Duschinsky rotation

It is very difficult to experimentally obtain the values of the scattering cross section. However, it is relatively easy to obtain the intensity of given normal mode k relative to that of another mode k. A simple expression relating the relative intensities has been derived for the special conditions of harmonic oscillators, no Duschinsky rotation, no change in normal mode frequencies, and pre-resonance (short time) condition spectra. Under these conditions the relative intensities of two modes is given by... 

In order to incorporate Duschinsky rotations and anharmonicity the potential energy difference is expanded as a power series in the excited electronic... 

Reimers, J. R., A practical method for the use of curvilinear coordinates in calculations of normal-mode-projected displacements and Duschinsky rotation matrices for large molecules, J. Chem. Phys., 115, 9103, 2001. 

The parameters 7G account for frequency shifts and for the so-called Duschinsky rotation. ... 

Including all 15 vibrational modes, however, was not possible within the linear model in a satisfactory manner. In particular, inclusion of the i/3 (Ai) mode provided problems, as it is also significantly coupled to the excitation. The problem was traced to the lack of second-order coupling. On calculating these parameters, it was found that the 1 2 and r-s modes were strongly coupled, and undergo Duschinsky rotation. No correction of the first order terms was then required and the resulting spectrum from the 15-mode model is shown in Fig. 5(c). ... 

Our calculations show that these second order terms are important for a quantitative description of nonadiabatic systems. This is demonstrated in the pyrazine S1/S2 system, where a reduced 4-mode model provides a qualitative picture with the main peaks of the spectrum in the correct places. The addition of second order terms and all degrees of freedom, results in the correct spectral envelope also being produced by the model. Also in the allene A/B system, the second order terms are required, not only for the correct description of the Duschinsky rotation in the excited state, but also for the high spectral density between the two bands. Even in the butatriene X/A system, in which second order terms play a minor role in the description of the spectral band, the inclusion of these terms means that the ab initio data could be taken with minimal adjustment, whereas a reduced dimensionality model required significant adjustment of the expansion parameters. 

Still in the harmonic approximation without Duschinsky rotation or frequency shifts, these quantities are simply (see details in the Online Resource 1)... 

In parentheses the fundamental modes of Si resulting from Duschinsky rotation between So vibrations. Average of four bands considered for estimation of root-mean-square (RMS) deviation. 

According to Equation 1.44, the normal coordinates of an excited electronic state q relative to those of the ground electronic state q are rotated (rotation matrix W) and displaced by the vector k. This rotation is called the Duschinsky rotation or Duschinsky mixing effect [41—44] (of the vibrational modes among each other). This mixing effect is subject to symmetry rules of the molecular symmetry group. Since in the most common instances vibrational modes of the same symmetry are mixed with each other (Equations 1.29-1.31 and 1.37), the matrix W assumes the... 

We shall begin our discussion with the derivation of line shape functions for optically allowed and vibronically induced transitions. This will permit us to demonstrate the influence of the Duschinsky rotation on optical spectra. As an example, we shall describe the spectrum of the p-terphenyl crystal from the same point of view and of the [Co(CN)2(tn)2]Cl3H20 complex in its crystal structure. These examples offer little more than a glimpse into the difficult, subtle, fascinating questions encountered in almost any attempt to interpret optical transitions. The discussion is extended in Section 7.2 to the analysis of phosphorescence spectra and the description of radiationless transition of aromatic molecules. Particular emphasis is placed on the mechanism of singlet-triplet relaxation in these molecules with nonbonding electrons. 

Table 7.7 Spectroscopic parameters and the corresponding density-of-state vibrational overlaps for selected DAPs (the Duschinsky rotation matrices for 2,4-DAP and 3,5-DAP can be derived directly from the Euler angles given in Table 7.5 the rotation matrix of 1,3-DAP is identical to that of 2,4-DAP).

Chapter 3 provides a detailed description of radiationless processes in a statistical large molecule embedded in an inert medium. In this chapter, we are for the first time able to express the vibrational overlap between the electronic states under consideration in terms of intramolecular distributions in the full harmonic approximation taking into account the effects of vibrational frequency distortion, potential surface displacement, and the Duschinsky rotation. 

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