1. 1

2. 2

3. Symmetry operation leaves everything unchanged Rotation Reflection Inversion Improper rotation Identity 3

4. Symmetry operation(element) leaves everything unchanged Rotation(axis)c n Reflection(plane)σ Inversion(center)i Improper rotation(axis+plane)s n IdentityE (Complete set of symmetry operations = point group) 4

5. Mo(CO) 6 5

6. Security First National Bank, California. United Banks of Colorado Pittsburgh National Bank, Woolmark C2 6

7. Security First National Bank, California. United Banks of Colorado Pittsburgh National Bank, Woolmark C2 C3 7

8. Chase Manhattan Bank First American National Bank, Tennessee C4 C5 Crocker Bank C6 8

9. 9

10. 10

11. 11

12. Structure Linear or low symmetry highly symmetric (octahedral, tetrahedral, icosahedral) Normal symmetry linear σ i i Yes D ∞h C ∞v CiCi C1C1 CsCs Yes No Linear Single Element Special Group Elements Sure? More than one C n axis of C 3 or higher? 6C 5 3C 4 3s 4 i IhIh I Yes No i OhOh O Yes No TdTd i ThTh T Yes No Yes Icosahedral Octahedral Tetrahedral High Symmetry Elements No Yes C 2  C n σhσh nσdnσd Yes No D nh D nd DnDn Yes No σhσh Yes nσvnσv S 2n Yes No C nh C nv CnCn S 2n Dihedral Single axis Normal Symmetry Elements (most common) 12

13. 1,3,5,7 -tetrafluoracyclooctatetrane 13

14. Boric acid [B(OH)3]Boric acid [B(OH)3] Boric acid [B(OH) 3 ] 14

15. C2 v 15

16. C3 v 16

17. 17

18. 18

19. 19

20. 20

21. 21

22. 22

23. 23

24. Point group C 2v 24

25. ++++ + ̶ + ̶ + ̶̶ + ++++ ++++ + ̶̶ + → a 1 → b 1 → b 2 2s 2p x 2p y 2p z → a 1 (1s+1s) (1s-1s) → b 2 O H-H 25

26. → a 1 → b 1 → b 2 2s 2p x 2p y 2p z → a 1 (1s+1s) (1s-1s) → b 2 O H-H Combine equal symmetries: a 1 :(2p z ) + (2s)nb (2p z ) + (1s+1s)b (2p z ) - (1s+1s)ab b 2 :(2p y ) + (1s-1s)b (2p y ) - (1s-1s)ab b 1 :(2p x )nb 26

27. a 1 :(2p z ) + (2s)nb (2p z ) + (1s+1s)b (2p z ) - (1s+1s)ab b 2 :(2p y ) + (1s-1s)b (2p y ) - (1s-1s)ab b 1 :(2p x )nb so what is the ground state of H 2 O? ↑↓ 27

28. so what is the ground state of H 2 O? (1a 1 ) 2 (2a 1 ) 2 (1b 2 ) 2 (3a 1 ) 2 (2b 1 ) 2 All pairs are a 1 : (a 1 X a 1 ) = 1 2. 1 2. 1 2. 1 2. = a 1 (b 1 X b 1 ) = 1 2. (-1) 2. 1 2. (-1) 2. = a 1 etc….  ground state symmetry: 28

29. 29