Point groups: sorting molecules by their symmetry

Molecules with the same symmetry share a point group

Intuition

Sort a bag of cups by how they can be turned and flipped. Cups that look the same under the same turns and flips go in the same pile. Point groups are those piles for molecules.

Molecules belong to the same point group if they are symmetric in the same way. NH$_3$, POCl$_3$ and HCo(CO)$_4$ share one. If you know a molecule’s point group, you know all its symmetry operations, and much more. The lecture’s examples of what symmetry decides are chirality, polarity (and so, for example, greenhouse gas activity) and chemical bonding.

Exam detail

Test 1 asks you to identify molecules with the same point group. The lecture says this can often be done visually, without working out all the operations or the group symbol. Match them by the shape of the molecule: same axes, same planes.

The groups come in five families

The lecture sorts point groups by the pattern of axes.

• Low symmetry. $C_1$ has only $E$ (CHFClBr, biomolecules). $C_s$ has only one mirror plane (HClC=O). $C_i$ has only an inversion centre (1,2-dichloro-1,2-difluoroethane). Any planar molecule has at least $C_s$.
• Linear. $C_{\infty v}$ for linear molecules without an inversion centre (HCl, N$_2$O). $D_{\infty h}$ for linear molecules with one (H$_2$, CO$_2$).
• High symmetry. Several high-order $C_n$ axes. $I_h$ is an icosahedron (C$_{60}$), $O_h$ an octahedron (SF$_6$) and $T_d$ a tetrahedron (CH$_4$).
• Cyclic. One $C_n$ axis with $n \ge 2$. $C_n$ has only the axis. $C_{nv}$ adds $n$ vertical mirror planes. $C_{nh}$ adds a $\sigma_h$. Examples: H$_2$O$_2$ is $C_2$, H$_2$O is $C_{2v}$, NH$_3$ is $C_{3v}$, B(OH)$_3$ is $C_{3h}$.
• Dihedral. $D_{nh}$ is like $C_{nv}$ with an extra mirror plane perpendicular to the principal axis (and $n$ additional $C_2$ axes perpendicular to it). $D_{nd}$ is like $C_{nv}$ with $n$ additional $C_2$ axes perpendicular to the principal axis. Examples: benzene $D_{6h}$, BF$_3$ $D_{3h}$, naphthalene $D_{2h}$, staggered ethane $D_{3d}$, ferrocene $D_{5h}$ (eclipsed) and $D_{5d}$ (staggered).

Compare

Dnh

Extra mirror plane perpendicular to the principal axis. Benzene D6h, BF3 D3h, naphthalene D2h.

Dnd

n extra C2 axes perpendicular to the principal axis. Staggered ethane D3d.

A flow chart finds the group in a few yes-or-no steps

Try it: worked example

Use the lecture decision path to assign NH3, H2O, CH4 and naphthalene.

Show answer and steps

AnswerNH3 is C3v. H2O (and H2CO) is C2v. CH4 is Td. Naphthalene is D2h.

1. H$_2$O and H$_2$CO. Linear? No. Any $C_n$ with $n \ge 2$? Yes. Highest order $C_2$, only one such axis. Two $C_2$ perpendicular to it? No. Two $\sigma_v$? Yes. So $C_{2v}$.
2. NH$_3$. Linear? No. $C_n$ exists? Yes, highest $C_3$, only one. Three $C_2$ perpendicular to $C_3$? No. Three $\sigma_v$? Yes. So $C_{3v}$.
3. CH$_4$. Linear? No. $C_n$ exists? Yes. More than one high-order axis? Yes. Axis counts: 4 $C_3$ axes. Inversion centre? No. Mirror planes? Yes. So $T_d$.
4. Naphthalene. Linear? No. $C_n$ exists? Yes, highest $C_2$, and there are three $C_2$ axes. $\sigma_h$? Yes. So $D_{2h}$.

The lecture’s flow chart for the groups $T$, $T_h$, $O$ and $I$ is complicated. It says you will never meet molecules in those groups unless you go looking, and it gives a practical flow chart instead. The slide figure of that chart did not extract, so the steps above come from the four worked examples.

Optional aside

Chemists use the Schönflies notation for molecules. For materials (Lectures 4 and 5) the unit uses International notation, also called Hermann–Mauguin. The slide marks this comment as non-examinable.

Groups follow four rules, which are beyond the course

The last slide gives the maths definition and marks it beyond scope. The set of a molecule’s operations forms a point group $G$ if four things hold. Closure: combining two operations gives another operation in the group. Identity: there is a unique $E$ with $A \diamond E = E \diamond A = A$. Associativity: $(A \diamond B)\diamond C = A \diamond (B \diamond C)$. Inverse: every operation has a unique inverse in the group. If $A \diamond B = B \diamond A$ the group is also called commutative or abelian. A subgroup is a subset that is itself a group.