Matrices Wigner

The matiix as well as the matrix ate called the Wigner matrices and are the subject of this section. Note that if we are interested in finding a relation between the adiabatic-to-diabatic transformation matrix and Wigner s matrices, we should mainly concentrate on the matiix. Wigner derived a fomiula for... 

For the three cases studied in Section V.A, the similarity to the three corresponding Wigner matrices is achieved in the following way ... 

The main difference between the adiabatic-to-diabatic transformation and the Wigner matrices is that whereas the Wigner matiix is defined for an ordinary spatial coordinate the adiabatic-to-diabatic transformation matrix is defined for a rotation coordinate in a different space. 

M is the projection of J on the space-fixed z-axis, 0 its projection on the body-fixed z-axis, which is chosen here along the r vector. The D Ijq are Wigner matrices and are angular functions in the coupled BF representation. 

We assumed that the magnetic tensors appearing in the spin Hamiltonian and the diffusion tensor have the same principal axis system. The are the Wigner matrices of rank /. The nuclear spin operators are expanded over the set of (21 +1) matrices Af j, I being the nuclear angular moment, defined via... 

This transformation law is quite simple, and on it relies the main advantages of using spherical tensors in problems involving rotations. The Wigner matrices defined by Eq. (B.2) provide a set complete and orthogonal in the space of Euler angles, thereby making it possible to use them as a suitable expansion basis set. 

Let us note also that for spedal values of their indices the Wigner matrices... 

The 1/j.p terms in the above equations contain products of the Wigner matrix elements and thus may be expressed as a hnear arrangement (up to the fourth order) of Wigner matrices [5]. The frequency sphtting due to the second-order quadrupolar interaction may be written as a sum of three terms ... 

The Wigner matrices multiply just like the rotations themselves. There is a one-to-one correspondence between the Wigner matrices of index l and the rotations R. These matrices form a representation of the rotation group. In fact, since the 2/ + 1 spherical harmonics of order / form an invariant subspace of Hilbert space with respect to all rotations, it follows that the matrices D ,m(R) form a (21 + 1) dimensional irreducible representation of the rotation R. Explicit formulas for these matrices can be found in books on angular momentum (notably Edmunds, 1957). 

It follows from the representation theory of groups that the direct product of two irreducible representations can be decomposed into direct sum of irreducible representations of the same group. In case of the SO(3) group, the direct product of two Wigner-matrices can be decomposed into a direct sum of Wigner-matrices in the form... 

For application of the perturbation theory, it is necessary to transform this expression to the laboratory frame, where the dominant Hamiltonian Hz is proportional do the spin operator 7. When such transformation is conducted, using Wigner matrices and Euler angles [10], the resulting form for Hq in the laboratory frame, assuming axial symmetry for the EFG tensor (q = 0) and using the secular approximation to keep only terms that commute with Hz, is ... 

The Maier-Saupe theory assmnes high symmetry for molecules forming liquid crystals. In reahty, this is usually not the case and the theory has been extended [3.18] to lath-like molecules. The order parameter tensor S is given by Eq. (3.8) for a biaxial molecule in a uniaxial phase. In the principal axis x y z) system of 5, only two order parameters, Szz and D = Sxx — Syy, are needed, which are related to the Wigner matrices according to Eq. (2.43) ... 

Thus, the director motion is solely described by the Euler angle / ". When small angle director fluctuations are assumed (i.e., / " is small such that sin0 = ), the only non-zero reduced Wigner matrices (see Appendix A) are... 

Some basic properties of the Wigner matrices are summarized as follows ... 

The product of two Wigner matrices of different ranks can be expressed in terms of the Clebsch-Gordon series ... 

We may assiune that the protein is immersed in an isotropic medium, so the equilibrium distribution is independent of and the total potential is just the interaction potential between the two processes for which we take the following expansion over Wigner matrices ... 

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