CHM 501 Lecture

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 Cn

An angular motion of  clockwise (by convention) about an axis (the axis may be defined in the symmetry element)
special cases: 0o 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 Cn operation, there are n-1 different rotations Cn, Cn2, Cn3, ..., Cnn-1

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 C2
xy = i×C2z
v = mirror containing bonds (usually) d = mirror between bonds (usually)
h = mirror perpendicular to highest rotation axis

4. Improper rotations Sn

A rotation followed by a mirror perpendicular to the rotation: Sn = Cn×h
Note that S2 = i and S1 =

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


ng = 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

(E) = 1
( (Cn) = +1)
(E) = 1
( (Cn) = -1)
(E) = 2
(E) = 3
t (sometimes f)
(E) = 4
(E) = 5
If an inversion element exists in the group
(i) > 0
add a g subscript
(i) < 0
add a u subscript
If there is a mirror element but no i
() > 0
add a ' superscript
() < 0
add a " superscript

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



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

ng = 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 IF5


C4v point group
x2+y2, z2
(Rx, Ry)
(xz, yz)
Total Rep

px, py, pz


n(a1) = [(1)(3)(1) + (2)(1)(1) + (1)(-1)(1) + (2)(1)(1) +(2)(1)(1)]/8 = 1
n(a2) = [(1)(3)(1) + (2)(1)(1) + (1)(-1)(1) + (2)(1)(-1) +(2)(1)(-1)]/8 = 0
n(b1) = [(1)(3)(1) + (2)(1)(-1) + (1)(-1)(1) + (2)(1)(1) +(2)(1)(-1)]/8 = 0
n(b2) = [(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 a1 + e a1 is pz, e is (px, py) 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 px, py, pz; x2+y2+z2 tell us s; z2 , x2-y2, xy, xz, yz tell the d orbitals

This simplifies things immensely at times. (note : Rx, Ry, Rz 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

a1 + b1 + b2 + 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)