SymmCD: Symmetry-Preserving Crystal Generation with Diffusion Models

TL;DR

SymmCD tackles the problem of generating crystalline materials with accurate and diverse symmetry by explicitly incorporating crystallographic structure into diffusion-based generation. It introduces a Wyckoff-position–based representation that models the asymmetric unit and the symmetry operations needed to unfold it, coupled with a symmetry-biasing, multi-component diffusion process and a dedicated denoiser. The method achieves competitive stability, novelty, and diversity on MP-20 while offering substantial computational efficiency and generalization across space groups, and it scales to larger MPTS-52 crystals. These advances enable more reliable and scalable discovery of symmetry-consistent crystals with predicted properties. Overall, SymmCD advances symmetry-preserving generative modeling in crystallography, with practical implications for accelerated materials discovery and design.

Abstract

Generating novel crystalline materials has the potential to lead to advancements in fields such as electronics, energy storage, and catalysis. The defining characteristic of crystals is their symmetry, which plays a central role in determining their physical properties. However, existing crystal generation methods either fail to generate materials that display the symmetries of real-world crystals, or simply replicate the symmetry information from examples in a database. To address this limitation, we propose SymmCD, a novel diffusion-based generative model that explicitly incorporates crystallographic symmetry into the generative process. We decompose crystals into two components and learn their joint distribution through diffusion: 1) the asymmetric unit, the smallest subset of the crystal which can generate the whole crystal through symmetry transformations, and; 2) the symmetry transformations needed to be applied to each atom in the asymmetric unit. We also use a novel and interpretable representation for these transformations, enabling generalization across different crystallographic symmetry groups. We showcase the competitive performance of SymmCD on a subset of the Materials Project, obtaining diverse and valid crystals with realistic symmetries and predicted properties.

Figures (10)

• Figure 1: Illustration of the SymmCD method.Left. Representation of the unit cell of a 2D crystal with $p4m$ symmetry where the asymmetric unit and the site symmetries of the atoms are highlighted. Leveraging symmetry results in a much more compact, yet complete representation. Right. Diffusion and denoising on the different components of the representation. For site symmetries and atom types, discrete diffusion is used. For the coordinate and asymmetric unit continuous diffusion is used. The diffusion and denoising processes preserve the space group symmetry.
• Figure 2: Crystal symmetry axes. The different axes describe the directions along which symmetry operations can occur. For each of the 15 axes, there are 13 possible symmetry operations.
• Figure 3: SymmCD training and sampling pipeline. For training (green), the crystal structures are pre-processed to find the space group $G$ along with the site symmetries $\mathbf{S}$ and a set of orbit representatives inside the asymmetric unit. The denoising model is a GNN with fully connected graphs, followed by a decoder. For sampling (blue), the positions are projected to the closest one compatible with their site symmetry. Then, the asymmetric unit is replicated to obtain the unit cell.
• Figure 4: Proportion of space group symmetries of the dataset, and each method. The width of each color segment represents the proportion of crystals with that symmetry. From left to right, the first few space groups are: P1, Fm$\bar{3}$m, Cm, P$\bar{1}$, C2/m, I4/mmm, Pm$\bar{3}$m, P$6_3$/mmc, and Pm.
• Figure 5: Proportion of space group symmetries of the MPTS-52 dataset. From left to right, the first few space groups are: Pnma, P2$_1$/c, Fm$\bar{3}$m, I4/mmm, P6$_3$/mmc, Pm$\bar{3}$m, C2/m, C2/c, $\bar{1}$, and Cmcm.