Centre of Invertion/Centre of Symmetry

Question

I would need a detailed explanation of centre of symmetry in organic chemistry stereochemistry. I understand it is a point that conects 2 groups or atomos at same distance of a point. However, what gets me is about the achiral classification of molecule when one finds this center. Every time you find this centre in a molecule you can certainly say that the molecule is achiral? In any molecule and geometry? Please, explain if possible with pictures. Thanks if someone can help.

Answer 1

A reasonable question to start with is what it means to map a specific structure onto its mirror image. Before you read on, take some time to consider how to express this mathematically.

What is a mirror operation through the $xy$ plane? This takes a point $(x,y,z)$ to a point $(x,y,-z)$.

Also useful to consider what a rotation looks like. A simple 180-degree rotation about the $z$-axis takes a point $(x,y,z)$ to a point $(-x,-y,z)$.

So what is a center of inversion? A center of inversion means that you can take point to the point diametrically opposite through the center, and it should be the same. In other words $(x,y,z)\rightarrow(-x,-y,-z)$ is a symmetry operation.

But if you look closely, this is equivalent to a rotation about the $z$-axis followed by a mirroring over the $xy$ plane.

So what you have just said is that having a center of inversion means you can rotate and reflect the object and make it look like itself. Certainly, then, the mirror image is superimposable.

To be more specific, an object is achiral if it possesses an improper axis of rotation. The improper axis of rotation $S_{n}$ is a symmetry element of rotation about an axis equal to an amount $\frac{1}{n}$ of a full rotation followed by reflection over a plane perpendicular to that axis. As described above, inversion is an $S_{2}$ operation. Since the molecule possesses such an improper axis of symmetry, it is achiral.