SymmCD: Symmetry-Preserving Crystal Generation with Diffusion Models

Daniel Levy; Siba Smarak Panigrahi; Sékou-Oumar Kaba; Qiang Zhu; Kin Long Kelvin Lee; Mikhail Galkin; Santiago Miret; Siamak Ravanbakhsh

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SymmCD: Symmetry-Preserving Crystal Generation with Diffusion Models

Daniel Levy, Siba Smarak Panigrahi, Sékou-Oumar Kaba, Qiang Zhu, Kin Long Kelvin Lee, Mikhail Galkin, Santiago Miret, Siamak Ravanbakhsh

arXiv:2502.03638v3Feb 5, 2025cond-mat.mtrl-scics.LG

TL;DR

Crystal generators often fail to reproduce realistic symmetry distributions or rely on database-derived symmetry information, despite symmetry’s role in crystal properties. SymmCD diffuses asymmetric units together with site-symmetry information using a binary-matrix representation. It produces stable, novel, and structurally diverse crystals while preserving desired symmetries and improving computational efficiency, with composition-conditioned structure prediction remaining more challenging.

Problem
Existing crystal generators can produce unrealistic or database-biased symmetry distributions, limiting generation of crystals with realistic crystallographic symmetries.
Method
SymmCD jointly diffuses asymmetric units and site-symmetry information, encoding symmetries with binary matrices that share information across space groups.
Results
SymmCD produces stable, novel, and structurally diverse crystals with desired symmetries and improved computational efficiency.
Takeaways & Limitations
The approach supports symmetry-preserving crystal generation across diverse space groups while retaining competitive stability performance.
Takeaways & Limitations
Composition-conditioned crystal structure prediction is more challenging because the framework samples a space group before sampling a composition conditioned on it.

Abstract

from arXiv · show

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.

1 INTRODUCTION

Crystal symmetry is central to material properties, yet existing generators often produce unrealistic or database-biased symmetry distributions. SymmCD addresses this by diffusing asymmetric units and symmetry information with a representation designed to generalize across symmetry groups.

Crystal symmetry influences stability and thermodynamic, electronic, and mechanical properties.
Existing generators frequently produce low-symmetry P1 crystals, while MatterGen recovers requested space groups only about 20% of the time and about 10% for more symmetric groups.
GNoME’s four most common generated space groups comprise 34% of crystals, despite each appearing in less than 1% of the ICSD.
SymmCD generates crystals from asymmetric units and site symmetries, making atomic positions consistent with the generated transformations.
Binary-matrix symmetry representations enable information-sharing and generalization across crystal and site symmetries.
The method is reported to generate stable structures on par with previous methods while improving computational efficiency.

2 RELATED WORK

Crystal-generation research spans many generative paradigms, but symmetry-aware methods face trade-offs between diversity, validity, and generalization. SymmCD is positioned against template-based and Wyckoff-label approaches that restrict novelty or cross-group generalization.

Prior crystal-generation methods include VAEs, GANs, reinforcement learning, diffusion, and flow-matching models.
DiffCSP++ uses predefined structural templates, which the authors report severely limits generated-sample diversity and novelty.
CrystalGFN constrains lattice parameters and composition by space group but does not guarantee symmetry-respecting atomic positions.
CrystalFormer and WyCryst predict atom symmetries using Wyckoff labels, limiting generation to space groups common in their datasets.
The authors state that SymmCD generalizes across groups and can generate valid crystals from groups rare in the dataset.

3 BACKGROUND

Crystal structures combine periodic lattices, atoms, and symmetry operations that preserve the crystal. Space groups classify these operations, while orbits, site symmetries, Wyckoff positions, and asymmetric units provide progressively useful structural classifications.

A crystal is modeled as a periodic lattice of identical unit cells containing atoms at fixed positions.
A space group is the set of Euclidean transformations that leave a crystal invariant while permuting atoms of the same type.
Three-dimensional crystals have 230 unique space-group types, with point groups obtained by retaining the operations’ linear parts.
A site symmetry group contains the space-group operations that leave a point fixed and is a subgroup of the crystal’s point group.
Wyckoff positions group points whose site-symmetry groups are related by conjugation and classify regions by symmetry type.
An asymmetric unit is a nonredundant part of a unit cell that generates the whole cell under space-group transformations and contains one atom from each orbit.

4 METHOD: SYMMETRIC CRYSTAL DIFFUSION (SYMMCD)

SymmCD represents crystals through their space group, asymmetric-unit orbit representatives, lattice parameters, atom types, and site symmetries, then models these components with diffusion while preserving symmetry during generation.

Symmetry-aware representation: Existing crystal representations do not encode crystal symmetry explicitly and provide no guarantee beyond the trivial P1 space group.
Symmetry-aware representation: SymmCD represents a crystal using its space group, lattice parameterization, asymmetric-unit coordinates, site symmetries, and atom types.The representation is C′ = (G, k, X′, S, A′), with orbit representatives replicated to reconstruct the crystal.
Symmetry-aware representation: Orbit representatives and their site symmetries identify crystallographic orbits that can be replicated under the space group to recover the full unit cell.The replication operation applies the space-group operations compatible with each site symmetry.
Symmetry-aware representation: Wyckoff-position supervision provides a physically motivated inductive bias because atoms preferentially occupy high-symmetry positions.
Symmetry representation: The space group is encoded similarly, but includes 26 operations because translations produce screw and glide transformations.
Diffusion model: During sampling, generated positions are projected onto site-symmetry-compatible positions before the asymmetric unit is replicated into the unit cell.
Diffusion model: SymmCD jointly diffuses lattice parameters, asymmetric-unit coordinates, atom types, and binary site-symmetry representations.Discrete diffusion handles categorical atom types and site symmetries, while continuous diffusion handles fractional coordinates and lattice parameters.

5 EXPERIMENTS

Experiments on MP-20 and MPTS-52 evaluate SymmCD’s symmetry and structural diversity, S.U.N. materials, proxy metrics, relaxation behavior, and computational efficiency. SymmCD generally matches or improves on baselines while scaling to larger crystals, though validity and seed variance remain concerns.

5.1 SYMMETRY AND STRUCTURAL DIVERSITY: SymmCD matches the dataset’s diverse space-group distribution, whereas CDVAE, DiffCSP, and FlowMM favor low-symmetry crystals.SymmCD and DiffCSP++ are reported as the only methods that accurately match the dataset distribution.
5.1 SYMMETRY AND STRUCTURAL DIVERSITY: SymmCD proposes the most unique and novel structural templates, unlike DiffCSP++, whose predefined templates prevent novel-template generation.The training dataset contains 3318 unique templates.
5.2 STABLE, UNIQUE AND NOVEL (S.U.N.) STRUCTURES: SymmCD performs on par with baselines on stable, unique, and novel crystal generation from 10,000 samples per method.Stability is assessed before and after CHGNet relaxation, with sampling over common space groups producing more stable structures.
5.3 VALIDATION WITH DENSITY FUNCTIONAL THEORY RELAXATIONS: SymmCD is significantly more successful at generating structures with lower maximum forces after electronic-structure relaxation, although convergence rates do not differ significantly.The electronic-structure evaluation uses 100 structures sampled from each model.
5.4 PROXY METRICS: SymmCD performs similarly to other methods on most proxy metrics but generates more structurally invalid crystals and shows substantial variation across random seeds.Each method is retrained with five seeds and evaluated using 10,000 generated samples per seed.
5.5 COMPUTATIONAL EFFICIENCY: SymmCD gains computational efficiency and reduces memory usage through its compact crystallographic-orbit representation, while scaling to crystals with up to 52 atoms per primitive unit cell.On MPTS-52, SymmCD produces valid, stable, and novel crystals matching the dataset distribution, although metrics are worse than on MP-20.

CONCLUSION

SymmCD generates crystals by combining asymmetric units with symmetry information in a diffusion framework, using representations designed to generalize across space groups. The framework produces stable, novel, and diverse crystals with improved computational efficiency, but makes composition-conditioned structure prediction more challenging.

CONCLUSION: SymmCD combines unconstrained asymmetric-unit generation with site-symmetry information in a diffusion model.The asymmetric unit is replicated through crystallographic symmetry operations to construct the full crystal.
CONCLUSION: The framework produces stable, novel, structurally diverse crystals with improved computational efficiency and precise symmetry properties.The conclusion reports these outcomes within the paper’s inorganic-crystal setting.
CONCLUSION: Crystal structure prediction from a composition is more challenging because the framework samples a space group before sampling a conditioned composition.The authors also identify polycrystalline-material generation as future work beyond single-crystal generation.
CONCLUSION: Predicted site symmetries are projected to valid crystallographic point groups before coordinates are adjusted to compatible Wyckoff positions.The nearest subgroup is selected using the Frobenius norm, followed by a PyXtal search for compatible coordinates.
CONCLUSION: The model diffuses representative fractional coordinates, atom types, site symmetries, and lattice parameters using continuous and discrete processes.Fractional coordinates use a Wrapped Normal process, atom types and site symmetries use categorical diffusion, and lattice parameters use a masked Gaussian process.

E ARCHITECTURE DETAILS

The denoising architecture uses a graph neural network conditioned on timestep, space-group, and site-symmetry representations. Experiments and implementation details specify the network configuration, training weights, and symmetry-aware inputs used for evaluation.

E ARCHITECTURE DETAILS: The graph neural network embeds timestep, space-group, and site-symmetry information before updating node representations.Site symmetries are embedded separately by axis and combined through a secondary MLP.
E ARCHITECTURE DETAILS: The denoising network predicts noise for representative coordinates, atom types, site symmetries, and lattice parameters from node embeddings.The outputs include ˆϵX′, ˆA′, ˆS, and ˆϵk.
E ARCHITECTURE DETAILS: The network has 8 layers, 1024-dimensional node representations, 128 distance frequencies, and a 10-dimensional timestep embedding.Layer normalization is applied at every GNN layer.
E ARCHITECTURE DETAILS: The selected loss coefficients are λk = 5, λX′ = 1, λA′ = 0.1, and λS = 10.These coefficients weight the lattice, coordinate, atom-type, and site-symmetry objectives.
E ARCHITECTURE DETAILS: Hyperparameter sweeps selected settings using structural validity, compositional validity, and dE performance.The architecture sweep evaluated 150 combinations after the loss-coefficient sweep.

F.1 MPTS 52 DATASET

MPTS-52 is a larger and more difficult Materials Project subset containing crystals with up to 52 atoms per primitive unit cell. SymmCD was trained and sampled on this dataset, and its generated atom-count distribution matched the training distribution.

F.1 MPTS 52 DATASET: MPTS-52 contains materials with up to 52 atoms per primitive unit cell and does not exclude radioactive elements.The dataset has 40,476 samples split chronologically into 27,380 training, 5,000 validation, and 8,096 test crystals.
F.1 MPTS 52 DATASET: SymmCD was trained on MPTS-52 for 1500 epochs and used to sample 10,000 crystals for proxy-metric and stability evaluation.Generated crystals were relaxed with CHGNet and checked for S.U.N. status.
F.1 MPTS 52 DATASET: The generated crystals’ number-of-atoms distribution matches the MPTS-52 training distribution.This evaluates whether the model learned multiplicities associated with site symmetries and space groups.

F.3 PROPERTY PREDICTION TASK

The property-prediction study evaluates formation-energy prediction using either a full multi-graph or SymmCD’s asymmetric unit, with or without node-level site-symmetry encoding. Site-symmetry information particularly helps when prediction uses only the asymmetric unit.

F.3 PROPERTY PREDICTION TASK: Formation energy per atom is used as the target property for prediction.The study evaluates whether symmetry-aware inputs improve this prediction task.
F.3 PROPERTY PREDICTION TASK: The ablation varies graph input format and whether site-symmetry information is encoded for each node.DimeNet++ is used as the base model.
F.3 PROPERTY PREDICTION TASK: Site-symmetry encoding has a minimal effect with the full multi-graph but substantially helps when only the asymmetric unit is available.The authors interpret this as recovering geometric information lost when using the asymmetric unit alone.
F.3 PROPERTY PREDICTION TASK: The study includes randomly sampled generated crystals with their chemical formulae and space-group symmetries.Figure 8 presents six generated examples.

F.5 PROXY METRICS

The evaluation uses established validity, coverage, and property-statistics metrics, with validity divided into structural and compositional checks. Five random-seed models are trained to measure metric variance.

Validity, coverage, and property statistics are used to compare the generated crystals.Structural validity requires no atom pair to be closer than 0.5 Å, while compositional validity requires charge balance under SMACT.
Structural validity excludes structures containing atom pairs closer than 0.5 Å.
Compositional validity checks whether charges are balanced according to SMACT, although MP-20 itself has only 92% compositional validity.
Five models with different random seeds are trained to estimate variance across metrics.

F.6 DENSITY FUNCTIONAL THEORY

The DFT evaluation relaxes generated structures with CP2K and examines atom displacement and maximum gradients as proxies for structural fidelity and proximity to local minima. SymmCD and DiffCSP++ show especially favorable relaxation behavior, while convergence remains difficult across methods.

F.6 DENSITY FUNCTIONAL THEORY: Relaxations use CP2K with the PBE functional, DZVP-MOLOPT-SR-GTH basis set, and GTH-PBE pseudopotential.The configuration settings, including convergence thresholds, are reported in Table 9.
F.6 DENSITY FUNCTIONAL THEORY: SymmCD and DiffCSP++ produce atom-displacement distributions peaking near zero, whereas other methods peak near 4.5 Å.Lower displacement indicates that generated structures require less optimization to reach relaxed configurations.
F.6 DENSITY FUNCTIONAL THEORY: Approximately 60% of relaxation trajectories fail to converge across all methods.The failure rate is measured using the portion of maximum-gradient distributions beyond the convergence threshold.
F.6 DENSITY FUNCTIONAL THEORY: The 10-space-group SymmCD treatment has a long tail toward low maximum-gradient values, indicating extremely high-fidelity samples.Lower maximum gradients correspond to relaxation endpoints closer to local minima.
F.6 DENSITY FUNCTIONAL THEORY: SymmCD tends to have a lower maximum gradient after relaxation than the other methods.Expected maximum-gradient values are obtained by integrating the empirical curves shown in Figure 10.