Symmetry and point groups

Turn an ammonia molecule by 120° about the axis through its N atom: every H lands exactly where another H was, and nobody could tell that it had moved. Turn it by 90° and the change is obvious. Chemists call such invisible moves symmetry operations, and all the symmetry operations of a molecule together make up its point group. The point group is an identity card that answers questions you would otherwise have to measure: can the molecule have a dipole moment? Can it exist as a pair of mirror-image twins that cannot be superimposed? Which of its vibrations absorb infrared light? How do the d orbitals of a complex split?

A symmetry operation is a move; a symmetry element is the geometric object the move is made about — an axis, a plane or a point. Molecules have only five kinds:

• Identity, E: do nothing. Every molecule has it, and it makes the set of operations a complete group.
• Rotation axis, $C_n$: turn by $360^\circ/n$ about the axis. Water has a $C_2$ axis, ammonia a $C_3$ axis, benzene a $C_6$ axis. The axis of highest $n$ is the principal axis, drawn vertical by convention.
• Mirror plane, σ: reflect through the plane. A plane perpendicular to the principal axis is $\sigma_h$ (horizontal); one containing the principal axis is $\sigma_v$ (vertical); one that contains the principal axis and bisects the angle between two neighbouring $C_2$ axes is $\sigma_d$ (dihedral).
• Centre of inversion, i: send every atom through the centre to the same distance on the other side. $\ce{SF6}$ and staggered ethane have one; $\ce{CH4}$ does not.
• Improper axis, $S_n$: turn by $360^\circ/n$, then reflect through the plane perpendicular to the axis. Neither step alone need be a symmetry operation; together they are. $\ce{CH4}$ has three $S_4$ axes but no $C_4$ axis.

$S_1$ is simply a mirror plane and $S_2$ a centre of inversion, so only improper axes with $n \ge 3$ are listed separately.

Symmetry elements do not come in arbitrary combinations: one symmetry operation followed by another always gives another symmetry operation of the same molecule. The full set of operations therefore forms a mathematical group, and because every element passes through one point (the centre of mass), it is called a point group; the number of operations, $h$, is the order of the group. Point groups are named with Schoenflies symbols: $C_{2v}$ means a $C_2$ axis with vertical mirror planes containing it; $D_{3h}$ means a principal $C_3$ axis, three $C_2$ axes perpendicular to it and a horizontal mirror plane; $T_d$ and $O_h$ are the full symmetry of the regular tetrahedron and octahedron.

You do not need every element to name the group — just ask a few questions in a fixed order:

• Is the molecule linear? With a centre of inversion it is $D_{\infty h}$ (like $\ce{CO2}$), without one $C_{\infty v}$ (like $\ce{HCN}$).
• Are there two or more axes with $n \ge 3$? Then it is one of the high-symmetry shapes: tetrahedron ($T_d$), octahedron ($O_h$) or icosahedron ($I_h$).
• Otherwise find the principal axis $C_n$ and ask: are there $n$ $C_2$ axes perpendicular to it? If so the group is a $D$ group; if not, a $C$ group.
• Finally look at mirror planes: a $\sigma_h$ adds the subscript h; with only $n$ planes containing the principal axis, a $D$ group gets d and a $C$ group gets v; with neither, nothing is added (a $C$ group should then be checked for an $S_{2n}$ axis).

The flowchart widget below lets you walk this path one question at a time; the right answer to every question is found by the program in the atom coordinates.

Polarity. A dipole moment is a vector fixed in the molecule. A symmetry operation leaves the molecule unchanged, so it cannot change the dipole either: the dipole can only point along a direction that every operation leaves alone. A $\sigma_h$, a centre of inversion and a $C_2$ axis perpendicular to the principal axis all reverse any vector along the principal axis, so only molecules in the groups $C_1$, $C_s$, $C_n$ and $C_{nv}$ (including $C_{\infty v}$) can have a dipole moment. $\ce{CH2Cl2}$ ($C_{2v}$, 1.60 D) has one; $\ce{BF3}$ ($D_{3h}$) and $\ce{CCl4}$ ($T_d$) do not — without knowing a single bond dipole. The formula below turns the rule into a calculation.

Take $\ce{NH3}$ ($C_{3v}$, $h = 6$): E contributes 3; each of the two $C_3$ turns contributes $1 + 2\cos 120^\circ = 0$; each of the three $\sigma_v$ contributes 1. $n_\mu = (3 + 0 + 3)/6 = 1$: the dipole has exactly one direction to point in — along the $C_3$ axis. For $\ce{BF3}$ ($D_{3h}$, $h = 12$): $3 + 2\times 0 + 3\times(-1) + 1 + 2\times(-2) + 3\times 1 = 0$, so no dipole is possible.

Chirality. A molecule is chiral when it cannot be superimposed on its mirror image. That happens exactly when it has no reflection-type operation — no σ, no i, no $S_n$ — that is, when its point group is $C_1$, $C_n$ or $D_n$. $\ce{CHFClBr}$ ($C_1$) is chiral; $\ce{CH2FCl}$ has a mirror plane ($C_s$) and is not. From 1911 Alfred Werner separated octahedral cobalt complexes into mirror-image pairs of optical isomers, and in 1914 he did the same for a complex containing no carbon at all — proof that chirality does not need carbon.