Franck harmonic 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 control scheme [13,17] that we have focused on, the time evolution of the interference terms plays an important role. We have already discussed more explicit forms of Eq. (7.75). One example is the Franck-Condon wave packet considered in Section 7.2.2 another example, which we considered above, is the oscillating Gaussian wave packet created in a harmonic oscillator by an (intense) IR-pulse. Note that the interference term in Eq. (7.76) becomes independent of time when the two states are degenerate, that is, AE = 0. The magnitude of the interference term still depends, however, on the phase S. This observation is used in another important scheme for coherent control [14]. 

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... 

The theoretical background which will be needed to calculate the excited state distortions from electronic and Raman spectra is discussed in this section. We will use the time-dependent theory because it provides both a powerful quantitative calculational method and an intuitive physical picture [42,46-50]. The method shows in a simple way the inter-relationship between Raman and electronic spectroscopy. It demonstrates that the intensity of a peak in a resonance Raman spectrum provides detailed information about the displacement of the excited state potential surface along the normal mode giving rise to the peak [42,48]. It can also be used to calculate distortions from the intensities of vibronic peaks in electronic spectra [49]. For harmonic oscillators, the time-dependent theory is mathematically equivalent to the familiar Franck-Condon calculation [48]. 

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... 

If we assume co = o), for a nontotally symmetric harmonic oscillator, Franck-Condon contributions to its Raman scattering cross section will vanish since Q = Qg by symmetry for such modes. The only source of Raman intensity is then vibronic coupling. In its simplest form, this mechanism can be described as due to the Q-dependence of the electronic transition moments in Eq. (19) ... 

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