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Diatomic molecule as a Rigid Rotator : Explanation of Rotational Spectra

The diatomic molecule as a Rigid Rotator: Let us start our topic with a question as asked below:

Obtain an expression for Rotational Energy of a rotating rigid diatomic molecule? Indicate how this expression is modified If the molecule is non-rigid.

 
Diatomic molecule as a Rigid Rotator
The simplest model of rotating diatomic molecule is rigid rotator.
Let us consider two atoms of masses M1 and M2 which are chemically bonded to form a diatomic molecule (fig. 1).


Diatomic molecule as a Rigid Rotator

Let “r” be the distance between two atoms and “r1” and “r2” the distance of atoms from the center of mass of the molecules. The moment of inertia of the system about an axis of rotation perpendicular to the axis joining two atoms is given by :
Diatomic molecule as a Rigid Rotator
By the definition of center of mass, we have :
Diatomic molecule as a Rigid Rotator
In order to determine, the possible energies of rotation of the molecule, we have to solve the Schrodinger wave equation which is as :
Diatomic molecule as a Rigid Rotator
The solving of rigid rotator Schrodinger wave equation gives the result that the eigenfunctions are single-valued, finite and continuous only for certain v as the value of E, given by :
Diatomic molecule as a Rigid Rotator
Rotational Spectrum :
 
Let us now investigate the spectrum expected from a rigid rotator. In terms of wave-number, the energy equation can be written as :
Diatomic molecule as a Rigid Rotator
Thus, we have a series of discrete rotational energy levels whose spacing increase with increasing J(fig 2.)
 
Diatomic molecule as a Rigid Rotator
When a transition takes place between an upper-level J’ and a lower level J” the wave number of absorbed or emitted radiation would be given by :
Diatomic molecule as a Rigid Rotator
Substituting J = 0, 1, 2, ….. we get,
v = 2B, 4B, 6B, 8B,…..
Thus the absorption spectra of the rigid rotator are expected to consists of a series of equidistant lines with constant separation 2B.
For non-rigid Rotator :
In practice, the rotational lines are ‘not’ exactly equidistant; the separations decrease
slightly with increasing J., This is attributed to the fact that the molecules are ‘not’ rigid.
They stretch while in the rotation, and the stretching (increase in bond-length) increases with increasing rotation. Taking this into account, the rotational terms come out to be:
F(J) = BJ(J+1)-DJ^2 (J + 1)^2 ,
where D is the ‘centrifugal distortion constants and is much smaller than B. Due to the
centrifugal term, the increase in spacing between successive rotational levels with
increasing J becomes slightly less rapid (Fig.) The wave numbers of the rotational lines
are now:
v = F(J + 1) – F(J) = 2B(J + 1) – 4D( J + 1)^3
This shows that the separation between the lines decreases slightly with increasing J, slightly because  D < < B. Increase in spacing between rotational levels are shown in fig. below.
Diatomic molecule as a Rigid Rotator
Fig. Increase in spacing between
rotational levels.
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