Groups character tables

Then, if we look through all the point group character tables in Appendix A to see if any of the translational symmetry species is totally symmetric, it is apparent that molecules belonging to only the following point groups have a permanent dipole moment ... 

J. A. SalthouSe and M. J. Ware, Point Group Character Tables and Related Data, p. 29, Cambridge University Press, 1972. 

Vibrations may be decomposed into three orthogonal components Ta (a = x, y, z) in three directions. These displacements have the same symmetry properties as cartesian coordinates. Likewise, any rotation may be decomposed into components Ra. The i.r. spanned by translations and rotations must clearly follow the appropriate symmetry type of the point-group character table. In quantum formalism, a transition will be allowed only if the symmetry product of the initial and final-state wave functions contains the symmetry species of the operator appropriate to the transition process. Definition of the symmetry product will be explained in terms of a simple example. 

Once these double-group character tables are known, the procedure is exactly the same as that followed in the preceding sections when single character tables were used. Let us now tackle the problem of understanding the spectrum given in Figure 7.9 for Sm + ions in YAB. 

It has turned out that the most concise description of the symmetry species compatible with a molecular point group, that still includes enough iirformation for useful predictions, is the group character table. The character table of a group is a list of the traces of sets of matrices that form groups isomorphic to the group or to one of its subgroups. 

In contrast, the n ==> Jt transition has a ground-excited state direct product of B2 x Bj = A2 symmetry. The C2V s point group character table clearly shows that the electric dipole operator (i.e., its x, y, and z components in the molecule-fixed frame) has no component of A2 symmetry thus, light of no electric field orientation can induce this n ==> Jt transition. We thus say that the n ==> 7t transition is El forbidden (although it is Ml allowed). 

Determine the point groups in which ferroelectricity is possible. [Hint. Check point-group character tables to see in which point groups at least one of x, y, or z form a basis for I. ]... 

Salthouse JA, Ware MJ (1972) Point group character tables and related data. Cambridge University Press, London (UK)... 

A simple example provides useful illustration of these observations. Consider the Csv point symmetry of the ammonia molecule. The group character table is... 

Table B.4. Double group characters table for the Ta point group. The numbers before the symmetry operations correspond to the number of geometrically different axes or symmetry planes. Some of the operations of the double group belong to the same class as those of the original group. When more than one IR is indicated, the first one corresponds to the notation of Mulliken [11], the second one to Koster et al. [9] and the one in parentheses to [3]...

The 0/, point group character table and the O (and O double group) multiplication table are given in the Appendix. The irreps are labelled Ti, T2... herein [72], corresponding to Ai, A2. The subscript g or u is usually not included in the labelling of 4fN crystal field states since it is even (odd) for even (odd) N, respectively [73]. 

The Oh point group character table and the 0 (and 0 double group) multiplication table are given in Tables A1 and A2. 

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