# Deducing orbital degeneracy in geometries apart from octahedral or tetrahedral

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?

If you look at the $D_\mathrm{3h}$ character table, you see that $(x^2 - y^2, xy)$ transform together as $E'$.
It isn't really a limitation of crystal field theory, either. Crystal field theory isn't about looking at how much orbitals overlap with each other and saying "oh, I think those look about the same, they're probably degenerate". If that was the case then in an octahedral complex you would not be able to tell why $\mathrm{d}_{z^2}$ and $\mathrm{d}_{x^2-y^2}$ are degenerate, since the former only really overlaps with two ligands and the latter overlaps with four.