On the validity of the born-oppenheimer separation and the accuracy of diagonal corrections in anharmonic molecular vibrations

The energies and wave functions of several lowest-lying vibrational states of FHF-, ClHCl-, and BrHBrhave been computed by a finite-difference method with and without the Born-Oppenheimer (BO) separation between the heavy (halogen) and light (hydrogen) particle motion. The so-called diagonal BO correction (DBOC), which includes the effect of the heavy particles' kinetic energy operator acting on the light particles' wave functions, has also been made to the energies. The errors caused by the BO approximation are found to be remarkably small (ca. 10^-5 au) and can be systematically and effectively reduced by the DBOC except for states excited in the heavy particle motion. When the bare mass of the light particle is used instead of the reduced mass in the BO approximation and, therefore, the translational degrees of freedom are not correctly decoupled, the errors in the BO treatment become greater by a factor of 2-7. However, these additional errors are almost completely erased by the DBOC. Analytical and numerical results suggest that the remaining errors in the BO and DBOC treatments be proportional to ε¹ and ε², where e is the mass ratio of the light to heavy particles, when the corrections are made to the potential energy surfaces and the wave functions for these surfaces are determined variationally. When the DBOC is applied in the first-order perturbation approximation, the remaining errors scale as ε^3/2.