Classical Theory of the Raman Effect: Molecular Polarizability

Classical Theory of the Raman Effect:- 

The basic concept of this spectroscopy depends on the polarizability of a molecule and the applied Electric field.When a molecule is put into a static electric field it suffers some distortion, the positively charged nuclei being attracted towards the negative pole of the field, the electrons to the positive pole.

Molecular Polarizability:

The size of the induced dipole μ, depends both on the magnitude of the applied field,
E, and on the ease with which the molecule can be distorted

μ = αE

Least Polarisibility:-

Electrons forming the bond are less easily displaced by the field across the bond axis

Greatest Polarisibility:-

Electrons forming the bond are more easily displaced by the field along the bond axis 

Polarizability Ellipsoid:-

The polarizability of a molecule in various directions is conventionally represented by  drawing a polarizability ellipsoid.

The polarizability of a molecule in various directions is conventionally represented by
drawing a polarizability ellipsoid.

In general a polarizability ellipsoid is defined as a three-dimensional surface whose distance from the electrical centre of the molecule in H2 this is also the (centre of gravity) is proportional to alfa. where α i the polarizability along the line joining a point i on the ellipsoid with the electrical centre. 

Points to remember:-

• The student must not make the mistake of confusing a polarizability ellipsoid
with electron orbitals or electron clouds.
• In a sense the polarizability ellipsoid is the inverse of an electron cloud-where
the electron cloud is largest the electrons are further from the nucleus and so are
most easily polarized.
• This, as we have seen, is represented by a small axis for the polarizability

Molecular Polarizability

When a molecules is subjected to a beam of radiation of frequency ‘ν’ the electric
field experienced by each molecule varies according to the equation
𝐸 = 𝐸0 𝑠𝑖𝑛 2𝜋νt
and thus the induced dipole also undergoes oscillations of frequency ν :
𝜇 = 𝛼𝐸
or, 𝝁 = 𝜶𝑬𝟎 𝒔𝒊𝒏 𝟐𝝅𝝂𝐭
This is the classical explanation of Rayleigh scattering. Here the oscillating dipole emits
radiation of its own oscillation frequency ν.

If, in addition, the molecule undergoes some internal motion, such as vibration or
rotation, which changes the polarizability periodically, then the oscillating dipole will
have superimposed upon it the vibrational or rotational oscillation.
𝛼 = 𝛼0 + 𝛽 𝑠𝑖𝑛2𝜋𝜈𝑣𝑖𝑏𝑡
𝜇 = 𝛼𝐸
𝜇 = (𝛼0 + 𝛽 𝑠𝑖𝑛2𝜋𝜈𝑣𝑖𝑏𝑡)𝐸0 𝑠𝑖𝑛 2𝜋𝜈𝑡
We know the trigonometric expression,
sin A sin B = 1 /2 {cos 𝐴 − 𝐵 − cos(𝐴 + 𝐵) 
𝜇 = (𝛼0 + 𝛽 𝑠𝑖𝑛2𝜋𝜈𝑣𝑖𝑏𝑡)𝐸0 𝑠𝑖𝑛 2𝜋𝜈𝑡
𝜇 = 𝛼0𝐸0 𝑠𝑖𝑛 2𝜋𝜈𝑡 +𝛽 𝑠𝑖𝑛2𝜋𝜈𝑣𝑖𝑏𝑡 · 𝐸0 𝑠𝑖𝑛 2𝜋𝜈𝑡
𝜇 = 𝛼0𝐸0 𝑠𝑖𝑛 2𝜋𝜈𝑡 + 1 2 𝛽𝐸0{cos 2𝜋(𝜈 − 𝜈𝑣𝑖𝑏) − cos 2𝜋(𝜈 + 𝜈𝑣𝑖𝑏) 𝑡}.

In order to be Raman active a molecular rotation or vibration must cause some
change in a component of the molecular polarizability. A change in polarizability is,
of course, reflected by a change in either the magnitude or the direction of the
polarizability ellipsoid.


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