Symmetry- and Property-Aware Crystal Generation with Reinforcement Learning for Inverse Materials Design

Ting-Wei Hsu; Arun Bansil; Qimin Yan

cond-mat.mtrl-sciSep 11, 2026

Symmetry- and Property-Aware Crystal Generation with Reinforcement Learning for Inverse Materials Design

Authors: Ting-Wei Hsu, Arun Bansil, Qimin Yan

Organizations: Department of Physics, Northeastern University, Boston, 02155, Massachusetts, USA. · Quantum Materials and Sensing Institute, Northeastern University, Burlington, 01803, Massachusetts, USA.

Abstract

The inverse design of crystalline materials ultimately seeks structures with desired physical properties. However, for many functional responses, a favorable numerical value is meaningful only when supported by the symmetry of the underlying crystal. Without the appropriate crystallographic constraints, an apparent response may be ill defined, accidental, or not symmetry protected. Here we introduce SPARC, a symmetry- and property-aware reinforcement learning framework that optimizes physical objectives while preserving the structural conditions required for their realization. We demonstrate SPARC on two complementary tasks. The first targets strong uniaxial dielectric anisotropy, a tensorial response that is well defined only within appropriate crystal classes. The second maximizes the spectroscopic limited maximum efficiency, a scalar device-level objective without a prescribed symmetry class, allowing the framework to identify favorable crystallographic motifs. These results show that symmetry is not merely an additional design constraint, but a physical foundation for generating candidates with meaningful, robust, and realizable functional properties.

Figures & tables

Figure 1: Schematic of the SPARC pipeline. Each reinforcement-learning iteration consists of five stages. (1) The policy samples a space group

G

from the proposal distribution

\pi^{(j)}

and the number of representative asymmetric-unit sites

M

from the space-group-dependent prior

p(M\mid G)

. (2) The diffusion policy

p_{\theta}

denoises the asymmetric-unit representation

\mathcal{M}^{\prime}=(\boldsymbol{A}^{\prime},\boldsymbol{F}^{\prime},\boldsymbol{\Sigma}^{\prime},\boldsymbol{k})

. (3) The full crystal

\mathcal{M}=(\boldsymbol{A},\boldsymbol{F},\boldsymbol{L})

is reconstructed by applying the symmetry operations of

G

. (4) The reconstructed structures are relaxed, filtered, and evaluated for the target property using a surrogate model. (5) Each resulting structure is assigned a terminal reward

r(\mathcal{M})

. The terminal reward drives two decoupled optimization processes. In the inner loop, the diffusion policy parameters

\theta

are fine-tuned over multiple steps using a reward-weighted policy gradient together with a KL-divergence penalty that anchors the policy to the pretrained generator

p_{\mathrm{pre}}

. In the outer loop, the space-group proposal

\pi^{(j)}(G)

is updated using rewards aggregated according to the realized space group, thereby increasing the sampling probability of productive symmetries relative to the dataset prior

\pi^{(0)}

. Throughout the figure panels, SG abbreviates space group.

Figure 2: Steering generation toward uniaxial dielectric anisotropy. (a) The SPARC loop specialized to the dielectric task. A space group $G$ is drawn from the current proposal $\pi^{(j)}$ , and the space-group-conditioned diffusion model $p_{\theta}(\mathcal{M}^{\prime}|G,M)$ generates a crystal. The TSENN surrogate predicts the static dielectric tensor $\boldsymbol{\varepsilon}$ and band gap $E_{g}$ . The uniaxial reward is defined as $r_{\mathrm{uni}}=\min[\Phi(q_{\mathrm{uni}}),\Phi(E_{g})]$ , where $q_{\mathrm{uni}}=g_{z}(1-m)$ , $m=|\varepsilon_{xx}-\varepsilon_{yy}|/(|\varepsilon_{xx}|+|\varepsilon_{yy}|)$ , and $g_{z}$ excludes cubic tensors. The band-gap term favors $E_{g}=0.3$ – $0.8$ eV. The resulting reward updates both the policy $p_{\theta}$ and the space-group proposal $\pi^{(j)}$ . Generated structures are filtered to stable, unique, and novel (S.U.N.) candidates before scoring. The worked example is a generated $\text{Cd}{\vphantom{\text{X}}}_{\smash[t]{\text{2}}}\text{SnBr}{\vphantom{\text{X}}}_{\smash[t]{\text{6}}}$ with $P\bar{3}1m$ symmetry whose predicted tensor is uniaxial, $\boldsymbol{\varepsilon}=\mathrm{diag}(6.2,6.2,4.7)$ with $\varepsilon_{xx}=\varepsilon_{yy}\neq\varepsilon_{zz}$ and vanishing off-diagonal entries. (b) Mean reward versus RL step during fine-tuning (raw per-step mean in grey, exponential moving average with span 10 in black), and the uniaxial-anisotropy score $r_{\mathrm{uni}}$ climbs from $0.17$ to $0.56$ . (c) The fifteen most frequent realized space groups among the generated structures (inset, full distribution over all 230 groups). Sampling concentrates on the hexagonal, trigonal, and tetragonal groups that admit a uniaxial dielectric tensor, led by $P\bar{3}1m$ and $P4_{2}/mnm$ , followed by $R\bar{3}$ , $P\bar{3}$ , and $P{3}_{1}$ . Together, these five groups account for approximately 44% of the generated structures.

Figure 3: Ablation of symmetry enforcement and pre-reward relaxation. SymmCD and DiffCSP are fine-tuned using the same uniaxial-anisotropy reward. For each backbone, MLIP relaxation is either enabled or disabled before reward evaluation. (a) Mean reward as a function of RL step. All runs improve during fine-tuning, and the reward trajectories overlap substantially. Similar reward values therefore do not indicate whether the targeted tensor relation is symmetry-enforced. (b) Distribution of the highest rotational axis among generated crystals for the four ablation settings. Without relaxation, DiffCSP generates almost exclusively

P1

P\bar{1}

structures. Relaxation recovers mainly twofold symmetry, with only rare threefold and fourfold structures and no sixfold structures. SymmCD retains substantial populations with threefold, fourfold, and sixfold axes because symmetry is imposed during generation. (c) Representative DiffCSP-generated

\text{CrTe}{\vphantom{\text{X}}}_{\smash[t]{\text{3}}}\text{W}

structure in space group 1 (

P1

). (d) SymmCD asymmetric unit for the same composition under the

P6_{3}/m

space-group operations. Dashed lines connect representative atoms to their symmetry images. (e) Full SymmCD crystal reconstructed from the asymmetric unit. The resulting structure belongs to space group 176 (

P6_{3}/m

) and contains a sixfold rotational axis.

Figure 4: RL-guided generation and DFT verification of dielectric-anisotropy candidates. (a) Distribution of generated structures in the plane of surrogate-predicted band gap

E_{g}

and uniaxial dielectric reward

r_{\mathrm{uni}}

. Kernel-density contours compare structures generated during the first 10 RL rounds with those generated during the last 10 rounds. The black and green circles indicate the corresponding population medians, and the arrow highlights the shift toward the finite-gap, high-reward region during fine-tuning. The dashed lines indicate the band-gap and reward thresholds defining the target region. The four numbered markers denote the candidates selected for first-principles verification:

\text{RbGePO}{\vphantom{\text{X}}}_{\smash[t]{\text{4}}}

\text{ZrTi}{\vphantom{\text{X}}}_{\smash[t]{\text{2}}}\text{O}{\vphantom{\text{X}}}_{\smash[t]{\text{6}}}

, blue phosphorus, and TeAs . (b) DFT-relaxed structures and calculated properties of the four selected candidates. The table reports the realized space group,

E_{\mathrm{hull}}

, PBE band gap

E_{g}

, and dielectric tensor

\boldsymbol{\varepsilon}_{0}

. The quantities in panel (a) are surrogate predictions before DFT validation, whereas those in panel (b) are obtained after DFT relaxation and may therefore differ. The three high-reward candidates,

\text{RbGePO}{\vphantom{\text{X}}}_{\smash[t]{\text{4}}}

(

P6_{3}

\text{ZrTi}{\vphantom{\text{X}}}_{\smash[t]{\text{2}}}\text{O}{\vphantom{\text{X}}}_{\smash[t]{\text{6}}}

(

P\bar{3}1m

), and blue phosphorus (

P3_{1}

), retain the symmetry-constrained form

\varepsilon_{xx}=\varepsilon_{yy}\neq\varepsilon_{zz}

. Blue phosphorus shows the largest dielectric anisotropy, with

\varepsilon_{\parallel}/\varepsilon_{\perp}\approx 3.46

. In contrast, orthorhombic TeAs (

Pbca

) has three distinct diagonal components and serves as a low-reward biaxial comparison.

Figure 5: Steering crystal generation toward high-efficiency single-junction absorbers. (a) Mean reward as a function of RL step during SPARC fine-tuning. The gray line shows the raw mean reward at each step, while the black line shows the exponential moving average with a span of 10 steps. The shaded regions indicate the prior window, defined by the first 10 RL steps, and the steered window, defined by the last 10 RL steps. (b) Spectroscopic limited maximum efficiency (SLME), $\eta$ , as a function of band gap $E_{g}$ for structures generated in the prior window (left, 149 structures) and the steered window (right, 279 structures). The points are colored by their kernel-density estimate, and the dashed curve indicates the Shockley–Queisser limit. The light-gray area marks the target region, defined by $E_{g}\approx 0.8$ – $1.9$ eV and $\eta\geq 25\%$ . The fraction of structures within this region increases from $11\%$ to $32\%$ after steering. (c) Periodic-table heatmaps showing the percentage of generated structures containing each element. The number of represented elements decreases from 53 in the prior population to 46 in the steered population. Steering redistributes the elemental frequencies and enriches sulfur- and antimony-containing structures. (d) Relative frequencies of the ten most commonly realized space groups in the prior and steered populations. The cubic $Fm\bar{3}m$ space group remains the most frequent symmetry after steering. The orthorhombic $Pnma$ and triclinic $P\bar{1}$ space groups become substantially more prevalent.

Appendix figures & tables16 assets

Supplementary material from the paper’s appendix.

Appendix

Setting	Shared	Dielectric RL	SLME RL
Initial policy	SymmCD MP-20 checkpoint	–	–
Reference policy	Frozen initial checkpoint	–	–
Optimizer	Adam	–	–
Initial learning rate	–	$3\times 10^{-5}$	$3\times 10^{-5}$
Learning-rate multiplier per round	–	$1.0$	$0.98$
Weight decay	$0$	–	–

Appendix

Supplementary Table 1: Reinforcement-learning optimization settings. Shared and task-specific settings used for SPARC fine-tuning. Both campaigns were run for

120

RL rounds.

Supplementary Figure 1: Symmetry-constrained decomposition of a crystal . On a two-dimensional toy example (wallpaper group

p4mm

, fictitious species X, Y, Z). (a) The asymmetric unit, comprising the representative atoms with their fractional coordinates, species, and site symmetries (X on a

4mm

site, Y on a mirror line, Z on a general site). (b) The corresponding space-group symmetry elements, the mirror lines and rotation centres. (c) Applying these operations to the asymmetric unit replicates it across the unit cell and recovers the full crystal. SPARC generates only the asymmetric unit in (a) and reconstructs the complete structure through (b) and (c) , so every generated crystal is symmetry-valid by construction.

Supplementary Figure 2: Site-symmetry marginal for space group 225 ( $Fm\bar{3}m$ ). Empirical site-symmetry marginal $p_{\mathrm{marg}}(\boldsymbol{\Sigma}^{\prime}\mid G{=}225)$ , aggregated over all training materials in $Fm\bar{3}m$ and shown as a $15\times 13$ heatmap over the $15$ crystallographic axes and $13$ site-symmetry categories. Color encodes probability.

Supplementary Figure 3: Discrete-diffusion transition toward the site-symmetry marginal (space group 225,

Fm\bar{3}m

, axis

u=1

). The forward process for the site-symmetry labels along one crystallographic axis, shown through the cumulative transition

\overline{\boldsymbol{Q}}_{t,u,G}

at diffusion steps

t=0,250,500,750,1000

. (a) The space-group marginal

\boldsymbol{m}_{\Sigma,u,G}

over the

13

site-symmetry classes, the absorbing prior the forward process relaxes onto; for this axis the mass concentrates on a few classes. (b) The noise-schedule weights

\alpha_{t}

and the cumulative

\bar{\alpha}_{t}

over diffusion time, with the five sampled timesteps marked;

\bar{\alpha}_{t}

decreases from

1

0

as the clean signal is replaced by the marginal. (c) The self-transition probability

\overline{\boldsymbol{Q}}_{t}[i,i]

, the probability of remaining in the initial class

i

, versus

t

for three representative classes; each decays from

1

toward its marginal weight. (d) The row distribution

\boldsymbol{e}_{i}^{\top}\overline{\boldsymbol{Q}}_{t}

for source class

i=12

at the five timesteps, showing how probability initially localized on class

12

spreads toward the marginal as

t

grows.

Crystal Family	Space Group No.	Lattice Shape	Constraint of Symmetric Bases
Triclinic	$1\sim 2$	No Constraint	No Constraint
Monoclinic	$3\sim 15$	$\alpha=\gamma=90^{\circ}$	$k_{1}=k_{3}=0$
Orthorhombic	$16\sim 74$	$\alpha=\beta=\gamma=90^{\circ}$	$k_{1}=k_{2}=k_{3}=0$
Tetragonal	$75\sim 142$	$\alpha=\beta=\gamma=90^{\circ}$ $a=b$	$k_{1}=k_{2}=k_{3}=0$ $k_{4}=0$
Trigonal/Hexagonal	$143\sim 194$	$\alpha=\beta=90^{\circ},\ \gamma=120^{\circ}$ $a=b$	$k_{2}=k_{3}=0,\ k_{1}=-\log(3)/4$ $k_{4}=0$
Cubic	$195\sim 230$	$\alpha=\beta=\gamma=90^{\circ}$ $a=b=c$	$k_{1}=k_{2}=k_{3}=0$ $k_{4}=k_{5}=0$

Appendix

Supplementary Table 2: Relationship between the lattice shape and the constraint on the symmetric-basis coefficients

\boldsymbol{k}

, where

a,b,c

and

\alpha,\beta,\gamma

denote the lengths and angles of the lattice bases, respectively (after DiffCSP++ Jiao et al. (2024) ).

Component	Value
Backbone	CSPNet message-passing graph neural network (fully connected over representatives)
Message-passing layers	8
Hidden / representation dim.	1024
Timestep embedding dim.	10
Coordinate Fourier frequencies	128
Activation	SiLU

Appendix

Supplementary Table 3: Denoising-network architecture hyperparameters (inherited from the SymmCD MP-20 model).

Hyperparameter	Value
Maximum spherical harmonic order $\ell_{\max}$	2
Embedding dimension	128
Number of layers	4
Channel multiplicity	16
Batch size	128

Appendix

Supplementary Table 4: Hyperparameters for the TSENN band-gap surrogate.

Supplementary Figure 4: Predicted versus DFT-computed band gap. Parity plots of the predicted band gap $E_{g}$ against the DFT reference, shown for the train, validation, and test splits (left to right). The dashed line indicates perfect agreement ( $y=x$ ). On the held-out test set, the model achieves $R^{2}=0.881$ with $\mathrm{MAE}=0.252\,\mathrm{eV}$ .

Hyperparameter	Value
Maximum spherical harmonic order $\ell_{\max}$	2
Embedding dimension	128
Number of layers	4
Channel multiplicity	64
Batch size	24

Appendix

Supplementary Table 5: Hyperparameters for the TSENN SLME surrogate.

Supplementary Figure 5: Predicted versus DFT-computed SLME. Parity plots of the predicted SLME $\eta$ against the value computed from the DFT dielectric spectrum, shown for the train, validation, and test splits (left to right). The dashed line indicates perfect agreement ( $y=x$ ). On the held-out test set, the model achieves $R^{2}=0.925$ with $\mathrm{MAE}=1.35\%$ (absolute efficiency percentage points).

Hyperparameter	Value
Maximum spherical harmonic order $\ell_{\max}$	2
Embedding dimension	64
Number of layers	2
Channel multiplicity	32
Batch size	16

Appendix

Supplementary Table 6: Hyperparameters for the TSENN dielectric-tensor surrogate.

Component	$R^{2}$	MAE
$\varepsilon_{xx}$	0.993	0.087
$\varepsilon_{yy}$	0.992	0.083
$\varepsilon_{zz}$	0.991	0.088
$\varepsilon_{xy}$	0.986	0.005
$\varepsilon_{xz}$	0.976	0.006
$\varepsilon_{yz}$	0.956	0.009

Appendix

Supplementary Table 7: Per-component performance of the TSENN dielectric-tensor surrogate on the held-out test split. MAE is reported in units of the relative dielectric constant.

Supplementary Figure 6: Predicted versus DFT-computed static-limit dielectric tensors across all three splits. Parity plots of each independent Cartesian component $\varepsilon_{ij}$ ( $i,j\in\{x,y,z\}$ ) of the dielectric tensor, evaluated on the train, validation, and test splits. The dashed line indicates perfect agreement ( $y=x$ ). The inset showing the MAE for per-component performance, and all the metrics are also reported in Supplementary Table 7 .

Supplementary Figure 7: Reward surfaces. Reward as a function of the predicted band gap $E_{g}$ and the task-specific property for (a) the uniaxial dielectric objective and (b) the SLME objective. In (a), the tensor-based property score is $q_{\mathrm{uni}}=g_{z}(1-m)$ , and the final reward is $r_{\mathrm{uni}}=\min[\Phi(q_{\mathrm{uni}};0,0.20),\Phi(E_{g};0.3,0.8)]$ . The vertical dotted line marks the onset of the band-gap contribution at $E_{g}=0.3$ eV. In (b), the final reward is $r_{\mathrm{SLME}}=0.8r_{\eta}+0.2r_{g,\mathrm{SLME}}$ , where $r_{\eta}=\Phi(\eta;0,0.35)$ and $r_{g,\mathrm{SLME}}$ is a symmetric tent centered at $E^{*}=1.3$ eV. The dashed curve shows the Shockley–Queisser detailed-balance limit $\eta_{\mathrm{SQ}}(E_{g})$ evaluated using the same AM1.5G spectrum.

Supplementary Figure 8: First-principles validation of generated uniaxial dielectric candidates. (a) Nine generated structures whose dielectric tensors

\boldsymbol{\varepsilon}_{0}

retain the targeted relation

\varepsilon_{xx}\approx\varepsilon_{yy}

with a distinct out-of-plane component. Each entry shows the DFT-relaxed structure, realized space group, calculated dielectric tensor, DFT band gap

E_{g}

, and

E_{\mathrm{hull}}

. The viewing direction is indicated below each structure, and

\odot

denotes an axis pointing out of the page. The hexagonal and tetragonal structures exhibit symmetry-enforced in-plane equality, while the monoclinic and orthorhombic structures exhibit approximate equality that is not required by symmetry. (b) Comparison cases illustrating three rejection mechanisms.

\text{Ti}{\vphantom{\text{X}}}_{\smash[t]{\text{2}}}\text{TeO}{\vphantom{\text{X}}}_{\smash[t]{\text{6}}}

retains an approximately uniaxial tensor but exceeds the stability cutoff,

\text{SrTiO}{\vphantom{\text{X}}}_{\smash[t]{\text{3}}}

has a nearly isotropic response and is suppressed by the out-of-plane contrast term, and

\text{TeI}{\vphantom{\text{X}}}_{\smash[t]{\text{2}}}

has a biaxial response that fails the in-plane isotropy requirement.

Supplementary Figure 9: First-principles optical response of eight generated candidates. For each material the left panel shows the relaxed crystal structure (ball-and-stick, unit cell outlined) viewed down the crystallographic axis marked in the lower-left corner. The

\odot

denotes a viewing axis pointing out of the page. Each structure panel associated to a spectra panel comparing the imaginary dielectric function

\varepsilon_{2}(\omega)

from first-principles DFT (QE, black solid) with the TSENN prediction (red dashed) over

0

–

10

eV. Annotations give the PBE indirect band gap

E_{g}

and the SLME evaluated following the parameters throughout the work describe in main text.

Explore similar work

Jun 1, 2026

Towards Automated Discovery: A Review of Generative Models, Multimodal Learning and Closed-Loop Workflows in Inverse Materials Design

Inverse materials design is shifting materials discovery from forward prediction to targeted proposal of candidates that satisfy objectives under physical constraints. Here, we review recent advances in generative crystal structure modeling, multimodal learning, and closed-loop design pipelines for crystalline solids. We survey how modern generators learn chemical-structural priors from large databases to enable controllable sampling of periodic structures, and compare leading model classes including variational autoencoders, normalizing flows, autoregressive formulations, and diffusion models. Particular attention is given to how feasibility constraints and physical priors are enforced across the workflow, through representation choices, training objectives, sampling-time guidance, and post-generation screening and relaxation. We also discuss how multimodal learning fuses diverse materials modalities, including crystal structures, thermodynamic, electronic information, microscopy, spectroscopy, processing context, and scientific text, to construct a more universal, transferable representation of chemical space. In addition, diverse inverse-design strategies are examined, particularly those that integrate conditional generation with latent optimization, Bayesian optimization, reinforcement learning, and active learning. Finally, we highlight recurring failure modes, such as surrogate exploitation, diversity collapse, distribution shift, and the stability-synthesizability gap, and outline discovery-grade evaluation practices based on staged reporting of validity, novelty, uniqueness, stability, and cost.

Anand Babu, Rogério Almeida Gouvêa, Gian-Marco Rignanese

Institute of Condensed Matter and Nanosciences, Université Catholique de Louvain, Louvain-la-Neuve, Belgium. · WEL Research Institute, Avenue Pasteur 6, Wavre, Belgium.

Crystal Structure Generation Autonomous Scientific Discovery Conditional Generative Modeling Inverse Design Multimodal Learning

Ask PDF

Oct 2, 2026

Reinforcement Learning on the Discrete Composition Channel of a Crystal Generator: Validated Gains and Reward Hacking

Inverse materials design is a long-standing goal of computational materials discovery. Generative models for crystalline materials are typically trained to match the distribution of a structure database, while nothing in their training objective points them at specific design goals such as targeted properties. We use group-relative policy optimization (GRPO) to align a generative model based on stochastic interpolants and discrete flow matching with general black-box reward functions through reinforcement learning. Atom types are generated by a discrete flow and the policy gradient of our generalization of GRPO directly acts on the likelihoods of the atom-type transitions, which differentiates our work from previous reinforcement-learning approaches for diffusion and flow-based generative models of crystalline materials. We introduce a reward function that raises the yield of metastable, unique and novel structures (mSUN) from 13.4% for the pretrained model to 45.5% for the reinforced model, as evaluated by a community benchmark. Our reward also improves the performance of a reinforcement learning framework for crystalline materials based on latent denoising diffusion models. At the same time, we find that directly reinforcing atom-type transition likelihoods enables reward exploitation that has to be prevented with explicit guards. The same analysis also exposes a gap in the community metric. Single-element structures in distinct packings are counted as metastable, unique and novel materials and inflate mSUN without yielding any new compounds. A stability claim is only as good as its reference hull. We report every result split by the number of reference phases behind it and argue that benchmarks should do the same.

Pawan Prakash, Philipp Höllmer, Addis Fuhr +4

University of Florida · New York University · Oak Ridge National Laboratory

Crystal Structure Generation Reward Hacking Group Relative Policy Optimization

Ask PDF

Feb 19, 2026

NextCrystal: a Symmetry-Driven Generative Framework for Crystal Structure Prediction

Crystal structure prediction (CSP), which aims to predict the 3D atomic arrangement of a crystal from its composition, is central to materials discovery and mechanistic understanding. Crystal symmetry plays a crucial role in CSP, but given the composition in a unit cell, existing methods either struggle with the NP-hard combinatorial challenge of enforcing symmetry rigorously or rely on retrieving known templates, inherently limiting both physical fidelity and the discovery of genuinely new materials. To address this challenge, we introduce NextCrystal, a symmetry-driven generative framework that employs large language models to encode chemical semantics and directly generate fine-grained Wyckoff site patterns from atomic stoichiometry, eliminating reliance on database lookups. To overcome the combinatorial complexity of site assignments, we incorporate domain knowledge via an efficient, linear-complexity heuristic beam search, rigorously enforcing algebraic consistency between site multiplicities and atomic stoichiometry. By integrating this symmetry-consistent template into a diffusion backbone, the framework constrains the stochastic generative trajectory to a physically plausible geometric manifold. NextCrystal achieves state-of-the-art performance on stability, uniqueness, and novelty (SUN) benchmarks, as well as superior structural matching, establishing a rigorous paradigm for exploring previously unexplored crystallographic space without relying on prior structural templates. As a representative application, first-principles screening of HfO2 candidates generated by NextCrystal identifies a previously unreported dynamically stable Pnma phase, 0.056~eV/atom lower in energy than the conventional high-pressure Pnma phase.

Jinming Mu, Lixin He, Xudong Zhu +1

Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, Anhui, China · Institute of Artificial Intelligence, Hefei Comprehensive National Science Center, Hefei 230088, Anhui, China

Crystal Structure Prediction Crystal Structure Generation

Ask PDF