The value that only exists in the bond

One quantity is missing from every electronegativity table. Write an atom’s energy as a function of its electron count and expand it: the first derivative is the negative of the chemical potential and comes out as half the sum of the ionisation energy and the affinity, which is Mulliken’s electronegativity. The second derivative is twice the hardness and comes out as half their difference. Both are fixed by the same two measurements, and only one of them is ever printed.

It ended by saying what to do with that. Hardness answers how much charge moves for a given difference; it does not answer what happens to the difference once the charge has moved, and the answer is that it closes.

That is Sanderson’s equalisation principle, and in this model it has a closed-form answer.

The solution, and why it is a weighted mean #

Each atom’s chemical potential is linear in its own charge,

$$\mu_i(q_i) = \mu_i^0 - 2\eta_i q_i,$$

with $\mu^0 = -(I + A)/2$ and $\eta = (I - A)/2$, both computed from the measured ionisation energy and affinity. Requiring every $\mu_i$ to be equal and the charges to sum to the molecule’s gives

$$\mu = \frac{\sum_i \mu_i^0 / 2\eta_i - Q}{\sum_i 1/2\eta_i}, \qquad q_i = \frac{\mu_i^0 - \mu}{2\eta_i}.$$

The equalised value is a hardness-weighted mean of the free-atom values, with a soft atom counting for more because it moves further for the same push. That is not the geometric mean Sanderson proposed, and it is not the arithmetic mean either.

The sign convention matters and is easy to get backwards. With q the charge — minus the excess of electrons — an atom that has given charge away has a lower chemical potential than it started with. Getting it the other way round makes hydrogen negative in methane, which is not what the Mulliken numbers say: hydrogen’s χ is 7.176 eV and carbon’s is 6.2615, so on this scale hydrogen is the more electronegative of the two, and carbon comes out positive.

That is not a defect of the arithmetic. It is the disagreement between electronegativity scales, showing up as a sign: Pauling puts carbon above hydrogen and Mulliken puts hydrogen above carbon, and the C–H bond is one of the six whose polarity direction is disputed.

One element, four compounds #

The point of solving for equalisation rather than subtracting table entries is that the answer is different for every molecule.

Carbon in methane carries +0.0692. In fluoromethane it carries +0.1259, in trifluoromethane +0.2449, in tetrafluoromethane +0.3074. The electronegativity it ends up sharing with its neighbours runs 6.954, 7.520, 8.710, 9.334 eV.

A table prints 6.2615 for carbon. That is the value it starts an iteration from, and no molecule has it.

The hydrogen result is sharper still. In methane hydrogen is −0.0173; in fluoromethane it is +0.0268; in trifluoromethane +0.1194. Adding a fluorine on the far side of the carbon reverses the C–H bond’s polarity, without changing either atom in it.

So a bond dipole is not a property of a bond. That is a conclusion the dipole-sum argument reaches from the other direction — a molecular dipole is not a vector sum of bond dipoles — and this is why: the bond dipoles themselves are not constants to be summed.

The same picture drawn for the other end of those bonds says the same thing from the receiving side, and it is worth having both because a charge that leaves one atom has to arrive somewhere.

The two rules disagree #

Sanderson’s proposal was that the equalised electronegativity is the geometric mean of the atoms’ values. That is a definite rule and it can be checked against the solution.

The disagreements are small in absolute terms and systematic in pattern. Ranking the twelve molecules by how far apart their atoms’ hardnesses are, and again by how far the two rules disagree, gives a rank correlation of 0.62 — which is the only thing a weighting could respond to, since the geometric mean is unweighted and the equalised value is weighted by nothing else.

Silane and boron trifluoride are the two worst, and both are cases where a soft central atom is surrounded by hard ligands. Ammonia is the best, at 0.0013 eV, because nitrogen and hydrogen have nearly the same hardness and the weighting has nothing to do.

There is a general moral in that pattern and it is not about Sanderson. An unweighted rule agrees with a weighted one exactly where the weights are alike, which means the rule’s long record of working is evidence that hardnesses are usually alike rather than evidence that the rule is right. Testing it on the cases where the weights differ is the only test that carries information, and there are two of them here.

So Sanderson’s rule is a good approximation to the answer of a model he did not have, and the places it fails are predictable from the quantity the tables leave out.

None of this resolves the disputed bonds. Electronegativity is not one quantity found six ordinary bonds whose polarity direction two of the four scales disagree about, and equalisation inherits whichever scale it is fed rather than adjudicating between them. What it adds is worse for the tables and not better: even within one scale the direction is not a property of the bond, because it depends on what else is in the molecule.

What equalisation does not do #

The model is a quadratic energy and a constraint, and it is worth being clear about what it leaves out, because the omissions are not small.

There is no distance in it. Every atom in the molecule is equalised against every other one directly, so a fluorine four bonds away is exactly as effective as one bonded to the carbon. That is the reason hydrogen’s sign flips so readily in the fluoromethanes above, and it overstates the effect: a real inductive effect falls off along a chain, and this one does not fall off at all.

There is no bond. The molecules are lists of atoms; nothing in the input distinguishes CH₃F from a random collection of one carbon, three hydrogens and a fluorine. Any two molecules with the same formula get the same answer, so isomers are indistinguishable.

And the charges are unbounded in principle. An atom with a very small hardness would take an arbitrary amount of charge, and the closed form divides by the hardness. Nothing here has one, and a hardness of zero is refused rather than answered.

What the numbers say about the received chemistry #

Three of the twelve molecules produce results a chemist would recognise and one produces a result a chemist would object to, and both are worth looking at.

Boron trifluoride puts +0.4824 on the boron and −0.1608 on each fluorine, which is the largest charge separation in the set. Boron is soft and electropositive, fluorine is hard and electronegative, and the two effects point the same way — so an electron-deficient centre surrounded by the most demanding ligand available is exactly where equalisation predicts the most transfer.

Silane puts +0.2412 on silicon and −0.0603 on each hydrogen, and methane puts +0.0692 on carbon and −0.0173 on each hydrogen. Both are positive at the centre, which is the Mulliken scale’s C–H direction again, and silicon’s is three and a half times carbon’s — which is the ordering every chemistry course teaches about Si–H and C–H reactivity, arrived at from two ionisation energies and two affinities.

Water puts −0.0196 on the oxygen and +0.0098 on each hydrogen, and this is the objectionable one. Every other account of water has the oxygen strongly negative. What has gone wrong is not the equalisation but the scale: the Mulliken value for hydrogen is 7.176 eV and for oxygen 7.540, so on this scale water is a very slightly polar molecule and the transfer is correspondingly tiny.

That is a limitation of the input rather than of the method, and it is the disagreement between scales arriving where it hurts. A method that equalises is only as good as the values it equalises, and the four common scales disagree about oxygen and hydrogen by more than enough to reverse water.

And the outlier is identifiable rather than merely suspected, because the disagreement about hydrogen is a disagreement one table has with the other three.

Two consequences worth stating plainly. Fed to a multipole calculation, water’s charges here would give a dipole an order of magnitude below the measured one — which is the sharpest available statement of how far the Mulliken input is from the measurement, and it is a statement about the input rather than about the constraint. And the model’s largest omission is that boron trifluoride’s geometry plays no part in any of this: three hard fluorines round a soft boron give the largest transfer in the set whether they are arranged in a plane, a pyramid or nothing at all.

The four scales, one more time #

Set beside the other readings, the equalisation adds a fourth kind of statement about electronegativity and it is worth listing all four.

Electronegativity is not one quantity: four scales, twelve discordant pairs out of a hundred and fifty-three, six ordinary bonds whose polarity direction is disputed.

A ranking is not a difference: two scales can agree on every ordering and disagree on every gap, so a rank correlation of one says nothing about whether a difference is meaningful.

A difference does not make a transfer: one difference moves 0.4472 of an electron at no repulsion and 0.0019 at a large one, a factor of 240, so a difference without a hardness beside it predicts nothing.

And now: an element’s electronegativity is a starting value rather than a property, because the quantity that matters is the one every atom in the molecule ends up sharing, and that depends on all the others.

Underneath all four is the same two-orbital problem overlap decides is built on: two atoms of different Coulomb integral, four electrons, and a difference that decides how much amplitude sits on each. What equalisation adds to it is one sentence — the difference that matters is not the free atoms’ difference. It is whatever is left of it after the charge has moved.

An iteration nobody has to run #

One small thing is worth recording because it is the reason the result is a closed form rather than an iteration.

Equalisation is usually described as an iterative process: charge moves, the electronegativities change, more charge moves, and so on until nothing changes. That description is right about the physics and wrong about the arithmetic. Because the chemical potential is linear in the charge — which is what a quadratic energy gives — the fixed point can be written down.

So there is no convergence to worry about, no starting guess, and no possibility of two answers. Every molecule has exactly one equalised state, and the whole of the calculation is a weighted mean and a subtraction.

That is a much stronger position than the self-consistent bond orders one field over, where the feedback is on the off-diagonal elements and the fixed point has to be found by iterating. The difference is entirely that one relation is linear and the other is not, and it is worth noticing which conveniences are worth what. Hybrids that were never orthogonal is another place where a linear relation makes a hard problem easy and the linearity is the whole of the reason.

What the standard repair buys, and the price it charges #

The distance the model has no room for has a standard repair, it is in daily use, and it is worth following through because what it fixes and what it breaks are both sharp.

The repair is to couple every pair of atoms by a distance-dependent term — a Coulomb interaction between their charges, softened at short range — instead of by nothing. The equalisation condition becomes a linear system rather than a single equation, and solving it gives a charge on every atom that depends on where the atoms are.

That buys exactly what the simple model lacks. Two isomers of one formula get different charges, because their atoms are arranged differently. A hydrogen next to a fluorine is polarised differently from one on the far side of the molecule. Charges change as a molecule moves, which is what makes such schemes usable in simulations of reacting systems, where a fixed charge on each atom is precisely what cannot be assumed.

The price is paid at the far end of the same coordinate, and it is not a small one.

Take the coupled model and separate two atoms without limit. The coupling between them goes to zero, so the equalisation condition returns to the uncoupled one — and the uncoupled one has a non-zero answer. For a sodium and a chlorine, with the ionisation energies and affinities that define the Mulliken quantities,

$$q = \frac{\chi_{\mathrm{Cl}} - \chi_{\mathrm{Na}}}{2(\eta_{\mathrm{Na}} + \eta_{\mathrm{Cl}})} = \frac{8.290 - 2.844}{2(2.296 + 4.678)} = 0.391.$$

The model says a sodium atom and a chlorine atom a metre apart exchange four tenths of an electron.

That is not a small error at a distance nobody cares about; it is the wrong answer to the one question about this pair whose answer is certain. Two atoms out of contact are two neutral atoms, and the hardness arithmetic says why for this pair specifically — making the ions costs 1.53 electronvolts more than it returns.

The failure has a single cause and it is the same one that makes the model tractable. The energy of an atom is written as a smooth function of its charge, so a fraction of an electron costs a fraction of the energy, and nothing in the expression knows that electrons come whole. At short range the neighbouring atom’s field supplies a reason to keep the charge small; remove the neighbour and the reason goes with it.

Which is why the schemes in practical use all carry a modification of some kind — charges constrained to fragments, transfers assigned to bonds rather than to atoms, or a penalty that grows once an atom’s charge exceeds what it could actually hold. Every one of those is a device for reintroducing the integer the smooth energy discarded.

So the trade is stated rather than hidden: coupling by distance buys isomers and costs dissociation, and a scheme that gets both needs something the quadratic energy does not contain.

What this cannot say #

The quadratic is a truncation. An atom’s energy is expanded to second order in its electron count, which is exact for a fractional charge in the sense of the derivative and is not a good description of an atom that has lost a whole electron. Charges here are all well below one, which is the regime the expansion is for.

Three of the affinities are negative. Beryllium, nitrogen and magnesium do not bind an extra electron, so their “affinities” are not measurements. None of the three is used above, and a comparison of the scales records which they are.

The molecules are formulas. No geometry, no connectivity, no distance dependence — see above.

No polarisability. An atom in a field has its own charge redistributed within it, and zero dipole is not no interaction is where that matters most; nothing here has an atom with an inside.

And nothing here is compared with a measurement. There is no experimental partial charge to check against, because a partial charge is not an observable: a weight that depends on how it is weighed applies to every population analysis ever published, and equalisation charges are one more convention among them.

What the model requires #

The charges sum to the molecule’s, exactly, in all twelve cases — the constraint the closed form was solved under, checked rather than trusted.

Every atom ends at the molecule’s chemical potential, to twelve decimal places, which is what equalisation means and is the check that the closed form is the solution.

A molecule of one element takes no charge at all, which is the tripwire: four carbons must move nothing between them.

Carbon’s charge is not one number across the fluoromethanes — a spread of 0.238 electrons — and neither is the electronegativity it shares, at 2.38 eV.

The two rules disagree by more than 0.2 eV somewhere, and the disagreement is largest where the hardnesses are furthest apart, at a rank correlation above 0.4.

And an atom with no hardness is refused rather than answered, because it would take an unbounded amount of charge.

Still open: an equalisation that knows about distance #

The obvious open question is the distance the model has no room for. An equalisation that fell off with separation — every pair coupled by something like a Coulomb term rather than by nothing — is the standard repair, and it turns the closed form into a matrix inversion. What it would buy is the thing this model most conspicuously lacks: an answer that distinguishes two isomers, and a hydrogen whose sign flips when a fluorine is put next to it rather than anywhere in the formula.

The nearer question is about the geometric mean. Sanderson’s rule agrees with the weighted mean to within a fiftieth of an electronvolt for the molecules where the hardnesses are alike, and the pattern of its failures is now measured. Whether it is the geometric mean specifically, or whether any unweighted mean would do as well, is a question one more column would answer — and if the geometric mean is no better than the arithmetic one, then the rule’s long survival is a fact about how rarely the hardnesses differ rather than about the rule.