The Franck-Condon Approximation

The FC or Born-Oppenheimer approximation is physically clear if the activation energy barrier is in the Dielectric Continuum. The reacting ion is activated by some collisional or vibrational-librational means from the classical Boltzmann thermal pool, so that the rate of activation is equal to the rate of arrival of energy, which is equal to a characteristic classical electrolyte frequency. The electron transfers when its energy exceeds that of the barrier due to the inertia of the solvent permanent dipoles. Marcus4149 consistently supposed that the medium may be regarded as a dense gas phase with a collision frequency, which in its 

Another evident mechanism for energy transfer to activated ions may be by bimolecular collisions between water molecules and solvated ion reactants, for which the collision number is n(ri+ r2)2(87tkT/p )l/2 where n is the water molecule concentration, ri and r2 are the radii of the solvated ion and water molecule of reduced mass p. With ri, r2 = 3.4 and 1.4 A, this is 1.5 x 1013 s 1. The Soviet theoreticians believed that the appropriate frequency should be for water dipole librations, which they took to be equal 10n s 1. This in fact corresponds to a frequency much lower than that of the classical continuum in water.78 Under FC conditions, the net rate of formation of activated molecules (the rate of formation minus rate of deactivation) multiplied by the electron transmission coefficient under nonadiabatic transfer conditions, will determine the preexponential factor. If a one-electron redox reaction has an exchange current of 10 3 A/cm2 at 1.0 M concentration, the extreme values of the frequency factors (106 and 4.9 x 103 cm 2 s 1) correspond to activation energies of 62.6 and 49.4 kJ/mole respectively under equilibrium conditions for adiabatic FC electron transfer. 

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At this stage we may distinguish between excitation involving different electronic states and excitation occurring within the same electronic (ground) state. Wlien the spectroscopic states are located in different electronic states, say the ground (g) and excited (e) states, one frequently assumes the Franck-Condon approximation to be applicable ... 

The Franck-Condon approximation (see Section 7.2.5.3) assumes that an electronic transition is very rapid compared with the motion of the nuclei. One important result is that the transition moment for a vibronic transition is given by... 

Along with the harmonic approximation, the Franck-Condon approximation is usually also used in calculating the probability of electron tunneling. In accordance with the latter approximation, the exchange matrix element V(q) in eqn. (20) is factorized outside the integral sign... 

Using the harmonic and the Franck-Condon approximations, it is possible to advance in calculating the sums of the (18) type using the method of generating functions developed by Kubo and Toyozawa [8, 9] for the calculation of the probabilities of optical and non-radiative transitions in the impurity centres in crystals. According to this method we can rewrite eqn. (18) as... 

First of all, consider the case when all normal vibrations are classical. This takes place if the condition a)k -4 T works well for all frequencies. In the classical case the probability of tunneling can be calculated with the help of the general formula (18) using the Franck-Condon approximation and the well-known [10] properties of quasi-classical wave functions. We will not dwell upon the details of transition from the quantum description to the... 

The traditional description of the elementary act of electron tunneling is based upon the adiabatic approximation and the Franck-Condon approximation. Estimations using various approaches lead to the conclusion that, for an accurate description of electron tunneling at large (R > 30 A) distances, one has often to go beyond the limits of the approximations referred to above. Regardless of the kind of approach used for these estimations, the final conclusion is that, with an increase in R, a decrease of the parameter y and, accordingly, an increase of the parameter a is possible. 

Chemical reaction can be treated as a quantum transition R -+ P which is governed by the term AH. The analysis can be carried out on the basis of the theory of quantum transitions and the amplitude of the transition can be written in the Franck-Condon approximation in the form... 

It is frequently the case that the electronic transition dipole matrix element is only weakly dependent on the nuclear coordinates R such that the Franck-Condon approximation [37] may be employed. Within this approximation,... 

We have here kept the description of the radiative decay rates fairly general and did not discuss their exact relations to the vibronic states (in particular within the Franck-Condon approximation). This description is sufficient for our purpose here and more adapted to the generalization to the case of MEF. It is nevertheless worth mentioning that there is an approximate symmetry between the transitions for absorption Sg(0) —> S 0) ) and those... 

Abstract The transition between electronic energy surfaces of molecules is usually described in the Franck-Condon approximation where the spatial variation of the coupling matrix elements is neglected. In this work we go beyond this approximation and explore the effects of such a variation using a simply parameterized interaction instead of the usually poorly known realistic variations. Moreover, we propose a model that allows us to steer molecular transitions by shaping the space-dependence of the coupling. 

The Franck-Condon approximation states that a molecular electronic transition is much faster than a molecular vibration. 

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