Franck overlap

An alternative perspective is as follows. A 5-frmction pulse in time has an infinitely broad frequency range. Thus, the pulse promotes transitions to all the excited-state vibrational eigenstates having good overlap (Franck-Condon factors) with the initial vibrational state. The pulse, by virtue of its coherence, in fact prepares a coherent superposition of all these excited-state vibrational eigenstates. From the earlier sections, we know that each of these eigenstates evolves with a different time-dependent phase factor, leading to coherent spatial translation of the wavepacket. 

The coefficients of the 5-fiinction in the sum are called Franck-Condon factors, and reflect the overlap of the initial state with the excited-state i at energy (see figure Al.6.13). Fonnally, equation (A1.6,88i... 

Figure Al.6.13. (a) Potential energy curves for two electronic states. The vibrational wavefunctions of the excited electronic state and for the lowest level of the ground electronic state are shown superimposed, (b) Stick spectrum representing the Franck-Condon factors (the square of overlap integral) between the vibrational wavefiinction of the ground electronic state and the vibrational wavefiinctions of the excited electronic state (adapted from [3]).

The last factor, the square of the overlap integral between the initial and final vibrational wavefunctions, is called the Franck-Condon factor for this transition. 

This example relates to the well known Franck-Condon principal of spectroscopy in which squares of overlaps between the initial electronic state s vibrational wavefunction and the final electronic state s vibrational wavefunctions allow one to estimate the probabilities ofpopulating various final-state vibrational levels. 

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. 

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... 

Both the weak- and strong-coupling results (2.82a) and (2.86) could be formally obtained from multiplying Aq by the overlap integral (square root from the Franck-Condon factor) for the harmonic q oscillator,... 

A qualitatively different approach to probing multiple pathways is to interrogate the reaction intermediates directly, while they are following different pathways on the PES, using femtosecond time-resolved pump-probe spectroscopy [19]. In this case, the pump laser initiates the reaction, while the probe laser measures absorption, excites fluorescence, induces ionization, or creates some other observable that selectively probes each reaction pathway. For example, the ion states produced upon photoionization of a neutral species depend on the Franck-Condon overlap between the nuclear configuration of the neutral and the various ion states available. Photoelectron spectroscopy is a sensitive probe of the structural differences between neutrals and cations. If the structure and energetics of the ion states are well determined and sufficiently diverse in... 

As we have reviewed here, the linear region is not fully repulsive, and transitions of the ground-state, linear conformer access vibrationally excited intermolecular levels that are delocalized in the angular coordinate. As depicted in Fig. 1, however, the internuclear distance is significantly longer in the excited state at the linear geometry. Consequently, there is favorable Franck-Condon overlap of the linear conformer with the inner-repulsive wall of the excited-state potential. It is therefore possible for the linear Rg XY conformers to be promoted to the continuum of states just above each Rg - - XY B,v ) dissociation limit. 

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. 

Note that the lability principle is formulated first of all for transferable electrons and atoms. An increase in their lability leads as a rule to an increase in the overlapping of the wave functions. For atoms the latter means a decrease in the Franck-Condon barrier. 

Flowever, there is a trade-off in using near-IR emissive lanthanides, in that luminescence lifetimes are shorter, and quantum yields lower, compared to complexes of Tb and Eu. This arises because the near-IR emissive lanthanides are quenched by lower harmonics of the O-H oscillator, increasing the Franck-Condon overlap with the metal excited state. For neodymium, matters are further complicated by the manifold of available metal-centered excited states, which leads to particularly effective quenching by C-H oscillators. Thus, complexes in which there are few C-H oscillators close to the metal are desirable if the luminescence lifetime is to be optimized (e.g. 44).76 97-101... 

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,... 

Radiative transitions may be considered as vertical transitions and may therefore be explained in terms of the Franck-Condon principle. The intensity of any vibrational fine structure associated with such transitions will, therefore, be related to the overlap between the square of the wavefunctions of the vibronic levels of the excited state and ground state. This overlap is maximised for the most probable electronic transition (the most intense band in the fluorescence spectrum). Figure... 

In Chapters 2 and 4, the Franck-Condon factor was used to account for the efficiency of electronic transitions resulting in absorption and radiative transitions. The efficiency of the transitions was envisaged as being related to the extent of overlap between the squares of the vibrational wave functions, /2, of the initial and final states. In a horizontal radiationless transition, the extent of overlap of the /2 functions of the initial and final states is the primary factor controlling the rate of internal conversion and intersystem crossing. 

Figure 5.3 The effect of energy gap in vibrational levels on Si VW> S0 internal conversion. Decreasing the vibrational energy gap leads to a radiationless transition in which the T overlap and Franck-Condon factor are reduced and the rate of internal conversion should be decreased...

An example of an application to actual spectroscopy is shown in Figure 6.9. This shows the states computed in the range of the 0 —> 3 CH spectral transition. The intensities were computed as a Franck-Condon overlap of the optically bright CH overtone state (Holme and Levine, 1989 Levine and Berry, 1989) with the relevant eigenstate. 

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