Guide to symmetry

Symmetry symbols are a way to describe different shapes, molecules in our case, by categorizing them based on mathmatical properties, such as how many mirror planes a shape has. You've already seen two of them, sigma-symmetry ("cylindrical symmetry around the internuclear axis") and pi symmetry ("above and below").

In certain specialist fields such as crystallography, learning these notations can be really useful. For most chemistry undergraduates these are boring and without much use. But you will be tested on them. You are also likely to come across one or two annoying teachers who describe the shapes of molecules by symmetry notation, even when words like "octahedral" or "tetrahedral" refer to the same thing.

Symmetry operation: An action, such as rotation through a certain angle, which leaves the molecule apparently unchanged.

Symmetry element: A point, line, or plane, in which the symmetry operation is performed in respect to.

Different symmetry operations are described below.

N-fold rotation (C_n)

Example, the rotation of H₂O:

The symmetry operation above is rotation of 180°. The symmetry element is the vertical line through the center of the molecule, called the n-fold rotational axis.

As you can see, the molecule is apparently unchanged. The central atom is considered not to be moved at all. All symmetry operations leave at least one point unmoved (the center of the molecule) which is why they are sometimes called operations of point group symmetry.

The C₂ symbol in the diagram is an example of an n-fold rotation symmetry operation, with the general formula C_n. This is an operation if the molecule appears unchanged after rotation by 360 / n degrees. So you can find n by dividing 360° by the angle the structure must be rotated. Easiest way to understand it is by diagrams:

The subscript describes the angle, the superscript describes how many rotations of that angle are done in the symmetry operation.

Many molecules have multiple n-fold rotational axes which exams will ask you to spot. Xenon tetrafluroride for example:

By convention, the highest order n-fold axis is set as the Z axis, and is typically drawn vertically.

You might reason that C₁, C₂², C₃³, and many others are all symmetry operations. This is correct. But there is no point mentioning them since anything rotated by 360 degrees is symmetrical, so writing it doesn't help us distinguish the molecule.

The one exception to this attitude is the next operation:

Identity operation (E)

This consists of doing nothing to the molecule. They all have at least this operation, and some have only this operation.

I don't know why people go through the effort of writing this. I doubt many undergrads know either. But it is important because many exams will ask you to describe symmetry elements in a molecule, and simply writing "E" can get you a mark.

My textbook describes the symmetry element of E as being "the whole of space". Presumably there is mathematical justification for that.

Reflection (σ)

This operation simply describes mirror planes, denoted with a sigma symbol σ. Water has two σ_v planes:

The subscript v or h refers to whether the planes are vertical or horizontal in respect to the principle axis of rotation - the axis with the highest number of turns.

For XeF₄:

The d in σ_d means "dihedral", refering to the fact that it intercepts two axes of rotation. Presumably this is just a way to distinguish it from σ_d - the plane which intercepts fluorine atoms.

Inversion (i)

To see this we imagine a point in the molecule, move it towards the center of the molecule, then move it outwards from the center the same distance. If the molecule appears unchanged at all points, it contains i symmetry.

This can be shown well with an octahedral shape:

In this case, the shape would only have inversion symmetry if all atoms were the same, or if 1 6, 3 and 5, 6 and 1, were equal to eachother. Tetrahedral shapes never have inversion symmetry.

Below is an ethene-shaped molecule, inversion is compared with a 2-fold rotation:

Improper rotation (S)

This is the most difficult to learn and to relearn, and probably often skipped in an exam question asking you to identify symmetry elements.

It involves rotation around a n-fold axis followed by reflection in the plane perpendicular to that axis:

The example above uses a 90 degrees rotation - equivalent to a 4-fold axis, so we denote this in subscript as S₄

A rotation of 360 degrees followed by reflection is equivalent to reflection alone, so we do not bother using S₁ since we can use σ.

A rotation of 180 followed by reflection is equivalent to inversion. So we do not use S₂ either, since we can use i.