Vibrational overlap

The quantity J dr is called the vibrational overlap integral, as it is a measure of the degree to which the two vibrational wave functions overlap. Its square is known as the Franck-Condon factor to which the intensity of the vibronic transition is proportional. In carrying out the integration the requirement that r remain constant during the transition is necessarily taken into account. 

Figure 7.21 illustrates a particular case where the maximum of the v = 4 wave function near to the classical turning point is vertically above that of the v" = 0 wave function. The maximum contribution to the vibrational overlap integral is indicated by the solid line, but appreciable contributions extend to values of r within the dashed lines. Clearly, overlap integrals for A close to four are also appreciable and give an intensity distribution in the v" = 0 progression like that in Figure 7.22(b). 

This specfmm is dominated by ftmdamenfals, combinations and overtones of fofally symmefric vibrations. The intensify disfribufions among fhese bands are determined by fhe Franck-Condon factors (vibrational overlap integrals) between the state of the molecule and the ground state, Dq, of the ion. (The ground state of the ion has one unpaired electron spin and is, therefore, a doublet state, D, and the lowest doublet state is labelled Dq.) The... 

In the conversion case, the solvent reorganization energy is very small and thus the one-mode expression [124] for the vibrational overlap factor G is generally adequate such that ... 

As discussed in Chapter 1, the probability of a nonradiative transition is proportional to the square of the vibrational overlap integral J xiXa drv ... 

Figure 5 shows a collection of S j -S0 R2PI spectra near the origin. The weak bands at low frequency are pure torsional transitions. We can extract the barrier height and the absolute phase of the torsional potential in S, from the frequencies and intensities of these bands. The bands labeled m7, wIq+, and are forbidden in the sense that they do not preserve torsional symmetry. In the usual approximation that the electronic transition dipole moment is independent of torsion-vibrational coordinates, band intensities are proportional to an electronic factor times a torsion-vibrational overlap factor (Franck-Condon factor). These forbidden bands have Franck-Condon factors m m") 2 that are zero by symmetry. Nevertheless, they are easily observed in jet-cooled spectra. They are comparably intense in many spectra, about 1-5% of the intensity of the allowed origin band. 

Certain features of light emission processes have been alluded to in Sect. 4.4.1. Fluorescence is light emission between states of the same multiplicity, whereas phosphorescence refers to emission between states of different multiplicities. The Franck-Condon principle governs the emission processes, as it does the absorption process. Vibrational overlap determines the relative intensities of different subbands. In the upper electronic state, one expects a quick relaxation and, therefore, a thermal population distribution, in the liquid phase and in gases at not too low a pressure. Because of the combination of the Franck-Condon principle and fast vibrational relaxation, the emission spectrum is always red-shifted. Therefore, oscillator strengths obtained from absorption are not too useful in determining the emission intensity. The theoretical radiative lifetime in terms of the Einstein coefficient, r = A-1, or (EA,)-1 if several lower states are involved,... 

The presence of the electron acceptor site adjacent to the donor site creates an electronic perturbation. Application of time dependent perturbation theory to the system in Figure 1 gives a general result for the transition rate between the states D,A and D+,A. The rate constant is the product of three terms 1) 27rv2/fi where V is the electronic resonance energy arising from the perturbation. 2) The vibrational overlap term. 3) The density of states in the product vibrational energy manifold. 

In the limit that Huty >> kgT, the rate constant for nonradiative decay is simply the product of the square of the vibrational overlap integral and an electronic term for the... 

The rhodium-hydride vibration disappears upon deuteration of the complex as the rhodium-deuteride vibration appears in the fingerprint region. The large frequency shift of the highest energy absorption is indicative of a trans-CO geometry [40]. In solution IR, the rhodium hydride vibration and the lowest energy CO vibration overlap, which results in only two absorptions. 

A possible explanation for this increase in lifetime is a reduction of the nonradiative processes. As pointed out by Robinson (95), these radiationless rates must depend upon the magnitudes of the product of the vibrational overlap integrals between the initial and final states. The substitution of deuterium for hydrogen results in lower vibronic amplitudes, yielding a smaller overlap product. 

To determine the vibrational structure of electronic transitions of polyatomics, we can make the same approximation [Equation (7.21)] as for diatomics of replacing Ptl by some sort of average Pel, which is independent of the Qi s. The intensities of vibrational bands in an electronic transition then depend on the vibrational overlap integral, which is like... 

Condon principle, and the square of the vibrational overlap integral (7.23) is the Franck-Condon factor for the transition. 

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. 

A much more interesting example of the intermediate case is encountered when the density of states is rather small but the vibronic coupling elements are large (due to favorable Franck-Condon vibrational overlap factors). The consequences of this type of intramolecular vibronic coupling are seen in the anomalously long radiative lifetimes of the first singlet states of N02, S02, and CS2 86-99 and in the many extra unexpected lines in the spectra of these molecules. 

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