Franck-Condon expression

The full PL envelope may be modeled using the Franck-Condon expression,... 

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. 

Following Eq. (75), the rate constant for spin conversion may be expressed as a product of the electronic matrix element V and the nuclear Franck-Condon... 

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. 

If solvent (or environment) relaxation is complete, equations for the dipole-dipole interaction solvatochromic shifts can be derived within the simple model of spherical-centered dipoles in isotropically polarizable spheres and within the assumption of equal dipole moments in Franck-Condon and relaxed states. The solvatochromic shifts (expressed in wavenumbers) are then given by Eqs (7.3) and (7.4) for absorption and emission, respectively ... 

We next consider the expression for k in the classical formalism. According to the Franck-Condon principle, internuclear distances and nuclear velocities do not change during the actual electron transfer. This requirement is incorporated into the classical electron-transfer theories by postulating that the electron transfer occurs at the intersection of two potential energy surfaces, one for the reactants... 

The probability of a particular vertical transition from the neutral to a certain vibrational level of the ion is expressed by its Franck-Condon factor. The distribution of Franck-Condon factors, /pc, describes the distribution of vibrational states for an excited ion. [33] The larger ri compared to ro, the more probable will be the generation of ions excited even well above dissociation energy. Photoelectron spectroscopy allows for both the determination of adiabatic ionization energies and of Franck-Condon factors (Chap. 2.10.1). 

The next important aspect to be considered is the electron-phonon interaction (lattice relaxation). Here, the effect of momentum conserving phonons, or promoting modes, can in principle be included in the electronic cross section this is discussed, for instance, by Monemar and Samuelson (1976) and Stoneham (1977). However, the configuration coordinate (CC) phonons (or accepting modes) are treated separately. The effect of these CC modes is usually expressed by the Franck-Condon factor dF c, where this factor is the same as the defined in our Fig. 16. Thus assuming a single mode,... 

The acetylene A <- X electronic transition is a bent <- linear transition that would be electronically forbidden ( - ) at the linear structure. The usual approximation is to ignore the possibility that the electronic part of the transition moment depends on nuclear configuration and to calculate the relative strengths of vibrational bands as the square of the vibrational overlap between the initial and final vibrational states (Franck-Condon factor). A slightly more accurate picture would be to express the electronic transition moment as a linear function of Q l (the fra/w-bending normal coordinate on the linear X1 state) in such a treatment, the transition moment would be zero at the linear structure and the vibrational overlap factors would be replaced by matrix elements of Qfl- Nevertheless, as long as one makes use of low vibrational levels of the A state, neglect of the nuclear coordinate dependence of the electronic excitation function is unlikely to affect the predicted dynamics or to complicate any proposed control scheme. 

Based on the expression for, a large increase in the useful NLO coefficient for a fixed wavelength is predicted in the case where the absorbance of the NLO dye lies between the fundamental and second harmonic. Residual absorption at the second harmonic is the limiting factor in the practical application of this technique, and has been addressed through the synthesis of new dyes. Improvement of lOx in reducing this absorbance has been achieved, and another factor of 5-lOx is estimated to be required before practical devices can be fabricated. Franck-Condon effects (vibronic structure) appear to be responsible for this residual absorption because small, rigid chromophores are often correlated with the lowest amounts of absorption. Chromophores based loosely on... 

A question that arises in consideration of the annihilation pathways is why the reactions between radical ions lead preferentially to the formation of excited state species rather than directly forming products in the ground state. The phenomenon can be explained in the context of electron transfer theory [34-38], Since electron transfer occurs on the Franck-Condon time scale, the reactants have to achieve a structural configuration that is along the path to product formation. The transition state of the electron transfer corresponds to the area of intersection of the reactant and product potential energy surfaces in a multidimensional configuration space. Electron transfer rates are then proportional to the nuclear frequency and probability that a pair of reactants reaches the energy in which they have a common conformation with the products and electron transfer can occur. The electron transfer rate constant can then be expressed as... 

When a normal mode is distorted but is not displaced, then, according to the selection rules, vibronic transitions with odd n, aO —> bn , are forbidden. For even n, Franck-Condon factors are expressed as follows ... 

This relation was first obtained by Forster and is usually called the Forster theory. Rather than expressing W,-. y in the spectral-overlap relation, lT, y can be expressed in terms of the Franck-Condon factors which can be calculated from the potential surfaces as was done for the photo-induced ET or radiationless transitions. 

If the electronic transition is allowed, is nonzero and the first term dominates the expression. This term can be viewed as a product of the electronic transition moment and the vibrational overlap integral, (v /v, v /v ), connecting the two vibrational wavefunctions, /v, in electronic states e and e". The Franck-Condon factors, which are the square of the vibrational overlap integral, determine the intensity distribution among the vibrational bands. The relative intensities of the band members within a vibrational progression is, therefore, given by the ratio of the Franck-Condon factors. If, through a symmetry restriction, the transition moment M°e vanishes, as in the present case, the band activity in the spectrum comes from the second term. When Qk is a nontotally symmetric... 

Here, Amn(cf) are the Franck-Condon factors, the expression of which are given by Eq. (C.15). 

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