The polarizability ellipsoids of other linear molecules are similar to those of H₂.  The ellipsoids are drawn to represent polarizability in a “reciprocal” manner.  When the distance from the center of the ellipsoid to a point on the surface is greatest, polarizability along that direction is least.  The radial distance of the ellipsoid to a  point i, r_i is proportional to 1/√ α_i where α_i is the polarizability along that axis.

If α₀ is the polarizability at equilibrium and ß the maximum extent of the change in polarizability during the vibration, then, the polarizability changes during molecular vibrations as   α =   α₀ + ß  sin 2π ν_vibt, where νvib  is the vibrational frequency.  Since the applied  electric field also has the sinusoidal component oscillating at a frequency ν, the oscillating dipole can be expressed as

μ =  (α₀ + ß  sin 2 π ν_vib t). E₀ sin 2πνt                                                                                (15.9)

Reexpressing the product of sines,

   μ = α₀ E₀ sin 2πνt + ½ ß E0 {cos 2π(ν-ν_vib) t + cos 2π(ν + ν_vib)t}                                        (15.10)

Here E₀ is the amplitude of the applied field.  We thus see that there are oscillating dipoles with frequency components ν ± ν_vib.

The selection rule for Raman transitions are governed by the changes in the components of molecular polarizability during a rotation or a vibration.

Let us consider the rotational Raman spectra of linear molecules. Without considering the centrifugal distortion constant D, the energy levels are given by

ε_J = B J(J + 1) cm^-1,  J = 0,1,2                                                                                          (15.11)

The rotational Raman Selection rule is,

ΔJ = 0, or ± 2 only  and                                                                                                    (15.12)
                       
Δε_J = E _J+2 - E_J = B (4J+6)                                                                                             (15.13)

Recall that for pure rotational spectra,  ΔJ =   ±1. From Fig.15.6(a),  it is seen that the rotation about the bond axis produces no change in the polarizability ellipsoid whereas the end over end rotation changes the ellipsoid.  In every complete rotation, the molecular ellipsoid has the same appearance twice.  The observed spectral lines appear at

ν_PR = ν_ex ± Δε =   ν_ex ± B(4J + 6) cm^-1                                                                        (15.14)

Here RR refers to rotational Raman.