Vibration Rotational Spectra

Pure vibration spectra are obtained only for liquids where rotational motions hindered due to interactions between the liquid molecules. However in case of gaseous molecules, as the energy required for rotation is much less than that required for vibration, the vibration motion is always accompanied by rotational motion and hence we do not have a pure vibration spectra ; instead we have a vibration-rotational spectra lying in the infrared region.

As rotational motion and vibration motion take place independent of the total energy of the molecule may be taken as the sum of the rotational energy and the vibration energy (which is much larger than the former). Thus

or it can be written as


Expressing in terms of wave numbers, it can be written as

But from eqn.


Substituting these values in eqn. we get
                   
Here represents the wave number of a rotational level with quantum number J in the vibration level with quantum number.

From eqn. it is clear that in the same vibration level (so that =constant), there will be a number of rotational levels corresponding to J= 0,1,2,3........

When a transition takes place form a level with quantum number and J to a level with quantum numbers and J, the energy change expressed in terms of wave numbers will be given by

Restricting to the transition form , we get

Putting (1-2x_e)  we can write

(i)    For transition with

(ii)    For transition with


Combining equations we get


where m = J+ 1 in eqn. and m = -(J' +1) in eqn. As J = 0,1,2,3......, this means that in eqn. m =  (+ sign for = +1  and -sign for = -1).

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