crystal-diffusion

Crystal Diffusion

Description

This skill covers de novo crystal structure generation using score-based diffusion and flow-matching generative models, including MatterGen (Microsoft Research), DiffCSP (Jiao et al.), and CDVAE (Xie et al.). These models learn the distribution of stable inorganic crystals from crystallographic databases and sample novel structures either unconditionally or conditioned on chemical composition, space group, or target properties. Invoke this skill when generating crystal structure candidates for materials discovery, when proposing hypothetical structures for a given formula, or when augmenting structural training sets with plausible synthetic crystals.

Domain Context

Crystal structure generation differs fundamentally from molecular generation because of three structural features that must be respected throughout the diffusion process:

• Periodic boundary conditions: Atom positions are fractional coordinates on a 3D torus [0, 1)³. A diffusion process operating on fractional coordinates must handle wraparound: coordinates 0.02 and 0.98 are separated by only 0.04 in fractional units, not 0.96. Naive Euclidean diffusion in fractional space destroys this periodicity.
• Lattice degrees of freedom: A crystal is fully specified by the 3×3 lattice matrix L (or equivalently the six scalars a, b, c, α, β, γ) together with the fractional coordinates and atom types. The lattice determines the physical scale and shape of the unit cell. Diffusion models must jointly denoise both the lattice and the atomic positions, and the parameterization of L affects equivariance properties.
• Discrete atom types: Element identities are discrete categorical variables. Most models represent them as continuous embeddings during diffusion and decode to discrete types at generation time, but the discretization step introduces additional approximation.

Physical concepts that determine model validity:

• SE(3)-equivariance: Crystal properties (energy, forces) are invariant under rotations and translations of the crystal. The generative model's score network or velocity field must be equivariant under these operations. Models based on MACE, NequIP, or e3nn backbones provide exact equivariance; models based on standard graph convolutions with distance-based features provide approximate invariance. [EXPERT REVIEW NEEDED: degree to which equivariance matters for generation quality vs. only for property prediction]
• Score matching and denoising diffusion: Score-based models learn the score function ∇_x log p_t(x) at each noise level t. At generation time, Langevin dynamics or an ODE/SDE solver follows the learned score backwards from noise to the data distribution. The score must be learned on the correct metric for each degree of freedom (Euclidean for fractional coordinates, SO(3)-like for the lattice, categorical for atom types).
• Flow matching: Learns a time-dependent velocity field v(x, t) that transports a simple prior (Gaussian noise) to the data distribution via an ODE. Flow matching on Riemannian manifolds (RFM) handles the non-Euclidean geometry of the lattice matrix and fractional coordinate torus more naturally than score matching. FlowMM and related models use this approach.
• Thermodynamic plausibility vs. stability: A generated structure that passes geometric validity checks (no overlapping atoms, reasonable bond lengths, charge balance) is not guaranteed to be thermodynamically stable. Stability requires DFT total-energy evaluation and comparison to competing phases on the convex hull. Pre-screening with a universal MLP (MACE-MP-0, CHGNet, M3GNet) is fast enough to apply to all generated structures before committing DFT compute.
• Composition space: Some models (CDVAE, DiffCSP) are conditioned on a fixed composition (the number and type of each element) provided by the user. Others (MatterGen) can generate composition jointly or condition on element-level properties. Composition-conditioned generation is more targeted for materials discovery but requires specifying a plausible composition.
• Crystal validity metrics: The community evaluates generative models using: (1) validity — fraction of generated structures passing geometric checks; (2) uniqueness — fraction that are structurally distinct from each other; (3) novelty — fraction not present in the training database; (4) FCD (Fréchet Crystal Distance) — distributional similarity between generated and reference structures in an embedding space; (5) match rate — fraction of generated structures that match DFT-relaxed experimental structures (for CSP benchmarks).