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Topology-Stratified Materials Discovery with A Flow-Based Generative Model

Jingyi Zhou, Oyshee Chowdhury, Noah Oyeniran, Chongze Hu

arXiv:2609.26547v1cond-mat.mtrl-scics.AI

TL;DR

Generating physically reliable, previously unknown crystals remains difficult amid complex structures and diverse compositions. UFO-MGen uses Wyckoff-representation topology in a flow-based generative model and outperforms existing models under multi-stability evaluation, with high SUN performance and extrapolation to unseen space groups.

  • Problem

    Physically reliable generation of unknown crystals remains limited by complex structures and diverse chemical environments, particularly for large unit cells.

    Source: §1 · p. 2

  • Method

    UFO-MGen uses a flow-matching generative model with a topology-aware Wyckoff representation to generate crystals across structural and chemical spaces.

    Source: §1 · p. 2, §2.1 · p. 3

  • Results

    UFO-MGen outperforms existing models under MSE and has the highest SUN rate; on MP20, it achieves a 45.3% extrapolation coverage rate by generating 24 unseen space groups.

    Source: §1 · p. 2, §2.5 · p. 6

  • Takeaways & Limitations

    The fine-tuning framework supports property-constrained inverse design, with generated crystals showing potential for mechanically robust materials.

    Source: §2.7 · p. 8

Abstract

from arXiv · show

Accurate generation of crystal structures is the foundation to the discovery of high-performance materials for extreme-environment applications, such as aerospace, additive manufacturing, and fusion energy systems. Although generative modeling has emerged as a promising approach for crystal design, its performance remains limited by the complex crystal structures and diverse chemical compositions. In this work, we develop UFO-MGen, a universal flow-based generative model that learns topological features of Wyckoff representations and leverages this information to accurately generate crystals across vast structural and chemical spaces. Compared with state-of-the-art generative models, UFO-MGen achieves the highest crystal generation success rate under a rigorous multi-stability evaluation framework, the highest SUN (stable, unique, novel) rate, and a remarkable extrapolation capability that has not been reported by previous models. Furthermore, a fine-tuning module is implemented to UFO-MGen for property-constrained crystal generation, enabling the inverse materials design toward target properties. The UFO-MGen opens a new avenue for accelerated materials discovery and providing a foundation for universal materials intelligence.

1 Introduction

UFO-MGen is a flow-matching model for generating crystals across broad crystallographic and chemical spaces. The work evaluates its structures for multiple forms of stability and introduces property-constrained generation for inverse materials design.

1 Introduction

  • Crystal generation remains challenging because of vast crystallographic configurations, diverse chemical environments, and physical constraints, while prior models have limited reliability and extrapolation.AI-driven discovery has largely expanded existing material systems, leaving unexplored systems as a potential source of materials with improved stability, properties, and performance.Source: §1 · p. 2
  • UFO-MGen outperforms existing crystal generative models under multi-stability evaluation and achieves the highest stable, unique, novel (SUN) rate.The evaluation considers thermodynamic, lattice-dynamic, and thermal stabilities.Source: §1 · p. 2
  • UFO-MGen demonstrates extrapolation beyond learned crystallographic space, a capability the authors report had not been shown by prior generative models.Source: §1 · p. 2
  • Compared with traditional diffusion models, UFO-MGen achieves higher accuracy and faster convergence through unified Wyckoff representation and a flow-matching generation engine.Source: §1 · p. 2
  • A fine-tuning module enables property-constrained crystal generation for inverse design toward targeted engineering applications.Source: §1 · p. 2

2 Results

UFO-MGen combines a topology-stratified Wyckoff representation with hierarchical flow-based generation, achieving strong stability, benchmark, and extrapolation results. Fine-tuning and scaffold selection further support inverse design of mechanically robust materials.

2.1 Unified Wyckoff representation

  • The unified Wyckoff representation separates discrete scaffold features from continuous fibers, using scaffold intrinsic dimension and compression ratio to stratify structural complexity.Special Wyckoff positions are treated as shared boundaries between fibers across different intrinsic dimensions; processing yielded 32,550 trainable structures spanning 35 space groups from the 154,875-structure Materials Project database.Source: §2.1 · pp. 3–4

2.2 Hierarchical UFO-MGen

  • UFO-MGen generates crystals through three stages: topology selection, chemical occupancy assignment, and flow-matching generation of lattice parameters and Wyckoff coordinates.The continuous variables are inserted into Wyckoff templates and space-group symmetry operations reconstruct the complete crystal structure.Source: §2.2 · p. 4

2.3 Multi-stability evaluation

  • The multi-stability evaluation requires thermodynamic stability, limited imaginary phonon modes, and retention of the original structure during 300 K molecular-dynamics simulations.Among 50,000 unconditional generations, all met ΔEf < 0; overall MSE rates were 25.5% with CHGNet, 45.5% with MACE, and 44.1% with MatterSim, while screened crystals covered nearly all elements and 34 of 35 training space groups.Source: §2.3 · pp. 4–5

2.4 Extrapolation capability and validation

  • UFO-MGen generated 5,921 fully stable crystals spanning 107 of 195 space groups absent from training, demonstrating extrapolation beyond the unified dataset.DFT calculations on representative interpolation and extrapolation crystals agreed with predicted lattice parameters and MSE screening, supporting the accuracy of its stability predictions.Source: §2.4 · p. 5

2.5 Performance comparison

  • UFO-MGen achieved the highest MSE success rate across individual stability criteria and overall, the highest SUN score at 31.2%, and 45.3% extrapolation coverage across 24 unseen space groups.The 10,000-structure comparison found substantial overlap between known- and unknown-space-group scaffold features in t-SNE, which may enable structural information transfer through continuous fibers.Source: §2.5 · p. 6

2.6 Key Mechanisms of UFO-MGen

  • Ablations show that combining the unified Wyckoff representation with flow matching generates physically reliable structures and converges more effectively than the three alternative models.For complex Al3CrP4 (Drep = 17), UFO-MGen generated a reliable initial structure and rapidly converged to low energy, whereas the ablations failed to generate initial structures or reach the correct configuration; ZnNi3 results showed a similar advantage.Source: §2.6 · pp. 6–7

2.7 Inverse materials design

  • Fine-tuned UFO-Mech shifted generated crystals toward higher mechanical extreme scores, with some predicted values comparable to cubic boron nitride, tungsten carbide, and titanium diboride.Applying a common scaffold to 50 materials improved average bulk, shear, and Young’s moduli by 45%–60% relative to their original structures; DFT validated predicted mechanical properties, and the framework can extend to other target properties.Source: §2.7 · pp. 7–8

S1 Wyckoff Representation of Crystal Structures

The section represents crystals through symmetry-defined Wyckoff orbits rather than redundant atom-coordinate descriptions, and uses scaffold parameters to quantify structural complexity. Analysis of the encoded Materials Project data shows that the trainable scaffold pool spans diverse crystal systems and complexity regimes.

S1.1 Original presentation of Wyckoff points

S1 Wyckoff Representation of Crystal Structures

S1.2 Crystal scaffolds and associated parameters

  • A scaffold records the space group, orbit count, Wyckoff letters, and orbit species; Drep counts remaining Wyckoff and lattice degrees of freedom as a measure of scaffold complexity.The representation shifts complexity from atom count to independent orbits and their remaining degrees of freedom.Source: S1.2 Crystal scaffolds and associated… · pp. 21–23

S1.3 Database analysis using scaffold parameters

  • Encoding 154,875 Materials Project structures covering 89 elements yielded 18,074 scaffolds; validity and support criteria, including ntrain ≥100, selected 32,550 structures across 119 trainable scaffold features.Scaffolds with insufficient examples were retained for dataset statistics and coverage analysis, but not used for primary scaffold-specific flow training.Source: S1.3 Database analysis using scaffold… · p. 23, Figure S1.3 · p. 25
  • The 119 trainable scaffolds span all seven crystal systems and broad Drep–ρ complexity, including high-complexity, weakly compressed monoclinic and triclinic regions, across 35 space groups.The selected space groups range from No. 1 to No. 230.Source: S1.3 Database analysis using scaffold… · pp. 23–24, Figure S1.3 · p. 25

S2 Stratified Wyckoff Space and Unification

Wyckoff crystal space is stratified rather than a single smooth manifold: symmetry-special positions change representation dimension, and physical constraints exclude invalid configurations. UFO-MGen unifies scaffold-specific continuous fibers and their boundary connections to model this structure.

S2.1 Wyckoff-stratified parameter space

  • Wyckoff coordinates approaching special positions increase site symmetry and reduce degrees of freedom, making the representation space a union of manifolds with different dimensions [14, 15].Special subsets include high-symmetry conditions such as x = 0 or x = y; collision subsets contain configurations where orbit distances are too short for atoms.Source: S2.1 Wyckoff-stratified parameter space · p. 26, Figure S2.1 · p. 27

S2.2 Unification of Wyckoff representation

  • 89% of sampled scaffold pairs in MP data have different representation dimensions, challenging fixed-dimensional smooth-manifold models.The analysis motivates Wyckoff unification for processing crystal data.Source: S2.2 Unification of Wyckoff… · p. 28
  • Each scaffold defines a continuous fiber, and special Wyckoff positions form lower-dimensional boundaries connecting fibers of different dimensions in a unified representation.The framework uses topological fiber bundle theory to connect continuous structural variations within scaffolds with discrete transitions between scaffolds.Source: S2.2 Unification of Wyckoff… · p. 28, Figure S2.2 · p. 29
  • Physical constraints alter each fiber’s topology, while a local Riemannian metric describes how lattice and Wyckoff-coordinate changes affect crystal geometry.Atomic overlap, geometric validity, and energy stability exclude physically inaccessible regions; the resulting topology changes are termed physically variable topology.Source: S2.2 Unification of Wyckoff… · p. 28
  • UFO-MGen significantly outperforms counterparts trained on MP databases without the unified Wyckoff representation.Source: S2.2 Unification of Wyckoff… · p. 28

S3 UFO-MGen Architecture and Training Objective · S3.1 Stage I: Hierarchical topology selection (HTS)

UFO-MGen separates crystal generation into topology selection, chemical occupancy, and continuous geometry generation. Stage I predicts a topology scaffold hierarchically from composition, reflecting the separation of discrete Wyckoff topology from later chemical assignment.

S3 UFO-MGen Architecture and Training Objective

  • UFO-MGen uses three stages: HTS predicts a topology scaffold, COM assigns chemical species, and SWG generates continuous geometry for crystal-structure prediction.Stage III combines lattice variables and Wyckoff free coordinates; its flow-matching model generates continuous fibers that are combined with Stage II scaffolds.Source: S3 UFO-MGen Architecture and Training… · pp. 30–31, Figure S3.1 · p. 30
  • HTS predicts the topology-only scaffold τ before species assignment, separating space group, orbit count, and Wyckoff letters from orbit-level chemistry.Stage II assigns species to Wyckoff orbits under chemical constraints including charge balance and neutrality.Source: S3 UFO-MGen Architecture and Training… · p. 30
  • The model first selects a discrete scaffold and its corresponding smooth fiber W0(c), then learns continuous flow within that fiber rather than across mixed dimensions and topologies.The fiber contains valid lattice and free-coordinate variables for the scaffold.Source: S3 UFO-MGen Architecture and Training… · p. 30
  • UFO-MGen factorizes p(τ, s1:K, ξ | y) into conditional topology, occupancy, and geometry distributions, with ξ combining lattice and orbit-coordinate representations.The input y includes crystal information such as chemical species and total atom count.Source: S3 UFO-MGen Architecture and Training… · p. 31

S3.1.1 Architecture of Stage I

  • A two-layer, 256-hidden-dimension MLP maps a 125-dimensional composition vector to topology features, learning a shared scaffold space that is more efficient than atom-by-atom learning.The unified Wyckoff space lets HTS distinguish crystals with different Drep values and learn their topological features.Source: S3.1.1 Architecture of Stage I · pp. 31–32
  • HTS predicts G, K, and ω1:K in hierarchical order, using canonicalized topology labels and an auxiliary route to separate scaffolds and improve prediction.Boundary configurations are re-encoded as low-dimensional scaffolds when criteria are met; otherwise, they are labeled boundary-ambiguous.Source: S3.1.1 Architecture of Stage I · p. 32

S3.1.2 Loss function of Stage I

  • Stage I loss sums G, K, and Wyckoff-letter losses with state, special-scaffold, and ambiguity terms.The state term distinguishes three states; the special-scaffold term maps to low-dimensional topology scaffolds, and the ambiguity term handles boundary structures.Source: S3.1.2 Loss function of Stage I · p. 32

S3.2 Stage II: Chemical occupancy module (COM)

Stage II COM completes a topology scaffold by assigning chemical species to Wyckoff orbits under composition, multiplicity, and chemical-prior considerations, then passes the completed scaffold to Stage III.

S3.2.1 Architecture of Stage II

  • COM autoregressively assigns species to Wyckoff orbits on Stage I’s topology scaffold, using multiplicities and residual composition to meet target composition and produce full scaffold c for Stage III.Source: S3.2.1 Architecture of Stage II · p. 33
  • Assignments integrate global composition, topology and orbit features, and assignment history, while hybrid element embeddings combine learned representations with fixed chemical descriptors as statistical priors, not quantum-mechanical constraints.Source: S3.2.1 Architecture of Stage II · pp. 33–34

S3.2.2 Loss functions of Stage II

S3.2.3 Chemical constrains

  • Oxidation-state balance is a relaxed prior, not a universal hard rule; its weight can be reduced or zeroed for metals, intermetallics, or strongly covalent systems.Source: S3.2.3 Chemical constrains · p. 34

S3.3 Stage III: Structured Wyckoff-aware generation (SWG)

Stage III generates scaffold-conditioned continuous Wyckoff parameters while keeping discrete crystal features fixed. Its masked flow model operates on scaffold-specific fibers, and decoding reconstructs symmetric crystal structures from lattice and orbit coordinates.

S3.3.1 Architecture of Stage III

  • Stage III uses scaffold-conditioned flow matching to generate continuous lattice and orbit coordinates while keeping the scaffold’s discrete features fixed.Scaffold-specific masks let one shared network handle different intrinsic dimensions: the orbit activity mask excludes inactive orbits, and the coordinate mask zeros invalid velocities and excludes padded coordinates from the loss.Source: S3.3.1 Architecture of Stage III · pp. 34–36

S3.3.2 Relation to the Wyckoff fiber

  • Unlike atom-wise generators, Stage III evolves only valid orbit-level free coordinates within the selected scaffold’s Wyckoff fiber, imposing symmetry during generation.The full crystal is reconstructed from Wyckoff coordinate templates and space-group operations.Source: S3.3.2 Relation to the Wyckoff fiber · p. 36

S3.3.3 Decoding for crystal structure prediction

  • The final lattice and orbit coordinates are decoded with the scaffold into physical lattice parameters and a full unit-cell structure.Lattice projection and symmetry refinement correct numerical deviations from finite-step ODE integration; they are cleanup, not additional generative stages.Source: S3.3.3 Decoding for crystal structure… · p. 36

S3.3.4 Loss functions of Stage III

  • Stage III trains by flow-matching velocity regression with equally weighted lattice and orbit losses and a masked orbit loss normalized across active degrees of freedom.The normalization uses ϵ = 10^-6; no additional coefficient is placed between the lattice and orbit terms.Source: S3.3.4 Loss functions of Stage III · pp. 36–37

S3.4 Training objective and loss decomposition

UFO-MGen’s hierarchical objective allows its three stages to be trained independently or jointly fine-tuned after pretraining, without requiring every run to backpropagate through all stages. The loss decomposition assigns distinct topology, occupancy, and flow-bridge objectives to the stages.

S3.4 Training objective and loss decomposition

  • The three-stage architecture supports independent training or joint fine-tuning after pretraining, so not every run must backpropagate through all stages.This applies to the overall LUFO objective in Equation (14).Source: S3.4 Training objective and loss… · p. 37
  • Stage I trains the topology selector with supervised losses for space group, orbit count, and Wyckoff sequence, plus an auxiliary route-role loss.The selector models πα(τ | y).Source: S3.4 Training objective and loss… · p. 37
  • Stage II trains the chemical occupancy distribution using species-assignment loss, composition-consistency penalties, and weak chemical priors.The occupancy model is pϕ(s1 | τ, y), and COM is trained with these objectives.Source: S3.4 Training objective and loss… · p. 37
  • Stage III uses a Gaussian base and Euclidean interpolation in normalized lattice space; this is an engineering choice, not a claimed geodesic on the full lattice manifold.The bridge sets ℓt = (1 − t)ℓ0 + tℓ1 and target velocity uℓ = ℓ1 − ℓ0.Source: S3.4 Training objective and loss… · p. 37
  • Stage III’s orbit bridge uses a uniform torus base, wrapped displacement, and interpolated orbit states and target velocities.The wrap-boundary discontinuity is a measure-zero issue and was not observed as a practical training problem.Source: S3.4 Training objective and loss… · pp. 37–38

S3.5 Sampling procedure

Given a composition-side condition, UFO-MGen generates a crystal through three stages: scaffold sampling or ranking, species assignment, and ODE-based structure generation. Optional filtering, hard composition masking, numerical cleanup, and implementation details constrain or refine the procedure without adding generative stages.

S3.5 Sampling procedure

  • UFO-MGen first samples or ranks topology-fixed scaffolds τ under a composition-side condition y.An auxiliary route role may support ranking or filtering, but is not part of the scaffold.Source: S3.5 Sampling procedure · p. 38
  • In Stage II, the model samples species assignments and updates the residual composition budget after each assignment.A hard composition mask can prevent choices that make the target composition impossible.Source: S3.5 Sampling procedure · p. 38
  • After the scaffold and species assignments are fixed, Stage III initializes and integrates an ODE to generate the structure variables.The current implementation uses 48-step explicit Euler integration with ∆t = 1/48.Source: S3.5 Sampling procedure · p. 38
  • After each integration step, coordinates are wrapped modulo one while inactive dimensions remain fixed by the mask.The decoded output is combined with scaffold c to form a crystal; lattice projection and symmetry refinement are optional numerical cleanup, not additional generative stages.Source: S3.5 Sampling procedure · p. 38

S3.6 Training and sampling pseudocode

UFO-MGen trains topology selection, scaffold construction, and flow-based coordinate generation in separate stages, then samples crystals by composing these components and refining the decoded structure.

S3.6 Training and sampling pseudocode

  • Training alternates topology prediction, teacher-forced scaffold modeling with residual-composition updates, and flow-velocity learning to fit parameters α, ϕ, and θ.The stages compute losses L_I, L_II, and L_III, respectively, and update each parameter set with AdamW.Source: S3.6 Training and sampling pseudocode · p. 39
  • Sampling selects or ranks a topology, builds its scaffold under a residual-composition budget, integrates the learned flow, and decodes and refines the resulting crystal.The flow starts from Gaussian lattice variables and uniform orbital coordinates, advances for T steps, and is followed by lattice projection and symmetry refinement.Source: S3.6 Training and sampling pseudocode · p. 39

S4 DFT Validation of UFO-MGen Prediction

DFT calculations validate UFO-MGen predictions through structural optimization, phonon, and molecular-dynamics tests on representative and additional crystals. The section also compares models under the multi-stability evaluation (MSE) framework.

S4 DFT Validation of UFO-MGen Prediction

S4.1 DFT structural optimization and thermodynamic stability

S4.2 DFT validation of lattice-dynamic stability

  • DFT phonon calculations found no imaginary frequencies in six representative generated crystals, agreeing with uMLIP predictions and confirming their lattice-dynamic stability.Structures with clean spectra or minor imaginary modes were considered stable; minor modes were defined by negative-region integrated phonon DOS below 0.05 phonon modes.Source: S4.2 DFT validation of lattice-dynamic… · p. 41, Figure S4.1 · p. 42

S4.3 AIMD validation of thermal stability

S4.4 DFT validation of additional generated crystals

S4.5 Comparison with Baseline Generative Models

S5 Evaluation Metrics

UFO-MGen is evaluated with established public crystal-generation metrics and separate space-group interpolation and extrapolation rates. The section reports its MP-20 benchmark results and defines the protocols used to assess validity, SUN, geometric error, and coverage.

S5.1 Public metrics

  • The public-metric evaluation reports structural and compositional validity, reference-set coverage, distributional agreement, SUN, and geometric error.Higher values are preferred for validity, coverage, and SUN; lower values are preferred for distribution distances and RMSD.Source: S5.1 Public metrics · p. 47

S5.1.1 Validity, coverage and distribution metrics

  • Public metrics follow benchmark protocols: Val-Struct and Val-Comp assess structural and chemical validity, COV-R and COV-P assess reference matching, and lower dρ, dE, delem, and dsg indicate better distributional agreement.The evaluation uses public evaluators and is separate from MSE validation and uMLIP/DFT screening; unavailable or protocol-mismatched baseline values are marked “/”.Source: S5.1.1 Validity, coverage and… · p. 47, S5.1 Public metrics · p. 47

S5.1.2 Stability, uniqueness and novelty (SUN) metric

  • SUN requires a generated crystal to be stable, unique within the generated set, and novel relative to training or reference structures.Stability follows the MatterGen-aligned relaxed near-hull protocol; uniqueness and novelty use the public evaluator’s structure matcher, and baseline SUN values are taken from published results rather than recomputed.Source: S5.1.2 Stability, uniqueness and… · p. 48

S5.1.3 RMSD and relaxed RMSD

  • RMSD measures geometric difference through matched local atomic coordination, with the benchmark matcher accounting for periodicity, atom ordering, species, and cell choices.The atom mapping pairs reference and generated coordinates; direct coordinate subtraction is not used.Source: S5.1.3 RMSD and relaxed RMSD · p. 48

S5.1.4 Comparison with MP-20 baseline

S5.2 Interpolation and extrapolation metrics

  • UFO-MGen has the highest interpolation and extrapolation coverage among evaluated models, demonstrating generation within and beyond training-set space groups.For fair comparison, UFO-MGen was retrained on MP20 using Stages II and III; Stage I fixes the scaffold and limits exploration of unseen space groups.Source: S5.2 Interpolation and extrapolation… · pp. 49–50, Table S5.2 · p. 50

S6 Ablation Studies of UFO-MGen Performance

Ablations show that UFO-MGen’s three stages govern distinct tradeoffs in stability, extrapolation, compositional novelty, and symmetry-preserving structure generation. Together, the findings support the model’s hierarchical three-stage design.

S6.1 Ablation test for Stage I: HTS

  • 52 previously unseen space groups pass MSE screening without Stage I, but overall stability decreases when its topology scaffold constraint is removed.Stage I guides generation toward stable, physically plausible structures, while training on 35 space groups limits exploration beyond the training domain.Source: S6.1 Ablation test for Stage I: HTS · p. 51, Table S6.1 · p. 52

S6.2 Ablation test for Stage II: COM

  • Without Stage II, crystal stability success rises, but generated structures tend to reproduce known structures rather than discover unique ones.Stage II enables diverse chemical occupancies and elemental compositions; its removal makes generation more deterministic and reduces chemical diversity and novelty.Source: S6.2 Ablation test for Stage II: COM · p. 51, Table S6.1 · p. 52

S6.3 Ablation test for Stage III: SWG

  • Removing Stage III reduces AIMD stability and can cause atomic drift and scaffold collapse, underscoring its role in preserving geometrically valid structures during relaxation.Thermodynamic and lattice-dynamic stabilities show only minor differences after relaxation. One stable structure is assigned to an unexplored space group after minor relaxation changes, but this is not true extrapolation.Source: S6.3 Ablation test for Stage III: SWG · pp. 51–52, Table S6.1 · p. 52

S7 Inverse Materials Design

UFO-MGen is fine-tuned across topology, elemental occupancy, and geometry for property-constrained generation. For mechanical targets, UFO-Mech’s predictions align with DFT, and imposing selected scaffolds improves mechanical properties.

S7.1 Fine-tuning architecture

  • Fine-tuning weights high-property structures across topology, elemental occupancy, and geometry; Stage II also uses COM-ALLOWED to restrict chemistry to elements linked to extreme properties.Property guidance is independently incorporated into the selection of topology scaffolds, elemental occupancy, and continuous geometric degrees of freedom.Source: S7.1 Fine-tuning architecture · p. 53

S7.2 Mechanical property datasets

  • UFO-Mech’s predictor uses 13,686 deduplicated crystals from JARVIS-DFT, Materials Project, and de Jong, split 10,948/1,369/1,369; targets are bulk, shear, and Young’s moduli and Poisson’s ratios.Structures with extreme mechanical properties receive higher training weights.Source: S7.2 Mechanical property datasets · p. 53

S7.3 DFT validation of UFO-Mech

  • For 33 generated crystals, UFO-Mech’s bulk, shear, and Young’s modulus predictions show strong linear agreement with DFT, while candidates span diverse stable structures.DFT validation calculates elastic constants from stress responses to small lattice distortions using the finite-difference method.Source: S7.3 DFT validation of UFO-Mech · pp. 54–55, Figure S7.2 · p. 55

S7.4 Scaffold effect on mechanical properties

  • Imposing three scaffolds improved the mechanical properties of tested crystals by 31.08%, 37.33%, and 32.08%, respectively: (P −1, 2i, 2i), (C2/c, 4e ∗4, 4f ∗4), and (P4/mbm, 2a, 4g, 4h, 8i).Each scaffold was imposed on 50 new crystal structures and compared with the corresponding original structures.Source: S7.4 Scaffold effect on mechanical… · p. 55, Figure S7.3 · p. 56
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