1
Rotational Spectroscopy
Definitions
Moment of Inertia: I   mi ri,2 where r is the Cartesian coordinate in the th direction.
i
Principal Rotation Axes: Three perpendicular rotation axes in a molecule defined as follows:
i) c axis – rotation axis that yield maximum moment of inertia
ii) b axis – rotation axis perpendicular to c axis that yields maximum moment of inertia
iii) a axis – rotation axis perpendicular to c axis and b axis
- Therefore by convention; Ic  Ib  Ia
Types of Rigid Rotors
Spherical Rotor (Top)
Spherical rotors have three equal principal moments of inertia. Ic  Ib  Ia
The Hamiltonian and energy for a spherical top are
2
Jˆ a2
Jˆ b2
Jˆ c2
Jˆ 2
H


 H
 E  J  J  1
2Ia 2I b 2I c
2I
2I
A spectroscopic term can be defined as the energy level divided by Planck’s constant and
the speed of light. Thus spectroscopic terms yield wavenumbers directly and thus are more
helpful to a spectroscopist.

E
hc
The spectroscopic term for a spherical rotor is
F  J   BJ  J  1 where B 

4cI b 4cI
Tetrahedral and octahedral molecules are spherical rotors.
Rotation of a spherical rotor can be characterized with quantum numbers J and M where M
specifies the molecules orientation in space (laboratory frame of reference).
Symmetric Rotor (Top)
Symmetric rotors have two equal principal moments of inertia and one that is different.
Two cases exist for symmetric rotors.
2
Prolate tops
In a prolate top, the c axis and b axis moments of inertia are equal, that is, Ic  Ib  Ia .
a
c
b
Prolate tops are “cigar-shaped” or “football-shaped”.
The Hamiltonian for a prolate top is
Jˆ 2
Jˆ 2
Jˆ 2
Jˆ 2 Jˆ 2  Jˆ 2 Jˆ 2  Jˆ 2
Jˆ 2  Jˆ 2  Jˆ 2
H  a  b  c  a  b c  a  a  a   b c
2Ia 2I b 2Ic 2Ia
2I b
2I a  2I b 2I b 
2I b
Jˆ a2
Jˆ a2  Jˆ 2b  Jˆ c2
Jˆ a2
Jˆ 2




2  Ia  I b 
2I b
2  I a  I b  2I b
Ĵa2 
2
K2 where K is quantum number describing the projection of total angular
momentum along “a” axis.
Therefore, the energy for a prolate top is
2
2
1 1
E     K2 
J  J  1
2  Ia I b 
2Ib
The spectroscopic term for a prolate top is
F  J, K   BJ  J  1   A  B K 2 where B 
4cI b
and A 
4cI a
Molecules like ammonia and methyl fluoride are prolate tops.
Rotations of symmetric tops are characterized by the J, K, and M quantum numbers.
- The wavefunctions of symmetrical tops, J K M , are linear combinations of spherical
harmonic functions.
- We will not consider explicit form of wavefunction because we know how the J K M
wavefunction behaves with the angular momentum operators.
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Oblate top
In an oblate top, the a axis and b axis moments of inertia are equal, that is, Ic  Ib  Ia .
c
a
b
Oblate tops are “disc shaped”
The Hamiltonian for an oblate top is
Jˆ a2
Jˆ 2b
Jˆ c2 Jˆ a2  Jˆ 2b Jˆ c2 Jˆ a2  Jˆ 2b  Jˆ c2
Jˆ c2  Jˆ c2
H








2Ia 2I b 2Ic
2I b
2I c
2I b  2I b 2I b  2I c
Jˆ a2  Jˆ 2b  Jˆ c2
Jˆ c2
Jˆ c2
Jˆ 2




2I b
2  Ic  I b  2I b 2  Ic  I b 
Ĵc2 
2
K2 where K is quantum number describing the projection of total angular
momentum along “c” axis.  Note difference in axis from prolate case
Therefore, the energy for an oblate top is
E
2
2Ib
J  J  1 
1 1 2
  K
2  Ic I b 
2
The spectroscopic term for an oblate top is
F  J, K   BJ  J  1   C  B K 2 where B 
4cI b
and C 
4cIc
Molecules like tetraamminenickel(II) and trans-diaquatetrachlorocobalt(III) are oblate tops.
OH2
NH3
H3N
H3N
Ni
NH3
Cl
Cl
Co
Cl
Cl
OH2
A prolate top “smooshed” along its “a” axis becomes a spherical top and upon further
compression, it becomes an oblate top.
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Linear Rotors
A linear rotor is a special case of a symmetric top where K = 0
Thus the energy and term of a linear rotor is
E
2
2I b
J  J  1
F  J, K   BJ  J  1
Asymmetric Rotors
Asymmetric Rotors have all three principal moments of inertia different. Ic  Ib  Ia
The Hamiltonian for the asymmetric top is
Jˆ 2
Jˆ 2
Jˆ 2
H a  b  c
2Ia 2I b 2I c
The energy levels and spectroscopic terms are very complicated. Analysis of the rotational
spectrum of an asymmetric top is very difficult. Usually an asymmetric top is
approximated as a prolate or oblate top. Then the analysis becomes tractable.
Deviations from the Rigid Rotor
Centrifugal Distortion
As a molecule rotates faster, the centrifugal force causes the bonds to stretch, thus changing
the moment of inertia of the molecule, thus changing its energy.
Spectroscopic term for centrifugal distortion in a diatomic molecule in vibrational ground
state.
F  J   D 0 J 2  J  1
2
If the vibrational potential energy is well approximated by a Morse potential, the centrifugal
distortion constant, D0, is
D0 
4B3e
 e2
Be – rotation constant for equilibrium bond lengths
e – fundamental frequency
e 
1 k
2 
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Harmonic Oscillator Interaction
As the molecule vibrates, the moment of inertia is constantly changing.
At low J and low , the vibration does not affect the rotational energy levels since the
rotation is slow compared to the vibrational motion. Thus as a molecule rotates, the bond
lengths oscillate several times in the course of one rotation. Therefore, using an average
bond length in the moment of inertia expression is appropriate.
r2  r
2
 re2
When J increases, the argument used above begins to fail. Likewise for increasing ,
2
r 2  r because the vibration is anharmonic.
Spectroscopic term of the harmonic oscillator interaction for a diatomic molecule is
1

F  J,     e J  J  1     .
2

If the vibrational potential energy is well approximated by a Morse potential, the vibrationrotation interaction constant, e, is
1
  B3  2 6B2
e  6  e e   e
e
 e 
e – “cubic” anharmonic constant
Selection Rules for Symmetric Rotors
Recall that the wavefunctions for the rigid rotor are spherical harmonics.
For a rotational transition to occur, YJM  ,   ˆ YJM   ,   must be nonzero.
In Cartesian coordinates,

  e x ˆi  y ˆj  z kˆ


In spherical polar coordinates,   er sin  cos  ˆi  sin  sin  ˆj  cos  kˆ

A recursion relationship exists between spherical harmonics that allows for a general solution
as to when the transition dipole matrix element is nonzero.
- Recall identical concept used to find selection rules of harmonic oscillator
wavefunctions.
Nonzero matrix elements occur when J  J 1 and J  J 1
Therefore selection rule is J  1
cos YJM  ,   
 J  1 Y M ,    J  1 Y M , 
 
 
 2J  1 J 1
 2J  1 J 1
Nonzero matrix elements occur when M  M  1 , M  M  1 and M  M
Therefore selection rule is M  0, 1
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1 i  i
1
e  e  cos    ei  e  i 

2i
2
Because we have not explicitly considered the K dependence of the wavefunctions, the origin
of the K selection rule is unclear. Nonetheless, the selection rule is K  0, 1 .
sin  
Stark Effect
Energy levels of rigid rotor are doubly degenerate (except J = 0)
- Clockwise rotation has same energy as counterclockwise rotation.
In polar molecules, energy levels can be made nondegenerate by applying an electric field.
- Dipoles aligned with field have lower energy.
- Dipoles aligned against field have higher energy.
 - permanent dipole moment
K
E – electric field
E
E cos 
J  J  1
 - angle between J and E
The angle  can also be defined by cos  
M
J  J  1
Thus the first-order Stark effect energy can be written as
KME
E
J  J  1
Nonpolar molecules have no Stark effect. (I.e., degenerate energy levels not affected by E.)
Also linear molecules exhibit no “first-order” Stark effect (since K = 0).
-Energy can be affected by E2, E3, etc … as well.
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Nuclear Spin and Rotational States
Review Pauli Exclusion Principle
When two fermions (S = ½, …) are exchanged, wavefunction must be antisymmetric.
When two bosons (S = 1, 2, …) are exchanged, wavefunction must be symmetric.
Assume we have homonuclear diatomic molecule in  g state. (Worry only about effect of
nuclei, electronic state will have no effect.)
Parity of spin state times parity of rotational state must equal total parity of state.
Parity of rotational state = (-1)J
Parity is odd (negative) when J = 1, 3, 5, …
Parity is even (positive) when J = 0, 2, 4, …
Parity of spin state, I
Number of positive parity spin states I  1

For fermions,
Number of negative parity spin states
I
Number of positive parity spin states
I

For bosons,
Number of negative parity spin states I  1
I
1
I 1 

I 1 2
Parity of I = ½ spin state
Parity of singlet state = -1
Parity of triplet state = +1
Example: Ortho-Para Hydrogen
Ortho-hydrogen – nuclear spins are parallel – triplet state
Parity of wavefunction = Parity of spin state  Parity of rotation
-1
=
1

?
Parity of rotation must be negative, thus J is only odd.
-i.e., in ortho-hydrogen, the even J states are missing.
Para-hydrogen – nuclear spins are paired – singlet state
Parity of wavefunction = Parity of spin state  Parity of rotation
-1
=
-1

?
Parity of rotation must be positive, thus J is only even.
-i.e., in para-hydrogen, the odd J states are missing.
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In the vibration-rotation spectrum of hydrogen, the peak heights reflect the fact that
hydrogen has three times more molecules in J = odd states (from nuclear spin triplet) as it has
molecules in J = even states (from nuclear spin singlet)
Consider the spectrum of C2H2. The rotational spectrum behaves like the H2 spectrum.
- 12C has a nuclear spin of zero and as such does not affect the ratio of intensities.
(Taken from L. Willard Richards, J. of Chemical Education, Vol 43, pp. 644-647 (1966))
The higher intensity lines come from odd J rotational states (negative parity).
The lower intensity lines come from even J rotational states (positive parity).
The ratio of the intensities between neighboring peaks is consistent with a ratio of 3 to 1
Number of positive parity spin states I  1

Number of negative parity spin states
I
I
1

2
I 1 3

I
1
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Consider the spectrum of C2D2. The rotational spectrum behaves like the D2 spectrum.
(Taken from L. Willard Richards, J. of Chemical Education, Vol 43, pp. 644-647 (1966))
The higher intensity lines come from even J rotational states (positive parity).
The lower intensity lines come from odd J rotational states (negative parity).
The ratio of the intensities between neighboring peaks is consistent with a ratio of 2 to 1
Number of positive parity spin states
I

Number of negative parity spin states I  1
I 1 
I
1

I 1 2
** These rotational spectra dramatically illustrate that the Pauli exclusion principle
and antisymmetry have measurable effects. The Pauli exclusion principle is not a
mathematical bookkeeping device for quantum particles. **