Automated Fitting of Neural Network Potentials at Coupled Cluster Accuracy: Protonated Water Clusters as Testing Ground

Highly accurate potential energy surfaces are of key interest for the detailed understanding and predictive modeling of chemical systems. In recent years, several new types of force fields, which are based on machine learning algorithms and fitted to ab initio reference calculations, have been introduced to meet this requirement. Here we show how high-dimensional neural network potentials can be employed to automatically generate the potential energy surface of finite sized clusters at coupled cluster accuracy, namely CCSD(T*)-F12a/aug-cc-pVTZ. The developed automated procedure utilizes the established intrinsic properties of the model such that the configurations for the training set are selected in an unbiased and efficient way to minimize the computational effort of expensive reference calculations. These ideas are applied to protonated water clusters from the hydronium cation, H$_3$O$^+$, up to the tetramer, H$_9$O$_{4}^{+}$, and lead to a common potential energy surface that describes all these systems at essentially converged coupled cluster accuracy with a fitting error of 0.06 kJ/mol per atom. The fit is validated in detail and separately for all clusters up to the tetramer and yields reliable results not only for stationary points, but also for reaction pathways, intermediate configurations, as well as different sampling techniques. Per design the NNPs constructed in this fashion can handle very different conditions including the quantum nature of the nuclei and enhanced sampling techniques covering very low as well as high temperatures. This enables fast and exhaustive exploration of the targeted protonated water clusters at essentially converged interactions. In addition, the automated process will allow one to tackle finite systems much beyond the present case.