Wigner rotation matrix elements

P indicates that the components are those for the principal axis system (PAS) of the tensor. The terms o(QpL(f)) are Wigner rotation matrix elements. They are functions of the set of Euler angles, Qpf (f), which relates the PAS of the chemical shift to the laboratory frame. Due to MAS, these angles are time dependent. A full treatment of the orientation dependence of the chemical shift requires the transformation between several different reference frames. 

Here D, is a second rank Wigner rotation matrix element and... 

The line shape of the ESR spectrum can be calculated using the spin Hamiltonian H (f) for the nitroxide radical. The time dependence of the spin Hamiltonian describes the orientation-dependent part which contains terms characterizing the anisotropy of the A- and g-tensors, and terms characterizing the rotational reorientation of the free radical as a classic stochastic process using the Wigner rotation matrix elements. A comprehensive theory of ESR spectra of nitroxides has been described step-by-step in contributions to two monographs and papers cited therein. Computer programs suitable for calculation of theoretical line shapes of ESR spectra measured in continuous wave (CW) and pulse experiments have become indispensable tools for practical applications. The ESR spectra of nitroxides are also described in chapters 1 and 3 in this volume. 

Molecular orientations are completely specified by the Euler angles Q (= a,/3,7), which relate the laboratory fixed frame to a molecular fixed frame. Any orientational dependent quantity of a molecule can be expressed in terms of the Wigner rotation matrix elements As seen in the... 

The Wigner rotation matrix elements are related to the modified (or normalized) spherical harmonics by... 

RE field along an axis of the rotating frame C = spin-rotation interaction tensor (Q) = reduced Wigner rotation matrix elements ... 

This latter form is more useful for our subsequent discussion. Based on the eigenstates JM) and JK) of a linear rigid rotor, our convention for the Wigner rotation matrix element is [36]... 

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