Local-Global Associative Frames for Symmetry-Preserving Crystal Structure Modeling

TL;DR

The paper addresses the challenge of achieving $SO(3)$ invariance in crystal property prediction without breaking crystal symmetry. It introduces SPFrame, a local-global frame that combines invariant local frames with a shared global frame to preserve symmetry while enforcing invariance, and integrates it into state-of-the-art crystal GNN backbones. Empirical results on JARVIS-DFT and Materials Project show that SPFrame, particularly its quaternion-based variant, improves predictive accuracy over traditional local frames and baselines, with modest computational costs. This work provides a principled framework for symmetry-preserving frame construction in materials modeling, with implications for more reliable materials discovery.

Abstract

Crystal structures are defined by the periodic arrangement of atoms in 3D space, inherently making them equivariant to SO(3) group. A fundamental requirement for crystal property prediction is that the model's output should remain invariant to arbitrary rotational transformations of the input structure. One promising strategy to achieve this invariance is to align the given crystal structure into a canonical orientation with appropriately computed rotations, or called frames. However, existing work either only considers a global frame or solely relies on more advanced local frames based on atoms' local structure. A global frame is too coarse to capture the local structure heterogeneity of the crystal, while local frames may inadvertently disrupt crystal symmetry, limiting their expressivity. In this work, we revisit the frame design problem for crystalline materials and propose a novel approach to construct expressive Symmetry-Preserving Frames, dubbed as SPFrame, for modeling crystal structures. Specifically, this local-global associative frame constructs invariant local frames rather than equivariant ones, thereby preserving the symmetry of the crystal. In parallel, it integrates global structural information to construct an equivariant global frame to enforce SO(3) invariance. Extensive experimental results demonstrate that SPFrame consistently outperforms traditional frame construction techniques and existing crystal property prediction baselines across multiple benchmark tasks.

Figures (4)

• Figure 1: An illustration with 2D plain group P4mm wang2024crystallinehahn1983internationaljiaospace. (a) The figure illustrates the lattice of the P4mm space group, visually demonstrating equivalent positions; the symmetry-equivalent positions are indicated by the same color. (b) The table depicts the Wyckoff positions present in this lattice.
• Figure 2: Using the 2D plane group P4mm as a running example, we demonstrate why local frame methods may disrupt the symmetry of crystal. For atoms $p$ and $q$ that belong to the same Wyckoff position type, their local structures can be related by a 90-degree rotation after graph construction. Since equivariant local frames are constructed solely based on local structural information, the resulting frames for $p$ and $q$ also exhibit a 90-degree rotational relationship, applying these frames eliminates the relative orientation between the two local structures. In contrast, SPFrame preserves these relative structural differences by incorporating global structural information during frame construction.
• Figure 3: Visual analysis. After graph construction, atoms at symmetry-equivalent positions may exhibit distinct local structures. Local frame methods tend to transform these local structures into identical representations, thereby removing the relative differences and making the atoms indistinguishable to the model. In contrast, SPFrame preserves these structural distinctions, enabling t he model to effectively differentiate between atoms located at symmetry-equivalent positions.
• Figure 4: The detailed architectures of eComFormer with SPFrame.