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I am a mathematician working in group theory and representation theory. When I try to explain these subjects to non-mathematicians, I often fail to provide a convincing reason to study these. So I am trying to learn how these theories are used outside pure mathematics. So how much group theory and representation theory do chemists really use in their research? I heard that there is something called "point group" that describes the symmetry of a molecule, but do you use any "advanced" group theory/representation theory results (something you don't expect to see in an undergraduate textbook) to study this point group thing or any other group that you use in chemistry? For example:

-Has anyone ever tried to use "modular representation" (representation over fields with positive characteristic) or "projective representation" (a group homomorphism into projective general linear group instead of the usual general linear group) in a chemistry research? If not, do you think chemistry will need such thing in the future?

-Do you ever need to think about representations with high dimension, or only work with at most 3-dimensional representations?

-How complicated are the groups you see in chemistry? Are they mostly finite subgroups of O(3) or other similar groups? Do you ever get an infinite group or a finite but complicated group, like being nonabelian and having long composition series, or having very large order?

-Can you recommend some high quality chemistry research papers involving a lot of group/representation theory, or some reputable journals where I can find such papers?

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    $\begingroup$ I am not a mathematician so I'm not sure if I understand all of the above. These are quite a few questions up there. Is probably too broad to be reasonably answered in our format. I think you should start with some introductory texts on quantum mechanics, physical and theoretical chemistry, and spectroscopy to get an overview. At the very least there's plenty of maths in there. Then follow the publications referenced within. This list will get you started: chemistry.stackexchange.com/q/37303/4945 $\endgroup$ Feb 19, 2023 at 22:52
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    $\begingroup$ Another hint I'd like to leave is to use the site search. There are about 100 questions with the tag group-theory. You'll probably then find chemistry.stackexchange.com/q/6231/4945, which I think would also be a good place to start, too. $\endgroup$ Feb 19, 2023 at 22:57
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    $\begingroup$ Some of the publications' titles indicate the relevance of group theory e.g. Raman Spectroscopy, Group Theory, and Computational Chemistry: A Physical Chemistry Laboratory Experiment on para-Difluorobenzene 2022JChemEduc2129, or that you can anticipate it is relevant to the article, one way, or an other, e.g. From Outside In: Stretching Students’ Conceptual Understanding of Molecular Symmetry with 2D and 3D Manipulatives, 2023JChemEduc1063. Both found with "Group theory" in ACS' search box. $\endgroup$
    – Buttonwood
    Feb 20, 2023 at 8:29
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    $\begingroup$ You need to look for info on Point Groups (for molecules) and Space Groups for crystals. Some more mathematical texts are D. Bishop 'Group Theory and Chemistry' (publ Dover), M. Tinkham 'Group theory and quantum mechanics' (publ Dover) and W. Borchardt-Orr 'Crystallography' (publ Springer) $\endgroup$
    – porphyrin
    Feb 20, 2023 at 11:23
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    $\begingroup$ In addition to the 119 questions with the "group-theory" tag here (which Martin pointed out), there's another 66 of them here and another 21 of them here. $\endgroup$ Feb 21, 2023 at 3:24

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I am a mathematician working in group theory and representation theory. When I try to explain these subjects to non-mathematicians, I often fail to provide a convincing reason to study these.

A brief discourse on history is usually sufficient to emphasize the importance of group theory in a talk or a lecture. So, you may have to start from the time when group theory helped to solve difficult chemical problems, especially in vibrational spectroscopy. The audience won't be convinced if you begin with a technically dry introduction. To be honest, most of those pioneers were hardcore physicists such as Mullikan, Placzek (never got credit for his work) Wigner and many others. You can also mention the severe opposition to group theory in early physics. Innovative ideas are always unwelcome by a certain group. Group theory users were called Gruppenpest (Group theory pests). I do not think you can find "pure" chemists (chemists who did not have mathematical training but were trained in chemical reactions) who woke up one day and started using it. Most of the work has been borrowed from traditional physics.

Here is the list of papers/books or works that you can start to convince the audience of the importance of group theory to chemical applications.

  1. Group Theory: And its Application to the Quantum Mechanics of Atomic Spectra by Eugene P. Wigner (Book by a Nobel Prize Winner).

  2. The “Gruppen Pest” Yesterday, Today, and Tomorrow, by Brian G. Wybourne (article)

  3. Zur Deutung der Spektren mehratomiger Molekule by L.Tisza, Again like Plakczek he got scant credit for his pioneering work.

Search the work of these authors. A couple of years ago I was compiling the history of group theory symbols used in the chemical literature out of curiosity. No claims to understanding Wigner's work.

From here, at least, you can search for their names and follow their citation chains to see the latest users of group theory.

P.S. Do not worry about a topic at an undergraduate level. Today's undergraduate curriculum is so superficially broad that students have now heard of everything, and they think they know everything, but often the reverse is true. Even some PhDs have this mentality. I had one undergraduate ask me if I knew string theory (of course not). So there is no need to start with a very esoteric or arcane application.

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I'm doing a PhD in organic electrochemistry. Let me first give you some literature that is usually used for studying chemistry here in Germany, probably in order of increasing complexity:

  • Ingo Peter Lorentz: Molekülsymmetrie und Spektroskopie (German) (ISBN 978-3110364927)
  • F. Albert Cotton: Chemical Applications of Group Theory (ISBN 978-0471510949)
  • Dirk Steinborn: Symmetrie und Struktur in der Chemie (German) (ISBN 978-3527284184)
  • Yves Jean: Molecular Orbitals of Transition Metal Complexes (ISBN 978-0198530930)
  • Thomas Albright: Orbital Interactions in Chemistry, 2. ed. (ASIN B010WF0GKG)

For a scientist, symmetry is the most important concept that exists. Symmetry can be found in all parts of science: chemistry, physics, astronomy, biology and much more. It's one of the few concepts that play an important role in both science and art as well. No other concept plays such an important role in chemistry.

Group theory is something that is often poorly understood (and poorly teached). As a pratical chemist who stands in the lab, I think about molecules and their symmetry all day. Since chemists prefer a visual type of thinking over strict group theory, we don't have group theory in our mind all the time. In general, we have the feeling that symmetry is our friend because it can really simplify a lot of complicated problems, both of practical or theoretical nature. Most important, no matter what spectroscopic method you use, the symmetry of a molecule has direct implications on the spectrum that you obtain from it.

Examples

Here are 10 examples where group theory is the indispensable foundation:

  • Let's start with a little quantum mechanics: Atomic orbitals and molecular orbitals are classified by group theory. Even if you are not a theoretical chemist, you should have learned that orbitals can only overlap if their symmetry and spacial orientation can match (and their energy is similar). Looking at the most simple molecule, the dihydrogen molecule $\rm H_2$, we know that each hydrogen atom has a single occupied 1s orbital. The overlap of both gives a bonding $\sigma$ orbital and an antibonding $\sigma^*$ orbital. The bonding $\sigma$ orbital is occupied by the two electrons, resulting in a lower energy of the system. This is the reason Helium prefers to stay as $\rm He$, not as $\rm He_2$: In that case, we would have 4 electrons and the antibonding orbitals would be occupied as well, resulting in a higher energy of the system. For more complex molecules, we need to make a symmetry adapted linear combination of orbitals. You can read more about that in "Molecular Quantum Mechanics" by Peter Atkins. An impressive example of how symmetry can simplify the discussion of orbitals is the molecular orbital diagram of ferrocene.

  • The most important spectroscopic method in all of chemistry is nuclear magnetic resonance spectroscopy (NMR spectroscopy). NMR spectroscopy has advanced chemistry more than any other method before or since. When organic chemists look at an NMR spectrum, we use the symmetry of a molecule to match the signals of the spectrum to the molecule. Doing so is standard in any review process when submitting a paper. If we look at isopropanol for example, we know it has $\rm C_{2v}$ symmetry, meaning that the 6 protons of the two $\rm CH_3$ groups will give us one singlet signal of height 6. The proton of the hydroxyl group gives us one singlet of height 1. Chemists don't think often about the point group, they think visually about the symmetry elements of the molecule.

  • Symmetry plays a major role in the synthesis of molecules. Just to give one example, a symmetrical ketone will give you the same enolate while an unsymmetrical ketone can give you two different ones. If two different ones can be obtained, it is important which one you obtain in order to synthesize the correct molecule. E.J. Corey and Robert B. Woodward pioneered this.

  • We also use Mulliken symbols: In coordination chemistry, we often deal with electronic spectra (UV-Vis). We learn for example that $\rm Mn^{2+}$ is almost colorless because Crystal Field Theory tells us it has the electronic configuration $\rm t_{2g}^3 eg^{2}$. This electronic state is classified by group theory as a $\rm {}^{6}A_{1g}$ state. In electronic transitions, the transition dipole moment must not be equa to zero. This means electronic transitions must be accompanied by a change in orbital symmetry. Because both $\rm t_{2g}$ and $\rm e_{g}$ are gerade (g), the transition is weak (LaPorte selectrion rule). Also, the electronic transition would require a change of spin of one electron, meaning spin multiplicity would change.

  • A molecule (inorganic or organic) can only be chiral if is does not contain an improper axis of rotation. In a chemists way of thinking, this means a carbon atom has 4 different substituents. The entire discussion of isomers is based on symmetry: If two molecules behave like mirror images of each others, they are called enantiomers.

  • In Infrared Spectroscopy, a molecular vibration can only be seen if the molecule has a permanent dipole moment. In Raman spectroscopy, a change in polarizability is required in order to see a molecular vibration in the Raman spectrum. Discussing which molecular vibration can be seen in which spectrum is done using group theory, which is very difficult to do for the average chemist. Symmetry also plays an important role in Mössbauer spectroscopy.

  • Solid state chemistry heavily relies on X-ray diffraction and other diffraction methods such as neutron diffraction. There are 230 space groups which can classify the crystal structure of a crystalline solid compound. Based on symmetry discussions, the reflexes in X-ray diffraction patterns can be interpreted. I know little about that, but I know it requires a lot of skill and experience to obtain good and correct results. (Note that diffraction methods are no spectroscopic methods, because spectroscopy is the energy-resolved analysis of the interaction of electromagnetic radiation with matter).

  • A very important thing in coordination chemistry is the Jahn-Teller-effect: Any molecular system which is not linear and has a degenerated electronic state will result in a breakdown of symmetry which changes the molecular structure and reduces the symmetry of the molecule. For example transition metal complex $\rm [Cr(H_2O)_6]^{3+}$ is not octahedral ($\rm O_h$), because its electronic ground state is degenerate according to Crystal Field Theory. The z-axis is elongated in the complex, meaning the symmetry reduces from $\rm O_h$ to $\rm D_{4h}$, because each orbital with a z-component is lowered in energy. The overall every of the complex is then lower. A similar case is $\rm [Ti(H_2O)_6]^{3+}$, but here the complex is compressed on the z-axis to resolve the degeneracy. This also lowers the energy by going from $\rm O_h$ symmetry to $\rm D_{4h}$ as well.

  • The Woodward–Hoffmann rules and the isolobal analogy are concepts which classify the symmetry of molecular orbitals to explain chemical reactions or structure of molecules. Hoffmann's Nobel lecture called "Building Bridges between Organic Chemistry and Inorganic Chemistry" is a must read for chemists.

  • The symmetry of orbitals and the symmetry of crystal structures can be combined to understand and explain properties of solids such as electronic or optical properties (electronic band structure). This plays a huge role for the understanding of semiconductors and other advanced materials. Also, the Goodenough-Kanamori-Anderson rules (GKA rules) are used to explain magnetism in solids based on symmetry and orientation of spin orbitals.

Summary

  • Chemists think about the structure of molecules and solids a lot. The structure of a molecule, its symmetry and the implications of this on the spectroscopic data you can obtain from the molecule is linked inseparable.
  • Chemists sometimes apply orbital theory if required (some more some less, depending on the field you are working in). The analysis of the symmetry of the molecule or the symmetry of the molecular orbitals is the foundation for this.
  • Symmetry is indespensable for chemistry, because entire fields of chemistry were founded based on symmetry considerations. A nice example is how Alfred Werner came up with coordination chemistry simply by making symmetry considerations. When Werner wrote down his ideas based on symmetry, he did not yet conduct one single experiment in the lab to prove any of this (!)
  • Symmetry plays a major role in the synthesis of molecules or solids.
  • Group theory is still poorly taught in my opinion. While there are plenty of good books about quantum mechanics or relativity, group theory remains a gap. I myself could construct the molecular orbital diagram of ferrocene based on symmetry analysis, but I could not tell you how to use the projection operator to obtain the molecular orbitals of the ammonia molecule for example.
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  • $\begingroup$ +10 for a solid answer! I gave a 300 point bounty here only because I'd already spent the bounty and it was the best answer out of the ones that came, but I was hoping the question could get a better answer, do you think you can write one? $\endgroup$ Feb 21, 2023 at 3:23
  • $\begingroup$ I understand that symmetry plays a very important role, but I still don't see how deep the mathematics involved is. But hopefully searching with these keywords will give some idea. Thank you! $\endgroup$
    – Absol
    Feb 24, 2023 at 6:17
  • $\begingroup$ @user1271772 I'm sorry but I don't think I can improve the answer given there. I am not a theoretical chemist. $\endgroup$ Mar 1, 2023 at 13:18
  • $\begingroup$ @Absol I think a strict treatment of group theory to understand of solve a problem in chemistry remains the job of a theoretical chemist. $\endgroup$ Mar 1, 2023 at 13:21

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