The quest for metal carbonyl bonding is going on here. As I mentioned before, I am working on it. This is another article by R.Hoffmann where you can read one of the best MO theory explanations and examples. He makes it so simple that, there is no way you can't learn MO theory from him. It's a quite long article, so I am not going to try to summarize it for you. But, I will mention one of the most interesting observations in bioinorganic chemistry and how he explains it.
As you know, hemoglobin binds to oxygen giving us oxyhemoglobin; and when unbound, deoxyhemoglobin. Oxyhemoglobin has a six-coordinate iron atom in the plane of the porphyrin ring, but iron is out of the plane in deoxyhemoglobin. Why? Simply, it's the spin state and ionic radius. Because, we know from the experiments that the spin state changes from low to high spin. You can read it in any biochemistry or bioinorganic textbook. Well, in this article, you can read it in more detail and how it is actually predicted. So, I guess, even if we hadn't observed this phenomenon, we would have guessed it.
So, assuming that the point group is C4v in these porphyrin complexes, for a d6 transition metal ion;
- low spin complexes will prefer a square pyramid and metal will be in the plane
- high spin ones will prefer to be out of the plane! (a1 orbital is occupied)