Symmetry in Chemistry - Group Theory

Symmetry inChemistry - Group Theory

Group Theory is one of the most prevailing mathematical tools used in Quantum Chemistry and Spectroscopy. It permits the user to forecast, interpret, reduce, and often abridge complex theory and data.
At its heart is the fact that the Set of Operations related with the Symmetry Elements of a molecule comprise a mathematical set known as a Group. This permits the application of the mathematical theorems allied with such groups to the Symmetry Operations.

All Symmetry Operations related with isolated molecules can be distinguished as Rotations:

(a) Proper Rotations: C_n^k ; k = 1,......, n 
When k = n, C_n^k = E, the Identity Operation 
n designate 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 axis followed by indication through a plane perpendicular to that axis.

Each of these Symmetry Operations is linked with a Symmetry Element which is a point, a line, or a plane concerning which the operation is performed such that the molecule's orientation and position prior to and after the operation are interchangeable.