Franck-Condon factors potential energy surface

Figure 2.1 Schematic representation of the ground and electronic excited potential energy surfaces (PESs) and the corresponding absorption spectra of the parent molecule, resulting from the reflection of different initial wavefunctions on a directly dissociative PES (a) absorption from a vibrationless ground state consists of a broad continuum and (b) absorption from a vibrationally excited state shows that extended regions are accessed, leading to a structured spectrum with intensities of the features being dependent on the Franck-Condon factors. Reproduced with permission from Ref. [34]. Reproduced by permission of lOP Publishing.

Figure 2.4. The energy-gap dependence of the nuclear Franck-Condon factor, which incorporates the role of the high-frequency intramolecular modes. Sc = A/2 is the dimensionless electron-vibration coupling, given in terms which reduce replacement (A) between the minimum of the nuclear potential surfaces of the initial and final electronic states. (Bixon and Jortner, 1999) Reproduced with permission.

Figure 13 Potential energy surfaces for electron transfer reactions. Hamionic oscillator potential energy functions for reactants and product are shown, including the nuclear wave functions, which are shaded. The dark shaded region indicates the magnitude of overlap of the nuclear wave functions, which is the Franck-Condon factor, (a) is the normal region, (b) is the activationless region and (c) is the inverted region as defined in the text. (Ref. 72. Reproduced by permission of Namre Publishing Group, www.nature.com)...

The spectra distribution of fluorescence is dictated by the Franck-Condon factors defined as the square of the second term in Eq, 1, and the position of the centre of pavity of the fluorescence depends upon any geometry changes between pound and excited states. The latter point is illustrated in Fig. 2 from which it can be seen that the most probable transition in absorption is to higher energies than that for fluorescence if the potential surface of the excited state undergoes some non-zero displacement nith respect to the ground state, and assuming that vibrational relaxation is... 

Let us consider the last point. The reader is already familiar with two important implications of the timescale separation between electronic and nuclear motions in molecular systems One is the Bom-Oppenheimer principle which provides the foundation for the concept of potential energy surfaces for the nuclear motion. The other is the prominent role played by the Franck-Condon principle and Franck-Condon factors (overlap of nuclear wavefunctions) in the vibrational structure of molecular electronic spectra. Indeed this principle, stating that electronic transitions occur at fixed nuclear positions, is a direct consequence of the observation that electronic motion takes place on a timescale short relative to that of the nuclei. 

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. 

Fig. 4.20 If the vibrational potential energy surfaces are the same in the ground and excited electronie states (A), the Franck-Condon factors are nonzero only fin vibionie transitions between corresponding vibrational levels, and these transitions all have the same entngy the absorption spectrum consists of a single line at this energy. If the minimum of the potential eneigy surface is displaced along the nuclear coordinate in the excited state, as in (B), Franck-Condon factors for multiple vibronic transitions are nonzero and the spectrum includes lines at multiple eneigies...

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