Franck-Condon coupling

Another conventional simplification is replacing the whole vibration spectrum by a single harmonic vibration with an effective frequency co. In doing so one has to leave the reversibility problem out of consideration. It is again the model of an active oscillator mentioned in section 2.2 and, in fact, it is friction in the active mode that renders the transition irreversible. Such an approach leads to the well known Kubo-Toyozawa problem [Kubo and Toyozava 1955], in which the Franck-Condon factor FC depends on two parameters, the order of multiphonon process N and the coupling parameter S... 

Both the weak- and strong-coupling results (2.82a) and (2.86) could be formally obtained from multiplying Aq by the overlap integral (square root from the Franck-Condon factor) for the harmonic q oscillator,... 

Si is often called the Huang-Rhys factor or coupling constant. The plot of the Franck-Condon factor versus v[ is also shown in Fig. 7. 

Figure 9. Schematic reprsentation of a classical trajectory moving on the Si and So energy surfaces of the H2—(CH) -NHt trans cis photoisomerization, starting near the planar Franck-Condon geometry. The geometric coordinates are (a) torsion of the C2—C3 and C3 C4 bonds and (b) asymmetric stretching coupled with pyramidalization. Both Si and So intersect at a conical intersection (Si/S0 Cl) located near the minimum of the Si surface (Min-C ) where the C2C3C4N5 torsion angle is 104°. [Reproduced with permission from [87], Copyright 2000 Amercian Chemical Society],...

R to P is slow, even when the isoenergetic conditions in the solvent allow the ET via the Franck-Condon principle. The TST rate for this case contains in its prefactor an electronic transmission coefficient Kd, which is proportional to the square of the small electronic coupling [28], But as first described by Zusman [32], if the solvation dynamics are sufficiently slow, the passage up to (and down from [33]) the nonadiabatic curve intersection can influence the rate. This has to do with solvent dynamics in the solvent wells (this is opposed to the barrier top description given above). We say no more about this here [8,11,32-36]. 

Distance The affects of electron donor-acceptor distance on reaction rate arises because electron transfer, like any reaction, requires the wavefunctions of the reactants to mix (i.e. orbital overlap must occur). Unlike atom transfer, the relatively weak overlap which can occur at long distances (> 10 A) may still be sufficient to allow reaction at significant rates. On the basis of work with both proteins and models, it is now generally accepted that donor-acceptor electronic coupling, and thus electron transfer rates, decrease exponentially with distance kji Ve, exp . FCF where v i is the frequency of the mode which promotes reaction (previously estimated between 10 -10 s )FCF is a Franck Condon Factor explained below, and p is empirically estimated to range from 0.8-1.2 with a value of p 0.9 A most common for proteins. 

In a semiclassical picture, the rate kda of nonadiabatic charge transfer between a donor d and an acceptor a is determined by the electronic coupling matrix element Vda and the thermally weighted Franck-Condon factor (f C) [25, 26] ... 

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