Non-Born-Oppenheimer corrections in an exactly solvable model of the hydrogen ion molecule

We have obtained some exact solutions for a solvable model for the hydrogen molecular ion where the Coulomb interaction between the electron and the nuclei is replaced by a harmonic potential. We have also obtained the solutions to this model within the ordinary Born-Oppenheimer (BO) approximation for the electronic and nuclear degrees of freedom. The model shows some of the qualitative features of the real system and allows for a detailed study of non-adiabatic effects on the electronic structure through the calculation of the first-order corrections to the BO approximation, resulting from the nuclear kinetic energy and the vibronic coupling. The results are of interest in the context of non-adiabatic treatment of molecules and for the description of molecules in external magnetic fields.