Simulating solvation and acidity in complex mixtures with first-principles accuracy: the case of CH$_3$SO$_3$H and H$_2$O$_2$ in phenol

We present a generally-applicable computational framework for the efficient and accurate characterization of molecular structural patterns and acid properties in explicit solvent using H$_2$O$_2$ and CH$_3$SO$_3$H in phenol as an example. In order to address the challenges posed by the complexity of the problem, we resort to a set of data-driven methods and enhanced sampling algorithms. The synergistic application of these techniques makes the first-principle estimation of the chemical properties feasible without renouncing to the use of explicit solvation, involving extensive statistical sampling. Ensembles of neural network potentials are trained on a set of configurations carefully selected out of preliminary simulations performed at a low-cost density-functional tight-binding (DFTB) level. The energy and forces of these configurations are then recomputed at the hybrid density functional theory (DFT) level and used to train the neural networks. The stability of the NN model is enhanced by using DFTB energetics as a baseline, but the efficiency of the direct NN (i.e., baseline-free) is exploited via a multiple-time step integrator. The neural network potentials are combined with enhanced sampling techniques, such as replica exchange and metadynamics, and used to characterize the relevant protonated species and dominant non-covalent interactions in the mixture, also considering nuclear quantum effects.