Translation group character table

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 ... 

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. 

The character table shows that the rotations and translations belong, respectively, to the TXg and TXu representations. After deleting these, we obtain the following list of genuine normal modes, grouped according to the activities of their fundamentals ... 

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,. ...

For all the calculator files on the CDROM, this display comprises the standard character table for the particular group and identifies the irreducible symmetries of translations and rotations about the origin of the coordinate system for molecular stmctures with this point symmetry. 

Table 4. Table of vector sums for the nine possible displacement vectors appearing in the structure of most common phyllosilicates. The individual vectors (v) are represented by their respective characters v and the result of summation should be taken modulo primitive hexagonal cell cf. Fig. 4). These nine vectors form a translation group with vector addition as the group operation. 

One must then remove the irreducible representation which result from the three translational and three rotational degrees of freedom (two for a linear molecule). These can be identified from the character table of the molecular point group the translational degrees of freedom transform as the functions x, y, and z (denoted Tx, Ty, or x, y, z in the character tables) and the rotational degrees of freedom transform as the components of an axial vector (denoted Rx, Ry, and R in the character tables). The... 

The character table allows identifying the active infrared modes. These modes are those with nonzero translation represented by T in the character table. For the C2v point group, the A and B modes are active in the infrared spectroscopy. Adding to this analysis the surface selection rule, only those modes with translation in the direction perpendicular to the surface will be active. Considering that the surface is included in the xy plane, only the Ai modes can be active, since only the Ai have translation in the z direction. The Ai mode is the symmetric species with respect to the rotation about the symmetry axis, as described above. Therefore, for the adsorbed N03 ion, only the symmetric modes will be active in the infrared spectroscopy. 

It is the purpose of this appendix to derive and present in convenient form tables of the characters for the molecular symmetry groups described in Chap. 5. (Certain additional information, such as the species (irreducible representations) of the molecular translational and rotational coordinates, and of the dipole moment and the polarizability, will also be indicated. Following the character tables proper, some information on the resolution of direct products will be given. The latter is useful in determining selection rules for overtone and combination transitions (see Chap. 7). Finally, the correlation tables mentioned in Chaps. 6 and 8 will be given. 

Table 17.1 A fragment of the character table of a one-dimensional translation group T (not to be confused with the point group T). That for a three-dimensional group would be superficially similar. For the latter case the k and t would be vectors (a direction has to be specified in the three-dimensional case) and so they have been given as vectors in the table. The complex exponents indicate travelling waves (see the text) because the power of the exponentials must be (complex) numbers and t is a vector in real space, k must be in reciprocal space in order that the implied dot (scalar) products in the exponents (k-t) be dimensionless. The t,. .. at the...

There are three translations and three rotations of the molecule as if it were a rigid body. For any molecule in the point group, the rigid body motions will have the same irreducible representations. In the standard character tables of Appendix 12 the symbols x, y, z andi , Ry, are written in the rightmost columns and can be used to identify the representations for rigid-body movement and rotation respectively. So, most of the time, it is just a matter of referring to the character table to find the irreducible representations that should be removed and so isolate the vibrational mode symbols. 

The factor-group analysis predicts a total of 98 lattice-vibration modes however, the character table for the point group Oh shows that the translation operations are of symmetry species Fiu only. One acoustic mode will also be found in this species and will be infrared-inactive. Therefore, this analysis predicts seventeen infrared-active lattice modes. In table 29.17, ten frequencies were listed for the RIG series. Twelve frequencies were listed for the RGaG series (table 29.19), and fifteen frequencies were listed for the RAG series (table 29.18). They suggest that the powder-transmission method does not yield all of the infrared-active modes. 

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