The Molecule as a Rigid Rotator : Expalaination of Rotational Spectra

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

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).
Fig 1. Atoms in diatomic molecule 
Let “r” be the distance between two atoms and “r1” and “r2” the distance of atoms from centre of mass of the molecules. 
The moment of inertia of the system about axis of rotation perpendicular to the axis joining two atoms is given by :        
By the definition of centre of mass, we have :

In order to determine, the possible energies of rotation of the molecule,we have to solve the Schrodinger wave equation which is as :
The solving of rigid rotator Schrodinger wave equation gives the result that the eigenfunctions are singke valued, finite and continuous only for certain v as lue of E, given by :

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 :
Thus, we have a series of discrete rotational energy levels whose spacing increase with increasing J(fig 2.)

Fig. Discrete rotational energy levels 
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 :

Subsituting J = 0, 1, 2, ….. we get, 
v = 2B, 4B, 6B, 8B,….. 
Thus the absorption spectra of rigid rotator is 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 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 constant, 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. 
Fig. Increase in spacing between
rotational levels. 
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