Symmetry operations and rotation axes

Symmetry means fewer numbers to store

Intuition

If a pattern repeats, you only need to describe one piece and say how to copy it. A tiled floor is one tile plus a rule. Symmetry does the same for molecules and crystals.

Lecture 2 opens with symmetry in nature, art, architecture, chemistry and biochemistry. In chemistry it shows up in molecular structure, electronic structure and supramolecular structure. Icosahedral viruses assemble from 60 or more units, and tobacco mosaic virus assembles 2130 protein units around an RNA strand.

The lecture’s reason that symmetry is useful is a counting argument. To model a C$_{60}$ gas molecule on a computer you need $x, y, z$ for each carbon, which is $3 \times 60 = 180$ numbers. For solid C$_{60}$, 12 g is Avogadro’s number of atoms, giving about $2 \times 10^{24}$ numbers. The lecture says the computer storage in the world is about $10^{21}$. A crystal is only manageable if symmetry lets you describe a small repeating piece.

Optional aside

The slide writes the solid-state count as $3 \times 6.022 \times 10^{22}$, but then gives $2 \times 10^{24}$. The result matches $3 \times 6.022 \times 10^{23}$, which is what Avogadro’s number gives. The exponent on the slide looks like a typo. The conclusion does not change.

An operation moves the molecule, an element is what it moves about

Intuition

Turn a square cushion a quarter turn and it looks the same. The turn is the operation. The imaginary pin through its middle that you turned it about is the element.

A symmetry operation is a transformation that maps a molecular geometry onto itself, moving atoms onto equivalent atoms. A symmetry element is the geometric thing (axis, plane or point) the operation is carried out about. The lecture lists four operations for molecules: rotation, reflection, inversion and improper rotation. It lists three types of element: axis of rotation, plane of reflection and inversion point. An improper rotation uses an axis and a plane.

Exam detail

Test 1 asks for the definitions of the symmetry operations. Learn the pair: operation (the move) and element (the axis, plane or point).

Rotation by 360 degrees over n gives n minus 1 new operations

Intuition

A three-bladed fan looks identical after a third of a turn. Two thirds of a turn also works. A full turn is the same as not turning at all.

The identity $E$ means do nothing. Every molecule has it. $C_n$ is a rotation through $2\pi/n$ (that is $360^\circ/n$) about a fixed axis, so the molecule has an $n$-fold rotation. NH$_3$ has a $C_3$ axis. Naphthalene has one $C_2$ axis perpendicular to the molecular plane and two $C_2$ axes in the plane.

When a molecule has several axes, the principal axis is the one with the highest $n$. By convention it is the $z$-axis. In the lecture’s triangle example the principal axis is $C_3$, with a $C_2$ axis lying in the plane.

Each $C_n$ generates more than one operation. There are $n - 1$ distinct rotations about an $n$-fold axis. A $C_3$ axis gives $C_3$ (120$^\circ$) and $C_3^2$ (240$^\circ$). Two further identities from the slide are $C_n^n \equiv E$ and $C_4^2 \equiv C_2$.

Try it: worked example

How many non-identity rotations does a C4 axis give, and what is C4 squared?

Show answer and steps

AnswerThree: C4, C4 squared and C4 cubed. C4 squared is the same as C2.

1. An $n$-fold axis gives $n - 1$ rotations. For $n = 4$ that is 3.
2. The rotations are $C_4$ (90$^\circ$), $C_4^2$ (180$^\circ$) and $C_4^3$ (270$^\circ$).
3. Two quarter turns make a half turn, so $C_4^2 \equiv C_2$, as on the slide.
4. A fourth quarter turn gives $C_4^4 \equiv E$.

Pitfall

Count the operations, not just the axes. One $C_3$ axis gives two rotation operations ($C_3$ and $C_3^2$). The identity is not counted among the $n - 1$.