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Packora: Systematic Design for Generative Molecular Crystal Structure Prediction

Nayoung Kim, Kiyoung Seong, Sungsoo Ahn

arXiv:2608.26962v1cs.LGcond-mat.mtrl-sci

TL;DR

Molecular crystal structure prediction needs comparable evaluation and models that handle varied chemical systems and available structural information. Packora jointly generates coordinates and lattices from molecular graphs, outperforming baselines across generation and ranking benchmarks.

  • Problem

    Existing molecular CSP methods differ in chemical scope, molecular inputs, candidate budgets, and evaluation pipelines, limiting direct comparison and leaving design effects incompletely understood.

  • Method

    Packora is an all-atom conditional flow generator that predicts crystal coordinates and unit cells from molecular graphs with optional templates, stereochemistry, and space-group conditioning.

  • Results

    Packora achieves the best matched-budget coverage across six generation benchmarks and improves downstream recovery, ranks, and convergence over baselines, including up to 77.6% relative improvement over CLARI-H.

  • Takeaways & Limitations

    The study provides a practical Packora design and evaluation recipe spanning diverse crystal systems, conditioning information, generation quality, and end-to-end ranking.

  • Takeaways & Limitations

    The selected recipe is not guaranteed globally optimal because one-at-a-time ablations omit interactions, and model selection primarily uses k=30 candidates.

Abstract

from arXiv · show

Molecular crystal structure prediction (CSP) is important in pharmaceuticals, agrochemicals, and organic electronics, where subtle differences in molecular conformation and packing can strongly affect material properties. We present Packora, a flow-based generative model for molecular CSP that jointly predicts atomic coordinates and the lattice from molecular graphs. Packora supports multi-component and organometallic crystals and can condition on any subset of molecular conformers, stereochemical labels, and space-group information within a single model. Inspired by the CCDC CSP blind test, we evaluate generation and ranking separately, using generation to isolate generator quality and ranking to measure end-to-end performance under a common relaxation and ranking pipeline. We also systematically study architecture, training, conditioning, inference, and scaling, identifying an effective design based on cacheable pairwise reasoning, training objective and numerical solver choices, conditioning dropout, and balanced scaling of pairwise and single representations. Packora outperforms the baselines on both structure generation and ranking benchmarks, achieving the best matched-budget coverage across all six generation benchmarks, as well as higher experimental-form recovery, lower experimental-form ranks, and faster convergence in ranking.

1 Introduction

Packora is an all-atom generative model for molecular crystal structure prediction that supports diverse crystal systems and optional structural conditioning. Its matched generation–ranking evaluation and systematic design studies show stronger coverage and downstream performance than prior methods.

  • Motivation: Molecular CSP is motivated by the strong dependence of crystal properties on molecular conformation and packing, with polymorphs exhibiting different functional properties.Applications include pharmaceuticals, agrochemicals, and organic electronics.
  • Model: Packora jointly predicts atomic coordinates and unit cells from molecular graphs while supporting multi-component, organometallic, hydrogen-containing, and optionally conditioned crystals.Conditioning information includes molecular templates, stereochemical labels, and space groups.
  • Evaluation: A matched two-track evaluation separates finite-budget generation coverage from downstream ranking under a common relaxation, filtering, deduplication, and lattice-energy pipeline.The generation track tests whether candidates contain the experimental structure, whereas the ranking track measures whether the experimental form is ranked near the top.
  • Design study: The study systematically evaluates architecture, training, conditioning, inference, and scaling to derive an evidence-backed design recipe for molecular CSP generation.It compares DiT with pair-bias attention, Pairformer, and Pairmixer, and examines time sampling, translation augmentation, and loss design.
  • Results: Packora-M or Packora-L achieves the best generation coverage across six benchmarks, with up to 77.6% relative improvement over CLARI-H.When coupled to common relaxation and ranking, Packora also achieves higher recovery, faster convergence, and lower experimental-form ranks than CLARI-H on single-form and polymorph FastCSP benchmarks.

2 Related Work

Molecular CSP spans search-based pipelines that separately generate, relax, and rank structures, and generative models that learn distributions over experimental crystals. Related generative approaches cover molecular, inorganic, and metal-organic-framework crystals, but differ in representations, conditioning, and physical-validity strategies.

  • Search-based molecular CSP: Search-based molecular CSP explores conformations and packings before geometry relaxation and energy ranking, with the seventh CCDC blind test evaluating generation and ranking separately (Reilly et al., 2016; Hunnisett et al., 2024a,b).Candidates are commonly generated across selected space groups by sampling unit-cell parameters.
  • Generative molecular CSP: Generative molecular CSP learns distributions over experimental structures using approaches based on all-atom diffusion, rigid-body geodesic flows, reinforcement-learning post-training, or pair-bias attention (Jin et al., 2025; Zeng et al., 2026; Subramanian et al., 2026; Lo et al., 2026).PackFlow uses MLIP energies and forces to improve physical validity and concentrate proposals in low-energy basins, while CLARI reports a 15–30× speedup over OXtal.
  • Generative CSP: Inorganic generative CSP methods jointly or separately model coordinates and lattices using variational, diffusion, and flow-matching formulations (Xie et al., 2022; Jiao et al., 2024; Miller et al., 2024; Luo et al., 2025).CDVAE combines variational autoencoding with diffusion-based coordinate generation, whereas DiffCSP conditions fractional coordinates and lattice on composition.
  • Generative CSP: MOF generation treats metal nodes and organic linkers as modular building blocks, with methods spanning rigid-body lattice and pose generation to building-block and linker-torsion generation (Kim et al., 2025a,b; Jiao et al., 2025; Kim et al., 2026).MOFFlow and MOF-BFN use Riemannian and Bayesian flows, while MOFFlow-2 additionally generates building blocks and linker torsions.

3 Method

Packora formulates molecular crystal structure prediction as conditional generation of atomic coordinates and a canonical lattice from molecular graphs, stoichiometry, formula-unit count, and optional structural information. Its variational flow architecture combines cacheable pairwise reasoning with noisy-state processing to predict coordinate and lattice endpoints.

  • Crystal representation: The crystal representation uses atom types, Cartesian coordinates, and a canonical lattice obtained through Niggli reduction, rotational invariantization, and Cholesky parameterization.These steps reduce equivalent lattice-basis and global-rotation ambiguities before expressing the cell as ℓ ∈ R6.
  • Molecular crystal structure prediction: Packora models pθ(X, ℓ|C) conditioned on molecular graphs, stoichiometry, formula-unit count Z, and optional templates, stereochemistry, or space-group labels.The crystal condition includes C = (G, r, Z, O), while the lattice is represented by a six-dimensional latent ℓ.
  • Training and sampling: Packora trains a variational flow-matching model to predict clean coordinate and lattice endpoints from noisy states, then generates samples by solving an ODE or SDE.A factorized Laplace variational posterior yields a component-weighted L1 objective for coordinates and lattice.
  • Model architecture: The architecture embeds conditions into single and pair representations, refines pairs with four Pairmixer blocks, caches condition-only computation, and injects noisy coordinates and lattice into a DiT trunk.The refined pair representation supplies attention biases, while separate heads predict coordinate and lattice endpoints.
  • Condition and noisy-state embeddings: Condition embeddings integrate atom identities, periodic-table descriptors, charges, optional templates, chirality, space groups, bonds, stereochemistry, and availability masks for missing optional inputs.Coordinate Fourier features, global lattice embeddings, and flow-time embeddings provide noisy-state and temporal information to the transformer.

4 Benchmark Evaluation

Packora is evaluated separately for unranked structure generation and downstream structure ranking across broad molecular-crystal benchmarks. It achieves the strongest reported generation coverage and improves experimental-form recovery, rank, and relaxation speed against CLARI baselines.

  • Evaluation protocol: Generation measures unranked recovery at fixed candidate budgets, whereas ranking evaluates experimental-form rank after relaxation and downstream deduplication.Generation uses collision-free candidates matching at least 8 of 15 molecules with RMSD15 < 2Å for standard SolC; ranking relaxes 1,000 candidates per CSD entry with UMA-S-1.2 and ASE BFGS.
  • 4.2 Structure generation: Packora achieves the highest solve rate on all six generation benchmarks at both 30- and 1,000-candidate budgets under standard and strict criteria, winning 23 of 24 settings.It ties only on the saturated CSP5 result; relative gains reach 54.5% on CSP7 standard SolC at 30 candidates and 188.0% on CSP6 strict Sol15/15 at 30 candidates.
  • 4.3 Structure ranking: Packora-L recovers 19 of 26 single-polymorph targets versus 16 for CLARI-H and lowers mean recovered rank from 4.25 to 2.05.Packora-M recovers 17 targets and reaches mean rank 2.00; the recovery advantage persists across evaluated rank cutoffs.
  • 4.3 Structure ranking: Packora-L reaches 50% relaxation convergence after 98 BFGS steps versus 198 for CLARI-H, while total relaxation takes 10.2 hours versus 16.8 hours.Packora-M completes relaxation in 11.4 hours on eight NVIDIA H200 GPUs.
  • 4.3 Structure ranking: In the multi-polymorph benchmark, Packora-L recovers 21 of 29 forms versus 18 for CLARI-H and lowers mean recovered rank from 102.17 to 54.62.Packora-M recovers 19 forms with mean recovered rank 47.11, and both Packora variants recover at least one polymorph for all eight systems versus seven for CLARI-H.

5 Building the Recipe

Packora’s final recipe is selected through controlled architecture, training, conditioning, inference, and scaling studies under a fixed evaluation protocol. The resulting choices favor Pairmixer, Beta–uniform time sampling, L1 loss, moderate template dropout, 400-NFE EDM–Heun sampling, and balanced width scaling.

  • Evaluation protocol: Controlled ablations train configurations for 300 epochs and evaluate checkpoints on 8,192 validation crystals using 30 candidates per target and a 200-step EDM–Heun sampler.Architecture, training, conditioning, and scaling studies vary one design choice at a time under this fixed protocol; inference studies vary sampler and function-evaluation budgets.
  • Architecture: Pairmixer outperforms Pairformer and DiT with pair-bias attention, while pre-entry injection slightly exceeds post-entry injection but sacrifices caching.SolC is 0.479 for Pairmixer, versus 0.462 for Pairformer and 0.421 for DiT; pre-entry scores 0.479 versus 0.474 post-entry.
  • Training: Random translation, L1 loss, a 10:1 coordinate-to-lattice weight ratio, Muon, and EMA decay 0.9999 are selected as the strongest training choices.Random translation raises SolC from 0.466 to 0.486; L1 reaches 0.486 versus 0.407 for L2, while auxiliary pair supervision adds only 0.004 at 1.37× training cost.
  • Conditioning: Moderate template dropout is most robust across conditioning settings and also improves performance when no optional conditioning is provided.ptpl = 0.5 achieves mean SolC 0.487 across five settings and 0.459 without optional conditioning, versus 0.442 when templates are never observed.
  • Inference: At 400 NFE, EDM–Heun outperforms flow SDE and flow ODE, while increasing to 1000 NFE provides only a small additional gain.EDM–Heun reaches SolC 0.482 versus 0.430 for flow SDE and 0.408 for flow ODE; increasing NFE adds only 0.005.
  • Scaling: Balanced joint scaling of single and pair widths yields a better sampling-budget trade-off than scaling either width alone.Single-only scaling eventually reduces crystal coverage, pair-only scaling is inconsistent, and joint scaling preserves single-sample success while recovering coverage.

6 Conclusion · A Model Architecture Details

Packora is a flexible all-atom variational-flow generator supporting multi-component and organometallic crystals with conditioning on templates, stereochemistry, and space groups. Its evaluated design combines cacheable pair reasoning with balanced representations and achieves leading generation and downstream ranking performance, while the appendix specifies the complete forward pass.

  • 6 Conclusion: Packora supports multi-component and organometallic crystals while conditioning on molecular templates, stereochemistry, and space-group information.The model is presented as a flexible all-atom generator based on variational flow matching.
  • 6 Conclusion: Controlled studies identify a practical recipe using cacheable Pairmixer representations, L1 endpoint regression, EDM–Heun sampling, and balanced single- and pair-track capacity.The scaling study reports that jointly scaling both widths reaches the sampling-budget Pareto frontier, whereas scaling either track alone does not consistently improve the trade-off.
  • 6 Conclusion: Across six benchmarks, Packora achieves the best matched-budget coverage and higher experimental-form recovery, lower ranks, and faster convergence under a shared relaxation and ranking pipeline.The two-track evaluation separates generator-only coverage from end-to-end performance under common downstream processing.
  • 6 Conclusion: The selected recipe is not guaranteed globally optimal because one-at-a-time ablations omit design interactions, and model selection primarily uses k=30 candidates despite budget-dependent trade-offs.Scaling results indicate a trade-off between small-budget success and larger-budget coverage.
  • A Model Architecture Details: The forward pass embeds conditions, refines pair features, injects noisy coordinates and lattice with time, applies pair-biased DiT blocks, and predicts coordinate and lattice endpoints.The algorithm takes a noisy state, flow time, condition, and valid-atom mask as input and returns posterior mean predictions.
  • A Model Architecture Details: Condition embedding combines atom, template, molecular-membership, bond, stereochemistry, and global space-group features, with masks restricting unavailable or invalid entries.Unavailable categorical conditions use learned null categories, while intramolecular pair features are refined independently before Pairmixer processing.
  • A Model Architecture Details: Pairmixer updates only pair representations through outgoing and incoming triangle multiplication followed by gated pair transitions over valid atoms.The pair track is then supplied as a bias to the DiT trunk, whose updates use time-dependent adaptive modulation and gated feed-forward transformations.
  • A Model Architecture Details: Noisy coordinates use fixed random Fourier features, the noisy lattice is broadcast across atoms, flow time uses sinusoidal features, and prediction heads output centered coordinates plus a pooled lattice.The final heads map the single representation to coordinate and lattice endpoints using valid-atom masking.

B Dataset Statistics

This section characterizes the CLARI training and validation splits used for benchmarking, including size changes after hydrogen completion and the 512-atom limit. It also summarizes composition, flexibility, and optional-conditioning availability using a simple rotatable-bond heuristic.

  • Dataset size: Hydrogen completion followed by the 512-atom limit reduces the training split from 917,014 to 912,807 structures and the validation split from 1,048 to 1,047.These statistics are reported before and after preprocessing for completeness.
  • Dataset properties: The reported split statistics cover chemical composition, molecular flexibility, and availability of optional conditioning information.These properties are summarized before and after 512-atom unit-cell filtering.
  • Flexibility definition: Structures are labeled rigid when every molecular component has at most three rotatable bonds and flexible when any component has more than three.The rule, similar to Jin et al. (2025), is a simple heuristic rather than a complete description of molecular flexibility.

C Training and Sampling Hyperparameters · D Structure Generation Evaluation

Packora-M and Packora-L share most training and sampling settings while differing in model capacity, and structure generation is evaluated with a formalized SolC protocol aligned across evaluators. The protocol accounts for collisions, COMPACK matching, and selection among multiple experimental references.

  • C Training and Sampling Hyperparameters: Packora-M and Packora-L differ only in width, head count, and parameter count, with architecture configurations specified in Table 7.Most other settings are shared between the two model sizes unless noted.
  • C Training and Sampling Hyperparameters: Training combines Muon for hidden matrices with AdamW otherwise, a 10^-4 learning rate, warmup, gradient clipping, and EMA; conditioning dropout is 0.5 for templates and stereochemistry and 0.9 for space groups.Augmentations include crystal translation plus template rotation and translation.
  • C Training and Sampling Hyperparameters: Both model sizes use the same sampling and conditioning configuration for benchmark generation.Training uses fixed-shape batches and a two-stage curriculum, assigning structures to atom-count buckets and padding them accordingly.
  • D Structure Generation Evaluation: The generation evaluation formalizes crystal solve rate SolC over targets and generated candidates, using collision status, COMPACK matching at packing size 15, and a selected experimental reference.The protocol defines whether each candidate solves its target through these criteria.
  • D Structure Generation Evaluation: A candidate is collision-free when all nonbonded heavy-atom pairs exceed the collision threshold relative to their van der Waals radii.Collision detection uses distances between nonbonded heavy-atom pairs and their corresponding van der Waals radii.
  • D Structure Generation Evaluation: For targets with multiple experimental references, evaluation selects the best-matching reference from the available reference set.This selection is part of the SolC evaluation definition.
  • D Structure Generation Evaluation: The released CLARI evaluator differs from OXtal in collision detection and experimental-reference selection, so CLARI-M, CLARI-L, and CLARI-H are re-evaluated using OXtal definitions.The re-evaluation enables consistent comparison across evaluators.

E Detailed Structure-Generation Results · F End-to-End Generation with Empirical Z Sampling · G Structure Overlay Visualizations

The appendix details structure-generation metrics, evaluates unknown Z through empirical sampling, and visualizes representative strict Packora-L matches against experiments. Empirical-Z generation remains competitive with supplied-Z generation across candidate budgets and evaluation criteria.

  • E Detailed Structure-Generation Results: Tables 11 and 12 provide detailed structure-generation results at 30 and 1,000 candidates, including standard and strict coverage plus collision, packing, recovery, and solve metrics.Table 11 includes resampling standard deviations at 30 candidates; Table 12 reports the full 30-candidate metric set for Packora-M and Packora-L.
  • F End-to-End Generation with Empirical Z Sampling: The end-to-end setting samples unknown Z from an empirical prior estimated from the CLARI training set rather than assuming a supplied formula-unit count.The prior is constructed by grouping identical molecular components by exact component SMILES and recovering Z and reduced stoichiometry.
  • F End-to-End Generation with Empirical Z Sampling: Inference truncates and renormalizes the empirical prior to Z values producing at most Nmax = 4096 atoms, then passes each sampled composition to Packora.All other inference settings match the main benchmark.
  • F End-to-End Generation with Empirical Z Sampling: Empirical Z sampling remains competitive with supplied Z across standard and strict criteria at effective 30-candidate and exact 1,000-candidate budgets.Figures 15–18 compare the two settings, while Figure 19 tracks their performance gap across candidate budgets on Rigid and Flexible benchmarks.
  • G Structure Overlay Visualizations: Eighteen representative strict matches overlay Packora-L predictions with experimental references, using gray experimental structures and green predictions while omitting hydrogen atoms.Figures 20 and 21 label each overlay with its CSD refcode and RMSD15.

H Detailed Structure-Ranking Results

This appendix provides complete tabular FastCSP structure-ranking results for single- and multi-polymorph benchmarks. It reports energy rank and RMSD30 for the best candidate matching each experimental form, marking unrecovered matches as NR.

  • Benchmark reporting: These tables provide the detailed tabular results underlying the FastCSP structure-ranking benchmarks reported in Section 4.3.The reported quantities are the energy rank and RMSD30 of the best candidate matching each experimental form.
  • Single-polymorph results: The appendix contains the complete FastCSP single-polymorph ranking results, reporting energy rank and RMSD30 (Å) for the best candidate matching each experimental form.NR indicates no recovery, for which RMSD30 is unavailable.
  • Multi-polymorph results: The appendix also contains complete FastCSP multi-polymorph results using the same energy-rank and RMSD30 (Å) reporting scheme for each experimental form.NR indicates no recovery, with the corresponding RMSD30 unavailable.

I Ablation Dataset Preprocessing

The ablation dataset was independently constructed through filtering, structure processing, benchmark exclusion, and deduplication, with stage-wise counts reported in Table 15. The pipeline enforces quality, standardizes structural representations, prevents benchmark leakage, and retains the lowest-R-factor representative among equivalent structures.

  • Preprocessing pipeline: The ablation dataset uses four stages—filtering, structure processing, benchmark exclusion, and deduplication—with remaining structure counts reported after each stage in Table 15.This dataset was constructed independently because public split refcodes were unavailable.
  • Filtering: Filtering retains eligible CSD entries deposited by May 1, 2025, requiring 3D coordinates, R-factor ≤9%, ambient-pressure single-crystal XRD, non-polymeric structures, known space groups, and at most 512 atoms.Crystals are Niggli-reduced and missing hydrogens are added using the CSD Python API, following OXtal (Jin et al., 2025).
  • Structure processing: Structure processing standardizes bonds and records coordinates, unit cells, atomic numbers, charges, stereochemistry, bonds, and space groups while generating independent molecular conformers with RDKit ETKDGv3 (Landrum et al., 2006; Wang et al., 2020).Missing bond types and aromatic or delocalized bonds are standardized with the CSD Python API.
  • Benchmark exclusion: Benchmark exclusion removes entries matching benchmark refcode families or containing benchmark molecular components from OXtal Rigid and Flexible, CSP5–7, and the CSD Teaching Subset.The cited benchmark sources are Jin et al. (2025), Bardwell et al. (2011), Reilly et al. (2016), Hunnisett et al. (2024b), Battle et al. (2010), and Lo et al. (2026).
  • Deduplication: Deduplication groups entries by six-letter CSD refcode family, clusters equivalent structures with pymatgen StructureMatcher, and retains the lowest-R-factor entry using ltol=0.2, stol=0.3, and angle_tol=5°.StructureMatcher is attributed to Ong et al. (2013).

J Detailed Architecture Ablation Results

The architecture ablation compares model size, compute, and best validation SolC across variants under a shared evaluation protocol. Config. (F), the cacheable post-entry design, is selected.

  • Architecture variants are evaluated by parameter count, GFLOPs, and best validation SolC.
  • The comparison uses a shared ≤300-atom, 30-candidate protocol with 200-step EDM–Heun sampling.
  • Config. (F) is the selected cacheable post-entry architecture.
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