Cl2O4 in the Stratosphere 

Group Theory and Vibrational Spectroscopy

 
  1. Character Tables

  2. Normal Modes of Vibration

  3. Vibrational Spectroscopy

  4. Polarized Raman Bands

  5. Example: The Geometry of the Sulfur Dioxide Molecule

 


Normal Modes of Vibration

The complex vibrations of a molecule are the superposition of relatively simple vibrations called the normal modes of vibration. Each normal mode of vibration has a fixed frequency. It is easy to calculate the expected number of normal modes for a molecule made up of N atoms.

Linear molecule of N atoms: # normal modes = 3N - 5
   
Nonlinear molecule of N atoms: # normal modes = 3N - 6

The symmetries of the normal modes can be classified by group theory. (In this project we won't go into how this is done).

As an example, water has a symmetrical bent structure of C2v symmetry. It has three atoms and three normal modes of vibration (3*3 - 6  =  3). Pictures of the three normal modes are shown here along with their symmetry types.

 
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Vibrational Spectroscopy

There are two types of spectroscopy that involve vibrational transitions. You should be very familiar with one of these from your Organic Chemistry course - infrared spectroscopy. During infrared spectroscopy experiments we observe transitions between vibrational energy levels of a molecule induced by the absorption of infrared (IR) radiation. The second type of vibrational spectroscopy is Raman spectroscopy. In Raman spectroscopy, vibrational transitions occur during the scattering of light by molecules.

At room temperature almost all molecules are in their lowest vibrational energy levels with quantum number n = 0. For each normal mode, the most probable vibrational transition is from this level to the next highest level (n = 0 -> 1). The strong IR or Raman bands resulting from these transitiions are called fundamental bands. Other transitions to higher excited states (n = 0 -> 2, for instance) result in overtone bands. Overtone bands are much weaker than fundamental bands.

Not all fundamental vibrational transitions can be studied by both IR and Raman specroscopy because they have different selection rules. Selection rules tell us if a transition is allowed or forbidden. An allowed transition has a high probability of occurring and will result in a strong band. Conversely a forbidden transition's probability is so low that the transition will not be observed. If a normal mode has an allowed IR transition, we say that it is IR active. Similarly if a normal mode has an allowed Raman transition, we say that it is Raman active.

If you know the point group of the molecule and the symmetry labels for the normal modes, then group theory makes it easy to predict which normal modes will be IR and/or Raman active. Look at the character table for the point group of the molecule.

If the symmetry label of a normal mode corresponds to x, y, or z, then the fundamental transition for this normal mode will be IR active.

If the symmetry label of a normal mode corresponds to products of x, y, or z (such as x2 or yz) then the fundamental transition for this normal mode will be Raman active.

Consider the character table for the C2v group shown at the right. We see that if a normal mode has A1, B1, or B2 symmetry then it will be both IR and Raman active. If a normal mode has A2 symmetry then it will be only Raman active.

 

 

In the example above, water has three normal modes: two of A1 symmetry and one of B2 symmetry. All of these are IR and Raman active.We would expect water to have three peaks corresponding to fundamental vibrations in the IR spectrum. There also would be three peaks in its Raman spectrum at the same frequencies as in the IR.

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Polarized and Depolarized Raman Bands 12

The assignment of Raman lines may be aided by measuring their intensity with a polarizing filter, first parallel and the perpendicular to the polarization of the incident radiation. If the polarization of the scattered beam is the same as that of the incident beam (intense only in the parallel direction), then the Raman line is said to be polarized. If the scattered light is intense in both the parallel and perpendicular direction, then the Raman line is depolarized.

Only totally symmetric vibrations (a normal mode with all characters = 1 in the character table) give rise to polarized lines.

In the water example above, two of the Raman lines correspond to a totally symmetric vibration (A1) and would be polarized. One would be depolarized.

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Example: The Geometry of the Sulfur Dioxide Molecule

Consider three possible geometries for the SO2 molecule, linear and bent, shown at the right with their point groups.  
The symmetry types for the normal modes of the three structures are shown here. For the Cs form 3 A' means that there are three different normal modes, all having the same symmetry (A'). Similarly, for the C2v form two of the three normal modes have the same symmetry (A1). These modes are not identical and do not have the same energy - they just happen to have the same symmetry.  

Now we need to look at the character tables to see which normal modes one would expect to be observed in the IR and Raman for each structure. The character tables for the three point groups are shown below.

Cs structure: 3 normal modes, all having A' symmetry
 
The Cs structure should have 3 IR active fundamental transitions. These three fundamental transitions also should be Raman active. We would expect to observe three strong peaks in the IR and three strong peaks in the Raman at the same frequency as in the IR.

All of the Raman lines would be polarized because they are totally symmetric (A' symmetry).

 
C2v structure: 3 normal modes, two with A1 symmetry, one with B2
 

The C2v structure should have 3 IR active fundamental transitions. These three fundamental transitions also should be Raman active.We would expect to observe three strong peaks in the IR and three strong peaks in the Raman at the same frequency as in the IR.

Two of the Raman lines are totally symmetric (A1 symmetry) and would be polarized. One Raman line would be depolarized.

 

 

The Dooh structure should have two IR active fundamental transitions. It will have one Raman active fundamental transition at a different frequency than either of the IR peaks..

The Raman line will be polarized.

The experimental infrared and Raman bands of liquid and gaseous sulfur dioxide have been reported in a book by Herzberg 7 . Only the strong bands corresponding to fundamental transitions are shown below. The polarized Raman bands are in red.

Fundamental 2 1 3
IR (cm-1) 519 1151 1336
Raman (cm-1) 524 1151 1336

Conclusion:

The existence of three experimental bands in the IR and Raman corresponding to fundamental transitions weighs strongly against the symmetrical linear (Dooh) structure. We usually do not expect more strong bands to exist than are predicted by symmetry.

Group theory predicts that both bent structures would have three fundamental transitions that are active in both the IR and Raman. However all three of the Raman lines would be polarized if the structure were unsymmetrical (Cs symmetry). The fact that one Raman line is depolarized indicates that the structure must be bent and symmetrical (C2v symmetry).

The sulfur dioxide molecule has C2v symmetry.

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Partial support for this work was provided by the National Science Foundation's Division of Undergraduate Education through grant DUE #9751605 and by CAChe Scientific through a Higher Education Program grant.

The PCOL community that partial support for this work was provided by the National Science Foundation's Division of Undergraduate Education through grant DUE #9950809. Additional support was provided by the Camille and Henry Dreyfus Foundation. PCOL faculty also acknowledge the National Science Teachers Association which awarded the PCOL Faculty Consortium the 1998 Gustav Ohaus Award for Innovation in College Science Teaching.

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This site created by David Whisnant (whisnantdm@wofford.edu).
This page was last updated on March 9, 2005
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