Raman Spectroscopy

To observe a transition between two non-degenerate states in a Raman spectrum, it is necessary that

Here, α_ij represents the components of the polarizability tensor α. In general, the six components (α_xx, α_yy, α_xy, α_xy, α_xz and α_yz) of this tensor differ from each other. The entries in the right column of the character table represent the components for a given point group and, in turn, the corresponding irreducible representation for α_ij is obtained. If one or both vibrational states are degenerate, the equation above is replaced by the more general statement

In many cases, the basic oscillation v = 0, which is total symmetric, represents the lower state: Γ(ψ_v'') = A. Therefore the conditions for a Raman transition between non-degenerate vibrations can be expressed in the following way:

Using the character table of point group C_2v, we find that all basic oscillations of the water molecule H₂O are allowed in Raman spectroscopy, as Γ (α_xx, α_yy, α_zz) = A₁ and Γ(α_yz) = B₁. Similarily, all basic oscillations of ammonia NH₃ (point group C_3v) are allowed too, as Γ( α_xx, α_yy, α_zz) = A₁ and Γ [(α_xx - α_yy, α_xy)(α_xx, α_yz)] = E.

The acetylene molecule (point group D_∞h) is treated in the same way. There are 3N-5 = 7 normal vibrational modes. Note that both bending oscillations are doubly degenerate. The table below characterizes the vibrational modes of acetylen.
 

Within the band 1 − 0 of this molecule, only the transitions 1₀¹, 2₀¹ and 4₀¹ are allowed. (In this notation, the number indicates the vibrational mode, the subscript the initial state and the superscript the final state after excitation.)

The oscillations of acetylene exemplify nicely the exclusion principle. If there is a centre of inversion within a molecule, the basic oscillations of even symmetry (g-oscillations) are Raman active but infrared inactive. Vice versa, oscillations of odd symmetry (u-oscillations) are infrared active but Raman inactive. Briefly, the conditions for both types of spectroscopy exclude each other.
 

The torsional oscillation ν4π of ethene

Nevertheless, there are vibrational transitions appearing neither in an infrared nor in a Raman spectrum. This is observed for the torsional oscillation of ethene (symmetry a_u). For the irreducible representation A_u of point group D_2h there is neither a translational component nor a corresponding component of the polarizability tensor.

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