Mirror planes, inversion and improper rotation

A mirror plane swaps left and right

Intuition

Stand a face in front of a mirror. One half matches the other. A mirror plane inside a molecule does the same job: cut it along the plane and each half is the mirror image of the other.

A mirror plane is an imaginary plane where the atoms on one side are an exact reflection of the other side. The operation is $\sigma$. A plane can be named by the Cartesian plane it lies in. By convention the principal axis is the $z$-axis.

Some molecules have more than one mirror plane. A vertical plane $\sigma_v$ contains the principal axis. A horizontal plane $\sigma_h$ is perpendicular to the principal axis.

Compare

σv (vertical)

Contains the principal axis. Looks like a wall standing along the axis.

σh (horizontal)

Perpendicular to the principal axis. Looks like a floor cutting across it.

Optional aside

One slide says $\sigma_{yz}$ is a mirror plane that coincides with the “xy” coordinate plane. The name says $yz$, so the wording on the slide looks like a typo. The lecture’s rule is that a plane is named after the coordinate plane it lies in, so read $\sigma_{yz}$ as the $yz$ plane.

Inversion sends every atom through the centre

Intuition

Pick the middle of the molecule. Point at any atom, then go the same distance straight through the middle and out the other side. If you land on an identical atom, every time, the molecule has inversion.

A point of inversion, or centre of symmetry $i$, is a point through which every atom can move in a straight line to an identical atom on the other side, at the same distance. The centre can be on an atom or on a bond. The lecture also says it can only be at the centre of mass. An octahedron has an inversion centre. A tetrahedron does not.

Improper rotation is a turn plus a flip

Intuition

Turn the molecule, then flip it across a horizontal mirror. Some molecules look unchanged only after both moves together, even though neither move works alone.

An improper rotation $S_n$ is a rotation by $360^\circ/n$ ($C_n$) followed by reflection in a perpendicular mirror plane $\sigma_h$, or the other way round. So $S_n \equiv \sigma_h \times C_n \equiv C_n \times \sigma_h$, and the order does not matter. If a molecule has $C_n$ and $\sigma_h$, it also has $S_n$. The lecture’s example is CH$_4$, where $C_4$ followed by $\sigma_h$ gives $S_4$.

The lecture also states $S_1 \equiv \sigma$ and $S_2 \equiv i$. The slide leaves the working blank as a worksheet task. Use the definition: $S_1$ is a $360^\circ$ turn, which does nothing, then a reflection. $S_2$ is a half turn then a reflection in the plane across the axis.

An inversion axis $\bar{n}$ combines an $n$-fold rotation with inversion through a centre point. The $\bar{4}$ axis is a quarter turn of $360^\circ/4 = 90^\circ$ and then an inversion. Many of these are redundant, because $E$, $C_n$, $\sigma$, $i$ and $S_n$ fully define the symmetry of a molecule. They matter for crystals, covered in Lectures 4 and 5.

Proper operations keep handedness, improper operations flip it

Intuition

Rotating your right glove never makes it a left glove. Reflecting it does. That is the whole idea behind proper and improper.

Proper operations are the identity $E$ and pure rotations $C_n$. They retain handedness. Improper operations involve a reflection or inversion and change handedness. A molecule with any improper symmetry element cannot be chiral, and a chiral molecule cannot have any improper element.

Pitfall

Do not shortcut to “chiral means no mirror plane and no inversion centre”. The lecture’s counterexample is 1,3,5,7-tetrafluorocyclooctatetraene. It lacks $\sigma$ and $i$ but is achiral because of an $S_4$ axis.

Exam detail

Test 1 asks for definitions of the symmetry operations, including those that bring in translation (screw axes and glide planes, in the next module). Know $E$, $C_n$, $\sigma$, $i$, $S_n$ and $\bar{n}$ and their proper or improper type.