Lecture 34
Rotational spectroscopy: intensities
(c) So Hirata, Department of Chemistry, University of Illinois at Urbana-Champaign. This material has
been developed and made available online by work supported jointly by University of Illinois, the
National Science Foundation under Grant CHE-1118616 (CAREER), and the Camille & Henry Dreyfus
Foundation, Inc. through the Camille Dreyfus Teacher-Scholar program. Any opinions, findings, and
conclusions or recommendations expressed in this material are those of the author(s) and do not
necessarily reflect the views of the sponsoring agencies.
Rotational spectroscopy


In the previous lecture, we have
considered the rotational energy levels.
In this lecture, we will focus more on
selection rules and intensities.
Selection rules and intensities
(review)

Transition dipole moment
m = òy x̂y i dt
*
f

Intensity of transition
Iµm
2
Rotational selection rules
Transition
moment
Oscillating electric field (microwave)
m fi = ò e f v f YJ
= ò YJ
f MJ , f
f MJ , f
x̂ e i viYJ M dt e dt v dt r
i
J ,i
( ò e v x̂e v dt dt ) Y
i i
i i
e
v
J i M J ,i
dt r
µpermanent dipole mev
= mev ò YJ
f MJ , f
x̂ YJ M dt r
i
J ,i
No electronic /
vibrational transition
Rotational selection rules

Gross selection rule:
nonzero permanent dipole

Does H2O have microwave spectra?
Yes
Does N2 have microwave spectra?
No
Does O2 have microwave spectra?
No





Quantum in nature
Microwave
spectroscopy
How could
astrochemists
know H2O exist
in interstellar
medium?
Public image
NASA
Selection rules of atomic
spectra(review)
TDM µ òò Yl*¢m¢ (Y10 or Y1±1 ) Ylm sinq dq dj ´ d m¢m
l

l
s s
From the mathematical properties of
spherical harmonics, this integral is zero
unless
Dl = l ¢ - l = ±1
Dml = ml¢ - ml = 0,±1
Dms = 0
Rotational selection rules

Specific selection rule:
DJ = ±1
DM J = 0,±1
m fi = mev ò YJ
f MJ , f
x̂ YJ M dt r
i
J ,i
Spherical & linear rotors

In units of wave number (cm–1):
F ( J )  BJ  J  1
F ( J )  F ( J  1)  2 BJ
Nonrigid rotor:
Centrifugal distortion
Diatomic molecule
( )
(
)
F  J   BJ  J F1J D=J BJ
J  JJ +1
1
2
2
DJ »
4B
n
3
2
Vibrational
frequency
B=
4p cI
mA mB 2
I=
R
mA + mB
Nonrigid rotor:
Centrifugal distortion
Nonrigid
F  J   BJ  J  1  DJ J
Rigid
2
 J  1
2
F ( J ) = BJ ( J +1)
Appearance of rotational spectra

Rapidly increasing
and then decreasing
intensities
2
Transition
moment2
Degeneracy
m J +1,J µ mev2
 2J 1

g J   2  2 J  1

2
  2 J  1
Boltzmann distribution
(temperature effect)
- EJ /kT
e
-hcBJ ( J +1)/kT
=e
Rotational Raman spectra


Gross selection rule:
polarizability changes by rotation
x2 + y2 + z2 ~ Y0,0
Specific selection rule:
xy, etc. are essentially Y0,0, Y2,0, Y2,±1, Y2,±2
Iµ
å
k
òY
*
J f ,M J , f
x̂Y
(0)
k
dt ò Y
(0)*
k
ŷYJ ,M dt
i
2
J ,i
E0(0) - Ek(0) ± hn
Linear rotors: ΔJ = 0, ±2
Spherical rotors: inactive (rotation
cannot change the polarizability)
Rotational Raman spectra

Anti-Stokes wing slightly less intense than
Stokes wing – why?

Boltzmann distribution (temperature effect)
Rotational Raman spectra

Each wing’s envelope is explained by the
competing effects of


Degeneracy
Boltzmann distribution (temperature effect)
H2 rotational Raman spectra

Why does the intensity alternate?
H2 rotational Raman spectra

Why does the intensity alternate?
Answer: odd J levels are triply degenerate
(triplets), whereas even J levels are singlets.
Nuclear spin statistics




Electrons play no role here; we are concerned
with the rotational motion of nuclei.
The hydrogen’s nuclei (protons) are fermions
and have α / β spins .
The rotational wave function (including
nuclear spin part) must be antisymmetric
with respect to interchange of the two nuclei.
The molecular rotation through 180°
amounts to interchange.
Para and ortho H2
Singlet (para-H2)
Sym.
Antisym.
Y ( r1 ,r2 ) µ {spatial part of rotation} {a (1)b (2) - b (1)a (2)}
Nuclear (proton) spins
Triplet (ortho-H2)
Sym.
ì
a (1)a (2)
ïï
Y ( r1 ,r2 ) µ {spatial part of rotation} í
b (1)b (2)
ï a (1)b (2) + b (1)a (2)
ïî
Antisym.
With respect to interchange (180° molecular rotation)
Spatial part of rotational wave
function

By 180 degree rotation, the wave function
changes sign as (–1)J (cf. particle on a ring)
Para and ortho H2
Singlet (para-H2)
Sym.
Antisym.ì
Sym.
a (1)a (2)
ïï
Y ( r1 ,r2 ) µ { J = odd } í
b (1)b (2)
ï a (1)b (2) + b (1)a (2)
ïî
Antisym.
Y ( r1 ,r2 ) µ { J = even} {a (1)b (2) - b (1)a (2)}
Triplet (ortho-H2)
Summary




We have learned the gross and specific
selection rules of rotational absorption and
Raman spectroscopies.
We have explained the typical appearance of
rotational spectra where the temperature
effect and degeneracy of states are
important.
We have learned that nonrigid rotors exhibit
the centrifugal distortion effects.
We have seen the striking effect of the
antisymmetry of proton wave functions in the
appearance of H2 rotational Raman spectra.