5.80 Lecture #24S
Fall, 2008
Page 1 of 9 pages
Lecture #24 Supplement: Energy levels of a Rigid Rotor
*
1
1
1
I
c
ω
c
2
=
P
a
2
+
P
b
2
+
P
c
2
W
r
=
2
I
a
ω
a
2
+
2
I
b
ω
b
2
+
2
2I
a
2I
b
2I
c
W
r
h
=
AP
a
2
+
BP
b
2
+
CP
c
2
where a, b, c denote the directions of the three principal axes of inertia (fixed in the molecule),
P
is the
total angular momentum vector, with components P
a
, P
b
, P
c
, and the labeling of the axes is chosen so that
I
a
< I
b
< I
c
,
or
A > B > C,
in terms of the rotational constants
A
=
8
π
h
2
I
a
,
B
=
8
π
h
2
I
b
, C
=
8
π
h
2
I
c
.
The magnitude of the total angular momentum vector is quantized, and
P
2
=
P
a
2
+
P
b
2
+
P
c
2
=
J(J
+
1)
2
J
=
0,1,2,
…
The orientation of
P
with respect to a space fixed z-axis is also quantized. The projection of
P
on this
z-axis can have only the values given by
P
z
= M
where, for each given value of J, the “azimuthal quantum number” M takes on the 2J + 1 values
M = J, J – 1, …, 0, …, –J.
It is convenient to distinguish the following types of rotors:
Moments of Inertia
Symmetry
Rotational Constants
Examples
I
a
= 0; I
b
= I
c
Linear
A =
∞
; B = C
HCl, CO
2
I
a
< I
b
= I
c
Prolate
A > B = C
CH
3
Cl, C
2
H
6
, Football
symmetric top
I
a
= I
b
< I
c
Oblate
A = B > C
CH
3
CF
3
, C
6
H
6
, Frisbee
symmetric top
I
a
= I
b
= I
c
Spherical top
A = B = C
CH
4
, SF
6
I
a
< I
b
< I
c
Asymmetric top
A > B > C
H
2
O, C
2
H
4
*
Professor Dudley Herschbach, Department of Chemistry, Harvard University