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

TL;DR

The paper tackles the challenge of obtaining DFT-quality molecular geometries at scale by removing iterative QM optimization from the workflow. It introduces GeoOpt-Net, a multi-branch SE(3)-equivariant geometry refinement network that performs single-shot refinement from low-cost conformers toward $\text{B3LYP}/\text{TZVP}$-level geometries, aided by a two-stage, multi-fidelity training regime and a fidelity-aware feature modulation ($\tilde{\mathbf{h}} = \mathbf{h} \odot (1 + \mathbf{g}_d) + \mathbf{b}_d$). The method achieves sub-milli-Å $RMSD$ and near-zero $\Delta E$ relative to DFT references, while substantially improving DFT convergence readiness and maintaining electronic observables such as dipole moments. Across external drug-like molecules, GeoOpt-Net shows robust scaling with molecular complexity and outperforms traditional conformer generators, semiempirical methods, and other ML refinement pipelines, enabling scalable, accurate, and reproducible DFT-ready geometries for high-throughput quantum chemistry.

Abstract

Accurate molecular geometries are a prerequisite for reliable quantum-chemical predictions, yet density functional theory (DFT) optimization remains a major bottleneck for high-throughput molecular screening. Here we present GeoOpt-Net, a multi-branch SE(3)-equivariant geometry refinement network that predicts DFT-quality structures at the B3LYP/TZVP level of theory in a single forward pass starting from inexpensive initial conformers generated at a low-cost force-field level. GeoOpt-Net is trained using a two-stage strategy in which a broadly pretrained geometric representation is subsequently fine-tuned to approach B3LYP/TZVP-level accuracy, with theory- and basis-set-aware calibration enabled by a fidelity-aware feature modulation (FAFM) mechanism. Benchmarking against representative approaches spanning classical conformer generation (RDKit), semiempirical quantum methods (xTB), data-driven geometry refinement pipelines (Auto3D), and machine-learning interatomic potentials (UMA) on external drug-like molecules demonstrates that GeoOpt-Net achieves sub-milli-Å all-atom RMSD with near-zero B3LYP/TZVP single-point energy deviations, indicating DFT-ready geometries that closely reproduce both structural and energetic references. Beyond geometric metrics, GeoOpt-Net generates initial guesses intrinsically compatible with DFT convergence criteria, yielding nonzero ``All-YES'' convergence rates (65.0\% under loose and 33.4\% under default thresholds), and substantially reducing re-optimization steps and wall-clock time. GeoOpt-Net further exhibits smooth and predictable energy scaling with molecular complexity while preserving key electronic observables such as dipole moments. Collectively, these results establish GeoOpt-Net as a scalable, physically consistent geometry refinement framework that enables efficient acceleration of DFT-based quantum-chemical workflows.

Figures (6)

• 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.
• 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 pre-trained 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 fidelity-aware feature modulation (FAFM) mechanism. (b) SE(3)-equivariant geometric encoding and coordinate refinement. Pairwise distances ($r_{ij}$), bond angles ($\theta_{ijk}$), and dihedral angles ($\varphi_{ijkl}$) are encoded as invariant scalar features ($\ell=0$) via radial basis functions and as directional features ($\ell\ge1$) 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.
• Figure 3: Benchmarking geometric and energetic accuracy against B3LYP/TZVP references. (a) Logarithmic distribution of all-atom RMSD values relative to B3LYP/TZVP-optimized geometries. (b) Logarithmic distribution of B3LYP/TZVP single-point energy deviations ($\Delta E$). (c--e) Decomposition of geometric errors into (c) bond length, (d) bond angle, and (e) dihedral angle deviations, evaluated with respect to the B3LYP/TZVP reference structures.
• Figure 4: Quality of molecular initial guess geometries and impact on DFT convergence. (a) Cumulative distribution functions (CDFs) of maximum force, RMS force, maximum displacement, and RMS displacement for initial geometries generated by different methods. 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 re-optimization performance starting from different initial geometries, quantified by the number of optimization steps (left) and total elapsed wall-clock time (right).
• 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.