Franck—Condon integrals

Palma, A., and Sandoval, L. (1988), The Nonabelian Two-Dimensional Algebra and the Franck-Condon Integral, IntlJ. Quant. Chem. S22, 503. 

Franck-Condon integral, 97,132 Franck-Condon principle, 68, 94, 169 quantum mechanical formulation of, 96... 

If the internuclear equilibrium distance of the excited electronic state (r s) shifts by the value A from the internuclear equilibrium distance of the ground state ( e). the Franck-Condon principle allows transitions to many excited vibrational levels. The shapes of the harmonic potentials also have an effect on the magnitude of the Franck-Condon integral. In this case, the theoretical intensities have been calculated as a function of A and B. The parameters B and A were varied until the theoretical intensities showed the closest match to the experimental intensities. In Fig. 21, the best fit for the progression obtained from the photoluminescence spectrum for the anchored vanadium oxidc/Si02 catalyst and theoretical Franck-Condon analysis is represented (725). 

Figure 2.6 Left Franck Condon integrals (, f) for IC are small for large energy gaps, because highly excited vibrational wavefunctions xt oscillate rapidly and their amplitude is small in the region of overlap with x - Right the density of vibrational levels in the final state increases with the energy gap...

The quantum mechanical formulation of this principle was given in Section 2.1.5 the intensity of a vibronic transition is proportional to the square of the Franck Condon integrals between the vibrational wavefunctions of the two states that are involved in the transition. Thus, the band shape of an electronic transition depends on the displacement of the excited electronic state relative to that of the ground state. This is illustrated for one vibrational degree of freedom of a given molecule in the schematic diagram in Figure 2.10. 

Electron-jump in reactions of alkali atoms is another example of non-adiabatic transitions. Several aspects of this mechanism have been explored in connection with experimental measurements (Herschbach, 1966 Kinsey, 1971). The role of vibrational motion in the electron-jump model has been investigated (Kendall and Grice, 1972) for alkali-dihalide reactions. It was assumed that the transition is sudden, and that reaction probabilities are proportional to the overlap (Franck-Condon) integral between vibrational wavefunctions of the dihalide X2 and vibrational or continuum wave-functions of the negative ion X2. Related calculations have been carried out by Grice and Herschbach (1973). Further developments on the electron-jump mechanism may be expected from analytical extensions of the Landau-Zener-Stueckelberg formula (Nikitin and Ovchinnikova, 1972 Delos and Thorson, 1972), and from computational studies with realistic atom-atom potentials (Evans and Lane, 1973 Redmon and Micha, 1974). 

Here, S y < 1 is the Franck-Condon integral of the intramolecular transition v<- w. Therefore, the transfer rate is less than for strong coupling, even with the same value of U. The product in Eqn. 2 may be regarded as the interaction energy between the vibronic transitions involved in the process. 

Here, i/td, .. . are electronic wavefunctions, while X D k,-.. are the corresponding nuclear wavefunctions. In the second equality, we have introduced the electronic coupling matrix element Tda and the Franck-Condon integrals Fd, and Faju of the donor and acceptor, respectively. This allows rewriting Eq. (4) as ... 

The transition moment thns depends on vibrational overlap. Since u is usually the lowest vibrational state (u = 0), it may be written as (HoolHi a The square of the latter is called the Franck-Condon integral. The width of the spectrum depends on how many vibrational levels in the ground state have overlapped with the vibrational levels in the excited state. If the excitation takes place in the gas phase and there are no obvious broadening mechanisms, the spectrum will be resolved in sharp vibration levels. Equation 4.80 is the quantum mechanical expression for the Franck-Condon principle. If the overlap is calculated in Eqnation 4.80, one obtains the highest intensity for the almost vertical transitions. 

Kupka, H., and Cribb, P. H. (1986] Multidimensional Franck-Condon integrals and Duschinsky mixing effects,/. Chem. Phys., 85,1303-1326. 

Santoro, R, Improta, R, Lami, A., Bloino, )., and Barone, V. (2007] Effective method to compute Franck-Condon integrals for optical spectra of large molecules in solution,/. Chem. Phys., 126, 084509/1-13. 

Finally, Chapter 8 deals with the evaluation of multidimensional Franck-Condon integrals. As an illustration of the complexity of the latter upon the normal mode rotation, a study of sequential two photon processes is presented. 

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