The Franck-Condon Factor

The probability of intramolecular energy transfer between two electronic states is inversely proportional to the energy gap, AE, between the two states. The value of the rate constant for radiationless transitions decreases with the size of the energy gap between the initial and final electronic states involved. This law readily provides us with a simple explanation of Kasha s rule and Vavilov s rule. 

The rate of internal conversion between electronic states is determined by the magnitude of the energy gap between these states. The energy gaps between upper excited states (S4, S3, S2) are relatively small compared to the gap between the lowest excited state and the ground state, and so the internal conversion between them will be rapid. Thus fluorescence is unable to compete with internal conversion from upper excited states. The electronic energy gap between Si and S0 is much larger and so fluorescence (Si — S0) is able to compete with Si(v = 0) So(v = n) internal conversion. 

In Chapters 2 and 4, the Franck-Condon factor was used to account for the efficiency of electronic transitions resulting in absorption and radiative transitions. The efficiency of the transitions was envisaged as being related to the extent of overlap between the squares of the vibrational wave functions, /2, of the initial and final states. In a horizontal radiationless transition, the extent of overlap of the /2 functions of the initial and final states is the primary factor controlling the rate of internal conversion and intersystem crossing. 

Adapted from C.E. Wayne and R.R Wayne, Photochemistry, Oxford Chemistry Primers 39 (Oxford University Press, 1996) 

1 Case (A) Both Electronic States Have a Similar 

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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]).

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 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. 

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. 

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... 

The height of the potential barrier separating the initial and final states of the nuclear subsystem decreases and, hence, the Franck-Condon factor increases (Fig. 6). In the classical limit, this results in a decrease of the activation free energy. 

Due to strong interaction of the reactants with the medium, the influence of the latter may not be reduced only to the widening of the vibrational levels of the proton in the molecules AH and BH. The theory takes into account the Franck-Condon factor determined by the reorganization of the medium during the course of the reaction. 

Thus, the inertia of the tunneling particle leads to two opposite effects a decrease of the transition probability due to the reorganization along the coordinate of the center of mass and an increase of the transition probability due to the increase of the Franck-Condon factor of the tunneling particle. Unlike the result in Ref. 66, it is found in Ref. 67 that for ordinary relationships between the physical parameters, the inertia leads to an increase of the transition probability. 

Internal conversion refers to radiationless transition between states of the same multiplicity, whereas intersystem crossing refers to such transitions between states of different multiplicities. The difference between the electronic energies is vested as the vibrational energy of the lower state. In the liquid phase, the vibrational energy may be quickly degraded into heat by collision, and in any phase, the differential energy is shared in a polyatomic molecule among various modes of vibration. The theory of radiationless transitions developed by Robinson and Frosch (1963) stresses the Franck-Condon factor. Jortner et al. (1969) have extensively reviewed the situation from the photochemical viewpoint. 

Si is often called the Huang-Rhys factor or coupling constant. The plot of the Franck-Condon factor versus v[ is also shown in Fig. 7. 

Figure 7. (a) Schematic descriptions of the Huang-Rhys factor and the Franck-Condon factors... 

Equation (49) contains the Franck-Condon factors that are the matrix elements of the translation operator involved in the canonical transformation (36) with k = 1 that are given for m > n by... 

The variations in efficiency (rate) of radiationless transitions result from differences in the Franck-Condon factor, visualised by superimposing the vibrational wavefunctions, / (or /2 - the probability distributions), of the initial and final states. We will consider three cases illustrated in Figure 5.2. 

Considering the deuterium effect on naphthalene, Ci0H8, in which all the hydrogen atoms are replaced by deuterium, the Franck-Condon factor is decreased and consequently the rate of internal conversion... 

In the quantum mechanical description (in continuation of Box 2.2), the wavefunction can be described by the product of an electronic wavefunction VP and a vibrational wavefunction / (the rotational contribution can be neglected), so that the probability of transition between an initial state defined by ViXa and a final state defined by TQ/b is proportional to electron coordinates, this expression can be rewritten as the product of two terms < f i M vP2> 2 Franck-Condon factor. Qualitatively, the transition occurs from the lowest vibrational state of the ground state to the vibrational state of the excited state that it most resembles in terms of vibrational wavefunction. 

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