Representations and Character Tables

14 For more thorough and mathematical developments in terms or group theory, see the books listed in Footnote t. 

Translations (and p orbitals) along the x and z axes in the water molecule conform to different symmetry patterns than the one just developed for the y axis. When the , C2, 7 (xz), and aj yz) operations are applied, in that order, to a unit vector pointing in the +x direction, the labels +1, —1, +1, and — 1 are generated. A vector pointing in the +z direction will be unchanged by any of the symmetry operations and thus will be described by the set +1, + 1, +1, +1. 

Many of the symmetry properties of a point group, including its characteristic operations and irreducible representations, are conveniently displayed in an array known as a character table. The character table for C2. is17 

The column headings are the classes of symmetry operations for the group, and each row depicts one irreducible representation. The + I and —1 numbers, which 

11 Character tables for other point groups can be found in Appendix Di 

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

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 relationships among symmetry operations, matrix representations, reducible and irreducible representations, and character tables are conveniently illustrated in a flowchart, as shown for C2v symmetry in Table 4.8. 

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