*The links between different STEM subjects is always a topic for discussion, and normally we just take peoples’ word for this. Remember at school a teacher saying, “this applies in Physics too” or “you’ll use this in Chemistry next year”, but just how far does this go and how deep are the connections between two scientific subjects? At degree level, skills and knowledge between different modules and courses intertwine all the time and learning about how the world works starts to make a lot more sense. Abstract algebra is typically a topic people steer clear from due to the mind-boggling concepts and extreme theorems, but a lot of these ideas help other models make a lot more sense. For example, and what we will be delving into, is the link between group theory in algebra and molecular rotation in quantum chemistry.*

## Group theory

In algebra we can have different classifications of numbers; some easy ones you will know are integers, natural numbers, and rational numbers. At degree level algebra we use the classification of numbers to assess their characteristics and patterns they follow, and we will see why this is useful in Chemistry after a few explanations.

Group theory is a study in modern algebra relating to the properties of elements in a set under a binary operation. The elements must follow certain ‘rules’ to be classed as a ‘group’. These require that the group be closed under the operation (the combination of any two elements produces another element of the group), that the binary operation obeys the associative law, that it contains an identity element (which, combined with any other element, leaves the latter unchanged), and that each element has an inverse (which combines with an element to produce the identity element). Here are the rules for $G$ to be a group with binary operation $\circ$ in mathematical notation:

$$\begin{align}

\text{(1) } & x \circ y \in G \text{ for all } x, y \in G, \\

\text{(2) } & (x \circ y) \circ z = x \circ (y \circ z) \text{ for all } x, y, z \in G, \\

\text{(3) } & \text{There exists } e \in G \text{ such that } x \circ e = x = e \circ x \text{ for all } x \in G, \\

\text{(4) } & \text{For every } x \in G \text{ there exists } y \in G \text{ such that } x \circ y = e = y \circ x.

\end{align}$$

For example, the integers (positive, negative, and $0$) form a group under addition, this is easy to verify that the sum of two integers is an integer, and for example that $0$ is the identity element. However, the natural numbers (the non-negative integers) do not form a group, since $1$ does not have an inverse, as that would be $-1$.

There are more rules that we can apply to further classify groups, and these more specific groups have more unique properties and applications. One type of group is dictated by geometric shapes and their rotations/reflections. Imagine any shape… how many ways can you transform the shape to get it back to its original position? These symmetries form the dihedral group.

So, if we take a square, what can we do to change this square back to a square? We can rotate it 90 degrees at a time, or any degrees, making sure they’re multiples of 90.

That’s an easy example. What about a 10-sided decagon? Well, we have the general rule that $2\pi / n$ will give the exterior angle of any regular polygon, and we know if you rotate a shape by that angle, you end up with the same as before. A massive help when forming a dihedral group!

Now let’s talk about reflections. You can ‘cut’ a shape in half and flip the shape on that axis and get the same shape again, the property for a dihedral group. We can use a square to visualise this again.

So now we know how to get the desired shapes, how do these form a group? They form the dihedral group and this is the set of all the rotations and reflections for the regular polygon, with the notation $D_n$. Here, $n$ denotes the number of vertices on the polygon. In a dihedral group the set contains: an identity element $\mathrm{id}$, which is how to get to the very original shape (i.e. a rotation of some multiple of $2 \pi$); $n-1$ other rotations denoted by $\rho^i$^{ }(where $i \in \{1, \ldots, n-1\}$); and $n$ reflections denoted by $\sigma_j$_{ }(where $j \in \{1, \ldots, n\}$).

Our finished dihedral group for a square is $D_4 = \{\mathrm{id}, \rho, \rho^2, \rho^3, \sigma_1, \sigma_2, \sigma_3, \sigma_4 \}$.

## Chemistry

Molecules are different combinations and amounts of atoms bonded together to create a new substance. For example, water (H_{2}O) is the bonding of 2 Hydrogen atoms and an Oxygen atom. Molecules make up everything and determine the characteristics of materials/ substances, this makes them quite important in understanding our universe. So how does this link with group theory – a very abstract part of Mathematics? Well, the symmetry of a molecule can tell us a lot about how it will behave in compounds and its potential for intramolecular bonding, and we can see from before that symmetries can be labelled and logged for each unique shape. If we can study and discover the symmetries for molecular orbitals, then we can determine the binding with other molecules. Binding is the attractive interaction between two molecules that results in a stable association in which the molecules are near each other. It is formed when atoms or molecules ‘bind’ together by the sharing of electrons. This binding allows for molecules to ‘clasp’ onto each other with different strengths and in different formations. To see how binding affects different objects, think about slime and a table: slime is almost a solid but is malleable and has movement, however a table is rigid and very hard. How did the inventors for slime know it would be all gooey and not as hard as a diamond? Well through this classification of molecules and known symmetries it would make it a lot easier to predict the outcome of their experiments! Now we have an understanding as to why understanding the symmetries of a molecule is important, just how does this link to algebra?

## Linking group theory and chemistry

By using the dihedral group theory, the shapes of molecules can be assessed and ‘labelled’. Molecules are 3D shapes and therefore the rotations and symmetries can be studied. Each rotation and symmetry can have its own distinct classification, like dihedral group shapes. Having a labelling system makes it available for scientists to predict and understand how a molecule will behave.

Symmetries play a huge role in chemistry, all due to molecular chirality and position in a compound. A chiral molecule cannot be superimposed on its mirror image, meaning its mirror image cannot be replicated by performing any rotations or inversions. An example of this is your hands: try and put your hands in the same plane or in the same position, they will never match! This is important in the chemical world as it can be the difference between a molecule being active (working) or not. LSD is a well know example of this; one chirality of the LSD molecule has no effect on the human body, but the other molecule will cause heavy hallucinations. A more sinister example of chirality with the human body is Thalidomide. Unfortunately, this drug’s issues with chirality caused very sad effects on a generation. Thalidomide was used to treat morning sickness in pregnant women, despite this it was later known this molecule’s two chiral partners had very different biological effects. One symmetry of the molecule would indeed cure morning sickness, however the other would cause mass deformations on the foetus when growing. When this was found out the drug was instantly removed for human consumption. This just shows the importance of now being able to label our drugs and molecules, so that no more disasters in the medical world can occur.

Leading on from this, the group theory labelling for symmetries in the pharmaceutical and drug research field is very useful. It saves time on analysing each molecule and running tests on their properties. It also means theorems can be developed for creating new substances, and then research will be easier for turning them into theories. The use of the symmetries in molecule synthesis is probably a chemist’s best friend!

Overall… the classification and prediction of molecules in chemistry means people can research chemical synthesis safely. Without the genius link to maths there would have to be a whole invention and agreements on a new categorization system for each molecule: very time consuming! The universal system allows chemists, biologists, physicists and mathematicians to learn a topic in common and communicate efficiently within the STEM community. The creation of new compounds, labelling them and letting other chemists use them with the safety via a mathematics theory is remarkable and a great achievement in the scientific research field.

Georgia is in her second year of Mathematics at UEA and will be completing her final year from September. She started her university career studying Chemical Physics, but found herself more partial to Maths. She loves the link between maths and science and how it explains the real world (just look at her articles!). When she's not studying she can be found out and about, eating and drinking in Norwich and London.