Deriving info from reducible representations

In summary, by counting the number of bonds left unchanged by each symmetry operation of a molecule, we can determine the reducible representation and extract the irreducible representations. This technique can be used to identify the number and symmetries of the normal modes of vibration in a molecule, as shown with the examples of water and SO3. In the case of SO3, the result indicates that there are three stretching vibrational modes, with symmetries and degeneracies that can be determined using group theory.
  • #1
ampakine
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Heres the reducible representation made by counting the number of bonds left unchanged by each symmetry operation of water:
[PLAIN]http://img808.imageshack.us/img808/704/red0.png
and here's the irreducible representations extracted from it:
http://imageshack.us/m/695/3829/red01l.png
in the book Inorganic Chemistry by Housecraft it states that "This result tells us that there are two non-degenerate stretching modes, one of A1 symmetry and one of B2 symmetry."

I don't understand how that result tells you anything about the number of stretching vibrational modes in the molecule. For a slightly more complex molecule SO3 here's the reducible representation you get from the bonds:
http://imageshack.us/m/821/323/red1t.png
and here's the IRs extracted from it:
http://imageshack.us/m/155/9027/red11.png
so what does this result tell you? Does that mean that SO3 has 3 stretching vibrational modes?
 
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  • #2
ampakine said:
Heres the reducible representation made by counting the number of bonds left unchanged by each symmetry operation of water:
[PLAIN]http://img808.imageshack.us/img808/704/red0.png
and here's the irreducible representations extracted from it:
http://imageshack.us/m/695/3829/red01l.png
in the book Inorganic Chemistry by Housecraft it states that "This result tells us that there are two non-degenerate stretching modes, one of A1 symmetry and one of B2 symmetry."

I don't understand how that result tells you anything about the number of stretching vibrational modes in the molecule.

The symmetry of a molecule is a fundamental property .. by definition, any normal mode vibration of the molecule must belong to one of the irreducible representations of the symmetry group for the molecule. By inspection, you might expect water to have two stretching modes .. one for each OH bond, that is called the local mode picture. The problem with the local mode picture is that those local vibrations involving motion of only one OH bond do not transform as one of the irreducible representations (or irreps) of the C2v point group. Thus we need to create normal modes by generating linear combinations of local modes that satisfy the correct symmetry properties. The method in the book you are reading is actually a shortcut for a more complicated and systematic way of determining the symmetries of the normal modes. It makes use of the fact that the only way a symmetry element can contribute to the character of the *reducible* representation for all possible vibrations of a molecule is if it leaves the positions of one or more atoms unchanged. Depending one how much group theory you have learned, that may not make much sense to you, but the technique does work (whether or not you know group theory ;).

For a slightly more complex molecule SO3 here's the reducible representation you get from the bonds:
http://imageshack.us/m/821/323/red1t.png
and here's the IRs extracted from it:
http://imageshack.us/m/155/9027/red11.png
so what does this result tell you? Does that mean that SO3 has 3 stretching vibrational modes?

Yes exactly ... can you figure out their symmetries and degeneracies? It makes sense that SO3 should have 3 stretching vibrations, since there are 3 identical SO bonds (by symmetry) whose individual vibrations are the local modes that are combined to form the normal modes .. there are always the same number of local and normal modes for a given type of vibration. Can you figure out how many bending vibrations SO3 has, and what their symmetries are?
 
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What is meant by "reducible representations" in scientific research?

Reducible representations refer to a mathematical technique used to simplify complex data sets or systems into smaller, more manageable components. It involves breaking down a larger system into its constituent parts to better understand its behavior or properties.

How is information derived from reducible representations?

Information can be derived from reducible representations by using mathematical operations, such as matrix algebra or Fourier transforms, to analyze and manipulate the simplified components. This allows researchers to gain insights and make predictions about the larger system based on the behavior of its smaller parts.

What are some common applications of deriving information from reducible representations?

Reducible representations are commonly used in fields such as physics, chemistry, and engineering to understand complex systems, such as molecular structures, electronic circuits, or physical forces. They can also be applied in data analysis, signal processing, and machine learning to extract meaningful patterns or relationships from large datasets.

What are the benefits of using reducible representations in scientific research?

Using reducible representations allows scientists to simplify and analyze complex systems more efficiently, saving time and resources. It also provides a deeper understanding of the underlying principles and relationships within a system, leading to more accurate predictions and insights.

Are there any limitations or challenges associated with deriving information from reducible representations?

One of the main challenges of using reducible representations is identifying the appropriate components to break down the system into. This requires a deep understanding of the system and its behavior. Additionally, the simplification process may result in some loss of information, so researchers must carefully consider the trade-offs between simplicity and accuracy.

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