Irreducible spherical tensor operators

As discussed in Ref. [1], we describe the rotation of the molecule by means of a molecule-fixed axis system xyz defined in terms of Eckart and Sayvetz conditions (see Ref. [1] and references therein). The orientation of the xyz axis system relative to the XYZ system is defined by the three standard Euler angles (6, (j), %) [1]. To simplify equation (4), we must first express the space-fixed dipole moment components (p,x> Mz) in this equation in terms of the components (p. py, p along the molecule-fixed axes. This transformation is most easily done by rewriting the dipole moment components in terms of so-called irreducible spherical tensor operators. In the notation in Ref. [3], the space-fixed irreducible tensor operators are... 

The same ideas can be readily extended to cover quantum mechanical operators. The formulation is slightly different because, although a transformation S turns a wave function spherical tensor operator) T T) of rank k is defined as an entity with (2k + 1) components, Tkp(T), which transform under rotations as... 

T ,k are the irreducible spherical tensor operators for the quadrupole interaction given by... 

The internal interactions, symbolically designated as X), which provides the wealth of information available from NMR, but which are also responsible for broadening of solid-state spectra of protons in solids may be written as a constant, Ca, times a product of coordinate space and spin space operators. In terms of the coordinate space irreducible spherical-polar tensor operators Rlm(0. < ). and the spin space irreducible spherical tensor operators... 

In recent years a more sophisticated approach using irreducible spherical tensor operators has been found advantageous [13,16-18]. Symmetric top matrix elements can be obtained by using the following extension of the Wigner-Eckart theorem to axially symmetric systems (within the Condon and Shortley phase convention)[16]... 

Philip and Kuchel described a graphical representation of the spin states of quadrupolar nuclei in NMR experiments. The spherical harmonics represent the irreducible spherical tensor operator basis set, and therefore any density matrix can be illustrated with computer graphics by converting it to a sum of... 

The averaging of nuclear spin Hamiltonians under rotations may be easily studied when it is expressed in terms of irreducible spherical tensor operators, TL,m and In liquid crystals, the main interest is in time averag-... 

Ri m will be non-zero only when L = 0 and 2. If i L,m is considered in its principal axis system, only components with m = 0, L are non-zero. The components of the irreducible spherical tensor operators will be denoted in their respective principal axis systems by which are given in Table 2.1 together with the constants C. Table 2.2 lists Ti m for each spin interaction. 

Table 2.1. The constants C and irreducible spherical tensor operators pL,m in a principal axis system.

The scalar product of 2 irreducible spherical tensor operators of rank k is defined by... 

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