Symmetry Properties of Atomic Polar Tensors

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 transfonnati Mi (4.22) will produce identical atomic 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. 

The site symmetry of a bond containing a representative atom of an equivaloit set 0 is a subgroup of die molecular point group G. If die representative atom is designated by a the respective polar tensor will have nine elements of the type 3p /9Xr ( ) where n = x, y, z. The following rule determines the structure of die 3 3 tensor if die p component and coordinate belong to the same symmetiy species of the site subgroup, the respective derivatives will differ from zero. In all other cases the elements will vanish. 

Decius and Mast [117] proposed a convenient notation to express die non-vanishing elements of atomic polar tensors. An AB4 (T ) molecule will have two types of atomic polar tensors for the two representative types of atoms A and B. The coordinate axes orientation is as shown in Fig. 4.1. Atom A is invariant under the full point group Tj. The non vanishing elements of the atomic polar tensors will be associated widi the triply degenerate symmetry class F2 and, also, dp /dx-x = dpyidyx = dpj/dzx elements of will be zero. The structure of the tensor will, 

The atom B situated along an AB bond axis widi site bond symmetry C3V will, in the present treatment, be the atom along the z direction (Fig. 4.1). The character table for T  

Character table and symmetry species for Cartesian coordinates fw T molecules 

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