Quantifying local point-group-symmetry order in complex particle systems

Crystals and other condensed phases are defined primarily by their inherent symmetries, which play a crucial role in dictating their structural properties. In crystallization studies, local order parameters (OPs) that describe bond orientational order are widely employed to investigate crystal formation. Despite their utility, these traditional metrics do not directly quantify symmetry, an important aspect for understanding the development of order during crystallization. To address this gap, we introduce a new set of OPs, called Point Group Order Parameters (PGOPs), designed to continuously quantify point group symmetry order. We demonstrate the strength and utility of PGOP in detecting order across different crystalline systems and compare its performance with that of commonly used bond-orientational order metrics. PGOP calculations for all finite point groups are implemented in the open-source package SPATULA (Symmetry Pattern Analysis Toolkit for Understanding Local Arrangements), written in parallelized C++ with a Python interface. The code is publicly available on GitHub at https://github.com/glotzerlab/spatula.