3.6 Total Energy
65
constant, D e , (3.48), the vibration–rotation interaction constant, α e , (3.40), and the
anharmonicity constant, ω e x e , which may be expressed as a function of the other
constants
ω e x e = B e
α e ω e
6B 2
e
+ 1
2
(3.50)
The typical order of magnitude is: ω e ≈ 10
4 GHz, B e ≈ 10 GHz, D e ≈ 40 kHz,
α e ≈ 200 MHz, and x e ≈ 0.02; see also Table 3.3.
The five parameters defining the rovibrational energy may be calculated using
the reduced mass μ which is a priori known and only three unknown parameters:
the equilibrium bond length, r e , the harmonic force constant k, and the Dunham
anharmonicity constant a 1 . It is important to know how the parameters α e and D e
depend on μ, k, and r e . From (3.40), one get
α e ∝ r
−4
μ
−3/2 k
−1/2 or
α e
B e
∝
1
√
kμ
(3.51)
and from (3.48)
D e = r
−6
μ
−2 k
−1 or
D e
B e
∝
1
kμ
(3.52)
When the mass of the molecule increases, α e and D e decrease faster than B e and
they decrease too when the force constant k increases. It is also possible to show that
γ e /α e ∝ B e /ω e .
It is also useful to check whether these parameters are independent. Indeed, the
product kr
2
e is approximately constant for a wide class of molecules and Badger
(1934) found a more accurate relationship in which the equilibrium bond length is a
simple function of the force constant k as well as two other constants depending on
the row of the Mendeleev classification. A similar empirical formula was also found
for the a i constants of the Dunham expansion. These empirical relations are not
very useful for a diatomic molecule, but they would be very useful for a polyatomic
molecule if it were possible to accurately determine force constants characteristic of
the bond, which is rarely the case. However, there are a few exceptions, in particular for the CH bond. Because the mass of the hydrogen atom is much smaller than
the masses of the other atoms, the frequency of the stretching vibration of the CH
bond is much higher than the other vibrations and the coupling between this vibration and other vibrations is often negligible. In conclusion, when there is only one
hydrogen atom in a molecule, there is an empirical relationship between the frequency
ν(CH stretch ), called isolated stretching frequency, and r e (CH). This relationship may
be used to determine r e (CH) with an accuracy of 0.2 pm, which is excellent for a
polyatomic molecule; see Sect. 8.7.1. When there are several hydrogen atoms in the
molecule, it is enough to replace them but one by deuterium. A similar relationship
was also used to determine the Au–Au and Ag-Ag bond lengths (Perreault 1992).
65
constant, D e , (3.48), the vibration–rotation interaction constant, α e , (3.40), and the
anharmonicity constant, ω e x e , which may be expressed as a function of the other
constants
ω e x e = B e
α e ω e
6B 2
e
+ 1
2
(3.50)
The typical order of magnitude is: ω e ≈ 10
4 GHz, B e ≈ 10 GHz, D e ≈ 40 kHz,
α e ≈ 200 MHz, and x e ≈ 0.02; see also Table 3.3.
The five parameters defining the rovibrational energy may be calculated using
the reduced mass μ which is a priori known and only three unknown parameters:
the equilibrium bond length, r e , the harmonic force constant k, and the Dunham
anharmonicity constant a 1 . It is important to know how the parameters α e and D e
depend on μ, k, and r e . From (3.40), one get
α e ∝ r
−4
μ
−3/2 k
−1/2 or
α e
B e
∝
1
√
kμ
(3.51)
and from (3.48)
D e = r
−6
μ
−2 k
−1 or
D e
B e
∝
1
kμ
(3.52)
When the mass of the molecule increases, α e and D e decrease faster than B e and
they decrease too when the force constant k increases. It is also possible to show that
γ e /α e ∝ B e /ω e .
It is also useful to check whether these parameters are independent. Indeed, the
product kr
2
e is approximately constant for a wide class of molecules and Badger
(1934) found a more accurate relationship in which the equilibrium bond length is a
simple function of the force constant k as well as two other constants depending on
the row of the Mendeleev classification. A similar empirical formula was also found
for the a i constants of the Dunham expansion. These empirical relations are not
very useful for a diatomic molecule, but they would be very useful for a polyatomic
molecule if it were possible to accurately determine force constants characteristic of
the bond, which is rarely the case. However, there are a few exceptions, in particular for the CH bond. Because the mass of the hydrogen atom is much smaller than
the masses of the other atoms, the frequency of the stretching vibration of the CH
bond is much higher than the other vibrations and the coupling between this vibration and other vibrations is often negligible. In conclusion, when there is only one
hydrogen atom in a molecule, there is an empirical relationship between the frequency
ν(CH stretch ), called isolated stretching frequency, and r e (CH). This relationship may
be used to determine r e (CH) with an accuracy of 0.2 pm, which is excellent for a
polyatomic molecule; see Sect. 8.7.1. When there are several hydrogen atoms in the
molecule, it is enough to replace them but one by deuterium. A similar relationship
was also used to determine the Au–Au and Ag-Ag bond lengths (Perreault 1992).