at arbitrary N and d. 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 N (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 N-2 volume derivatives of the phonons in terms of the order N 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 rock-salt structure at N=3, demonstrating a massive speedup compared to popular finite displacement methods in the literature.