Franck-Condon spectrum

Although theoretical techniques for the characterization of resonance states advanced, the experimental search for reactive resonances has proven to be a much more difficult task [32-34], The extremely short lifetime of reactive resonances makes the direct observation of these species very challenging. In some reactions, transition state spectroscopy can be employed to study resonances through "half-collision experiments," where even very short-lived resonances may be detected as peaks in a Franck-Condon spectrum [35-38]. Neumark and coworkers [39] were able to assign peaks in the [IHI] photodetachment spectrum to resonance states for the neutral I+HI reaction. Unfortunately, transition state spectroscopy is not always feasible due to the absence of an appropriate Franck-Condon transition or due to practical limitations in the required level of energetic resolution. The direct study of reactive resonances in a full collision experiment, such as with a molecular beam apparatus, is the traditional and more usual environment to work. Unfortunately, observing resonance behavior in such experiments has proven to be exceedingly difficult. The heart of the problem is not a... 

A semiclassical description is well established when both the Hamilton operator of the system and the quantity to be calculated have a well-defined classical analog. For example, there exist several semiclassical methods for calculating the vibrational autocorrelation function on a single excited electronic surface, the Fourier transform of which yields the Franck-Condon spectrum. ° In particular, semiclassical methods based on the initial-value representation of the semiclassical propagator,which circumvent the cumbersome root-search problem in boundary-value based semiclassical methods, have been successfully applied to a variety of systems (see, for example, the reviews Refs. 85, 86 and references therein). These methods cannot directly be applied to nonadiabatic dynamics, though, because the Hamilton operator for the vibronic coupling problem [Eq. (1)] involves discrete degrees of freedom (discrete electronic states) which do not possess an obvious classical counterpart. 

Fig. 1. Franck-Condon spectrum for a two-dimensional uncoupled local mode Hamiltonian. The potential is of the form V(x) + V(y), where V is a Morse potential. The Franck-Condon factors (vertical lines) were computed for a wavepacket displaced along the x=y axis, i.e., along the symmetric stretch", which is an unstable mode of motion for this local mode system. The intensity pattern of Franck-Condon factors is such that, by smoothing the spectrum, we get a series of peaks at the energies of the normal mode "slice" of the potential, l/(s) E 2Y(s//l). Lesson The "pluck" we give the system is one thing, the intrinsic potential, quite another.

It is actually fairly unusual for a bound-to-bound Franck-Condon spectrum to be nearly devoid of structure, as is carbon suboxide. There is only so much "damage" the promotion of a single electron can do, and it is uncommon to see large displacements of several modes. The more structure there is in a spectrum, the more one can potentially extract about the regional potential surface. Even carbon suboxide has some structure spaced at about 620 cm , so we can be sure that (t) makes at least one pass near after... 

Fig. 9. Overtone bands of full Franck-Condon spectrum (inset) and selected bands, computed quantum mechanically (converged basis set calculation) and classically, using the Wigner method.

Figure Al.6.11. Idealized UV absorption spectrum of CO2. Note the regular progression of intemiediate resolution vibrational progression. In the frequency regime this structure is interpreted as a Franck-Condon...

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]).

Section 6.13.2 and illustrated in Figure 6.5. The possible inaccuracies of the method were made clear and it was stressed that these are reduced by obtaining term values near to the dissociation limit. Whether this can be done depends very much on the relative dispositions of the various potential curves in a particular molecule and whether electronic transitions between them are allowed. How many ground state vibrational term values can be obtained from an emission spectrum is determined by the Franck-Condon principle. If r c r" then progressions in emission are very short and few term values result but if r is very different from r", as in the A U — system of carbon monoxide discussed in Section 7.2.5.4, long progressions are observed in emission and a more accurate value of Dq can be obtained. 

This model permits one to immediately relate the bath frequency spectrum to the rate-constant temperature dependence. For the classical bath (PhoOc < 1) the Franck-Condon factor is proportional to exp( —with the reorganization energy equal to... 

Note in passing that the common model in the theory of diffusion of impurities in 3D Debye crystals is the so-called deformational potential approximation with C a>)ccco,p co)ccco and J o ) oc co, which, for a strictly symmetric potential, displays weakly damped oscillations and does not have a well defined rate constant. If the system permits definition of the rate constant at T = 0, the latter is proportional to the square of the tunneling matrix element times the Franck-Condon factor, whereas accurate determination of the prefactor requires specifying the particular spectrum of the bath. 

Another conventional simplification is replacing the whole vibration spectrum by a single harmonic vibration with an effective frequency co. In doing so one has to leave the reversibility problem out of consideration. It is again the model of an active oscillator mentioned in section 2.2 and, in fact, it is friction in the active mode that renders the transition irreversible. Such an approach leads to the well known Kubo-Toyozawa problem [Kubo and Toyozava 1955], in which the Franck-Condon factor FC depends on two parameters, the order of multiphonon process N and the coupling parameter S... 

This approximation is not valid, say, for the ohmic case, when the bath spectrum contains too many low-frequency oscillators. The nonlocal kernel falls off according to a power law, and kink interacts with antikink even for large time separations. We assume here that the kernel falls off sufficiently fast. This requirement also provides convergence of the Franck-Condon factor, and it is fulfilled in most cases relevant for chemical reactions. 

Comparison between flame-sampled PIE curves for (a) m/z = 90 (C H ) and (b) m/z = 92 (C Hg) with the PIE spectra simulated based on a Franck-Condon factor analysis and the cold-flow PIE spectrum of toluene. Calculated ionization energies of some isomers are indicated. (From Hansen, N. et al., /. Phys. Chem. A, 2007. With permission.)... 

Figure 9. Photodissociation spectra of the insertion intermediate of the FeO + CH4 reaction. Top [HO—Fe—CDs], middle [HO—Fe—CHs], bottom (dashed) Franck-Condon simulation of the [HO—Fe—CHs] spectrum. The spectrum shows a long progression in the Fe-C stretch (Vii = 478 cm ) and short progressions in the Fe—O stretch (vg = 861 cm ) and O—Fe—C bend (V14 = 132 cm ).

The emission spectrum observed by high resolution spectroscopy for the A - X vibrational bands [4] has been very well reproduced theoretically for several low-lying vibrational quantum numbers and the spectrum for the A - A n vibrational bands has been theoretically derived for low vibrational quantum numbers to be subjected to further experimental analysis [8]. Related Franck-Condon factors for the latter and former transition bands [8] have also been derived and compared favourably with semi-empirical calculations [25] performed for the former transition bands. Pure rotational, vibrationm and rovibrational transitions appear to be the largest for the X ground state followed by those... 

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,... 

Evaluation of the Work Term from Charge Transfer Spectral Data. The intermolecular interaction leading to the precursor complex in Scheme IV is reminiscent of the electron donor-acceptor or EDA complexes formed between electron donors and acceptors (21). The latter is characterized by the presence of a new absorption band in the electronic spectrum. According to the Mulliken charge transfer (CT) theory for weak EDA complexes, the absorption maximum hv rp corresponds to the vertical (Franck-Condon) transition from the neutral ground state to the polar excited state (22). 

As shown in Fig. 6, there is a correlation between absorption spectrum and emission spectrum. Taking into consideration the Franck-Condon principle, which states that there is no motion of the atoms during an electronic transition, one has to differentiate between the two following possibilities in the one the geometry of the excited state is similar to the one of the ground state (same interatomic distances),... 

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. 

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. 

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... 

For some aromatic hydrocarbons such as naphthalene, anthracene and pery-lene, the absorption and fluorescence spectra exhibit vibrational bands. The energy spacing between the vibrational levels and the Franck-Condon factors (see Chapter 2) that determine the relative intensities of the vibronic bands are similar in So and Si so that the emission spectrum often appears to be symmetrical to the absorption spectrum ( mirror image rule), as illustrated in Figure B3.1. 

---