Molecular Symmetry and Group Theory

Lecture 2 — Point groups

Lecture Summary

Conventions for Axes

The axis of the highest order rotation is known as the principal axis. By convention the principal axis of a molecule is chosen to be the z-axis (assumed to be vertical). If several axes have the same order, the z-axis is the one passing through the most atoms. Thus for C₂H₄, the C₂ axis passing through both C atoms is the z-axis.
For planar molecules with the z-axis in the plane, the x-axis is perpendicular (^) to the plane.
If the z-axis is perpendicular (^) to the plane, the x-axis passes through the maximum number of atoms.
Mirror planes are related to the principal axis as follows:

σ_h — ‘horizontal’ planes perpendicular (^) to the principal axis
σ_v — ‘vertical’ planes // to the principal axis
σ_d — ‘dihedral’ planes // to the principal axis, bisecting C₂ axes

Some molecules e.g. XeF₄ have two types of dihedral planes: those that bisect the F—Xe—F bonds are labelled σ_d; those that contain Xe—F bonds σ_v.
The H₂O molecule has one C₂ axis (z-axis) and two mirror planes designated σ_xz and σ_yz.

Point Group Determination

The symmetry of any molecule, i.e. the type and number of symmetry elements it possesses, can be classified by a point group. A molecule can be assigned to its point group by following the ‘route map’ outlined below: