Quantifying Local Point-Group-Symmetry Order in Complex Particle Systems

TL;DR

The paper addresses the challenge of directly quantifying local point-group symmetry during crystallization by introducing $PGOP$, a continuous, bounded order parameter computed from Gaussian representations of neighboring particles and overlaps under symmetry operations. It implements $PGOP$ in the open-source SPATULA toolkit and demonstrates superior sensitivity to local symmetry across simple and complex crystals and Lennard–Jones nucleation, outperforming conventional bond-orientational order parameters. The framework supports extensions to other symmetry operations and to local BOODs, enabling interpretable, per-point symmetry metrics in both disordered and crystalline states. It also outlines pathways toward differentiable formulations and use as a collective variable for enhanced sampling in future work.

Abstract

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 to commonly used bond-orientational order metrics. PGOP calculations for all non-infinite 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.

Figures (6)

• Figure 1: Steps for computing the point group order parameter (PGOP) for a given query point (red sphere). (A) The system of semi-transparent white points with the query point highlighted in red. (B) The nearest neighbors of the query point (blue) are identified. (C) The chosen point group, $\mathrm{C_{2v}}$, with its symmetry elements: the two-fold rotation axis $\hat{C}2$ (green) and the mirror planes $\hat{\sigma}_{xz}$ (light red) and $\hat{\sigma}_{yz}$ (light blue). (D–F) The initial neighborhood configuration (blue) is symmetrized with respect to each symmetry element, yielding symmetrized neighborhoods colored according to the generating operator: $\hat{C}2$ (green, D), $\hat{\sigma}_{xz}$ (light red, E), and $\hat{\sigma}_{yz}$ (light blue, F). The 1D Gaussian cartoons represent the 3D Gaussians actually used in the calculation. Overlaps between each symmetrized neighborhood and the original neighborhood are shown in tan. The PGOP value is obtained as the average sum of these overlaps.
• Figure 2: The measure of order using MSM ($l=6$), PGOPs, or PGOP-BOODs on increasingly noisy FCC and HCP crystal data. The solid blue line indicates the mean values of each OP for a given crystal and noise amount ($\Sigma$), while the black lines indicate the value for an ideal gas. The brown filled areas indicate the spread of data associated with standard deviation, while the light blue areas indicate the area enclosed by the first and last quantiles for the distribution of OP values. Panels A and D are computed for MSM for FCC and HCP crystals respectively. For the FCC crystal (top panel), PGOP-BOOD (middle, B) and PGOP (right, C) values are computed for the point group $\mathrm{O_h}$. For the HCP crystal (bottom panel), PGOP-BOOD (middle, E) and PGOP (right, F) values are computed for the point group $\mathrm{D_{3h}}$.
• Figure 3: The application of PGOP for simple crystal identification. Each color corresponds to a specific neighborhood, and remains consistent across this figure: FCC in magenta, BCC in green, SC in gray, and HCP in yellow-green. A: Distribution of PGOP values (semi transparent points) for point groups $\mathrm{D_{3h}}$ and $\mathrm{O_h}$ for 4 different simple crystals (FCC, BCC, SC, HCP) constructed with Gaussian noise. The values for perfect crystals are given by square marks, while the ideal gas is denoted by a black square mark. B The change in PGOP $\mathrm{D_{3h}}$ value for perfect crystals as a function of Gaussian width ($\sigma$). Perfect crystal values are given with full lines, while ideal gas is denoted by dashed black line. C: The perfect simple crystals SC, FCC, BCC, HCP with particles colored by their environment. Every particle has a perfect value of symmetry for their respective crystals: perfect $\mathrm{D_{3h}}$ symmetry for HCP and perfect $\mathrm{O_h}$ symmetry for FCC, BCC and SC.
• Figure 4: Using PGOP to identify local environments found in complex crystals. The particles within a noisy (A) A15 crystal and a noisy (B) $\mathrm{\gamma}$-brass crystal plotted by PGOP for two point groups found within the specified crystal ($\mathrm{T_h}$ and $\mathrm{D_{2d}}$ for A15; $\mathrm{C_{3v}}$ and $\mathrm{C_{2v}}$ for $\mathrm{\gamma}$-brass). PGOP plots are paired with snapshots of the perfect crystal colored by its Wyckoff site designation. These colors are consistent between the PGOP plots and the perfect crystal snapshot. For both crystals, the values for the perfect crystal are represented by squares, while circles represent values from the noisy crystals. The ideal gas value is represented by a black square.
• Figure 5: Impact of the neighbor list choice on PGOP results for pyrochlore crystal. Four panels display the particle values of PGOP for different neighbor lists: SANN (A), RAD (B), Voronoi (C) and radial cut neighbor list (D). The points represent the values of PGOP for $\mathrm{C_2}$ and $\mathrm{C_3}$ point groups for pyrochlore crystal with Gaussian noise. The squares represent the values for the perfect crystal and ideal gas (black). The crystalline points and squares are colored according to their Wyckoff site designation. A perfect pyrochlore crystal is shown in E. Particles are colored according to their Wyckoff site, with their corresponding colors matching across the plots.