Euler angles Wigner rotation matrices

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. 

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... 

Here is the Wigner rotation matrix, and denotes the set of Euler angles for overall rotation that connects the laboratory and molecular frames. Equations (2) and (5) then yield ... 

Other than linear molecules. If molecules of symmetry other than axial are considered, it is not possible to describe their orientation by an azimuthal and polar angle, Euler angles, Q = a pi, y and Wigner rotation matrices are then needed as Eq. 4.8 suggests. In that case, besides the set of parameters X, X2, A, L that has been used for linear molecules, two new parameters, u, with i = 1,2, occur that enter through the rotation matrices. These must be chosen so that the dipole moment is invariant under any rotation belonging to the molecular symmetry group. The rotation matrix is expressed as a linear combination of such... 

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