A Cross-Domain Graph Learning Protocol for Single-Step Molecular Geometry Refinement

中文 / Bilingual · English version

Chengchun Liu, Wendi Cai, Boxuan Zhao, and Fanyang Mo*

Cite This: https://doi.org/10.1021/acs.jctc.6c01080

ABSTRACT: Accurate molecular geometries are indispensable for predictive quantum chemistry, yet iterative density functional theory (DFT) optimization remains a major computational bottleneck in large-scale screening. Here, we introduce GeoOpt-Net, a deterministic, SE(3)-equivariant single-step geometry refinement network that maps inexpensive force-field conformers directly to B3LYP/TZVP-quality structures. Trained using a two-stage multifidelity protocol with theory-aware feature modulation, GeoOpt-Net learns transferable geometric priors and calibrates them to target-level quantum accuracy in a single forward pass. Under strictly matched B3LYP/TZVP conditions, GeoOpt-Net achieves structural deviations on the order of ${10}^{-4}\text{ Å }$ and single-point energy deviations on the order of ${10}^{-4}\mathrm{{kcal}}{\mathrm{{mol}}}^{-1}$ , outperforming classical, semiempirical, and neural potential-based approaches. Notably, its predicted geometries satisfy all standard DFT convergence criteria for 65.0% (loose) and 33.4% (default) of molecules, whereas baseline methods remain near zero, substantially reducing subsequent optimization effort. By replacing iterative relaxation with deterministic single-step refinement, GeoOpt-Net offers a scalable and physically consistent protocol for accelerating high-throughput quantum-chemical workflows.

INTRODUCTION

Accurate three-dimensional molecular geometries form the foundation of computational chemistry, ${}^{1,2}$ underpinning reliable predictions of reaction pathways, ${}^{3,4}$ spectroscopic signatures, ${}^{5 - 9}$ and the rational design of catalysts, ${}^{10}$ functional materials, ${}^{11}$ and drug candidates. ${}^{{12},{13}}$ At the core of these applications lies the exploration of the potential energy surface (PES), a high-dimensional hypersurface defined by the Born-Oppenheimer approximation, whose stationary points correspond to stable conformers and transition states. ${}^{14}$ Quantummechanical (QM) methods, ranging from Hartree-Fock ${\left( \mathrm{{HF}}\right) }^{{15},{16}}$ to Kohn-Sham density functional theory (DFT), ${}^{17}$ evaluate energies and forces through the self-consistent solution of the electronic Schrödinger equation, with computational cost scaling steeply with system size ${}^{18}$ (Figure 1, left).

In practical quantum-chemical workflows, hybrid density functionals combined with triple- $\zeta$ basis sets, such as B3LYP/ TZVP, are widely adopted as a balanced compromise between accuracy and computational cost. ${}^{{19} - {22}}$ Extensive benchmarks have shown that B3LYP/TZVP geometries are close to basis-set-converged structures for typical organic molecules, with residual geometric differences exerting only minor influence on relative energies, spectroscopic properties, and qualitative chemical trends. ${}^{{23},{24}}$ As a result, B3LYP/TZVP has become a de facto reference level for geometry optimization in large-scale and high-throughput DFT applications.

In routine applications, DFT geometry optimization is rarely initiated from arbitrary coordinates. Instead, established pipelines are commonly employed to generate chemically reasonable initial structures prior to QM refinement. Classical distance-geometry approaches, exemplified by RDKit’s ETKDG sampler, ${}^{25}$ are widely used to produce force-field-level conformer ensembles. More automated workflows, such as Auto3D (ANI-2x, AIMNET), ${}^{26}$ combine conformer enumeration with machine-learned interatomic potentials and force-driven numerical optimization to identify low-energy structures. Semiempirical quantum methods, including GFN2- xTB, ${}^{27}$ further accelerate this process by providing fast preoptimized geometries at reduced computational cost. Despite their widespread use, these pipelines share a common characteristic: DFT-quality geometries are obtained only after iterative, gradient-based optimization on an explicit or implicit PES. Consequently, the computational cost and robustness of the overall workflow remain tied to the convergence behavior of numerical optimizers, motivating the search for alternative strategies that can reduce or eliminate this bottleneck.

A parallel line of research has focused on molecular conformation generation directly from molecular graphs, using distance-geometry formulations, torsion-based construction, or stochastic generative modeling. Representative examples include distance-geometry-based methods such as GraphDG, ${}^{28}$ graph-conditioned refinement approaches such as CGCF, ${}^{29}$ torsion-driven construction models such as Geo-Mol, ${}^{30}$ as well as score-based, flow-based, and diffusion-based generative methods, including ConfGF, ${}^{31}$ DMCG, ${}^{32}$ and GeoDiff. ${}^{33}$ These approaches are primarily designed to sample conformational diversity or explore chemical space, rather than to serve as drop-in replacements for practical DFT geometry-optimization workflows. In particular, these approaches can yield nonphysical geometries that violate basic metric constraints or fail distance-matrix realizability, leading to premature termination or nonconvergence before any self-consistent electronic-structure or gradient evaluation is performed. Such failures preclude their direct use in standard quantum-chemical optimization workflows and are quantified in Table S1 and Table S2 of the Supporting Information.

Received: May 30, 2026

Revised: July 1, 2026

Accepted: July 2, 2026

Figure 1. Conceptual comparison of geometry optimization paradigms. Left: conventional quantum-chemical optimization on an ab initio potential energy surface. Middle: iterative relaxation driven by a learned machine-learning force field. Right: GeoOpt-Net directly refines molecular geometries in a single forward pass without explicit potential energy surface construction or iterative optimization.

Another prominent strategy for accelerating geometry optimization involves machine-learned interatomic potentials (MLFFs), which learn differentiable approximations to the PES and enable rapid evaluation of energies and atomic forces. ${}^{{34} - {36}}$ Representative MLFF architectures include SchNet, ${}^{34}$ SpookyNet, ${}^{36}$ PaiNN, ${}^{37}$ DimeNet, ${}^{38}$ GemNet, ${}^{39}$ and LEFTNet, ${}^{40}$ which differ in their treatment of geometric information and target properties but share a data-driven PES paradigm, in which energies and atomic forces are learned from reference electronic-structure calculations and final geometries are obtained via force-based numerical optimization (Figure 1, middle). Recent advances, exemplified by UMA, ${}^{41}$ have demonstrated state-of-the-art accuracy by training MLFFs directly on hybrid-DFT data. Nevertheless, MLFF-based approaches fundamentally predict energies and atomic forces on a learned potential energy surface and therefore require external, iterative force-based optimization to obtain final geometries. This distinction in task definition and inference paradigm-predicting forces versus directly predicting refined coordinates-is summarized in Table S3.

Here, we introduce GeoOpt-Net, a multiscale SE(3)- equivariant geometry refinement framework designed to directly bridge low-cost initial conformers and DFT-level accuracy at the B3LYP/TZVP level of theory. In contrast to PES-based machine-learning potentials and conformer-generation models, GeoOpt-Net targets the specific task of single-shot local refinement toward DFT-converged geometries within a given conformational hypothesis. It is therefore not intended to replace conformer search or conformer selection; rather, it refines chemically reasonable force-field-level conformers into DFT-ready geometries. By explicitly decoupling bond lengths, bond angles, and dihedral torsions into dedicated equivariant graph streams and fusing them through a lightweight Transformer decoder, GeoOpt-Net predicts refined atomic coordinates in a single forward pass (Figure 1, right). To account for systematic differences across electronic-structure methods, GeoOpt-Net is trained using a two-stage, multifidelity strategy, in which transferable geometric priors are first learned from large-scale pretraining and subsequently fine-tuned to higher levels of theory via fidelity-aware feature modulation (FAFM).

By combining explicit geometric inductive biases, SE(3)- equivariance, and theory-aware fine-tuning, GeoOpt-Net addresses a critical gap in current molecular modeling pipelines: the absence of a fast, robust, and physically grounded route to DFT-ready geometries at the B3LYP/ TZVP level, a widely adopted accuracy-cost balance for organic and drug-like molecules, without iterative quantummechanical optimization. Beyond accuracy gains, GeoOpt-Net reshapes the role of machine learning in quantum chemistry by transforming local geometry refinement from a costly iterative procedure into a deterministic, single-step prediction problem, enabling scalable and reproducible quantum-chemical work-flows targeting DFT-level accuracy for large molecular systems and high-throughput applications.

DATA SETS AND COMPUTATIONAL DETAILS

Data Sets

GeoOpt-Net was trained, fine-tuned, and evaluated using multiple quantum-chemical data sets spanning increasing molecular size, elemental diversity, and theoretical levels.

Pretraining Data Sets

The initial pretraining of GeoOpt-Net was performed jointly on the QM9 and QM40 data sets. QM9 ${}^{42}$ comprises approximately 130k small organic molecules containing up to nine heavy atoms $\left( {\mathrm{C},\mathrm{N},\mathrm{O}\text{ , and }\mathrm{F}}\right)$ , while QM ${40}^{43}$ extends the chemical space to larger and more drug-like systems, covering 160k molecules with elements commonly found in pharmaceutical compounds (C, N, O, S, F, and Cl), representing approximately 88% of the FDA-approved drug chemical space. Together, these two data sets provide a combined training corpus of approximately 290k molecules spanning a broad range of molecular sizes and compositions. All molecular geometries and associated quantum-mechanical properties in both QM9 and QM40 were computed consistently at the B3LYP/6-31G(2df, p) level of theory, ensuring a uniform theoretical framework during the pretraining stage.

Figure 2. Two-stage training and SE(3)-equivariant geometric framework of GeoOpt-Net. (a) Two-stage training across quantum-chemical fidelity levels. The model is pretrained on large-scale geometries at the B3LYP/6-31G(2df, p) level and subsequently fine-tuned on target-level B3LYP/ TZVP data using a warm-start strategy with the fidelity-aware feature modulation (FAFM) mechanism. (b) SE(3)-equivariant geometric encoding and coordinate refinement. Pairwise distances $\left( {r}_{ij}\right)$ , bond angles $\left( {\theta }_{ijk}\right)$ , and dihedral angles $\left( {\varphi }_{ijkl}\right)$ are encoded as invariant scalar features $\left( {l = 0}\right)$ via radial basis functions and as directional features $\left( {l \geq 1}\right)$ via spherical harmonics. Equivariant features are coupled through Clebsch-Gordan tensor products and decoded by a transformer-based module to produce SE(3)-equivariant coordinate updates.

Fine-Tuning Data Set

Model fine-tuning was performed using the QMe14S data set, which consists of 180k small organic molecules featuring substantially increased elemental and functional-group diversity. QMe ${14}{\mathrm{\;S}}^{22}$ includes 14 elements (H, B, C, N, O, F, Al, Si, P, S, Cl, As, Se, and Br) and spans 47 functional groups. All molecular geometries in QMe14S were obtained from density functional theory calculations at the B3LYP/TZVP level, enabling GeoOpt-Net to adapt to a higher basis set accuracy and broader chemical environments.

External Validation Data Set

To further assess extrapolation performance beyond both the training and fine-tuning domains, an additional validation set was constructed from the ZINC20 database. ${}^{44}$ From the $\sim 1\mathrm{\;B}$ “ZINC20” collection, 1000 molecules were selected based on combined constraints on heavy-atom count and rotatable-bond count, ensuring coverage of larger and more conformationally flexible compounds not present in the QM9, QM40, or QMe14S data sets. These molecules provide a stringent test of GeoOpt-Net under realistic, drug-like structural complexity. To improve reproducibility, the exact SMILES list used for the external evaluation is provided as data/selected_zin-c20_external_set.txt, and the corresponding filtering script is provided as scripts/select_zin-c20_external_set.py in the public repository.

Detailed statistics of data set splits, along with distributions of heavy-atom counts, rotatable-bond counts, and elemental compositions, are reported in Figures S1-S3 of the Supporting Information.

Reference Geometry Computations

Initial 3D structures for all molecules were generated via RDKit’s ETKDG algorithm ${}^{25}$ and preoptimized with the MMFF94 force field. ${}^{45}$

• External “ZINC20” test sets: final reference geometries and single-point energies were computed at the B3LYP/ TZVP level using Gaussian 16 software, with default convergence settings.

Data Preprocessing

Coordinates were centered and standardized (zero mean, unit variance per axis) before the model input. Atom types were one-hot encoded; bond adjacency and angular/dihedral graphs were constructed via RDKit v2024.03.5. Data set splits follow an 80:10:10 train/val/test ratio stratified by heavy-atom count; external sets are used only for final evaluation.

Inference and Evaluation Protocol

During inference, GeoOpt-Net receives the molecular graph $G$ and the initial Cartesian coordinates ${\mathrm{R}}^{\left( 0\right) }$ . With a frozen checkpoint ${\theta }^{ * }$ , the refined geometry is generated as

For structural evaluation, the predicted coordinates $\widehat{\mathbf{R}}$ are compared with the corresponding DFT-optimized reference geometry ${\mathrm{R}}^{\mathrm{{DFT}}}$ after Kabsch alignment eq 4

Kabsch-aligned RMSD values can be computed using scripts/compute aligned rmsd.py, which reads XYZ files, preserves atom ordering, checks atom counts and element ordering, and supports both single-pair and batch evaluations.

IR RESULTS AND DISCUSSION

Training Protocol and Model Architecture

GeoOpt-Net is a single-pass, SE(3)-equivariant geometry refinement model trained under a two-stage, multifidelity protocol (Figure 2). The framework integrates transferable geometric pretraining with equivariant coordinate decoding to achieve target-level quantum-chemical accuracy in a single forward pass.

Two-Stage Multifidelity Training

As illustrated in Figure 2a, GeoOpt-Net is trained using a two-stage strategy across quantum-chemical fidelity levels. In stage 1, the network is pretrained on large-scale data sets computed at a lower level of theory (B3LYP/6-31G(2df, p)) to learn transferable geometric priors that capture generic bonding patterns, angular preferences, and torsional statistics. In stage 2, the pretrained weights are warm-started and fine-tuned on target-level B3LYP/TZVP data, enabling systematic correction of element- and basis-set-dependent deviations while preserving the global geometric structure learned during pretraining.

To condition geometric refinement on the target theoretical level, we introduce a fidelity-aware feature modulation (FAFM) mechanism

where ${\mathrm{g}}_{d}$ and ${\mathrm{b}}_{d}$ are learned scale and shift parameters derived from a domain embedding $d$ . This mechanism enables theory-specific geometric calibration without retraining the entire network, facilitating efficient transfer across quantum-chemical levels.

SE(3)-Equivariant Geometric Architecture

The SE(3)-equivariant model architecture is shown in Figure 2b. Starting from an RDKit ETKDG v3 + MMFF94 conformer, atomic Cartesian coordinates are projected into a 256-dimensional latent space using a linear embedding followed by LayerNorm and GELU activation. ${}^{46}$

Molecular geometry is decomposed into three complementary equivariant streams corresponding to pairwise distances ${r}_{ij}$ , three-body bond angles ${\theta }_{ijk}$ , and four-body dihedral torsions ${\varphi }_{ijkl}$ . Scalar geometric features $\left( {l = 0}\right)$ are encoded via radial basis expansions $\phi \left( {r}_{ij}\right)$ , while directional information $\left( {l \geq 1}\right)$ is represented using real spherical harmonics ${Y}^{\left( l\right) }\left( {\widehat{\mathbf{r}}}_{ij}\right)$ .

Equivariant message passing is performed via Clebsch-Gordan tensor products ${}^{47}$

which guarantees exact SE(3) equivariance by construction. Nonlinear activations are applied exclusively to scalar channels, whereas vector channels undergo only learned linear updates and gated residual additions, following established equivariant architectures. ${}^{{37},{48} - {51}}$

The fused equivariant representation is decoded by a lightweight transformer-based module to predict an equivariant coordinate refinement

yielding refined atomic coordinates that are invariant to global translations and equivariant under rotations.

Equivariance Verification

To empirically validate SE(3) equivariance, we applied random global rotations and translations to 1000 test structures and compared the predicted coordinates under “rotate-then-predict” and “predict-then-rotate” protocols. The maximum deviation between the two procedures remained below ${1.1} \times \; {10}^{-4}\text{ Å }$ , with a mean deviation of ${3.262} \times {10}^{-5}\text{ Å }$ , confirming numerical SE(3) equivariance within floating-point precision (see Figure S4).

Training Objective

Model optimization minimizes a composite geometry-aware loss that jointly constrains global and local structural accuracy relative to the DFT-optimized training target. Specifically, the loss includes a global RMSD penalty on Cartesian coordinates $\left( {L}_{\text{ rmsd }}\right)$ , mean-squared-error losses on bond lengths, bond angles, and dihedral torsions $\left( {L}_{\text{ bond }}\right.$ , $\left. {{L}_{\text{ angle }},{L}_{\text{ dihedral }}}\right)$ , as well as a soft bond-length range constraint $\left( {L}_{\text{ bond range }}\right)$ to suppress unphysical distortions. The DFT-optimized coordinates are used only as supervised targets for these loss terms during training and are not required during inference. All loss definitions, weights, and ablation analyses are provided in Tables S4-S6.

Training is performed using AdamW (initial learning rate 1 $\times {10}^{-3}$ , weight decay $1 \times {10}^{-5}$ ), with learning-rate decay at epochs 50 and 75 and gradient clipping on mini-batches of 32 molecules. This training protocol achieves submilli-Å accuracy in a single forward pass while preserving strict SE(3) equivariance.

Such a multibranch geometric design is critical. Bond-only models neglect angular and torsional constraints, angle-only models miss long-range torsional flexibility, and point-cloud approaches treat atoms as orderless, discarding chemical connectivity. ${}^{{30} - {33},{52}}$ In contrast, GeoOpt-Net explicitly models pairwise, three-body, and four-body geometric interactions, ensuring that each atomic representation encodes its full local coordination environment and thereby enhancing robustness and transferability across diverse chemical families.

Geometric and Energetic Fidelity Relative to DFT References

We benchmark GeoOpt-Net against representative structure-generation paradigms, including classical conformer construction (RDKit), semiempirical quantum methods (GFN2-xTB), and neural interatomic potentials trained at the B3LYP/TZVP level of theory (SchNet@TZVP, PaiNN@TZVP, and UMA@ TZVP). To eliminate cross-theory discrepancies, all neural baselines are trained or fine-tuned using identical B3LYP/ TZVP geometries, energy data, and data set splits. SchNet@ TZVP and PaiNN@TZVP are trained from scratch, whereas UMA@TZVP is obtained via fine-tuning of a pretrained equivariant model. Detailed training configurations are provided in Tables S7-S10.

Table 1. Representative Examples Comparing Optimized Molecular Geometries ${}^{a}$

${}^{a}$ RMSD values are computed with respect to the DFT reference structures after Kabsch alignment, and ${\Delta E}$ denotes the single-point energy difference at the B3LYP/TZVP level of theory.

Figure 3. Benchmarking geometric and energetic fidelity relative to B3LYP/TZVP references. (a) Logarithmic distributions of all-atom RMSD values and B3LYP/TZVP single-point energy deviations ( ${\Delta E}$ ) with respect to DFT-optimized geometries. Decomposition of geometric errors into (b) bond length, (c) bond angle, and (d) dihedral angle deviations, evaluated relative to B3LYP/TZVP reference structures.

Importantly, these methods differ not only in architecture but also in inference paradigm, as summarized in Table S3. RDKit and GFN2-xTB obtain geometries through classical or semiempirical iterative relaxation, while SchNet@TZVP, PaiNN@TZVP, and UMA@TZVP predict energies and/or forces that require an external optimizer. By contrast, GeoOpt-Net directly maps $G$ and ${\mathrm{R}}^{\left( 0\right) }$ to $\widehat{\mathbf{R}}$ in a single forward pass, without external iterative geometry optimization. The inference script is provided in the public repository, and additional checks on the inference workflow are summarized in Table S11.

Figure 4. Quality of molecular initial guess geometries and impact on DFT convergence. (a) Cumulative distribution functions (CDFs) of four DFT geometry-optimization convergence quantities: maximum force, RMS force, maximum displacement, and RMS displacement. Dashed lines indicate the corresponding DFT geometry optimization convergence thresholds. (b) “All-YES” convergence rates, defined as the fraction of molecules simultaneously satisfying all four convergence criteria under loose and default thresholds. (c) DFT reoptimization performance starting from different initial geometries, quantified by the number of optimization steps (left) and total elapsed wall-clock time (right).

Representative comparisons are summarized in Table 1. Across all cases, GeoOpt-Net reproduces the reference geometries with submilli-Å RMSD values and near-zero single-point energy deviations. Additional per-molecule RMSD examples using the same Kabsch alignment protocol are provided in Table S12. In contrast, the baseline methods exhibit progressively increasing structural distortions. Among the neural potentials trained at identical theory level, UMA@ TZVP achieves RMSD values predominantly in the ${10}^{-3} - {10}^{-2}$ Å regime, PaiNN@TZVP concentrates in the ${10}^{-2} - {10}^{-1}$ Å range, and SchNet@TZVP shifts further toward larger deviations. These differences are accompanied by systematic increases in energy errors, indicating that even modest geometric distortions translate into measurable energetic discrepancies.

The full statistical distributions are shown in Figure 3a. To make the distribution-level comparison more explicit, percentile-level RMSD statistics, including ${Q}_{5},{Q}_{25},{Q}_{50},{Q}_{75}$ , ${Q}_{90},{Q}_{95},{Q}_{99}$ , and the mean, are provided in Table S13. GeoOpt-Net yields a sharply peaked RMSD distribution centered at $\log \left( \mathrm{{RMSD}}\right) \approx - 4$ , corresponding to sub-0.001 Å accuracy for the vast majority of molecules. In contrast, all baseline methods exhibit substantially broader distributions shifted toward larger deviations, even when trained or fine-tuned at the identical B3LYP/TZVP level of theory. A similar ordering is observed for single-point energy deviations: GeoOpt-Net remains tightly concentrated near zero, whereas neural interatomic potentials and semiempirical or rule-based methods display progressively wider spreads extending into the subkcal ${\mathrm{{mol}}}^{-1}$ and multikcal ${\mathrm{{mol}}}^{-1}$ regimes. The strong geometric-energetic correspondence indicates that the energetic fidelity of GeoOpt-Net originates from accurate structural refinement rather than explicit energy fitting (see Figure S5 for correlation analysis).

Figure 5. Energy scaling with molecular complexity. DFT single-point energy deviations ( ${\Delta E}$ , B3LYP/TZVP) evaluated as a function of molecular complexity, quantified by the number of rotatable bonds (left) and heavy atom counts (right), for different molecular geometry initialization approaches.

To further elucidate the origin of these improvements, Figure 3b-d decomposes the geometric errors into bond length, bond angle, and dihedral angle components. GeoOpt-Net achieves near-numerical-precision bond lengths, subdegree bond angle errors, and dihedral angle errors reduced by one to 2 orders of magnitude relative to all baseline methods. Additional bond-length diagnostics on CO, including refinement from RDKit/MMFF-generated coordinates and diagnostic short- and long-bond perturbations, are provided in Table S14. The pronounced reduction in torsional errors is particularly significant, as dihedral distortions are a dominant source of conformational and energetic inaccuracies in flexible molecules. The most pronounced separation appears in torsional degrees of freedom: dihedral errors for SchNet@ TZVP and PaiNN@TZVP commonly span the ${10}^{ \circ }$ scale, whereas UMA@TZVP narrows this distribution, and GeoOpt-Net reduces torsional deviations by one to 2 orders of magnitude relative to all baselines.

Collectively, these results demonstrate that even when trained at an identical DFT level and under matched data conditions, existing neural interatomic potentials exhibit nontrivial geometric drift. GeoOpt-Net, by contrast, performs deterministic single-step refinement that preserves fine-grained internal coordinates at DFT fidelity, yielding consistent improvements across bond, angular, and torsional degrees of freedom.

Impact of Initial Geometry Quality on DFT Convergence Efficiency

We assess the practical utility of initial guess geometries produced by GeoOpt-Net and baseline methods (UMA@ TZVP, PaiNN@TZVP, SchNet@TZVP, GFN2-xTB, and RDKit) by directly measuring how close these structures are to standard B3LYP/TZVP geometry optimization convergence conditions. We consider four quantities used in routine DFT optimizations: the maximum force, RMS force, maximum displacement, and RMS displacement.

Figure 4a reports the cumulative distribution functions (CDFs) for these four metrics, where dashed vertical lines denote the corresponding convergence thresholds (maximum force: ${4.50} \times {10}^{-4}$ hartree/Bohr; RMS force: ${3.00} \times {10}^{-4}$ hartree/Bohr; maximum displacement: ${1.80} \times {10}^{-3}$ Bohr; RMS displacement: ${1.20} \times {10}^{-3}$ Bohr). Across all metrics, GeoOpt-Net produces substantially more “near-converged” initial structures, with its CDF curves shifted markedly toward the threshold region. In contrast, the baseline methods exhibit distributions largely separated from the convergence thresholds, particularly for displacement-based criteria, indicating that their initial geometries typically require substantial additional optimization before satisfying standard DFT convergence conditions.

This separation translates into qualitatively different convergence behavior under the stringent “All-YES” criterion, which requires simultaneously satisfying all four convergence conditions (Figure 4b). GeoOpt-Net achieves high all-criteria pass rates of 65.0% (loose) and 33.4% (default). By comparison, UMA@TZVP reaches only 5.0% (loose) and 0.1% (default), while PaiNN@TZVP and SchNet@TZVP are near-zero ( ${0.3}\% /{0.0}\%$ and ${0.1}\% /{0.0}\%$ , respectively), and xTB/ RDKit yield 0.0% under both thresholds. These results indicate that only GeoOpt-Net frequently produces initial geometries that are already compatible with standard DFT convergence conditions.

To quantify the downstream computational impact, we perform full B3LYP/TZVP reoptimizations starting from each method’s initial geometries (Figure 4c). GeoOpt-Net consistently reduces the required optimization steps and the total wall-clock time, reflecting initialization closer to the converged manifold and thus reduced search effort for the electronic-structure optimizer. Overall, GeoOpt-Net provides initial structures that substantially improve the convergence efficiency of practical DFT workflows, enabling both higher immediate convergence rates and lower reoptimization cost. In addition to reducing downstream DFT reoptimization cost, the neural-network inference itself introduces only a small computational overhead; a single-molecule runtime benchmark is provided in Table S15.

Figure 6. Geometric proximity and DFT convergence analysis. DFT single-point energy deviation ( ${\Delta E},$ B3LYP/TZVP) plotted against the remaining number of geometry-optimization steps required to reach convergence, illustrating how initial structures generated by different methods relate to the DFT convergence process.

Robustness, Scaling Behavior, and Physical Consistency beyond the Training Domain

We further evaluate the robustness and physical reliability of GeoOpt-Net by examining how DFT single-point energy deviations ( ${\Delta E},\mathrm{B}3\mathrm{{LYP}}/\mathrm{{TZVP}}$ ) scale with molecular complexity. All analyses are conducted on 1000 molecules uniformly sampled from a billion-scale drug-like molecular library, yielding a chemically diverse evaluation set that extends beyond the training regime in both size and flexibility.

Figure 5 summarizes the dependence of ${\Delta E}$ on two standard complexity descriptors: the number of rotatable bonds (left) and the heavy-atom count (right), with out-of-training-range bins explicitly highlighted. Across both axes, all methods show increasing ${\Delta E}$ with growing complexity, reflecting the greater difficulty of producing DFT-consistent geometries for larger and more flexible molecules. However, the magnitude and growth rate differ substantially. In particular, GeoOpt-Net consistently maintains the smallest ${\Delta E}$ across the full range, with a smooth and gradual degradation as complexity increases, and without abrupt error escalation in the out-of-range regimes. By contrast, the baseline methods exhibit progressively larger deviations and steeper scaling, reaching the kcal ${\mathrm{{mol}}}^{-1}$ and higher regimes for increasingly complex molecules. These trends indicate that GeoOpt-Net retains robust generalization to molecular sizes and flexibilities beyond those seen during training.

Beyond static complexity measures, Figure 6 connects ${\Delta E}$ to the remaining number of B3LYP/TZVP optimization steps required to reach convergence, providing an operationally meaningful proxy for geometric proximity to DFT convergence. Initial structures produced by RDKit, xTB, and the neural interatomic potentials typically exhibit larger ${\Delta E}$ and require substantially longer remaining optimization trajectories, consistent with reduced convergence efficiency. In contrast, GeoOpt-Net concentrates near small ${\Delta E}$ while requiring markedly fewer additional optimization steps, demonstrating that its predicted geometries translate directly into accelerated DFT convergence behavior.

Finally, we assess whether improved geometries also yield improved electronic observables. As shown in Table 2, dipole moments computed from GeoOpt-Net geometries closely match the DFT reference, with a ${\Delta \mu }$ of only +0.001 D. UMA@TZVP remains relatively close (+0.017 D), whereas PaiNN@TZVP, SchNet@TZVP, xTB, and RDKit display larger deviations (up to -0.319 D), indicating that their geometric errors propagate into measurable electronic-structure discrepancies. Together, these results support that GeoOpt-Net improves not only energetic proximity and convergence efficiency but also the physical consistency of downstream quantum-chemical observables beyond the training domain.

Table 2. Comparison of Dipole Moments Computed from Molecular Geometries Optimized by GeoOpt-Net and Baseline Methods ${}^{a}$

${}^{a}$ All dipole moments $\left( \mu \right)$ are evaluated at the B3LYP/TZVP level of theory using the corresponding optimized geometries. ${\Delta \mu }$ denotes the deviation from the reference value obtained from the DFT-converged geometry at the B3LYP/TZVP level of theory.

CONCLUSION

We have presented GeoOpt-Net, a single-pass SE(3)- equivariant geometry refinement framework that bridges inexpensive conformer generation and hybrid DFT accuracy at the B3LYP/TZVP level of theory. Unlike classical conformer generators, semiempirical quantum methods, or iterative machine-learning interatomic potentials, GeoOpt-Net directly learns a theory-aware geometric mapping that refines bond lengths, bond angles, and torsional degrees of freedom in a unified equivariant architecture. The combination of multibranch geometric modeling, SE(3) equivariance, and multifidelity fine-tuning enables systematic correction of basis-set- and element-dependent deviations while preserving transferable structural priors.

Under strictly matched B3LYP/TZVP training conditions, GeoOpt-Net consistently delivers submilli-A RMSD and tightly bounded single-point energy deviations in a single forward pass. Error decomposition reveals that the dominant gains arise from substantially improved torsional accuracy, reducing dihedral deviations by 1 to 2 orders of magnitude relative to baseline methods. These geometric improvements translate directly into practical computational benefits: Geo-Opt-Net produces initial geometries that satisfy standard DFT convergence criteria at nonzero rates (65.0% loose; 33.4% default), dramatically shortening subsequent reoptimization trajectories and reducing wall-clock cost.

Importantly, GeoOpt-Net retains a robust behavior beyond its training domain. Across increasing molecular size and conformational flexibility, including out-of-training-range regimes, energy deviations scale smoothly without an abrupt escalation. Improved geometric fidelity further propagates to electronic observables, with dipole moments closely matching DFT references. Together, these results demonstrate that deterministic, equivariant geometry refinement offers a scalable and physically consistent route toward generating DFT-ready structures, enabling the efficient acceleration of large-scale quantum-chemical workflows.

A ASSOCIATED CONTENT

Data Availability Statement

The computational models, data sets, trained checkpoints, inference scripts, runtime benchmark scripts, and external validation resources reported in this work are publicly available on GitHub at https://github.com/ChengchunLiu/GeoOpt_ Net.

@ Supporting Information

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.jctc.6c01080.

Details on the construction of the data set (including data extraction and data set composition), experimental settings and training procedures, and additional experimental details (PDF)

AUTHOR INFORMATION

Corresponding Author

Fanyang Mo - School of AI for Science, AI for Science (AI4S)- Preferred Program, School of Advanced Materials, and Guangdong Provincial Key Laboratory of Nano-Micro Materials Research, Peking University Shenzhen Graduate School, Shenzhen 518055, China; State Key Laboratory of Advanced Waterproof Materials, School of Materials Scienceand Engineering, Peking University, Beijing 100871, China; Corcid.org/0000-0002-4140-3020;

Authors

Chengchun Liu - School of AI for Science and AI for Science (AI4S)-Preferred Program, Peking University Shenzhen Graduate School, Shenzhen 518055, China; State Key Laboratory of Advanced Waterproof Materials, School of Materials Science and Engineering, Peking University, Beijing 100871, China; (C) orcid.org/0009-0002-5550-4145

Wendi Cai - School of AI for Science and AI for Science (AI4S)-Preferred Program, Peking University Shenzhen Graduate School, Shenzhen 518055, China

Boxuan Zhao - School of AI for Science and AI for Science (AI4S)-Preferred Program, Peking University Shenzhen Graduate School, Shenzhen 518055, China; State Key Laboratory of Advanced Waterproof Materials, School of Materials Science and Engineering, Peking University, Beijing 100871, China

Complete contact information is available at: https://pubs.acs.org/10.1021/acs.jctc.6c01080

Author Contributions

F.M. and C.L. conceived the project, designed the method, and analyzed the results. C.L. organized and prepared structural data. W.C. and B.Z. participated in the discussion and provided key suggestions. F.M. and C.L. wrote the paper. F.M. supervised the project.

Notes

The authors declare no competing financial interest.

ACKNOWLEDGMENTS

We thank Peking University Shenzhen Graduate School and Shenzhen Government for the start-up funding support. We thank the High-Performance Computing Platform of Peking University for machine learning model training. We appreciate Dr. Jianning Zhang for his constructive suggestions and insightful discussions.

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