1. 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, C 3, C 3 2, σ 1, σ 2, σ 3 } = C 3v

2. { E, i } = C i { E, C 3, C 3 2, 3C 2, σ h, S 3, S 3 2, 3σ v } = D 3h { E, 4C 3, 4C 3 2, 3C 2, i, 3σ h, 4S 6, 4S 6 5 } = T d

3. The Types of point groups If an object has no symmetry (only the identity E) it belongs to group C 1 Axial Point groups or C n class C n = E + n C n (set has n operations) C nh = E + n C n +  h ( “ “ 2n operations) C nv = E + n C n + n  v ( “ “ 2n operations) Dihedral Point Groups or Dn class D n = C n + nC2 (  ) D nd = C n + nC2 (  ) + n  d D nh = C n + nC2 (  ) +  h Sn groups: S 1 = C s S 2 = C i S 3 = C 3h S 4, S 6 forms a group S 5 = C 5h

4. Linear Groups or cylindrical class C∞v and D∞h = C∞ + infinite  v = D∞ + infinite  h Cubic groups or the Platonic solids.. T: 4C 3 and 3C 2, mutually perpendicular T d (tetrahedral group): T + 3S 4 axes + 6  v O: 3C 4 and 4C 3, many C 2 O h (octahedral group): O + i + 3  h + 6  v Icosahedral group: I h : 6C 5, 10C 3, 15C 2, i, 15  v

5. See any repeating relationship among the Cubic groups ? T: 4C 3 and 3C 2, mutually perpendicular T d (tetrahedral group): T + 3S 4 axes + 6  v O: 3C 4 and 4C 3, many C 2 O h (octahedral group): O + i + 3  h + 6  v Icosahedral group: I h : 6C 5, 10C 3, 15C 2, i, 15  v

6. See any repeating relationship among the Cubic groups ? T: 4C 3 and 3C 2, mutually perpendicular T d (tetrahedral group): T + 3S 4 axes + 6  v O: 3C 4 and 4C 3, many C 2 O h : 3C 4 and 4C 3, many C 2 + i + 3  h + 6  v Icosahedral group: I h : 6C 5, 10C 3, 15C 2, i, 15  v 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?

8. C4C4 C4C4 C4C4 C3C3 C3C3

10. C3C3

16. C4C4 C4C4 C3C3 C3C3

17. C4C4 C4C4 C4C4 C3C3 C3C3

21. C3C3 C 4 is now destroyed!

22. OhOh