Model calculations testing the adiabatic Born-Oppenheimer approximation and its non-adiabatic corrections

Abstract

Tests of the accuracy of the Born-Oppenheimer approximation, with and without non-adiabatic corrections are made on a model system of two coupled harmonic oscillators representing electronic and nuclear motion. Two couplings are considered: bilinear and biquadratic in the displacements. For the energies of sublevels of the ground state, the transition dipole moments of the lowest vibrational transition, and the transition moment in a vibrationally perturbed forbidden electronic transition, the Born-Oppenheimer result including non-adiabatic corrections agrees with the exact result up to terms quadratic in the coupling constant. In a model of near-resonance of the coupled systems, “level crossing”, the adiabatic approximation fails, as is well-known. Even with inclusion of the non-adiabatic corrections the result does not agree to quadratic terms. However near resonance, with physically reasonable values, the methods give results that are very close. In all cases inclusion of the non-adiabatic terms is essential.