C2NP: A Benchmark for Learning Scale-Dependent Geometric Invariances in 3D Materials Generation

TL;DR

This work introduces C2NP, a systematic benchmark that tests how well generative models learn scale-dependent geometric invariances when transitioning from bulk unit cells to finite nanoparticles. The authors construct a large, DFT-validated dataset by carving spherical nanoparticles from a $20\times20\times20$ supercell across radii $R$ in $[6,30]$ Å and apply stratified rotational augmentations on SO$(3)$ to create train, in-distribution, and out-of-distribution splits, framing two tasks: Unit Cell to Nanoparticle generation and Nanoparticle to Unit Cell lattice inference. Across multiple state-of-the-art models (CDVAE, DiffCSP, FlowMM, MatterGen-MP, ADiT), results show that low training loss does not guarantee geometric fidelity, with many methods failing under distribution shift while CDVAE demonstrates superior geometric consistency on Task 1 but Task 2 remains unresolved for joint lattice and symmetry recovery. C2NP thus provides a rigorous, reproducible framework for diagnosing scale-generalization failures in crystalline matter generation, with immediate relevance to nanoparticle design and materials discovery, and sets a path for future work to broaden crystallographic coverage and incorporate more realistic surface phenomena.

Abstract

Generative models for materials have achieved strong performance on periodic bulk crystals, yet their ability to generalize across scale transitions to finite nanostructures remains largely untested. We introduce Crystal-to-Nanoparticle (C2NP), a systematic benchmark for evaluating generative models when moving between infinite crystalline unit cells and finite nanoparticles, where surface effects and size-dependent distortions dominate. C2NP defines two complementary tasks: (i) generating nanoparticles of specified radii from periodic unit cells, testing whether models capture surface truncation and geometric constraints; and (ii) recovering bulk lattice parameters and space-group symmetry from finite particle configurations, assessing whether models can infer underlying crystallographic order despite surface perturbations. Using diverse materials as a structurally consistent testbed, we construct over 170,000 nanoparticle configurations by carving particles from supercells derived from DFT-relaxed crystal unit cells, and introduce size-based splits that separate interpolation from extrapolation regimes. Experiments with state-of-the-art approaches, including diffusion, flow-matching, and variational models, show that even when losses are low, models often fail geometrically under distribution shift, yielding large lattice-recovery errors and near-zero joint accuracy on structure and symmetry. Overall, our results suggest that current methods rely on template memorization rather than scalable physical generalization. C2NP offers a controlled, reproducible framework for diagnosing these failures, with immediate applications to nanoparticle catalyst design, nanostructured hydrides for hydrogen storage, and materials discovery. Dataset and code are available at https://github.com/KurbanIntelligenceLab/C2NP.

Figures (3)

• Figure 1: C2NP data generation and evaluation pipeline. The first stage (green) constructs finite nanoparticles from periodic crystallographic inputs: starting from each DFT-optimized unit cell $\mathcal{U}$ with lattice parameters $\Lambda = (a,b,c,\alpha,\beta,\gamma)$, a $20\times20\times20$ supercell $\mathcal{T}$ is built to ensure sufficient spatial extent, after which a spherical cluster $\mathcal{P}_R$ is carved by retaining atoms within radius $R$ of a chosen center $\mathbf{x}_0$. The second stage (blue) applies rotational augmentation over $\mathrm{SO}(3)$ using a geodesic separation constraint on unit quaternions to generate three stratified orientation grids: a sparse training set $\mathcal{Q}_{\mathrm{train}}$ with threshold $\theta_{\mathrm{train}}=15^\circ$, an interpolative in-distribution set $\mathcal{Q}_{\mathrm{ID}}$ with $\theta_{\mathrm{ID}}=12^\circ$ and exclusion margin $\delta_{\mathrm{ID}}=6^\circ$, and a dense extrapolative out-of-distribution set $\mathcal{Q}_{\mathrm{OOD}}$ with $\theta_{\mathrm{OOD}}=9^\circ$ and margin $\delta_{\mathrm{OOD}}=4.5^\circ$. Deterministic quaternion seeding and fixed offset rotations $R_{\mathrm{ID}}$ and $R_{\mathrm{OOD}}$ enforce non-overlapping geodesic neighborhoods between train, ID, and OOD orientations while decorrelating orientation manifolds. The third stage (orange) defines two complementary benchmark tasks: a forward problem mapping unit-cell parameters and target radius $(\mathcal{U},R)$ to the corresponding nanoparticle geometry $\mathcal{P}_R$, and an inverse problem recovering lattice parameters and symmetry $(\Lambda,\Gamma)$ from a nanoparticle $\mathcal{P}_R$, providing a unified testbed for both generative and inverse lattice--nanoparticle modeling.
• Figure 2: Representative crystal-to-nanoparticle transformations in the C2NP dataset, derived from DFT-validated crystallographic reference structures. Columns (left to right) correspond to ZrCdH$_3$, LiBeH$_3$, LiCaH$_3$, BaMnH$_3$, and LiBeH$_3$. Rows show the reference unit cell (top), a nanoparticle carved at $R=10$,Å (middle), and a nanoparticle carved at $R=30$,Å (bottom). Each panel reports the total atom count ($N$) and elemental composition. Across the dataset, particle sizes span more than four orders of magnitude, from a minimum of $N=4$ atoms for the smallest particles to a maximum of $N=18{,}123$ atoms for the largest configurations. Atom colors follow the standard CPK scheme: H (white), Li (purple), Be (dark green), Ca (green), Ba (light green), Mn (purple), Zr (dark gray), and Cd (light gray).
• Figure 3: Comprehensive analysis of the C2NP dataset. (a--d) Lattice-level statistics of the crystallographic unit cells, including anisotropy ratios and marginal distributions of lattice parameters $a$, $b$, and $c$, highlighting controlled symmetry constraints and directional distortions. (e--h) Unit-cell volume statistics and pairwise correlations between lattice parameters, revealing strong coupling induced by crystallographic structure. (i--l) Radius-dependent nanoparticle properties across $R=6$--30 Å, including convex-hull scaling, density and packing behavior, atom counts, and global shape descriptors. Together, these panels demonstrate physically consistent geometric scaling alongside substantial structural variability, establishing C2NP as a rigorous benchmark for both forward nanoparticle generation and inverse lattice recovery.