Configurational Franck-Condon factor

There are cases where the variation of the electtonic ttansition moment with nuclear configuration caimot be neglected. Then it is necessary to work with equation (B 1.1.6) keeping the dependence of on Q and integrating it over the vibrational wavefiinctions. In most such cases it is adequate to use only the tenns up to first-order in equation (B 1.1.7). This results in modified Franck-Condon factors for the vibrational intensities [12]. 

If the equilibrium position of the excited state C is located outside the configurational coordinate curve of the ground state, the excited state intersects the ground state in relaxing from B to C, leading to a nonradiative process. As described above, the shape of an optical absorption or emission spectrum is decided by the Franck-Condon factor and also by the electronic population in the vibrational levels at thermal equilibrium. For the special case where both ground and excited states have the same angular frequency, the absorption probability can by calculated with harmonic oscillator wavefunctions in a relatively simple form ... 

The next important aspect to be considered is the electron-phonon interaction (lattice relaxation). Here, the effect of momentum conserving phonons, or promoting modes, can in principle be included in the electronic cross section this is discussed, for instance, by Monemar and Samuelson (1976) and Stoneham (1977). However, the configuration coordinate (CC) phonons (or accepting modes) are treated separately. The effect of these CC modes is usually expressed by the Franck-Condon factor dF c, where this factor is the same as the defined in our Fig. 16. Thus assuming a single mode,... 

The acetylene A <- X electronic transition is a bent <- linear transition that would be electronically forbidden ( - ) at the linear structure. The usual approximation is to ignore the possibility that the electronic part of the transition moment depends on nuclear configuration and to calculate the relative strengths of vibrational bands as the square of the vibrational overlap between the initial and final vibrational states (Franck-Condon factor). A slightly more accurate picture would be to express the electronic transition moment as a linear function of Q l (the fra/w-bending normal coordinate on the linear X1 state) in such a treatment, the transition moment would be zero at the linear structure and the vibrational overlap factors would be replaced by matrix elements of Qfl- Nevertheless, as long as one makes use of low vibrational levels of the A state, neglect of the nuclear coordinate dependence of the electronic excitation function is unlikely to affect the predicted dynamics or to complicate any proposed control scheme. 

The protonated cluster ions were observed to be the major product ions in the CMS of ammopia, water, etc., although unprotonated cluster ions have also been observed in these cases, depending on the choice of carrier gas, method of ionization, and electron energy utilized in the investigation (Shinohara et al. 1985, 1986). The inability to observe unprotonated cluster ions is usually attributed to poor Franck-Condon factors for the vertical ionization transitions. These poor Franck-Condon factors arise from the large differences in the configuration of the neutral and ionic clusters (Stace 1987a). 

Equation [48] gives the Franck-Condon factor that defines the probability of finding a system configuration with a given magnitude of the energy gap between the upper and lower CT free energy surfaces. It can be directly used to define the solvent band shape function of a CT optical transition in Eq. [134]... 

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 ... 

When I - (M) < , the ion-pair configuration does not couple the initial and final channel, and the threshold will be at much higher energies. When the bond lengths are too different, the Franck-Condon factors are unfavourable for a transition A+ + M- -> A + M in the interaction region.2,6,15 Consequently the excitation cross section will be small. 

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. 

The data for CH4 (Fig. 9) show a very low probability for forming the ground state of the CH4 ion. The reason is that the equilibrium configuration of the ion is considerably distorted from the tetrahedral form by virtue of the Jahn-Teller effect " and therefore the Franck-Condon factor for the transition to the vibrational ground state of the ion is very small. 

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