Analytical nuclear gradients for the CC2 method using pair-natural orbitals based on localized virtual molecular orbitals for ground states

Analytical nuclear energy gradients are indispensable for geometry optimizations and potential-energy-surface explorations, yet their development for pair-natural-orbital (PNO)-based methods remains limited. In this work, we present the analytical nuclear gradient theory for the ground-state PNO-based second-order approximate coupled cluster (CC2) model using localized virtual molecular orbitals (LVMOs). The gradient theory is formulated within the Lagrangian framework as an extension of our previously developed LVMO-based PNO-MP2 analytical gradient theory. Although the use of LVMOs requires additional localization constraints in the Lagrangian, orthonormal LVMOs simplify the coupled-perturbed equations and energy gradient expressions, and these advantages are exploited in the PNO-CC2 analytical gradient formulation. The implementation is extended to the spin-component-scaled variant. To facilitate the development and implementation, we extended our previously developed automated symbolic derivation framework, which systematically generates the derivative expressions for gradient calculations. Benchmark calculations on phenylalkane chains demonstrate favorable scaling of the analytical gradient evaluation. Gradients vary smoothly along a bond-dissociation coordinate of a ketene derivative, indicating that PNO orbital truncation does not introduce noticeable artifacts. Geometry optimizations of difluorodiene reproduce the characteristic elongation of electron-rich bonds observed for the CC2 theory relative to the MP2 theory. Applications to intermolecular complexes show that the PNO-CC2 framework provides optimized structures and binding energies comparable with those from the PNO-MP2 framework. Single-point gradient calculations for realistic molecular systems of around one hundred atoms demonstrate applicability of the implementation. This work establishes the analytical gradient formulation for ground-state PNO-CC2 theory and provides a foundation for future extensions to excited states.