Franck-Condon factors distributions

Developed into a power series in R 1, where R is the intermolecular separation, H exhibits the dipole-dipole, dipole-quadrupole terms in increasing order. When nonvanishing, the dipole-dipole term is the most important, leading to the Forster process. When the dipole transition is forbidden, higher-order transitions come into play (Dexter, 1953). For the Forster process, H is well known, but 0. and 0, are still not known accurately enough to make an a priori calculation with Eq. (4.2). Instead, Forster (1947) makes a simplification based on the relative slowness of the transfer process. Under this condition, energy is transferred between molecules that are thermally equilibriated. The transfer rate then contains the same combination of Franck-Condon factors and vibrational distribution as are involved in the vibrionic transitions for the emission of the donor and the adsorptions of the acceptor. Forster (1947) thus obtains... 

The variations in efficiency (rate) of radiationless transitions result from differences in the Franck-Condon factor, visualised by superimposing the vibrational wavefunctions, / (or /2 - the probability distributions), of the initial and final states. We will consider three cases illustrated in Figure 5.2. 

Fig. 2.2. Electron ionization can be represented by a vertical line in this diagram. Thus, ions are formed in a vibrationaUy excited state if the intemuclear distance of the excited state is longer than in the ground state. Ions having internal energies below the dissociation energy D remain stable, whereas fragmentation will occur above. In few cases, ions are unstable, i.e., there is no minimum on their potential energy curve. The lower part schematically shows the distribution of Franck-Condon factors, fyc, for various transitions.

The probability of a particular vertical transition from the neutral to a certain vibrational level of the ion is expressed by its Franck-Condon factor. The distribution of Franck-Condon factors, /pc, describes the distribution of vibrational states for an excited ion. [33] The larger ri compared to ro, the more probable will be the generation of ions excited even well above dissociation energy. Photoelectron spectroscopy allows for both the determination of adiabatic ionization energies and of Franck-Condon factors (Chap. 2.10.1). 

If we use an ns probe pulse, we can tune its wavelength resonant to one particular vibronic transition. In this case, the LIF signal reflects the population of a single vibrational level involved in the WP. By scanning the wavelength of the probe pulse, we can observe the population distribution of the eigenstates involved in the WP. The peak intensities of the LIF signal are influenced by the Franck-Condon factors and the probe laser intensities, so that the relevant corrections are necessary to obtain the population distribution. 

Here /rLE/Jo, Mct/s0> / ct>0 are the z independent transition moment matrix elements in terms of zero order states, but I ct/soI l/kn-vwl- The Franck-Condon factors in Eq. (35) are assumed to be z independent since LE and CT have similar vibrational spectra. It follows simply that each z value contributes the following element to the spectrum for an arbitrary distribution P(z. t),... 

Utilizing ionization efficiency curves to determine relative populations of vibrationally excited states (as in the photoionization experiments) is a quite valid procedure in view of the long radiative lifetime that characterizes vibrational transitions within an electronic state (several milliseconds). However, use of any ionization efficiency curve (electron impact, photon impact, or photoelectron spectroscopic) to obtain relative populations of electronically excited states requires great care. A more direct experimental determination using a procedure such as the attenuation method is to be preferred. If the latter is not feasible, accurate knowledge of the lifetimes of the states is necessary for calculation of the fraction that has decayed within the time scale of the experiment. Accurate Franck -Condon factors for the transitions from these radiating states to the various lower vibronic states are also required for calculation of the modified distribution of internal states relevant to the experiment.991 102... 

They also measured a LIF spectrum of the CS radical, which showed that the vibrational distribution peaks at v" 1 3 and decreases at higher vibrational levels. These measurements only extended up to v" = 6 because of the low Franck-Condon factors for transitions from higher vibrational levels of CS(X) in the band sequence they measured. 

If the electronic transition is allowed, is nonzero and the first term dominates the expression. This term can be viewed as a product of the electronic transition moment and the vibrational overlap integral, (v /v, v /v ), connecting the two vibrational wavefunctions, /v, in electronic states e and e". The Franck-Condon factors, which are the square of the vibrational overlap integral, determine the intensity distribution among the vibrational bands. The relative intensities of the band members within a vibrational progression is, therefore, given by the ratio of the Franck-Condon factors. If, through a symmetry restriction, the transition moment M°e vanishes, as in the present case, the band activity in the spectrum comes from the second term. When Qk is a nontotally symmetric... 

This spectrum is dominated by fundamentals, combinations and overtones of totally symmetric vibrations. The intensity distributions among these bands are determined by the Franck Condon factors (vibrational overlap integrals) between the Sl state of the molecule and the ground state, D0, 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 D0.) The... 

Even with PI, theoretically one of the simplest ionization processes, the internal energy distribution, P(E), of the molecular ion cannot be predicted on the basis of Franck—Condon factors alone. Autoionization is well-known as being important [15, 177, 637, 640, 800], as is the more recently recognised effect of shape resonance [220, 803, 906]. It has also been shown that the onset of a decomposition can affect the energy distribution, P(E), [801, 802]. The latter effect is possibly a consequence of competition between neutral and ionic decompositions. 

Studies of electron-impact ionization of molecular nitrogen and oxygen near threshold [141, 142] have demonstrated that a Franck-Condon distribution of vibrational levels is not obtained because many ions are formed indirectly, via autoionizing states. The importance of autoionization can be seen in the case of where the Franck-Condon factors for transitions... 

This means that the total emission intensity depends only on the purely electronic transition dipole moment. Thus, the electronic allowedness represents the source of intensity which is distributed according to the Franck-Condon factor to the different vibrational satelHtes. With respect to the symmetry of the Franck-Condon active vibrations, it is remarked that this factor can only be nonzero for totally symmetric modes (if it is referred to fundamentals), since the vibrational ground state n(v = 0) is totally symmetric (e.g.see [154,p. 113]). 

Franck-Condon factors for the photodissociation of HjS via 454 predissociated levels of the Bj excited state calculated. Partial Franck-Condon factors for the product vibrational distributions presented... 

The spectra distribution of fluorescence is dictated by the Franck-Condon factors defined as the square of the second term in Eq, 1, and the position of the centre of pavity of the fluorescence depends upon any geometry changes between pound and excited states. The latter point is illustrated in Fig. 2 from which it can be seen that the most probable transition in absorption is to higher energies than that for fluorescence if the potential surface of the excited state undergoes some non-zero displacement nith respect to the ground state, and assuming that vibrational relaxation is... 

To clarify the question of the chemical reaction heat distribution in the vibrational degrees of freedom of the product, let us compare the matrix elements of the transition from the fundamental initial state to various final vibrational states, assuming for the sake of definiteness that the transition is nonadiabatic. Applying the known expressions for the Franck-Condon factors of harmonic oscillators, we obtain... 

Since the Franck-Condon factors for all vibrational levels of the excited state add up to unity, the total intensity of a transition is given by the electronic dipole transition moment The resulting intensity distribution of the vibrational fine structure is depicted in Figure 1.13 for some typical cases. 

A similar situation existed in the case of CO. The envelope of the observed transition agreed quite well with Nicholls 98 Franck-Condon factors except that again the population of the higher vibrational states was larger than expected. More recently, Moore" has reexamined the N2 and CO problem by exciting these molecules with 1-3 keV H +, and N+ ions. Again the distribution of the vibrational transitions in the electronic transition ->a ng was greater than predicted by the Franck-Condon... 

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