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Our example will the O cation, which has point group
symmetry, as shown in Figure 1. The Cartesian
coordinates of this molecule are given in Table
1, and the character
table for is given in Table 2. From the table,
we can see that there are eight distinct symmetry operations for this
point group: the identity (), three different rotation axes, a
center of inversion (), and three mirror planes (). You can
easily verify that O possesses all of these symmetry properties.
These symmetry operations form the columns of the table. There are
also eight rows, or irreducible representations, labeled ,
, , . The 1's and 1's in the table indicate
whether the irreducible representation (or irrep, for short) is symmetric
or antisymmetric for that symmetry operation.
Figure 1:
O Cation

Table 1:
O Cartesian Coordinates

Table 2:
Character Table for Point Group












1 
1 
1 
1 
1 
1 
1 
1 



1 
1 
1 
1 
1 
1 
1 
1 



1 
1 
1 
1 
1 
1 
1 
1 



1 
1 
1 
1 
1 
1 
1 
1 



1 
1 
1 
1 
1 
1 
1 
1 



1 
1 
1 
1 
1 
1 
1 
1 



1 
1 
1 
1 
1 
1 
1 
1 



1 
1 
1 
1 
1 
1 
1 
1 


Next: How Many Vibrational Modes
Up: grpthyvib
Previous: Introduction
David Sherrill
20100720