Triple excitations in nuclear–electronic orbital coupled cluster theory for multiple quantum protons

Within the nuclear–electronic orbital (NEO) framework, specified nuclei, typically protons, are treated quantum mechanically on the same level as the electrons. This framework allows for nuclear quantum effects, such as anharmonic zero-point energy, to be included in quantum chemical calculations in a computationally efficient manner. NEO coupled cluster (NEO-CC) methods provide a promising strategy for producing accurate ground-state properties of moderately sized molecular systems. Herein, the inclusion of triple excitations in NEO-CC methods is explored for systems with multiple quantum protons. Full and perturbative treatments of electron–electron–proton and electron–proton–proton triple excitations are investigated. The perturbative treatment agrees quantitatively with the full treatment for proton affinity calculations and is much more computationally efficient, especially for systems with multiple quantum protons. The NEO-CCSD(T) method, which includes single, double, and perturbative electron–electron–proton, electron–proton–proton, and electron–electron–electron triple excitations, reproduces experimentally measured proton affinities within experimental uncertainty using a complete basis set extrapolation. Moreover, application of the NEO-CCSD(T) method to protonated water tetramers, with all nine protons treated quantum mechanically, incorporates the essential anharmonic zero-point energy with only a single-point energy calculation. NEO-CC methods offer an accurate and computationally practical approach for inclusion of nuclear quantum effects in molecular systems and may serve as a benchmark for lower-level NEO methods.