Classical-path approximation, nonadiabatic

As long as no approximation is introduced, it is clear that the equations of motion are equivalent in the diabatic and adiabatic representations. This is no longer true, however, once the classical-path approximation is employed the resulting classical-path equations of motion in the adiabatic representation are not equivalent to the diabatic equations of motion. Depending on whether the approximation is employed in the diabatic or in the adiabatic representation, the resulting classical-path Hamiltonian contains identical first-order nonadiabatic couphngs but different... 

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. 

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