Born-Oppenheimer and non-Born-Oppenheimer contributions to time-dependent electron momenta

Using a model system for a coupled electron-nuclear motion, we calculate time-dependent expectation values of the electronic momentum operator. Whereas, within the velocity form, this quantity vanishes if the Born-Oppenheimer (BO) approximation is applied, it differs from zero if the calculation employs the length form of the expectation value. Using the adiabatic expansion of the total wave function, it is analyzed which terms contribute to the mean electronic momentum. For an adiabatic motion, where the BO approximation holds, it is shown that in the length form, the BO wave function yields an excellent estimate of the momentum. On the other hand, in the velocity form, it is necessary to include non-BO terms to calculate its value. This illustrates the different convergence behavior of the matrix elements in the two formulations. In the diabatic limit where the electron density does only marginally change upon the nuclear motion, both approaches converge to a vanishing mean electronic momentum.