64
3 Diatomic Molecules
ξ =
r − r e
r e
≈
2B e
ω e
2
J (J + 1)
(3.45)
Example: H
35 Cl
With B e = 317587 MHz and ω e = 2891 cm
−1 , r – r e = 0.7 pm for J = 10 and
5.95 pm for J = 30.
The Hamiltonian of the non-vibrating molecule is
H =
P
2
2μr 2 + V =
P
2
2μr 2
e
(1 − 2ξ ) + V
(3.46)
Using the expression of r – r e , 3.44, the potential energy may be written in unit
of frequency (i.e., dropping h)
V =
1
2
k(r − r e )
2
≈ 4
B
3
e
ω 2
e
J
2
(J + 1)
2
= D e J
2
(J + 1)
2
(3.47)
D e is called quartic centrifugal distortion constant.
Finally, the rotational energy, without vibration, is
E R = B e J (J + 1) − D e J
2
(J + 1)
2
(3.48)
For the centrifugal distortion, the harmonic vibration is a good approximation in
most cases.
3.6 Total Energy
The total energy may finally be written (in frequency unit) using (3.10, 3.16, 3.22,
3.39, and 3.48)
E
h
=
υ +
1
2
ω e −
υ +
1
2
2
ω e x e +
υ +
1
2
3
ω e y e + · · ·
+ B e J (J + 1) − D e J
2
(J + 1)
2
+ H e J
3
(J + 1)
3
+ · · ·
− α e
υ +
1
2
J (J + 1) + γ e
υ +
1
2
2
J (J + 1) + · · ·
(3.49)
The energy is mainly defined by five parameters (higher-order parameters are
also given in (3.49): ω e y e , H e , and γ e ): the harmonic vibrational frequency ω e , (3.7),
the equilibrium rotational frequency, B e , (3.23), the quartic centrifugal distortion