Franck-Condon factor density

Let us consider how independent /i(i ) 2 effects contribute to the v E) for the hydrogen halides, HX (X = I, Br, and Cl). The curves shown on Fig. 7.6 correspond to relativistic adiabatic potential energy curves (respectively 0 dotted, 0+ dashed, 1 and 2 solid) for HI obtained after diagonalization of the electronic plus spin-orbit Hamiltonians (see Section 3.1.2.2). The strong R-dependence of the electronic transition moment reflects the independence of the relative contributions of the case(a) A-S-Q basis states to each relativistic adiabatic II state. The independent experimental photodissociation cross sections are plotted as solid curves in Fig. 7.7 for HI and HBr. Note that, in addition to the independent variations in the A — S characters of each fl-state caused by All = 0 spin-orbit interactions, all transitions from the X1E+ state to states that dissociate to the X(2P) + H(2S) limit are forbidden in the separated atom limit because they are at best (2Pi/2 <— 2P3/2) parity forbidden electric dipole transitions on the X atom. In the case of the continuum region of an attractive potential, the energy dependence of the dissociation cross section exhibits continuity in the Franck-Condon factor density (see Fig. 7.18 Allison and Dalgarno, 1971 Smith, 1971 Allison and Stwalley, 1973). 

Density of states weighted Franck-Condon factor Deoxyribonucleic acid Barrier height for the adiabatic hole motion Difference in ionization potentials of adenine-thymine and guanine-cytosine base pairs... 

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. 

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. 

For a dye-sensitized electrode with a large number of accessible acceptor levels, the summation over all of the terms of the Franck-Condon factor, (EC), reduces to the unweighted density of the final states [28]. The rate constant, ka, of the... 

Franck-Condon factors, rather than energy-level density, are primarily responsible for the variation in nonradiative rates. 

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. 

Cf. Example 1.8.) This result shows that in the 3x3 model, the spin-orbit coupling vector depends on three factors the coefficient Q + of the in-phase (/d + /d) character of the singlet state, the spin-orbit coupling parameter (heavy atom effect), and the spatial disposition of the orbitals Xa and Xb-actual intersystem crossing rate will also depend on the Franck-Condon-weighted density of states. (Cf. the Fermi golden rule. Section S.2.3.)... 

First consider the case of low temperatures in which si can be taken as the vibrationless component of the electronic state Boltzmann factors p(si) are negligible for all but the lowest vibrational level. It has been popular to make the crude factorization of (39) into a product of the electronic factor, an average Franck-Condon factor F2), and an effective density of states p, (AE) 6,7,9,21,22,47 ... 

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