Franck Condon approximation

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,... 

If the electronic matrix element does not vary significantly with the vibrational coordinate ), nuclear and electronic factors can be separated in the sense of a Franck-Condon approximation. The line width T may then be estimated by... 

Recent ab initio calculations of the relevant Na2 potentials by Konowalow, Rosenkrantz and Olson (14), henceforth abbreviated as KRO, permit such an approach. In the quasistatic or classical Franck-Condon approximation the binary contribution to k(v,T) is given by (15.16)... 

In the case of MCD spectroscopy, for a fully allowed electronic transition based on the rigid shift, Bom-Oppenheimer, and Franck Condon approximations, the intensity equation, according to the modified conventions recommended by Stephens, Piepho and Schatz, is... 

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... 

The evaluation made in the preceding section shows the possibility of a natural explanation of the regulation of low-temperature solid-state reactions in a model accounting for barrier parameter oscillations resulting from intermolecular vibrations. A consistent analysis of such a model is required. A mathematical body used for this purpose is, conceptually, close to the common theory of nonradiative transitions, but unlike the latter it enables us to exceed the limits of the one-dimensional Franck-Condon approximation which is inapplicable in treatment of heavy-particle transfer. 

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