2. Time independent H o  o = E o  o Time dependent [H o + V(t)]  = iħ  /  t Harry Kroto 2004 Time dependent Schrödinger [H o + V(t)]  = iħ  /  t

4. Atoms Molecules Basically only electronic transitions >10000 cm -1 Harry Kroto 2004

5. C We have to solve the Time independent problem H o  o = E o  o Harry Kroto 2004

6. Atoms Molecules Basically only electronic transitions >10000 cm -1 electronic transitions E > 10000 cm -1 Vibrational transitions E = 100-10000 cm -1 Rotational transitions E = 0.1 – 100 cm -1 Harry Kroto 2004

7. The Born-Oppenheimer Separation H  = E  H = H el + H vib + H rot + … Harry Kroto 2004

8. The Born-Oppenheimer Separation H  = E  H = H el + H vib + H rot + …  =  el  vib  rot …  =  i  i Harry Kroto 2004

9. The Born-Oppenheimer Separation H  = E  H = H el + H vib + H rot + …  =  el  vib  rot …  =  i  i E = E el + E vib + E rot +… E=  i E i Harry Kroto 2004

10. The Born-Oppenheimer Separation H  = E  H = H el + H vib + H rot + …  =  el  vib  rot …  =  i  i E = E el + E vib + E rot +… E=  i E i We shall often use Dirac notation  m   m  and  m *   n  Harry Kroto 2004

12. Time independent H o  o = E o  o Harry Kroto 2004

13. Time independent H o  o = E o  o Stationary States  m o   m  Harry Kroto 2004

14. Time independent H o  o = E o  o Stationary States  m o   m   m   o  Harry Kroto 2004

15. D Selection Rules Need to solve the Time Dependent Problem Harry Kroto 2004

16. Time independent H o  o = E o  o Stationary States  m o   m  Time dependent [H o + V(t)]  = iħ  /  t  m   o  Harry Kroto 2004

17. Time independent H o  o = E o  o Stationary States  m o   m  Time dependent [H o + V(t)]  = iħ  /  t V(t) = -E e (t)  e  m   o  Harry Kroto 2004

18. Time independent H o  o = E o  o Stationary States  m o   m  Time dependent [H o + V(t)]  = iħ  /  t V(t) = -E e (t)  e E e (t) = E e o cos 2  t E e (t) Radiation field  e Electric dipole moment  m   o  Harry Kroto 2004

19. Time independent H o  o = E o  o Stationary States  m o   m  Time dependent [H o + V(t)]  = iħ  /  t V(t) = -E e (t)  e E e (t) = E e o cos 2  t E e (t) Radiation field  e Electric dipole moment  =  m a m (t)  m   m   o  Harry Kroto 2004

23. Fermi’s Golden Rule IoIo I xx l Harry Kroto 2004

24. Fermi’s Golden Rule IoIo I xx l Beer Lambert Law I= I o e -  l Harry Kroto 2004

25. Fermi’s Golden Rule IoIo I xx l Beer Lambert Law I= I o e -  l Harry Kroto 2004

26. Fermi’s Golden Rule Beer Lambert Law I= I o e -  l IoIo I xx l Harry Kroto 2004

27. Fermi’s Golden Rule Beer Lambert law I= I o e -  l IoIo I xx l Harry Kroto 2004

28. Fermi’s Golden Rule Beer Lambert law I= I o e -  l  is the absorption coefficient  = (8  3 /3hc)  n  e  m   2 (N m -N n )  (  o -  ) IoIo I xx l Harry Kroto 2004

29.  = (4  /3ħc)  n  e  m  2  (N m -N n )  (  o -  ) Harry Kroto 2004

30.  = (4  /3ħc)  n  e  m  2  (N m -N n )  (  o -  ) ① 1.Square of the transition moment  n  e  m  2 Harry Kroto 2004

31.  = (4  /3ħc)  n  e  m  2  (N m -N n )  (  o -  ) ① ② 1.Square of the transition moment  n  e  m  2 2.Frequency of the light  Harry Kroto 2004

32.  = (4  /3ħc)  n  e  m  2  (N m -N n )  (  o -  ) ① ② ③ 1.Square of the transition moment  n  e  m  2 2.Frequency of the light  3.Population difference (N m - N n ) Harry Kroto 2004

33.  = (4  /3ħc)  n  e  m  2  (N m -N n )  (  o -  ) ① ② ③ ④ 1.Square of the transition moment  n  e  m  2 2.Frequency of the light  3.Population difference (N m - N n ) 4.Resonance factor - Dirac delta function  (0) = 1 Harry Kroto 2004

34. C Solution > Energy Levels For the H atom we shall just use the Bohr result E(n) = - R/n 2 DSelection Rules  n no restriction  l = ±1 ETransition Frequencies  E = - R[ 1/n 2 2 – 1/n 1 2 ] Harry Kroto 2004

43. Hot gas cloud – the famous Orion Nebulae At the centre is the Trapezium Cluster of very hot new stars Harry Kroto 2004

44. Collisions in the Interstellat Medium ISM In space the pressures are low Very low If n = number of molecules per cc (mainly H) then  2b = 10 3 /n yrs per collision  3b = 10 23 /n 2 yrs per collision Number densities are anything from n = 1-1000 Harry Kroto 2004

45. B n<-m Einstein Coefficients nn mm Harry Kroto 2004

46. B n<-m B n->m nn mm Einstein Coefficients Harry Kroto 2004

47. A n->m / B n->m = 8  h  3 /c 3 B n<-m B n->m A n->m nn mm Einstein Coefficients Harry Kroto 2004

48. A = 1.2 x 10 -37  3  n  e  m  2 transitions per sec Spontaneous emission lifetime   (sec) = 1/A = 10 37 /  3 sec B n<-m B n->m A n->m nn mm Einstein Coefficients Harry Kroto 2004

49.  (sec) = 10 37 /  3  (cm -1 )  (Hz)  3 (Hz 3 )  (sec) H (1420 MHz) 21cm 0.05 1.5x10 9 3x10 27 10 10 * H 2 CO rotations 1cm 1 3 x 10 10 3x10 31 10 6 CO 2 vibrations 10  10 3 3 x 10 13 3 x 10 40 10 -3 Na D electronic 500nm2x10 4 1.5 x 10 14 6 x 10 44 10 -7 H Lyman  100nm 10 5 3 x 10 15 3 x 10 46 10 -9 Calculations assume  e = 1Debye 1yr = 3 x 10 7 sec * magnetic dipole Harry Kroto 2004

51. a n = a o n 2 a o = 0.05 nm Bohr radius Harry Kroto 2004

52. a n = a o n 2 a o = 0.05 nm Calculate a 10, a 100 and a 300 in cm Bohr radius Harry Kroto 2004

53. a n = a o n 2 a o = 0.5 Å (1Å = 10 -8 cm) a 300 = 0.5x10 -3 cm = 0.005 mm Bohr radius Harry Kroto 2004

55. Nitrosoethane Harry Kroto 2004

56. What can molecules do Harry Kroto 2004

57. What can molecules do 2 Harry Kroto 2004

58. What can molecules do 2 Harry Kroto 2004