Cartesian representation, tensor properties

We will use Cartesian sums and tensor products to build and decompose representations in Chapters 5 and 7. Tensor products are useful in combining different aspects of one particle. For instance, when we consider both the mobile and the spin properties of an electron (in Section 11.4) the state space is the tensor product of the mobile state space defined in Chapter 3)... 

For example the dipole polarizability given in (2.33) has spherical tensor components OQQdl) and a2K(11) The dipole-quadrupole polarizability (A .gY in Cartesian notation), which describes the quadrupole moment induced by an electric field or the dipole moment induced by an electric field gradient, has components 0 (12), 021(02) and 02k(12). The polarizabilities are even (g) or odd (u) under inversion according as 1+1 is even or odd. This information is then sufficient, with the help of Table 3, to determine the transformation properties in the molecular symmetry group. Any component which transforms according to the totally symmetric representation may have a non-zero value. 

Note that Equation [51] represents formally the tensorial relationship, while Equation [52] expresses this relation by the (Cartesian) components of P and E and some coefficients whereas Equation [53] states this relation by using Einstein s summation convention. Here, the coefficients are the components of the electric susceptibility tensor which is a tensor of rank 2. The tensor % is an example of what is usually called a property tensor or matter tensor. Strictly speaking, property tensors describe physical properties of the static crystal which belong to the totally symmetric irreducible representation of the relevant point group. Properties, however, that depend on vibrations of the crystal lattice are described by tensors which belong to the different irreducible representations. The corresponding tensors are then often designated as tensorial covariants. 

Here the entries TW, , , denote the Cartesian components of the tensor of rank [m] where i = 1, 2, 3 has to be taken into account. Equations [56] to [58] describe the transformation properties of an arbitrary tensor of rank [m] with respect to the orthogonal group 0(3, R), respectively. The matrix group Ml ] is the w-fold tensor representation of 0(3, R), where M g) is a real 3-dimensional matrix representation of g s 0(3, R), which implies that MM defines a real 3 -dimensional 0(3, R)-represen-tation. Moreover, note that in Equation [59] Einstein s summation convention has been used. Finally, one has to distinguish carefully between polar and axial tensors of rank [m] since their inherent transformation properties with respect to 0(3, R),... 

Atomic polar tensors, defined widi respect to an aibitraty Cartesian system, nn be transformed into quantities referring to a bond axis system using Eq. (4.22). If a molecule has sets of equivalent atmns the transfonnatiatomic polar tensors, provided the local refermce systems are chosen in a consistent way. With such representations it is clear that the number of independent atomic polar tensor elements is smaller for molecules with higher symmetiy. Decius and Mast [117] analyzed in detail the site symmetiy properties of atomic polar tensors exjnessed in tenns of bond axis system. The treatment covers molecules with sufficient symmetiy so that the directions of transitional dipole moments are uniquely determined. Molecules with symmetiy point grotq> G = C2h> Ci, C, Q and Ci are excluded from fire analysis. As already discussed, all elements of the Pq matrix for such molecules cannot be determined from experiment. 

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