# 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? – Vlad Oct 8 '16 at 2:36
• yeah of course. It just doesn't have a matching function next to them, so I was unsure of how to figure out how many orbitals from my ligand set match those symmetry representations. This is also a more general question, though, but that's a good example of my problem. – gannex Oct 8 '16 at 2:37
• T refers to triply degenerate set, E refers to doubly degenerate set, A and B are singly degenerate. The total number of orbitals should add up to the total number of atomic orbitals on your ligands. – Vlad Oct 8 '16 at 2:39
• Oh, OK that actually answers my question. – gannex Oct 8 '16 at 3:24
• got it. I'm more concerned about getting the number right. The energy order is just a detail. Based on what V. Vladimirov said about T=triply degenerate, E=doubly, etc., my problem is solved. – gannex Oct 8 '16 at 17:04

## 1 Answer

T refers to triply degenerate set, E refers to doubly degenerate set, A and B are singly degenerate. The total number of orbitals should add up to the total number of atomic orbitals on your ligands. 1,2,g,u are symmetry labels (look up Mulliken symbols). One thing to remember when constructing you MO diagram is that only orbitals of similar energy and the exact same type and symmetry can mix. So, you can only mix A with A, B with B, E with E and T with T. Also, only g with g, u with u, 1 with 1 and 2 with 2.