From Energies to Structures: PNO-F12 Geometry Optimization of Large Molecules for Near-Spectroscopic Accuracy

Geometry optimization at the coupled-cluster level remains a major bottleneck in high-accuracy quantum chemistry, particularly for explicitly correlated and local-correlation approaches for which analytical gradients are not available. Here we present a general and scalable framework that enables routine geometry optimizations at the PNO-F12 level by combining numerical gradients with a dual-coordinate strategy: geometry updates are performed in generalized internal coordinates, while gradients are evaluated via finite differences along normal modes. A key aspect of the present development is that the optimization of mode-dependent displacement step sizes is essential to control not only the usual truncation and noise errors of finite differences, but also additional numerical effects arising from explicit correlation and from the geometry dependence of pair-specific orbital domains. This allows numerical gradients to reach the precision required for submilliangstrom structural refinements. The resulting protocol delivers equilibrium geometries of near-spectroscopic accuracy for both benchmark molecules and challenging large systems such as cytosine tautomers, corannulene, and coronene, while retaining affordable computational times and excellent parallel scalability. Comparison with near-canonical composite schemes shows that the use of PNO-F12 numerical gradients preserves coupled-cluster structural accuracy. The present work demonstrates that geometry optimization is no longer a limiting factor for bringing explicitly correlated coupled-cluster accuracy from energies to equilibrium structures, opening the way to routine high-accuracy structural and spectroscopic studies for medium-to-large molecular systems.