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arXiv:2603.20419

Analytic Gradients and Geometry Optimization for Orbital-Optimized Pair Coupled Cluster Doubles

Saman Behjou, Iulia Emilia Brumboiu, Katharina Bogusławski

Published Mar 20, 2026• 0 citations
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.2603.20419

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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.
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.
Orbital optimization within the variational quantum eigensolver (VQE) framework is useful for improving active-space quality and accurately capturing electron correlation with shallow quantum circuits. The orbital-optimized unitary coupled cluster (OO-UCC) method [W. Mizukami et al., Phys. Rev. Res. 2, 033421 (2020)], in its original formulation, employs the orbital gradient and Hessian in a Newton--Laphson procedure. In this study, we propose sequential orbital-optimized unitary coupled cluster doubles (SOO-UCCD), extending the gradient-free sequential minimal optimization approach [K. M. Nakanishi et al., Phys. Rev. Res. 2, 043158 (2020)] to orbital rotation parameters. Based on the frequency structure implied by the algebraic properties of the orbital rotation generators, our method determines each orbital rotation angle from a nine-point energy profile. The energy landscape is evaluated by classically transforming the Hamiltonian integrals and contracting them with the reduced density matrices already measured for the VQE energy evaluation. This allows for orbital optimization without evaluating the orbital gradient or Hessian. By building on the core benefits of the OO-UCC framework, our method provides a gradient-free and hyperparameter-free orbital-optimization procedure. State-vector simulations for LiH in the STO-3G basis show that SOO-UCCD reproduces UCCSD-level accuracy along the bond-stretching curve and exhibits macro-iteration convergence comparable to that of L-BFGS-based orbital optimization.
We derive analytic energy gradients of the driven similarity renormalization group (DSRG) multireference second-order perturbation theory (MRPT2) using the method of Lagrange multipliers. In the Lagrangian, we impose constraints for a complete-active-space self-consistent-field reference wave function and the semicanonical orthonormal molecular orbitals. Solving the associated Lagrange multipliers is found to share the same asymptotic scaling of a single DSRG-MRPT2 energy computation. A pilot implementation of the DSRG-MRPT2 analytic gradients is used to optimize the geometry of the singlet and triplet states of p-benzyne. The equilibrium bond lengths and angles are similar to those computed via other MRPT2s and Mukherjee's multireference coupled cluster theory. An approximate DSRG-MRPT2 method that neglects the contributions of three-body density cumulant is found to introduce negligible errors in the geometry of p-benzyne, lending itself to a promising low-cost approach for molecular geometry optimizations using large active spaces.
PyCC is a Python-based quantum chemistry package that emphasizes coupled cluster and related methods. Developed over the last five years, PyCC's purpose is to provide a clear reference implementation that will serve as a validation suite for production code, a platform for rapid prototyping and testing of new methods, and an educational resource for newcomers to the field. The package covers both closed- and open-shell single-determinant references in spatial- and spin-orbital formulations for the coupled cluster hierarchy through triples-including methods such as CC3, as well as for MP2 and CI methods. PyCC provides a broad range of molecular properties such as analytic first and second energy derivatives including gradients, Hessians, polarizabilities and atomic polar tensors, as well as non-energy-derivative properties, such as velocity-gauge atomic polar tensors and diagonal Born-Oppenheimer corrections, plus dynamic linear response functions, and excitation energies. The package includes a number of unique capabilities, such as real-time CC propagation (up through CC3), atomic axial tensors for simulating vibrational circular dichroism spectra at the MP2 and CI levels, and local correlation methods including perturbation-aware correlation domains. The code is designed around a shared Wavefunction base and a method-agnostic derivative layer that enables a new method to inherit the full property set once it supplies its reduced densities and their first-order responses. PyCC is fully open source and available on GitHub under the BSD 3-clause license.
Wave functions constructed from electron-pair states can accurately model strong electron correlation effects and are promising approaches especially for larger many-body systems. In this article, we analyze the nature and the type of electron correlation effects that can be captured by wave functions restricted to electron-pair states. We focus on the pair-coupled-cluster doubles (pCCD) ansatz also called the antisymmetric product of the 1-reference orbital geminal (AP1roG) method, combined with an orbital optimization protocol presented in Boguslawski et al. [Phys. Rev. B 89, 201106(R) (2014)], whose performance is assessed against electronic structures obtained form density-matrix renormalization-group reference data. Our numerical analysis covers model systems for strong correlation: the one-dimensional Hubbard model with a periodic boundary condition as well as metallic and molecular hydrogen rings. Specifically, the accuracy of pCCD/AP1roG is benchmarked using the single-orbital entropy, the orbital-pair mutual information, as well as the eigenvalue spectrum of the one-orbital and two-orbital reduced density matrices. Our study indicates that contributions from singly occupied states become important in the strong correlation regime which highlights the limitations of the pCCD/AP1roG method. Furthermore, we examine the effect of orbital rotations within the pCCD/AP1roG model on correlations between orbital pairs.
macroqc is a quantum chemistry software package designed for high-accuracy electronic structure computations and large-scale molecular simulations. The macroqc framework provides efficient implementations of energies and analytic gradients for a wide range of many-body perturbation theory and coupled-cluster (CC) methods. In particular, the analytic gradient implementations of second-order Møller-Plesset perturbation theory (MP2), coupled-cluster singles and doubles (CCSD), and CCSD with perturbative triples [CCSD(T)] exhibit excellent computational performance, even in comparison with established commercial quantum chemistry programs. In addition to conventional correlated methods, macroqc incorporates several advanced capabilities, including molecular fragmentation techniques for ground and excited state molecular systems and density-fitted analytic gradients. The software also supports interoperability with selected external packages through a flexible plugin framework, allowing developers to construct customized interfaces and extensions. The notable features of macroqc are the linear-scaling systematic molecular fragmentation (LSSMF) and the cluster-in-molecule (CIM) approaches, which enable the application of high-level post-Hartree-Fock approaches to large molecular systems that would otherwise be computationally prohibitive. Furthermore, the new machine learning (ML) library enables the integration of data-driven approaches into quantum chemical workflows, providing efficient tools for developing and applying ML-based models for molecular property prediction and electronic structure computations. With its combination of efficient implementations, advanced theoretical methods, and extensible design, macroqc is expected to serve as a useful platform for a broad range of computational chemistry applications.