3.3: Molecular 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 .
- 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.
Figure \(\PageIndex{1}\) steps for determining a molecule's overall point group
Example \(\PageIndex{1}\)
Find the point group of benzene (C 6 H 6 ).
- Answer
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Solution
1. Benzene is neither high or low symmetry -
2. Highest order rotation axis: C
6
3. There are 6 C 2 axes perpendicular to the principal axis
4. 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
| 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 |