A Point Group describes all the symmetry operations that can be performed on a molecule that result in a conformation indistinguishable from the original. Point groups are used in Group Theory, the mathematical analysis of groups, to determine properties such as a molecule's molecular orbitals.

Assigning Point Groups

While a point group contains all of the symmetry operations that can be performed on a given molecule, it is not necessary to identify all of these operations to determine the molecule's overall point group. Instead, a molecule's point group can be determined by following a set of steps which analyze the presence (or absence) of particular symmetry elements.

Steps for assigning a molecule's point group:

Determine if the molecule is of high or low symmetry.

If not, find the highest order rotation axis, C_{n}.

Determine whether the molecule has any C_{2} axes perpendicular to the principal C_{n} axis. If so, then there are n such C_{2} axes, and the molecule is in the D set of point groups. If not, it is in either the C or S set of point groups.

Determine whether the molecule has a horizontal mirror plane (σ_{h}) perpendicular to the principal C_{n} axis. If so, the molecule is either in the C_{nh} or D_{nh} set of point groups.

Determine whether the molecule has a vertical mirror plane (σ_{v}) containing the principal C_{n} axis. If so, the molecule is either in the C_{nv} or D_{nd} set of point groups. If not, and if the molecule has n perpendicular C_{2} axes, then it is part of the D_{n} set of point groups.

Determine whether there is an improper rotation axis, S_{2n}, collinear with the principal C_{n} axis. If so, the molecule is in the S_{2n} point group. If not, the molecule is in the C_{n} point group.

Example \(\PageIndex{1}\)

Find the point group of benzene (C_{6}H_{6}).

Answer

Solution

1. Benzene is neither high nor low symmetry

2. Highest order rotation axis: C_{6}

3. There are 6 C_{2} axes perpendicular to the principal axis