Vibrational spectroscopy.pptx

Important in the study of analytical chemistry

1. Part 2.10: Vibrational Spectroscopy 1

2. Single Atom No Rotation A point cannot rotate 3 Degrees of Freedom (DOF) 0 Vibrations Vibration- Atoms of a molecule changing their relative positions without changing the position of the molecular center of mass. No Vibartion “It takes two to vibrate” Translation Can move in x, y, and/or z 2

3. Diatomic Molecule 6 DOF - 3 Translation - 2 Rotation 1 Vibration 2 atoms x 3 DOF = 6 DOF Translation Rotation For a Linear Molecule # of Vibrations = 3N-5 3

4. Linear Triatomic Molecule 3 atoms x 3 DOF = 9 DOF For a Linear Molecule # of Vibrations = 3N-5 9 DOF - 3 Translation - 2 Rotation 4 Vibration Argon (1% of the atmosphere)- 3 DOF, 0 Vibrations 4

5. Nonlinear Triatomic Molecule 3 atoms x 3 DOF = 9 DOF 3 Translation 3 Rotation Linear non-linear 5

6. Nonlinear Triatomic Molecule 3 atoms x 3 DOF = 9 DOF 9 DOF - 3 Translation - 3 Rotation 3 Vibration For a nonlinear Molecule # of Vibrations = 3N-6 R z R x R y Trans z Trans y Trans x 6

7. Nonlinear Triatomic Molecule 3 atoms x 3 DOF = 9 DOF 9 DOF - 3 Translation - 3 Rotation 3 Vibration For a nonlinear Molecule # of Vibrations = 3N-6 7

8. Molecular Vibrations Atoms of a molecule changing their relative positions without changing the position of the molecular center of mass. Even at Absolute Zero! In terms of the molecular geometry these vibrations amount to continuously changing bond lengths and bond angles. Center of Mass Reduced Mass 8

9. Molecular Vibrations Hooke’s Law Assumes- It takes the same energy to stretch the bond as to compress it. The bond length can be infinite. k = force constant x = distance 9

10. Molecular Vibrations Vibration Frequency ( n ) Related to: Stiffness of the bond (k). Atomic masses (reduced mass, m ). 10

11. Molecular Vibrations Classical Spring Quantum Behavior Sometimes a classical description is good enough. Especially at low energies. 11

12. 6 Types of Vibrational Modes Symmetric Stretch Twisting Assymmetric Stretch Wagging Scissoring Rocking 12

13. What kind of information can be deduced about the internal motion of the molecule from its point-group symmetry? Each normal mode of vibration forms a basis for an irreducible representation of the point group of the molecule. 1) Find number/symmetry of vibrational modes. 2) Assign the symmetry of known vibrations. 3) What does the vibration look like? 4) Find if a vibrational mode is IR or Raman Active. Vibrations and Group Theory 13

14. 1) Finding Vibrational Modes Assign a point group Choose basis function (three Cartesian coordinates or a specific bond) Apply operations -if the basis stays the same = +1 -if the basis is reversed = -1 -if it is a more complicated change = 0 Generate a reducible representation Reduce to Irreducible Representation Subtract Translational and Rotational Motion 14

15. Example: H 2 O C 2v point group Basis: x 1-3 , y 1-3 and z 1-3 E: 3 + 3 + 3 = 9 Assign a point group Choose basis function Apply operations -if the basis stays the same = +1 -if the basis is reversed = -1 -if it is a more complicated change = 0 E Atom 1: x 1 = 1 y 1 = 1 z 1 = 1 Atom 2: x 2 = 1 y 2 = 1 z 2 = 1 Atom 3: x 3 = 1 y 3 = 1 z 3 = 1 Atom: 1 2 3 15

16. Example: H 2 O C 2v point group Basis: x 1-3 , y 1-3 and z 1-3 E: 3 + 3 + 3 = 9 C 2 : 0 + -1 + 0 = -1 Assign a point group Choose basis function Apply operations -if the basis stays the same = +1 -if the basis is reversed = -1 -if it is a more complicated change = 0 C 2 Atom 1: x 1 = 0 y 1 = 0 z 1 = 0 Atom 2: x 2 = -1 y 2 = -1 z 2 = 1 Atom 3: x 3 = 0 y 3 = 0 z 3 = 0 Atom: 1 2 3 16

17. Example: H 2 O C 2v point group Basis: x 1-3 , y 1-3 and z 1-3 E: 3 + 3 + 3 = 9 C 2 : 0 + -1 + 0 = -1 s xz : 0 + 1 + 0 = 1 Assign a point group Choose basis function Apply operations -if the basis stays the same = +1 -if the basis is reversed = -1 -if it is a more complicated change = 0 s xz Atom 1: x 1 = 0 y 1 = 0 z 1 = 0 Atom 2: x 2 = 1 y 2 = -1 z 2 = 1 Atom 3: x 3 = 0 y 3 = 0 z 3 = 0 Atom: 1 2 3 17

18. Example: H 2 O C 2v point group Basis: x 1-3 , y 1-3 and z 1-3 E: 3 + 3 + 3 = 9 C 2 : 0 + -1 + 0 = -1 s xz : 0 + 1 + 0 = 1 Assign a point group Choose basis function Apply operations -if the basis stays the same = +1 -if the basis is reversed = -1 -if it is a more complicated change = 0 s yz : 1 + 1 + 1 = 3 s yz Atom 1: x 1 = -1 y 1 = 1 z 1 = 1 Atom 2: x 2 = -1 y 2 = 1 z 2 = 1 Atom 3: x 3 = -1 y 3 = 1 z 3 = 1 Atom: 1 2 3 18

19. Example: H 2 O Assign a point group Choose basis function Apply operations -if the basis stays the same = +1 -if the basis is reversed = -1 -if it is a more complicated change = 0 Generate a reducible representation G 9 3 -1 1 C 2v point group Basis: x 1-3 , y 1-3 and z 1-3 E: 3 + 3 + 3 = 9 C 2 : 0 + -1 + 0 = -1 s xz : 0 + 1 + 0 = 1 s yz : 1 + 1 + 1 = 3 Atom: 1 2 3 19

20. Assign a point group Choose basis function Apply operations -if the basis stays the same = +1 -if the basis is reversed = -1 -if it is a more complicated change = 0 Generate a reducible representation Reduce to Irreducible Representation Reducible Rep. Irreducible Rep. C 2v point group Basis: x 1-3 , y 1-3 and z 1-3 G 9 3 -1 1 Example: H 2 O 20

21. Decomposition/Reduction Formula h = 1 + 1 + 1 + 1 = 4 order (h) a A 1 = + ( 1 )( -1 )( 1 ) + ( 1 )( 1 )( 1 ) + ( 1 )( 3 )( 1 ) 1 4 [ ] ( 1 )( 9 )( 1 ) = 3 = 12 4 Example: H 2 O G 9 3 -1 1 G = 3A 1 + A 2 + 2B 1 + 3B 2 21

22. Assign a point group Choose basis function Apply operations -if the basis stays the same = +1 -if the basis is reversed = -1 -if it is a more complicated change = 0 Generate a reducible representation Reduce to Irreducible Representation Subtract Rot. and Trans. C 2v point group Basis: x 1-3 , y 1-3 and z 1-3 Example: H 2 O G = 3A 1 + A 2 + 2B 1 + 3B 2 3 atoms x 3 DOF = 9 DOF 3N-6 = 3 Vibrations Trans = A 1 + B 1 + B 2 Rot = A 2 + B 1 + B 2 Vib = 2A 1 + B 2 22

23. Assign a point group Choose basis function Apply operations -if the basis stays the same = +1 -if the basis is reversed = -1 -if it is a more complicated change = 0 Generate a reducible representation Reduce to Irreducible Representation Subtract Rot. and Trans. C 2v point group Basis: x 1-3 , y 1-3 and z 1-3 Example: H 2 O Vibration = 2A 1 + B 2 23

24. B 2 A 1 A 1 Start with a drawing of a molecule Draw arrows Use the Character Table Predict a physically observable phenomenon Example: H 2 O C 2v point group Basis: x 1-3 , y 1-3 and z 1-3 Vibrations = 2A 1 + B 2 All three are IR active but that is not always the case. 24

25. 1) Find number/symmetry of vibrational modes. 2) Assign the symmetry of known vibrations. 3) What does the vibration look like? 4) Find if a vibrational mode is IR or Raman Active. Vibrations and Group Theory 25

26. 2) Assign the Symmetry of a Known Vibrations Assign a point group Choose basis function (stretch or bend) Apply operations -if the basis stays the same = +1 -if the basis is reversed = -1 -if it is a more complicated change = 0 Generate a reducible representation Reduce to Irreducible Representation Vibrations = 2A 1 + B 2 Stretch Stretch Bend Bend Stretch 26

27. Assign a point group Choose basis function Apply operations -if the basis stays the same = +1 -if the basis is reversed = -1 -if it is a more complicated change = 0 C 2v point group Basis: Bend angle Example: H 2 O s xz E: 1 C 2 : 1 s xz : 1 s yz : 1 E C 2 s yz 27

28. Assign a point group Choose basis function Apply operations -if the basis stays the same = +1 -if the basis is reversed = -1 -if it is a more complicated change = 0 Generate a reducible representation Reduce to Irreducible Representation Reducible Rep. Irreducible Rep. C 2v point group Basis: Bend angle G 1 1 1 1 Example: H 2 O 28

29. 2) Assign the Symmetry of a Known Vibrations Vibrations = A 1 + B 2 Stretch Stretch Bend A 1 Assign a point group Choose basis function (stretch) Apply operations -if the basis stays the same = +1 -if the basis is reversed = -1 -if it is a more complicated change = 0 Generate a reducible representation Reduce to Irreducible Representation Stretch 29

30. Assign a point group Choose basis function Apply operations -if the basis stays the same = +1 -if the basis is reversed = -1 -if it is a more complicated change = 0 C 2v point group Basis: OH stretch Example: H 2 O s xz E: 2 C 2 : 0 s xz : 0 s yz : 2 E C 2 s yz 30

31. Assign a point group Choose basis function Apply operations -if the basis stays the same = +1 -if the basis is reversed = -1 -if it is a more complicated change = 0 Generate a reducible representation Reduce to Irreducible Representation Reducible Rep. Irreducible Rep. G 2 2 Example: H 2 O C 2v point group Basis: OH stretch 31

32. Decomposition/Reduction Formula h = 1 + 1 + 1 + 1 = 4 order (h) a A 1 = + ( 1 )( )( 1 ) + ( 1 )( )( 1 ) + ( 1 )( 2 )( 1 ) 1 4 [ ] ( 1 )( 2 )( 1 ) = 1 = 4 4 Example: H 2 O G 2 2 G = A 1 + B 2 a B 2 = + ( 1 )( )(- 1 ) + ( 1 )( )(- 1 ) + ( 1 )( 2 )( 1 ) 4 ( 1 )( 2 )( 1 ) = 1 = 4 1 [ ] 4 32

33. 2) Assign the Symmetry of a Known Vibrations Vibrations = A 1 + B 2 Stretch Stretch Bend A 1 3) What does the vibration look like? By Inspection By Projection Operator 33

34. 1) Find number/symmetry of vibrational modes. 2) Assign the symmetry of known vibrations. 3) What does the vibration look like? 4) Find if a vibrational mode is IR or Raman Active. Vibrations and Group Theory 34

35. 3) What does the vibration look like? G = A 1 + B 2 By Inspection G 1 1 1 1 G 1 1 -1 -1 A 1 B 2 35

36. 3) What does the vibration look like? Projection Operator Assign a point group Choose non-symmetry basis ( D r 1 ) Choose a irreducible representation (A 1 or B 2 ) Apply Equation - Use operations to find new non-symmetry basis ( D r 1 ) - Multiply by characters in the irreducible representation Apply answer to structure 36

37. 3) What does the vibration look like? Projection Operator Assign a point group Choose non-symmetry basis ( D r 1 ) Choose a irreducible representation (A 1 ) Apply Equation - Use operations to find new non-symmetry basis ( D r 1 ) - Multiply by characters in the irreducible representation C 2v point group Basis: D r 1 s xz E C 2 s yz For A 1 37

38. 3) What does the vibration look like? Assign a point group Choose non-symmetry basis ( D r 1 ) Choose a irreducible representation (A 1 ) Apply Equation - Use operations to find new non-symmetry basis ( D r 1 ) - Multiply by characters in the irreducible representation s xz E C 2 s yz Projection Operator C 2v point group Basis: D r 1 For A 1 38

39. 3) What does the vibration look like? Assign a point group Choose non-symmetry basis ( D r 1 ) Choose a irreducible representation (B 2 ) Apply Equation - Use operations to find new non-symmetry basis ( D r 1 ) - Multiply by characters in the irreducible representation s xz E C 2 s yz Projection Operator C 2v point group Basis: D r 1 For B 2 39

40. 3) What does the vibration look like? Assign a point group Choose non-symmetry basis ( D r 1 ) Choose a irreducible representation (B 2 ) Apply Equation - Use operations to find new non-symmetry basis ( D r 1 ) Multiply by characters in the irreducible representation Apply answer to structure Projection Operator C 2v point group Basis: D r 1 B 2 A 1 40

41. 3) What does the vibration look like? Bend Symmetric Stretch Asymmetric Stretch A 1 B 2 A 1 A 1 A 1 B 2 Molecular Structure + Point Group = Find/draw the vibrational modes of the molecule Does not tell us the energy! Does not tell us IR or Raman active! 41

42. 1) Find number/symmetry of vibrational modes. 2) Assign the symmetry of known vibrations. 3) What does the vibration look like? 4) Find if a vibrational mode is IR or Raman Active. Vibrations of C60 42

43. Vibrations of C60 43

44. Vibrations of C60 44

45. Vibrations of C60 45

46. 1) Find number/symmetry of vibrational modes. 2) Assign the symmetry of known vibrations. 3) What does the vibration look like? 4) Find if a vibrational mode is IR or Raman Active. Vibrations and Group Theory Next ppt ! 46

47. Kincaid et al . J . Phys. Chem. 1988 , 92, 5628. C 2v : 20A 1 + 19B 2 + 9B 1 Side note: A Heroic Feat in IR Spectroscopy 47