Franck-Condon analysis

The relation of the preceding method to the common Franck-Condon analysis can be shown as follows. The integrand of Eq. (30) for any component a when... 

The interpretation of band progressions by the time dependent procedure is therefore identical with the Franck-Condon analysis and, in the low temperature limit, to the method of molecular distributions as well. The line shape function obtained on the basis of Eq. (52) (for E = hv) differs under this condition from that of Eq. (12) only in the line shape function of each vibrational member in the progression which in Eq. (52) is the delta function and in Eq. (12) has a Lorentz type distribution. 

Fig. 5.5-2. The Franck-Condon analysis of the 5 ionization peak for Mo2(02CCH3)4, showing how the Mo-Mo bond length change is estimated. By permission from Ref. 172...

If the internuclear equilibrium distance of the excited electronic state (r s) shifts by the value A from the internuclear equilibrium distance of the ground state ( e). the Franck-Condon principle allows transitions to many excited vibrational levels. The shapes of the harmonic potentials also have an effect on the magnitude of the Franck-Condon integral. In this case, the theoretical intensities have been calculated as a function of A and B. The parameters B and A were varied until the theoretical intensities showed the closest match to the experimental intensities. In Fig. 21, the best fit for the progression obtained from the photoluminescence spectrum for the anchored vanadium oxidc/Si02 catalyst and theoretical Franck-Condon analysis is represented (725). 

Fk5. 21. Progression of the vibrational tine structure of the photoluminescence spectr um of vanadium oxide catalyst anchored to SiOj at 77 K. Theoretical results from a Franck-Condon analysis (striped bars) and experimental results (solid bars) [reproduced with permission from Patterson et al. (725)]. 

Relationship between the Results of the Franck-Condon Analysis and Photoreduction of the Catalyst with CO... 

The Franck-Condon analysis of the vibrational fine structure of the photoluminescence spectrum of the anchored vanadium oxide observed at 77 K indicates that the equilibrium V-0 bond distance of the vanadyl group is elongated in the charge-transfer excited state by 0.013 nm compared with the ground state value (725). UV irradiation of the anchored vanadium oxides at 280 K in the presence of CO led to the photoformation of CO2. Since the photoformation of CO2 from CO is accompanied by the removal of oxygen from the oxide (i.e.. the photoreduction of the oxide), such an elongation of the equilibrium nuclear distance of the V-0 bond in the excited state is closely associated with the facile photoformation of CO2 on the anchored vanadium oxides. In other words, the O hole trapped centers in the electron-hole pair state of the (V -0 ) complex exhibit a high reactivity similar to 0 anion radicals 66). 

The most widely used method of obtaining excited state distortions from spectroscopic data is the well-known Franck-Condon analysis [16,17]. This... 

All of the analysis in this chapter will be carried out from the time-dependent theoretical point of view. This method of analysis gives results which are identical with those of the Franck-Condon analysis. However, thinking in the time domain offers a new point of view from which the unity between electronic and resonance Raman spectroscopy and the interpretation of the spectroscopic effects of multiple mode distortions can more readily be seen. Thus, before we discuss specific molecules, we will introduce the important aspects of time-dependent theory. 

The Franck-Condon analysis of the Bg state shows that progressions of both the totally symmetrical V3 C-C stretch and the asymmetric HCH bending vibrations can contribute to the observed progression of 1170 cm (cf. the discussion of the UV bands below). This accounts for the first, structured, PE band whose maximum is at about 12.0 eV. The two other maxima at 12.7 and 13.4 eV then belong to the two Ag states whose interaction is likely to be one of the causes of the irregular, diffuse character of these bands. 

The bond length and angle of the excited 3BX state of S02 have been calculated to be 1.491 0.002 A and 126.1 0.1° from a modified Franck-Condon analysis of the absorption and emission spectra.118 Photoabsorption of solid COa 119 and rotational analysis of the 593.3 and 800—900 nm bands of N02120 have been reported. The application of the laser spectroscopy of supersonic molecular beams to the NOa spectrum has been discussed.121 The 2 + Jt2U... 

The cationic Re-H and Re-D stretching frequencies are lowered in each case in comparison to those corresponding to the neutral molecule. This experimental information can be used to evaluate quantitatively the changes in Re-H bonding and equilibrium bond distance upon removal of an electron. For all three of these molecules, Franck-Condon analysis of the observed vibrational structure indicates that the Re-H or Re-D equilibrium bond distance is lengthened by about 0.25 A on ionization. The deduction that the Re-H bond orbital is localized is confirmed by theoretical calculations. 

From a Franck-Condon analysis of the PES spectrum together with the known geometry for CIO2. Three other geometries are consistent with the experimental data but the above is the preferred stmcture. 

From a Franck-Condon analysis of vibrational progressions in the stretching vibration. ) Rydberg states from a 2 r —> Cx promotion. 

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