Radiationless transitions Franck-Condon factor

Internal conversion refers to radiationless transition between states of the same multiplicity, whereas intersystem crossing refers to such transitions between states of different multiplicities. The difference between the electronic energies is vested as the vibrational energy of the lower state. In the liquid phase, the vibrational energy may be quickly degraded into heat by collision, and in any phase, the differential energy is shared in a polyatomic molecule among various modes of vibration. The theory of radiationless transitions developed by Robinson and Frosch (1963) stresses the Franck-Condon factor. Jortner et al. (1969) have extensively reviewed the situation from the photochemical viewpoint. 

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. 

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...

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. 

This relation was first obtained by Forster and is usually called the Forster theory. Rather than expressing W,-. y in the spectral-overlap relation, lT, y can be expressed in terms of the Franck-Condon factors which can be calculated from the potential surfaces as was done for the photo-induced ET or radiationless transitions. 

In conclusion, we can say, that with respect to the Franck-Condon factors, a radiationless transition from ip to i//° will be very slow when the net positive overlap is poor, i.e. (y, Xf) is nearly 0 and fast with (y, Xf) > 0 [85, 86]. 

Deuterium substitution leads to a decrease in the non-radiative rate by several orders of magnitude. The application of radiationless transition theory indicates that the large isotope effect is due to a large decrease in Franck-Condon factors which more than overcomes an increased density of states. 

Figure 5.7. Franck-Condon factors for radiationless transitions between different potential energy curves of a diatomic molecule a) for a large and b) for a small energy gap, such as those observed, for instance, between S and S, or between S, and S respectively, and c) for the case that the potential energy curves (e.g., S, and T,) cross.

In 1976 TET was first applied to H abstractions [53]. One year later Suhnel [54] used TET to explain radiationless transitions in indigoid compounds, and Phillips [55] tested the harmonic approximation used by the theory in H abstractions. CT interactions [56] and substituent effects [57] in H abstractions were also addressed, as well as H abstractions by uranyl ion [58]. Support for TET also came from the demonstration [59] that in radiationless transitions theories, some Franck-Condon factors may be expressed by a nuclear tunneling formula like the TET one. 

Siebrand W. (1967), Radiationless transitions in polyatomic molecules. I. Calculation of Franck-Condon factors , /. Chem. Phys. 46, 440-447. 

In the quantum-mechanical theories the intersection of the potential energy surfaces is deemphasized and the electron transfer is treated as a radiationless transition between the reactant and product state. Time dependent perturbation theory is used and the restrictions on the nuclear configurations for electron transfer are measured by the square of the overlap of the vibrational wave functions of the reactants and products, i.e. by the Franck-Condon factors for the transition. Classical and quantum mechanical description converge at higher temperature96. At lower temperature the latter theory predicts higher rates than the former as nuclear tunneling is taken into account. 

Equation (1.5) is the simplest of all Golden Rule expressions because both the electronic matrix element and the Franck-Condon factor are taken as averages over all interacting vibronic states. A better model employs a Franck-Condon weighted density of states, Pf(F), in order to account for the fact that not all states in the dense manifold couple with i> with the same probability. In any case, the variation in the Franck-Condon factors with electronic energy gap, AE = Ej - Ef, determines the relative rates of radiationless transitions in compounds that contain the same chromophores and hence exhibit similar values of Ujf. The relative magnitudes of the Franck-Condon factors for different vibrational modes also determines the nature of the accepting modes populated preferentially by the radiationless transition. 

Contrary to what happens with the related expression (1.5) for radiative transitions, the Franck-Condon term in (1.7) is made up of a single overlap integral for each vibrational mode, corresponding to the unique isoenergetic transition. In order to discuss the role of the Franck-Condon factor in radiationless transitions, it is worthwhile considering the two archetypal simations shown in Fig. 1.10. 

In intermediate situations (minima nested but with crossing points not too far from the excited-state minimum) it may be more convenient for the molecule to go through the crossing point, because of a more favorable Franck-Condon factor, despite the substantial activation energy required. In these cases, the rates of radiationless transitions may become very sensitive to temperature. 

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