Nonadiabatic electron wavepackets along branching paths

2 Nonadiabatic electron wavepackets along branching paths 

It is actually very difficult to solve the entire scheme down to Eq. (6.5) for systems of chemical interest, even if a very good set of /) is available. (Note that electronic structure theory (quantum chemistry) can handle far larger molecular systems within the Born-Oppenheimer approximation) than the nuclear dynamics based on Eq. (6.5) can do.) This is because the short wavelength natme of nuclear matter wave blocks accurate computation and brings classical nature into the nuclear dynamics, in which path (trajectory) representation is quite often convenient and useful than sticking to the wave representation. Then what do the paths of nuclear dynamics look like on the occasion of nonadiabatic transitions, for which it is known that the nuclear wavepackets bifurcate, reflecting purely quantum nature. 

When we talk about wavepacket bifurcation in nonadiabatic transitions, on the other hand, not only the nuclear wavepackets but the electronic counterparts should also undergo branching. 

Therefore we explore below a theoretical framework in which electronic wavepackets propagate in time along bifurcating nuclear paths. 

To formulate the electron wavepacket dynamics on a clear basis, we begin by representing the total Hamiltonian operator in a basis set h/(R)) R) such that [422, 423] 

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It is obvious that the path is not branched by the nonadiabatic interaction, and therefore the important feature of quantum entanglement caused by nonadiabatic transition is not captured correctly in this formalism. Furthermore, in a more rigorous mathematical context, the above naive treatment of electron wavepackets along with classical paths is foimd to be not quite correct. Both of Eqs. (4.41) and (4.43) need corrections, which should be referred in [493]. 

One possible way to treat such a case is to use an approximated approach of the nonadiabatic electron wavepacket theory, the phase-space averaging and natural branching (PSANB) method [493], or the branching-path representation, in which the wavepackets propagate along non-Born-Oppenheimer branching paths. 

As discussed above, the nuclear paths can branch due to the presence of nonadiabatic interactions. Since the electronic wavepackets are to be carried over along those branching paths, the electronic packets should also branch accordingly. Such branchings can be viewed as follows. 

Then the recent notion of nonadiabaticity in electron d3mamics is introduced. To be consistent with the wavepacket bifurcation, we introduce the method of electron wavepacket d3mamics that undergoes bifurcation while being carried along the so-called non-Born-Oppenheimer paths, which also branch due to nonadiabatic interactions. We will further proceed to the discussion about the interaction of molecular nonadiabatic states with intense laser fields. In this way, we penetrate on one hand into unknown domains of molecular properties such as (1) electron-nuclear quantmn entanglement due to nonadiabatic transitions and its experimental observation, (2) coherence and decoherence of electron and nuclear wavepackets, which qualitatively dominate the quantmn mechanical probabiUties of quantum transition dynamics, (3) characteristic phenomena arising from the time-dependent fluctuation of molecular electronic states, (4) the physics of interference between the nonadiabatic djmamics and external fields, and so on. 

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