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I came across this compound: $\ce{[Pd(η^2-C2H4)Cl2(H2O)]}$
What does that '$\eta^2$' sign mean? I am guessing that it means that the ligand is bidentate because if it wasn't, I don't think it is possible (well it would be rare) for a metal to be only co-ordinates by 3 ligands.
If this is the case, how can the carbon atom in ethene bond to Pd as it doesn't have a lone pair? Does the double break and hence the 2 carbons bond to the metal?
Ethylene is a monodentate ligand and there are 4 ligands total: ethylene, two chlorides, and water.
Regarding hapticity, you can find a great answer here: What does the eta notation mean in the naming of a transition metal complex?
While a chapter can be dedicated to the nature of organometallic bonding, I think it is practical to say that it is the two electrons in the pi-bond that are involved in bonding to the metal center.
It is not carbon that is donating a pair - instead, it is a $\pi$ bond as a whole. Its two electrons, which were shared between two carbons, are now shared between them and palladium in a kind of multicentered bond.
There is a huge class of compounds where organic molecules are attached to a metal in the same way. With larger conjugated $\pi$-systems, we may have ligands of $\eta^5$ coordination (look up ferrocene) and more, all the way to $\eta^8$ and maybe beyond that.
