Which symmetry operation projects each atom through a single point and out the same distance on the other side?

The correct answer is the symmetry operation known as inversion. Inversion involves a transformation where each point in the molecule is reflected through a central point, known as the inversion center. Specifically, if you have a molecule with an atom at coordinates (x, y, z), after the inversion operation, the atom's new coordinates would be (-x, -y, -z). This means that every atom is effectively pushed through the central point of inversion to an equal distance on the opposite side, reflecting through that point.

This operation is significant in chemistry and crystallography because it helps determine the symmetry characteristics of molecules and crystal structures. A molecule with an inversion center generally has specific properties, including certain selection rules in spectroscopic transitions.

On the other hand, other symmetry operations such as improper rotation and reflection do not exhibit this kind of behavior. Improper rotation combines rotation with a reflection across a plane perpendicular to the axis of rotation, while reflection merely flips the structure across a defined plane but does not involve a central point projection. The identity operation leaves the molecule unchanged and therefore does not involve moving atoms through a point in any manner.