Franck-Condon matrix element

Fig. 13 Franck-Condon matrix elements Mom for weak (g = 0.1, squares), intermediate (g = 1, triangles), and strong (g = 10, circles) electron-vibron interaction. Lines are the guides for eyes.

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. 

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. 

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... 

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... 

One expects the impact of the electronic matrix element, eqs 1 and 2, on electron-transfer reactions to be manifested in a variation in the reaction rate constant with (1) donor-acceptor separation (2) changes in spin multiplicity between reactants and products (3) differences in donor and acceptor orbital symmetry etc. However, simple electron-transfer reactions tend to be dominated by Franck-Condon factors over most of the normally accessible temperature range. Even for outer-... 

In a semiclassical picture, the rate kda of nonadiabatic charge transfer between a donor d and an acceptor a is determined by the electronic coupling matrix element Vda and the thermally weighted Franck-Condon factor (f C) [25, 26] ... 

In line with the Franck-Condon principle, the electron transfer occurs at the seam of the crossing between diabatic (localized) states of donor and acceptor. The electronic coupHng is the off-diagonal matrix element of the Hamiltonian defined at the crossing point. 

Fig. 2. Pump and probe scheme within a tiers picture (schematic). The zeroth order bright state which is not Franck-Condon (FC) active in the electronic transition is excited via the near IR- laser pulse. FC-active modes m later tiers having no population at t=0 are probed and their time dependent population is a measure for IVR (Vj being matrix elements connecting zeroth order states) in the molecule giving rise to an enhancement of the electronic absorption.

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. 

Here /rLE/Jo, Mct/s0> / ct>0 are the z independent transition moment matrix elements in terms of zero order states, but I ct/soI l/kn-vwl- The Franck-Condon factors in Eq. (35) are assumed to be z independent since LE and CT have similar vibrational spectra. It follows simply that each z value contributes the following element to the spectrum for an arbitrary distribution P(z. t),... 

Along with the harmonic approximation, the Franck-Condon approximation is usually also used in calculating the probability of electron tunneling. In accordance with the latter approximation, the exchange matrix element V(q) in eqn. (20) is factorized outside the integral sign... 

It is also interesting to estimate the maximum value of the frequency factor in the case of purely quantum nuclear motion. This can be done with the help of the formula W 2nV2Sp, where V2 exp(—2yR) is the exchange matrix element, S is the Franck-Condon factor, p 1 jco is the density of the vibrational levels, and co 1000 cm-1 is the characteristic vibrational frequency of the nuclei. In the atomic unit system, the multiplier 2np has the order 103 and the atomic unit of frequency is 4.13 x 1016s-1 consequently, in the usual unit system, the frequency factor is of the order 4 x 1019Ss-1. The frequency factor reaches its maximum value when S 1. Thus, in the case of purely quantum nuclear motion, the maximum value of the frequency factor is also 1019-102°s-1. 

Tunneling Matrix Elements J0 and Franck-Condon Factors for Diffusion of Light Impurities in Metals... 

Significant electronic coupling V between donor and bridge as well as between bridge and acceptor, represented by the matrix element (x u y), must exist. In other words, the transition between the initial (neutral), intermediate (locally excited) and final (charge separated) state must be Franck-Condon allowed... 

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