IR / Raman factor group analysis (site and factor group determination)

I'm currently trying to interpret some IR and Raman spectra of solid state compounds. They show a simple molecular unit of which the expected bands are split due to the coordination in the crystal structure. For a complete overview I'd like to add a factor group analysis, too.


I was reading through the original paper but it's a bit too mathematical for me. So instead I looked at different publications that already did a similar compilation of point group, site group and factor group and tried to recreate their solutions. I think I'm fairly sure it matches their results but since I haven't found an exact statement in literature, that would describe the procedure as I do it, I'd like to get some second thoughts on my resoning.


What isn't entirely clear to me is the description of the site and of the factor group. To my understanding the site group is a non-trivial subgroup of the molecule's point group. Say I have a tetrahedral molecular unit in my crystal structure of point group $T\_{d}$ the possible non-trivial subgroups contain for example $D\_{2d}$, $S\_{4}$, $C\_{2v}$, etc. So I would focus on how the coordination of the molecular unit to the next cations in the crystal structure would reduce the symmetry elements and therefore also reduce the point group to a subgroup.


For the factor group I understand that you have to consider the crystal structure and the optical phonons, which is no problem as services like the Bilbao Crystallographic Server have options to calculate this. The optical phonons determined are also split into the different Wyckoff-sites.
To my reasoning, I would simply take the Wyckoff-sites that are occupied by the central ion of my molecular unit, since it's the point I also define my point group and my site group symmetry on.


I am also pretty certain that I read this definition in a publication but at the moment I'm only able to find examples where the Brillouin zone is discussed. So I'm now a bit sceptical about my approach.


Could someone please give me a hint if I'm doing a mistake somewhere that just happened to work for the given examples?


