Classes: Two symmetry elements belong to the same class if one symmetry operation can be changed into a second symmetry operation by application of a third symmetry operation
Representations: these are the set of matrices that can be used to do the mathematics of group theory and label objects
There are two types of representations (reps): reducible and irreducible:
Reducible representation are linear combinations of irreducible reps
Irreducible reps can be used to generate groups, just as symmetry elements
# of irred. reps = # of classes of symmetry operations
We rarely need to use matrices but can work with only the character of a matrix. Characters are found by summing the diagonal elements of the representation matrix.
Order of a group = h = number of symmetry operations
The order of the group is related to characters and representations
or
n_{g} = the number of symmetry elements in the class
= the character
R = symmetry element
= irreducible representation
g = symmetry class index
Irreducible reps are orthogonal: (inner product)
These equations allow determination of all the characters associated with any irred. rep. without the need for generating any matrices.
Labeling of irred. reps.
Labels are determined by the nature of the character associated with
symmetry elements in the group; thus, the irred. rep tells us something
about the symmetry of the molecule
The character of the irred. rep associated with the E symmetry
is the dimensionality of the rep


( (C_{n}) = +1) 

( (C_{n}) = 1) 





















If more than one ired. rep. fits a set of criteria, then differentiate
by arbitrarily adding subscripts 1, 2, 3, ... ; one exception: the irred.
rep with all +1 characters (the totally symmetric representation) is always
a gets 1 as the subscript
Character Tables denote the relationship between symmetry elements (a
group), irreducible reps (also a group), and the characters. We can use
the properties of the classes, irred. reps, and symmetry operations to
generate characters or use the characters to go from symmetry operations
to irreducible reps.
Organization of Character Tables:
Point Group Symmetry Operations grouped by class irred reps
Characters
Character Tables are used to label objects (degrees of freedom, vibrations, orbitals, electronic states) with irred. reps.; the irred. reps. can be used to determine orbital overlaps, spectral selection rules, reactivity, etc.
Irred reps are found from red. reps. The characters for the red reps are found using symmetry elements: the symmetry operation from each class in the point group is applied to the set of objects under consideration. The character associated with the red rep for each class of symmetry element is the sum of the projection of the portion of the initial object remaining at the same position after application of the symmetry operation.
Once the red rep (also called the total representation) is found, it can be reduced to the linear combination of irreducible reps using the reduction formula:
n_{} = number of times the th irred rep appears in the total rep
h = the order of the group
n_{g} = number of operations in the symmetry class
g = class index of the point group
_{R} = character of the Rth symmetry operation for the total rep
_{} = character of the Rth symmetry operation for the irred. rep
Find the irred. reps for the p orbital on the I atom in IF_{5}
IF_{5}
C_{4v} point group
C_{4v} 






a_{1} 






x^{2}+y^{2}, z^{2} 
a_{2} 







b_{1} 





x^{2}y^{2}  
b_{2} 





xy  
e 





(R_{x}, R_{y}) 
(xz, yz) 
Total Rep
p_{x}, p_{y}, p_{z} 





n(a_{1}) = [(1)(3)(1) + (2)(1)(1)
+ (1)(1)(1) + (2)(1)(1) +(2)(1)(1)]/8
= 1
n(a_{2}) = [(1)(3)(1) + (2)(1)(1)
+ (1)(1)(1) + (2)(1)(1) +(2)(1)(1)]/8
= 0
n(b_{1}) = [(1)(3)(1) + (2)(1)(1)
+ (1)(1)(1) + (2)(1)(1) +(2)(1)(1)]/8
= 0
n(b_{2}) = [(1)(3)(1) + (2)(1)(1)
+ (1)(1)(1) + (2)(1)(1) +(2)(1)(1)]/8
= 0
n(e) = [(1)(3)(2) + (2)(1)(0) + (1)(1)(2)
+ (2)(1)(0) +(2)(1)(0)]/8 = 1
(numbers in bold are the class number, numbers in italics are the characters for the reducible representation, unhighlighted numbers are the characters for the irreducible representation found in the character table)
Thus, the p orbitals transform as a_{1} + e a_{1} is p_{z}, e is (p_{x}, p_{y}) degenerate pair
Note in all character tables there are two columns to the right that have things like, x, y, x, xy, etc
These are called basis functions (basis functions can be used to generate irred. reps). More importantly, the basis functions tell us how orbitals located at the center of the symmetry group transform: x, y, z tell us p_{x}, p_{y}, p_{z}; x^{2}+y^{2}+z^{2} tell us s; z^{2} , x^{2}y^{2}, xy, xz, yz tell the d orbitals
This simplifies things immensely at times. (note : R_{x}, R_{y}, R_{z} denote the irred reps for the rotational degrees of freedom)
Thus, we can read directly from the character table that the d orbitals on I transform as
a_{1} + b_{1} + b_{2} + e
We can make no comments directly from the table about the F orbitals
because they are not located at the center point of the symmetry group
(in a molecule like benzene, we can make no deductions of how the orbitals
transform because none of the atoms lie at the center of symmetry)