Franck-Condon intensity

Quite apart from the necessity for Franck-Condon intensities of vibronic transitions to be appreciable, it is essential for the initial state of a transition to be sufficiently highly populated for a transition to be observed. Under equilibrium conditions the population 1, of any v" level is related to that of the u" = 0 level by... 

Zhang, J.Z.H. and Miller, W.H. (1990). Photodissociation and continuum resonance Raman cross sections and general Franck-Condon intensities from 5-matrix Kohn scattering calculations with application to the photoelectron spectrum of H2F- + hv —> H2 + F, HF + H+e, J. Chem. Phys. 92, 1811-1818. 

Studied extensively using Hg-Rg electronic transitions. Kaya and co-workers observed two band systems near the Hg P, <- Sq resonance line, using LIF and dispersed fluorescence spectra of Hg-Rg (Rg = Ne, Ar, Kr, and Xe) complexes [177-180]. Analyses of vibrational level spacings and Franck-Condon intensity patterns led to a determination of the interatomic potentials for the electronically excited (A and B) and ground states [52, 54, 178-180]. LIF spectra of Rg = He, Ne, and Xe complexes were also reported by Soep and co-workers [52-54]. Recently, rotational structures of the A <- X and B <- X systems of Rg = He, Ne, and Ar complexes were observed by Yamanouchi et al., although the spectra were found to be congested by contributions from the six Hg isotopes [181]. Rotational resolution of the A <- X and B <- X systems indicated that the A and B states are 0" and 1, respectively, in the Hund s case (c) representation. 

The non-Franck-Condon intensities observed in the lower ( 113/2) spin-orbit state spectrum were attributed to autoionization involving another Rydberg series which converges to a higher... 

Section BT1.2 provides a brief summary of experimental methods and instmmentation, including definitions of some of the standard measured spectroscopic quantities. Section BT1.3 reviews some of the theory of spectroscopic transitions, especially the relationships between transition moments calculated from wavefiinctions and integrated absorption intensities or radiative rate constants. Because units can be so confusing, numerical factors with their units are included in some of the equations to make them easier to use. Vibrational effects, die Franck-Condon principle and selection mles are also discussed briefly. In the final section, BT1.4. a few applications are mentioned to particular aspects of electronic spectroscopy. 

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]. 

There are cases where the variation of the electtonic ttansition moment with nuclear configuration caimot be neglected. Then it is necessary to work with equation (B 1.1.6) keeping the dependence of on Q and integrating it over the vibrational wavefiinctions. In most such cases it is adequate to use only the tenns up to first-order in equation (B 1.1.7). This results in modified Franck-Condon factors for the vibrational intensities [12]. 

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). 

The Franck-Condon principle reflected in tire connection between optical and tliennal ET also relates to tire participation of high-frequency vibrational degrees of freedom. Charge transfer and resonance Raman intensity bandshape analysis has been used to detennine effective vibrational and solvation parameters [42,43]. 

Franck-Condon factors". Their relative magnitudes play strong roles in determining the relative intensities of various vibrational "bands" (i.e., peaks) within a partieular eleetronie transition s speetrum. 

In electronic spectra there is no restriction on the values that Au can take but, as we shall see in Section 1.2.53, the Franck-Condon principle imposes limitations on the intensities of the transitions. 

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. 

Resonance Raman Spectroscopy. If the excitation wavelength is chosen to correspond to an absorption maximum of the species being studied, a 10 —10 enhancement of the Raman scatter of the chromophore is observed. This effect is called resonance enhancement or resonance Raman (RR) spectroscopy. There are several mechanisms to explain this phenomenon, the most common of which is Franck-Condon enhancement. In this case, a band intensity is enhanced if some component of the vibrational motion is along one of the directions in which the molecule expands in the electronic excited state. The intensity is roughly proportional to the distortion of the molecule along this axis. RR spectroscopy has been an important biochemical tool, and it may have industrial uses in some areas of pigment chemistry. Two biological appHcations include the deterrnination of helix transitions of deoxyribonucleic acid (DNA) (18), and the elucidation of several peptide stmctures (19). A review of topics in this area has been pubHshed (20). 

Excited-State Relaxation. A further photophysical topic of intense interest is pathways for thermal relaxation of excited states in condensed phases. According to the Franck-Condon principle, photoexcitation occurs with no concurrent relaxation of atomic positions in space, either of the photoexcited chromophore or of the solvating medium. Subsequent to excitation, but typically on the picosecond time scale, atomic positions change to a new equihbrium position, sometimes termed the (28)- Relaxation of the solvating medium is often more dramatic than that of the chromophore... 

In retrospect, by inspecting the literature, we find a confirmation of this variance (see for instance Ref. [67]). Peak intensities of bands originally assigned to Franck-Condon components of the excilonic emission have random relative intensities. This would not be possible if the bands were intrinsically vibronic. Since we know that the excilonic emission, as it is observed in single crystals, is rather sharp at low temperatures, we were forced to reconsider the assignment of the fluorescence of thin films. From the temperature dependence of the fluorescence effi-... 

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. 

Assignment of such spectra and fitting of model potential parameters to the observed band frequencies yields the magnitudes of V3 and V6 in the two electronic states involved, either S, and S0, or D0 and S,. Franck-Condon modeling of relative band intensities then yields the relative phase of the potentials in the two electronic states. We typically fix the absolute phase of the potentials from ab initio electronic structure calculations on both the S0 and D0 states. This provides an overall consistency check as well. 

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,... 

If the experimental lineshapes do not exhibit sub-bands and are asymmetric, or if they involve sub-bands, but with intensity anomalies with respect to the Franck-Condon progression law, then, together with the dephasing mechanism, damping of the slow mode ought to be also considered as occurring in a sensitive competitive way. 

Such CR bands, which have been observed for many radical cations, usually manifest themselves by intense, broad bands in the visible or NIR part of the spectrum. The reason for the broadness is that, upon excitation of an electron from 7T+ to 7r, the antibonding interaction is greatly enhanced. Consequently, the equilibrium distance of the 7r-systems in the excited state is significantly larger than in the ground state of the radical cation (or that of the neutral molecule) which results in a Franck-Condon envelope for the EA band which may be even broader than that for the corresponding PE band. 

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