center of inversion

Imagine standing at the exact middle of a molecule and, for every atom you see in one direction, finding an identical atom the same distance away in the precisely opposite direction. Draw a straight line from any atom through that central point and continue an equal distance on the other side — you land on a matching atom. When this works for every atom at once, the molecule has a center of inversion, written i. It is the symmetry of a molecule that looks the same after being turned inside-out through its middle.

The inversion operation takes an atom at coordinates (x, y, z) and sends it to (-x, -y, -z), measuring from the center. Unlike a rotation, you cannot physically do this to a rigid object; it is a point reflection through a single point. Molecules that have it are called centrosymmetric. Octahedral SF6, square-planar PtCl4 (2-), the staggered ethane skeleton, trans-difluoroethene, and benzene all have a center of inversion; tetrahedral CH4, pyramidal NH3, and bent water do not.

The presence or absence of i is one of the most useful facts you can know about a molecule. A center of inversion immediately forbids the molecule from being chiral, and it forces a strict rule on spectroscopy: in a centrosymmetric molecule, no vibration (and no electronic transition between orbitals of the same parity) can be both infrared- and Raman-active — this is the rule of mutual exclusion, and it underlies the Laporte selection rule that makes many transition-metal d-d transitions weak. Orbitals in a centrosymmetric environment also get labeled g (gerade, symmetric under inversion) or u (ungerade, antisymmetric), a tag that runs all through molecular-orbital and ligand-field diagrams.