# Accessible statement on the intuition and usefulness of group theory to chemistry

I am writing a lesson plan for high school students who are studying group theory, and I am having difficulty in relating the subject matter to the real world. I have quite a few examples using the Rubik's cube, but I would like to use some statement that might provide some intuition to students of how group theory is useful for chemists. I understand that group theory is used with spectroscopy, but I am not sure how to relate this back to the theory.

I have seen some good reasonable non rigorous attempts to relate group theory to the insolubility of quintic polynomials, but I think it would great to have a second more concrete example, such as chemistry.

I understand this may be a big thing to ask, and would be happy with just a broad brush strokes answer without being too technical or trivial.

Individual molecules rely on point group symmetry - that is, the symmetry about a point. Crystal structures rely on space group symmetry including the translational symmetry in a crystal.

Many molecules, particularly important ones, are highly symmetric. The different symmetry operations (reflections, rotations, etc.) form a finite group.

There are a number of interesting group theory results, but a simple statement comes down to two things:

• Understanding Orbitals, Bonding, Vibrations..: Using point groups, any function of the molecule corresponds to a linear combination of irreducible representations. This means I can quickly understand vibrations of a molecule, spectroscopy (as you mentioned) and orbital diagrams and bonding using just pencil and paper.
• Faster Calculations: Using symmetry, you can greatly reduce the amount of time required for quantum calculations. For one, you only need to consider the symmetry-unique atoms and bonds. For another, you can show that certain types of integrals and calculations will always be zero and can be completely ignored. This leads to immense speedups.

For example, chemists label orbitals with symmetry labels based on the point group. Using the direct product we can realize that certain interactions will never occur.

Similarly, vibrations have symmetry labels, and we can determine what types of spectroscopy will interact with those vibrations, or whether they are "silent."

• THanks for that answer Geoff, that gives some nice insight without getting too technical. – Michael T Mckeon Sep 26 '14 at 2:03

I realize my answer renders this off-topic for Chemistry, but what about general relativity? The fact that there is no "special" point in the universe is made concrete by saying that general relativity is invariant under the group of rigid motions (rotations and translations). That's an interesting real world group on its own.

• Thanks for that all coments are helpfull, I know this is a very difficult topic. – Michael T Mckeon Sep 25 '14 at 13:49