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

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 \AA{} (0.01 \AA{}) for bond lengths or less than 1$^\circ$ for bond angles from CCSD(F12c)(T*) (MP2) reference structures.