Symmetry and the point groups

1. Symmetry and the point groups

3. Symmetry Elements and Symmetry Operations • Identity • Proper axis of rotation • Mirror planes • Center of symmetry • Improper axis of rotation

4. Symmetry Elements and Symmetry Operations • Identity => E

5. Symmetry Elements and Symmetry Operations • Proper axis of rotation => Cn • where n = 2, 180o rotation • n = 3, 120o rotation • n = 4, 90o rotation • n = 6, 60o rotation • n = , (1/)o rotation • principal axis of rotation, Cn

6. 2-Fold Axis of Rotation

7. 3-Fold Axis of Rotation

8. Rotations for a Trigonal Planar Molecule

9. Symmetry Elements and Symmetry Operations Mirror planes sh => mirror plane perpendicular to a principal axis of rotation sv => mirror plane containing principal axis of rotation sd => mirror plane bisects dihedral angle made by the principal axis of rotation and two adjacent C2 axes perpendicular to principal rotation axis

10. Mirrors svsv Cl Cl sh I sd sd Cl Cl

11. Rotations and Mirrors in a Bent Molecule

12. Benzene Ring

13. Symmetry Elements and Symmetry Operations • Center of symmetry => i

14. Center of Inversion

15. Inversion vs. C2

16. Symmetry Elements and Symmetry Operations • Improper axis of rotation => Sn • rotation about n axis (360° /n) followed by reflection through a plane perpendicular to the axis

17. Improper Rotation in a Tetrahedral Molecule

18. S1 and S2 Improper Rotations

19. Successive C3 Rotations onTrigonal Pyramidal Molecule

20. Linear Molecules

21. Selection ofPoint Group from Shape • first determine shape using Lewis Structure and VSEPR Theory • next use models to determine which symmetry operations are present • then use the flow chart to determine the point group

23. Decision Tree

27. Selection ofPoint Group from Shape 1. determine the highest axis of rotation 2. check for other non-coincident axis of rotation 3. check for mirror planes

28. H2O and NH3

31. Geometric Shapes

32. E, S4, C2 Point Groups with improper axes S2n (n ≥ 2) 1,3,5,7-tetrafluorocycloocta-1,3,5,7-tetraene (S4)

33. 2) Point Groups of high symmetry (cubic groups) In contrast to groups C, D, and S, cubic symmetry groups are characterized by the presence of several rotational axes of high order (≥ 3). Cases of regular polyhedra: • Td (tetrahedral) BF4‑ , CH4 Symmetry elements:E, 4C3, 3C2, 3S4, 6sd Symmetry operations: E, 8C3, 3C2, 6S4, 6sd If all planes of symmetry and i are missing, the point group is T (pure rotational group, very rare); If all dihedral planes are removed but 3 sh remain, the point group is Th ( [Fe(py)6]2+ )

34. 3) Point Groups of high symmetry • Oh (octahedral)TiF62‑, cubane C8H8 Symmetry elements:E, i, 4S6, 4C3, 3S4, 3C4, 6C2, 3 C2, 3sh, 6sd Symmetry operations: E, i, 8S6, 8C3, 6S4, 6C4, 6C2, 3 C2, 3sh, 6sd Pure rotational analogue is the point group O (no mirror planes and no Sn; very rare)

35. 4) Point Groups of high symmetry Th group (symmetry elements: E, i, 4S6, 4C3, 3C2, 3sh) can also be considered as a result of reducing Ohgroup symmetry (E, i, 4S6, 4C3, 3S4, 3C4, 6C2, 3 C2, 3sh, 6sd ) by eliminating C4, S4 and some C2 axes andsd planes

36. Point Groups of high symmetry • Ih (icosahedral) B12H122‑, C20 Symmetry elements: E, i, 6S10, 6C5, 10S6, 10C3, 15C2, 15s Pure rotation analogue is the point group I (no mirror planes and thus no Sn, very rare)

37. Enantiomer Pairs

38. Enantiomer Pairs

39. Polarimeter

40. (b) Chirality • A chiral molecule : not superimposed on its mirror image. • optically active (rotate the plane of polarized) A molecule may be chiral only if it does not posses an axis of improper rotation Sn. All molecules with center of inversion are achiral  optically inactive. S1=s any molecule with a mirror plane is achiral.