Why bother? The point group as a fortune-teller
In the previous two guides of this rung you learned to hunt for symmetry elements and then collapse the whole list into a single label — the molecule's point group. That label might feel like a stamp in a passport: tidy, but what is it good for? This guide is the first big payoff. Two physical properties that real chemists measure in the lab — optical activity and the permanent dipole moment — can be predicted from the point group alone, before you write down a single energy or do any spectroscopy. You do not even need to know which elements the molecule is made of. You only need its symmetry.
The logic behind both rules is the same, and it is worth holding onto: a symmetry operation, by definition, maps a molecule onto something indistinguishable from where it started. So any physical property attached to the molecule — its mirror image, its dipole arrow — must survive every operation in the group. If an operation would have to flip a property or wipe it out, then that property is simply not allowed to exist. Chirality and polarity each forbid a particular kind of symmetry. Find that element, or prove it is absent, and you have your answer.
Chirality: the molecule that cannot be superimposed on its mirror image
A molecule is chiral if it cannot be superimposed on its own mirror image, the way your left hand will never lie perfectly on top of your right no matter how you twist it. The two non-superimposable mirror forms are called enantiomers. They are subtle twins: nearly every ordinary property is identical, but they rotate the plane of polarized light in opposite directions — one clockwise, one counterclockwise — which is what we mean by optical activity. You met this idea concretely back in the coordination rung as optical isomerism in complexes, for example a tris-chelate like [Co(en)3]3+ that comes in left- and right-handed propeller forms. Symmetry tells us exactly when this is possible.
Here is the rule, and it is sharper than the one you may have memorized in introductory organic chemistry. A molecule is chiral if and only if it has no improper rotation axis, Sn — no operation that rotates and then reflects. This single statement quietly swallows two special cases. A plain mirror plane is just S1 (rotate by nothing, then reflect), and a center of inversion is just S2 (rotate by 180 degrees, then reflect). So a molecule with any mirror plane, or any inversion center, automatically owns an improper axis and is therefore achiral. The improper axis is the master criterion; mirror plane and inversion center are simply its two most common faces.
Why does an improper axis kill chirality? Because the Sn operation literally produces the mirror image and then rotates it back onto the original. If that operation is a genuine symmetry of the molecule, the molecule is indistinguishable from its own reflection — the two "enantiomers" are really the same object wearing different coats, so there is nothing to rotate light. Translating this into the point-group statement: groups that contain only proper rotations (the Cn and Dn families, plus the rotational groups T, O, I) permit chirality; the moment a group includes any sigma, i, or Sn — that is, the Cnv, Cnh, Dnh, Dnd, Sn, Td, Oh, Ih groups — chirality is forbidden.
Polarity: where a permanent dipole is allowed to live
Now the second payoff. Two guides back in the bonding rung you decided polarity by adding up bond-dipole arrows; here we get the same answers from pure symmetry, with no arithmetic. A permanent dipole moment is a vector — an arrow with a fixed length and direction frozen into the molecule. The question "is this molecule polar?" becomes "can the molecule's symmetry tolerate such an arrow?" And the answer follows from the survival principle: the dipole arrow must be left completely unchanged by every operation in the group.
Think about what each operation does to that arrow. A mirror plane that the arrow does not lie inside will flip it, so a dipole can only survive a mirror if it lies entirely within that plane. A center of inversion sends every vector to its exact opposite — it reverses any arrow through the center — so an inversion center destroys a dipole outright; a molecule with i is always nonpolar. A rotation axis leaves a vector unchanged only if the vector lies along the axis. Put these together: the dipole has to lie along whatever rotation axes survive, and inside whatever mirror planes survive, all at once. That is a very narrow keyhole.
The clean conclusion, the heart of the symmetry-and-polarity rule: a molecule can be polar only if it belongs to one of the groups C1, Cs, Cn, or Cnv. In Cn the dipole sits along the single rotation axis; in Cnv it sits along that axis, which is the one line shared by all the vertical mirror planes; in Cs it lies in the lone mirror plane; in C1 there is no symmetry to forbid anything, so any direction is fine. Every other group — anything with more than one independent rotation axis (the D groups, T, O, I), or with a horizontal mirror perpendicular to the main axis, or with an inversion center — has no surviving direction for the arrow, and the molecule is necessarily nonpolar.
Working it out: a two-question checklist
In practice you do not re-derive the theory each time; you assign the point group using the flowchart from the previous guide, then run two quick questions against the group. Here is the whole procedure.
- Assign the point group as you learned to do — locate the highest rotation axis, then check for perpendicular C2 axes, a horizontal mirror, vertical mirrors, and so on, until you land on a label like C2v, D3h, Td, or C3.
- Chirality question: does the group contain any improper element — a mirror plane sigma, an inversion center i, or an improper axis Sn? If yes, the molecule is achiral. If the group has only proper rotations (C1, Cn, or Dn), the molecule is chiral and optically active.
- Polarity question: is the group one of C1, Cs, Cn, or Cnv? If yes, the molecule is polar, and the dipole points along the rotation axis (or lies in the mirror plane, for Cs). Any other group — in particular anything containing an inversion center, a horizontal mirror, or multiple rotation axes — means the molecule is nonpolar.
molecule point group improper elements? chiral? polar? ---------------------------------------------------------------------- H2O C2v sigma_v (2) no YES NH3 C3v sigma_v (3) no YES CO2 D(inf)h i, sigma_h no no BF3 D3h sigma_h, S3 no no CHFClBr C1 none YES YES cis-[CoCl2(en)2]+ C2 none YES YES trans-[CoCl2(en)2]+ C2h i, sigma_h no no
Look hard at the last two rows, because they show why this is worth learning. The cis and trans isomers of [CoCl2(en)2]+ are built from exactly the same atoms and bonds, yet symmetry alone splits them apart. The cis isomer falls in C2 — only a proper rotation, no improper element — so it is both chiral and polar. The trans isomer sits in C2h, which carries an inversion center and a horizontal mirror; that inversion center alone forbids both a dipole and chirality. Same formula, opposite verdict on two measurable properties, decided in seconds by the point group. That is the power you have just unlocked.
Honest fine print: what the rules do and do not promise
These rules are exact statements about the idealized, rigid, single structure you drew — and that is exactly where the caveats hide. First, symmetry only tells you whether a dipole is allowed, never how big it is. A molecule in Cnv is permitted a dipole, but if its bonds are barely polar the moment may be tiny; the magnitude still comes from electronegativity and shape, the machinery you used two guides ago. Symmetry sets the gate; chemistry decides how much walks through it.
Second, the chirality rule is about the static shape on paper, but real molecules wriggle. A floppy molecule can flip rapidly between mirror-image conformations, so even if a single frozen snapshot looks chiral, the time-averaged molecule may show no optical activity — the two forms interconvert too fast to separate. Conversely, two enantiomers that are locked rigid (a tris-chelate, a propeller-shaped complex) can be isolated and will rotate light. When chemists ask "is it chiral?" they usually mean "can I bottle one enantiomer?", which folds in this question of rigidity that the bare point group cannot see.
Step back and notice what just happened. With nothing but the point-group label you assigned in the previous guide, you read off two experimental properties — optical activity and the permanent dipole — that earlier rungs reached only through dipole arrows or model-building. This is the recurring promise of group theory: the symmetry classification is not bookkeeping, it is a prediction engine. The next guide turns the same machinery loose on the character table, where these survival arguments become the formal language that will later sort molecular orbitals and predict which vibrations show up in the infrared and Raman spectra.