Herzberg-Teller corrections

It has been found that the above theory is quite useful in predicting the correct overall behavior of matter in the presence of electromagnetic radiation. However, in order to predict correctly Raman intensities, it has been found necessary to refine the theory by accounting for small deviations in the electronic wave functions with nuclear motion. In the framework of the Herzberg-Teller theory, it is assumed that the corrected electronic wave functions may be obtained by the use of a first-order perturbation expansion as a linear combination of the complete set of zero-order Born-Oppenheimer functions, discussed above. Since here we are mainly interested in the normal Raman effect, we shall consider only corrections to the second term in Eq. (41). If we first examine corrections to the state X, the resulting expression for the derivative of the transition moment with motion along a normal mode is  

This matrix element represents the coupling of states K and S through terms in the Hamiltonian, which has previously been neglected in the Bom-Oppenheimer approximation. Since we have expanded the dipole moment matrix element, which is responsible for the intensities of optical transitions, we can see that the effect of this expansion is to allow states to borrow intensity from one another. A transition which may be forbidden in the zero-order theory may obtain intensity from a nearby state to which a transition is allowed. Note that in Eq. (46), the sum runs over all the excited states S, except the state K. The are the zero-order energies, and we may write the denominator as hiws - o k) - Clearly, efficient borrowing 

On substituting Eq. (46) into Eq. (41), note that the result involves sums over pairs of excited states, and since the wave functions are assumed real, so that Mics = [see Eq. (38)], we may see that in this summation there will be pairs of terms which differ only by interchange of K and We may then take advantage of the identity  

It was this equation that was derived by Albrecht in his definitive study of Raman intensities. Note that by considering 

This term was not considered important by Albrecht since he was considering only molecular states, and thus only transitions from the ground state, in which case the latter expression would have been inconsequential. This is due to the rather large energy gap between ground and first excited states for molecules. In our case, however, where we will allow the possibility that the intermediate state is a metal state and these states may lie close to the molecular ground state, then this term may not be ignored. 

---

In the present simulation we apply a simple model consisting of two vibrational modes in the relevant system. The first one will contribute to the Stokes shift as well as to the Herzberg-Teller correction, while the second one only to the Stokes shift. Next, we will assume a mutual coupling of the modes in the bilinear form Q Qi for the excited state. This type of coupling is usually omitted in literature. The Hamiltonian 77s is written as... 

Burland and Robinson 41) have presented semi-quantitative arguments (in the case of internal conversion) to determine the ratios of the electronic matrix elements 5 in what they call the Herzberg-Teller scheme and the ABO approximation. Their arguments may be qualtitativeley correct only if the CBO approximation, is employed instead of the Herzberg-Teller scheme15). 

The correction of the CA approximation performed above is known as vibronic coupling and the wavefunction (1.24a) is sometimes designated as the Herzberg-Teller approximation. In this approximation, the corrected molecular eigenfunction... 

---