Development of nuclear basis sets for multicomponent quantum chemistry methods

The nuclear-electronic orbital (NEO) framework provides a practical approach for directly incorporating nuclear quantum effects and non-Born-Oppenheimer effects of specified nuclei, typically protons, into quantum chemistry calculations. Multicomponent wave function based methods, such as NEO coupled cluster singles and doubles, and multicomponent density functional theory (DFT), such as NEO-DFT, require the appropriate selection of electronic and nuclear basis sets. Although a wide array of electronic basis sets are available, systematically developed nuclear basis sets that balance accuracy and efficiency have been lacking. Herein, a series of nuclear basis sets are developed and shown to be accurate and efficient for describing both ground and excited state properties of multicomponent systems in which electrons and specified protons are treated quantum mechanically. Three series of Gaussian-type nuclear basis sets, denoted PB4, PB5, and PB6, are developed with varying levels of angular momentum. A machine-learning optimization procedure relying on the Gaussian process regression method is utilized to accelerate the optimization process. The basis sets are validated in terms of predictions of ground state energies, proton densities, proton affinities, and proton vibrational excitation energies, allowing the user to select the desired balance between accuracy and efficiency for the properties of interest. These nuclear basis sets will enhance the tractability of NEO methods for applications to a wide range of chemical systems.