On the calculation of time-dependent electron momenta within the Born-Oppenheimer approximation

In the case of an adiabatic motion in molecules, electrons adjust to the smoothly changing geometry of the nuclei. Although then the Born-Oppenheimer (BO) approximation is valid, it fails in predicting the time-dependence of electron momenta because, within its product ansatz for the wave function, the respective expectation values are zero. It is shown that this failure can be circumvented using the Ehrenfest theorem. Here we extend our former work [T. Schaupp et al., Eur. Phys. J. B 91, 97 (2018)] and regard models in higher dimensions and for more particles. We solve the time-dependent Schrödinger equation for the combined nuclear-electronic motion and compare the results to those derived from BO wave functions. For all situations, it is found that the time-dependent BO electronic momenta are in excellent agreement with the numerically exact results.