Franck-Condon factors systems

Theoretically this kj -value is again given by an electronic matrix element and a Franck-Condon factor. Systems with the same free enthalpy change should have the similar Franck-Condon factors, since the latter is a function of the former. However, the yields are scattered, which should be reduced to the difference of the electronic matrix elements. Small yields were observed for the systems with a component molecule having carbonyl group or heteroatom. In these molecules, an n-Tr triplet level is placed between or near the initial electron transfer and the final triplet states, which may accelerate the intersystem crossing. The kj -values increase and the formation yield of ion radicals is reduced. This interpretation is consistent with the result that the yield is determined by the chemical properties of the component molecules. 

The ZEKE-PE process shown in Figure 9.50(c) can be modified as shown by changing the wavenumber Vj of the first laser to excite the molecule to an excited vibrational level of M. Then the Franck-Condon factors for the band system are modified. This can allow... 

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. 

Here the average describes the correlation of the mixing terms, Py is the probability of the system to be found in the vibrational state v, and the generalized Franck-Condon factor is defined by... 

The temperature sensitivity arises due to disposition of T2 state with respect to S, state. If T2 is considerably above S, transfer to T, is less probable because of unfavourable Franck-Condon factor. As a consequence, fluorescence is the easiest way for deactivation and fluorescence yield is nearly unity. No dependence on temperature is expected. On the other hand, if T2 is sufficiently below S so that the density of state is high at the crossing point, fluorescence quantum yield should be less than unity as triplet transfer is fecilitated. Again no temperature dependence is observed. But if T2 is nearly at the same energy as S, a barrier to inter-system crossing is expected and fluorescence yield will show temperature dependence. 

We study two adiabatic schemes that, use a sequence of time-delayed transform limited pulses. The first one, known as STIRAP (Stimulated Raman adiabatic passage) controls the population transfer between three vibrational states. The frequency of the first pulse (t)[ is tuned in resonance with the transition from 4> (x) to the intermediate state (f>i0 x), and the frequency of the second pulse [ 2(t)] is resonant with the transition from i0 x) to 4>q x) i0 x) is the intermediate state that maximizes the Franck-Condon factors for both transitions at the same time, working as an efficient wave function bridge between the initial and target wave functions [5]. Using counterintuitive pulses, such that (t) precedes x (t), the wave function of the system has the interesting form [3]... 

It is also interesting to estimate the maximum value of the frequency factor in the case of purely quantum nuclear motion. This can be done with the help of the formula W 2nV2Sp, where V2 exp(—2yR) is the exchange matrix element, S is the Franck-Condon factor, p 1 jco is the density of the vibrational levels, and co 1000 cm-1 is the characteristic vibrational frequency of the nuclei. In the atomic unit system, the multiplier 2np has the order 103 and the atomic unit of frequency is 4.13 x 1016s-1 consequently, in the usual unit system, the frequency factor is of the order 4 x 1019Ss-1. The frequency factor reaches its maximum value when S 1. Thus, in the case of purely quantum nuclear motion, the maximum value of the frequency factor is also 1019-102°s-1. 

If really good wavefunctions can be employed, then the results are convincing. Wolniewicz,175 with very accurate wavefunctions for H2, has calculated transition probabilities for the B-X,C-X and E,F-B systems. He has even considered individual vibrational and rotational lines and has shown that owing to significant variation of the electronic moments with intemuclear distance, the use of Franck-Condon factors is not permissible. 

The vibrational population in the excited state n(v ) is determined by the vibrational population in the ground state n(v), if the electron impact excitation from the ground state is the most dominant excitation mechanism. The application of the Franck-Condon principle for electron impact excitation allows a calculation of n(v ) from n(v) based on the Franck-Condon factors between ground and excited state. Figure 4.2 illustrates this scheme for the three states involved in the Fulcher transition upper and lower state, d3nu and a3A)] respectively, in the triplet system and the ground state... 

The Franck-Condon factors determination is of special interest when the two electronic states, involved in the transition exhibit very different geometries. This is especially the case of electronic transition in the valence shell such as n — tt, which induces conjugation change, as well as geometrical change, in the molecular system. This phenomenon was studied in the fluorescence spectra of acetaldehyde and acetone [62,63], and in the phosphorescence spectra of thioacrolein and thioacetaldehyde [64,65] and thioacetone [66]. 

Figure 4 Hierarchy of reaction coordinates in deriving the Franck-Condon factor from the system Hamiltonian.

Equations [41]-[50] provide an exact solution for the CT free energy surfaces and Franck-Condon factors of a two-state system in a condensed medium with quantum electronic and classical nuclear polarization fields. The derivation does not make any specific assumptions about the off-diagonal matrix elements of the Hamiltonian. It, therefore, includes the off-diagonal... 

Equation [48] gives the Franck-Condon factor that defines the probability of finding a system configuration with a given magnitude of the energy gap between the upper and lower CT free energy surfaces. It can be directly used to define the solvent band shape function of a CT optical transition in Eq. [134]... 

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