# 1.2: Molecular Point Groups

- Page ID
- 183432

## 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.

- 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}).

**Solution**-
- Highest order rotation axis: C
_{6} - Benzene is neither high or low symmetry
- There are 6 C
_{2}axes perpendicular to the principal axis - There is a horizontal mirror plane (σ
_{h})

Benzene is in the D

_{6h}point group. - Highest order rotation axis: C

## Low Symmetry Point Groups

Low symmetry point groups include the C_{1}, C_{s}, and C_{i} groups

Group | Description | Example |
---|---|---|

C_{1} |
only the identity operation (E) | CHFClBr |

C_{s} |
only the identity operation (E) and one mirror plane | C_{2}H_{2}ClBr |

C_{i} |
only the identity operation (E) and a center of inversion (i) | C_{2}H_{2}Cl_{2}Br |

## High Symmetry Point Groups

High symmetry point groups include the T_{d}, O_{h}, I_{h}, C_{∞v}, and D_{∞h} groups. The table below describes their characteristic symmetry operations. The full set of symmetry operations included in the point group is described in the corresponding character table.

Group | Description | Example |
---|---|---|

C_{∞v} |
linear molecule with an infinite number of rotation axes and vertical mirror planes (σ_{v}) |
HBr |

D_{∞h} |
linear molecule with an infinite number of rotation axes, vertical mirror planes (σ_{v}), perpendicular C_{2} axes, a horizontal mirror plane (σ_{h}), and an inversion center (i) |
CO_{2} |

T_{d} |
typically have tetrahedral geometry, with 4 C_{4} axes, 3 C_{2} axes, 3 S_{4} axes, and 6 dihedral mirror planes (σ_{d}) |
CH_{4} |

O_{h} |
typically have octahedral geometry, with 3 C_{4} axes, 4 C_{3} axes, and an inversion center (i) as characteristic symmetry operations |
SF_{6} |

I_{h} |
typically have an icosahedral structure, with 6 C_{5} axes as characteristic symmetry operations |
B_{12}H_{12}^{2-} |

## D Groups

The D set of point groups are classified as D_{nh}, D_{nd}, or D_{n}, where n refers to the principal axis of rotation. Overall, the D groups are characterized by the presence of n C_{2} axes perpendicular to the principal C_{n} axis. Further classification of a molecule in the D groups depends on the presence of horizontal or vertical/dihedral mirror planes.

Group | Description | Example |
---|---|---|

D_{nh} |
n perpendicular C_{2} axes, and a horizontal mirror plane (σ_{h}) |
benzene, C_{6}H_{6} is D_{6h} |

D_{nd} |
n perpendicular C_{2} axes, and a vertical mirror plane (σ_{v}) |
propadiene, C_{3}H_{4} is D_{2d} |

D_{n} |
n perpendicular C_{2} axes, no mirror planes |
[Co(en)_{3}]^{3+} is D_{3} |

## C Groups

The C set of point groups are classified as C_{nh}, C_{nv}, or C_{n}, where n refers to the principal axis of rotation. The C set of groups are characterized by the absence of n C_{2} axes perpendicular to the principal C_{n} axis. Further classification of a molecule in the C groups depends on the presence of horizontal or vertical/dihedral mirror planes.

Group | Description | Example |
---|---|---|

C_{nh} |
horizontal mirror plane (σ_{h}) perpendicular to the principal C_{n} axis |
boric acid, H_{3}BO_{3} is C_{3h} |

C_{nv} |
vertical mirror plane (σ_{v}) containing the principal C_{n} axis |
ammonia, NH_{3} is C_{3v} |

C_{n} |
no mirror planes | P(C_{6}H_{5})_{3} is C_{3} |

## S Groups

The S set of point groups are classified as S_{2n}, where n refers to the principal axis of rotation. The S set of groups are characterized by the absence of n C_{2} axes perpendicular to the principal C_{n} axis, as well as the absence of horizontal and vertical/dihedral mirror planes. However, there is an improper rotation (or a rotation-reflection) axis collinear with the principal C_{n} axis.

Group | Description | Example |
---|---|---|

S_{2n} |
improper rotation (or a rotation-reflection) axis collinear with the principal C_{n} axis |
12-crown-4 is S_{4} |