symmetry and chirality
/ ky-RAL-it-ee /
Your left and right hands are mirror images that you can never quite line up — no matter how you twist one, it will not lie perfectly on the other. Molecules can have the same property: a chiral molecule and its mirror image are two distinct, non-superimposable forms (called enantiomers), like a left and right glove. The astonishing thing is that symmetry alone tells you, with a single clean test, whether a molecule has this handedness.
Here is the rule, and it is exact: a molecule is chiral if and only if it has no improper rotation axis (Sn) of any order. Because a mirror plane is just S1 and a center of inversion is just S2, the rule automatically rules out any molecule that has any mirror plane or any inversion center. So in practice you scan the molecule for any sigma, any i, and any higher Sn (the easy-to-miss S4 in particular). Find even one, and the molecule is achiral. Find none, and it is chiral, and its two mirror-image forms will rotate plane-polarized light in opposite directions.
In inorganic chemistry this is more than a stereochemistry footnote. Octahedral metal complexes with bidentate (chelating) ligands, such as tris(ethylenediamine)cobalt(III), [Co(en)3]3+, come in left- and right-handed propeller forms (labeled lambda and delta) precisely because they belong to a point group (D3) with no improper axis. Alfred Werner famously resolved such complexes into their two mirror forms, proving that metal complexes are three-dimensional — a landmark argument that rested entirely on this symmetry rule.
[Co(en)3]3+ (three ethylenediamine chelates around cobalt) is chiral: it has only proper rotation axes (point group D3) and no Sn, sigma, or i, so its left- and right-handed propeller forms cannot be superimposed.
[Co(en)3]3+ is chiral — no improper axis means left and right forms differ.
The correct test is "no improper axis Sn," not just "has no asymmetric carbon" or "no mirror plane." A molecule can lack a mirror plane and a center of inversion yet still be achiral if it has a higher Sn (such as S4) — so always check for Sn explicitly.