Franck-Condon principle factor

The Franck-Condon principle says that the intensities of die various vibrational bands of an electronic transition are proportional to these Franck-Condon factors. (Of course, the frequency factor must be included for accurate treatments.) The idea was first derived qualitatively by Franck through the picture that the rearrangement of the light electrons in die electronic transition would occur quickly relative to the period of motion of the heavy nuclei, so die position and iiioiiientiim of the nuclei would not change much during the transition [9]. The quaiitum mechanical picture was given shortly afterwards by Condon, more or less as outlined above [10]. 

The synnnetry selection rules discussed above tell us whether a particular vibronic transition is allowed or forbidden, but they give no mfonnation about the intensity of allowed bands. That is detennined by equation (Bl.1.9) for absorption or (Bl.1.13) for emission. That usually means by the Franck-Condon principle if only the zero-order tenn in equation (B 1.1.7) is needed. So we take note of some general principles for Franck-Condon factors (FCFs). 

Forster, Th 211, 278, 282, 285 Forster resonance energy transfer, 282 Forster singlet energy transfer, 378 Franck-Condon factors, 23 Franck-Condon principle, 5 Franck-Condon transition, 5 French, C. S., 555 Friedman, G., 353 Fritzsche, J., 37 Frosch, R. P 252, 267, 269 Fumaronitrile, photodimerization in solid state, 478... 

These selection rules are affected by molecular vibrations, since vibrations distort the symmetry of a molecule in both electronic states. Therefore, an otherwise forbidden transition may be (weakly) allowed. An example is found in the lowest singlet-singlet absorption in benzene at 260 nm. Finally, the Franck-Condon principle restricts the nature of allowed transitions. A large number of calculated Franck-Condon factors are now available for diatomic molecules. 

The next important aspect to be considered is the electron-phonon interaction (lattice relaxation). Here, the effect of momentum conserving phonons, or promoting modes, can in principle be included in the electronic cross section this is discussed, for instance, by Monemar and Samuelson (1976) and Stoneham (1977). However, the configuration coordinate (CC) phonons (or accepting modes) are treated separately. The effect of these CC modes is usually expressed by the Franck-Condon factor dF c, where this factor is the same as the defined in our Fig. 16. Thus assuming a single mode,... 

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

The vibrational overlap factor (6t 6t) is also known as the Franck-Condon factor and represents the probability of finding a common nuclear geometry in the initial and final states. If the nuclei do not move during the electronic transition, such a transition can take place only in such a common nuclear geometry. Figure 3.12, p. 39, provides an illustration of this Franck-Condon principle electronic radiative transitions are vertical, non-radiative transitions are horizontal but they are followed by vertical vibrational deactivation. 

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... 

In the Born-Oppenheimer approximation, the relative importance of channels (la) and (lb), together with their dependence on wavelength would depend upon the matrix elements for the transition between the electronic states, the Franck-Condon factors, the Honl-London factors, and upon the probabilities for spontaneous dissociation of the excited state formed. In principle, except for the last one, these are well known quantities whose product is the transition probability for that particular absorption band of Cs. When multiplied by the last quantity, and with an adjustment of numerical constants i becomes the cross section for the photolysis of Cs into Cs + Cs. It is the measurement of this cross section that lies at the focus of this work. 

Fluorescence is defined simply as the electric dipole tranation from an excited electronic state to a lower state, usually the ground state, of the same multiplicity. Mathematically, the probability of an electric-dipole induced electronic transition between specific vibronic levels is proportional to R f where Rjf, the transition moment integral between initial state i and final state f is given by Eq. (1), where represents the electronic wavefunction, the vibrational wavefunctions, M is the electronic dipole moment operator, and where the Born-Oppenheimer principle of parability of electronic and vibrational wavefunctions has been invoked. The first integral involves only the electronic wavefunctions of the stem, and the second term, when squared, is the familiar Franck-Condon factor. 

Feldberg plot 23, 24, 35 Franck-Condon principle 5, 6, 10 Franck-Condon factor 10, 53, 54 FRASTA 160 Freeze-thaw test 244 Friction factor 135 Fullerene 107, 347... 

Let us consider the last point. The reader is already familiar with two important implications of the timescale separation between electronic and nuclear motions in molecular systems One is the Bom-Oppenheimer principle which provides the foundation for the concept of potential energy surfaces for the nuclear motion. The other is the prominent role played by the Franck-Condon principle and Franck-Condon factors (overlap of nuclear wavefunctions) in the vibrational structure of molecular electronic spectra. Indeed this principle, stating that electronic transitions occur at fixed nuclear positions, is a direct consequence of the observation that electronic motion takes place on a timescale short relative to that of the nuclei. 

One important rule for excitation in atom-molecule collisions comes from the Franck-Condon principle. When only small amounts of kinetic energy are converted into vibrational energy it seems reasonable that the non-adiabatic transition is governed by the Franck-Condon factor in the transition region. However, due to the interaction of the heavy particles these factors can be quite different from those for the undisturbed molecule. Bauer et al.15 show that the interaction matrix element which determines the transition probability can be written as a product of the electronic part of the matrix element and the square root of the Franck-Condon factor. 

Certain features of the results are quite interesting. The cross sections show a strong dependence on the vibrational quantum number for both reactant electronic states. If the Franck-Condon principle were valid for the nonadiabatic transitions which occur in this system, then the charge transfer cross section would be independent of the reactant vibrational level [19]. It is well known that the Franck-Condon principle breaks down badly at low collision energies for most charge transfer systems. The most remarkable result seen in Fig. 4 is the very small cross section for N2+ (X v = 0) + Ar at all three collision energies its maximum value is 1.6 A2 at 20 eV. (By comparison the cross sections for other N2+ (X v) -I- Ar states are at least 14 A2.) This occurs even though there is a product state, Ar+(2P3/2) + N2(v = 0), which is only 0.18 eV away thus, this. In addition, the Franck-Condon factor for the transition N2 (X v = 0) - N2 (v = 0) is 0.92 ... 

The principle factors contributing to the Franck Condon factor are the free energy of reaction (AGda, the contributions of low-frequency vibrational modes to the reorganizational free energy (xs), and the contributions of high-frequency vibrational modes and displacements jhvn and Ah). The contributions of these factors have been extensively discussed, " ... 

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