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Journal of Computational Chemistry

Integrating Coupled‐Cluster Theory Within AI Workflows for Accurate Molecular Geometries

Shahzad Akram, Konstantinos D. Vogiatzis

Published Aug 3, 2026• 0 citations
Publisher

Abstract

ABSTRACT We introduce a hybrid data‐driven coupled‐cluster (DDCC) framework that enables efficient coupled‐cluster singles and doubles (CCSD)‐level geometry optimizations through direct prediction of wavefunction parameters. In conventional CCSD, each optimization step requires solving the nonlinear ‐amplitude equations and the linear ‐equations, both scaling as , making geometry optimizations computationally demanding. The data‐driven CC (DDCC) scheme replaces these iterative solvers with neural network predictions of the and amplitudes, thereby removing the dominant computational bottleneck. The model utilizes a physically grounded representation of electron correlation, combining excitation‐based descriptors with a molecular orbital (MO) framework defined through orbital centroids. This design enables accurate recovery of the CC wave function parameters, which in turn yields reliable nuclear gradients throughout the optimization process. Across a diverse set of thermally perturbed geometries and out‐of‐sample molecules, DDCC reproduces CCSD geometries with near‐quantitative agreement, while reducing computational cost by over an order of magnitude. These results establish DDCC as an effective and transferable strategy for accelerating high‐level electronic structure calculations, providing a practical route toward routine CCSD‐quality geometry optimization in larger and more complex molecular systems.

DOI
10.1002/jcc.70475

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ABSTRACT We introduce a hybrid data‐driven coupled‐cluster (DDCC) framework that enables efficient coupled‐cluster singles and doubles (CCSD)‐level geometry optimizations through direct prediction of wavefunction parameters. In conventional CCSD, each optimization step requires solving the nonlinear ‐amplitude equations and the linear ‐equations, both scaling as , making geometry optimizations computationally demanding. The data‐driven CC (DDCC) scheme replaces these iterative solvers with neural network predictions of the and amplitudes, thereby removing the dominant computational bottleneck. The model utilizes a physically grounded representation of electron correlation, combining excitation‐based descriptors with a molecular orbital (MO) framework defined through orbital centroids. This design enables accurate recovery of the CC wave function parameters, which in turn yields reliable nuclear gradients throughout the optimization process. Across a diverse set of thermally perturbed geometries and out‐of‐sample molecules, DDCC reproduces CCSD geometries with near‐quantitative agreement, while reducing computational cost by over an order of magnitude. These results establish DDCC as an effective and transferable strategy for accelerating high‐level electronic structure calculations, providing a practical route toward routine CCSD‐quality geometry optimization in larger and more complex molecular systems.
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Solving the coupled-cluster (CC) equations is a cost-prohibitive process that exhibits poor scaling with system size. These equations are solved by determining the set of amplitudes ( t ) that minimize the system energy with respect to the coupled-cluster equations at the selected level of truncation. Here, a novel approach to predict the converged coupled-cluster singles and doubles (CCSD) amplitudes, thus the coupled-cluster wave function, is explored by using machine learning and electronic structure properties inherent to the MP2 level. Features are collected from quantum chemical data, such as orbital energies, one-electron Hamiltonian, Coulomb, and exchange terms. The data-driven CCSD (DDCCSD) is not an alchemical method because the actual iterative coupled-cluster equations are solved. However, accurate energetics can also be obtained by bypassing solving the CC equations entirely. Our preliminary data show that it is possible to achieve remarkable speedups in solving the CCSD equations, especially when the correct physics are encoded and used for training of machine learning models.
Computational cost limits the applicability of post-Hartree–Fock methods such as coupled-cluster on larger molecular systems. The data-driven coupled-cluster (DDCC) method applies machine learning to predict the coupled-cluster two-electron amplitudes ( t 2 ) using data from second-order perturbation theory (MP2). One major limitation of the DDCC models is the size of training sets that increases exponentially with the system size. Effective sampling of the amplitude space can resolve this issue. Five different amplitude selection techniques that reduce the amount of data used for training were evaluated, an approach that also prevents model overfitting and increases the portability of data-driven coupled-cluster singles and doubles to more complex molecules or larger basis sets. In combination with a localized orbital formalism to predict the CCSD t 2 amplitudes, we have achieved a 10-fold error reduction for energy calculations.
Density functional theory (DFT) is the most widely used method for calculating molecular properties; however, its accuracy is often insufficient for quantitative predictions. Coupled-cluster (CC) theory is the most successful method for achieving accuracy beyond DFT and for predicting properties that closely align with experiment. It is known as the ''gold standard'' of quantum chemistry. Unfortunately, the high computational cost of CC limits its widespread applicability. In this work, we present the Molecular Orbital Learning (MōLe) architecture, an equivariant machine learning model that directly predicts CC's core mathematical objects, the excitation amplitudes, from the mean-field Hartree-Fock molecular orbitals as inputs. We test various aspects of our model and demonstrate its remarkable data efficiency and out-of-distribution generalization to larger molecules and off-equilibrium geometries, despite being trained only on small equilibrium geometries. Finally, we also examine its ability to reduce the number of cycles required to converge CC calculations. MōLe can set the foundations for high-accuracy wavefunction-based ML architectures to accelerate molecular design and complement force-field approaches.
We introduce a reusable geometry-optimization engine in PyBEST for analytic, gradient-driven molecular structure optimization, with particular emphasis on orbital-optimized pair coupled-cluster doubles (OOpCCD/AP1roG). The engine interfaces PyBEST with the \texttt{geomeTRIC} optimizer, combining analytic electronic-structure gradients from PyBEST with the translation-rotation-internal coordinate (TRIC) framework, step control, and convergence machinery provided by \texttt{geomeTRIC}. Specifically, we present the first implementation of analytic OOpCCD nuclear gradients within a Lagrangian formalism. Our approach and implementation are generally applicable to any seniority-zero wavefunctions that feature orbital optimization and allow for the evaluation of response one- and two-particle reduced density matrices. Owing to the seniority-zero structure of pCCD and the orbital stationarity of the optimized reference, the resulting gradient equations are compact, minimizing the storage of the full two-particle reduced density matrix, and avoiding finite-difference differentiation of wavefunction parameters. Validation on representative closed-shell systems shows that the OOpCCD-based PyBEST-\texttt{geomeTRIC} workflow converges robustly and reproduces reference equilibrium geometries and energies within tight tolerances. Most importantly, OOpCCD produces structural parameters that deviate by approximately 0.02 Å (0.01 Å) for bond lengths or less than 1∘^\circ∘ for bond angles from CCSD(F12c)(T*) (MP2) reference structures.
Variational quantum eigensolver ansätze hold considerable promise for ground-state energy calculations on near-term quantum hardware, yet most promising ansatz designs currently strongly depend on how well the molecular orbital basis captures the electronic correlation of the system. Computing optimized orbital coefficients via classical routines is computationally expensive and must be performed independently for each molecular geometry -- a bottleneck that limits scalability across chemical space. We present a graph neural network framework that predicts optimized orbital coefficients directly from molecular geometry and pair-wise bonding structure. Trained on hydrogenic systems of modest size (H4H_4H4​ and H6H_6H6​) across tens of thousands of geometries, our model transfers to larger, unseen systems (H8H_8H8​, H10H_{10}H10​ and H12H_{12}H12​) without retraining -- demonstrating strong out-of-distribution generalization with respect to system size. When evaluating on structured and random configurations, and comparing against energies obtained with full classical optimization, our model reaches mean absolute energy errors O(102)\mathcal{O}(10^2)O(102) and O(10)\mathcal{O}(10)O(10) milli-Hartrees, respectively. Beyond energy estimation, the predicted orbitals serve as high-quality warm-start initializations that substantially reduce optimizer iterations to ground-state energy convergence. These results establish graph neural networks as an effective and scalable strategy for accelerating orbital optimization in hybrid quantum-classical workflows, directly reducing the classical pre-processing overhead that currently limits the practical deployment of variational quantum eigensolver on near-term quantum hardware.