Rotational Spectroscopy

Rotational Spectroscopy
Rotational spectroscopy refers to the study of transition between rotational energy levels. In order to calculate the rotational energy it is assumed that during rotation while all the nuclei in a molecule move the distance between the nuclei (i.e. bond distances) remain fixed. Such a rotating molecule with a fixed inter-nuclear distance is called a rigid rotator.

1. The Rigid Rotator
The kinetic energy (T) of an atom of mass, m, is given by:


….(1)
where, are the polar co-ordinates. For a rigid rotator,

……(2)

Consider, a diatomic molecule in which two atoms of mass m1 and m2 are at a distance r1 and r2 from the centre of mass (origin) of the system. For the diatomic molecule the kinetic energy is sum of kinetic energy of the nuclei. For a rigid rotator, the potential energy is zero. So the total energy ER is just equal to the kinetic energy. For the diatomic molecule the co-ordinates ? and ? are same for the two atoms. Thus one may write:
..(3)
where I is the moment of inertia of the diatomic molecule. I may be written as:

………(4)
Where, ? is the reduced mass of the molecule and r0= r1+r2, is the bond distance. Comparing equation (2) with equation (3) it is evident that a rigid rotator behaves like a particle of mass I placed at a distance unity (i.e. r=1) from the origin (centre of mass) of the system. Thus for a rigid rotator, using mass=I and energy=ER, and noting potential energy V=0, the Schr?dinger equation is:

…..(5)
Solution of equation (5) gives rotational energy (ER) and rotational wave-function (?R). It may be shown that the rotational energy corresponding to the J-th level, EJ is given by:

……(6)

The rotational quantum number J may be zero or any positive integer e.g. J=0, 1, 2, 3… Each rotational is degenerate with 2J+1 levels.




2. Rotational Transitions
Consider, transition from a rotational level of quantum number J to an upper level of quantum number J . In this case, the energy of the photon absorbed is equal to EJ -EJ so that:

…(7)

The frequency ?- of the photon absorbed, expressed in wavenumber, (cm-1) is given by:

…..(8)

, is known as the rotational constant of the molecule and is expressed in cm-1. The main aim of rotational spectroscopy is to determine B which gives I and finally the bond distance, r0.