Groups , and character tables

SymApps converts 2D structures From the ChemWindow drawing program into 3D representations with the help of a modified MM2 force field (see Section 7.2). Besides basic visualization tools such as display styles, perspective views, and light source adjustments, the module additionally provides calculations of bond lengths, angles, etc, Moreover, point groups and character tables can be determined. Animations of spinning movements and symmetry operations can also he created and saved as movie files (. avi). 

F. A. Cotton, Chemical Applications of Group Theory, Third Edition, Wiley -Interscience, New York, 1990 M. Orchin, H. H. Jaffe, Symmetry. Point Groups, and Character Tables I, Symmetry Operations and Their Importance for Chemical Problems. J. Chem. Educ. 1970, 47, 372-377. 

Find the reducible representation for the vectors, using the appropriate group and character table, and find the irreducible representations that combine to form the reducible representation. 

This appendix briefly summarizes molecular symmetry (symmetry elements, point groups and character tables) of a molecule so as to facilitate readers understanding of the molecular structures and molecular orbitals presented in this chapter. We use some speciflc examples such as NH3 with Csv symmetry, CH3 radical with >3 symmetry (see Fig. 5.18) and c-C4Fg radical with D4h symmetry (Fig. 5.3) to introduce and illustrate some important aspects of the molecular symmetry. 

Acetylene (HC=CH) belongs to the point group whose character table is given in Table A.37 in Appendix A, and its vibrations are illustrated in Figure 6.20. Since V3 is a vibration and T T ) = 2"+, the 3q transition is allowed and the transition moment is polarized along the z axis. Similarly, since Vj is a vibration, the 5q transition is allowed with the transition moment in the xy plane. 

What has been mentioned up to now allows us to infer that the relevant information needed for a representation is given by the characters of its matrices. In fact, the full information for a given group is given by its character table. This table contains the character files of a particular set of representations the irreducible representations. Table 7.2 shows the character table of the Oh point group. A character table, such as Table 7.2, contains the irreducible representations (10 for the Oh group) and their characters, the classes (also 10 for the Oh group), and the set of basis functions. 

Table 12.3. Multiplication table, factor tables, and character tables for the point group C,.

Table 17.14. Jones symbols and character tables for InSb (space group 216) at A and X.

As an example, consider IR activity of the six normal vibrations of the NH3 molecule, which are classified into 2A and 2E species of C3V point group. The character table shows that fiz belongs to the A and the pair of (px, fiy) belongs to the E species. Thus, all six normal vibrations are IR-active. 

Next, we discuss representations and character tables. A group of order h can be represented by h matrices, each of dimensions hxh. Flowever, there exist so-called irreducible representations for each group, which are block-diagonal submatrices, which span the space. Table 7.6 presents the character tables for all 32 crystallographic point groups, and some other groups as well. Table 7.7 is an abbreviated form of Table 7.6. 

The numbers in the table, the characters, detail the effect of the symmetry operation at the top of the colurrm on each representation labelled at the front of the row. The mirror plane that contains the H2O molecule, a (xz), leaves an orbital of bi symmetry unchanged while a Ci operation on the same basis changes the sign of the wavefimction (orbital representations are always written in the lower case). An orbital is said to span an irreducible representation when its response upon operation by each symmetry element reproduces the same characters in the row for that irreducible representation. For atoms that fall on the central point of the point group, the character table lists the atomic orbital subscripts (e.g. x, y, z as p , Pj, p ) at the end of the row of the irreducible representation that the orbital spans. A central s orbital always spans the totally synunetric representation (aU characters = 1). For the central oxygen atom in H2O, the 2s orbital spans ai and the 2px, 2py, and 2p span the bi, b2, and ai representations, respectively (see (25)). If two or more atoms are synunetry equivalent such as the H atoms in H2O, the orbitals must be combined to form symmetry adapted hnear combinations (SALCs) before mixing with fimctions from other atoms. A handy mathematical tool, the projection operator, derives the functions that form the SALCs for the hydrogen atoms. 

Three of the representations for C2 , labeled Aj, Bj, and B2 below, have been determined so far. The fourth, called A2, can be found by using the properties of a group described in Table 4-7. A complete set of irreducible representations for a point group is called the character table for that group. The character table for each point group is unique character tables for the common point groups are included in Appendix C. 

Groups (symmetries), and character tables. AI3-A20 Group theory, and symmetry,... 

Symmetry Elements and Symmetry Operations 46 Point Groups and Molecular Symmetry 53 Irreducible Representations and Character Tables 59 Uses of Point Group Symmetry 63 Crystallography 74... 

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