# Group-theoretical high-order rotational invariants for structural representations: Application to linearized machine learning interatomic potential

## TL;DR

This paper introduces high-order rotational invariants derived from group theory to improve the accuracy of machine learning interatomic potentials, demonstrated on elemental aluminum.

## Contribution

It develops a systematic method to generate high-order rotational invariants up to sixth order for crystal structures using group theory, enhancing MLIP accuracy.

## Key findings

- High-order invariants improve MLIP predictive power.
- The method effectively reduces the invariants using group-theoretical projectors.
- Application to aluminum shows increased accuracy in structure-property predictions.

## Abstract

Many rotational invariants for crystal structure representations have been used to describe the structure-property relationship by machine learning. The machine learning interatomic potential (MLIP) is one of the applications of rotational invariants, which provides the relationship between the energy and the crystal structure. Since the MLIP requires the highest accuracy among machine learning estimations of the structure-property relationship, the enumeration of rotational invariants is useful for constructing MLIPs with the desired accuracy. In this study, we introduce high-order linearly independent rotational invariants up to the sixth order based on spherical harmonics and apply them to linearized MLIPs for elemental aluminum. A set of rotational invariants is derived by the general process of reducing the Kronecker products of irreducible representations (Irreps) for the SO(3) group using a group-theoretical projector method. A high predictive power for a wide range of structures is accomplished by using high-order invariants with low-order invariants equivalent to pair and angular structural features.