4.2 Classical Kinetic Energy of the Rigid Rotor
81
rotation. The kinetic energy T of the molecule may be written
2T =
N
α=1
m α V
2
α =
N
α=1
m α
˙
R+ω × r α
2
(4.4)
2T = ˙
R
2
N
α=1
m α +
N
α=1
m α (ω × r α )
2
+ 2 ˙
R · ω
N
α=1
m α r α
(4.5)
N is the number of atoms, the first term is the energy of translation, and the last
term is null because the origin is the center of mass:
N
a=1 m a r a = 0
The kinetic energy of rotation is
2T R =
N
a=1
m a (ω × r a )
2
=
N
a=1
m a
ω
2 r
2
a − (ω · r a )
2
(4.6)
Or
2T R = ω
2
x
N
a=1
m a
y
2
a + z
2
a
+ ω
2
y
N
a=1
m a
x
2
a + z
2
a
+ ω
2
z
N
a=1
m a
y
2
a + x
2
a
− 2ω x ω y
N
a=1
m a x a y a − 2ω y ω z
N
a=1
m a y a z a − 2ω x ω z
N
a=1
m a x a z a
(4.7)
This equation may be rewritten using the moment of inertia tensor I which is a
3 × 3 symmetric matrix
I =
⎛
⎝
I xx I xy I xz
I xy I yy I yz
I xz I yz I zz
⎞
⎠
(4.8)
Its diagonal elements are (with g, g
, g
= x, y, z by cyclic permutation)
I gg =
N
a=1
m a
g
2
a + g
2
a
(4.9)
and the non-diagonal elements
I gg = −
N
a=1
m a g a g
a
(4.10)
Finally,