Raman Spectroscopy
To observe a transition between two nondegenerate 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 nondegenerate 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 H_{2}O are allowed in Raman spectroscopy, as Γ (α_{xx}, α_{yy}, α_{zz}) = A_{1} and Γ(α_{yz}) = B_{1}. Similarily, all basic oscillations of ammonia NH_{3} (point group C_{3v}) are allowed too, as Γ( α_{xx}, α_{yy}, α_{zz}) = A_{1} 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
3N5 = 7 normal vibrational modes. Note that both bending
oscillations are doubly degenerate. The table below characterizes the
vibrational modes of acetylen.
Normal vibrational modes of acetylene 

Name 
Motions of nuclei 
Mode 
ν/cm^{1} 
irred. Repr. 
Infrared 
Raman 
CHstretching 

ν_{1} 
3374 
σ_{g}^{+} 
no 
yes 
CCstretching 

ν_{2} 
1974 
σ_{g}^{+} 
no 
yes 
CHstretching 

ν_{3} 
3287 
σ_{u}^{+} 
yes 
no 
transbending 

ν_{4} 
612 
π_{g} 
no 
yes 
cisbending 

ν_{5} 
729 
π_{u} 
yes 
no 
Within the band 1 − 0 of this molecule, only the transitions 1_{0}^{1}, 2_{0}^{1} and 4_{0}^{1} 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 (goscillations) are Raman
active but infrared inactive. Vice versa, oscillations of odd
symmetry (uoscillations) are infrared active but Raman inactive.
Briefly, the conditions for both types of spectroscopy exclude each
other.
The torsional oscillation ν_{4}π 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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