Can two objects with the same center of mass oscillate about that center?

WolfOfTheSteps

1. The problem statement, all variables and given/known data

Explain why there is no optical phonon in the dispersion curve for the one-dimensional monatomic chain of atoms.

3. The attempt at a solution

I am completely confused. I know that optical phonons get their name because when an array of 2 atoms of different charge oscillate they are eloctramagnetically active and can absorbe or emit infrared radiation.

So I could say optical phonons are impossible for monatomic chains because you would need two atoms of different charge to get phonons.

But, I think that is wrong. Because elsewhere I read that optical phonons are described by oscillations about a center of mass, while acoustic phonons are described by a translation of the center of mass. This must imply that monatomic chains cannot have oscillations about the atom pair's centers of mass. So now I'm thinking there must be some reason why two atoms of the same mass cannot oscillate about their center of mass....

But I have no idea what that reason would be! Just imagining two balls connected by a spring, it seems that they can oscillate about this center.

Any help would be greatly appreciated!