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Orbitol: Crystal Generation on Wyckoff Orbits

Ivan Žugec, Joshua Rosenthal, Ivor Lončarić, Jan-Willem van de Meent, Tin Hadži Veljković

arXiv:2610.30086v1cs.LG

TL;DR

Unconditional crystal generation offers no direct space-group control, although symmetry constrains which responses are allowed. Orbitol generates crystals from Wyckoff-orbit tokens with valid structures invariant under a requested group. On MP-20, 13.8% of its attempts yielded novel, unique, stable or metastable structures retaining that group, compared with 10.8% for SGFM.

  • Problem

    Unconditional generators lack direct control over a crystal’s space group, which constrains whether responses such as piezoelectricity are symmetry-allowed.

    Source: §1 · p. 1

  • Method

    Orbitol uses Wyckoff-orbit tokens to generate orbit atom types and free coordinates plus a compatible lattice, making every valid decoded structure invariant under the requested group.

    Source: §1 · p. 1

  • Results

    On MP-20, 13.8% of Orbitol attempts were MSUN structures retaining the requested group, versus 10.8% for SGFM; screened piezoelectric candidates had nine times the median figure of merit of Materials Project DFPT piezoelectrics.

    Source: §1 · p. 2

  • Takeaways & Limitations

    At the evaluated generation speed, generation is no longer the bottleneck in discovery campaigns, making reinforcement-learning fine-tuning affordable.

    Source: §6 · p. 14

    The empirical prior samples only skeletons seen in training, and benchmark energies are MLIP estimates, whereas piezoelectric candidates are DFT verified.

    Source: §6 · p. 14

Abstract

from arXiv · show

Generative models propose crystalline materials by learning composition and geometry, but unconditional sampling gives no direct control over the space group, which constrains whether responses such as piezoelectricity are symmetry-allowed. We present Orbitol, a crystal generator built on the Crystalite diffusion Transformer that replaces atom tokens with Wyckoff-orbit tokens. Given a space group and a collection of occupied Wyckoff positions, Orbitol generates the atom type and free coordinates of each orbit together with a compatible lattice, so every valid structure is invariant under the operations of the requested group. It generates structures two to three orders of magnitude faster than the symmetry-aware baselines at the evaluated settings. On MP-20, 13.8% of Orbitol's attempts yielded stable or metastable, unique and novel structures that retained the requested space group after relaxation, compared with 10.8% for SGFM, the next best model. We also introduce SPARC, a score that combines retention of the raw detected space group with agreement between the retained-group and training distributions. Finally, we used Orbitol to generate candidates in piezoelectric crystal classes. Of the structures selected by internal screening, 89% passed our density functional theory checks, and 79% also matched no Materials Project or Alexandria entry with the same space group. Their median figure of merit was nine times that of the piezoelectric materials computed in the Materials Project, and 24% of the new candidates ranked in that set's top 10%.

1 Introduction

Unconditional crystal generators do not directly control space group, although symmetry determines which responses are allowed. Orbitol generates crystals from Wyckoff-orbit tokens, with valid decoded structures invariant under a requested group; its MP-20 results include strong discovery yield and targeted piezoelectric candidates.

1 Introduction

  • Unconditional generators lack direct space-group control, while a group label alone does not guarantee that generated structures are invariant under that group’s operations.Space-group symmetry constrains allowed responses: bulk piezoelectricity requires a noncentrosymmetric class other than 432, while pyroelectricity and conventional ferroelectricity require a polar class.Source: §1 · p. 1
  • Orbitol replaces atom tokens with Wyckoff-orbit tokens and, given a space group and occupied rows, generates orbit atom types, free coordinates, and a compatible lattice.Each valid decoded structure is invariant under the requested group’s operations; its geometric attention accounts for all members of each orbit.Source: §1 · p. 1
  • Figure 1 illustrates the FeSe P4/nmm skeleton and plots MP-20 discovery yield against relaxed-group Jensen–Shannon divergence for ten symmetry-aware models.For FeSe, the skeleton fixes rows 2a and 2c; Orbitol predicts orbit atom types, Se’s free height, and lattice parameters a and c.Source: §1 · p. 1, Figure 1 · p. 2
  • Orbitol samples skeletons at training frequencies and evaluates symmetry after unconstrained relaxation, when the detected group can change.SPARC combines retention of the raw detected space group with agreement between retained-group and training distributions; high retention alone can result from sampling a few groups that rarely change.Source: §1 · p. 2
  • 13.8% of Orbitol attempts yielded stable or metastable, unique and novel structures retaining the requested group, versus 10.8% for SGFM; overall MSUN was 19.0%, versus 18.1% for SGEquiDiff.The introduction also reports that 89% of internally screened piezoelectric candidates passed DFT checks, 79% matched no MP or Alexandria entry with the same group, their median figure of merit was nine times the Materials Project DFPT piezoelectrics, and 24% ranked in that set’s top 10%.Source: §1 · p. 2

2 Related Work

Related work is organized by the symmetry information models receive or generate and whether their construction guarantees invariance under the chosen operations. Orbitol is an enforced generator that receives Wyckoff assignments and generates only orbit free coordinates.

2 Related Work

  • Crystal generators differ in whether they specify or generate symmetry information and whether their construction guarantees invariance under the chosen group’s operations.The paper classifies the discussed models along these dimensions in Table 2 and Figure 8.Source: §2 · p. 3
  • Agnostic models specify no space group for an individual sample; examples include atom-coordinate diffusion models, flow generators, and Crystalite’s diffusion Transformer on atom tokens.MatterGen’s base model and DiffCSP diffuse ambient coordinates with the lattice, while Crystalite uses interatomic distances as attention biases.Source: §2 · p. 3
  • Conditioned models may receive or declare a group without constraining both coordinates and lattice, so expanding positions by group operations does not guarantee whole-structure invariance.MatterGen’s adapter conditions on a group label; CrystaLLM variants and PLaID++ declare groups and expand positions, but leave the lattice unconstrained.Source: §2 · p. 3
  • Enforced generators build invariance into construction, differing in whether they generate Wyckoff assignments, receive them, or infer compatible rows implicitly.CrystalFormer and SGEquiDiff generate orbit choices; DiffCSP++ and SGFM receive assignments, while SymmCD predicts site-symmetry information and projects representatives onto compatible positions.Source: §2 · p. 3
  • Orbitol receives Wyckoff assignments like DiffCSP++ and SGFM but generates only orbit free coordinates; it also samples and fixes skeletons from training data.Unlike those coordinate models, Orbitol replaces Crystalite’s atom tokens with orbit tokens and reduces to Crystalite in P1.Source: §2 · p. 3
  • The detected group can exceed the enforced group, so the paper evaluates agreement with the requested group separately from retention of the detected group through relaxation.Source: §2 · p. 3

3 Orbitol

Orbitol replaces atom tokens with one token per occupied orbit, conditioning generation on a space-group and Wyckoff-row skeleton. It generates orbit atom types, free coordinates, and lattice entries jointly.

3.2 Orbit and lattice representation

  • Given a skeleton, Orbitol generates one atom type and free coordinates per orbit, plus only the lattice entries left free by the crystal family.The row charts expand each orbit's representative into its symmetry-related members.Source: §3 · p. 3, §3.2 · p. 5

3.1 Crystals and symmetry

  • A Wyckoff row groups symmetry-related points into an orbit, and a skeleton records the space group and occupied rows, including repeats.Each orbit has one atom type; fixed rows occur at most once, while multiple distinct orbits may occupy a row with free coordinates.Source: §3.1 · pp. 4–5

3.3 Diffusion formulation

  • Orbitol measures free-coordinate noise by the resulting atomic displacement, averaging over group rotations so orbit members receive equal displacement.The coordinate-noise metric is fixed and does not depend on the generated lattice.Source: §3.3 · p. 6

3.4 Tokens and network

  • A Crystalite Transformer processes one token per orbit and a lattice token; orbit embeddings use representative positions, while row and group embeddings identify their geometry and symmetry.The embedding informs the network, but the row maps and lattice chart—not the embeddings—ensure valid decoded structures have the requested symmetry.Source: §3.4 · pp. 6–7

3.5 Denoising targets

  • For periodic free coordinates, Orbitol trains with a posterior-mean target that averages equivalent coordinate images by their Gaussian probabilities.The sine skip also respects periodicity by shifting the prediction when the noisy input is shifted by an integer.Source: §3.5 · p. 7

3.6 Geometry in orbit attention

  • Orbitol contracts atom-pair geometry biases into orbit-pair attention biases, retaining one representative query row and summing key-orbit attention weights.The log-sum-exp preserves the attention output of an expanded layer whose members share each orbit's key and value; averaging biases would change total attention assigned to an orbit.Source: §3.6 · p. 8, Figure 3 · p. 8

3.7 Sampling

  • For de novo generation, Orbitol samples skeletons from their empirical training distribution, so each generated structure shares its skeleton with a training structure.The MP-20 training set has 3,290 distinct skeletons; 5.2% of validation and 4.9% of test structures have skeletons absent from training.Source: §3.7 · p. 8

3.8 Equivalence to Crystalite in P1

  • In P1, Orbitol can reproduce Crystalite's representation and network exactly when the type encodings, weights, execution modes, sampler settings, objective, and embeddings are matched.This equivalence uses objective M, disables GEM, and sets row and group embeddings to zero.Source: §3.8 · p. 9

4 Experiments

Orbitol was evaluated on MP-20 and Alex-MP-20 for discovery yield, symmetry after relaxation, and sampling cost. On MP-20, it led the evaluated symmetry-aware models in joint yield, while SPARC assessed symmetry retention together with distribution agreement.

4.1 Setup

  • The evaluation used 2,500 attempts per model, counted generation and structure-check failures, and assessed symmetry after unconstrained relaxation and detection at 0.1 Å.MP-20 and Alex-MP-20 training data were restricted to cells with at most 20 atoms.Source: §4.1 · p. 9

4.2 Evaluation metrics

  • SPARC combines raw-group retention with agreement between the retained-group and training distributions, since either retention or coverage alone can mislead.Source: §4.2 · pp. 9–10

4.3 Discovery yield

  • On Alex-MP-20, Orbitol reached 34.2% MSUN and retained its requested group in 71.6% of attempts.Source: §4.3 · p. 10
  • 13.8% joint yield was Orbitol’s highest observed result among evaluated symmetry-aware models on MP-20, versus 10.8% for SGFM.Orbitol also reached 19.0% MSUN, versus 18.1% for SGEquiDiff.Source: Table 1 · p. 10, §4.3 · p. 10

4.4 Symmetry after relaxation

  • Among enforced models, Orbitol and SGFM had the lowest relaxed P1 shares and JSDs; Orbitol retained its requested group in 74.7% of attempts.The JSD difference was within the finite-sample spread of an ideal sampler, and more sampling seeds would be needed to resolve it.Source: §4.4 · p. 11
  • PLaID++ retained the most declared groups but had a relaxed JSD of 0.257 bits; SPARC scored Orbitol 73.8 and PLaID++ 39.7.SPARC combines raw-group retention with agreement between retained groups and the training distribution.Source: §4.2 · p. 10, §4.4 · p. 11

4.5 Sampling cost

  • Orbitol sampled about 920 structures per second in large batches, about 310 times faster than SGEquiDiff, using a 25-step sampler and implementation optimizations.These included one token per orbit, packing crystals without padding, compiled CUDA graphs, and vectorized decoding.Source: §4.5 · p. 12

5 Piezoelectric Candidates

Orbitol was used to search piezoelectric crystal classes, with selected candidates evaluated by DFT. The 55 candidates passing both stability checks had a higher median figure of merit than the Materials Project comparison set, and most matched no MP or Alexandria entry with the same space group.

5 Piezoelectric Candidates

  • Orbitol samples from piezoelectric crystal classes, then retains structures whose space group survives relaxation and whose energy is within 0.1 eV/atom of the Alex-MP-20 DFT hull.Of 62 structures selected by a machine-learned screening pipeline, four with PBE band gaps at most 0.03 eV were excluded from response calculations, leaving 58 candidates.Source: §5 · p. 12
  • The transverse energy-harvesting figure of merit d31g31 = d2 31/(ε0εT 33) was maximized over orientations, with εT evaluated at fixed stress.The comparison used the 702 Materials Project materials with a non-zero piezoelectric tensor.Source: §5 · pp. 12–13
  • 55 candidates passed both stability checks; 54 lay within 0.1 eV/atom of the MP and Alexandria hulls, and 49 matched no entry with the same space group.The 55 were the structures without imaginary optical Γ-point modes that were also mechanically stable.Source: §5 · p. 13
  • 1.23 × 10−12 m2 N−1 was the candidates’ median figure of merit, versus 0.133 for MP; 13 of 55 exceeded MP’s 90th percentile of 3.09.The candidates were preselected for high predicted figure of merit; 12 of the 49 that matched no Alexandria structure exceeded MP’s 90th percentile.Source: §5 · p. 13

6 Conclusion

Orbitol generates symmetry-constrained crystals from Wyckoff-orbit tokens, and its conclusion reports fast sampling and strong piezoelectric candidates. The appendices classify generator approaches and detail Orbitol’s row, lattice, coordinate, and network design choices.

6 Conclusion

  • Orbitol builds the requested symmetry into every valid decoded structure; on MP-20 it had the highest MSUN and joint yield among evaluated symmetry-aware models, with fidelity comparable to the best baselines and much faster sampling.For piezoelectric candidates, most screened structures passed DFT checks, most matched no MP or Alexandria entry with the same space group, and their median figure of merit was about nine times the Materials Project’s DFPT piezoelectrics.Source: §6 · p. 13

A Classification of crystal generators

B Orbitol definitions and implementation

  • The appendix describes Orbitol’s definitions, implementation choices, network inputs, conditioning, training losses, orbit attention, sampling, and reduction to Crystalite in P1.These passages map the scope of the implementation appendix rather than report new benchmark results.Source: B Orbitol definitions and implementation · p. 21

B.1 Detailed example of Wyckoff rows

  • Rutile uses Wyckoff rows 2a and 4f, while CrF4 uses three rows for four orbits, including two independently parameterized F orbits on 4g.The four CrF4 orbit tokens have five free coordinates total, and its lattice token specifies tetragonal lengths a and c.Source: §B.1 · p. 21, Table 3 · p. 21

B.2 Wyckoff tables, preprocessing and lattice charts

  • Wyckoff tables use PyXtal’s default Hall setting, with trigonal groups in hexagonal axes; preprocessing admits structures only after orbit expansion passes a round-trip check.The lattice chart encodes crystal-family constraints, while fixed rows receive neither coordinate noise nor coordinate loss.Source: §B.2 · pp. 21–22

B.3 Chart metric, noise and preconditioning

  • Orbitol uses a group-invariant metric for displacement and chart noise because fractional rotations need not preserve Euclidean lengths.The coordinate noise and readout are defined on active chart coordinates, and fixed rows receive no coordinate noise.Source: §B.3 · p. 22

B.4 Network inputs, conditioning and coordinate variant

  • Orbitol-X is the main-text model; its representative-coordinate input is periodic under integer chart lifts but does not identify every additional fractional translation in Λw.The appendix also describes a variant with a different coordinate input and readout, while both variants use hybrid row and group embeddings.Source: §B.4 · p. 23

B.5 Periodic target and training losses

The appendix specifies Orbitol’s periodic-coordinate training target, orbit-aware attention, sampling and decoding, and implementation checks. It also records the scope of the experimental details supporting Section 4.

B.5 Periodic target and training losses

  • The periodic-posterior target P averages periodic images with conditional Gaussian weights, unlike objective M’s nearest-image residual; the authors do not compare trained M and P models.At high noise, the wrapped density becomes uniform and the target approaches uσ without preferring a chart origin.Source: §B.5 · pp. 23–24
  • Coordinate losses average over active real orbit tokens, while type error sums over 16 features and lattice error averages across six slots and structures.The periodic target uses adaptive image sums at low noise and a dual-lattice Fourier series at higher noise, with an identity limit when its correction is negligible.Source: §B.5 · p. 24

B.6 Geometry in orbit attention

  • Orbit attention sums key-member weights so tied orbit values preserve the expanded weighted attention output up to rounding.With exact distances in a symmetry-compatible cell, changing the query representative permutes each key orbit’s distance multiset; finite image searches need not preserve this invariance for arbitrary noisy cells.Source: §B.6 · p. 25

B.7 Sampling, decoding and valid outputs

  • Sampling uses Crystalite’s EDM schedule, stochastic churn, and Heun correction; the decoder checks lattice compatibility and whether group operations map atoms to same-type periodic images.Invalid or nonfinite states, incompatible structures, and failed group containment are rejected and retained in output metadata.Source: §B.7 · pp. 25–26

B.8 P1 reduction and verification

  • In P1, Orbitol matches Crystalite’s denoiser, loss, and sampler when encodings, weights, execution modes, and sampler settings match, with specified symmetry features disabled.Tests include zero-tolerance comparisons of model outputs and sampler states, alongside integration tests that trained and sampled under the implemented coordinate objectives.Source: §B.8 · p. 26

C Experimental details and supporting results

  • The appendix supports Section 4 with evaluated configurations, scoring definitions, SPARC derivation, full results, coordinate-configuration comparisons, sampler settings, and sampling timings.The appendix describes these as experimental details and supporting results.Source: C Experimental details and supporting… · p. 26

C.1 Data, training and evaluated configurations

Orbitol samples space-group and Wyckoff-row skeletons from a training cache, then generates types and geometry rather than copying them from templates. The section records the evaluated configurations and checkpoint and sampling choices.

C.1 Data, training and evaluated configurations

  • Orbitol samples skeletons from the training cache, which supplies the space group and untyped Wyckoff rows; its empirical prior does not generate unseen skeletons.The MP-20 orbit cache contains 27,136 structures, while the Alex orbit cache retains 540,159 structures.Source: §C.1 · p. 26
  • The main MP-20 checkpoint was selected by highest observed in-training MSUN, whereas Alex-MP-20 uses the checkpoint at 650k steps.MP-20 screens used 10,000 attempts and relaxed constructed structures for 500 FIRE steps; the selection score counts failures as zero and differs from the LeMat benchmark.Source: §C.1 · pp. 26–27
  • Orbitol-X uses representative inputs and projection readout, while Orbitol-I uses chart-coordinate inputs and direct readout; both use width 512, distance GEM, target P, and EMA weights.Each sampling seed gives 2,500 attempts.Source: Table 5 · p. 27
  • Independent training replicates are unavailable.The main MP-20 run used compiled training steps and compiled GEM geometry, batched periodic targets, device-side type encoding, and fused AdamW/EMA updates.Source: §C.1 · pp. 26–27

C.2 Evaluation and failure accounting

The evaluation defines energy, novelty, and symmetry measurements with explicit attempt denominators and failure handling. It also specifies the model pools and relaxation and detection procedures used to produce those measurements.

C.2 Evaluation and failure accounting

  • LeMat thresholds the arithmetic mean of available ORB, MACE, and UMA energies above their respective hulls at 0 and 0.1 eV/atom for SUN and MSUN, respectively.A structure with no available model value counts as neither stable nor metastable.Source: §C.2 · p. 28
  • Evaluated models contribute attempts 0 through 2499, retaining generation, decoding, and structure-check failures; yield and requested-group retention use all 2,500 attempts.PLaID++ rates describe its first 2,500 released decoded rows, not necessarily all original language-model draws, because upstream filtering and losses are unknown.Source: §C.2 · p. 28
  • Space groups are detected with spglib 2.7.0 using 5° angle tolerance and positional tolerances of 0.1 Å and 0.01 Å.Relaxed symmetry excludes failed or implausible endpoints, including those with relaxed-to-raw volume ratios outside [0.5, 2] or interatomic distances below 0.5 Å.Source: §C.2 · p. 28
  • Requested-group retention requires exact equality between the requested and detected groups, while own-group strict retention measures whether a raw-detected group survives relaxation.Own-group strict retention uses raw-detected structures as its denominator; it is not interchangeable with requested-group retention’s attempt denominator.Source: §C.2 · p. 29
  • Joint yield divides the count of LeMat SUN and disjoint metastable structures that retain symmetry by all 2,500 attempts; retention among MSUN attempts instead divides by the MSUN count.These metrics are averaged over the three Orbitol-X MP-20 seeds.Source: §C.2 · p. 29

C.3 SPARC

SPARC combines raw detected-group retention with agreement between retained-group and training distributions. The section defines its interpretation and compares raw-group scoring with a requested- or declared-group variant.

C.3 SPARC

  • SPARC is raw-group retention Rraw multiplied by the squared Bhattacharyya coefficient between training and retained-group distributions.If no attempt is retained, SPARC is zero, and the raw detected group makes the score applicable to agnostic models.Source: §C.3 · p. 29
  • For Orbitol-X on MP-20 at 0.1 Å, raw-group retention is 76.6% and requested-group retention is 74.7%.These rates use different definitions; SPARC uses raw detected groups rather than requested groups.Source: §C.3 · p. 29
  • A generator retaining only P1 scores 1.7 on MP-20 even if it retains every structure’s group; SPARC reaches 100 only with full retention and a matching retained-group distribution.The score uses every group without a minimum count, and all 2,500 attempts remain in the denominator.Source: §C.3 · p. 29
  • An ideal sampler’s expected SPARC at 2,500 attempts is 97.9 on MP-20 and 98.3 on Alex-MP-20, estimated from 2,000 simulated sets.These are reference means, not hard upper bounds for individual samples; the three Orbitol-X seeds have mean SPARC 73.8 and average 95% percentile intervals of 70.6–74.0.Source: §C.3 · p. 29
  • Using requested or declared groups instead gives SPARC values of 72.1 for Orbitol-X, 73.5 for CrystaLLM, and 70.6 for SGFM on MP-20, swapping Orbitol-X and CrystaLLM’s order relative to raw SPARC.The reported model table distinguishes valid, unique, novel, SUN, and MSUN attempt percentages, alongside raw P1 and relaxed symmetry results.Source: §C.3 · p. 29, Table 7 · p. 30

C.4 Additional results and reported comparisons

The appendices extend the symmetry and generation comparisons while qualifying how relaxation variability, detection tolerance, and evaluation settings affect interpretation. They also report coordinate-configuration comparisons, but do not establish a general yield advantage for those changes.

C.4 Additional results and reported comparisons

  • Tables 7 and 8 provide generation and symmetry diagnostics on MP-20 and Alex-MP-20, while Table 8 defines retention, SPARC, P1, and JSD reporting conventions.Table 8 uses 2,500 attempts, symprec = 0.1 Å, and counts failed relaxations and implausible endpoints as retention and SPARC losses.Source: §C.4 · p. 30, Table 8 · p. 32
  • Up to 18 of 500 fixed structures changed detected group across compute nodes, so 1–3-point retention differences can fall within relaxation variability.This repeat-relaxation check measures variability for fixed inputs, not variation across independent training or sampling seeds.Source: §C.4 · p. 30
  • At 0.01 Å detection tolerance, Figure 9 repeats the main symmetry comparison, but the values depend on tolerance and relaxed geometry and do not establish dynamical stability.The figure uses the same models, samples, relaxation, and endpoint exclusions as Figure 4.Source: §C.4 · p. 30, Figure 9 · p. 31

C.5 Coordinate-configuration comparisons

  • The coordinate-configuration pairs do not establish yield advantages for orbit representation over Crystalite or for P over M at matched training.The MP-20 pair changes coordinate skip; the Alex pair changes both input and readout, and its higher observed MSUN is not a statistically established chart-input benefit.Source: §C.5 · p. 31, Table 11 · p. 34
  • The ideal-sampler JSD spread at 2,500 draws is 0.0114–0.0160 bits, making the Orbitol–SGFM JSD difference too small to resolve without more samples.The ideal-sampler mean is 0.0137 bits.Source: §C.5 · p. 32
  • 4.7–13.4 percentage points higher measured MSUN rates were obtained for the six symmetry-aware models in the evaluation and leaderboard comparison.Some leaderboard scores predate the benchmark’s MACE checkpoint fix, and the GEODE MSUN column excludes SUN.Source: §C.5 · pp. 33–34

C.4 Additional results and reported comparisons

  • Tables 9 and 10 report additional LeMat diagnostics, but scores are comparable only within their respective blocks.LeMat-Bulk includes Alexandria, so its novelty reference overlaps the training source; the GEODE and leaderboard blocks use distinct evaluation settings.Source: §C.4 · p. 31, Table 9 · p. 33, Table 10 · p. 34

C.6 Sampler settings and sensitivity

The sampler sweep varies settings for the main MP-20 Orbitol-X checkpoint, with each setting evaluated on 10,000 attempts under a common in-house relaxation and scoring setup. The observations support the chosen setting without establishing statistical equivalence between samplers.

C.6 Sampler settings and sensitivity

  • 25-step and 150-step settings have similar observed yield, while 50 steps change little from 25 and 10 steps reduce observed MSUN.The screen uses 10,000 attempts per setting, NequIP-OAM-L relaxation for 500 FIRE steps, and MSUN over all attempts.Source: §C.6 · p. 35
  • At 25 steps, raising σmin from 0.002 to 0.01 improves one-shot MSUN and lowers median initial force, while raising it to 0.05 reduces one-shot and relaxed MSUN.The selected setting is supported by these observations, but the comparisons do not establish statistical equivalence between samplers.Source: §C.6 · p. 35

C.7 Sampling throughput and implementation

The throughput account combines sampler settings with packing, compilation, and decoding optimizations, measured under a specified single-GPU timing procedure. The implementation reduces token and attention overhead, but the timings do not isolate each component’s contribution.

C.7 Sampling throughput and implementation

  • Timing measures prior draw through in-memory pymatgen structures per 1,000 attempts, including failures, but excludes loading, compilation, file writes, and relaxation.Runs used one RTX 6000 Ada after warm-up; Orbitol timings average two complete pools, while each baseline value is one measurement.Source: §C.7 · p. 35
  • Orbitol averages 4.70 orbits versus 10.4 atoms per MP-20 stored cell, roughly halving token count and quartering attention pairs.Packing groups whole crystals into fixed 8,192-token chunks; at 256 attempts, the 25-step sampler takes 3.4 seconds per 1,000.Source: §C.7 · p. 35
  • At width 512, compilation captures the fixed-shape transformer trunk as a reusable CUDA graph, while embeddings, GEM geometry, and output heads remain outside it.Compilation and graph capture take about 17 seconds per process and are excluded from timings.Source: §C.7 · p. 35
  • The fast decoder agrees with the reference decoder on atom types, masks, and lattices bitwise, with fractional coordinates within 10^-7 on tested outputs.Its fixed cost makes it slower for a single structure, and fallback is used for uncertified reductions or structures near numerical boundaries.Source: §C.7 · p. 36
  • The 150-step configuration is about 60 times faster than SGEquiDiff, but the 5.40-to-1.09-second comparison does not isolate a step-count speedup.Complete-pool timing includes prior draws, decoding, and structure construction; timings combine sampler choices with implementation choices.Source: §C.7 · pp. 35–36, Table 13 · p. 37
  • Packing and compilation preserve computed quantities but can change floating-point sampling trajectories enough to yield different decoded structures.Decoder agreement on fixed sampled states therefore does not imply bitwise agreement between sampling trajectories.Source: §C.7 · p. 36

D Piezoelectric screening and DFT validation

The study selects generated piezoelectric candidates, evaluates their electronic, mechanical, and vibrational stability, and computes response figures of merit from DFT tensors. The results include novel structures, but Γ-point stability does not establish stability throughout the Brillouin zone, and the largest figures of merit occur near elastic instability.

D Piezoelectric screening and DFT validation

  • The 62 selected structures were screened for high predicted figure of merit, and four with PBE band gaps at most 0.03 eV were excluded from response calculations.The DFT settings use PBE and VASP; magnetic orders other than ferromagnetic were not considered for Mn-, Fe-, and Co-containing materials.Source: D Piezoelectric screening and DFT… · p. 37
  • Displacing atoms along this mode (largest displacement 0.05 Å) in the same 18-atom cell and relaxing the cell and atoms with the same settings yielded P21, 0.42 meV/atom below Cmc21, with no unstable Γ mode.Source: D Piezoelectric screening and DFT… · p. 38
  • The largest figures of merit, 308 and 109 in 10^-12 m2 N^-1, belong to structures near elastic instability, where a nearly singular elastic tensor inflates d = e C^-1.Their smallest elastic-tensor eigenvalues are 0.72 and 1.25 GPa.Source: D Piezoelectric screening and DFT… · p. 38
  • Γ-point phonons identify two candidates with instability, while two candidates fail positive-definite elastic stability; the Γ test does not establish full dynamical stability.One candidate also fails at Γ, and the two mechanically unstable candidates have undefined piezoelectric strain coefficients.Source: D Piezoelectric screening and DFT… · p. 38
  • The figure of merit is computed from d = e C^-1 and maximized over polarization/electrode direction p and in-plane strain axis u perpendicular to p.The formulation uses the transverse open-circuit response of a thin plate and neglects fringing fields.Source: D Piezoelectric screening and DFT… · p. 38
  • The Materials Project reference comprises 702 materials with nonzero piezoelectric tensors, after excluding two class-432 materials from 704 tensor-bearing materials.The comparison uses two DFT datasets with similar but not identical settings.Source: D Piezoelectric screening and DFT… · p. 38
  • 46 of 55 candidates match no Materials Project or Alexandria entry even at default matcher tolerances; six have identical Alexandria structures and three match entries only in different space groups.The same-structure criterion requires matching space groups and a stricter StructureMatcher match.Source: D Piezoelectric screening and DFT… · p. 39

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