Tensor irreducible, spherical

Table 1 NMR interactions in irreducible spherical tensor form ...

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... 

A spherical tensor is called irreducible spherical tensor if its (2m + 1) components transform under a rotation R of the coordinate system according to... 

By using the multipole expansion, we in fact replace the exact radial expansion coefficients A (7 ) in Eq. (1-124) by the approximate coefficients A poi(-K), which are power series in R 1. Closed expressions for the latter have been given149 161 in terms of the irreducible spherical tensors of multipole moments and polarizabilities. 

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 irreducible spherical tensor operator (usually abbreviated to spherical tensor operator) T T) of rank k is defined as an entity with (2k + 1) components, Tkp(T), which transform under rotations as... 

Since a second-rank cartesian tensor Tap transforms in the same way as the set of products uaVfj, it can also be expressed in terms of a scalar (which is the trace T,y(y), a vector (the three components of the antisymmetric tensor (1 /2 ) Tap — Tpaj), and a second-rank spherical tensor (the five components of the traceless, symmetric tensor, (I /2)(Ta/= + Tpa) - (1/3)J2Taa). The explicit irreducible spherical tensor components can be obtained from equations (5.114) to (5.118) simply by replacing u vp by T,/ . These results are collected in table 5.2. It often happens that these three spherical tensors with k = 0, 1 and 2 occur in real, physical situations. In any given situation, one or more of them may vanish for example, all the components of T1 are zero if the tensor is symmetric, Yap = Tpa. A well-known example of a second-rank spherical tensor is the electric quadrupole moment. Its components are defined by... 

In our own work on both diatomic and polyatomic molecules, we have found it valuable to have a summary of the most important results from irreducible spherical tensor algebra, particularly those relating to the evaluation of matrix elements in various angular momentum coupling schemes. We now provide a summary of those results detailed derivations are, of course, to be found in the main body of the text. 

To obtain the quadrupole Hamiltonian of a spin in a magnetic field the Hamiltonian needs to be transformed from the PAS to the LAB frame, keeping only those terms that commute with L. This is called truncation of a Hamiltonian and is only valid when Hq << Hz (the high field approximation). To perform the transformation it is much more convenient if second-rank irreducible spherical tensors are used. The Cartesian and spherical tensor elements (T) can be related (see Schmidt-Rohr and Spiess 1994 and Eq. 8, in Man 2000), with two of the more common elements being... 

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

Local interactions in NMR can be conveniently described as scalar products of spatial and spin irreducible spherical tensors ... 

With Ki a. spherical tensor of arbitrary rank, either spatial or spin, irreducible spherical tensors are defined by their commutation relations with linear Cartesian operators in the following manner... 

Using the generic coupling tensor representing the different interactions A, the generic Hamilton operator is expressed in the notation of irreducible spherical tensors... 

Here the Hamiltonian responsible for relaxation (A. = D for the dipole-dipole interaction) has been written in the notation of irreducible spherical tensors introduced in Section 3.1.2. For isotropic motion, the ensemble average reduces to... 

We begin our review by describing collision-induced light scattering mechanisms in the language of Cartesian tensors. We continue our description by the way of irreducible spherical tensors showing that the irreducible spherical tensors approach is indispensable for the spectral lineshape computation. 

Some time ago we contributed to the development of the irreducible spherical tensor multipolar theory of light scattering [13,28], According to Ref. 13, the M component of the A th-rank dipole-arbitrary order multipole linear polarizability of a pair of interacting molecules A and B reads as... 

When induction operators of high-order multipoles are taken into account intensity calculations tend to become very cumbersome [30,31]. We propose a relatively easy way of performing these calculations using the irreducible spherical tensor theory of multipole light scattering [e.g., Eqs. (6) and (7)] together with symbolic calculations of the Wigner coefficients by computer. 

Using Eq. (6) and decoupling procedures for irreducible spherical tensors [32], we easily derive the following general form of the autocorrelation function... 

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... 

These components transform under rotation like the spherical harmonic functions T]m. In general, an irreducible spherical tensor Tim transforms like the function Ylm. The spherical components of a second-rank tensor are again collected in Table 1.13. 

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