Symmetry is a concept introduced at school and never really revisited. Its importance isn’t grasped and often remains in the form of trivial ‘Spot the Difference’ puzzles on the bottom page of a newspaper. Even on the smallest scale of life, symmetry is relevant. It is a tool that can be used to differentiate and classify shapes. In mathematics, understanding an object through its axes and symmetry planes allows them to be grouped together.

The easiest way to explain this, is visually. A series of symmetry elements can be performed on this triangle. The tips of the triangle are labelled to easily follow these changes.

The triangle can be rotated clockwise or anticlockwise. Without getting into the details of notation – an E label signifies that the triangle is unchanged.

The second triangle (C_{3}) is the result of one rotation anti-clockwise. Applying an additional anti-clockwise rotation to this shape shifts the triangle to give the last shape. The double anti-clockwise rotation can be described as equivalent to one clockwise rotation, i.e. two turns in one direction gives you the same outcome as one turn in the opposite direction.

The triangle can also be reflected through the following three axes:

As tedious as this explanation might be – symmetry is a powerful tool. The rotations and reflections are combined to monitor the triangle’s movements.

This practice, called ‘group theory’, is applied in chemistry. Using symmetry, molecules too can be classified. This is because all molecules release energy in the form of vibrations. Molecules don’t just vibrate – they rotate and move up and down axes. By determining the symmetry of a molecule, the motions within are described.

The amount of energy released by a molecule can be thought of in terms of energy levels. The transition from one energy state to another can be compared to a ladder, climbed one step at a time. As a molecule jumps up this ladder, energy is released in the form of vibrations. A large excess of energy gives the molecule the potential to reach a high energy state. As each transition occurs, the shape of the molecule changes. By performing various symmetry operations, we can determine which atoms in a molecule change position when affected by this vibration.

A set of rules dictate which levels are allowed or not. Not every molecule will stretch its legs far enough to reach the next steps of the ladder. The relationship between the symmetry and the vibrations of a complex predicts the allowed transition states of a molecule. Those which reveal the highest degree of symmetry will vibrate the least. Analysis by infrared spectroscopy can confirm this mathematical approach.

It’s important to stress the elegance of group theory. No complicated software or experiment is required, simply a pen and paper. It is often said that the vibrations of any molecule could be scribbled down on the back of a napkin in a café.

Group theory offers a basis for understanding symmetry in chemistry. The axes and planes which build a molecule describe how a molecule relates to others. Aside from predicting what a spectrum will look like, symmetry can be the key to distinguishing a toxic drug from a pharmacologically active one. The difference in one configuration can be the difference between an orange scent or a lemon one. This basic ‘spot the difference’ game becomes essential in chemical applications.

Although this article is simplified, it is an attempt to present symmetry in a new context. Symmetry goes beyond geometry classes in secondary school and lies at the heart of the smallest, most complex systems.

Sources:

References:

D. C. Harris, M. D. Bertolucci, *Symmetry and Spectroscopy: An Introduction to Vibrational and Electronic Spectroscopy*, Dover Publications, New York, 1989.

Book: M. S. Dresselhaus, G. Dresselhaus, A. Jorio, *Group Theory: Application to the physics of Condensed Matter*, 2008, Springer-Verlag, Berlin, Heidelberg. ISBN 978-3-540-32897-1. https://link-springer-com.manchester.idm.oclc.org/chapter/10.1007/978-3-540-32899-5_1

K.C. Molloy, *Group Theory for Chemists: Fundamental Theory and Applications*, Woodhead Publishing, Oxford, 2011.

Thank you! Fascinating article..’spot the difference’ puzzles will not be the same!

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