Franck-Condon approximation calculation

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... 

Using the harmonic and the Franck-Condon approximations, it is possible to advance in calculating the sums of the (18) type using the method of generating functions developed by Kubo and Toyozawa [8, 9] for the calculation of the probabilities of optical and non-radiative transitions in the impurity centres in crystals. According to this method we can rewrite eqn. (18) as... 

First of all, consider the case when all normal vibrations are classical. This takes place if the condition a)k -4 T works well for all frequencies. In the classical case the probability of tunneling can be calculated with the help of the general formula (18) using the Franck-Condon approximation and the well-known [10] properties of quasi-classical wave functions. We will not dwell upon the details of transition from the quantum description to the... 

Recent ab initio calculations of the relevant Na2 potentials by Konowalow, Rosenkrantz and Olson (14), henceforth abbreviated as KRO, permit such an approach. In the quasistatic or classical Franck-Condon approximation the binary contribution to k(v,T) is given by (15.16)... 

The dipole transition moment is rather large for both PMI and PDI (8 and 9 Debye units, respectively) and does not change significantly upon distortion of the molecule from the So to the 5i structure. Hence, the Franck-Condon approximation is appropriate for calculating the vibrational substructure of the absorption and emission bands. 

Trace in Eq. (17) operates only with respect to the quantum states of the heavy particles at fixed states of the electrons. Calculating trace in the coordinate representation, one has, in general, to take into account that i and depend on the coordinates of the heavy particles. If this dependence is weak, they may be taken out of the integral at the point corresponding to the maximum of the product of the density matrices p,p/. Just this procedure may be called the Franck-Condon approximation for chemical reactions. 

Also shown in Fig. 5 are results for the total ionization yield obtained by the semiclassical Franck-Condon approximation introduced in Eqs. (60) and (58). As discussed above, this theory is only valid for sufficiently short probe pulses. While this condition is well satisfied for probe pulse up to 20 fs duration [panels (a) and (b)], the approximation is seen to introduce spurious structures in the case of 32 fs pulses [panel (c)]. Since the Franck-Condon approximation reduces the cost of explicit pmnp>-probe simulations to the cost of a standard time-dependent wave-packet propagation, one obtains an overall computational speed-up of about two orders of magnitude compared to the full nonperturbative calculation. 

The Franck-Condon approximation is used to calculate the intensities. The bound state energies and wave functions are obtained by numerically solving the Schrodinger equation ... 

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)... 

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. 

The Gj(t) functions of Eq. (15) have been calculated by Lin [60] when summing over Franck-Condon factors obtained from all possible (infinite) wavefunctions in the harmonic oscillator approximation. These Gj(t) are rather complicated functions of the frequencies arf, co and reduced masses M j, M which are attributed to the corresponding normal coordinates Qf and Q j. They are collected in parameters describing the frequency relation ft2 and the potential minimum shift Aj of the excited state with respect to the ground state... 

In the crude Born-Oppenheimer approximations, the oscillator strength of the 0-n vibronic transition is proportional to (FJ)2. Furthermore, the Franck-Condon factor is analytically calculated in the harmonic approximation. From the hamiltonian (2.15), it is clear that the exciton coupling to the field of vibrations finds its origin in the fact that we use the same vibration operators in the ground and the excited electronic states. By a new definition of the operators, it becomes possible to eliminate the terms B B(b + b ), BfB(b + hf)2. For that, we apply to the operators the following canonical transformation ... 

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