Duschinsky transformation

This transformation is a good approximation when the molecule does not undergo a noticeable distortion during the transition. An extensive discussion of the transformation between the normal coordinates including a possible distortion of the molecule and the limitations of the Duschinsky transformation, as well as the necessity of minimizing the rotation effects, has been done by several authors [35-38]. 

A first and simple approximation is to neglect the mode mixing and consider a one-to-one relation between the modes of the initial and final states, with the Duschinsky transformation matrix J equal to the identity matrix (notice that VG and AS models belong by definition to this approximation). The interest of this approximation, called parallel-mode approximation, is that the multidimensional FC integrals can be calculated as products of one-dimensional integrals using the relation... 

The fitting parameters in the transform method are properties related to the two potential energy surfaces that define the electronic resonance. These curves are obtained when the two hypersurfaces are cut along the/th normal mode coordinate. In order of increasing theoretical sophistication these properties are (i) the relative position of their minima (often called the displacement parameters), (ii) the force constant of the vibration (its frequency), (iii) nuclear coordinate dependence of the electronic transition moment and (iv) the issue of mode mixing upon excitation—known as the Duschinsky effect—requiring a multidimensional approach. 

We adopt harmonic approximation and Q, are Q/ are the column vectors representing the sets of normal coordinates of states, ) and 4>f). According to Duschinsky [65], the following linear transformation holds ... 

Finally, the calculation of the matrix elements of the electric dipole moment operator requires a relation between the normal modes of the lower and higher states, which, as a general rule, are different. The standard approach refers to the linear transformation between the two sets of normal modes proposed by Duschinsky [276],... 

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