# How do I determine the number and order of ligand orbitals?

I am trying to create interaction diagrams for transition metal species by reducing the representations for both $\sigma$ and $\pi$ ligand orbital sets and mixing them with the metal orbitals. The order of metal orbitals is obvious because I know I have my 5 degenerate $n\text{d}$ orbitals, then my $(n+1)\text{s}$ orbital, which is higher in energy due to its principal quantum number, then my $(n+1)\text{p}$ orbital, which is higher in energy due to penetration (it's a p orbital); and the number of metal orbitals is apparent because my character table lists the each representation's matching function, so I can tell $\mathrm{A}_{1g}$ will have s-symmetry and $\mathrm{E}_g$ will match my $\mathrm{d}_{x^2-y^2}$ and my $\mathrm{d}_{z^2}$ orbitals, for example.

I don't see exactly how I should do the same for my ligand orbitals, though. For example, I know that my ligand $\pi$ orbital irreducible representation is $\Gamma_{\pi} = \mathrm{T}_{1g} + \mathrm{T}_{2g} + \mathrm{T}_{1u} + \mathrm{T}_{2u}$, and I know from my table's matching function list that there will be three $\mathrm{T}_{1u}$ $(x,y,z)$ and three $\mathrm{T}_{2g}$ $(xy, xz, yz)$ orbitals, but I don't know how to identify that there are also three $\mathrm{T}_{1g}$ and $\mathrm{T}_{2u}$ orbitals, since my character table doesn't list the matching functions for those representations.

Also; my understanding of how to decide on the ligand orbitals' relative energy levels is dubious. Yves Jean's Molecular Orbitals of Transition Metal Complexes says "one must analyze the bonding or antibonding character" (p. 43), but I am not getting it right every time. Can anyone give me some tips?

• Are you looking at the Oh group character table as it does contain the T1g and T2u representations?