Back-bonding is two interactions

Carbon monoxide is the most-studied ligand there is, and the reason is that it reports on its own bonding. It has a stretching vibration near 2,100 wavenumbers, that vibration is intense and easy to measure, and its position moves by hundreds of wavenumbers depending on what the molecule is attached to.

Free carbon monoxide stretches at 2,143 cm⁻¹. Bound to chromium in the neutral hexacarbonyl it stretches at 2,000. Bound to titanium in a dianion it stretches at 1,748. The obvious reading is that binding weakens the C–O bond and that more binding weakens it more.

That reading is half right, and the other half is on the same figure. One of the six species stretches at 2,204 — above free carbon monoxide. No account in which a ligand can only be weakened by binding produces a number on that side of the reference.

The two interactions #

The ligand and the metal are related by two separate orbital interactions, and they have opposite consequences for the C–O bond.

Donation. Carbon monoxide’s highest occupied orbital is a lone pair localised mostly on carbon, which is why the molecule binds through carbon. That orbital is slightly C–O antibonding: it has a node between the atoms, and it is occupied. Donating from it into an empty metal orbital therefore removes antibonding density from the C–O bond, and strengthens it.

Back-donation. The metal’s filled lower set of d orbitals has the right symmetry to overlap the ligand’s empty π* orbitals, which are strongly C–O antibonding. Donating into them puts antibonding density back, and weakens the bond.

Both interactions raise the metal–carbon bonding. They disagree only about what they do to the carbon–oxygen bond inside the ligand, which is why the stretching frequency is the quantity that separates them.

Where those two orbitals come from #

The argument depends entirely on two claims about carbon monoxide’s own orbitals, and both are worth substantiating rather than stating, because they are the sort of claim that is easy to state backwards.

The highest occupied orbital is on carbon and is slightly antibonding. Carbon monoxide is a heteronuclear diatomic: the oxygen orbitals lie lower than the carbon ones, so every bonding combination is weighted towards oxygen and every antibonding combination towards carbon. The orbital in question is the antibonding partner of a σ combination — mixed with other orbitals of the same symmetry, which reduces its antibonding character but does not remove it — so it is carbon-heavy and a little destabilising to the C–O bond. That is why the molecule binds through carbon rather than through the more electronegative atom, which is the first surprise a student of this ligand meets.

The lowest empty orbital is π and it is low.* A triple bond has two π bonding orbitals and two π* antibonding ones, and in carbon monoxide the π* pair lies low enough to be reached by a metal d orbital. That accessibility is what makes the molecule a π acceptor at all; dinitrogen has the same orbital arrangement with a π* that is considerably higher, and it is a correspondingly poor ligand.

Both claims are about which combination is which, and molecular orbital and valence bond and overlap decides argue at length that those are computed properties of a pair of atoms rather than conventions. What the present model takes from them is only the sign: donating out of an antibonding orbital helps a bond, filling an antibonding orbital hurts it.

What is computed and what is stated #

The two population transfers are computed here and their consequences for the bond order are not, so it is worth separating those clearly.

Computed. Each interaction is a two-level problem solved exactly — the same arithmetic the antibonding level goes up more uses — and the quantity taken from it is how much of the filled orbital now sits on the other fragment. With the default parameters that is 0.386 of an electron donated from σ and 0.504 donated back into π*.

Stated. How much each of those does to the C–O bond order is a pair of coefficients, and they are unequal: emptying the 5σ orbital helps a little because it is only slightly antibonding, and filling π* hurts a lot because it is fully so. The coefficients used are 0.15 and 0.8 per electron, and one of them was chosen so that a neutral hexacarbonyl comes out near its measured 2,000 cm⁻¹.

Derived from a stated proportionality. The frequency follows from the bond order through $\nu \propto \sqrt{k}$ with $k$ proportional to order, which is a harmonic relation with the reduced mass held fixed. That is checkable against something outside the model: a carbon–oxygen double bond, of order two, stretches near 1,700 cm⁻¹ in a ketone, and $\sqrt{2/3} \times 2143$ is 1,750.

So what the model is asked to reproduce is not the magnitude — it has been shown one — but the sign of each term and the ordering of the series, neither of which was fitted to anything. That is the same division as in Hückel with a heteroatom: the model supplies the structure of the answer and somebody else’s instrument supplies the scale.

The π overlap that back-donation runs through falls away faster with separation than the σ overlap does, which is why back-bonding is sensitive to the metal–ligand distance in a way donation is not. That asymmetry is one more reason the two halves cannot be summarised by a single interaction strength.

The series that decides it #

Five of the six species in the figure are isoelectronic: a metal with eighteen electrons, six carbonyls, differing only in the charge on the complex.

The frequency falls by 456 cm⁻¹ from one end of the series to the other, monotonically, as the complex is made more negative. That is a single-variable experiment of a kind chemistry rarely provides: the geometry is the same, the ligand is the same, the electron count is the same, and what changes is how much electron density the metal has to give away.

More negative charge means more available density, means more back-donation into π*, means a weaker C–O bond and a lower frequency. The direction is exactly what the second interaction predicts and is unrelated to anything the first can do.

And the dication is above free CO. With a doubly positive metal there is very little density to back-donate, so the σ donation term is left as the dominant effect — and σ donation strengthens the bond. The frequency accordingly goes the other way.

Why the frequency, and not something else #

There is a caution to attach here, because there is an essay warning against exactly the inference being made.

The frequency is not the bond strength argues that a stretching frequency measures a force constant — the curvature of the potential at its minimum — and that a bond dissociation energy measures its depth, and that the two are different quantities which happen to correlate. Nothing in the present essay repairs that. What is claimed is narrower: within one ligand, with the same two atoms and the same reduced mass throughout, a change in the stretching frequency is a change in the force constant, and that is a statement about the C–O bond and not about the metal–carbon one.

That narrowness is what makes the carbonyl series useful. Every comparison is between two molecules containing the same ligand, so the mass factor cancels exactly and the anharmonicity is nearly the same. Comparing the C–O stretch of a carbonyl with the C–O stretch of a ketone, as the calibration above does, is a much weaker comparison and is used here only as a sanity check on the order-to-frequency relation.

What this does to the electron count #

The eighteen-electron count follows from a σ-only reduction: six bonding orbitals and three non-bonding, the non-bonding ones being the metal d orbitals no ligand σ combination can reach.

Back-bonding changes their status. A ligand with empty π* orbitals gives those three metal orbitals something to interact with, and they stop being non-bonding: they become genuinely bonding, stabilised by donating into the ligand.

That is why the eighteen-electron rule works so much better for carbonyls and phosphines than for halides and oxides. The rule is a statement about the size of a gap, and back-donation both lowers the top of the filled set and widens what is above it.

Back-donation acts on the block a σ-only count labels non-bonding, which is where the metal’s own d electrons sit — so the same interaction that reports itself in the C–O stretching frequency also changes what those three orbitals are. The count does not notice, and the frequency does.

The connection runs the other way too, and is measurable. The strength of a metal’s back-donation can be read off the carbonyl frequency, and the same metal fragment’s ligand field splitting can be read off its electronic spectrum, and the two agree about which complexes are strong-field. Two independent instruments, one parameter.

Where the ligand sits in the series #

The spectrochemical ordering of the previous essay is the same argument seen from the ligand field side.

Ligands ordered by their π parameter run from strong donors at one end to strong acceptors at the other, and carbon monoxide sits at the acceptor extreme. Its position there and its stretching frequency in a complex are the same fact reported twice, which is why the series is ordered by an interaction rather than by a charge.

A π-acceptor ligand is one with low-lying empty orbitals of π symmetry, and being a good acceptor is precisely what puts a ligand at the strong-field end of the spectrochemical series. Carbon monoxide is at the top of that series and at the extreme of the acceptor scale, and those are not two facts.

The check on these figures tests the connection in the direction that could fail: turn back-donation up in the model and the predicted frequency must fall, turn it off and the frequency must rise above free CO. Both are checked, and the second is the one that would catch a model in which the two interactions had been given the same sign.

What the model leaves out #

Three omissions, each of which matters for a different reason.

There is no calculation of the metal–carbon bond. The model computes what happens to C–O and says nothing quantitative about M–C, though the two are complementary: the interaction that weakens the first strengthens the second, so a low carbonyl frequency goes with a strong metal–carbon bond. Confirming that would need a calculation not done here.

The orbital energies are parameters. Where the ligand’s σ and π* levels sit relative to the metal d is what fixes how much density moves, and those positions are inputs. A more electronegative metal, a different oxidation state, a different co-ligand all move them, and the model has to be told.

There is no electron repulsion. The populations are computed from a one-electron picture, which systematically overestimates how much charge will move — real electrons resist piling up on one centre. That is the caution the whole applied field carries, and the size of the effect is what the smallest many-electron calculation measures on a system small enough to solve exactly.

Part of the blue shift is not bonding at all #

The species stretching above free carbon monoxide is the strongest evidence in this essay, and it deserves a complication, because charge does two things at once and only one of them is an orbital interaction.

The family is real and it extends well past the one member here. Carbonyls of gold, platinum and mercury in high positive oxidation states stretch at 2,235, 2,261 and 2,280 wavenumbers against free carbon monoxide’s 2,143 — a hundred and forty above, in the last case. They are sometimes called non-classical carbonyls precisely because the classical account cannot produce a number on that side.

The orbital explanation is the one this essay gives and it is sound: donation comes from an orbital that is slightly C–O antibonding, so removing electrons from it strengthens the bond, and a metal with no density to give back leaves that strengthening unopposed.

There is a second mechanism and it needs no orbital overlap whatever. A carbon monoxide molecule in an electric field changes its stretching frequency, because the field polarises the bond and the polarisation depends on the bond length. That is the vibrational Stark effect, it is measured directly on carbon monoxide, and it is used as a ruler for the electric fields inside enzyme active sites for exactly that reason.

The size is right for it to matter. A unit positive charge two ångström from the carbon produces a field of a few volts per ångström, and carbon monoxide’s measured response to a field of that order is tens to a hundred wavenumbers — the same size as the shifts being attributed to back-donation.

So the series in charge that this essay uses to settle the competition is varying two things together. Adding positive charge to the metal reduces the density available for back-donation, and it also puts the ligand in a stronger field. Both raise the frequency, both scale with the charge, and the frequency alone cannot separate them.

That does not overturn the argument, because the two mechanisms agree about direction at every point in the series, so the ordering is safe. It does mean the sizes are not all back-donation, and it identifies the experiment that would separate them: an isoelectronic pair at the same charge with the ligand at different distances, or a neutral complex in an applied external field, where the electrostatic term can be turned without touching the orbitals.

The same two interactions elsewhere #

Carbon monoxide is the clearest case because it reports on itself, and the picture was not invented for it.

Alkenes. An alkene donates from its filled π orbital and accepts into its π*. Both halves weaken the C=C bond — the donation this time is out of a bonding orbital — so a coordinated alkene has a longer, weaker double bond, measurably so, and in the limit of very strong back-donation the ligand is better described as a metallacycle with two single bonds. That progression is a continuum rather than two cases, which is the same observation where two-centre bonding stops makes about bonding in general.

Dinitrogen. Isoelectronic with carbon monoxide, and a far worse ligand: its donor orbital is lower and its π* is higher, so both interactions are weaker. The stretching frequency of coordinated dinitrogen falls from 2,331 cm⁻¹ in the free molecule to around 2,000 in complexes that manage to bind it, and getting that number lower is essentially the problem of nitrogen fixation.

Phosphines. The acceptor orbitals are σ* combinations on the phosphorus substituents rather than a π* on the ligand’s own multiple bond, so the acceptor strength can be tuned by changing the substituents without changing the donor atom — which is why phosphine ligands are the workhorse of catalysis design and why their electronic properties are catalogued by, of all things, the carbonyl frequency of a standard nickel complex they are put into.

That last practice is worth pausing on. The accepted measure of how electron-donating a phosphine is, used across organometallic chemistry, is a C–O stretching frequency measured on a different ligand in the same complex. The carbonyl is being used as a meter for what the metal has left to give, which is exactly what this essay’s series demonstrates and is a rare case of a spectroscopic quantity becoming an industrial parameter.

The competition implied by that — one metal, several ligands, a finite amount of density to distribute — is the subject of the trans influence is an overlap argument, where two ligands on opposite sides of a metal are made to compete for the same orbital and the loser’s bond order is computed.

Who found it, and when #

The two-interaction picture is Michael Dewar’s, from 1951, written for the bonding of alkenes to silver; Joseph Chatt and Leonard Duncanson generalised it to platinum–alkene complexes in 1953, and the same account was applied to carbonyls immediately. It is universally known by the three names.

What is worth noticing about its history is how much it was asked to explain and how little it contains. The picture is two orbital interactions with opposite consequences, no numbers, and no calculation; and it accounts for the carbonyl frequency series, the spectrochemical position of π-acceptor ligands, the stability of low-valent metals in carbonyl complexes, and the lengthening of a coordinated alkene’s double bond. A model that cheap explaining that much is usually a sign that it has identified the right variable.

The variable is the direction of electron flow. Chemistry before 1951 had one direction — a ligand is a donor and a metal is an acceptor — and the observation that makes carbonyls work is that the flow goes both ways at once, through orbitals of different symmetry, and that the two flows can be counted separately because they touch different bonds.