Cyclic and helical symmetry-informed machine learned force fields: Application to lattice vibrations in carbon nanotubes

TL;DR

This paper introduces a symmetry-informed machine learning force field framework for accurately modeling lattice vibrations in carbon nanotubes, leveraging cyclic and helical symmetries to improve efficiency and precision over traditional ab initio methods.

Contribution

The authors develop a formalism that incorporates cyclic and helical symmetries into machine learned force fields, enabling efficient and accurate simulations of nanostructures like carbon nanotubes.

Findings

Achieved root mean square errors of 1.4e-4 Ha/atom in energy and 4.8 cm$^{-1}$ in phonon frequencies.

Validated the MLFF against DFT and density functional perturbation theory calculations.

Identified unique torsional modes and established laws for phonon frequency variations in CNTs.

Abstract

We present a formalism for developing cyclic and helical symmetry-informed machine learned force fields (MLFFs). In particular, employing the smooth overlap of atomic positions descriptors with the polynomial kernel method, we derive cyclic and helical symmetry-adapted expressions for the energy, atomic forces, and phonons (describe lattice vibration frequencies and modes). We use this formulation to construct a symmetry-informed MLFF for carbon nanotubes (CNTs), where the model is trained through Bayesian linear regression, with the data generated from ab initio density functional theory (DFT) calculations performed during on-the-fly symmetry-informed MLFF molecular dynamics simulations of representative CNTs. We demonstrate the accuracy of the MLFF model by comparisons with DFT calculations for the energies and forces, and density functional perturbation theory calculations for the phonons, while considering CNTs not used in the training. In particular, we obtain a root mean square error of $1.4 \times 10^{-4}$ Ha/atom, $4.7 \times 10^{-4}$ Ha/Bohr, and 4.8 cm$^{-1}$ in the energy, forces, and phonon frequencies, respectively, which are well within the accuracy targeted in ab initio calculations. We apply this framework to study phonons in CNTs of various diameters and chiralities, where we identify the torsional rigid body mode that is unique to cylindrical structures and establish laws for variation of the phonon frequencies associated with the ring modes and radial breathing modes. Overall, the proposed formalism provides an avenue for studying nanostructures with cyclic and helical symmetry at ab initio accuracy, while providing orders-of-magnitude speedup relative to such methods.