Vibronic intensities, evaluation

It should be mentioned here that attempts to use the matrix sum rule to evaluate vibronic intensities introduces unnecessary errors. The sum rule can only he applied when one is dealing with a complete set of wave functions. The use of approximate wave functions, incomplete sets, etc., invalidates the matrix closure rules. This statement applies to both the electronic and vibrational problems separately and jointly (see, e.g., reference 1). 

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. 

The contributions of the high frequency vibrational modes to A,r, hereafter written Xh, appear as nuclear tunnehng contributions to k t, and as vibronic side bands in CT spectroscopy. The intensity of a vibronic side band relative to the intensity of the fhndamental component is equal to A,h// Vh If the high frequency components can be resolved in the emission spectra, then A,h can be determined and the nuclear tunneling contributions to can be evaluated for each such component. 

Warshel, A. and Huler, E. (1974) Theoretical Evaluation of Potential Surfaces, Equilibrium Geometries and Vibronic Transition Intensities of Excimers The Pyrene Crystal Excimer, Chem. Phys. 6, 463—468. 

As the intensity of the transitions are proportional to the square of the dipole moments (see eqn (8.6)), the Franck-Condon factors weight each of the vibronic transitions. To evaluate these terms we write... 

To compute the intensity of the vibronic transition, S->-N, all we need do is evaluate the vibronic perturbation AVgj, it is assumed that we have experimental or theoretical knowledge of the energies AE j, hv S —> N), and hv TN) and the electronically allowed oscillator strength f TN). [Experimentally, the latter may be obtained by the evaluation of the area under the absorption curve when its ordinate is the extinction coefficient. e v), and its abscissa is the frequency of absorption, r(cm ), according to the formula / = 4.33 Theoretically, it... 

The classic cases of the Herzberg-Teller mechanism relate to coupling between two electronic states of different symmetry. An important example of this case occurs when electric dipole transitions of one of the two states are forbidden (e.g., the Laporte-forbidden d-d and f-f transitions). In this case, the forbidden transition may acquire absorption intensity by Herzberg-Teller mixing with an allowed transition via a nontotaUy symmetric mode of appropriate symmetry (the irreducible representation of the active mode must be contained in the direct product of the irreducible representations for the two states coupled by the Herzberg-Teller mechanism). We shall illustrate our results in Chapter 7 by evaluating the vibronic induced d-d transitions in transition metal complexes. 

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