Rotation matrix, Wigner

XII. The Adiabatic-to-Diabatic Transformation Matrix and the Wigner Rotation Matrix... 

Xn. THE ADIABATIC-TO-DIABATIC TRANSFORMATION MATRIX AND THE WIGNER ROTATION MATRIX... 

The obvious way to form a similarity between the Wigner rotation matrix and the adiabatic-to-diabatic transformation mabix defined in Eqs. (28) is to consider the (unbreakable) multidegeneracy case that is based, just like Wigner rotation matrix, on a single axis of rotation. For this sake, we consider the particular set of T matrices as defined in Eq. (51) and derive the relevant adiabatic-to-diabatic transfonnation matrices. In what follows, the degree of similarity between the two types of matrices will be presented for three special cases, namely, the two-state case which in Wigner s notation is the case, j =, the tri-state case (i.e.,7 = 1) and the tetra-state case (i.e.,7 = ). 

It is expected that for a certain choice of paiameters (that define the x matrix) the adiabatic-to-diabatic transformation matrix becomes identical to the corresponding Wigner rotation matrix. To see the connection, we substitute Eq. (51) in Eq. (28) and assume A( o) to be the unity matrix. 

C2H-molecule (1,2) and (2,3) conical intersections, 111-112 H3 molecule, 104-109 Wigner rotation matrix and, 89-92 Yang-Mills field, 203-205 Aharonov-Anandan phase, properties, 209 Aharonov-Bohm effect. See Geometric phase effect... 

As previously, the B pr) and B pr) tensors can be expressed in terms of Bkn in P frame using the Wigner rotation matrix ... 

The ZFS is assumed to be cylindrically symmetric (only the /q component is different from zero) and of constant magnitude. The static part of the Hzfs is obtained by averaging the Wigner rotation matrix Dq q[ pm(0] over the anisotropic distribution function, Pip pj. The principal axis of the static ZFS is, in addition, assumed to coincide with the dipole-dipole IS) axis. Eq. (48) becomes equivalent to Eq. (42), with the /q component scaled by Z)q q[ 2pm(0] The transient part of the Hzfs can be expressed in several ways, the simplest being 92) ... 

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. 

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