Franck-Condon factor theory

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. 

Due to strong interaction of the reactants with the medium, the influence of the latter may not be reduced only to the widening of the vibrational levels of the proton in the molecules AH and BH. The theory takes into account the Franck-Condon factor determined by the reorganization of the medium during the course of the reaction. 

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. 

An additional concern arises in regard to any differences which may exist between the classical theory and the quantum-mechanical approach in the calculation of the Franck-Condon factors for symmetrical exchange reactions. In fact, the difference is not very large. For a frequency of 400 cm for metal-ligand totally symmetric vibrational modes, one can expect... 

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... 

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. 

The theory for this intermolecular electron transfer reaction can be approached on a microscopic quantum mechanical level, as suggested above, based on a molecular orbital (filled and virtual) approach for both donor (solute) and acceptor (solvent) molecules. If the two sets of molecular orbitals can be in resonance and can physically overlap for a given cluster geometry, then the electron transfer is relatively efficient. In the cases discussed above, a barrier to electron transfer clearly exists, but the overall reaction in certainly exothermic. The barrier must be coupled to a nuclear motion and, thus, Franck-Condon factors for the electron transfer process must be small. This interaction should be modeled by Marcus inverted region electron transfer theory and is well described in the literature (Closs and Miller 1988 Kang et al. 1990 Kim and Hynes 1990a,b Marcus and Sutin 1985 McLendon 1988 Minaga et al. 1991 Sutin 1986). 

Bednorz-Muller theory Beer-Lambert law Bose-Einstein statistics Debye-Huckel theory Diels-Alder reaction Fermi-Dirac statistics Fischer-Tropsch effect Fisher-lohns hypothesis Flory-Huggins interaction Franck-Condon factor Friedel-Crafts reaction Geiger-Miiller effect... 

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. 

The rate constant for ET can mathematically be regarded as the optical spectrum of a localized electron in the limit where the photon energy to be absorbed or emitted approaches zero. Erom the theory of radiative transitions [10, 12] and r / -b 1) = / for a positive integer /, we see that the factor multiplied to on the right-hand side of Eq. 27 represents the thermally renormalized value of the Franck-Condon factor [i.e., the squared overlap integral between the lowest phonon state in Vy(Q) and the ( AG /te)-th one in piQ)] for ET. The renormalization manifests itself in the Debye-Waller factor exp[—,vcoth( / (y/2)], smaller than e which appears also in neutron or X-ray scattering 12a]. Therefore, yen in Eq- 27 represents the effective matrix element for electron tunneling from the lowest phonon state in the reactant well with simultaneous emission of i AG /liw) phonons. 

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. 

Most LRET processes in biological systems are nonadiabatic. In quantum-mechanical electron-transfer theory, the rate constant for nonadiabatic ET from a donor to acceptor can be expressed as the product of the square of an electronic coupling matrix element (Hp ) and a nuclear Franck-Condon factor (FC) kpy = (27t/h)[Hpj2(FC).The [HpJ is a measure ofthe... 

It displays a superposition of lines that correspond to the excitation of different numbers of vibrational quanta during the electronic transition (hence the name multiphonon transition rate). The relative line intensities are determined by the corresponding Franck-Condon factors. The fact that the lines appear as <5 functions results from using perturbation theory in the derivation of this expression. In reality each line will be broadened and simplest theory (see Section 9.3) yields a Lorentzian lineshape. 

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. 

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