Franck-Condon envelope

In principle, refined and relatively reliable quantum-theoretical methods are available for the calculation of the energy change associated with the process of equation 2. They take into account the changes in geometry, in electron distribution and in electron correlation which accompany the transition M(1 fio) — M+ (2 P/-), and also vibronic interactions between the radical cation states. Such sophisticated treatments yield not only reliable predictions for the different ionization energies 7 , 77 or 7 , but also rather precise Franck-Condon envelopes for the individual bands in the PE spectrum. However, the computational expenditure of these methods still limits their application to smaller molecules. We shall mention them later in connection with examples where such treatments are required. 

FIGURE 13. Franck-Condon envelopes of the low-energy part of the PE spectra of (a) butatriene 158 and (b) tetradeuteriobutatriene 159. The central feature labeled 1 is the mystery band mentioned in the text... 

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. 

In contrast, if the initial and final electronic states have very different geometries and/or vibrational frequencies along some mode, a very broad Franck-Condon envelope peaked at high-vf will result as shown below ... 

Franck-Condon Envelope.—An absorption band system including all... 

In common with other aza-naphthalenes, 1,6-naphthyridine displays a progression in the angle-bending mode. Along the a axis this progression displays a normal Franck-Condon envelope but when based on the false origins 347 and 484/8 polarized parallel to the b and c axes, this is certainly not the case. 

Franck-Condon envelopes for the individual bands in the PE spectrum. However, the computational expenditure of these methods still limits their application to smaller molecules. We shall mention them later in connection with examples where such treatments are required. 

A direct consequence of the observation that Eqs. (12.55) provide also golden-rule expressions for transition rates between molecular electronic states in the shifted parallel harmonic potential surfaces model, is that the same theory can be applied to the calculation of optical absorption spectra. The electronic absorption lineshape expresses the photon-frequency dependent transition rate from the molecular ground state dressed by a photon, g) = g, hco ), to an electronically excited state without a photon, x). This absorption is broadened by electronic-vibrational coupling, and the resulting spectrum is sometimes referred to as the Franck-Condon envelope of the absorption lineshape. To see how this spectrum is obtained from the present formalism we start from the Hamiltonian (12.7) in which states L and R are replaced by g) and x) and Vlr becomes Pgx—the coupling between molecule and radiation field. The modes a represent intramolecular as well as intermolecular vibrational motions that couple to the electronic transition... 

Equations (12.60) and (12,61) are expressions for the low temperature (i.e. ksT < hota) electronic absorption lineshape. The frequency dependence originates from the individual transition peaks, that in reality are broadened by intramolecular and intermolecular interactions and may overlap, and from the Franck-Condon envelope... 

This Franck-Condon envelope characterizes the broadening of molecular electronic spectra due to electronic-vibrational coupling. 

The characteristic Franck-Condon envelopes and vibronic signatures of polyene transitions observed in the fluorescence spectra of molecules such as... 

Electronic transitions within the valence shell of atoms and molecules appear in the energy-loss spectrum from a few electron volts up to, and somewhat beyond, the first ionization energy. Valence-shell electron spectroscopy employs incident electron energies from the threshold required for excitation up to many kiloelectron volts. The energy resolution is usually sufficient to observe vibrational structure within the Franck-Condon envelope of an electronic transition. The sample in valence-shell electron energy-loss spectroscopy is most often in the gas phase at a sufficiently low pressure to avoid multiple scattering of the... 

Figure 7 The time auto correlation function and the corresponding spectrum for a Gaussian wave packet propagating on an excited harmonic potential energy surface, (a) The short time decay of C(/) (cf. Eq. (17)) and the broad spectrum (= the Franck Condon envelope (cf. (18)). (b)The longer time dependence of C(r) and the corresponding, vibrationally resolved, spectrum.

As a rule of thumb the Franck-Condon envelopes of the bands 0 can be interpreted as follows ... 

Both semi-classical and quantum-mechanical calculations show " that the resulting Franck-Condon envelope of the band system (D, should consist of three (overlapping) maxima. The position of the second maximum is predicted to be close... 

The reader should be warned that this seeming simplicity of the C2s band system of hydrocarbon PE spectra is spurious, and largely a consequence of the low (inherent and instrumental) resolution of the experiment. Theoretical treatments including the effects of electronic relaxation and electron correlationshow that the observed shapes of the individual C2s bands are the result of the superposition of a whole series of Franck-Condon envelopes, each corresponding to one of many cation states involving multiply excited configurations. However, the positions of these individual bands tend to cluster around the one band (dominantly) associated with the Koopmans state due to simple... 

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