Multicomponent Orbital-Optimized Perturbation Theory with Density Fitting: Anharmonic Zero-Point Energies in Protonated Water Clusters

Nuclear quantum effects such as zero-point energy are important in a wide range of chemical and biological processes. The nuclear-electronic orbital (NEO) framework intrinsically includes such effects by treating electrons and specified nuclei quantum mechanically on the same level. Herein, we implement the NEO scaled-opposite-spin orbital-optimized second-order Møller-Plesset perturbation theory with electron-proton correlation scaling (NEO-SOS′-OOMP2) using density fitting. This efficient implementation allows applications to larger systems with multiple quantum protons. Both the NEO-SOS′-OOMP2 method and its counterpart without orbital optimization predict proton affinities to within experimental precision and relative energies of protonated water tetramer isomers in agreement with previous NEO coupled cluster calculations. Applications to protonated water hexamers and heptamers illustrate that anharmonicity is critical for computing accurate relative energies. The NEO-SOS′-OOMP2 approach captures anharmonic zero-point energies at any geometry in a computationally efficient manner and hence will be useful for investigating reaction paths and dynamics in chemical systems.