6.2 Diatomic Molecules
281
Centrifugal Distortion and Vibration–Rotation Interaction Corrections
The Euler–Maclaurin corrections (typically for temperatures T < 10 rot ) discussed
in the previous subsection address the differences between the quantum mechanical
sum in Eq. (6.2.59) and the classical approximation of Eq. (6.2.61a) for symmetry
number σ = 1. They still assume the rigid-rotor model. However, we also know
that because the chemical bonds in diatomic molecules are not truly rigid, they
will elongate slightly for rapidly rotating molecules simply due to the centrifugal
forces associated with acceleration in the circular orbits of the two atoms about
their (common) centre-of-mass. The centrifugal distortion energy can be shown via
perturbation theory to have the functional form
cd (j ) = −D e [j (j + 1)]
2
+ H e [j (j + 1)]
3
+ · · · ,
(6.2.67)
in which D e and H e , both positive, are called the quartic and sextic centrifugal
distortion constants, respectively. Sextic and higher-order centrifugal distortion
constants may play significant roles in the accurate determination of spectral
transitions, but for the present discussion it will suffice simply to examine the role
of the quartic distortion constant in greater detail.
At the same time that centrifugal distortion contributions to the rotational energy
are being considered, we should also consider that rotational energies may depend
upon the specific vibrational state of the rotating molecule: this is referred to as the
vibration–rotation interaction energy, and is represented empirically in molecular
spectroscopy by the introduction of an energy term having the form α(v+
1
2 )j (j +1).
The total vibration–rotation internal energy int for a nonrigid diatomic molecule
can be written as
int = vib + rot ,
(6.2.68)
in which vib is the anharmonic vibrational energy, as given, for example, by
Eq. (6.2.29), and rot is the rotational energy for a nonrigid diatomic molecule, given
by
rot = (B e −
1
2 δ)j (j + 1)[1 − γj (j + 1) − vδ] .
(6.2.69)
In this expression, γ and δ are, respectively, the ratios of the quartic centrifugal
distortion constant and the vibration–rotation interaction constant to the rotational
constant, i.e., γ ≡ D e /B e and δ ≡ α/B e . Expression (6.2.69) for the rotational
energy neglects second-order terms in γ, δ, while the vibrational energy expression
(6.2.29) neglects the second-order term x 2
e ω e v(v − 1). The characteristic rotational
temperature can now be introduced into the argument ββ rot via
ββ rot = j (j + 1)[1 − γj (j + 1) − vδ]
rot
T
,
(6.2.70)
281
Centrifugal Distortion and Vibration–Rotation Interaction Corrections
The Euler–Maclaurin corrections (typically for temperatures T < 10 rot ) discussed
in the previous subsection address the differences between the quantum mechanical
sum in Eq. (6.2.59) and the classical approximation of Eq. (6.2.61a) for symmetry
number σ = 1. They still assume the rigid-rotor model. However, we also know
that because the chemical bonds in diatomic molecules are not truly rigid, they
will elongate slightly for rapidly rotating molecules simply due to the centrifugal
forces associated with acceleration in the circular orbits of the two atoms about
their (common) centre-of-mass. The centrifugal distortion energy can be shown via
perturbation theory to have the functional form
cd (j ) = −D e [j (j + 1)]
2
+ H e [j (j + 1)]
3
+ · · · ,
(6.2.67)
in which D e and H e , both positive, are called the quartic and sextic centrifugal
distortion constants, respectively. Sextic and higher-order centrifugal distortion
constants may play significant roles in the accurate determination of spectral
transitions, but for the present discussion it will suffice simply to examine the role
of the quartic distortion constant in greater detail.
At the same time that centrifugal distortion contributions to the rotational energy
are being considered, we should also consider that rotational energies may depend
upon the specific vibrational state of the rotating molecule: this is referred to as the
vibration–rotation interaction energy, and is represented empirically in molecular
spectroscopy by the introduction of an energy term having the form α(v+
1
2 )j (j +1).
The total vibration–rotation internal energy int for a nonrigid diatomic molecule
can be written as
int = vib + rot ,
(6.2.68)
in which vib is the anharmonic vibrational energy, as given, for example, by
Eq. (6.2.29), and rot is the rotational energy for a nonrigid diatomic molecule, given
by
rot = (B e −
1
2 δ)j (j + 1)[1 − γj (j + 1) − vδ] .
(6.2.69)
In this expression, γ and δ are, respectively, the ratios of the quartic centrifugal
distortion constant and the vibration–rotation interaction constant to the rotational
constant, i.e., γ ≡ D e /B e and δ ≡ α/B e . Expression (6.2.69) for the rotational
energy neglects second-order terms in γ, δ, while the vibrational energy expression
(6.2.29) neglects the second-order term x 2
e ω e v(v − 1). The characteristic rotational
temperature can now be introduced into the argument ββ rot via
ββ rot = j (j + 1)[1 − γj (j + 1) − vδ]
rot
T
,
(6.2.70)