3.2 The Vibrational Energy
59
Table 3.3 Some typical values a of the Dunham constants
Y 01 ≈ B e /MHz
Y 02 = −
D e /kHz
Y 11 = −
α e /MHz
Y 10 ≈
ω e /cm −1
Y 20 = −
ω e x e /cm −1
H 35 Cl
317,580.97
−15945.8
−9208.96
2990.9664
−52.8364
H 79 Br
253,850.6
−10381.5
−6997.58
2649.624
−45.444
12 C 16 O
57,898.3404
−183.5202
−524.7536
2169.812593
−13.28794
205 Tl 35 Cl
2740.01367
−1.1294530
−11.927006
284.71102
−0.86123
a Source Tiemann (1982, 1992); Hübner (1998)
it is mainly used for the expansion of the stretching potential of bonds in polyatomic
molecules. Other expansion parameters are discussed in Le Roy (2011).
The higher-order terms in the series expansion can be treated by a perturbation
calculation, and the energy can be written as
E V = ω e
υ +
1
2
− ω e x e
υ +
1
2
2
(3.16)
In this equation, all the parameters are in cm
−1 , the unit normally used in infrared
spectroscopy. The coefficient ω e x e is called anharmonicity constant. Higher-order
terms may have to be introduced in (3.16); see (3.49). ω e x e is always positive and
much smaller than ω e . For instance, for HCl, ω e = 2991 cm
−1 and ω e x e = 52.8 cm
−1 .
Some typical values are given in Table 3.3.
3.3 The Rigid Rotor
The rotation of any system is conveniently treated using the angular velocity Ω and
the moment of inertia I. The distance between the two atoms, assumed to be point
masses, is the bond length r = r A + r B . From the definition of the center of mass:
m A r A = m B r B . The moment of inertia may be written as
I = m A r
2
A + m B r
2
B =
m A m B
m A + m B
r
2
= μr
2
(3.17)
In classical mechanics, the Hamiltonian is written as
H =
P
2
2I
with P = I Ω
(3.18)
Ω is the angular velocity not to be confused with the harmonic vibrational frequency
ω e.
In quantum mechanics, the Hamiltonian has the same form
59
Table 3.3 Some typical values a of the Dunham constants
Y 01 ≈ B e /MHz
Y 02 = −
D e /kHz
Y 11 = −
α e /MHz
Y 10 ≈
ω e /cm −1
Y 20 = −
ω e x e /cm −1
H 35 Cl
317,580.97
−15945.8
−9208.96
2990.9664
−52.8364
H 79 Br
253,850.6
−10381.5
−6997.58
2649.624
−45.444
12 C 16 O
57,898.3404
−183.5202
−524.7536
2169.812593
−13.28794
205 Tl 35 Cl
2740.01367
−1.1294530
−11.927006
284.71102
−0.86123
a Source Tiemann (1982, 1992); Hübner (1998)
it is mainly used for the expansion of the stretching potential of bonds in polyatomic
molecules. Other expansion parameters are discussed in Le Roy (2011).
The higher-order terms in the series expansion can be treated by a perturbation
calculation, and the energy can be written as
E V = ω e
υ +
1
2
− ω e x e
υ +
1
2
2
(3.16)
In this equation, all the parameters are in cm
−1 , the unit normally used in infrared
spectroscopy. The coefficient ω e x e is called anharmonicity constant. Higher-order
terms may have to be introduced in (3.16); see (3.49). ω e x e is always positive and
much smaller than ω e . For instance, for HCl, ω e = 2991 cm
−1 and ω e x e = 52.8 cm
−1 .
Some typical values are given in Table 3.3.
3.3 The Rigid Rotor
The rotation of any system is conveniently treated using the angular velocity Ω and
the moment of inertia I. The distance between the two atoms, assumed to be point
masses, is the bond length r = r A + r B . From the definition of the center of mass:
m A r A = m B r B . The moment of inertia may be written as
I = m A r
2
A + m B r
2
B =
m A m B
m A + m B
r
2
= μr
2
(3.17)
In classical mechanics, the Hamiltonian is written as
H =
P
2
2I
with P = I Ω
(3.18)
Ω is the angular velocity not to be confused with the harmonic vibrational frequency
ω e.
In quantum mechanics, the Hamiltonian has the same form