Rotational spectroscopy: intensities

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
y
* f y i d t
 Intensity of transition
I
m 2

Rotational selection rules
Oscillating electric field (microwave)
Transition moment m fi
=
e
=
Y
J f f v
M f
J , f
Y
J f J , f
ˆ e
(
M
e i v i
ˆ x e i v i i v i
Y
J i d t
M
J , i e d t d t v
) e d t
Y
J i v d t
M
J , i r d t r
µ permanent dipole m ev
= m ev
Y
J f
M
J , f
ˆ Y
J i
M
J , i d t r
No electronic / vibrational transition

Rotational selection rules
 Gross selection rule: nonzero permanent dipole






Does H
2
O have microwave spectra?
Yes
Does N
2
No have microwave spectra?
Does O
2
No have microwave spectra?

Quantum in nature
Microwave spectroscopy
How could astrochemists know H
2
O exist in interstellar medium?
Public image
NASA

Selection rules of atomic spectra(review)
TDM
µ
Y
* l
¢ m l
¢
(
Y
10
or Y
1
±
1
)
Y lm l sin q d q d j ´ d m
¢ s m s
 From the mathematical properties of spherical harmonics, this integral is zero unless
D m l
D l
= ¢ l
= ±
1
= m l
¢
D m s
m l
=
0
=
0,
±
1

Rotational selection rules
 Specific selection rule:
J
1
M
J
0,
1 m fi
= m ev
ò
Y
J f
M
J , f
ˆ Y
J i
M
J , i d t r

Spherical & linear rotors
 In units of wave number (cm
–1
):
( )


1
( )

(
 
2 BJ

Nonrigid rotor:
Centrifugal distortion
   
Diatomic molecule
 ( )
=
J
2
BJ J
+
1 1

2
D
J
»
4 B
3 n
2
Vibrational frequency
B
=
4 p cI
I
= m m
A m
B
A
+ m
B
R
2

Nonrigid rotor:
Centrifugal distortion
Nonrigid
    
D J
J
2

J

1

2
F J
Rigid
( )
=
BJ J
+
1

Appearance of rotational spectra
 Rapidly increasing and then decreasing intensities
Transition moment 2
Degeneracy g
J
 m






J
+
1, J

2 µ
2 J

1
J

2 J

1

1 m
2 ev
2

Boltzmann distribution
(temperature effect)
e
-
E
J
/ kT = e
hcBJ ( J
+
1)/ kT

Rotational Raman spectra
 Gross selection rule: polarizability changes by rotation
 Specific selection rule: x 2 + y 2 + z 2 ~ Y
0,0 xy , etc. are essentially Y
0,0
, Y
2,0
, Y
2, ± 1
, Y
2, ± 2
I
µ å k
ò
Y
J
* f
, M
J , f
ˆ Y k
E
0
(0)
(0) d t ò
-
E k
(0)
Y (0)* k
± h n
J i
, M
J , i d t
Linear rotors: Δ J = 0, ± 2
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)

H
2 rotational Raman spectra
 Why does the intensity alternate?

H
2 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 H
2
Singlet ( paraH
2
)
Y r
1
, r
2
Sym.
( )
µ
{ spatial part of rotation
Antisym.
} { a
(1) b
(2)
b
(1) a
(2)
}
Nuclear (proton) spins
Triplet ( orthoH
2
)
Antisym.
Y
( )
µ
{ spatial part of rotation
}
¢
¢
Sym.
a
(1) a
(2) b
(1) b
(2) a
(1) b
(2)
+ b
(1) a
(2)
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 H
Singlet ( paraH
2
)
Y
2
( )
µ
Antisym.
{
J
= odd
}
¢
¢
Y r
1
, r
2
Sym.
( )
µ
{
J
= even
} { a
(1) b
(2)
b
(1) a
(2)
}
Sym.
a
(1) a
(2) b
(1) b
(2) a
(1) b
(2)
+ b
(1) a
(2)
Triplet ( orthoH
2
)

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 H
2 rotational Raman spectra.