Raman Spectroscopy
To observe a transition between two non-degenerate states in a Raman spectrum, it is necessary that
Here, αij represents the components of the polarizability tensor α. In general, the six components (αxx, αyy, αxy, αxy, αxz and αyz) 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 αij 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(ψv'') = 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 C2v, we find that all basic oscillations of the water molecule H2O are allowed in Raman spectroscopy, as Γ (αxx, αyy, αzz) = A1 and Γ(αyz) = B1. Similarily, all basic oscillations of ammonia NH3 (point group C3v) are allowed too, as Γ( αxx, αyy, αzz) = A1 and Γ [(αxx - αyy, αxy)(αxx, αyz)] = 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.
Normal vibrational modes of acetylene |
||||||
Name |
Motions of nuclei |
Mode |
ν/cm-1 |
irred. Repr. |
Infrared |
Raman |
CH-stretching |
|
ν1 |
3374 |
σg+ |
no |
yes |
CC-stretching |
|
ν2 |
1974 |
σg+ |
no |
yes |
CH-stretching |
|
ν3 |
3287 |
σu+ |
yes |
no |
trans-bending |
|
ν4 |
612 |
πg |
no |
yes |
cis-bending |
|
ν5 |
729 |
πu |
yes |
no |
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 au). For the irreducible representation Au of point group D2h there is neither a translational component nor a corresponding component of the polarizability tensor.
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