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.

1. Determine if the molecule is of high or low symmetry.
2. If not, find the highest order rotation axis, C_n.
3. Determine if the molecule has any C₂ axes perpendicular to the principal C_n 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 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.
5. 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 C₂ axes, then it is part of the D_n set of point groups.
6. 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.

Flowchart for assigning point groups

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

Benzene, C₆H₆

Example: Finding the point group of benzene (C₆H₆)

1. Benzene is neither high or 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 (σ_h)

Benzene is in the D_6h point group.

Low Symmetry Point Groups

Low symmetry point groups include the C₁, C_s, and C_i groups.

Low Symmetry Groups

Group	Description	Example
C1	only the identity operation (E)	CHFClBr
Cs	only the identity operation (E) and one mirror plane	C2H2ClBr
Ci	only the identity operation (E) and a center of inversion (i)	C2H2Cl2Br

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 (σv)	HBr
h	linear molecule with an infinite number of rotation axes, vertical mirror planes (σv), perpendicular C2 axes, a horizontal mirror plane (σh), and an inversion center (i)	CO2
Td	typically have tetrahedral geometry, with 4 C4 axes, 3 C2 axes, 3 S4 axes, and 6 dihedral mirror planes (σd)	CH4
Oh	typically have octahedral geometry, with 3 C4 axes, 4 C3 axes, and an inversion center (i) as characteristic symmetry operations	SF6
Ih	typically have an icosahedral structure, with 6 C5 axes as characteristic symmetry operations	B12H122-

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 C₂ 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
Dnh	n perpendicular C2 axes, and a horizontal mirror plane (σh)	benzene, C6H6 is D6h
Dnd	n perpendicular C2 axes, and a vertical mirror plane (σv)	propadiene, C3H4 is D2d
Dn	n perpendicular C2 axes, no mirror planes	[Co(en)3]3+ is D3

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 C₂ 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
Cnh	horizontal mirror plane (σh) perpendicular to the principal Cn axis	boric acid, H3BO3 is C3h
Cnv	vertical mirror plane (σv) containing the principal Cn axis	ammonia, NH3 is C3v
Cn	no mirror planes	P(C6H5)3 is C3

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 C₂ 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
S2n	improper rotation (or a rotation-reflection) axis collinear with the principal Cn axis	12-crown-4 is S4

See Also

Symmetry Gallery