Transition moment vibronic

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. 

In 1928, Condon treated the intensities of vibronic transitions quantum mechanically. The intensity of a vibronic transition is proportional to the square of the transition moment which is given by (see Equation 2.13)... 

If vibrations are excited in either the lower or the upper electronic state, or both, the vibronic transition moment corresponding to the electronic transition moment Rg in Equation (7.115), is given by... 

The Franck-Condon approximation (see Section 7.2.5.3) assumes that an electronic transition is very rapid compared with the motion of the nuclei. One important result is that the transition moment for a vibronic transition is given by... 

Nevertheless, 1,4-difluorobenzene has a rich two-photon fluorescence excitation spectrum, shown in Figure 9.29. The position of the forbidden Og (labelled 0-0) band is shown. All the vibronic transitions observed in the band system are induced by non-totally symmetric vibrations, rather like the one-photon case of benzene discussed in Section 7.3.4.2(b). The two-photon transition moment may become non-zero when certain vibrations are excited. 

The transition moment // between the vibronic states and where 0... 

Substitution of Eq. (6.3) into Eq. (6.2) leads to the following expression for the vibronic transition moment... 

The parameterized, analytical representations of fi, ., fiy, fifi determined in the fitting are in a form suitable for the calculation of the vibronic transition moments V fi V") (a—O, +1), that enter into the expression for the line strength in equation (21). These matrix elements are computed in a manner analogous to that employed for the matrix elements of the potential energy function in Ref. [1]. 

As detailed in Section 2, we have derived and programmed the expression for line strengths of individual rotation-vibration transitions of XY3 molecules the line strengths depend on the vibronic transition moments entering into equation (70). With the theory of Section 2, we can simulate rotation-vibration absorption spectra of XY3 molecules. In computing the transition wavenumbers, line strengths, and intensities we use rovibronic wavefunctions generated as described in Ref. [1]. 

It is seen from Equation 19 that the electronic transitions take place without changing the equilibrium positions of the nuclei, and the electronic component of the dipole transition moment is non-zero only if there is no change of the vibronic state during this transition. Dg is non-zero only if the transitions occur between the vibronic states within one electronic state, and the selection rules of Equation 16 are derived from the conditions for a non-vanishing matrix element in Dg. ... 

In the Herzberg-Teller method [22] for accounting for induced vibronic intensities, the transition moment M°e between the ground e" and excited e electronic states is expanded as a Taylor s series in the normal coordinate Q for anitsymmetric mode k... 

For small-amplitude vibrations in which the transition moments are relatively smooth, the expansion can be terminated at the linear second term. A vibronic transition between a set of e v e"v" electronic/vibrational levels is defined by the transition moment... 

The evaluation of the transition moment is straightforward now. Even though the energy difference between a5 B and ll A2 is small and the Tx level is strongly perturbed, the dipole transition to the ground state is forbidden as long as the molecule remains planar because of the dipole selection rules this transition would require an operator of A2 symmetry, but x, y, and z transform like B1, B2, and A, respectively. The transition may gain some intensity due to second-order spin-vibronic interactions, however. 

In Fig. 7 we recapitulate the spin-averaged Einstein coefficients for the Vegard-Kaplan emission from the lowest vibrational state of the triplet as well as the corresponding values reported by Piper [89]. The relative transition probabilities for different vibronic phosphorescence bands are quite good [26]. The absolute and the relative intensities of the higher vibrations v" are very sensitive to the transition moment curve... 

From the largest CAS calculation ( CAS-2 in ref. [26]) a lifetime of 2.58 s was predicted for one of the 0 = 1 spin-sublevels (Ms = 1) of the lowest triplet vibrational level. In a rotationless molecule the 0 = 0 sublevel does not radiate, so the observed spin-averaged radiative lifetime of the Vegard-Kaplan emission is equal to (3/2)rn=i = 3.87 s. It should be compared with Piper s value, 2.37 s [89]. We can see from Fig. 7 that for the lower v" vibronic quantum numbers the agreement is much better than for the most intensive transitions to the upper vibronic levels. The underestimation of the transition moment in the range 1.3-1.4 A is more serious for the small active spaces. In the valence type CAS ( CAS-1 in ref. [26]) an averaged lifetime equal to 5.48 s was obtained. 

The electric dipole T-S transition moments calculated as functions of the internuclear distances were used for the estimation of the vibronic transition probabilities by a vibrational averaging procedure. The calculated Einstein coefficients for emission from... 

The mechanisms described above for benzene as well as a few others have been computed by direct MCQR calculations of vibronic phosphorescence intensity in Ref. [83]. For the Born-Oppenheimer wave functions the transition moment between the singlet ground state So with v -th vibrational excitation and the first triplet T excited state... 

Hamiltonian is taken into account. In this case only vibronic phosphorescence bands (u = 0—m/=1) appear through non-totally symmetrical b g and ezg vibrations as follows from the scheme given in Fig. 7 and as discussed in the previous section. Their transition moments are equal to (in a.u.) ... 

---