symmetry n group theory
DESCRIPTION
Symmetry n Group TheoryTRANSCRIPT
Nonaxial symmetriesThis class includes C1, Ci, and Cs, which have no proper or improper rotation axis.
C1 GroupC1 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.
Ci GroupCi 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,C2H2F2Cl2 has an inversion center.
Figure 2.2 C2H2F2Cl2. Point group is Ci.
.
Cs GroupCs 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
Cyclic symmetriesThis 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, Cn
2, … , Cnn-1}
The order of Cn group is n.
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.
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 Cn
m(m=1, … ,n-1)}
The order of Cnh group is 2n.
For example, point group of C2H2F2 is C2h.
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 S3are the same as C3h, e.g. B(OH)3.
S3={E, S3, S32, S3
3, S34, S3
5} ={E, S3, C32, σh, C3, S3
5}= 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.
Dihedral symmetriesThis class includes Dn, Dnh, and Dnd, which have one proper rotation Cn axis and n C2 axis perpendicular to Cn axis.
Dihedral group: Dn groupFor 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.
Prismatic group: D nh group For Dnh group, symmetry elements are E, Cn, σh,and nC2 (σCn).
And symmetry operations are {E, Cnk(k=1, … ,n-1), ?h, Sn
m(m=1, … ,n-1), nC2, nσv}
The order of Dnh group is 4n.
For example, the point group of benzene is D6h.
Antiprismatic group: D nd group For Dnd group, symmetry elements are E, Cn, σd, and nC2 (σCn).
And symmetry operations are {E, Cnk(k=1, … ,n-1), S2n
m(m=1, … ,2n-1), nC2, nσd}
The order of Dnd group is 4n.
For example, pinot group of C2H6 is D3d.
Polyhedral symmetriesThis 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, 4S6
5, 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.
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.
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.
I h 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, 12S10
3, 20S6, 15σ}
The order of Ih group is 120.
For example, the point group of C60 is Ih.
Linear groupsThis 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.