Donation strengthens the C–O bond, back-donation weakens it

One of the figures on a d shell in a field: What a set of ligands does to five degenerate orbitals — computed twice, from an integrated point-charge potential and from an angular overlap matrix, which agree on every ratio.

Two essays draw this figure, each at the values its own argument needs rather than at the setting shown above. What each one uses it to show is below, in the words of its own caption.

Back-bonding is two interactions

The two interactions and the six measurements. On the left, what the model moves: 0.386 of an electron donated out of the ligand’s σ orbital, worth +0.058 on the C–O bond order, and 0.504 donated back into π*, worth −0.403. On the right, six stretching frequencies, five of them isoelectronic species differing only in charge.

The same pair of interactions with the ligand’s empty orbital pushed much higher. Donation is nearly unchanged and back-donation has almost stopped, so the two halves respond to quite different parameters — which is the essay’s claim in the form the model makes cheapest to check.

And with it pushed low, where back-donation dominates. The C–O bond order falls below the free molecule’s and the stretching frequency with it, which is the regime a strong-field carbonyl actually sits in — and it is reached by moving one parameter, not by adding a term.

The count that is not always eighteen

Both halves of the interaction as two-level problems, and the acceptor half measured independently beside them. Donation out of the ligand’s σ orbital moves 0.39 of an electron and is worth 0.06 on the C–O bond order; donation back into π* moves 0.5 and is worth −0.4, six times as much and in the opposite direction. To the right, five isoelectronic carbonyl complexes and free CO: the C–O stretching frequency falls as the metal’s charge falls, because more density is pushed into that same empty π orbital. It is the interaction whose sign this essay’s whole result turns on, seen in a spectrum rather than in a parameter.