Franck - Condon rates

Bondybey and Miller (1978) and Katayama, et al, (1979) proposed that the rates of electronically inelastic processes in the gas phase should follow a Franck-Condon rate law,... 

Section BT1.2 provides a brief summary of experimental methods and instmmentation, including definitions of some of the standard measured spectroscopic quantities. Section BT1.3 reviews some of the theory of spectroscopic transitions, especially the relationships between transition moments calculated from wavefiinctions and integrated absorption intensities or radiative rate constants. Because units can be so confusing, numerical factors with their units are included in some of the equations to make them easier to use. Vibrational effects, die Franck-Condon principle and selection mles are also discussed briefly. In the final section, BT1.4. a few applications are mentioned to particular aspects of electronic spectroscopy. 

This model permits one to immediately relate the bath frequency spectrum to the rate-constant temperature dependence. For the classical bath (PhoOc < 1) the Franck-Condon factor is proportional to exp( —with the reorganization energy equal to... 

Note in passing that the common model in the theory of diffusion of impurities in 3D Debye crystals is the so-called deformational potential approximation with C a>)ccco,p co)ccco and J o ) oc co, which, for a strictly symmetric potential, displays weakly damped oscillations and does not have a well defined rate constant. If the system permits definition of the rate constant at T = 0, the latter is proportional to the square of the tunneling matrix element times the Franck-Condon factor, whereas accurate determination of the prefactor requires specifying the particular spectrum of the bath. 

Sethna [1981] considered two limiting cases. The calculation of action in the fast flip approximation (a>j CO ) proceeds by utilizing the expansion exp ( — cu,-1t ) 1 — cu t. After substituting the first term, i.e. the unity, in (5.72) we get precisely the quantity which yields the Franck-Condon factor in the rate constant. The next term cancels the adiabatic renormalization and changes KM)... 

Following Eq. (75), the rate constant for spin conversion may be expressed as a product of the electronic matrix element V and the nuclear Franck-Condon... 

Developed into a power series in R 1, where R is the intermolecular separation, H exhibits the dipole-dipole, dipole-quadrupole terms in increasing order. When nonvanishing, the dipole-dipole term is the most important, leading to the Forster process. When the dipole transition is forbidden, higher-order transitions come into play (Dexter, 1953). For the Forster process, H is well known, but 0. and 0, are still not known accurately enough to make an a priori calculation with Eq. (4.2). Instead, Forster (1947) makes a simplification based on the relative slowness of the transfer process. Under this condition, energy is transferred between molecules that are thermally equilibriated. The transfer rate then contains the same combination of Franck-Condon factors and vibrational distribution as are involved in the vibrionic transitions for the emission of the donor and the adsorptions of the acceptor. Forster (1947) thus obtains... 

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]. 

In Chapters 2 and 4, the Franck-Condon factor was used to account for the efficiency of electronic transitions resulting in absorption and radiative transitions. The efficiency of the transitions was envisaged as being related to the extent of overlap between the squares of the vibrational wave functions, /2, of the initial and final states. In a horizontal radiationless transition, the extent of overlap of the /2 functions of the initial and final states is the primary factor controlling the rate of internal conversion and intersystem crossing. 

The variations in efficiency (rate) of radiationless transitions result from differences in the Franck-Condon factor, visualised by superimposing the vibrational wavefunctions, / (or /2 - the probability distributions), of the initial and final states. We will consider three cases illustrated in Figure 5.2. 

Considering the deuterium effect on naphthalene, Ci0H8, in which all the hydrogen atoms are replaced by deuterium, the Franck-Condon factor is decreased and consequently the rate of internal conversion... 

Figure 5.3 The effect of energy gap in vibrational levels on Si VW> S0 internal conversion. Decreasing the vibrational energy gap leads to a radiationless transition in which the T overlap and Franck-Condon factor are reduced and the rate of internal conversion should be decreased...

Further proof of the importance of Franck-Condon factors is shown by the dramatically increased triplet-state lifetimes of aromatic hydrocarbons that have been deuterated. The effect of this deuteration is to decrease the rate of Ti A/W> S0 intersystem crossing, which is accompanied by a corresponding increase in triplet-state lifetime (Table 5.1). 

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-... 

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... 

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