# Symmetry Point Groups

Symmetry is very important.

### Nonaxial symmetries

This class includes C1, Ci, and Cs, which have no proper or improper rotation axis.

### C1 Group

C1 has only one symmetry operation, {E}. The order of C1 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 C1. This picture is drawn by ACD Labs 11.0.

#### Ci Group

Ci has 2 symmetry operations, {E, i}. The order of Ci group is 2. Molecules in this group have low symmetry, an inversion center. For example, C2H2F2Cl2 has an inversion center.

Figure 2.2  C2H2F2Cl2. Point group is Ci. This picture is drawn by MacMolPlt.

#### Cs Group

Cs has 2 symmetry operations, {E, σ}. The order of Cs group is 2. Molecules in this group have low symmetry, a mirror plane. For example, CH2BrCl has a mirror plane.

Figure 2.3 CH2BrCl. Point group is Cs. This picture is drawn by MacMolPlt.

### Cyclic symmetries

This class includes Cn, Cnh, Cnv, and Sn, which have only one proper or improper rotation axis.

#### Cyclic group: Cn group

Cn (nσ2)

symmetry elements, E and Cn.

And n symmetry operations, {E, Cn1, Cn2, … , Cnn-1}

The order of Cn group is n.

Figure 2.4 C2H4Cl2. Point group is C2. This picture is drawn by MacMolPlt.

#### Pyramidal group: Cnv group

For Cnv group, symmetry elements are E, Cn, and nσv.

And symmetry operations are {E, Cnk(k=1, … ,n-1), nσv }

The order of Cnv group is 2n. For  example, NH3 has a C3 axis and three mirror planes σv.

Therefore, the point group of NH3 is C3v.

Figure 2.5 NH3. Point group is C3v. This picture is drawn by MacMolPlt.

Now we can generate a group multiplication table for NH3:

Table 2.2 Group multiplication table of symmetry operation of NH3 molecule

This C3v group, as what is mentioned before, has all the properties of a group in mathmetics. And all the molecules that have one C3 axis and 3 mirror planes such as NH3 molecule can be assigned to this C3v 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: Cnh group

For Cnh group, symmetry elements are E, Cn, σh, and Sn.

And symmetry operations are {E, Cnk(k=1, … ,n-1), σh, σh Cnm(m=1, … ,n-1)}

The order of Cnh group is 2n.

For example, point group of C2H2F2 is C2h.

Figure 2.6 C2H2F2. Point group is C2h. This picture is drawn by MacMolPlt.

Improper rotation group: Sn group

If n=1, S1=Cs

If n=2, S2=Ci

If n=odd number, Sn (n=3, 5, 7 …) = Cnh

For example, operations in S3 are the same as C3h, e.g. B(OH)3.

S3={E, S3, S32, S33, S34, S35} ={E, S3, C32, σh, C3, S35}= C3h

Figure 2.7 B(OH)3. Point group C3h. This picture is drawn by MacMolPlt.

Therefore, for Sn group, n can only be 4, 6, 8 …..

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

For example, the point group of 1,3,5,7 -tetrafluoracyclooctatetrane is S4.

Figure 2.8 1,3,5,7 -tetrafluoracyclooctatetrane. Point group is S4. The left

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

### Dihedral symmetries

This class includes Dn, Dnh, and Dnd, which have one proper rotation Cn axis and n C2 axis perpendicular to Cn axis.

#### Dihedral group: Dn group

For Dn group, symmetry elements are E, Cn, and nC2 (σCn).

And symmetry operations are {E, Cnk(k=1, … ,n-1), nC2}

The order of Dn group is 2n.

For example, the point group of [Co(en)3]3+ is D3.

Figure 2.9 [Co(en)3]3+.Point group is D3. The figure is drawn by ACD Labs 11.0.

#### Prismatic group: Dnh group

For Dnh group, symmetry elements are E, Cn, σh,and nC2 (σCn).

And symmetry operations are {E, Cnk(k=1, … ,n-1), ?h, Snm(m=1, … ,n-1), nC2, nσv}

The order of Dnh group is 4n.

For example, the point group of benzene is D6h.

Figure 2.10 Benzene. Point group D6h. This picture is drawn by MacMolPlt.

#### Antiprismatic group: Dnd group

For Dnd group, symmetry elements are E, Cn, σd, and nC2 (σCn).

And symmetry operations are {E, Cnk(k=1, … ,n-1), S2nm(m=1, … ,2n-1), nC2, nσd}

The order of Dnd group is 4n.

For example, pinot group of C2H6 is D3d.

Figure 2.11  C2H6. Point group D3d.This picture is drawn by MacMolPlt.

### Polyhedral symmetries

This class includes T, Th, Td, O, Oh, I and Ih, which have more than two high-order axes.

Cubic groups: T, Th, Td, O, Oh

These groups do not have a C5 peoper rotation axis.

T group

For T group, symmetry elements are E, 4C3, and 3C2.

And symmetry operations are {E, 4C3, 4C32, 3C2}

The order of T group is 12.

Th group

For Td group, symmetry elements are E, 3C2, 4C3, i, 4S6 and 3σh.

And symmetry operations are {E, 4C3, 4C32, 3C2, i, 4S6, 4S65, 3σh}

The order of Td group is 24.

Td group

For Td group, symmetry elements are E, 3C2, 4C3, 3S4 and 6σd.

And symmetry operations are {E, 8C3, 3C2, 6S4, 6σd}

The order of Td group is 24.

For example, the point group of CCl4 is Td.                                    Figure 2.12 CCl4. Point group is Td. The figure is drawn by ACD Labs 11.0.

O group

For O group, symmetry elements are E, 3C4, 4C3, and 6C2.

And symmetry operations are {E, 8C3, 3C2, 6C4, 6C2}

The order of O group is 24.

Oh group

For Oh group, symmetry elements are E, 3S4, 3C4, 6C2, 4S6, 4C3, 3?h, 6σd, and i.

And symmetry operations are {E, 8C3, 6C2, 6C4, 3C2, i, 6S4, 8S6, 3σh, 6σd}

The order of Oh group is 48.

For example, the point group of SF6 is Oh.

Figure 2.13 SF6. Point group is Oh. The figure is drawn by ACD Labs 11.0.

Icosahedral groups: I, Ih

These groups have a C5 peoper rotation axis.

#### I group

For I group, symmetry elements are E, 6C5, 10C3, and 15C2.

And symmetry operations are {E, 15C5, 12C52, 20C3, 15C2}

The order of I group is 60.

#### Ih group

For Ih group, symmetry elements are E, 6S10, 10S6, 6C5, 10C3, 15C2 and 15σ.

And symmetry operations are {E, 15C5, 12C52, 20C3, 15C2, i, 12S10, 12S103, 20S6, 15σ}

The order of Ih group is 120.

For example, the point group of C60 is Ih.

Figure 2.14 C60. Point group is Ih. The figure is drawn by ACD Labs 11.0.

#### Linear groups

This class includes C?v and D?h, which are the symmetry of linear molecules.

#### C∞v group

For C∞v group, symmetry elements are E, C and ∞σv.

such as CO, HCN, NO, HCl.

#### D∞h group

For Dσh group, symmetry elements are E, C ∞σv , σh, i, and ∞C2.

such as CO2, O2, N2.

Figure 2.16 O2. Point group is D?h. This picture is drawn by MacMolPlt

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