[论文解读] Analytic Gradients and Geometry Optimization for Orbital-Optimized Pair Coupled Cluster Doubles

Saman Behjou, Iulia Emilia Brumboiu+1 more|arXiv (Cornell University)|Mar 20, 2026

Advanced Chemical Physics Studies被引用 0

一句话总结

简要：在 PyBEST 中引入一个由解析梯度驱动的几何优化引擎，用于轨道优化的对偶 CC 双子（OOpCCD/AP1roG），与 geomeTRIC 和 TRIC 坐标接口，并通过参考几何进行验证。

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 exttt{geomeTRIC} optimizer, combining analytic electronic-structure gradients from PyBEST with the translation-rotation-internal coordinate (TRIC) framework, step control, and convergence machinery provided by exttt{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- exttt{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.

研究动机与目标

• Motivate accurate geometry optimization for strongly correlated systems where orbital optimization is essential.
• Develop and implement analytic nuclear gradients for OOpCCD within a Lagrangian framework.
• Provide a reusable geometry-optimization workflow that combines PyBEST gradients with geomeTRIC/TRIC for robust convergence.
• Demonstrate reduced storage and stable performance due to seniority-zero pCCD structure and orbital optimization.

提出的方法

• Formulate analytic gradients for OOpCCD using a Lagrangian (Z-vector) approach with orbital optimization eliminating explicit orbital-response terms.
• Represent gradients as contractions of derivative one- and two-electron integrals with response 1- and 2-RDMs (sparse, pair-structured).
• Evaluate derivative integrals in AO MO-transformed basis (mixed AO-MO approach) to balance cost and sparsity.
• Transform Hamiltonian to an orthonormal MO (OMO) basis and apply symmetric-connection derivatives for integrals.
• Utilize TRIC internal coordinates and geomeTRIC for geometry updates, step control, and convergence in a gradient-driven optimization loop.
• Benchmark against diatomic and small organic molecules using cc-pVDZ/cc-pVTZ with RHF and OOpCCD gradients.

Figure 1: Molecular test set used for benchmarking the performance of the analytic-gradient-driven geometry optimization engine within OOpCCD. The molecular structures are taken from Ref. 25 and re-optimized for OOpCCD. All molecular structures are visualized using the Jmol package. 1

实验结果

研究问题

• RQ1 Can analytic OOpCCD gradients be implemented via a Lagrangian framework without explicit orbital-response terms?
• RQ2 How accurately do OOpCCD-optimized geometries reproduce reference bond lengths and angles compared to MP2 and CCSD(F12c)(T*)?
• RQ3 Is the PyBEST–geomeTRIC workflow robust for optimizing ground- and transition-state structures using OOpCCD gradients?
• RQ4 What are the computational savings and numerical stability benefits of using a mixed AO-MO gradient evaluation for OOpCCD?
• RQ5 How do internal-coordinate optimizations (TRIC) perform for systems with soft intermolecular modes in the OOpCCD context?

主要发现

Molecule	Basis set	Numeric PES fit: E(r_e) [Eh]	Numeric PES fit: r_e [Å]	Analytic: E_e [Eh]	Analytic: r_e [Å]
BN	cc-pVDZ	-79.029999	1.2688	-79.029999	1.2688
BN	cc-pVTZ	-79.065094	1.2582	-79.065094	1.2582
C2	cc-pVDZ	-75.549523	1.2387	-75.549522	1.2387
C2	cc-pVTZ	-75.582371	1.2213	-75.582369	1.2213
CN+	cc-pVDZ	-91.803418	1.1657	-91.803774	1.1630
CN+	cc-pVTZ	-91.842591	1.1499	-91.842591	1.1499
CO	cc-pVDZ	-112.855529	1.1231	-112.855529	1.1231
CO	cc-pVTZ	-112.911671	1.1156	-112.911671	1.1156
F2	cc-pVDZ	-198.855477	1.5186	-198.857066	1.5190
F2	cc-pVTZ	-198.949747	1.4624	-198.949747	1.4624
N2	cc-pVDZ	-109.062713	1.1016	-109.062706	1.1016
N2	cc-pVTZ	-109.127740	1.0867	-109.127740	1.0867

• Analytic OOpCCD gradients are correctly implemented within a Lagrangian framework, yielding gradients equal to partial derivatives at stationarity.
• The OOpCCD gradient evaluation relies on sparse, pair-structured 1-RDM and 2-RDM, enabling efficient MO-based contraction with derivative integrals.
• The mixed AO-MO approach efficiently computes one-body terms in AO basis and two-body terms in MO basis, balancing storage and cost.
• Validation shows OOpCCD geometry optimizations reproduce reference equilibrium geometries and energies within tight tolerances, with bond lengths typically within ~0.02 Å (0.01 Å in some cases) of CCSD(F12c)(T*)/MP2 references.
• Diatomic bond lengths from analytic optimizations agree with those obtained from PES fits, indicating reliable gradient correctness.
• Transition-state optimizations and standard ground-state geometries are successfully targeted using the PyBEST–geomeTRIC interface.

Figure 2: RMSEs of bond lengths (Å, blue bars) and bond angles (degrees, orange bars) for OOpCCD equilibrium geometries relative to CCSD(F12c)(T*) reference structures, with all OOpCCD calculations performed using the cc-pVDZ basis set. The right vertical axis refers to angular deviations (degrees),