Irreducible Cartesian tensors

Coope JAR, Sinder RF, McCourt FR (1965) Irreducible Cartesian tensors. J Chem Phys 43 2269-2275... 

Coope JAR (1970) Irreducible Cartesian tensors. III. Clebsch-Gordan reduction. J Math Phys 11 1591-1612... 

The generalization of Schwartz s result to D dimensions requires the use of angular momentum algebra in D dimensions which can be developed using the method of irreducible Cartesian tensors. 

It is readily seen that the 2" rank Cartesian tensor that we have derived in equation (7) is symmetric and traceless and so Fj j is an irreducible Cartesian tensor of symmetry type 11 2. We have already discussed how to generate a tensor with a specified symmetry type from a given tensor using Young operators, but the question remains how to project out the traceless component of a tensor The example... 

Table 17.3. Irreducible representations of the point group Oh = 0 <8> C, and their Cartesian tensor bases (forprincipal axis along ky).

Table 9 Irreducible Cartesian Compound Tensor Operators Resulting from a Product of Two First-Rank Tensor Operators 3 and jT ...

The operator V,, b is physically interpreted as representing the interaction of the instantaneous moment with respect to center A with the instantaneous 2>B moment with respect to center B and can be expressed in terms of irreducible spherical or reducible Cartesian tensor operators of multipole moments. The operator V,, can be written 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... 

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... 

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... 

The quantities relevant to the rotationally averaged situation of randomly oriented species in solution or the gas phase must necessarily be invariants of the rotational symmetry. Accordingly, they must transform under the irreducible representations of the rotation group in three dimensions (without inversion), R3, just like the angular momentum functions of an atom. The polarisability, po, is a second-rank cartesian tensor and gives rise to three irreducible tensors (5J), (o), a(i),o(2), corresponding in rotational behaviour to the spherical harmonics, with / = 0,1,2 respectively. The components W, - / < m < /, of the irreducible tensors are given below. 

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. 

Ordinary Cartesian tensors can be reduced into irreducible tensorial sets, example, the second-rank Cartesian tensor T... 

Components of a second-rank tensor Irreducible Cartesian components... 

Although we have introduced irreducible spherical tensors, we do not yet have a formalism which admits ready generalization to D dimensions. This can be accomplished by transforming the spherical tensor to a Cartesian tensor. The spherical components of a tensor are related by a unitary transformation to Cartesian components [11,12,13]. For example consider a spherical harmonic of / = 1 (a spherical tensor of rank 1) written as a three component vector... 

The orientation of a molecule with respect to a space-fixed direction can be described in several different ways. The formulation of the spin hamiltonian using irreducible tensor notation leads to a description of the degree of orientation in terms of the averages of Wigner rotation matrices. In much work using Cartesian tensors the molecular orientation is described in terms of products of direction cosines. The ordering matrix is then defined by... 

In concluding this section we explore the relationship between the anisotropic properties of a nematic and the orientational order parameters. It is convenient to do this using irreducible spherical tensors since they have particularly straightforward transformation properties under rotation, as we shall discover. We begin with the tensor components set in the molecular frame which are necessarily independent of the molecular orientation in the laboratory frame. For a tensor of rank L there are (2L + 1) components which are distinguished by the label m, taking values L, L - I,. .. -L + 1, -L these are denoted by T [2]. They are simply linear combinations of the components of the symmetric Cartesian tensor of the same rank. The components of the tensor in the laboratory frame are T, and are related to T by... 

Although the irreducible spherical tensor approach is valuable fix the definition of the (xdo ing tensors and for their manipulation under rotation it does not always provide a ready undostanding of the physical significance of the various components. This is sometimes available from a Cartesian representation of the ordering tensor. The most familiar example is the Saupe ordering matrix which represents the orientational ordering at the second rank level [8]. It is defined by... 

Comparing (2.3.34) and (2.2.40), we note that the three components of the spin-orbit operator are treated alike in the Cartesian form (2.2.40) but differently in the spin-tensor form (2.3.34). The spin-tensor representation (2.3.34), on the other hand, separates the spin-orbit operator into three terms, each of which produces a well-defined change in the spin projection. From the discussion in this section, we see that the singlet and triplet excitation operators (in Cartesian or spin-tensor form) allow for a compact representation of the second-quantization operators in the orbital basis. The coupling of more than two elementary operators to strings or linear combinations of strings that transform as irreducible spin tensor operators is described in Section 2.6.7. 

The Cartesian operators may be expressed through the components (q = -1, 0, +1) of the first-rank spherical irreducible tensor Lnamely,... 

Relation Between Irreducible Spherical and Cartesian Components of a Symmetric Tensor of Rank 2... 

In the case of a scalar field, the irreducible matrix D is a unit matrix, and drops out of. I1. For rotation through an angle S9t about the Cartesian axis ek, the rotational submatrix of the Lorentz matrix is given by Xkx = ()Hkekl]x], where el]k is the totally antisymmetric Levi-Civita tensor. For the one-electron Schrodinger field f, Noether s theorem defines three conserved components of a spatial axial vector,... 

As an example, consider the product of two arbitrary first-rank tensor operators 0 and It is nine-dimensional and can be reduced to a sum of compound irreducible tensor operators of ranks 2, 1, and 0, respectively. Operators of this type play a role in spin-spin coupling Hamiltonians. In terms of spherical and Cartesian components of 0 and J2, the resulting irreducible tensors are given in Tables 8 and 9, respectively.70... 

The first term is a tensor of rank zero involving only spin variables. It does not contribute to the multiplet splitting of an electronic state but yields only a (small) overall shift of the energy and is, henceforth, neglected. The operator 7 is a traceless (irreducible) second-rank tensor operator, the form of which in Cartesian components is... 

Figure 2.7-6 A Assignment of the Cartesian coordinate axes and the symmetry operations of a planar molecule of point group C2,.. B Character table, 1 symbol of the point group after Schoen-flies 2 international notation of the point group 3 symmetry species (irreducible representations) 4 symmetry operations 5 characters of the symmetry operations in the symmetry species +1 means symmetric, -1 antisymmetric 6 x, y, z assignment of the normal coordinates of the translations, direction of the change of the dipole moment by the infrared active vibrations, R, Ry, and R stand for rotations about the axes specified in the subscript 7 x, xy,. .. assign the Raman active species by the change of the components of the tensor of polarizability, aw, (Xxy,. ...

Let us use these selection rules for investigating the main features of the Rayleigh and pure rotational Raman scattering by spherical-top molecules in the lowest vibronic states (Ogurtsov et al., 1978). The polarizability tensors dif2i and can be expanded into components of irreducible tensor operators that in cubic groups transform as E, T2, and T, respectively. Here the behavior of the operators dir 71 with respect to time reversal 0 has to be taken into consideration. To do this, we use the explicit form of the operator djj(a>) in Cartesian coordinates ... 

A fundamental transition will be Raman active only if the normal mode involved belongs to the same representation as one or more of the components of the polarizability tensor of the molecule. For example for NH3 molecule the charactertable for C3v group is used to obtain the following irreducible representations for the quadratic and binary cartesian coordinates. 

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