# Point Group Assignment Practice Makes

Three tools for teaching symmetry in the context of an upper-level undergraduate or introductory graduate course on the chemical applications of group theory are presented. The first is a collection of objects that have the symmetries of all the low-symmetry and high-symmetry point groups and the point groups with rotational symmetries from 2-fold to 6-fold. Students can sort the objects by point group and can also see similarities and differences between point groups. The second is a magnet-backed mirror onto which are placed molecular models that have been modified to rest on the mirror and represent atoms and bonds that are on the mirror plane. The third is a frame to show the perpendicular *C*_{2} axes that are a feature of all the *D*_{nh}, *D*_{nd}, and *D*_{n} point groups.

## Point Groups

Chemists classify molecules according to their symmetry. The collection of symmetry elements present in a molecule forms a “group”, typically called a *point group*. Why is it called a “point group”? Because all the symmetry elements (points, lines, and planes) will intersect at a single point.

So let's look at a specific example, say water. What symmetry elements does water possess? Identity, E; two reflection planes, σ_{xz} and σ_{yz}; and one 2-fold rotation axis, C_{2}. In the common notation (aka Schoenflies notation), this is known as the C_{2v} point group.

Another molecule that also belongs to the C_{2v} point group is cyclohexane in the boat conformation. Look at the two figures below and see that they do contain the identical set of symmetry elements, even though their overall shapes are quite different.

## Determining Point Groups

So how does one determine the point group of a molecule? One possible approach is simply to find *all* the symmetry elements and then look at a set of tables (something called character tables will work) until you find a matching set. While this would not be hard for something as simple as the example above, molecules like methane that contain 24 symmetry elements would be more tedious!

So chemists have developed various flowcharts that make the process as simple as answering a serious of yes/no questions. If you would like some practice in this process, you can go to the Symmetry Challenge page of this web site and find out how it works.

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