Franck-Condon factors dynamics

Equation (A 1.6.94) is called the KHD expression for the polarizability, a. Inspection of the denominators indicates that the first temi is the resonant temi and the second temi is tire non-resonant temi. Note the product of Franck-Condon factors in the numerator one corresponding to the amplitude for excitation and the other to the amplitude for emission. The KHD fonnula is sometimes called the siim-over-states fonnula, since fonnally it requires a sum over all intennediate states j, each intennediate state participating according to how far it is from resonance and the size of the matrix elements that coimect it to the states i. and The KHD fonnula is fiilly equivalent to the time domain fonnula, equation (Al.6.92). and can be derived from the latter in a straightforward way. However, the time domain fonnula can be much more convenient, particularly as one detunes from resonance, since one can exploit the fact that the effective dynamic becomes shorter and shorter as the detuning is increased. 

Equation (33) assumes that IV// is large compared to 2J (i.e., no electronic and vibrational recurrences). In addition, Eq. (33) deals only with population dynamics Interferences between different Franck-Condon factors are neglected. These interferences do influence the rate, and the interplay between electronic and vibrational dynamics can be quite complex [25], Finally, as discussed by Jean et al. [22], Eq. (33) does not separate the influence of pure dephasing (T-T) and population relaxation (Ti). These two processes (defined as the site representation [22]) can have significantly different effects on the overall rate. For example, when (T () becomes small compared to Eq. (33) substantially overestimates the rate compared to... 

The acetylene A <- X electronic transition is a bent <- linear transition that would be electronically forbidden ( - ) at the linear structure. The usual approximation is to ignore the possibility that the electronic part of the transition moment depends on nuclear configuration and to calculate the relative strengths of vibrational bands as the square of the vibrational overlap between the initial and final vibrational states (Franck-Condon factor). A slightly more accurate picture would be to express the electronic transition moment as a linear function of Q l (the fra/w-bending normal coordinate on the linear X1 state) in such a treatment, the transition moment would be zero at the linear structure and the vibrational overlap factors would be replaced by matrix elements of Qfl- Nevertheless, as long as one makes use of low vibrational levels of the A state, neglect of the nuclear coordinate dependence of the electronic excitation function is unlikely to affect the predicted dynamics or to complicate any proposed control scheme. 

Suppose that the two potential surfaces are dissimilar. Then the Franck-Condon factors are less than imity and you get different probabilities for making transitions to final v" vibrational levels depending on the vibrational overlap. We shall make repeated use of the Franck-Condon principle in imderstanding which vibrational levels are populated in various dynamical processes. In Section 9.2 we will generalize the principle so that it also applies to excitation as a result of a collision (where we need not be in the sudden limit). 

The So Si internal conversion step excites So nonrandomly. A microcanonical ensemble of states is not prepared, although So may relax to this ensemble after efficient and complete IVR. Thus, to accurately simulate the intramolecular and unimolecular dynamics of the excited So molecule, it is necessary to choose correct initial conditions for So- The specific vibrational excitations on So have probabilities proportional to Ajj, where i is the initial vibrational level on Si and j is the vibrational level on So. " The term includes a Franck-Condon factor so that only certain types of So mode excitations have high probabilities and therefore the excitation of So may be highly... 

The first 15 wave functions were used in a dynamical simulation of the Liouville equation for the density operator. The Franck-Condon factors were generated by assuming that the ground-state vibrational wave function is a Gaussian centered at —1 A, with a width of 0.1 A. The central frequency of the light pulse was chosen... 

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