# 12: Group Theory - Exploiting Symmetry

- Page ID
- 204003

- 12.2: The Symmetry of Molecules
- A symmetry operation is an action that leaves an object looking the same after it has been carried out. For example, if we take a molecule of water and rotate it by 180° about an axis passing through the central O atom (between the two H atoms) it will look the same as before. Each symmetry operation has a corresponding symmetry element, which is the axis, plane, line or point with respect to which the symmetry operation is carried out.

- 12.3: Symmetry Operations Define Groups
- Now that we have explored some of the properties of symmetry operations and elements and their behavior within point groups, we are ready to introduce the formal mathematical definition of a group.

- 12.4: Symmetry Operations can be Represented by Matrices
- Matrices can be used to map one set of coordinates or functions onto another set. Matrices used for this purpose are called transformation matrices. In group theory, we can use transformation matrices to carry out the various symmetry operations discussed previously. As a simple example, we will investigate the matrices we would use to carry out some of these symmetry operations on a vector in 2D space (x,y).

- 12.6: Character Tables
- The trace of a matrix representative Γ(g) is usually referred to as the character of the representation under the symmetry operation g . The characters of a matrix representation are often more useful than the matrix representatives themselves. Characters have several important properties. The character of a symmetry operation is invariant under a similarity transform Symmetry operations belonging to the same class have the same character in a given representation.