# 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:
1. Determine if the molecule is of high or low symmetry.
2. If not, find the highest order rotation axis, Cn.
3. Determine whether the molecule has any C2 axes perpendicular to the principal Cn 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.
4. Determine whether the molecule has a horizontal mirror plane (σh) perpendicular to the principal Cn axis. If so, the molecule is either in the Cnh or Dnh set of point groups.
5. Determine whether the molecule has a vertical mirror plane (σv) containing the principal Cn axis. If so, the molecule is either in the Cnv or Dnd set of point groups. If not, and if the molecule has n perpendicular C2 axes, then it is part of the Dn set of point groups.
6. Determine whether there is an improper rotation axis, S2n, collinear with the principal Cn axis. If so, the molecule is in the S2n point group. If not, the molecule is in the Cn point group.
##### Example $$\PageIndex{1}$$

Find the point group of benzene (C6H6).

###### 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 (σh)

Benzene is in the D6h point group.