Degeneracy - University of Michigan
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Nils Walter: Chem 260 Degeneracy Energy only determined by J ß ß ß all m J = -J,…,+J share the same energy ß ß ß 2J+1 degeneracy Selection rule: ∆ ∆ ∆m J = 0, ± ± ±1 J=3 J=2 m J = m J = m J = J, J-1,…, -J
Transcript of Degeneracy - University of Michigan
Nils Walter: Chem 260
DegeneracyEnergy only
determined by J����
all mJ = -J,…,+Jshare the same
energy����
2J+1 degeneracy
Selection rule: ∆∆∆∆mJ = 0, ±±±±1
J=3
J=2
mJ =mJ =
mJ = J, J-1,…, -J
Nils Walter: Chem 260
Total relativepopulation∆∆∆∆J = ±±±±1
The allowed rotational transitions of arigid linear rotor and their intensity
2J + 1degenerate
states ����increasinglydegenerate
B large B small
2J + 1
E = hBJ(J+1)
Boltzmanndistribution
kTE
lower
upper eNN ∆−
=
Nils Walter: Chem 260
Rotational Raman spectroscopy
Experimentalsetup:
laser
Gross selection rule:anisotropic polarization
(example: H-H!)
Specific selection rules:∆∆∆∆J = +2 (Stokes lines)
∆∆∆∆J = -2 (anti-Stokes lines)
Nils Walter: Chem 260
The vibration of molecules
Morse potential
νhE ��
���
� +=21vv
Solution for the harmonicoscillator:
���
����
�=
µπν k
21
2)(21
eRRkV −=
BA
BA
mmmm
+=µeffective
(reduced) mass
A B B
zero point energy
springconstant
Nils Walter: Chem 260
���
����
�=
µπν k
21
���� 300-3000 cm-1 = Infrared
Vibrational transitions:Infrared spectroscopy
Gross selection rule:the electric dipole
moment of themolecule must changeduring the vibration
Nils Walter: Chem 260
Specific selection rulesIn reality:
anharmonic oscillator
D0
Deq
ev xhhE νν2
21v
21v �
�
���
� +−��
���
� +=
xe = anharmonicity constant
∆∆∆∆v = ±±±±1, ±±±±2, ±±±±3,...
∆∆∆∆v = ±±±±1
harmonic oscillator
hνννν
Nils Walter: Chem 260
The vibrating rotor
Born-Oppenheimer approximation:The energies of rotations and vibrations are so
different that Etotal = Erot. + Evib.
∆ ∆ ∆ ∆J = -1 ∆ ∆ ∆ ∆J = +1
Nils Walter: Chem 260
Vibrations of polyatomic molecules:How many are there?Each atom can move along one of three axes:
���� 3N possible displacements (= degrees of freedom)����
Three of these degrees of freedom correspond to translational motion:���� 3N - 3 degrees of freedom left
����
Three (/two) degrees of freedom correspond to rotations:���� 3N - 6 (3N - 5 for linear molecule) degrees of freedom left for vibrations
