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```Lecture 33
Rotational spectroscopy: energies
Rotational spectroscopy


In this lecture, we will consider the
rotational energy levels.
In the next lecture, we will focus more on
selection rules and intensities.
Diatomic molecules

A rigid rotor in 3D =
the particle on a sphere
l(l +1)
El =
2
2mr
J (J +1)
EJ =
2I
2
2
mA mB 2
r
Moment of inertia I = mr =
mA + mB
2
Polyatomic molecules
Moment of inertia
I xx = å m x
2
i i
i
æ I
xx
ç
I = ç I yx
ç
çè I zx
I xy
I yy
I zy
I xz ö
÷
I yz ÷
÷
I zz ÷ø
COM x
åm x
=
åm
i i
i
i
i
Moments of inertia
æ I
xx
ç
ç I yx
ç
çè I zx
æ I
xx
ç
ç I yx
ç
çè I zx
æ I
xx
ç
ç I yx
ç
çè I zx
I xy
I yy
I zy
I xy
I yy
I zy
I xy
I yy
I zy
I xz ö æ ax
÷ç
I yz ÷ ç a y
÷ç
I zz ÷ø çè az
æ a ö
ö
x
ç
÷
÷
÷ = Ia ç ay ÷
ç
÷
÷
ç
÷ø
è az ÷ø
Principal axis of rotation
Principal moment of inertia
æ b ö
I xz ö æ bx ö
x
÷ç
÷
ç
÷
I yz ÷ ç by ÷ = I b ç by ÷
÷ç
÷
ç
÷
ç
÷
ç
I zz ÷ø è bz ø
è bz ÷ø
æ c ö
I xz ö æ cx ö
x
÷ç
÷
ç
÷
I yz ÷ ç c y ÷ = I c ç c y ÷
÷ç
÷
ç
÷
ç
÷
ç
I zz ÷ø è cz ø
è cz ÷ø
When faced with a symmetric matrix,
always look for eigenvalues/eigenvectors.
Linear rotors
mAmB 2
I a = Ib =
R
mA + mB
One moment of inertia (Ic) equal to zero
Spherical rotors
8
I a = I b = I c = mH R2
3
I a = I b = I c = 4mF R
Three equal moments of inertia
2
Symmetric rotors
I a = 2mH R 2 (1- cosq )
mH mN
I b = I c = mH R (1- cosq ) +
R 2 (1+ 2cosq )
3mH + mN
2
Two equal moments of inertia
The rotational energy levels

n
(in cm–1) = v / c = hv / hc = E / hc
(
) 2I = hcBJ ( J +1)
EJ = J J +1
B=
2
hc2I
=
2
4p cI
The rotational constant B is given in cm–1
Quantum in nature
Microwave
spectroscopy
or
computational
quantum
chemistry
How could
chemists know
HF bond
length is
0.917Å?
Wikipedia
mH mF 2
B=
;I=
RHF
4p cI
mH + mF
Spherical & linear rotors

In units of wave number (cm–1):
(
) 2I = hcBJ ( J +1)
EJ = J J +1
2
F ( J )  BJ  J 1
Symmetric rotors

Classically,
l +l
l
l -l
l
l2 æ 1
1 ö 2
E=
+
=
+
=
+ç
la
÷
2I b
2I a
2I b
2I a 2I b è 2I a 2I b ø
 Quantum-mechanical rotational terms are
2
b
2
c
(
2
a
)
2
2
a
(
2
a
) (
)
F J , K = BJ J + 1 + A - B K
2
J = 0,1,2,…
K = 0,±1,… ,±J
4p cI a
4p cI b
Symmetric rotors
(
)
(
) (
)
F J , K = BJ J + 1 + A - B K 2
J = 0,1,2,…
K = 0,±1,… ,±J
K acts just like MJ. The
only distinction is that K
refers to rotation around
the principal axis, whereas
MJ to externally fixed axis
Degeneracy




F ( J, K ) = BJ ( J +1) + ( A - B) K 2
Linear rotors
 (2J+1)-fold degeneracy (mJ = 0,±1,…, ±J)
Symmetric rotors
 (2J+1)-fold degeneracy (mJ = 0,±1,…, ±J)
 another 2-fold degeneracy (K = ±1,…, ±J)
[(A – B) ≠ 0 and K and –K give the same energy]
Spherical rotors
 (2J+1)-fold degeneracy (mJ = 0,±1,…, ±J)
 another (2J+1)-fold degeneracy (K = 0,±1,…,
±J)
[(A – B) = 0 and energy does not depend on K]
K for shape, mJ for orientation.
Zeeman effect
mJ-degeneracy can be lifted by
applying a static magnetic field
along z axis.
(0)* ˆ ˆ
(0)
Y
h×
l
Y
dt
z
ò
Zeeman effect
First-order
perturbation theory
(0)* ˆ ˆ
(0)
Y
h×
l
Y
dt
z
ò
E ( J, M J ) = hcBJ ( J +1) - ghM J
Stark effect
mJ-degeneracy can be lifted by
applying a static electric field along
z axis.
E  J , M J   hcBJ  J  1  a  J , M J   2 E 2
aJ,MJ  
J  J  1  3M J2
2hcBJ  J  1 2 J  1 2 J  3
Stark effect
Second-order
perturbation theory
E  J , M J   hcBJ  J  1  a  J , M J   2 E 2
aJ,MJ  
J  J  1  3M J2
2hcBJ  J  1 2 J  1 2 J  3
Linear rotors
2J+1 degenerate
No rotation
around the axis
No degeneracy
Zeeman
effect
1ST order
PT
Stark effect
2ND order PT
Spherical rotors
2J+1 degenerate
2J+1 degenerate
Zeeman
effect
1ST order
PT
Stark effect
2ND order PT
Symmetric rotors
2J+1 degenerate
Doubly degenerate
Zeeman
effect
1ST order
PT
Stark effect
2ND order PT
Willis Flygare
From UIUC Department of Chemistry website
“Willis Flygare earned his bachelor's degree
from St. Olaf College in 1958 and his
doctorate from the University of California at
Berkeley in 1961. He was a professor of
chemistry at Illinois from 1961 until his
death. During that period he developed a
new experimental method involving the
molecular Zeeman effect, and with it, he
measured most of the known molecular
susceptibility anisotropies. He developed a
highly sensitive microwave spectrometer by
combining molecular beams with Fourier
transform techniques.”
Summary




We have learned the rotational energy levels
of molecules in the rigid-rotor approximation.
We have classified rigid rotors into linear
rotors, spherical rotors, symmetric rotors, and
the rest.
We have discussed the energy levels and
their degeneracy of these rotors.
We have learned the Zeeman and Stark
effects.
```