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ChemWiki: The Dynamic Chemistry Hypertext > Theoretical Chemistry > Symmetry > Common Point Groups for Molecules > Symmetry Point Groups

Symmetry Point Groups

This class includes C_{1}, C_{i}, and C_{s}, which have no proper or improper rotation axis.

C_{1} has only one symmetry operation, {E}. The order of C_{1} group is 1. Molecules in this group have no symmetry, which means we can not perform rotation, reflection of a mirror plane, etc. And the only symmetry operation is identity, E.

Figure 2.1 HCFBrCl. Point group is C_{1}. This picture is drawn by ACD Labs 11.0.

C_{i} has 2 symmetry operations, {E, i}. The order of C_{i} group is 2. Molecules in this group have low symmetry, an inversion center. For example, C_{2}H_{2}F_{2}Cl_{2} has an inversion center.

Figure 2.2 C_{2}H_{2}F_{2}Cl_{2}. Point group is C_{i}. This picture is drawn by MacMolPlt.

C_{s} has 2 symmetry operations, {E, σ}. The order of C_{s} group is 2. Molecules in this group have low symmetry, a mirror plane. For example, CH_{2}BrCl has a mirror plane.

Figure 2.3 CH_{2}BrCl. Point group is C_{s}. This picture is drawn by MacMolPlt.

This class includes C_{n}, C_{nh}, C_{nv}, and S_{n}, which have only one proper or improper rotation axis.

C_{n} (nσ_{2})

symmetry elements, E and C_{n}.

And n symmetry operations, {E, C_{n}^{1}, C_{n}^{2}, … , C_{n}^{n-1}}

The order of C_{n} group is n.

Figure 2.4 C_{2}H_{4}Cl_{2.} Point group is C_{2}. This picture is drawn by MacMolPlt.

For C_{nv} group, symmetry elements are E, C_{n}, and nσ_{v}.

And symmetry operations are {E, C_{n}^{k}(k=1, … ,n-1), nσ_{v} }

The order of C_{nv} group is 2n. For example, NH_{3} has a C_{3} axis and three mirror planes σ_{v}.

Therefore, the point group of NH_{3} is C_{3v}.

Figure 2.5 NH_{3}. Point group is C_{3v}. This picture is drawn by MacMolPlt.

Now we can generate a group multiplication table for NH_{3}:

Table 2.2 Group multiplication table of symmetry operation of NH_{3} molecule

This C_{3v }group, as what is mentioned before, has all the properties of a group in mathmetics. And all the molecules that have one C_{3 }axis and 3 mirror planes such as NH_{3} molecule can be assigned to this C_{3v} group. In the same way, the operations in the following groups also have all the properties of a mathmetical group and can generate a multiplicaiton table.

__Reflection group: C _{nh }group __

For C_{nh} group, symmetry elements are E, C_{n}, σ_{h}, and S_{n}.

And symmetry operations are {E, C_{n}^{k}(k=1, … ,n-1), σ_{h}, σ_{h} C_{n}^{m}(m=1, … ,n-1)}

The order of C_{nh} group is 2n.

For example, point group of C_{2}H_{2}F_{2} is C_{2h}.

Figure 2.6 C_{2}H_{2}F_{2}. Point group is C_{2h}. This picture is drawn by MacMolPlt.

__Improper rotation group: S _{n }group __

If n=1, S_{1}=C_{s}

If n=2, S_{2}=C_{i}

If n=odd number, S_{n} (n=3, 5, 7 …) = C_{nh}

For example, operations in S_{3 }are the same as C_{3h}, e.g. B(OH)_{3}.

S_{3}={E, S_{3}, S_{3}^{2}, S_{3}^{3}, S_{3}^{4}, S_{3}^{5}} ={E, S_{3}, C_{3}^{2}, σ_{h}, C_{3}, S_{3}^{5}}= C_{3h}

Figure 2.7 B(OH)_{3}. Point group C_{3h}. This picture is drawn by MacMolPlt.

Therefore, for S_{n }group, n can only be 4, 6, 8 …..

The symmetry elements are E and S_{n}. And symmetry operations are {E, S_{n}^{k}(k=1, … ,n-1)}. The order of S_{n} group is n.

For example, the point group of 1,3,5,7 -tetrafluoracyclooctatetrane is S_{4}.

Figure 2.8 1,3,5,7 -tetrafluoracyclooctatetrane. Point group is S_{4}. The left

picture is drawn by MacMolPlt. The animated figure is drawn by ACD Labs 11.0.

This class includes D_{n}, D_{nh}, and D_{nd}, which have one proper rotation C_{n} axis and n C_{2} axis perpendicular to C_{n} axis.

For D_{n} group, symmetry elements are E, C_{n}, and nC_{2} (σC_{n}).

And symmetry operations are {E, C_{n}^{k}(k=1, … ,n-1), nC_{2}}

The order of D_{n} group is 2n.

For example, the point group of [Co(en)_{3}]^{3+} is D_{3}.

Figure 2.9 [Co(en)_{3}]^{3+}.Point group is D_{3}. The figure is drawn by ACD Labs 11.0.

For D_{nh} group, symmetry elements are E, C_{n}, σ_{h},and nC_{2} (σC_{n}).

And symmetry operations are {E, C_{n}^{k}(k=1, … ,n-1), ?_{h}, S_{n}^{m}(m=1, … ,n-1), nC_{2}, nσ_{v}}

The order of D_{nh} group is 4n.

For example, the point group of benzene is D_{6h}.

Figure 2.10 Benzene. Point group D_{6h}. This picture is drawn by MacMolPlt.

For D_{nd} group, symmetry elements are E, C_{n}, σ_{d}, and nC_{2} (σC_{n}).

And symmetry operations are {E, C_{n}^{k}(k=1, … ,n-1), S_{2n}^{m}(m=1, … ,2n-1), nC_{2}, nσ_{d}}

The order of D_{nd} group is 4n.

For example, pinot group of C_{2}H_{6} is D_{3d}.

Figure 2.11 C_{2}H_{6}. Point group D_{3}d.This picture is drawn by MacMolPlt.

This class includes T, T_{h}, T_{d}, O, O_{h}, I and I_{h}, which have more than two high-order axes.

__Cubic groups: T, T _{h}, T_{d}, O, O_{h} __

These groups do not have a C_{5} peoper rotation axis.

__T group __

For T group, symmetry elements are E, 4C_{3}, and 3C_{2}.

And symmetry operations are {E, 4C_{3}, 4C_{3}^{2}, 3C_{2}}

The order of T group is 12.

__T _{h} group __

For T_{d} group, symmetry elements are E, 3C_{2}, 4C_{3}, i, 4S_{6 }and 3σ_{h}.

And symmetry operations are {E, 4C_{3}, 4C_{3}^{2}, 3C_{2}, i, 4S_{6}, 4S_{6}^{5}, 3σ_{h}}

The order of T_{d} group is 24.

__T _{d} group __

For T_{d} group, symmetry elements are E, 3C_{2}, 4C_{3}, 3S_{4 }and 6σ_{d}.

And symmetry operations are {E, 8C_{3}, 3C_{2}, 6S_{4}, 6σ_{d}}

The order of T_{d} group is 24.

For example, the point group of CCl_{4} is T_{d}. Figure 2.12 CCl_{4}. Point group is T_{d}. The figure is drawn by ACD Labs 11.0.

__O group __

For O group, symmetry elements are E, 3C_{4}, 4C_{3}, and 6C_{2}.

And symmetry operations are {E, 8C_{3}, 3C_{2}, 6C_{4}, 6C_{2}}

The order of O group is 24.

__O _{h} group __

For O_{h} group, symmetry elements are E, 3S_{4}, 3C_{4}, 6C_{2}, 4S_{6}, 4C_{3}, 3?_{h}, 6σ_{d}, and i.

And symmetry operations are {E, 8C_{3}, 6C_{2}, 6C_{4}, 3C_{2}, i, 6S_{4}, 8S_{6}, 3σ_{h}, 6σ_{d}}

The order of O_{h} group is 48.

For example, the point group of SF_{6} is O_{h}.

Figure 2.13 SF_{6}. Point group is O_{h}. The figure is drawn by ACD Labs 11.0.

__Icosahedral groups: I, I _{h} __

These groups have a C_{5} peoper rotation axis.

For I group, symmetry elements are E, 6C_{5}, 10C_{3}, and 15C_{2}.

And symmetry operations are {E, 15C_{5}, 12C_{5}^{2}, 20C_{3}, 15C_{2}}

The order of I group is 60.

For I_{h} group, symmetry elements are E, 6S_{10}, 10S_{6}, 6C_{5}, 10C_{3}, 15C_{2} and 15σ.

And symmetry operations are {E, 15C_{5}, 12C_{5}^{2}, 20C_{3}, 15C_{2}, i, 12S_{10}, 12S_{10}^{3}, 20S_{6}, 15σ}

The order of I_{h} group is 120.

For example, the point group of C_{60} is I_{h}.

Figure 2.14 C_{60}. Point group is I_{h}. The figure is drawn by ACD Labs 11.0.

This class includes C_{?v} and D_{?h}, which are the symmetry of linear molecules.

For C_{∞v} group, symmetry elements are E, C_{∞} and ∞σ_{v}.

such as CO, HCN, NO, HCl.

For D_{σh} group, symmetry elements are E, C_{∞} ∞σ_{v} , σ_{h}, i, and ∞C_{2}.

such as CO_{2}, O_{2}, N_{2}.

Figure 2.16 O_{2}. Point group is D_{?h}. This picture is drawn by MacMolPlt

Last modified

01:18, 6 Feb 2015

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This material is based upon work supported by the National Science Foundation under Grant Number 1246120

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