Molecular Quantum States

1
Molecular Quantum States Spectroscopy
(Transition Probability) Fred J.
Grieman
2
Molecular Quantum States
Electronic (Diatomic Molecule)   He -½ ?i? i 2
?i (1/riA) ?i (1/riB)?i?ji1 (1/rij)
1/R   He ?e Ee?e
See handout on webpage for diatomic molecular
term symbols
3
Vibration Nuclear Hamiltonian H?N -½
?Nme/MN?N2 Ee?N Etotal?N Harmonic Oscillator
Model Ee ½ kr2 r R - Re Solved Ev h?
(v½) v 0,1,2,3, ? ½?(k/?)½
?e (v½) ?e (cm-1) h?/hc k stiffness of
bond second derivative ? curvature
4
Rotation   V 0 , Rigid Rotor   Erot
(?2/2I)J(J1) J 0,1,2, 2J1 degenerate
Be J(J1) Be(cm-1) ?2/(2Ihc) I
?r2
5
Total Energy Eint Ee Ev Er Etot Etrans
Eint ? ?e?N ?e?v?r How do we see these
states? One way interaction with light
spectroscopy Rotation microwave Vibration
infrared Electronic visible/ultraviolet
6
I2 electronic spectrum with vibration
sub-structure
7

• I ? Probability ? Population
• ) Population Boltzmann Equilibrium
• Ni (Ntot/q) gi e-?i/kT q
  partition function (later)
• gi degeneracy
• k Boltzmanns constant
• Relative populations
• (Ni/Nj) (gi/gj) e-(?i-?j)/kT
  (Proved Later)
• 2) Probability from Q.M. ? Selection
  Rules/Line Strengths
• Transition Probability
  via
• Time Dependent Perturbation
  Theory

8
Light is time dependent phenomenon Spectroscopy
transition between two states via light Consider
General Case
?2 E
h? h?
?1
Complete wave function ?2 ?2e-iE2t/?
?2e-i?2t ?1 ?1e-iE1t/? ?1e-i?1t
9
Time Dependent Schroedinger Equation H?j i?
??j/?t System wave function ? c1(t) ?1
c2(t) ?2 Hamiltonian H H? H (e.g., H
-? ? E ) (H? H) c1(t) ?1 c2(t)
?2 i? ? c1(t) ?1 c2(t) ?2 /?t
c1H??1 c2H??2 c1H?1
c2H?2 c1 i? ??1/?t c2 i? ??2/?t
i?( ?1?c1/?t ?2?c2/?t)
because H??i i? ??i/?t
10
then c1H?1 c2H?2 i?( ?1?c1/?t
?2?c2/?t) We want P2 c22 probability
of being in state 2 x ?2 on left and
integrate c1lt?2H?1gt c2lt?2H?2gt i?
?c2/?t Assume t -?, c1 1, c2 0 t t , c1
? 1 c2 ? 0 i? ?c2/?t lt?2H?1gt
(space integral)
e-i(?1-?2)t lt?2H?1gt
11
Solve for c2 c2 (1/i?)?-t? lt?2H??1gt
e-i(?12)t dt ?12 ?1 - ?2 Now for light
specifically H -? ? E electric field
interaction ? permanent dipole (rotation)
change in dipole (vibration) electronic
dipole (electronic) E from light Assume at
first x polarized Ex Ex? cos?t (Ex?/2)
(ei?t e-i?t) Then H? -?x Ex
-?x(Ex?/2)(ei?t e-i?t) f(x)f(t)
i? ?c2/?t e-i(?1-?2)t lt?2H??1gt
12
c2 (-1/2i?) Ex? lt?2?x?1gt ?-t?
(ei?te-i?t) e-i(?12)t dt (-Ex?/2i?)
Rx21 sin(??t/2)/(??/2) ??
?-?12 c22 (Ex?/2?)2 Rx212
sin2(??t/2)/(??/2)2
Intensity Line strength Line
shape of light selection
rules
Look at Rx212 lt?2?x?1gt2
(?21)x2 (text) transition
dipole next time