Symmetry and Group Theory

Definition of terms and procedures using Group Theory

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

χ = the character

R = symmetry element

Γ = 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

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_Γ = number of times the Γth 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

χ_R = character of the Rth symmetry operation for the total rep

χ_Γ = character of the Rth symmetry operation for the Γ irred. rep

Example

Find the irred. reps for the p orbitals on the I atom in IF₅

IF₅

C_4v point group

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 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 correspond to p_x, p_y, p_z; x²+y²+z² correspond to the s orbital; z² , x²-y², xy, xz, yz correspond to 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 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)