Linear momentum of atoms of a molecule and their frequencies

Question

This exerpt on "Normal Modes of a Diatomic Molecule" is from Introduction to Mechanics Kleppner and Kolenkow:

Suppose we have a polyatomic molecule model with N masses and several springs coupling them. We now look for special solutions of the form

$x_ i = a_i sin(ωt + φ)$

$ i = 1, . . . , N$

The phase factor φ is also the same for each mass.where ai is the vibration amplitude of the ith mass. Note that in the special solution we are looking for, each mass vibrates with the same angular frequency ω. We justify the existence of such a solution by arguing that if the masses were vibrating with different frequencies, it would not be possible to conserve linear momentum for an isolated molecule."

The bolded line explains that the atoms of polyatomic molecules move with same frequency; otherwise, their linear momentum wouldn't be conserved. I searched but didn't find any relation between frequency and linear momentum conservation. Could anyone please explain on what basis were those lines written?

Answer 1

They are just arguing that if the atoms vibrated with different frequencies, then the center of mass of the atom would be oscillating as a function of time. Since this could only occur if an external force was acting on the molecule, which there is not, they conclude that the atoms must all have the same vibration frequency. Another way to say this is that the overall momentum of the molecule must be conserved (force is the change in momentum with time).

For a concrete example, consider a diatomic molecule like O$_2$. The only way to have one of the atoms oscillating without oscillating the center of mass is to have the other atom moving with the symmetry to exactly counteract the momentum of the first (either with an equal and opposite vibration, or they could be rotating together about the center of mass).