Harmonic oscillator. Franck-Condon factor

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,... 

If the equilibrium position of the excited state C is located outside the configurational coordinate curve of the ground state, the excited state intersects the ground state in relaxing from B to C, leading to a nonradiative process. As described above, the shape of an optical absorption or emission spectrum is decided by the Franck-Condon factor and also by the electronic population in the vibrational levels at thermal equilibrium. For the special case where both ground and excited states have the same angular frequency, the absorption probability can by calculated with harmonic oscillator wavefunctions in a relatively simple form ... 

In the case of the harmonic oscillator, the Franck-Condon factors (i.e. the squared overlap integrals of the wave functions with quantum numbers n and m) can be easily calculated using the tabulated integrals [19]... 

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 ... 

A Maple worksheet for a similar ealenlation of Franek-Condon factors for harmonic and Morse oscillators is available from the Maple Application Center at www.maple-apps.com see Estimation of Franck-Condon Factors with Model Wave Functions by G. J. Fee, J. W. Nibler, and J. F. Ogilvie (2001). A Mathcad calculation for a harmonic oscillator is described by T. J. Zielinski, J. Chem. Educ. 75, 1189 (1998). 

A Mathematica calculation of Franck-Condon factors that determine electronic transition intensities of I2 is presented in Chapter III, and program statements for this are illustrated for I2 in Fig. III-6. In this fignre, note the dramatic differences between the intensity patterns predicted for the harmonic oscillator and Morse cases and compare these patterns with those seen in your absorption spectra. If yon have access to this software, yon might examine the changes in the harmonic-oscillator and Morse-oscillator wavefnnctions for different v, v" choices. A calcnlation of the relative emission intensities from the v = 25, 40, or 43 level conld also be done for comparison with emission spectra obtained with a mercury lamp or with a krypton- or argon-ion laser, hi contrast to the smooth variation in the intensity factors seen in the absorption spectra, wide variations are observed in relative emission to v" odd and even valnes, and this can be contrasted with the calcnlated intensities. Note that, if accnrate relative comparisons are to be made with experimental intensities, the theoretical intensity factor from the Mathematica program for each transition of wavennmber valne v shonld be mnltiphed by v for absorption and for emission. ... 

Figure 6.1-1 Resonance Raman scattering via the A-term as given in Eq. 6.1-8. For a non-displaced electronic state (harmonic oscillator model) there is always one vanishing Franck-Condon factor which brings the A-term contribution to zero. A displacement (A) of the excited electronic state in respect to the ground state allows nonvanishing vibrational overlaps for both, the upward and downward transitions (Adapted from Asher, 1988).

The Franck-Condon factor is given by the squared overlap integral of displaced harmonic oscillator functions (Hermite functions). It can be related [154, p. 113] to the so-called Huang-Rhys parameter (or factor) S according to... 

To clarify the question of the chemical reaction heat distribution in the vibrational degrees of freedom of the product, let us compare the matrix elements of the transition from the fundamental initial state to various final vibrational states, assuming for the sake of definiteness that the transition is nonadiabatic. Applying the known expressions for the Franck-Condon factors of harmonic oscillators, we obtain... 

Figure 13 Potential energy surfaces for electron transfer reactions. Harmonic oscillator potential energy functions for reactants and product are shown, including the nuclear wave functions, which are shaded. The dark shaded region indicates the magnitude of overlap of the nuclear wave functions, which is the Franck-Condon factor, (a) is the normal region, (b) is the activationless region and (c) is the inverted region as defined in the text. (Ref. 72. Reproduced by permission of Nature Publishing Group, www.nature.com)...

When using the model functions [ j to represent the basis set r/t (q,, the time evolution in equation (9) is essentially determined by Franck-Condon factors involving the overlap between harmonic-oscillator functions for different diabatic electronic attractors. Their actual calculation can be done with the help of the powerful methods developed by Palma [17]. 

The Franck-Condon factors are weighted by the density-of-states factor if the fragment is treated as a rigid rotor-harmonic oscillator, /> (e) is given by... 

FIGURE 1.1 Semilogarithmic plot of the Franck-Condon factor, F, as a function of lower state vibrational quantum number, v, for several values of the displacement parameter, y. The displacement parameter is related to the frequency of the vibration, (O, and the displacement, Q, of a harmonic oscillator by y=coQW2. (From W. Siebrand in Modem Theoretical Chemistry Vol. 1, Plenum Press (Kluwer), 1976. With permission.)... 

The Franck-Condon factor of an electronic transition from the vibrational level v = 0 of the electronic ground state to the corresponding level v = 0 of the electronic excited state has a simple analytical solution when the vibrational motion is described by a harmonic oscillator. In this case, the vibrational wavefunctions of the ground vibrational levels is described by a normal (Gaussian) distribution function... 

Figure 15.10 Franck-Condon factors calculated using Morse or harmonic oscillators for the X - g to a Ag transition in Oj (white bars), to transition in N2 (black bars) and X 2g to...

Chang, J.L. (2008) A new method to calculate Franck-Condon factors of multidimensional harmonic oscillators including the Duschinsky effect./. Chem. Phys., 128, 174111. 

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