Symmetry in Chemistry - Group Theory

Group Theory is one of the most powerful mathematical tools used in Quantum Chemistry and Spectroscopy. It allows the user to predict, interpret, rationalize, and often simplify complex theory and data.

At its heart is the fact that the Set of Operations associated with the Symmetry Elements of a molecule constitute a mathematical set called a Group. This allows the application of the mathematical theorems associated with such groups to the Symmetry Operations.

All Symmetry Operations associated with isolated molecules can be characterized as Rotations:

(a) Proper Rotations: C_n^k ; k = 1,......, n
When k = n, C_n^k = E, the Identity Operation
n indicates a rotation of 360/n where n = 1,....
(b) Improper Rotations: S_n^k , k = 1,....., n
When k = 1, n = 1 S_n^k = s , Reflection Operation
When k = 1, n = 2 S_n^k = i , Inversion Operation

In general practice we distinguish Five types of operation:

(i) E, Identity Operation
(ii) C_n^k , Proper Rotation about an axis
(iii) s, Reflection through a plane
(iv) i, Inversion through a center
(v) S_n^k, Rotation about an an axis followed by reflection through a plane perpendicular to that axis.

Each of these Symmetry Operations is associated with a Symmetry Element which is a point, a line, or a plane about which the operation is performed such that the molecule's orientation and position before and after the operation are indistinguishable.

The Symmetry Elements associated with a molecule are:

(i) A Proper Axis of Rotation: C_n where n = 1,....
This implies n-fold rotational symmetry about the axis.
(ii) A Plane of Reflection: s
This implies bilateral symmetry about the plane.
These planes are further classified as:
s_h - Horizontal Plane which is perpendicular to the Principal Axis of Rotation (i.e. Axis with highest value of n). If no principal axis exists s_h is defined as the molecular plane.
s_v or s_d - Vertical Plane which contains the Principal Axis of Rotation and is perpendicular to a s_h plane, if it exists. When both s_v and s_d planes are present, the s_v planes contain the greater number of atoms, the s_d planes contain bond angle bisectors. If only one type of vertical plane is present, s_v or s_d may be used depending on the total symmetry of the molecule.
(iii) A Center of Inversion - i
This is a central point through which all C_n and s elements must pass. If no such common point exists there is no center of symmetry.
(iv) Improper Axis: S_n
This is made up of two parts: C_n and s_h both of which may or may not be true symmetry elements of the molecule. If both the C_n and the s_h are present then S_n must also exist. The following relations are helpful in this regard:
(a) If n is even, S_nⁿ = E
(b) If n is odd, S_nⁿ = s and S_n²ⁿ = E
(c) If m is even, S_n^m = C_n^m when m < n
S_n^m = C_n^m-n when m > n
(d) If S_n with even n exists then C_n/2 exists.
(e) If S_n with odd n exists then both C_n and s perpendicular to C_n exist.

The key to applying Group Theory is to be able to identify the "Point Group" of the molecule i.e. its characteristic set of Symmetry Operations. The possible Symmetry Operations associated with a molecule are determined by the Symmetry Elements possessed by that molecule. Therefore the first step in applying Group Theory to molecular properties is to identify the complete set of Symmetry Elements possessed by the molecule. This requires the individual to visually identify the elements of symmetry in a 3-dimensional object. Experience has shown that this is often the most difficult step for a beginner.

In the following we will use "GIF" images, appropriately annotated to help the beginner become competent at identifying the symmetry elements in a molecule. The user can also access the Molecular Visualization Program utilizing "RASMOL v 2.5" to view 3-Dimensional images of the molecules.

Molecules can be categorized as: