Exploring Vibrational Energy Levels of Harmonic Oscillator

2. Vibrational Energy Levels Harmonic Oscillator G(v) = ω(v + ½) cm-1

3. Vibrational Energy Levels Harmonic Oscillator G(v) = ω(v + ½) cm-1 Remember the Potential Energy for a SHO is

4. Vibrational Energy Levels Harmonic Oscillator G(v) = ω(v + ½) cm-1 Remember the Potential Energy for a SHO is V(x) = - k x2

5. symmetric stretch of a homonuclear diatomic molecule such as H2

6. symmetric stretch of a homonuclear diatomic molecule such as H2

7. symmetric stretch of a homonuclear diatomic molecule such as H2 re

8. symmetric stretch of a homonuclear diatomic molecule such as H2 re

9. symmetric stretch of a homonuclear diatomic molecule such as H2 r re

10. symmetric stretch of a homonuclear diatomic molecule such as H2 r re The vibrational coordinate is thus x = r – re

12. G(r) re

13. G(r) re is called the equilibrium bond length re

14. G(r) r – re is the vibrational coordinate re r – re

15. Vibrational Energy Levels Harmonic Oscillator G(v) = ω(v + ½) cm-1

16. G(r) re r – re

17. G(r) G(v) = ω(v + ½) re r – re

18. G(r) G(v) = ω(v + ½) v = 0 ½ ω re r – re

19. G(r) v = 1 1½ ω G(v) = ω(v + ½) v = 0 ½ ω re r – re

20. G(r) v = 2 2½ ω v = 1 1½ ω G(v) = ω(v + ½) v = 0 ½ ω re r – re

21. G(r) v = 3 3½ ω v = 2 2½ ω v = 1 1½ ω G(v) = ω(v + ½) v = 0 ½ ω re r – re

22. G(r) v = 4 4½ ω v = 3 3½ ω v = 2 2½ ω v = 1 1½ ω G(v) = ω(v + ½) v = 0 ½ ω re r – re

23. G(r) 5½ ω v = 5 v = 4 4½ ω v = 3 3½ ω v = 2 2½ ω v = 1 1½ ω G(v) = ω(v + ½) v = 0 ½ ω re r – re

24. v = 6 6½ ω G(r) 5½ ω v = 5 v = 4 4½ ω v = 3 3½ ω v = 2 2½ ω v = 1 1½ ω G(v) = ω(v + ½) v = 0 ½ ω re r – re

25. v = 6 6½ ω G(r) 5½ ω v = 5 v = 4 4½ ω v = 3 3½ ω v = 2 2½ ω v = 1 1½ ω G(v) = ω(v + ½) v = 0 ½ ω re r – re

26. v = 6 6½ ω G(r) 5½ ω v = 5 v = 4 4½ ω v = 3 3½ ω v = 2 2½ ω v = 1 1½ ω G(v) = ω(v + ½) v = 0 ½ ω re r – re

27. v = 6 6½ ω G(r) 5½ ω v = 5 v = 4 4½ ω v = 3 3½ ω v = 2 2½ ω Notice that the energy levels are equidistantly space byω v = 1 1½ ω v = 0 ½ ω re r – re

30. Harry Kroto 2004

31. Nuclear Energies H + H E(r) Chemical Energies v=3 2 1 0 r  Harry Kroto 2004

32. Harry Kroto 2004

33. Harry Kroto 2004

34. Harry Kroto 2004

35. - gif - www.files.chem.vt.edu/chem-ed/quantum/graphic...

36. Harry Kroto 2004