Normal Coordinates and Duschinsky Effect

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 

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. 

Let us now return to Equation 1.29 for the potential energy surface of the ath electronic state and reformulate it in a more suitable (canonical) form  

Inderiving(1.39),wehavemadeuseoftherelationA = A for A being orthogonal. (The inverse of the matrix is its transpose A A = E.) The linear term in q in 

Equation 1.41 pertains to the normal coordinates in the ground electronic state an analogous expression holds for any electronic state a, where again A = A F (A ) = diag (XJjXj. X ) and A is the transformation matrix to mass-weighted coordinates, defined by 

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