Spectrum of hydrogen atom having a harmonically bound nucleus: Beyond the Born-Oppenheimer approximation

Spectral features of a quantum mechanical model system are determined within the formalisms of degenerate and nondegenerate perturbation theory. These features include energy level shifts, vibronic intensities, and interactions between members of degenerate manifolds not predicted by the application of the Born-Oppenheimer approximation to the model system. Time-dependent processes resulting in vibrationally induced pre-ionization are also investigated. The model system consists of a proton subject to a Hooke's law potential (constrained to quantum mechanical motion in one dimension) and an electron attracted to the proton by a Coulombic potential. The equilibrium position of the proton is the origin of an imposed coordinate frame. Such a system may be compared with a diatomic molecule such as rubidium hydride. The energy of the ground state of the model system as calculated by perturbation theory deviates from the Born-Oppenheimer approximation prediction by 2.6% of the zero-point energy of the proton.