Point Groups (section 3.4, p. 66)

Point group: a set of symmetry operations that completely describe the

symmetry of an object/molecule.

Going back to our example set:{E, C3, C3

2, σ1, σ2, σ3} = C3v

{ E, i } = Ci

{ E, C3, C32, 3C2, σh, S3, S3

2, 3σv } = D3h

{ E, 4C3, 4C32, 3C2, i, 3σh, 4S6, 4S6

5 } = Td

The Types of point groups

If an object has no symmetry (only the identity E) it belongs to group C1

Axial Point groups or Cn class Cn = E + n Cn (set has n operations)Cnh= E + n Cn + sh ( “ “ 2n operations)Cnv = E + n Cn + n sv ( “ “ 2n operations) 

 Dihedral Point Groups or Dn class Dn = Cn + nC2 (^)

Dnd = Cn + nC2 (^) + n sd

Dnh = Cn + nC2 (^) + sh

 Sn groups:

S1 = Cs

S2 = Ci

S3 = C3h

S4 , S6 forms a groupS5 = C5h

 Linear Groups or cylindrical class

C∞v and D∞h= C∞ + infinite sv

= D∞ + infinite sh

 

Cubic groups or the Platonic solids..

T: 4C3 and 3C2, mutually perpendicularTd (tetrahedral group): T + 3S4 axes + 6 sv

 O: 3C4 and 4C3, many C2

Oh (octahedral group): O + i + 3 sh + 6 sv

Icosahedral group:Ih : 6C5, 10C3, 15C2, i, 15 sv

See any repeating relationship among the Cubic groups ?

T: 4C3 and 3C2, mutually perpendicularTd (tetrahedral group): T + 3S4 axes + 6 sv

 O: 3C4 and 4C3, many C2

Oh (octahedral group): O + i + 3 sh + 6 sv

Icosahedral group:Ih : 6C5, 10C3, 15C2, i, 15 sv

See any repeating relationship among the Cubic groups ?

T: 4C3 and 3C2, mutually perpendicularTd (tetrahedral group): T + 3S4 axes + 6 sv

 O: 3C4 and 4C3, many C2

Oh : 3C4 and 4C3, many C2 + i + 3 sh + 6 sv

Icosahedral group:Ih : 6C5, 10C3, 15C2, i, 15 sv

How is the point symmetry of a cube related to an octahedron?

…. Let’s see!

How is the symmetry of an octahedron related to a tetrahedron?

C4

C4

C4 C3

C3

C3

C4

C4

C3

C3

C4

C4

C4 C3

C3

C3

C4 is now destroyed!

Oh