Rotation matrix symmetry relations

The operators so and ss are compound tensor operators of rank zero (scalars) composed of vector (first-rank tensor) operators and matrix (second-rank tensor) operators. We will make use of this tensorial structure when it comes to selection rules for the magnetic interaction Hamiltonians and symmetry relations between their matrix elements. Similar considerations apply to the molecular rotation and hyperfine splitting interaction... 

If the group is rotational or helical and ij> is not 5-type, then the />, on each site become linear combinations of basis functions related by the rotation matrix of the appropriate angular momentum and the appropriate rotational or helical step angle [27]. It is traditional to use Cartesian-Gaussian orbital basis sets in quantum-chemical calculations [28], but solid-spherical-harmonic Gaussians [29] are best for symmetry adaption and matrix element evaluation. Including an extra factor of (-)M in the definition of the solid spherical harmonics [30]... 

Since we have axial symmetry, we can take the axis to be in the plane of H and the z axis, thus making Hy — 0. To diagonalize the energy matrix for the Zeeman energy, we shall rotate our spin-coordinate system such that the new z axis makes an angle a> with the z axis of the molecule. Spin operators in this new coordinate system are related to those in the molecular system by the equations... 

From the WET, Eq. [166], it is obvious that the reduced matrix element (RME) depends on the specific wave functions and the operator, whereas it is independent of magnetic quantum numbers m. The 3/ symbol depends only on rotational symmetry properties. It is related to the corresponding vector... 

For a linear molecule, the position of the symmetry axis (the molecule-fixed. z-axis) in space is specified by only two Euler angles, / and 7, which are respectively identical to the spherical polar coordinates 6 and (see Fig. 2.4). The third Euler angle, a, which specifies the orientation of the molecule-fixed x- and y-axes, is unaffected by molecular rotation but appears explicitly as an O- dependent phase factor in the rotational basis functions [Eq. (2.3.41)]. Cartesian coordinates in space- and molecule-fixed systems are related by the geometrical transformation represented by the 3x3 direction cosine matrix (Wilson et al., 1980, p. 286). The direction cosine matrix a given by Hougen (1970, p. 18) is obtained by setting a = 7t/2 (notation of Wilson et al, 1980 6 fi,4)=, x = oi 7t/2). The direction cosine matrix is expressed in terms of sines and cosines of 9 and 4>. Matrix elements (J M O la JMQ), evaluated in the JMQ) basis, of the direction cosines, are expressed in terms of the J, M, and quantum numbers. The direction cosine matrix elements of Hougen (1970, p. 31), Townes and Schawlow (1955, p. 96), and Table 2.1 assume the basis set definition derived from Eq. (2.3.40) and the phase choice a = 7t/2 ... 

There are three geometrically distinct variants of twin-related TlAl/TlAl interfaces which cam be described, using the notation of Fig. 1, as follows (1) Twin in which the [110] direction of the matrix is Emtiparallel to the [110] direction of the twin, (ii) 120 rotational fault for which the [110] direction of the matrix is p allel to the [101] direction of the twin, (lil) Pseudot win for which the (110] direction of the matrix is Emtiparallel to the [101] direction of the twin. It follows from the symmetry that structural variants with the (111) APB type displacement are equi ent to the undisplaced ones for the 12()" rotational fault Emd the pseudotwin. However, the twin with the APB type shift is not equivalent to the ordered twin. In fact, both coherent and shifted ordered twin boundEiries have been observed (Rlcolleau, et Ed. 1994) and it is the latter boundsuy which will be discussed in more details below. 

As already discussed, an important step towards reducing the experimental absorption intensities into molecular intensity parameters is die evaluation of the matrix Ps [Eq (3.3)]. Dipole moment derivatives with respect to symmetry coordinates nuty, however, contain contributions fiom molecular rotation for certain vibrations. These contributicms can be eliminated by using the following relation... 

If experimental data are treated, Vx is calculated from die relation (4.93). hi Eq. (4.93) Ps is the array of dipole nuMiient derivatives with respect to symmetry coordinates [Eq. (3.4)]. As underlined earlier, Vx refers to a molecule-fixed reference cocHrdinate system as also do the experimental dp /dQj derivatives ( = x, y, z). The array Vx may contain implicidy contributions originating from die compensatory molecular rotation in the case of polar molecules. These contributions are also present in the P matrix. RotatirmaUy corrected P may, however, be used to derive a fully rotation-free atomic polar tensor matrix. This is achieved through the equation... 

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