The third function fills the floor

A σ orbital built from one s function on each of two identical atoms has a momentum profile along the bond that is exactly zero at q = π/R: the two atoms contribute equal amplitudes and the phase between them is reversed there. A floor no charge transfer explains took the same construction to two different atoms and found the zero gone even at equal coefficients. Carbon monoxide’s minimum sits at 1.35 per cent of the profile’s peak, because carbon’s and oxygen’s radial functions are different shapes and cannot cancel at one momentum whatever their weights. Cauchy–Schwarz makes that floor irreducible at one function per atom.

The obvious objection was that one function per atom is not how a real σ bond is built. Carbon monoxide’s σ orbitals mix 2s and 2p on both atoms. And a p function’s momentum distribution is zero at the origin and peaks further out, so mixing some into the diffuse atom — carbon — should shift that atom’s contribution towards the compact atom’s shape. The shape mismatch might then be reducible after all. That would make the floor a property of the minimal basis rather than of a polar bond, and the question was left there.

It has a clean answer, and it is the opposite of the hope.

At π/R an s and a p on different atoms cannot see each other #

In momentum space an atomic function of angular momentum l carries a factor of $(-i)^l$: an s function is real, a p function is imaginary. Put two functions on atoms a distance R apart and their contributions to the profile interfere through the phase $e^{iqR}$ between the two centres. If the two functions’ l have the same parity, the product of their factors is real and the interference goes as cos qR; if opposite, it is imaginary and the interference goes as sin qR.

At q = π/R the cosine is −1 and the sine is zero. So at exactly the momentum where the floor is measured, an s function on one atom and a p function on the other do not interfere at all. Only pairs of the same parity — s with s, p with p — can cancel there, and whether they cancel or reinforce depends on the relative sign of the pair.

The four terms for carbon monoxide at its measured length of 1.1283 Å, with hydrogenic functions at Slater’s charges of 3.25 and 4.55 as in the one-function calculation: the s–s term is −0.0300, the two s–p terms are zero to the last digit of a double, and the p–p term, for the pair whose overlap is largest, is +0.4554. The s pair is what makes the minimum. The p pair works the other way, fifteen times as hard.

The rule is the heteronuclear form of one the homonuclear profiles already showed: there, a σg orbital mixing a 2s bond with a 2p bond carried a cosine combination plus a sine combination, and their cross term vanished at π/R. Here the atoms are different, the functions are different sizes, and the coefficients are free, and the rule survives all three because it is about the phase each function carries, not about the functions being the same.

p on the diffuse atom only adds #

With the rule in hand the lead’s proposal can be computed directly. Carbon’s function becomes a hybrid, $\sqrt{1-w}$ times its 2s plus $\sqrt{w}$ times its 2p along the bond, oxygen’s stays 2s, and the coefficients stay equal.

The carbon p function’s only contact with oxygen at π/R is through the sine, which is zero, and with carbon’s own s it has no cross term at any momentum, because two functions on one centre with l of opposite parity are 90° apart in phase everywhere. So its contribution at π/R is its own density there, nothing more. The computed numerator agrees with the s-only numerator plus the p function’s own density to eleven significant figures. Meanwhile the s part that did the cancelling is weighted down by 1 − w. Every p weight on carbon fills the minimum further: 1.35 per cent at none, 2.17 at a tenth, 3.63 at a quarter, 6.96 at a half.

The proposal was about shape, and the calculation above holds the coefficients equal, so it might seem the p function could still help at some other ratio. It cannot. The Cauchy–Schwarz floor of the one-function bond was the smallest depth over all coefficient ratios, 1.23 per cent, at a ratio of 1.44 in carbon’s favour. With a tenth p on carbon the smallest over all ratios is 2.17, with a quarter 3.35, with a half 4.86. No choice of weights lets a p function on the diffuse atom take the minimum below the s-only floor. The sign of the p function does not matter either; flipping it changes the profile elsewhere but not at π/R or at zero, and the two depths agree to ten decimals.

The hope rested on a true statement about shape: a p function does push momentum density outward, towards the compact atom’s distribution. What the hope left out was phase. Two contributions cancel only if they are the same shape and opposite in sign at the same momentum, and a p function on one atom has no sign relation to an s function on the other at π/R. Making the shapes more alike does nothing if the two things cannot interfere.

A bonding p pair fills it most #

p on oxygen alone does slightly more damage than p on carbon — 3.97 per cent at a quarter against 3.63 — because oxygen’s tighter 2p has more density at π/R. That is the same quadrature argument from the other side.

The interesting case is p on both atoms, because then the p pair can interfere, and which way depends on the sense. Carbon monoxide’s σ bond is made with the two hybrids pointing at each other, the sense that maximises their overlap. With the atoms placed as here that sense has every term of the hybrids’ overlap positive, and it is the sense in which the pσ pair is a bonding pair. And a bonding pair of pσ functions is a sine combination, not a cosine: it is zero at q = 0 and largest near π/R. So the p pair of a real σ bond does not cancel at the minimum; it peaks there.

At a tenth p on both atoms the minimum is 5.07 per cent of the peak; at a quarter, 12.5; at a half, 34.9. The same p weights in the opposite relative sense — the pair with the smaller overlap — leave the minimum almost where it was, 1.48 per cent at a quarter. Of the two ways to put p on both atoms, the one that makes the bond is the one that fills the minimum.

The reading becomes a reading of the hybridisation #

That changes what the minimum can be used for. The one-function essay’s conclusion was that the depth of the minimum, with the computable floor subtracted, reads the bond’s polarity — monotone in the chemical direction, with a floor that free-atom radial functions supply before any measurement.

Swept across polarities from −0.44 to +0.55, the s-only bond’s minimum moves between 1.24 and 2.61 per cent, a range of 1.4. A tenth p on both atoms lifts the whole curve by 3.7 per cent and a quarter by 11.2. The polarity still moves the minimum in the chemical direction on every curve, and by a little more with p in it than without — 2.1 per cent at a tenth, 3.8 at a quarter — but the offset the hybridisation adds is several times the whole range the polarity covers. For a bond whose σ orbitals mix s and p, a reading of the minimum is a reading of the p content first and of the polarity second. Subtracting a free-atom floor does not rescue it, because the p content is a property of the molecule’s orbitals and not of its free atoms. It would have to come from the same calculation the measurement was supposed to replace.

That is not a small correction to the one-function result. It restricts the proposed polarity measure to bonds whose σ orbitals are nearly pure s, which in the first row means little beyond the alkali-metal bonds — lithium’s valence shell is one s bond. It is the same shape of difficulty that a charge nobody measured keeps running into: the quantity is well defined, and it answers a different question from the one it was wanted for.

What would lower the floor #

The quadrature rule says what kind of function could lower the floor: one of the same parity as oxygen’s s, so that it interferes through the cosine. A second s function on carbon, of a tighter exponent, is the simplest.

It does lower it. A second function of effective charge 3.5 takes the floor from 1.35 to 0.89 per cent; 4.0 takes it to 0.23; 4.25 to 0.06; and at 4.55, oxygen’s own, it is zero to the precision of the arithmetic. But the mixture that does this is not a mixture. At every exponent the best combination is almost entirely the second function — slightly more than all of it, with a small negative admixture of carbon’s own, which the two functions’ overlap of 0.92 to 0.99 allows. The floor falls only by replacing carbon’s function with one more like oxygen’s, and vanishes only when it is oxygen’s.

That answers the lead’s question in its own terms. The floor is not a residue of a minimal basis that more functions would dissolve while leaving carbon as carbon. It is the difference between the two atoms’ functions, and a basis flexible enough to remove it is one that lets the diffuse atom borrow the compact atom’s shape. That is the same borrowing a counterpoise correction exists to undo when two molecules’ basis sets overlap in a calculation of their interaction, and the same reason a function that is already there adds less than it seems to. A variational calculation would decide how much borrowing carbon actually does, and nothing here estimates that.

How the profiles were built #

Each atom carries a hydrogenic 2s, and where asked a hydrogenic 2p along the bond, at Slater’s effective charge: 3.25 for carbon and 4.55 for oxygen, the charges the one-function calculation used. The atoms sit at ∓R/2 with R = 1.1283 Å. Each function’s radial momentum transform is tabulated once, and the orbital’s momentum amplitude on the plane $p_z = q$ is assembled as a complex number: the s parts real, the p parts carrying −i and the factor cos θ = q/p of a function along the bond, each atom’s part multiplied by the phase of its centre. The profile is the squared modulus integrated over the plane, by a Gauss rule on $\rho = R u^2$ that crowds points towards the axis.

The norm is computed separately, from position-space overlaps between every pair of functions, and every profile integrated over q is checked against it to one part in a thousand; a wrong phase between s and p fails that check, and does so by more than a per cent, which is the refusal. With no p and no second s the depth reproduces the one-function calculation’s to 10⁻⁶. “Maximum overlap” means the relative sign of the two p functions for which every term of the hybrids’ overlap is positive, and its norm is checked to be the larger of the two senses at every weight. The smallest depth over coefficients is a grid search in the ratio refined by golden section. The second s function is mixed with carbon’s own, normalised using their same-centre overlap, and the mixing angle is optimised the same way.

What must hold, and is checked: the reduction to the one-function depth; every profile’s integral against its norm; both s–p cross terms at π/R zero to 10⁻¹²; the s pair negative and the maximum-overlap p pair positive there; the p-on-carbon value at π/R equal to the s-only value plus the p function’s own density; the depth unchanged by the p function’s sign; every increase in p weight, on carbon, on oxygen or on both, filling the minimum further; no coefficient ratio letting p on carbon below the s-only floor; a quarter p on both filling it by more than five times the floor and by more than the s-only bond’s whole polarity range; and the second s function’s floor falling with its exponent and vanishing at oxygen’s.

What one orbital with hybrids still leaves out #

The p weights are chosen, not computed. Nothing here says how much p carbon monoxide’s σ orbitals actually carry. The sweep shows what each weight does, and the argument needs only that a real σ bond carries some. How much is a variational calculation’s business, and so is how carbon’s s function would reshape if it were free.

One orbital is not the molecule. Carbon monoxide has ten valence electrons in several occupied orbitals, and a filled pair contributes a residue of its own at π/R. The sum is what a measurement would see. This essay says that even the one orbital the polarity argument was about is dominated at π/R by its p content. It says nothing about how that combines with the other three.

A directional profile, not an averaged one. Everything is along the bond. A gas measurement averages over orientations, and the averaged profile keeps a residue even for the homonuclear bond, so the question of a floor is posed here in the variable where it is sharpest, not in the one a gas experiment reports.

Hydrogenic functions at Slater’s charges. The 2s functions have a radial node, as hydrogenic 2s functions do; Slater’s own functions do not. The quadrature rule does not depend on that, since it is a statement about angular momentum. The size of every number does.

Phase before shape #

The one-function essay separated the homonuclear zero into two conditions: a phase that reverses at π/R, and two amplitudes that are equal because the atoms are the same. A polar bond loses the second, and the natural repair was to make the amplitudes more alike by giving the diffuse atom a more flexible shape. That repair works on the wrong condition. A p function changes the shape and brings no phase with it that can act on an s partner at π/R. The pair of p functions that does bring a phase brings the one that reinforces, because a bonding pσ pair is a sine combination.

Cancellation needs both conditions, and adding a function can supply one while destroying the other. A floor that comes from unequal amplitudes can be lowered only by a function of the partner’s parity. For carbon monoxide the one available is a second s, and it lowers the floor by making carbon’s function oxygen’s. The minimum a polar σ bond’s momentum profile shows is then set mostly by how much p its orbitals carry. That is a property of the molecule’s electronic structure, and it can be read from the profile only once it is already known.

Still open: how much p, and the orbital that is not a bond #

The obvious open question is the p weight itself. Carbon monoxide’s σ system has three occupied σ orbitals of mixed s and p character, and how much p each carries is the output of a calculation, not an input. A two-function-per-atom variational treatment — the same two-level problem with overlap as before, grown to four functions — would give the weights, and with them a prediction of the minimum that the quadrature rule says is dominated by the bonding p pair. That calculation would say whether the minimum of carbon monoxide’s σ bond is near the 1.35 per cent of the s-only picture or near the 12.5 of a quarter p, and so whether the polarity measure survives at all for the molecule it was built on.

The nearer question is the lone pair. Carbon monoxide’s highest occupied orbital is a σ orbital that is mostly an sp hybrid on carbon pointing away from oxygen, and in the maximum-overlap language it is on the other side of the sign choice: its p part points the wrong way to bond. The second relative sense above, which left the minimum nearly untouched, is the sense a lone pair’s hybrid has towards its partner. Whether carbon monoxide’s lone-pair orbital shows a deeper minimum than its bonding σ orbital is a computation this construction can already make.