The lone pair sits where the floor was

An orbital’s momentum distribution is the same orbital seen in the other variable, and along a bond it carries a fringe. A σ orbital built from one s function on each of two identical atoms has a profile along the bond that is exactly zero at q = π/R, because the two atoms’ contributions are equal and reversed in phase there; the zero follows parity, not bonding, and a σ orbital built from p functions carries a sine and sits near its peak at the same momentum. Two different atoms cannot cancel exactly, and carbon monoxide’s s-only σ bond has a floor at 1.35 per cent of its peak — deeper or shallower with the bond’s polarity, which made the floor look like a measure of polarity.

A third function then filled it. At π/R an s function on one atom and a p function on the other are in quadrature and cannot interfere, so p on one atom only adds its own density, and a p pair in the bonding sense adds fifteen times as strongly as the s pair cancels. A quarter p on both atoms took the minimum to 12.5 per cent — eight times the whole range through which polarity moved the s-only bond.

That essay swept the p weight and said so. How much p carbon monoxide’s σ orbitals actually carry is the output of a calculation, and the lead it left was that calculation: grow the two-level problem to two functions on each atom, take the weights variationally, and see whether the molecule’s own σ bond sits near 1.35 per cent or near 12.5. Nearer, it asked whether the carbon lone pair — whose hybrid points the non-bonding way — shows a deeper minimum than the bond.

Both questions turn out to have a premise the molecule does not share. Carbon monoxide has no single σ bond to read, and its lone-pair orbital has no minimum at π/R at all.

Four functions, three occupied orbitals #

The calculation is the smallest one that can answer the question honestly. Each atom carries a hydrogenic 2s and 2pσ function at Slater’s effective charge — 3.25 for carbon, 4.55 for oxygen, the charges the earlier essays used — the two atoms sit at carbon monoxide’s measured 1.1283 Å, and the four functions’ overlaps are the ones the swept picture already computed. The Hamiltonian is the extended Hückel one: each function’s diagonal element is its valence-state ionisation energy as Hoffmann tabulated it — carbon −21.4 and −11.4 eV, oxygen −32.3 and −14.8 — and each off-diagonal element is the Wolfsberg–Helmholz rule, 1.75 times the overlap times the average of the two diagonal elements. The generalised eigenproblem is solved through the overlap matrix’s inverse square root.

That gives four σ levels. With ten valence electrons, four of them in the two π orbitals, the lowest three σ levels are occupied: 3σ at −34.8 eV, 4σ at −17.6, 5σ at −14.3, and the fourth is empty far above. Carbon monoxide’s photoelectron spectrum puts the corresponding ionisations near 14.0 and 19.7 eV for 5σ and 4σ, and 3σ’s in a broad inner-valence band in the high thirties where the one-electron picture itself breaks down — the order is right and the spacing is roughly right, which is as much as a method that treats orbital energies as ionisation energies should be asked for.

Every orbital is then exactly one of the swept picture’s bonds, with its four coefficients set by the calculation instead of by hand, and its profile along the bond is computed by the same momentum-space construction — the same (−i)ˡ phases, the same check that the profile integrates to the orbital’s norm. All six occupied σ orbitals below, carbon monoxide’s and nitrogen’s, integrate to one within a part in ten thousand.

The weights the sweep assumed, computed #

The swept picture drew a single polar σ bond with equal s coefficients on the two atoms and some p mixed into each. None of the three orbitals looks like that.

3σ is three quarters oxygen 2s — 74.8 per cent — with 22.8 per cent carbon 2s and almost no p. It is the orbital closest in character to the s-only bond, and it is not a polar bond with equal coefficients; it is oxygen’s 2s shell, lent a little of carbon’s.

4σ spreads over all four functions: 40 per cent carbon 2s, 14 per cent carbon 2p, a little oxygen 2s and 38.5 per cent oxygen 2p. Its population is 54 per cent on carbon.

5σ is shared nearly equally between the atoms, 49 per cent on carbon, and carbon’s share is almost all 2p. Its carbon hybrid has the non-bonding sense the lead named: in the swept picture’s convention the maximum-overlap hybrid has its s and p coefficients of opposite sign, and 5σ’s have the same sign, so the hybrid points away from oxygen. That is the orbital chemistry calls carbon monoxide’s carbon lone pair, the one that donates to a metal.

A caution about labels belongs here. Textbook accounts describe 4σ as largely an oxygen lone pair and 5σ as largely carbon’s, and in this minimal basis the Mulliken populations do not say that — 4σ carries slightly more carbon than 5σ does. What does match the textbook picture is the direction of 5σ’s carbon hybrid, which is a statement about the sign of two coefficients rather than about how a population is divided, and dividing a population between atoms is a choice the profiles below do not depend on.

The orbital with the lone pair’s sense has no minimum #

The swept picture had two senses for a p pair and found that the non-bonding one left the minimum almost untouched. The lead expected the lone pair, which has that sense, to show a deeper minimum than the bond. It shows none.

The profile of 5σ starts at the origin, dips to a small minimum at 0.29 π/R, rises to its peak at 0.83 π/R, and at π/R stands at 85 per cent of that peak — 2.58 times its value at the origin. It is the sine-like shape the parity rule gives a σ orbital built from p functions, because 5σ is mostly p on both atoms, and a p pair on two centres interferes through the cosine with the sign that adds at π/R.

The earlier expectation came from reading the non-bonding sense as the one that leaves the minimum alone. That was true of a bond built mostly from s functions with a quarter p mixed in, where the s pair supplies the minimum and the p pair decides only whether to fill it. 5σ is not that bond. Its carbon part is 82 per cent p and its oxygen part almost entirely p, so there is no s pair left to make a minimum for the p pair to fill or spare, and what remains is a p–p interference that peaks near π/R whichever way the carbon hybrid points.

There is a sharper way to see why the lone-pair sense did not matter, and it is in the signs of 5σ’s four coefficients. The lone-pair sense is a relation between carbon’s own s and p: they share a sign, so the hybrid points away from oxygen. What decides the profile at π/R is a different relation — between carbon’s p and oxygen’s p, the only pair in this orbital large enough to interfere — and in 5σ that pair has opposite signs in the swept picture’s convention, which is exactly the maximum-overlap sense that the quadrature rule showed adds at π/R. One orbital can have the non-bonding sense on one atom and the bonding sense between the atoms, because the two statements are about different pairs of coefficients. The lead read the first as if it were the second.

So the lone pair does not sit below the bond’s floor. It sits where the floor was, at the top of its own profile.

4σ is the only minimum, and it is not the floor #

Of the three, one orbital does have a minimum at π/R: 4σ, whose profile rises from the origin to a peak at 0.40 π/R and falls to a minimum at 0.975 π/R before a second maximum beyond. At π/R it reads 8.4 per cent of its own peak — about six times the s-only floor, and still below the quarter-p value of 12.5.

It would be easy to call that the answer: carbon monoxide’s σ minimum sits at 8.4 per cent, between the two swept values, closer to the p-rich one. But 4σ is not the polar σ bond either. Its population is a mixture of all four functions with carbon’s s and oxygen’s p the largest parts, and it carries two of the molecule’s six σ electrons. The minimum is a property of one orbital out of three; the measurement a Compton experiment makes is of all of them at once, and 3σ, whose floor is the nearest thing to the s-only one — an s pair, the kind whose cancellation is clean only at zero overlap — at 2.52 per cent of its peak — nearly twice 1.35, because its tenth of p on carbon adds without cancelling — carries another two.

The σ electrons together have no minimum #

A momentum-space measurement does not select an orbital. It sees the density of every electron, and the σ part of that density is two electrons in each of 3σ, 4σ and 5σ.

It has no minimum anywhere from the origin to 2π/R. It falls monotonically: from its value at the origin, where three quarters of it is 3σ’s, to 39.5 per cent of that value at π/R, where 85.6 per cent of what is left belongs to 5σ. The minimum that the polarity measure was built on is not visible in the σ total at all — π/R is a point on a smooth shoulder, and the orbital supplying almost all of the profile there is the one whose own profile peaks there.

That is the answer to the question the swept essay posed. Whether carbon monoxide’s σ minimum is near 1.35 per cent or 12.5 per cent cannot be decided, because the molecule’s σ density has no minimum to place. The floor was a property of a single bond made of s functions, and carbon monoxide’s σ electrons are not in one.

Nitrogen reads higher #

A measure of polarity needs a zero, and the natural zero is a non-polar molecule with the same electrons. An earlier essay made that comparison for s-only bonds; here nitrogen goes through exactly the same calculation — Slater’s charge of 3.9 on both atoms, Hoffmann’s −26.0 and −13.4 eV, 1.0977 Å.

Nitrogen’s lowest σ orbital, 2σg, is a nearly pure s pair and reads 0.04 per cent at π/R — the parity zero, filled only by the one per cent of p its s–p mixing admits. Its highest, 3σg, is 99 per cent p and reads 86 per cent of its peak at π/R, almost exactly as carbon monoxide’s 5σ does. The structure repeats; only the division into atoms differs.

In each molecule’s own units the two σ totals very nearly coincide. At π/R carbon monoxide reads 39.5 per cent of its value at the origin and nitrogen 41.6. The polar molecule reads lower — the opposite of what a floor produced by polarity would give, since the non-polar molecule was supposed to be the one that cancels. The difference is two percentage points on a quantity of forty, and its sign is set by how the p-rich upper orbital compares in the two molecules rather than by charge moving between the atoms.

So the difference between isoelectronic molecules, which the earlier lead offered as the fallback if the total hid the polarity, does not rescue the measure either. It exists and it can be computed; it does not have the sign or the origin of a polarity signal.

The verdicts against the constant #

Extended Hückel has one free number that is not a tabulated energy: the Wolfsberg–Helmholz constant, conventionally 1.75. A result that depended on it would be a result about the constant.

From 1.5 to 2, 5σ reads between 3.4 and 2.1 times its value at the origin — above one at every constant, so no minimum. 3σ moves between 3.3 and 2.3 per cent, always above the s-only floor. 4σ moves most, from 20 to 46 per cent of its value at the origin. The σ total barely moves at all, 39.8 to 39.4 per cent, and nitrogen’s stays above carbon monoxide’s at every constant. None of the verdicts depends on the constant; only the middle orbital’s number does.

How the claims can fail #

Each statement is checked where its figures are drawn. Every occupied σ orbital of both molecules must integrate to one in momentum space. Carbon monoxide must have three bound σ levels below an empty one, the lowest mostly oxygen 2s; its highest must be shared nearly equally between the atoms with carbon’s share nearly all p, and its carbon hybrid must have the non-bonding sign. The lowest must read above the s-only floor, the middle several times above it, and the highest more at π/R than at the origin. 4σ must be the one orbital with a minimum at π/R. Neither σ total may have a minimum out to 2π/R, and at every constant the polar molecule’s total must read lower at π/R than nitrogen’s.

The refusal is the parity cancellation itself. Nitrogen’s 2σg has equal s coefficients on its two atoms by symmetry, and with its small p part removed it must read exactly zero at π/R — to twelve decimal places. If it did not, the construction would have lost the cancellation that made the s-only floor a floor, and every number above would be measuring the arithmetic rather than the molecules.

Where the model stops #

Extended Hückel is not self-consistent. Its orbitals do not respond to the charge they place, and a polar molecule is exactly where that matters: the oxygen end of carbon monoxide is more negative than the neutral-atom ionisation energies know. A self-consistent calculation would move weight between 4σ and 5σ, and the middle orbital’s minimum — the one number here that moved a factor of two with the constant — would move with it.

Four functions, and hydrogenic ones. No 3s, no 3d, no polarisation functions; hydrogenic radial functions where extended Hückel normally uses Slater-type ones, so each 2s here has a radial node a Slater 2s lacks; and the charges fixed at their atomic values. The quadrature rule that governs π/R is exact for any basis, so the qualitative reading — p–p interference peaks there, s–s cancels there — survives any basis; the proportions between the orbitals do not have to.

Only the σ electrons. Carbon monoxide’s four π electrons contribute to the profile along the bond too, through a p pair across the bond that carries the cosine. They are left out because the question was about σ, and including them would add a further smooth contribution to a total that already has no minimum.

And one determinant. The zero at π/R for an s pair belongs to one determinant and is filled by correlation; the profiles here are single-determinant ones, so they are the most favourable case for a minimum rather than the least.

A measure built on the wrong object #

The minimum at π/R was a clean object while it was the property of one bond. It was exactly zero for a homonuclear s pair, it had a floor for a heteronuclear one, the floor moved with polarity, and a third function showed that p character swamped it. Each step asked what the minimum does as the bond changes. None asked whether a real molecule has one σ bond whose minimum could be read.

Carbon monoxide does not. Its σ electrons are in three orbitals of very different character, the one that dominates at π/R is the one with the least bonding sense, and the sum has no minimum at all. That is the same thing the hybrid picture runs into whenever it is asked to be a measurement rather than a description: a hybrid is a way of writing a set of orbitals, and a quantity defined on one hybrid bond need not correspond to anything the molecule’s density contains.

Still open: the π electrons, and a molecule with one σ bond #

The obvious open question is the full profile. The π electrons have a known parity — a p pair across the bond carries the cosine and has its own zero at π/R — so the complete directional profile along the bond is σ total plus a π part that does vanish there. Whether adding it creates a minimum in the total that the σ electrons alone do not have, and whether that minimum’s depth then says anything about polarity, is one more orbital pair through the same construction.

The nearer question is where the measure could still work: a molecule whose occupied σ space really is one bond. A hydride such as lithium hydride or hydrogen fluoride has a single σ bond with no second σ pair on the heavy atom to swamp it at π/R, or — for hydrogen fluoride — with its other σ orbital nearly a pure fluorine 2s. The swept picture’s floor might be readable there, and the four-function calculation here would say so with one change of atoms.