Diagonal Born-Oppenheimer Corrections within the Nuclear-Electronic Orbital Framework

The nuclear-electronic orbital (NEO) method treats specified nuclei, typically protons, quantum mechanically on the same level as the electrons. This approach invokes the Born-Oppenheimer separation between the quantum and classical nuclei, as well as the conventional separation between the electrons and classical nuclei. To test the validity of this additional adiabatic approximation, herein the diagonal Born-Oppenheimer correction (DBOC) within the NEO framework is derived, analyzed, and calculated numerically for a set of eight molecules. Inclusion of the NEO DBOC is found to change the equilibrium bond lengths by only ∼10^-4 Å and the heavy atom vibrational stretching frequencies by ∼1-2 cm^-1 per quantum proton bonded to an atom participating in the vibrational mode. These results imply that the DBOC does not significantly impact molecular properties computed with the NEO approach, although it can be included when necessary. Understanding the physical characteristics and quantitative contributions of the DBOC has broad implications for applications of multicomponent density functional theory and wave function methods.