How do you determine the crystal field splitting pattern when given the structure of a complex?
It's quite simple for structures like octahedral or tetrahedral, but I find that the reasoning behind the patterns for the trigonal bipyramidal structure, for example, is somewhat arbitrary.
Continuing the example of a trigonal bipyramidal structure, the $d_{xy}$ and the $d_{x^2-y^2}$ orbitals are supposed to be degenerate, but the $d_{x^2-y^2}$ fully overlaps with one of the ligands on the complex, while the $d_{xy}$ partly overlaps with two, and I don't think we have enough information to assume that the two "parts" add up to one.
Do I just have to accept that this is how it works (as in, is this just a "limitation" of the crystal field theory)? Or is there some aspect that I am missing/misunderstanding?