Raman Effect: Classical Theory and Polarizability

Raman Effect: Classical Theory and Polarizability
According to the classical theory, any scattered light arises from an oscillating dipole. The electric field of the incident light produces an induced dipole moment ? which is given by:
?=?E …..(1)
where ? is the polarizability of the molecule. The electric field E of a radiation of frequency ? is given by: E= E0 cos2??t.
?=? E0 cos2??t …..(2)
According to equation (2) the induced dipole oscillates at a frequency ?. If the polarizability (?) of a molecule changes during vibrational and rotational motion.
a. Vibrational Raman Effect
The vibrational Raman effect arises from the change of ? with the vibrational co-ordinate, q. The essential condition of Raman effect is:

…(3)
One can expand ? as a Taylor’s series:
…(4)
Where and A is the amplitude of vibration. For simple harmonic vibration.
…..(5)
where ?0 denotes the vibrational frequency. Thus:
….(6)
Substituting the value of ? in equation (2):
…..(7)
…...(8)

The last equation gives a classical explanation of the Raman effect. Evidently, in equation (8) the first, second and third terms respectively, gives rise to the Rayleigh, Stokes and anti-Stokes lines. It is also easy to see that according to this equation the selection rule for Raman effect is ?n=±1.
b. Rotational Raman Effect
The classical theory of Raman effect is almost identical to that for the vibrational Raman effect. For rotational Raman effect, the essential condition is that the molecule must be anisotropic i.e. its polarizability changes with direction or orientation of the molecule. For rotational Raman, the polarizability changes because during molecular rotation the orientation of the molecule with respect to the electric field E of incident light changes. For a diatomic molecule if the rotational frequency is ?R the time dependent polarizability is given by:
…(9)
In this case the factor 2?R arises because during a complete rotation (by 2?) the molecule assumes same orientation twice (for rotation by ? and 2?).
Then in analogy to equation (8) for rotational Raman one can write that:
…(10)
According to this equation the frequencies of Stokes and anti-Stokes lines are ?-2?R and ?+2?R respectively. In other words, the selection rule for Rotational Raman is ?J=±2.

IR and Raman Spectra for centrosymmetric molecules: Mutual Exclusion
Usually a molecule has many vibrational degrees of freedom (3N-6 for a non-linear molecule). All the vibrational modes are not observed in simple Infra Red (IR) absorption and Raman spectra. According to quantum theory, for a centrosymmetric molecule if a particular vibrational mode is IR active (i.e. those for which IR transition is allowed), it can not display Raman effect. Conversely, a vibration which gives rise to Raman line (i.e. Raman active) is IR inactive. Thus it is said that for a centrosymmetric molecule, IR and Raman spectra are mutually exclusive.