Geometry Relaxation and Transition State Search throughout Chemical Compound Space with Quantum Machine Learning

We use energies and forces predicted within response operator based quantum machine learning (OQML) to perform geometry optimization and transition state search calculations with legacy optimizers. For randomly sampled initial coordinates of small organic query molecules we report systematic improvement of equilibrium and transition state geometry output as training set sizes increase. Out-of-sample S$_\mathrm{N}$2 reactant complexes and transition state geometries have been predicted using the LBFGS and the QST2 algorithm with an RMSD of 0.16 and 0.4 \r{A} -- after training on up to 200 reactant complexes relaxations and transition state search trajectories from the QMrxn20 data-set, respectively. For geometry optimizations, we have also considered relaxation paths up to 5'500 constitutional isomers with sum formula C$_7$H$_{10}$O$_2$ from the QM9-database. Using the resulting OQML models with an LBFGS optimizer reproduces the minimum geometry with an RMSD of 0.14~\r{A}. For converged equilibrium and transition state geometries subsequent vibrational normal mode frequency analysis indicates deviation from MP2 reference results by on average 14 and 26\,cm$^{-1}$, respectively. While the numerical cost for OQML predictions is negligible in comparison to DFT or MP2, the number of steps until convergence is typically larger in either case. The success rate for reaching convergence, however, improves systematically with training set size, underscoring OQML's potential for universal applicability.