Born-Oppenheimer approximation, phase-space

Readers who can remember what their first course in chemistry was like may recall how hard it was to learn how to visualize molecular structures and from there to visualize electronic wavefunctions. It is fair to say that an even more difficult challenge awaits those who attempt to visualize the phase space of reactive classical molecular dynamics within the Born-Oppenheimer approximation. The term phase space seems to have been originally coined by Gibbs. 2 But what is phase space And if it is so hard to visualize, why should we want to visualize it Obviously, such an endeavor is worthwhile to the extent that it yields new insights about molecular motion and helps us to interpret experiments. The concept of phase space will gradually emerge from the discourse in the remainder of this section. 

If one is interested in spectroscopy involving only the ground Born Oppenheimer surface of the liquid (which would correspond to IR and far-IR spectra), the simplest approximation involves replacing the quantum TCF by its classical counterpart. Thus pp becomes a classical variable, the trace becomes a phase-space integral, and the density operator becomes the phase-space distribution function. For light frequency co with ho > kT, this classical approximation will lead to substantial errors, and so it is important to multiply the result by a quantum correction factor the usual choice for this application is the harmonic quantum correction factor [79 84]. Thus we have... 

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. 

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