Franck-Condon factor Boltzmann weighted

The quantity (FC) is the Franck-Condon factor it is a sum of products of overlap integrals of the vibrational and solvation wavefunctions of the reactants with those of the products, suitably weighted by Boltzmann factors. The value of the Franck-Condon factor may be expressed analytically by considering the effective potential energy curves, of both the initial and the final states, as a function of their nuclear configurations. Relatively simple relationships can be derived if the appropriate curves are harmonic with identical force constants. Under these conditions ... 

To make quantitative statements about the product internal distribution a computer program is utilized to simulate the observed excitation spectrum [10]. As input for the calculations we estimate the relative vibrational and rotational populations. Each line is weighted by the population of the initial (v, J ) level, by the Franck-Condon factor and the rotational line strength of the pump transition. At each frequency, the program convolutes the lines with the laser bandwidth and power to produce a simulated spectrum such spectra are compared visually with the observed spectra and new estimates are made for the (v ,J") populations. Iteration of this process leads to the "best fit" as shown in the lower part of Fig. 3. For this calculated spectrum all vibrational states v" = 0...35 are equally populated as is shown in the insertion. The rotation, on the other hand, is described by a Boltzmann distribution with a "temperature" of 1200 K. With such low rotational energy no band heads are formed for v" < 5 in the Av = 0 sequence and for nearly all v" in the Av = +1 sequence (near 5550 A). 

The absorption strength at a given frequency depends on the sum of the Boltzmann-weighted Franck-Condon factors for all the vibronic transitions in which the change in the total energy (electronic plus vibrational) matches the photon energy hw (Box 4.14). 

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