Symmetry and Group Theory
Symmetry is associated with the idea that certain geometrical transformations
leave an object unchanged. The geometric operations that do this for molecules
are called symmetry elements or symmetry operations. The set of all symmetry
elements associated with a molecule obeys the properties associated with
a mathematical group so the mathematics of group theory can be applied
to molecules. This math allows us to characterize and label some properties
of molecules such as molecular orbitals or spectroscopic transitions. In
this course, we will use group theory to label molecular orbitals, which
will be used to construct molecular orbital diagrams and understand bonding
in molecules.
4 classes of symmetry elements:
1. inversion
2. rotations
Point groups (1 + 2) 
3. translations
Space groups (1 + 2 + 3) 
4. time inversion
Magnetic space groups (all 4) 
Symmetry Elements
1. Inversion i
The signs of the Cartesian coordinates are made negative leaving the
object invariant in appearance (absolutely identical)
2. Rotations C_{n}
An angular motion of
clockwise (by convention) about an axis (the axis may be defined in the symmetry element)
special cases: 0^{o} rotation is given the symbol E and acts as the identity element in group theory
C_{¥} an infinitely small rotation  appropriate for linear molecules
For any given C_{n} operation, there are n1 different
rotations C_{n}, C_{n}^{2}, C_{n}^{3},
..., C_{n}^{n1}
3. mirrors or planes of symmetry
Reflection through a plane that leaves an object invariant
All mirrors are composite symmetry elements  combinations of i
and C_{2}
_{xy} = i×C_{2z}
_{v} = mirror containing
bonds (usually) _{d} = mirror between bonds (usually)
_{h} = mirror perpendicular to highest rotation axis
4. Improper rotations S_{n}
A rotation followed by a mirror perpendicular to the rotation: S_{n}
= C_{n}×_{h}
Note that S_{2} = i and S_{1}
=
Point Groups
The set of all symmetry elements associated with a molecule is called the point group.
Each point group can be generated from a smaller set of symmetry operations known as
generators; we only need to be able to identify the generators in order to identify the point
group. Knowing the point group is essential to using group theory but also is an indicator
of molecular structure!
Summary flow chart for finding Point Groups:
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) 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
_{G} = character of the Rth symmetry
operation for the irred. rep
Example: 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)