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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

 = (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

 = (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

 = (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

 = (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

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

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

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 = /n 2 yrs per collision Number densities are anything from n = Harry Kroto 2004

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

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

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

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

 (sec) = /  3  (cm -1 )  (Hz)  3 (Hz 3 )  (sec) H (1420 MHz) 21cm x10 9 3x * H 2 CO rotations 1cm 1 3 x x CO 2 vibrations 10  x x Na D electronic 500nm2x x x H Lyman  100nm x x Calculations assume  e = 1Debye 1yr = 3 x 10 7 sec * magnetic dipole Harry Kroto 2004

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

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

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

Nitrosoethane Harry Kroto 2004

What can molecules do Harry Kroto 2004

What can molecules do 2 Harry Kroto 2004

What can molecules do 2 Harry Kroto 2004