Non-Born-Oppenheimer electronic and nuclear wavepacket dynamics.

A practical quantum theory for unifying electronic and nuclear dynamics, which were separated by the Born-Oppenheimer approximation, is proposed. The theory consists of two processes. Nonadiabatic (quantum) electron wavepacket dynamics on branching (non-Born-Oppenheimer) nuclear paths are first constructed. Since these paths are not the classical trajectories, most of the existing semiclassical theories to generate quantum wavepacket do not work. Therefore, we apply our own developed semiclassical wavepacket theory to these generated non-Born-Oppenheimer paths. This wavepacket is generated based on what we call the action decomposed function, which does not require the information of the so-called stability matrix. Thus, the motion of nuclei is also quantized, and consequently the total wave function is represented as a series of entanglement between the electronic and nuclear wavepackets. In the last half of the article, we show the practice to demonstrate how these independent theories can be unified to give electron-nuclear wavepackets in a two-state model. The wavepackets up to the phases and resultant transition probabilities are compared to the full quantum-mechanical counterparts. It turns out that the lowest level approximation to the wavepacket approach already shows a good agreement with the full quantum quantities. Thus, the present theoretical framework gives a basic method with which to study non-Born-Oppenheimer electronic and nuclear wavepacket states relevant to ultrafast chemical events.