1. Rigid Diatomic molecule
I = r  = m1m2/(m1+m2)
B (MHz) = /I (u Å 2)
B (cm-1) = /I (u A2)
Harry Kroto 2004

2. Rotational Spectroscopy of Linear Molecules
F(J) = BJ(J+1)
0 2B 6B 12B 20B 30B…
F(J) = 2B(J+1)
2B 4B 6B 8B 10B 12B…
Harry Kroto 2004

3. Rotational Spectroscopy of Linear Molecules
F(J) = BJ(J+1)
2B 6B 12B 20B 30B…
F(J) = 2B(J+1)
2B 4B 6B 8B 10B 12B… 1 2B
Harry Kroto 2004

4. Rotational Spectroscopy of Linear Molecules
F(J) = BJ(J+1)
0 2B 6B 12B 20B 30B…
F(J) = 2B(J+1)
2B 4B 6B 8B 10B 12B… 2 6B 1 2B
Harry Kroto 2004

5. Rotational Spectroscopy of Linear Molecules
F(J) = BJ(J+1)
0 2B 6B 12B 20B 30B…
F(J) = 2B(J+1)
2B 4B 6B 8B 10B 12B… 3
12B 2 6B 1 2B
Harry Kroto 2004

6. Rotational Spectroscopy of Linear Molecules
F(J) = BJ(J+1)
0 2B 6B 12B 20B 30B…
F(J) = 2B(J+1)
2B 4B 6B 8B 10B 12B… 4
20B 3
12B 2 6B 1 2B
Harry Kroto 2004

7. Rotational Spectroscopy of Linear Molecules
F(J) = BJ(J+1)
0 2B 6B 12B 20B 30B…
F(J) = 2B(J+1)
2B 4B 6B 8B 10B 12B… 5
30B 4
20B 3
12B 2 6B 1 2B
Harry Kroto 2004

8. Rotational Spectroscopy of Linear Molecules
F(J) = BJ(J+1)
0 2B 6B 12B 20B 30B…
F(J) = 2B(J+1)
2B 4B 6B 8B 10B 12B… 6
42B 5
30B 4
20B 3
12B 2 6B 1 2B
Harry Kroto 2004

9. Rotational Spectroscopy of Linear MoleculesJ 7
56B
F(J) = BJ(J+1)
0 2B 6B 12B 20B 30B…
F(J) = 2B(J+1)
2B 4B 6B 8B 10B 12B… 6
42B 5
30B 4
20B 3
12B 2 6B 1 2B
Harry Kroto 2004

10. A Classical Description > E = T + V E = ½I2 V=0
B QM description > the Hamiltonian H J  = E J  H = J2/2I
C Solve the Hamiltonian > Energy Levels F (J) = BJ(J+1)
D Selection Rules > Allowed Transitions J = ±1
E Transition Frequencies > F B(J+1)
F Intensities > THE SPECTRUM
J Analysis > Pattern recognition; assign J numbers
H Experimental Details > microwave spectrometers
I More Advanced Details: Centrifugal distortion, spin effect
J Information obtainable: structures, dipole moments etc
Harry Kroto 2004

11. Rotational Spectroscopy of Linear MoleculesJ 7
56B
F(J) = BJ(J+1)
2B 6B 12B 20B 30B…
F(J) = 2B(J+1)
2B 4B 6B 8B 10B 12B… 6
42B 5
30B 4
20B 3
12B 2 6B 1 2B
Harry Kroto 2004

12. Rotational Spectroscopy of Linear MoleculesJ 7
56B
F(J) = BJ(J+1)
2B 6B 12B 20B 30B…
F(J) = 2B(J+1)
2B 4B 6B 8B 10B 12B… 6
42B 5
30B 4
20B 3
12B 2 6B 1 2B
Harry Kroto 2004

13. Rotational Spectroscopy of Linear MoleculesJ 7
56B
F(J) = BJ(J+1)
2B 6B 12B 20B 30B…
F(J) = 2B(J+1)
2B 4B 6B 8B 10B 12B… 6
42B 5
30B 4
20B 3
12B 2 6B 1 2B
Harry Kroto 2004

14. Rotational Spectroscopy of Linear MoleculesJ 7
56B
F(J) = BJ(J+1)
2B 6B 12B 20B 30B…
F(J) = 2B(J+1)
2B 4B 6B 8B 10B 12B… 6
42B 5
30B 4
20B 3
12B 2 6B 1 2B
Harry Kroto 2004

15. Rotational Spectroscopy of Linear MoleculesJ 7
56B
F(J) = BJ(J+1)
2B 6B 12B 20B 30B…
F(J) = 2B(J+1)
2B 4B 6B 8B 10B 12B… 6
42B 5
30B 4
20B 3
12B 2 6B 1 2B
Harry Kroto 2004

16. Rotational Spectroscopy of Linear MoleculesJ 7
56B
F(J) = BJ(J+1)
2B 6B 12B 20B 30B…
F(J) = 2B(J+1)
2B 4B 6B 8B 10B 12B… 6
42B 5
30B 4
20B 3
12B 2 6B 1 2B
Harry Kroto 2004

17. Rotational Spectroscopy of Linear MoleculesJ 7
56B
F(J) = BJ(J+1)
2B 6B 12B 20B 30B…
F(J) = 2B(J+1)
2B 4B 6B 8B 10B 12B… 6
42B 5
30B 4
20B 3
12B 2 6B 1 2B
Harry Kroto 2004

18. Rotational Spectroscopy of Linear MoleculesJ 7
56B
F(J) = BJ(J+1)
2B 6B 12B 20B 30B…
F(J) = 2B(J+1)
2B 4B 6B 8B 10B 12B… 6
42B 5
30B 4
20B 3
12B 2 6B 1 2B
Harry Kroto 2004

19. A Classical Description > E = T + V E = ½I2 V=0
B QM description > the Hamiltonian H J  = E J  H = J2/2I
C Solve the Hamiltonian > Energy Levels F (J) = BJ(J+1)
D Selection Rules > Allowed Transitions J = ±1
E Transition Frequencies > F (J) = 2B(J+1)
F Intensities > THE SPECTRUM
J Analysis > Pattern recognition; assign J numbers
H Experimental Details > microwave spectrometers
I More Advanced Details: Centrifugal distortion, spin effect
J Information obtainable: structures, dipole moments etc
Harry Kroto 2004

20. B(J+1)(J+2)
J+1
BJ(J+1) J
F(J) = 2B(J+1)
Harry Kroto 2004

21. Rotational Spectroscopy of Linear MoleculesJ 7
56B
F(J) = BJ(J+1)
2B 6B 12B 20B 30B…
F(J) = 2B(J+1)
2B 4B 6B 8B 10B 12B…
14B 6
42B
12B 5
30B
10B 4
20B 8B 3
12B 6B 2 6B 4B 1 2B 2B
Harry Kroto 2004

22. A Classical Description > E = T + V E = ½I2 V=0
B QM description > the Hamiltonian H J  = E J  H = J2/2I
C Solve the Hamiltonian > Energy Levels F (J) = BJ(J+1)
D Selection Rules > Allowed Transitions J = ±1
E Transition Frequencies > F B(J+1)
F Intensities > THE SPECTRUM
J Analysis > Pattern recognition; assign J numbers
H Experimental Details > microwave spectrometers
I More Advanced Details: Centrifugal distortion, spin effect
J Information obtainable: structures, dipole moments etc
Harry Kroto 2004

23. Far infrared rotational spectrum of CO J= 12 15 20B10
Far infrared rotational spectrum of CO
J= 12 15
20B
23.0 cm-1
61.5 cm-1
Line separations 2B
Approximately 61.5 – 23 = 38.5 cm-1 = 20B
2B = B = cm-1 (
50/3.85 = = 13 so line at 50cm-1 is J=12
B = / I I = / B
I = 8.76 uA2 I =  r2
 = m1m2/(m1+m2)= 16x12/28 = 6.86
8.76/6.86 = = r2
r = 1.277½ = A ( acc B value 1.921)
Harry Kroto 2004

24. Approximately 61.5 – 23 = 38.5 cm-1 = 20B 2B = 3.85 B = 1.925 cm-1 ( 10
J= 12 15
20B
23.0 cm-1
61.5 cm-1
Line separations 2B
Approximately 61.5 – 23 = 38.5 cm-1 = 20B
2B = B = cm-1 (
50/3.85 = = 13 so line at 50cm-1 is J=12
B = / I I = / B
I = 8.76 uA2 I =  r2
 = m1m2/(m1+m2)= 16x12/28 = 6.86
8.76/6.86 = = r2
r = 1.277½ = A ( acc B value 1.921)
Harry Kroto 2004

26. My ABC System of Spectroscopy
Harry Kroto 2004

27. Nuclear Energies H + H E(r) Chemical Energies r  v=3 2 1
r 
Harry Kroto 2004

28. Rotational Spectroscopy
Harry Kroto 2004

29. Nuclear Energies H + H E(r) Chemical Energies Rotational levels r 
r 
Harry Kroto 2004

30. A Classical Description > E = T + V E = ½I2 V=0
B QM description > the Hamiltonian H J  = E J  H = J2/2I
C Solve the Hamiltonian > Energy Levels F (J) = BJ(J+1)
D Selection Rules > Allowed Transitions J = ±1
E Transition Frequencies > F B(J+1)
F Intensities > THE SPECTRUM
J Analysis > Pattern recognition; assign J numbers
H Experimental Details > microwave spectrometers
I More Advanced Details: Centrifugal distortion, spin effect
J Information obtainable: structures, dipole moments etc
Harry Kroto 2004

31. m2 m1 Rotational Spectra Linear Molecules Rigid Diatomic molecule
E = ½I2
Rigid Diatomic molecule
Angular velocity  m2 m1
I = r2
= m1m2/(m1+m2)
Harry Kroto 2004

32. Rotational Energy Linear Diatomic Molecules
Rigid Diatomic molecule
Angular velocity  m2
Rotational Energy Linear Diatomic Molecules
E = ½I2 m1
I = r2
= m1m2/(m1+m2)
Harry Kroto 2004

33. A Classical Description > E = T + V E = ½I2 V=0
B QM description > the Hamiltonian H J  = E J  H = J2/2I
C Solve the Hamiltonian > Energy Levels F (J) = BJ(J+1)
D Selection Rules > Allowed Transitions J = ±1
E Transition Frequencies > F B(J+1)
F Intensities > THE SPECTRUM
J Analysis > Pattern recognition; assign J numbers
H Experimental Details > microwave spectrometers
I More Advanced Details: Centrifugal distortion, spin effect
J Information obtainable: structures, dipole moments etc
Harry Kroto 2004

34. Rotational Spectra Linear Molecules
E = ½I2  J2/2I (J = I )
E = ½ mv2  p2/2m (p = mv)
H = J2/2I (Note V= 0)
Harry Kroto 2004

35. Rotational Spectra Linear Molecules
E = ½I2  J2/2I (J = I )
E = ½ mv2  p2/2m (p = mv)
H = J2/2I (Note V= 0)
Harry Kroto 2004

36. H = J2/2I J J2 J  = ħ2 J(J+1) E(J) = (ħ2/2I) J(J+1)
F(J) = B J(J+1)
B = ħ2/h2I MHz
B = ħ2/hc2I cm-1
J J2 J   J* J2 Jd
Harry Kroto 2004

37. A Classical Description > E = T + V E = ½I2 V=0
B QM description > the Hamiltonian H J  = E J  H = J2/2I
C Solve the Hamiltonian > Energy Levels F (J) = BJ(J+1)
D Selection Rules > Allowed Transitions J = ±1
E Transition Frequencies > F B(J+1)
F Intensities > THE SPECTRUM
J Analysis > Pattern recognition; assign J numbers
H Experimental Details > microwave spectrometers
I More Advanced Details: Centrifugal distortion, spin effect
J Information obtainable: structures, dipole moments etc
Harry Kroto 2004

38. H = J2/2I J J2 J  = ħ2 J(J+1) E(J) = (ħ2/2I) J(J+1)
F(J) = B J(J+1)
B = ħ2/h2I MHz
B = ħ2/hc2I cm-1
J J2 J   J* J2 Jd
Harry Kroto 2004

39. H = J2/2I J J2 J  = ħ2 J(J+1) E(J) = (ħ2/2I) J(J+1)
F(J) = B J(J+1)
B = ħ2/h2I MHz
B = ħ2/hc2I cm-1
J J2 J   J* J2 Jd
Harry Kroto 2004

40. H = J2/2I J J2 J  = ħ2 J(J+1) E(J) = (ħ2/2I) J(J+1)
F(J) = B J(J+1)
B = ħ2/h2I MHz
B = ħ2/hc2I cm-1
J J2 J   J* J2 Jd
Harry Kroto 2004

41. H = J2/2I J J2 J  = ħ2 J(J+1) E(J) = (ħ2/2I) J(J+1)
F(J) = B J(J+1)
B = ħ2/h2I MHz
B = ħ2/hc2I cm-1
J J2 J   J* J2 Jd
Harry Kroto 2004

42. Approximately 61.5 – 23 = 38.5 cm-1 = 20B 2B = 3.85 B = 1.925 cm-1 (
Line separations 2B
Approximately 61.5 – 23 = 38.5 cm-1 = 20B
2B = B = cm-1 (
50/3.85 = = 13 so line at 50cm-1 is J=12
B = / I I = / B
I = 8.76 uA2 I =  r2
 = m1m2/(m1+m2)= 16x12/28 = 6.86
8.76/6.86 = = r2
r = 1.277½ = A ( acc B value 1.921)
Harry Kroto 2004

43. A Classical Description > E = T + V E = ½I2 V=0
B QM description > the Hamiltonian H J  = E J  H = J2/2I
C Solve the Hamiltonian > Energy Levels F (J) = BJ(J+1)
D Selection Rules > Allowed Transitions J = ±1
E Transition Frequencies > F B(J+1)
F Intensities > THE SPECTRUM
J Analysis > Pattern recognition; assign J numbers
H Experimental Details > microwave spectrometers
I More Advanced Details: Centrifugal distortion, spin effect
J Information obtainable: structures, dipole moments etc
Harry Kroto 2004

44. Approximately 61.5 – 23 = 38.5 cm-1 = 20B 2B = 3.85 B = 1.925 cm-1 (
Line separations 2B
Approximately 61.5 – 23 = 38.5 cm-1 = 20B
2B = B = cm-1 (
50/3.85 = = 13 so line at 50cm-1 is J=12
B = / I I = / B
I = 8.76 uA2 I =  r2
 = m1m2/(m1+m2)= 16x12/28 = 6.86
8.76/6.86 = = r2
r = 1.277½ = A ( acc B value 1.921)
Harry Kroto 2004

45. Approximately 61.5 – 23 = 38.5 cm-1 = 20B 2B = 3.85 B = 1.925 cm-1 (
J= 12
23.0 cm-1
61.5 cm-1
Line separations 2B
Approximately 61.5 – 23 = 38.5 cm-1 = 20B
2B = B = cm-1 (
50/3.85 = = 13 so line at 50cm-1 is J=12
B = / I I = / B
I = 8.76 uA2 I =  r2
 = m1m2/(m1+m2)= 16x12/28 = 6.86
8.76/6.86 = = r2
r = 1.277½ = A ( acc B value 1.921)
Harry Kroto 2004

46. Approximately 61.5 – 23 = 38.5 cm-1 = 20B 2B = 3.85 B = 1.925 cm-1 ( 10 15
Line separations 2B
Approximately 61.5 – 23 = 38.5 cm-1 = 20B
2B = B = cm-1 (
50/3.85 = = 13 so line at 50cm-1 is J=12
B = / I I = / B
I = 8.76 uA2 I =  r2
 = m1m2/(m1+m2)= 16x12/28 = 6.86
8.76/6.86 = = r2
r = 1.277½ = A ( acc B value 1.921)
Harry Kroto 2004

47. A Classical Description > E = T + V E = ½I2 V=0
B QM description > the Hamiltonian H J  = E J  H = J2/2I
C Solve the Hamiltonian > Energy Levels F (J) = BJ(J+1)
D Selection Rules > Allowed Transitions J = ±1
E Transition Frequencies > F B(J+1)
F Intensities > THE SPECTRUM
J Analysis > Pattern recognition; assign J numbers
H Experimental Details > microwave spectrometers
I More Advanced Details: Centrifugal distortion, spin effect
J Information obtainable: structures, dipole moments etc
Harry Kroto 2004

48. Radiotelescope in Canada
Harry Kroto 2004

49. A Classical Description > E = T + V E = ½I2 V=0
B QM description > the Hamiltonian H J  = E J  H = J2/2I
C Solve the Hamiltonian > Energy Levels F (J) = BJ(J+1)
D Selection Rules > Allowed Transitions J = ±1
E Transition Frequencies > F B(J+1)
F Intensities > THE SPECTRUM
J Analysis > Pattern recognition; assign J numbers
H Experimental Details > microwave spectrometers
I More Advanced Details: Centrifugal distortion, spin effect
J Information obtainable: structures, dipole moments etc
Harry Kroto 2004

50. B(J+1)(J+2) – D(J+1)2(J+2)2 J+1 BJ(J+1) – DJ2(J+1)2 J
F(J) = 2B(J+1) – 4D(J+1)3
Harry Kroto 2004

51. A Classical Description > E = T + V E = ½I2 V=0
B QM description > the Hamiltonian H J  = E J  H = J2/2I
C Solve the Hamiltonian > Energy Levels F (J) = BJ(J+1)
D Selection Rules > Allowed Transitions J = ±1
E Transition Frequencies > F B(J+1)
F Intensities > THE SPECTRUM
J Analysis > Pattern recognition; assign J numbers
H Experimental Details > microwave spectrometers
I More Advanced Details: Centrifugal distortion, spin effect
J Information obtainable: structures, dipole moments etc
Harry Kroto 2004

52. Approximately 61.5 – 23 = 38.5 cm-1 = 20B 2B = 3.85 B = 1.925 cm-1 ( 10 15
Line separations 2B
Approximately 61.5 – 23 = 38.5 cm-1 = 20B
2B = B = cm-1 (
50/3.85 = = 13 so line at 50cm-1 is J=12
B = / I I = / B
I = 8.76 uA2 I =  r2
 = m1m2/(m1+m2)= 16x12/28 = 6.86
8.76/6.86 = = r2
r = 1.277½ = A ( acc B value 1.921)
Harry Kroto 2004

53. Approximately 61.5 – 23 = 38.5 cm-1 = 20B 2B = 3.85 B = 1.925 cm-1 ( 10 15
Line separations 2B
Approximately 61.5 – 23 = 38.5 cm-1 = 20B
2B = B = cm-1 (
50/3.85 = = 13 so line at 50cm-1 is J=12
B = / I I = / B
I = 8.76 uA2 I =  r2
 = m1m2/(m1+m2)= 16x12/28 = 6.86
8.76/6.86 = = r2
r = 1.277½ = A ( acc B value 1.921)
Harry Kroto 2004

54. Harry Kroto 2004