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 GroupsEdit
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})Edit
- 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 GroupsEdit
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 GroupsEdit
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 |
_{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 GroupsEdit
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 GroupsEdit
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 GroupsEdit
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} |