Relating molecules, atoms to number of optical branches (modes)

[war485]

I'm stumped as to how many optic branches (modes) are present if there are X number of molecules in a unit cell and Y number of atoms in each molecule.
I know that each individual atom has 3 degrees of freedom and more generally in a single unit cell, there are 3 acoustic branches (modes) and 3*atoms-3 number of optic branches (modes). There should be a total of 3*atoms branches (modes) due to the degrees of freedom. I'm struggling to understand how to generalize this further to include molecules into the picture.
Logically speaking, 3*molecules for all the degrees of freedom in a molecule, and 3*atoms for all the degrees of freedom of an atom. Maybe each molecule can be thought of as an individual unit cell inside a larger unit cell. Would there be a total number of 3*X*Y-3*X optical branches (modes)?

 

[M Quack]

The total number of branches has to be the same, if you group your atoms into molecules or not.

Let's assume that the interactions within the molecule(s) are stronger than the interactions between molecules. Like for example in a crystal of bucky balls.

The 3 acoustic branches will correspond to the long-wavelength displacement of entire molecules.

Then there should be branches of intermediate frequency corresponding to molecules moving out of phase, similar to optical modes when you put a simple atom in the place of the molecule. If there are X molecules in the unit cell, then you should get 3X-3 bands like this.

Finally, there should be relatively flat optical bands that correspond more or less to the vibrational modes of the free molecules.
If there are Y atoms in the molecule, then each molecule has 3Y-3 modes, giving X(3Y-3) modes total that will be very close to each other (because vibrations of one molecule will likely have very little effect on the vibrations of the other molecules).

That makes a total of 3+3X-3 + 3X(Y-1) = 3XY bands, which is exactly 3 times the number of atoms - as it should be.