Rotational averaging

The concept of defining monodomains in the way shown above includes (with the exception of pathologic cases) that, in the assembly of monodomains, each orientation occurs with the same probability. So it should be possible to construct a rotationally averaged representative monodomain. 

The square of the structure amplitude in radial direction is shown in Fig. 17 for segment lengths 5 and 10. Even for short segment lengths, drops rapidly against zero due to the conformational and rotational average within the... 

To derive the pseudocontact shift in solution S1 we must now take the rotational average of Eq. (IV.8). By recalling that... 

The vibrationally adiabatic approximation should be made for the fast HCl vibrational coordinate r. Solving the one-dimensional Schrodinger equation with (R,9) fixed leads to the parameterized form

[Pg.155]

The expressions for Raman and ROA intensities depend on products of the various transition tensors. For ROA, isotropic samples are of interest and rotational averages of the products of these tensors are therefore required. The rotational averages can be expressed by double contractions of the isotropic (is), the anisotropic (anis), and the antisymmetric (a) part, which for a second rank tensor T are defined as... 

In a molecule with a one-dimensional conjugation path for the n-electrons one can assume that the tensor element with all indices in this backbone direction (x) dominates over all other tensor elements. The rotational average for these lD-molecules simplify to... 

The rotational average of the second-order hyperpolarizability in third har-... 

Similar as for third-harmonic generation, in the rotational average of the second-order hyperpolarizability yf0FWM = y(-co,co,-co,coj) the indices with... 

For a two-dimensionally conjugated molecule in solution two different notations are possible. In the rotational average both notation are equivalent. In the first one, the permutation symmetry of the spatial indices is sacrified in order to retrieve simpler expressions of the tensor elements. 

The second-order hyperpolarizability of the tetrasubstituted TEEs are analyzed with the three-level model of the sum-over-states expression. The rotational average of the second-order hyperpolarizability y (Eq.(49)) in the case of THG of a two-dimensional, planar molecule is... 

Figure 3. Integral (rotationally averaged) excitation cross sections (left panel) and vibrational state-to-state rate constants (right panel) for the LiH-He scattering system.

The orientational order parameter is the second Legendre polynomial of cos9 and, where the chemical shift dispersion is reduced by rotational averaging, is given by... 

This rotational averaging approach has the advantage of sampling the entire population distribution as opposed to the RIS model which assumes that only stable... 

Tam et al. [141] attempted to determine how reliable and accurate CC and DFT/ TDDFT calculations are for this conformationally flexible molecule. In addition, they explored the sensitivity of the chiroptical response to two different factors. One was the accuracy of the mole fractions, and another was how different were the ORs of individual rotamers calculated at different levels of theory. It was found that with DFT, at the B3LYP/aug-cc-pVDZ level, the optical rotations were overestimated while CC yielded better agreement with experiment [141, 142], The predicted gas phase optical rotation, averaged by CC or DFT mole fractions, were not in good agreement with either gas or solution phase experimental measurements. The DFT calculated optical rotations differed between 15 and 65% from experiment. 

Some examples which show the dynamic effects of alkyl and silyl substituents on barriers to rotation in allyllithium compounds 26 to 32 are listed in Table 7. These results were obtained from proton NMR line shape data. The procedure for compounds 26, 28, 29 and 32, which exhibited rotation around their C — CH2 bonds, is diagrammed by structure 33 in which hydrogens A, B and X are all nonequivalent and each couples to the others. Rotation averages the A and B shifts as well as the coupling constant between them and 3./(Ha.Hx) averages with 37(HB,HX), (Figure 15). At the same time, the Hx resonance... 

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