# 5.2: Point Groups

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- Contributed by Kathryn Haas
- Associate Professor (Chemistry) at Saint Mary's College, Notre Dame, IN

*Click here to see a lecture on this topic*.

## 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 if 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 if 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 if 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 if 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**-

1. Benzene is neither high or low symmetry**Solution** - 2. Highest order rotation axis: C
_{6}3. There are 6 C

_{2}axes perpendicular to the principal axis4. There is a horizontal mirror plane (σ

_{h})Benzene is in the D

_{6h}point group.

## See Also