point groups (section 3.4, p. 66) point group: a set of symmetry operations that completely describe...
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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