Incorporating advantages of time-dependent dynamics in time-independent collision methods: Early asymptotic analysis

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Abstract
Transition amplitudes that may be written as matrix elements of a Green’s operator are quite usefully cast as temporal correlation function of localized wave packets. In many situations, the correlation function is nonzero only for a short time while the wave packet is close to its initial position. The transition amplitude is only sensitive to the potential near the initial location of the wave packet. We derive time-independent analogs of the above described features of time-dependent collision theory for matrix elments of the time-independent Green’s function. Transition amplitudes are shown to be completely independent of the potential outside the Franck–Condon region when the true wave function can be approximated by a (primitive or uniform) semiclassical form in the outer region. The reaction coordinate can be separated into a strongly interacting Franck–Condon region, to be treated by standard close coupling methods, and an asymptotic region for which no dynamical calculations are required in the semiclassical limit. Because this is a spatial, rather than temporal, separation, our semiclassical method is successful in situations, such as predissociation, where time-dependent methods fail. Also, we can evaluate total cross sections by using basis sets that are optimal for the Franck–Condon region without ever considering the transition to asymptotic target states. Numerical illustrations are provided for predissociations in Na2 and Br2.