Harmonic oscillator. Franck-Condon

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) ... 

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,... 

Flowever, there is a trade-off in using near-IR emissive lanthanides, in that luminescence lifetimes are shorter, and quantum yields lower, compared to complexes of Tb and Eu. This arises because the near-IR emissive lanthanides are quenched by lower harmonics of the O-H oscillator, increasing the Franck-Condon overlap with the metal excited state. For neodymium, matters are further complicated by the manifold of available metal-centered excited states, which leads to particularly effective quenching by C-H oscillators. Thus, complexes in which there are few C-H oscillators close to the metal are desirable if the luminescence lifetime is to be optimized (e.g. 44).76 97-101... 

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 ... 

Fig. 1. Schematic of vertical (a) and non-vertical (b) electronic transitions with the electronic energies represented by displaced harmonic potentials. The initial state is the vibrational ground state on Vi. The wave packet on V2 in (a) is the Franck-Condon wave packet and the dashed arrows mark the positions of the turning points for the oscillation this wave packet will undergo under field-free conditions.

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]. 

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... 

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)...

---