Space Group Equivariant Crystal Diffusion

TL;DR

This work introduces SGEquiDiff, a crystal generative framework that enforces space group constraints throughout generation by combining an $SE(3)$-invariant lattice sampler, a permutation-invariant Wyckoff/element transformer, and a space group equivariant diffusion process for atomic coordinates. The authors prove that space group equivariant vector fields naturally live on Wyckoff manifolds (Theorem 1) and that SGWN scores are invariant/equivariant under space group actions (Theorem 2), enabling manifold-respecting diffusion without projection. Empirically, SGEquiDiff achieves state-of-the-art performance on MP20 and MPTS52 datasets, with high stable-unique-novel (S.U.N.) rates and favorable symmetry metrics, validated by DFT relaxations. The approach demonstrates the practicality of symmetry-aware generative modeling for rapid crystal discovery, with code and resources available for reproducibility and further exploration.

Abstract

Accelerating inverse design of crystalline materials with generative models has significant implications for a range of technologies. Unlike other atomic systems, 3D crystals are invariant to discrete groups of isometries called the space groups. Crucially, these space group symmetries are known to heavily influence materials properties. We propose SGEquiDiff, a crystal generative model which naturally handles space group constraints with space group invariant likelihoods. SGEquiD-iff consists of an SE(3)-invariant, telescoping discrete sampler of crystal lattices; permutation-invariant, transformer-based autoregressive sampling of Wyckoff positions, elements, and numbers of symmetrically unique atoms; and space group equivariant diffusion of atomic coordinates. We show that space group equivariant vector fields automatically live in the tangent spaces of the Wyckoff positions. SGEquiDiff achieves state-of-the-art performance on standard benchmark datasets as assessed by quantitative proxy metrics and quantum mechanical calculations. Our code is available at https://github.com/rees-c/sgequidiff.

Figures (5)

• Figure 1: (a) The asymmetric unit ($\Pi$) and special Wyckoff positions labeled by letter in the conventional unit cell of space group 10. (b-c) Histograms of occupied space groups and Wyckoff dimensionalities by crystals in the MP20 cdvaematerialsprojecticsd training dataset. Space groups and Wyckoff positions were determined by the SpaceGroupAnalyzer module in pymatgenpymatgenspglib using tolerances of 0.1 Å and 5$^\circ$. These tolerances help account for the moderate convergence criteria of the Materials Project DFT relaxations.
• Figure 2: Illustration of our crystal generation process.
• Figure 3: Special Wyckoff positions and the asymmetric unit in the conventional unit cell of hexagonal space group 192. Closed asymmetric unit boundary edges and facets ($\partial\Pi$) are in orange.
• Figure 4: S.U.N. crystals generated by SGEquiDiff trained on MP20.
• Figure 5: S.U.N. crystals generated by the +LDiff variant of SGEquiDiff trained on MP20.