# Advanced Inorganic Chemistry/Molecular Point Groups

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.
The steps for determining a molecule's overall point group are shown in the included flowchart.

### Example: Finding the point group of benzene (C$_6$H$_6$)

* Benzene is neither high or low symmetry
* Highest order rotation axis: C$_6$
* 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.

## Low Symmetry Point Groups

Low symmetry point groups include the C$_1$, C$_s$, and C$_i$ groups.
Low Symmetry 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.
High Symmetry Groups :
| Group | Description | Example |
| C$_{∞v}$ | linear molecule with an infinite number of rotation axes and vertical mirror planes (σ$_v$) | HBr |
| $_h$ | 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.
D Groups :
| 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.
C Groups :
| 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.
S Groups :
| Group | Description | Example |
| S$_{2n}$ | improper rotation (or a rotation-reflection) axis collinear with the principal C$_n$ axis | 12-crown-4 is S$_4$ |

## See Also

Symmetry Gallery