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I've drawn MO diagrams for both molecules and they are isoelectronic. The main difference is that $\ce{C}$ is closer in electronegativity to $\ce{N}$ than it is to $\ce{O}$ so there will be less stabilisation of the bonding orbitals for $\ce{CO}$ due to the increased energy difference in the AO's.

Does this difference lead to $\ce{CO}$ having its $\pi^*$ orbital (LUMO) lower than that of $\ce{CN^{-1}}$ so it more readily accepts $\pi$ backdonation from a metal centre? Does this alone account for why $\ce{CO}$ is higher in the spectrochemical series? What about the difference between the $\sigma$ orbitals (HOMO) in $\ce{CO}$ and $\ce{CN^{-1}}$?

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You're on the right track. It has to do with the energies of the frontier orbitals. As you rightly said, both species are isoelectronic, and the orbital energies in CO are lower than those in CN.

The lower HOMO energy means that CO is a poorer σ donor orbital towards the metal than CN. Likewise the lower LUMO makes it a better π acceptor.

These two factors are conflicting: stronger π acceptors are stronger-field ligands, but poorer σ donors are weaker-field ligands. Empirically, since CO is higher in the spectrochemical series, it stands to reason that the π acceptor effect is the most important factor here. This isn't really a surprise, as π-effects tend to outweigh σ-effects when it comes to the spectrochemical series.

I don't pretend that the above is a rigorous proof, but short of crunching the numbers, I don't think there's a much better explanation.

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