Franck-Condon approximation convergence

This approximation is not valid, say, for the ohmic case, when the bath spectrum contains too many low-frequency oscillators. The nonlocal kernel falls off according to a power law, and kink interacts with antikink even for large time separations. We assume here that the kernel falls off sufficiently fast. This requirement also provides convergence of the Franck-Condon factor, and it is fulfilled in most cases relevant for chemical reactions. 

Initial conditions for the wavepacket calculations correspond to the Franck-Condon geometry, and the effective-mode expansion is defined with respect to this reference geometry. Figure 15.3 shows the time-dependent diabatic S2 populations and autocorrelation functions C(f) = (V (0) vf(f)) generated from the successive spectral density approximants J (o), M = 1,...,3. All orders agree over the shortest time scale ( 5 fs), and the orders M = 2,3 are found to be very close over the complete observation interval. The M = 3 result is virtually indistinguishable from the result obtained for the reference spectral density and can be considered converged. 

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