Franck-Condon factor reactions

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. 

Due to strong interaction of the reactants with the medium, the influence of the latter may not be reduced only to the widening of the vibrational levels of the proton in the molecules AH and BH. The theory takes into account the Franck-Condon factor determined by the reorganization of the medium during the course of the reaction. 

One expects the impact of the electronic matrix element, eqs 1 and 2, on electron-transfer reactions to be manifested in a variation in the reaction rate constant with (1) donor-acceptor separation (2) changes in spin multiplicity between reactants and products (3) differences in donor and acceptor orbital symmetry etc. However, simple electron-transfer reactions tend to be dominated by Franck-Condon factors over most of the normally accessible temperature range. Even for outer-... 

An additional concern arises in regard to any differences which may exist between the classical theory and the quantum-mechanical approach in the calculation of the Franck-Condon factors for symmetrical exchange reactions. In fact, the difference is not very large. For a frequency of 400 cm for metal-ligand totally symmetric vibrational modes, one can expect... 

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 the previous section, we alluded to the Franck Condon factors (FCF) in controlling electron transfer rates. For this topic, detailed reviews of theory and experiment are provided elsewhere. In sum, it is now well known that the reaction free energy required to transfer charge can be reduced by the reaction free energy, AG°, as summarized in the famous Marcus equation AG = (AG° — where X, the reorganization energy, is related to the degree of... 

A number of techniques have been used previously for the study of state-selected ion-molecule reactions. In particular, the use of resonance-enhanced multiphoton ionization (REMPI) [21] and threshold photoelectron photoion coincidence (TPEPICO) [22] has allowed the detailed study of effects of vibrational state selection of ions on reaction cross sections. Neither of these methods, however, are intrinsically capable of complete selection of the rotational states of the molecular ions. The TPEPICO technique or related methods do not have sufficient electron energy resolution to achieve this, while REMPI methods are dependent on the selection rules for angular momentum transfer when a well-selected intermediate rotational state is ionized in the most favorable cases only a partial selection of a few ionic rotational states is achieved [23], There can also be problems in REMPI state-selective experiments with vibrational contamination, because the vibrational selectivity is dependent on a combination of energetic restrictions and Franck-Condon factors. 

The results obtained from thermal spin equilibria indicate that AS = 1 transitions are adiabatic. The rates, therefore, depend on the coordination sphere reorganization energy, or the Franck-Condon factors. Radiationless deactivation processes are exothermic. Consequently, they can proceed more rapidly than thermally activated spin-equilibria reactions, that is, in less than nanoseconds in solution at room temperature. Evidence for this includes the observation that few transition metal complexes luminesce under these conditions. Other evidence is the very success of the photoperturbation method for studying thermal spin equilibria intersystem crossing to the ground state of the other spin isomer must be more rapid than the spin equilibrium relaxation in order for the spin equilibrium to be perturbed. 

The possibility of the anomalous isotope effect for electron tunneling reactions was first noted by Ulstrup and Jortner [7]. This effect becomes possible when the reorganization energy is approximately equal to the reaction exothermicity. If, in this case, for example, the relationship Er - J + a) — 0 is satisfied, where co is the vibrational frequency for a heavy isotope, then from the viewpoint of the activation energy [see eqn. (42)1, the transition (0 - 1) is optimal for the heavy isotope. Compared with this transition for the heavy isotope, both the transitions (0 - 0)and(0 - 1) for the light isotope contain the additional activation multiplier. In this situation the anomalous isotope effect will be observed, provided that the Franck Condon factor for the transition (0 -> 1) of the heavy isotope is not too small compared with that of the light isotope. An example of the electron tunneling reaction for which the anomalous isotope effect is observed experimentally will be considered in Chap. 7, Sect. 4. 

If we approximate the dependence of the probability of tunneling on 7 by the function exp( - 2yR), then the dependence of the energy with which the electron tunnels on the quantum numbers of the vibrational degrees of freedom can reveal itself in the appearance of the dependence of the parameter y on R. Let us assume, for example, that at small values of R the values of the Franck-Condon factors are such that it is more favourable to dissipate all the reaction exothermicity on the donor, i.e. the transitions without any changes of the vibrational quantum number of the acceptor,... 

Due to the complicated kinetics for both processes no attempt was made in ref. 83 to treat the data quantitatively. It was estimated, however, that the back electron transfer reaction is slower by about 3 orders of magnitude than that of the forward electron transfer. At the same time, the free energy change for the forward reaction (AG° = - 0.4 eV) is smaller than that for the back electron transfer (AG° = — 1.7 eV). This decrease of the reaction rate at large exothermicity was attributed [83] to the decrease of the Franck-Condon factors with increasing J in the situation when J > Er (see Chap. 3, Sect. 5). 

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