I have been going through this topic in my coordination chemistry chapter. I read that in metal carbonyls, -CO group acts as a ligand and donates its lone pair of electrons into a vacant orbital of the metal to form a sigma bond. Also, there's a pi bond formation by the donation of electrons from the filled d orbital of the metal into pi* MO of the -CO molecule, and this pi bond strengthens the metal to ligand bonding.
But filling of electrons in the antibonding MO means decrease of bond order and correspondingly decrease of bond strength and bond lengths. It sounds contradictory to what's written in my book.
Where am I making a mistake?