Machine-learning surrogate models for particle insertions and element substitutions.

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Two machine-learning-aided thermodynamic integration schemes to compute the chemical potentials of atoms and molecules have been developed and compared. One is the particle insertion method, and the other combines particle insertion with element substitution. In the former method, the species is gradually inserted into the liquid and its chemical potential is computed. In the latter method, after the particle insertion, the inserted species is substituted with another species, and the chemical potential of this new species is computed. In both methods, the thermodynamic integrations are conducted using machine-learned potentials trained on first-principles datasets. The errors of the machine-learned surrogate models are further corrected by performing thermodynamic integrations from the machine-learned potentials to the first-principles potentials, accurately providing the first-principles chemical potentials. These two methods are applied to compute the real potentials of proton, alkali metal cations, and halide anions in water. The applications indicate that these two entirely different thermodynamic pathways yield identical real potentials within statistical error bars, demonstrating that both methods provide reproducible real potentials. The computed real potentials and solvation structures are also in good agreement with past experiments and simulations. These results indicate that machine-learning surrogate models enabling particle insertion and element substitution provide a precise method for determining the chemical potentials of atoms and molecules.