Tensors rotational averages

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... 

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 rotational averages for the odd order terms in a and y come from the first term in w and are ot = and y i yajj + liffi) which is the scalar part of the tensor. 

We now calculate the perturbation to the Zeeman field due to the quadrupolar interaction by means of average Hamiltonian theory.This is accomplished by transforming TYq to the Zeeman interaction frame and then applying the spherical tensor rotation properties to the spin elements 72,The resulting quadrupolar Hamiltonian TTq in the rotating frame is given by ... 

The quantities relevant to the rotationally averaged situation of randomly oriented species in solution or the gas phase must necessarily be invariants of the rotational symmetry. Accordingly, they must transform under the irreducible representations of the rotation group in three dimensions (without inversion), R3, just like the angular momentum functions of an atom. The polarisability, po, is a second-rank cartesian tensor and gives rise to three irreducible tensors (5J), (o), a(i),o(2), corresponding in rotational behaviour to the spherical harmonics, with / = 0,1,2 respectively. The components W, - / < m < /, of the irreducible tensors are given below. 

The molecular motion of redox couples within polymer-coated electrodes has recently been investigated by making use of both nitroxide spin probes and various cationic spin probes [94-97]. Spin probes, such as the nitroxide probe TEMPO (see Sect. 2.1.2) and its derivatives, have well-defined electrochemistry and their ESR spectra in viscous media exhibit effects due to incomplete rotational averaging of the g and hyperfine coupling constant tensors. Analysis of the spectra [98] allows deductions to be made concerning the molecular rotation. Such analysis has been performed for spin probes incorporated into various polymer films. 

Measurements of optical rotation are almost exclusively carried out on liquid samples (though gas-phase measurements recently became possible (Muller et al. 2000)) and we wUl, therefore, be concerned primarily with the rotational average of the G tensor, which is... 

The direct spin-spin couplings Djcl are purely anisotropic and vanish in the rotational averaging, while for the shielding and indirect spin-spin coupling tensors the averaging leads to what is commonly referred to as the nuclear shielding constants (Tk and the reduced indirect nudear spin-spin coupling constants Kkl-... 

Although the nuclear shielding parameter, a, is a tensor quantity, in an NMR experiment on a nonviscous sample only its average value is measured. Thus we need only to be concerned with rotationally averaged values of the terms in eq. 4 when considering high resolution NMR spectra. Nevertheless it should be mentioned that in experiments where molecules are aligned, such as those on solids and nematic phases, it is possible to measure the tensor components of a and thus to estimate its anisotropy. 

Figure 7-16 and 7-17) [71]. This rotation averages both of the chemical shift and quadrupolar interactions, such that the NMR powder pattern becomes axially symmetrical (t)q = X]q = 0), the principal axes of the chemical shift tensor and quadrupolar tensor coincide, and the Cq value decreases from 4.7 MHz in the crystalline P2 phase to 1.7 MHz in the liquid crystalline region (Figure 7-15 and Table 7-3). 

NMR studies of the motions of the aromatic amino acids of the fd bacteriophage protein were made, and the results are shown in Fig. 8.12. The experimental spectrum fits the calculated flip-averaged spectrum better than it fits the rotationally averaged spectrum. Therefore the rings are executing 180 flips in less than 1 ns, and probably in less than 0.01 ns. Additionally, the experimental spectrum is narrower than expected from the static chemical-shift tensor, and so additional molecular motion of the main chain is also involved. 

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