Franck Condon factors

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]).

Equation (A 1.6.94) is called the KHD expression for the polarizability, a. Inspection of the denominators indicates that the first temi is the resonant temi and the second temi is tire non-resonant temi. Note the product of Franck-Condon factors in the numerator one corresponding to the amplitude for excitation and the other to the amplitude for emission. The KHD fonnula is sometimes called the siim-over-states fonnula, since fonnally it requires a sum over all intennediate states j, each intennediate state participating according to how far it is from resonance and the size of the matrix elements that coimect it to the states i. and The KHD fonnula is fiilly equivalent to the time domain fonnula, equation (Al.6.92). and can be derived from the latter in a straightforward way. However, the time domain fonnula can be much more convenient, particularly as one detunes from resonance, since one can exploit the fact that the effective dynamic becomes shorter and shorter as the detuning is increased. 

The observation of a bend progression is particularly significant. In photoelectron spectroscopy, just as in electronic absorption or emission spectroscopy, the extent of vibrational progressions is governed by Franck-Condon factors between the initial and final states, i.e. the transition between the anion vibrational level u" and neutral level u is given by... 

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. 

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]. 

If we can use only the zero-order tenn in equation (B 1.1.7) we can remove the transition moment from the integral and recover an equation hrvolving a Franck-Condon factor ... 

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). 

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. 

Here, I(co) is the Fourier transform of the above C(t) and AEq f is the adiabatic electronic energy difference (i.e., the energy difference between the v = 0 level in the final electronic state and the v = 0 level in the initial electronic state) for the electronic transition of interest. The above C(t) clearly contains Franck-Condon factors as well as time dependence exp(icOfvjvt + iAEi ft/h) that produces 5-function spikes at each electronic-vibrational transition frequency and rotational time dependence contained in the time correlation function quantity <5ir Eg ii,f(Re) Eg ii,f(Re,t)... 

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... 

The ZEKE-PE process shown in Figure 9.50(c) can be modified as shown by changing the wavenumber Vj of the first laser to excite the molecule to an excited vibrational level of M. Then the Franck-Condon factors for the band system are modified. This can allow... 

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... 

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,... 

Sethna [1981] considered two limiting cases. The calculation of action in the fast flip approximation (a>j CO ) proceeds by utilizing the expansion exp ( — cu,-1t ) 1 — cu t. After substituting the first term, i.e. the unity, in (5.72) we get precisely the quantity which yields the Franck-Condon factor in the rate constant. The next term cancels the adiabatic renormalization and changes KM)... 

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. 

Suppose now that both types of vibrations are involved in the transition. The symmetric modes decrease the effective tunneling distance to 2Q, while the antisymmetric ones create the Franck-Condon factor in which the displacement 2Qq now is to be replaced by the shorter tunneling distance 2Q, [Benderskii et al. 1991a]... 

Thus the promoting vibrations reduce the Franck-Condon factor itself, which is not reflected in the spin-boson model (5.55), (5.67). As an illustration, three-dimensional trajectories for various interrelations between symmetric (Ws) and antisymmetric (oja) vibration frequencies, and odo are shown in fig. 33. 

Franck-Condon factors 83 Free energy, interfacial 137 Fuzzy coatings 151... 

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.)... 

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... 

To simulate the vibrational progression, we obtain the Franck—Condon factors using the two-dimensional array method in ref 64. We consider 1 vibrational quantum v = 0) from the EC stationary point and 21 vibrational quanta (z/ = 0, 1,. .., 20) from the GC stationary point. The Franck— Condon factors are then calculated for every permutation up to 21 quanta over the vibrational modes. It is necessary in order to get all Franck—Condon factors of the EC stationary point with respect to each three alg vibrational state (Figure 6) of the GC to sum to one. One obtains a qualitative agreement between the calculated and the experimental emission profiles (Figure... 

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