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However, it would be too lengthy and beyond the scope of this course to cover this material rigorously.

|Godfrey H. Hardy1 In the preceding chapter, we proved the Great Orthogonality Theorem, which is a statement about the orthogonality between the matrix ele-ments corresponding to difierent irreducible representations of a group.

Character Tables •Character tables contain, in a highly symbolic form, information about how something of interest (an bond, an orbital, etc.) Character Tables List of the complete set of irreducible representations (rows) and symmetry classes (columns) of a point group. Understanding using matrix; References; Practice; Every molecule has a point group associated with it, which are assigned by a set for rules (explained by Group theory). •Each point group has a unique character table, which is organized into a matrix. Character tables for common point groups are given in Appendix B. C2h EC2 i σh linear quadratic Ag 11 1 1R z x2, y 2, z , xy Bg 1-1 1 -1R x, R y xz, yz Au 1 1 -1 -1 z Bu 1-1 -1 1x, y irreducible representations This lists the character tables for the more common molecular point groups used in the study of molecular symmetry.These tables are based on the group-theoretical treatment of the symmetry operations present in common molecules, and are useful in molecular spectroscopy and quantum chemistry.Information regarding the use of the tables, as well as more extensive lists of them, can be … Once the point group of a molecule is known, we can begin to make use of character tables. Character Tables: 1 The Groups C1, Cs, Ci 3 •Column headings are the symmetry operations , which are grouped The group C nv.A molecule belongs to the group C nv if in addition to the identity E and a C n axis, it has n vertical mirror planes σ v.Examples: H 2O molecule belongs to the C 2v group as it has the symmetry elements E, C 2, and two vertical mirror planes which are called σ v and σ0 v. The NH 3 molecule belongs to the C 3v group as it has the symmetry elements E, C If the symmetry label of a normal mode corresponds to products of x, y, or z (such as x 2 or yz) then the fundamental transition for this normal mode will be Raman active.
A simple way to determine the characters of a representation. Look at the character table for the point group of the molecule. If you know the point group of the molecule and the symmetry labels for the normal modes, then group theory makes it easy to predict which normal modes will be IR and/or Raman active. The table below describes their characteristic symmetry operations. Look at the character table for the point group of the molecule. Luckily, when each basis function transforms as a 1D irreducible representation (which is true in many cases of … Introduction to Character Tables The Character Table for C2v What Makes Up a Character Table Character tables contain information about how functions transform in response to the operations of the group Five parts of a character table 1 At the upper left is the symbol for the point group 2 The top row shows the operations of the point group, High symmetry point groups include the T d, O h, I h, C ∞v, and D ∞h groups.

As an example, we will look at the character table for the C 2v point group. Character Table.
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