The channel that points at the metal

Three essays of this argument have ended at the same wall. The denominator could not account for the trend and failed in opposite directions on the two sides of the series; the computed overlaps grew where the argument needed them to shrink; the metal’s contraction and then the donor’s were both too small and the second pointed the wrong way. In every one of them, two of the five ligands were not merely missed but unreachable: ammonia’s fitted π parameter is exactly zero and cyanide’s is −0.10, and the model’s quantity is a quotient of two squared overlaps, which is positive whatever goes into it.

A quantity that cannot be negative is not a quantity that has been fitted badly. It is a model with a term missing, and the missing term has a name.

The interaction the derivation does not have #

The angular overlap model’s derivation is second-order perturbation theory with one interacting pair. A metal d orbital mixes with a filled ligand orbital below it, the mixing pushes the metal orbital up by the square of the interaction over the gap, and the interaction is taken proportional to the overlap. Everything about the model follows: $e_\sigma$ goes as $S_\sigma^2$, $e_\pi$ as $S_\pi^2$, and their ratio is a ratio of squares.

There is a second interaction and the chemistry has known about it since before the model was written. A ligand with an empty antibonding π orbital above the metal level mixes with the metal’s d in exactly the same way and in the opposite direction: the metal orbital is the lower of the two now, so the mixing pushes it down. The two interactions were separated several essays ago in the context of electron counting, where what mattered was that they move charge opposite ways. Here what matters is that they move an orbital energy opposite ways, so their difference has no fixed sign.

Writing both down gives one expression with two terms:

The donor channel contributes $S_\pi^2$ over the gap from the ligand’s filled π up to the metal, and the acceptor channel subtracts $S_{\pi^*}^2$ over the gap from the metal up to the ligand’s empty π*. A ligand with no π* has only the first term and is a π donor, which is every halide. A ligand with both has a difference, and a difference can be negative.

That is not a new idea and the point of computing it is not to announce it. The point is that until now nothing in this argument had a π* overlap in it, so the acceptor channel was a name rather than a number, and the two ligands it is supposed to explain were being reported as failures of a model that had never been asked the question.

What has to be measured, and what can be computed #

The two channels need four quantities each and only two of them are integrals.

The levels are measured. A filled π’s position is a vertical ionisation energy: 16.91 electronvolts for carbon monoxide’s 1π band and 16.98 for dinitrogen’s. An empty π*'s position is an attachment energy, read off the resonance a slow electron is captured into, and that measurement was already in use on the denominator side: 1.50 electronvolts above the vacuum for carbon monoxide, 2.30 for dinitrogen.

Cyanide is a substitution and it is named rather than buried. Neither of cyanide’s two levels can be measured on the free anion — an extra electron on an anion is not a resonance anybody reads, and its ionisation is an electron detachment rather than a band. Hydrogen cyanide stands in for both, at 13.61 and 2.26 electronvolts, which is the same π system with a proton on the nitrogen instead of a metal. It is the weakest link in the arithmetic and it is the same weak link the essay before it already declared, now carried on both sides.

The metal’s level is the one thing nothing here measures, so it is swept rather than chosen, across the range that keeps a filled π below it and an empty π* above it.

And the overlaps are computed, which is what this essay is for. A π* has amplitude on both atoms of a diatomic ligand and they sit at different distances from the metal — the near one at the metal–ligand bond length, the far one at that plus the ligand’s own bond. So each channel’s overlap is two integrals rather than one, weighted by the coefficients the ligand’s own π system gives.

Those coefficients come from a two-level problem with the overlap kept: two p functions at the ligand’s measured bond length, their diagonal elements the two atoms’ valence ionisation energies, their coupling from the overlap by Wolfsberg–Helmholz. The model supplies the shape and the measurement supplies the level, and keeping those apart is the whole of the honesty here. The energies that fall out of the two-level problem are a caricature — two atomic ionisation energies pushed apart by a fitted constant — and they are not used for anything. Only the coefficients are.

Where each orbital keeps its amplitude #

The coefficients turn out to be the finding, which was not the expectation.

In a heteronuclear ligand the bonding combination lies towards the more electronegative atom and the antibonding one lies away from it. That is elementary and it is usually mentioned in connection with where the electrons are. Here it decides something else: the metal touches only the near atom, so it reaches whichever orbital is leaning towards it.

Cyanide’s filled π puts 0.394 on its donor carbon and 0.809 on the nitrogen beyond. Its empty π* puts 0.971 on the carbon and −0.667 on the nitrogen. The near-atom coefficients differ by a factor of 2.46, which is 6.06 in the square — and the square is what a second-order mixing is built from.

Carbon monoxide is the same shape with a smaller asymmetry: 0.435 against 0.939 on its carbon, a factor of 4.67 in the square.

So the empty orbital’s overlap with the metal is several times the filled one’s in both ligands, not because of anything about the metal and not because of the distance, but because the ligand keeps its bonding density at the far end and its antibonding density at the near one. The channel that wins is the one whose orbital is pointing at the metal.

The far lobe is a correction rather than the mechanism, and it is worth saying so because it is the explanation one reaches for first. A π* is a difference of two p functions and a π is their sum, so the far atom subtracts from one overlap and adds to the other. It does — and it is between 4.7 and 9.6 per cent of the near contribution, which modifies the comparison by a tenth. The factor of four is the near-atom coefficient and nothing else.

The gaps are pointing the other way #

With both overlaps in hand the two terms can be compared, and the comparison is not what the usual account suggests.

A π acceptor is normally explained by its low-lying π*, and the word low-lying does real work in that sentence: the denominator is small, so the term is large. Computed against these measured levels, it is not small enough. At a metal level of −4.75 electronvolts, cyanide’s filled π sits 8.86 electronvolts below the metal and its π* 7.01 above — so the acceptor gap is smaller, by a quarter. But carbon monoxide’s π is 12.16 below against 6.25 above and dinitrogen’s 12.23 against 7.05, and in all three the ratio of gaps is under two.

If the two overlaps were equal, every ligand here would come out a π donor at some metal levels and be near a tie at the rest. They are not equal: the acceptor’s overlap squared is 4.4 times the donor’s in cyanide and 3.7 times in carbon monoxide, which is more than enough to reverse a factor of two in the denominators.

That is a reassignment of the mechanism rather than a new result about it. Back-bonding is real and the model now produces it; what produces it here is the polarisation of the ligand’s own π system, with the low-lying π* contributing about a third of the effect and the geometry of the ligand the rest.

What the sign does across the sweep #

The metal level is swept because nothing measures it, and the sweep is where the two useful statements are.

Both fitted acceptors are negative everywhere. Across forty-one metal levels from the vacuum down to −10.05 electronvolts, cyanide and carbon monoxide come out with a negative π parameter at every one. That is the refusal lifted: the single-channel derivation could not produce a negative number at any radial function, any bond length and any basis, and this one produces it without being asked and without a choice being made about where the metal sits.

And dinitrogen changes sign inside the range. It is negative near the vacuum and positive below −8.25 electronvolts, so eight of the forty-one levels make it a π donor. The model cannot say what dinitrogen is.

Moving the level rather than reading the sweep makes the two statements one picture. The donor term shortens and the acceptor term lengthens as the metal drops, because the donor gap widens while the acceptor gap narrows, and the net is their difference — so a ligand whose two terms are close in size is the one whose sign is decided by where the metal sits. Dinitrogen’s acceptor term is the least dominant of the three at every level — 5.9 times its donor term near the vacuum against cyanide’s 17 and carbon monoxide’s 24, and 0.69 times it at the bottom — and it is the only one of the three whose ratio falls below one inside the range. That is the whole of why it is the ligand that crosses: the sign of a difference is decided by the smaller of the two quotients, and dinitrogen has it everywhere.

That is not a weak verdict, it is none — and it is the case where the mechanism is switched off. Dinitrogen is homonuclear, so its π and π* coefficients are equal by symmetry: 0.625 on each atom in the bonding combination and 0.833 on each in the antibonding one, with no lean in either direction. Its two overlaps differ by 1.21 where the polarised ligands’ differ by 3.7 and 4.4, and with nothing to reverse the denominators the answer is left to a quantity nobody has measured.

So the one ligand the model cannot classify is the one its own mechanism does not apply to. That is a stronger control than a third heteronuclear case would have been, because it was not chosen for the purpose: dinitrogen is in the sweep because its π* resonance is measured and its π parameter is not, which made it the prediction rather than the test.

The one ratio the series fits twice #

Two ligands in this set have a fitted π parameter, which gives exactly one number the model can be checked against without fitting anything: their ratio, 1.80.

The model over-predicts it everywhere, from 1.90 at the closest to 4.46 at the far end. That is the honest headline and it is the same direction every other comparison in this argument has gone.

What is worth the sweep is the shape. The over-prediction falls to 5.8 per cent at a metal level of −4.75 electronvolts and rises again on both sides, so the curve has a minimum inside the range rather than running to a boundary. Six per cent is nearer than anything else in this argument has come — the earlier essays were out by factors of two to six — and the minimum sits at a metal level that is not absurd for a chromium 3d orbital.

It costs one swept quantity and that has to be said beside it. A one-parameter family passing within six per cent of a single number is a weak claim; the same family passing through a minimum at a physically ordinary value is slightly less weak, and a family whose best is 1.058 rather than 1.000 is still a family that misses. What would turn it into a measurement is a third acceptor with a fitted scale, which would make the target a trend instead of a ratio — and nitrosyl and isocyanide are both in the series and both have measured π* resonances.

Ammonia is refused again #

The fifth ligand is the one this essay was expected to reach and does not.

Ammonia’s fitted π parameter is exactly 0.00, and a difference of two positive terms can be zero, so the arithmetic that refused it before no longer does. It is refused by the chemistry instead. Ammonia has no π level and no π*: its nitrogen’s other two p functions are in N–H bonds, and there is no antibonding combination for a metal to donate into. The acceptor term is not small, it is absent.

And the donor term is not absent. The model computes 1.39 × 10⁻² for it, on a bare p orbital placed where the molecule’s own structure has none spare — which is exactly what the earlier sweeps kept reporting as an ordinary computed value of 0.355 for a ligand whose fitted parameter is zero.

That is a better refusal than the one it replaces. Before, ammonia failed because the model’s expression had the wrong range; now it fails because the model’s inputs describe a ligand ammonia is not. The first is a defect in the algebra and could be repaired by algebra. The second says the zero in the table is a decision about the molecule’s own electronic structure, and no amount of second-order perturbation theory on a p orbital that is not there will produce it.

What was computed, and how #

Every metal–ligand overlap is a quadrature over Slater radial functions: chromium’s 3d at the effective charge Slater’s rules give chromium(III), the ligand’s 2p functions at their own, at the measured metal–ligand bond length and at that length plus the ligand’s own bond for the far atom. The ligand’s internal p–p overlap is the same quadrature at the ligand’s bond length.

Thirteen things are checked. That each filled π is a sum of its two p functions and each empty π* their difference, which is what makes them a bonding and an antibonding pair rather than two arbitrary roots. That the far atom contributes between two and twenty per cent of the near one’s overlap, so it is real and is not the mechanism. That the empty orbital’s overlap with the metal is the larger in every ligand. That the homonuclear ligand’s coefficients are equal to twelve decimals, so its polarisation is switched off rather than merely small, and that its two channels differ by under half what the polarised ones’ do. That both fitted acceptors are negative at every level of the sweep and the homonuclear one is not. That the fitted ratio is over-predicted throughout, that its closest approach is within a tenth, and that the approach is a minimum inside the range rather than an endpoint. That both gaps stay positive at four separate metal levels, since a negative gap would be reporting an acceptor as a donor and would deliver a negative answer for entirely the wrong reason. And that ammonia has a donor term and no acceptor term at all.

Where this model stops #

The two-channel expression is still second-order perturbation theory with the overlaps entering as squares, and every reservation the earlier essays raised about that survives. The proportionality of an interaction to an overlap is an assumption; the denominators are one-electron gaps standing in for many-electron ones; the radial functions are Slater’s. What has changed is only that the expression now has the term the chemistry requires.

The π* coefficients are the shakiest computed quantity here. A two-level Wolfsberg–Helmholz model of a triple-bonded diatomic is crude, and the polarisation it produces — carbon monoxide’s π* at 0.94 on carbon — is in the range quoted from real calculations rather than identical to any of them. The finding that depends on it is a factor of four, and a factor of four does not turn into a factor of one under a modest change in a coefficient. The finding that would not survive such a change is dinitrogen’s crossing point, which sits at −8.25 electronvolts and would move.

And the sweep’s free parameter is doing real work, which the six per cent should not be allowed to obscure. The metal level is not merely unmeasured here; it is unmeasurable in the form the model wants it, because a d orbital’s energy in a complex is not an observable the way an ionisation energy is. That is why the sweep is reported as a range and the minimum as a minimum, rather than a level being chosen and a single number quoted.

The generalisation #

The habit is to ask what a model’s expression can produce before asking whether its inputs are right, and then — when the answer is that it cannot produce the observed value — to ask what term is missing rather than what parameter is wrong.

Four essays of this argument swept parameters. Each was a legitimate question and each returned a clean refusal, and two of the five ligands were untouched by all four because the defect was never in a parameter. The tell was available from the first: a fitted quantity coming out with a sign the model cannot produce is not a poorly determined parameter but the fit reporting a mechanism the model has no term for. Reading it that way turns four sweeps into one structural question, and the structural question is cheaper than any of them.

The corollary is about explanations that name the right effect for the wrong reason. A π acceptor has a low-lying π*, is true, and it is offered as the explanation of back-bonding in every account of it. Computed, the low-lying π* supplies about a third of the effect and the ligand’s own polarisation supplies the rest — and the case that separates them is the homonuclear ligand, where the polarisation is zero by symmetry and the model promptly stops being able to say anything at all.

Who found it, and when #

Back-bonding is Dewar, Chatt and Duncanson, 1951 and 1953. The angular overlap model is Schäffer and Jørgensen, 1965, and its one-channel derivation is part of its original statement; two-channel versions of it are standard and older than this argument. The ionisation and attachment energies are quoted, with hydrogen cyanide’s standing in for cyanide’s. Everything else — the π and π* coefficients, both metal–ligand overlaps per ligand, the two terms, the sweep and the ratio’s minimum — is computed here.

The number worth carrying is not 5.8 per cent. It is 6.06: the factor by which cyanide’s empty π* out-reaches its filled π at the atom the metal touches, in a comparison where the energy denominators are pulling the other way.

Still open: a third acceptor, and what the first channel was hiding #

The obvious open question is the third acceptor with a fitted scale. Everything quantitative above rests on one ratio of two numbers, and a ratio of two numbers has no redundancy in it at all — the minimum at 1.058 could be the model working or could be two errors of similar size. Nitrosyl and isocyanide are both in the spectrochemical series, both have measured π* resonances, and either would turn the target from a ratio into a trend, at which point the over-prediction could be read as a slope and the direction of the failure established rather than inferred from one pair.

The nearer question is what the donor channel does to the halides now that it is not alone. Every earlier essay computed a π overlap for chloride, fluoride and water on a bare p orbital, as though the donor’s π electrons were in an atomic function — and the two-channel treatment has just shown, for the diatomic ligands, that a ligand’s π density is not where its donor atom is. A halide is monatomic and has no such correction, but water’s donor oxygen has its π-type lone pair in a molecule with two O–H bonds, and ammonia has none at all. Whether the three donors’ overlaps should be computed on molecular orbitals rather than atomic ones is the same correction applied to the other end of the series, and it would move the three ligands whose windows were found to sit above their fitted values by between 1.5 and 2.3.