Franck-Condon factor matrix elements

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. 

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] ... 

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),... 

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... 

These overlaps, which are matrix elements of the translation operator, are the well-known Franck-Condon factors ... 

Undamped time-dependent situations corresponding to representations // and ///. Damped time-dependent situations corresponding to representations // and ///. Matrix elements of the translation operator (Franck-Condon factors). 

Equations [41]-[50] provide an exact solution for the CT free energy surfaces and Franck-Condon factors of a two-state system in a condensed medium with quantum electronic and classical nuclear polarization fields. The derivation does not make any specific assumptions about the off-diagonal matrix elements of the Hamiltonian. It, therefore, includes the off-diagonal... 

In the Born-Oppenheimer approximation, the relative importance of channels (la) and (lb), together with their dependence on wavelength would depend upon the matrix elements for the transition between the electronic states, the Franck-Condon factors, the Honl-London factors, and upon the probabilities for spontaneous dissociation of the excited state formed. In principle, except for the last one, these are well known quantities whose product is the transition probability for that particular absorption band of Cs. When multiplied by the last quantity, and with an adjustment of numerical constants i becomes the cross section for the photolysis of Cs into Cs + Cs. It is the measurement of this cross section that lies at the focus of this work. 

Note that the wave functions for the initial and final states (, and / include both donor and acceptor. This eqnation is nsnally simplified by making the Bom-Oppenheimer approximation for the separation of nnclear and electron wave functions, resulting in equation (12), in which V is the electronic matrix element describing the coupling between the electronic state of the reactants with those of the product, and FC is the Franck Condon factor. 

The first term of the right hand side of Eq. (7) represents the squared electronic dipole matrix element and specifies the intensity of the purely electronic transition. The second term is the Franck-Condon factor, that is discussed below in more detail. It leads to the well-known Franck-Condon progression of vibrational satellites that progress in the spectrum by the energy Vq of the normal mode under consideration. 

The rate constant for ET can mathematically be regarded as the optical spectrum of a localized electron in the limit where the photon energy to be absorbed or emitted approaches zero. Erom the theory of radiative transitions [10, 12] and r / -b 1) = / for a positive integer /, we see that the factor multiplied to on the right-hand side of Eq. 27 represents the thermally renormalized value of the Franck-Condon factor [i.e., the squared overlap integral between the lowest phonon state in Vy(Q) and the ( AG /te)-th one in piQ)] for ET. The renormalization manifests itself in the Debye-Waller factor exp[—,vcoth( / (y/2)], smaller than e which appears also in neutron or X-ray scattering 12a]. Therefore, yen in Eq- 27 represents the effective matrix element for electron tunneling from the lowest phonon state in the reactant well with simultaneous emission of i AG /liw) phonons. 

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