A difference does not make a transfer

Two essays about the scales stop short of asking what the scales are for. Electronegativity is not one quantity shows that four definitions give four different rankings and that two scales can correlate at 0.99 and still put hydrogen on the wrong side of carbon. A ranking is not a difference shows that the same four agree about order and disagree about size by a factor of sixty-two.

Both of those are complaints about the scales. This one is about what a difference between two of them is supposed to predict.

The complaint here is different in kind from those two, and it is worse. Those essays argue that the scales disagree with each other. This one argues that even a scale everyone agreed on would not predict the thing it is used to predict, because the prediction needs a second number that no scale carries.

What a difference is supposed to buy #

The use of an electronegativity difference is to say how polar a bond is: how much charge has moved from the less electronegative atom to the more. That is a real, definable quantity — the difference between an atom’s electron count in the molecule and in isolation — and it is what a dipole moment is mostly made of.

The prediction rests on a two-level argument every course gives. Two orbitals at energies differing by Δε, coupled by t, produce a bonding combination weighted towards the lower one, and the charge it carries there is

$$\Delta n = \frac{\Delta\varepsilon}{\sqrt{\Delta\varepsilon^{2} + 4t^{2}}}$$

which is a monotonic function of the difference, running from nothing at zero to a whole electron when the difference swamps the coupling. Feed in an electronegativity difference and read off a polarity.

Everything in that formula is one-electron. Δε is a difference of orbital energies, t is a coupling between orbitals, and the electrons are placed in the resulting levels two at a time without any of them noticing the others.

The model this is tested in #

The argument above has no repulsion in it. Both electrons of the bond are treated as moving in the same one-electron potential, and nothing costs anything extra when they end up on the same atom.

The smallest model that puts a price on that is the one used wherever repulsion has to be in the picture: two sites, two electrons, hopping t, and an energy U charged whenever both electrons sit on the same site. The smallest many-electron calculation is the essay that introduces it, and its point is that adding a single term takes the problem out of the reach of any one-electron picture and keeps it exactly solvable.

What is new here is that the two sites are given different energies. A difference of Δε between them is an electronegativity difference, expressed in the only currency this model has, and the whole calculation is exact: the four-dimensional matrix is diagonalised, the one-particle density matrix is built from the ground state, and the charge on each site is read off its diagonal.

Every number in this essay comes from the lowest state of a two-site spectrum that is small enough to write down in full, and there is nothing else in the model for a number to come from. That is the point of using it: the quantity being computed is exact, so a disagreement with an electronegativity scale is a fact about the scale rather than about an approximation.

What the repulsion does #

The top curve is the one every course draws. The others are the same curve with a price attached to double occupancy, and the effect is not a shift or a softening — it is a collapse.

A factor of two hundred and forty-two, at a fixed difference, from a quantity that appears on no scale.

The exactness matters here. Every number in this essay comes from diagonalising a four-by-four matrix whole, with no configuration left out and no truncation anywhere — which is what makes a factor of two hundred and forty a measurement rather than an artefact of an approximation that has broken down.

Two limits, and the crossover between them #

The two ends of the range are worth separating because they are two different chemistries and the model contains both.

At small repulsion the transfer follows the two-level formula and the bond is as ionic as the difference says. That is the regime where the one-electron picture is a good description and where a table of electronegativities does what it claims.

At large repulsion the transfer goes to zero and the two electrons sit one on each atom whatever the difference. That is a covalent bond in the strictest sense — one electron from each partner — and it arrives not because the atoms are alike but because moving charge has become too expensive.

The crossover between the two is where Δε and U are comparable, and it is not sharp: the curves bend rather than break. What the figure shows is that ordinary chemistry sits in the crossover rather than at either end, which is the least convenient place for a rule with one input.

Why it happens, in one sentence #

Transferring charge means putting both electrons on one atom some of the time, and that costs U. So the gain from moving charge downhill is Δε and the cost is U, and the transfer is decided by their ratio rather than by Δε alone.

At large U the answer goes to zero however large Δε is. The two electrons stay one to a site, one on each atom, and the bond is covalent no matter how unequal the partners are — because the alternative is to pay a price larger than the whole difference.

That limit has a name in this collection already. It is the same physics as the insulator band theory cannot see, where a half-filled band describes an insulator because the electrons stay apart, and the same as where molecular orbital theory dissociates — see also the hole that is not repulsion for what the electrons do to each other before any repulsion is switched on — where the one-electron description puts both electrons on one atom half the time at every bond length including infinite. All three are the same failure of an independent-electron description, met in three settings.

The quantity that is missing has a name #

What decides the answer alongside the difference is how much an atom resists having its electron count changed, and that quantity has a name and a definition.

An atom’s electronegativity in the Mulliken sense is half the sum of its ionisation energy and its electron affinity — the average of what it costs to take an electron away and what is gained by adding one. The difference between those two, halved, is the other combination, and it measures the curvature rather than the slope: how sharply the energy rises as the electron count moves away from neutrality. It is called the hardness, and it plays exactly the role U plays in the model above.

So the scales report one of a pair of quantities and the other is left out. A pair of atoms with a large difference and large hardnesses behaves like a pair with a small difference and small ones, and no table of electronegativities distinguishes them.

Every one of the four standard scales reports a position on a single axis, which is exactly the quantity the argument above shows to be insufficient. There is no second column in any of them for the thing that decides how much charge a given difference moves, and adding one would make it something other than an electronegativity scale.

What survives, and it is not nothing #

The conclusion is not that electronegativity is useless, and the evidence is in the shapes of the curves rather than in their heights.

Every curve in the figure is monotonic and every one of them passes through zero at zero difference. So the sign of the polarity is right on every one of them: the more electronegative atom does get the charge, and identical atoms transfer none. A difference of zero moves no charge at any repulsion, exactly, and that is checked to 10⁻¹².

Nor is the ordering wrong. At fixed U, a larger difference always moves more charge. So a table of electronegativities ranks the bonds of a series correctly, provided the atoms in the series are comparable in hardness.

What fails is the quantitative reading: taking a difference of 1.4 in some scale to mean a definite fraction of an electron, or comparing bonds between chemically unlike partners on the strength of the difference alone. That is precisely the use a ranking is not a difference was already suspicious of, for a different reason — the scales disagree about size by a factor of sixty-two — and the two objections stack rather than overlap.

The occupations say the same thing from another direction #

The charge on a site is one reading of the ground state. Another is the natural occupations — the eigenvalues of the one-particle density matrix — and they tell the same story in the language of correlation.

With no repulsion the ground state is a single determinant: one orbital holds two electrons and the other holds none, and the bonding orbital is the one weighted towards the more electronegative atom. As the repulsion rises the occupations move towards one and one, which is a state no single determinant has at all.

So the collapse of the charge transfer and the failure of the single-determinant description are the same event seen twice.

Two kinds of correlation, and only one is small is the essay about that second reading, and its subject — the difference between an error in the energy and an error in the description — is exactly the difference between a polarity that is slightly wrong and one that is wrong by a factor of two hundred.

Where the second axis would come from #

If a scale needs two numbers rather than one, the obvious question is whether anybody has built one, and the answer is that the ingredients have been on the table since the definition of the first scale.

The Mulliken definition uses the ionisation energy and the electron affinity, and it uses their mean. Their difference is available from the same two measurements at no extra cost, and it is the quantity the argument above needs. Nothing prevents a table with two columns rather than one; what prevents it is that a single column can be printed along the top of a periodic table and a pair cannot.

The cost of that convenience is the whole of this essay. A one-dimensional scale invites subtraction, subtraction produces a number, and the number looks like a prediction.

The measurement that would separate them #

The natural question is whether the two quantities can be told apart by an experiment, and they can, in a way that says something about which is being measured.

A dipole moment measures the transfer, which is the thing that depends on both. An ionisation energy and an electron affinity measure the two combinations directly — their mean is the electronegativity and half their difference is the hardness — so a pair of atoms whose bond is more covalent than its electronegativity difference suggests should show up as a pair with large affinities and ionisation energies far apart.

That is a real and checkable pattern, and it is why the elements at the top right of the table — small, hard atoms — form bonds less ionic than their differences imply, and the heavy elements below them, which are soft, form bonds more ionic than theirs do. The correction runs the same way down every group.

Why the rule survives being wrong #

A rule that can be out by two orders of magnitude does not usually stay in every textbook, and the reason this one does is worth stating, because it is not that nobody has checked.

Electronegativity differences are used almost entirely for comparison within a series: is C–F more polar than C–Cl, is O–H more polar than S–H. Within such a series the atoms being compared are chemically similar, so their hardnesses are similar too, and the neglected quantity is very nearly a common factor. Dividing it out of both sides leaves the ranking intact.

The rule fails where the series is not homogeneous — comparing a bond between two small hard atoms with one between two large soft ones — and that is exactly the comparison a scale invites, because a scale is a single axis and a single axis invites comparison of anything on it with anything else.

That is the same shape of failure as the one the scales began with. A scale is a device for ordering things, and its numbers acquire, by being numbers, an appearance of being differences that can be subtracted and multiplied. Electronegativity is not one quantity is about the same appearance in a different place.

What a stretched bond does, and the molecule that settles it #

The model contains a prediction about bond length that is worth extracting, because chemistry has a strong intuition about it and the intuition is right only under a stated condition.

Stretching a bond weakens the hopping and leaves the site energies alone, so it raises both ratios at once: the difference against the hopping, which favours charge transfer, and the repulsion against the hopping, which opposes it. Which wins is decided by the two quantities that do not depend on the length — and the condition is simply whether the difference exceeds the repulsion.

If the difference is larger, the ionic arrangement is the lower one and stretching drives the bond towards it, ending in a pair of ions.

If the repulsion is larger, the covalent arrangement is lower and stretching drives the bond away from ionicity, ending in two neutral atoms.

Sodium chloride settles which case a real bond is in, and it does so without any calculation. In the gas phase it dissociates to a sodium atom and a chlorine atom, not to a pair of ions, and the reason is one subtraction: making the ions costs sodium’s ionisation energy of 5.14 electronvolts and returns chlorine’s electron affinity of 3.61, so the ionic products lie 1.53 electronvolts above the neutral ones.

That is the model’s second case, in the most ionic bond anybody would name. The condition for it is met by essentially every heteronuclear bond, because ionisation energies exceed electron affinities by several electronvolts for every element pair, and the difference in electronegativity never comes close.

The distance at which the two arrangements exchange places follows from the same numbers. The ionic pair is stabilised by its own Coulomb attraction, which equals 1.53 electronvolts when the atoms are 14.4/1.53 ≈ 9.4 ångström apart — so a sodium chloride molecule is ionic inside about nine ångström and covalent outside it, and its bond at equilibrium sits deep inside that radius.

So the intuition is exactly inverted, and the inversion has a length attached. A heteronuclear bond is at its most ionic near its equilibrium separation and becomes less so as it is pulled apart, until at nine ångström it stops being ionic at all.

What is left #

The model is two sites and two electrons, which is the smallest system in which the question can be posed at all, and every number above is exact for it. What it is not is a calculation of any molecule: there is one orbital per atom, no core, no geometry, and the two parameters are stated rather than derived.

The parameter that is hardest to map onto a real bond is U itself. In the model it is the cost of putting two electrons on one site, which for a real atom is roughly the difference between its ionisation energy and its electron affinity — ten to fifteen electronvolts for a light main-group element, and a good deal less for a heavy one. Against a hopping of a few electronvolts, that puts real bonds well into the region where the curves have already collapsed, which is the direction the argument points but is not a calculation of any actual polarity.

The geometry is absent too, and it matters: a real bond’s polarity depends on the bond length through the coupling, and lengthening a bond weakens t, which raises U/t and pushes the pair towards the covalent limit — the opposite of the usual intuition that a longer bond between unlike atoms is a more ionic one. The section above takes that as far as the crossing point of two asymptotes, which is a statement about the two ends of the curve rather than about its shape between them; the shape is what a scan over the coupling would give and is not computed here.

And the transfer computed here is the charge on a site, which is not an observable: partitioning a molecule’s electron density between its atoms requires a convention, and what a dipole cannot tell apart is the essay about how many distributions share one moment. The model’s site charges are well defined because the model has sites; a molecule’s atomic charges are not.