Vibronic integrals

Just the linear vibronic integral is nonzero for any point group the direct product of the IRs of the electronic wave functions is a reducible representation that necessarily contains the IR of a symmetry coordinate... 

As an effect of the linear and quadratic vibronic integrals the adiabatic potential surface stays no longer paraboloid-shaped. It exhibits an additional warping with several local minima and saddle points out of the reference high-symmetry configuration Q0. 

The vibronic integrals Vs and Vst contain a radial part and an angular part. The angular part can be determined with the help of the group theory and the remainder (the reduced matrix element) is taken as a parameter depending only on the symmetry type (Xe and Xee). Considering the quadratic approximation to the E-e vibronic coupling the vibronic matrix becomes expressed as follows [88-90] ... 

Figure 7.35b shows how the value of the vibronic integral [Eq. (7.11.10)] accumulates as the integration proceeds from R = 0 to larger R. Results for the N and 1F)N isotopic molecules are shown. After rapid, large-amplitude oscillation, the integral eventually stabilizes to a very small value at an inter-nuclear distance where Xv=4 is zero. The calculated predissociation rates are displayed on Fig. 7.33 and agree well with the experimental isotopic dependence. 

This last transition moment integral, if plugged into equation (B 1.1.2). will give the integrated intensity of a vibronic band, i.e. of a transition starting from vibrational state a of electronic state 1 and ending on vibrational level b of electronic state u. 

Figure 3, Wavepacket dynamics of the photodissociation of NOCl, shown as snapshots of the density (wavepacket amplitude squared) at various times, The coordinates, in au, are described in Figure b, and the wavepacket is initially the ground-state vibronic wave function vertically excited onto the 5i state. Increasing corresponds to chlorine dissociation. The density has been integrated over the angular coordinate. The 5i PES is ploted for the geometry, 9 = 127, the ground-state equilibrium value,...

An example of an El forbidden but "vibronically allowed" transition is provided by the singlet n ==> ti transition of H2CO that was discussed earlier in this section. As detailed there, the ground electronic state has Ai symmetry, and the n ==> 71 state is of 1A2 symmetry, so the El transition integral... 

The quantity J dr is called the vibrational overlap integral, as it is a measure of the degree to which the two vibrational wave functions overlap. Its square is known as the Franck-Condon factor to which the intensity of the vibronic transition is proportional. In carrying out the integration the requirement that r remain constant during the transition is necessarily taken into account. 

The intensity of a vibronic transition depends upon the square of the overlap integral of the vibrational wave functions,... 

The term III scattering (equation 8) is the weakest in the three scattering mechanisms, as shown by two derivative terms (M ) in the electronic transition integrals. Clearly, for a dipole forbidden transition (M° = 0) the only non-zero term is term III. The term in scattering results in binary overtone and combination transitions of vibronically active modes. It is noted that no fundamental transition survives. 

To perform a PO analysis of nonadiabatic quantum dynamics, we employ a quasi-classical approximation that expresses time-dependent quantities of a vibronically coupled system in terms of the vibronic POs of the system [123]. Considering the quasi-classical expression (16) for the time-dependent expectation value of an observable A, this approximation assumes that the integrable islands in phase space represent the most significant contributions to the dynamics of the observables considered [236]. As a consequence, the short-time dynamics of the system is determined by its shortest POs and can be approximated by a time average over these orbits. Denoting the A th PO with period 7 by qk t),Pk t) we obtain [123]... 

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