5.2: Point Groups

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Introduction

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 C₂ axes perpendicular to the principal C_n axis. If so, then there are n such C₂ 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 C₂ 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.

Figure \(\PageIndex{1}\): Decision tree for determining a molecule's point group (CC-BY-NC-SA; Kathryn Haas)

Example \(\PageIndex{1}\)

Find the point group of benzene (C₆H₆).

Answer

Solution

1. Benzene is neither high nor low symmetry

2. Highest order rotation axis: C₆

3. There are 6 C₂ axes perpendicular to the principal axis

4. There is a horizontal mirror plane (σ_h)

Benzene is in the D_6h point group.

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Introduction

Assigning Point Groups

Steps for assigning a molecule's point group:

Example \(\PageIndex{1}\)

Solution

See Also