The quantity no scale prints

An electronegativity table is a column of numbers, one per element, and it is used by subtraction. Take the difference, and the larger it is the more polar the bond is supposed to be.

There is no such column — there are four of them, defined from four unrelated measurements, and they disagree about the direction of ordinary bonds. And the numbers in any one of them are ranks wearing the costume of a scale: the intervals between them are not comparable, so the difference of two of them is not a quantity in any unit. Put a difference into a model with repulsion in it and the charge it moves is not determined by the difference at all.

This essay says what the missing quantity is, and it turns out to be sitting in the same two measurements the scale was built from.

Two derivatives, and only one of them is tabulated #

Write the energy of an atom as a function of how many electrons it has, and expand it about the neutral atom. Two derivatives follow, and both are fixed by two numbers that have been measured for every element in this table.

The first derivative — how much the energy falls when an electron arrives — comes out as half the sum of the ionisation energy and the electron affinity. That is Mulliken’s electronegativity, arrived at from a direction Mulliken did not take.

The second derivative comes out as half their difference. It has a name, hardness, and it is not in any electronegativity table anywhere.

The vertical axis is the whole subject of this essay. If hardness were nearly the same for every element it could be absorbed into the units of the scale, and everything a chemist does with an electronegativity difference would be safe. It is not nearly the same: it runs from 1.92 electronvolts for potassium to 7.30 for nitrogen.

The scale, rebuilt from its own inputs #

Before anything is computed from the two derivatives, the table they come from has to be checked against the table already in use, because they are supposed to be the same thing.

Half the sum of the quoted ionisation energy and the quoted electron affinity should reproduce the quoted Mulliken value, element for element. For fifteen of the eighteen it does, to 0.021 electronvolts at worst, which is the precision the values were printed to.

The three failures are the interesting part, and the check behind this figure is written in two halves for that reason: the bound ones must agree, and the unbound ones must not. A check on the agreement alone would pass on a table that had been quietly patched, and the patch is the finding.

A beryllium anion is unstable: an electron approaching a beryllium atom is not bound by it, so there is no detachment energy to measure. The compilers of the scale needed a number anyway, and what went in was a valence-state or fitted value that the table does not mention. So beryllium’s tabulated electronegativity is a different kind of number from fluorine’s, and nothing on the page says so.

That is a small instance of a recurring pattern: a quantity defined by a choice, quoted as though it were measured.

What actually flows #

Bring two atoms together and let charge move until neither of them can lower its energy by giving up or taking more. Each atom’s energy is quadratic in the charge it has gained, to second order, so the condition is linear and the amount transferred falls out:

$$\Delta N = \frac{\chi_B - \chi_A}{2(\eta_A + \eta_B)}$$

The numerator is the electronegativity difference. The denominator is the sum of two hardnesses, and it is not a constant: across the bonds below it runs from 6.14 electronvolts to 13.72, a factor of 2.2.

The shape of that expression is worth a moment. It is a ratio of a first derivative to a second, which is Newton’s step towards a minimum — the transfer is one step of an optimisation, and the hardness is the curvature that decides how far a step goes. The same structure appears wherever a site computes how far a system moves under a driving force: the distortion a filling chooses is a competition between a first-order gain and a second-order cost, and a distortion needs two states for the same reason a transfer needs two derivatives.

This expression is Parr and Pearson’s, from 1983, and it is quoted rather than derived here. It contains no distance, no orbital, no overlap and no bond. It is an answer to which way, and roughly how much; the same question asked with a bond present is what a bond model computes, and the two answers are worth holding side by side.

Thirty-five bonds, ranked twice #

Rank a list of bonds by electronegativity difference, then rank the same list by the charge the equalisation model moves, and join each bond’s two positions.

The individual movements are worth reading rather than the count.

B–F falls from second to sixth. Its electronegativity difference, 6.12 electronvolts, is beaten only by Li–F. Boron and fluorine are both hard — 4.01 and 7.01 — so the denominator is 11.02 and the transfer comes out at 0.278 of an electron.

Lithium iodide climbs from eighth to fifth, on a difference of 3.75 electronvolts, which is sixty-one per cent of B–F’s. Lithium and iodine are both soft, the denominator is 6.08, and the transfer is 0.308 — more than B–F moves on nearly twice the difference.

K–Br goes from fourth to first. Potassium is the softest atom in the table; the pair’s denominator is 6.14, the smallest in the list, and 5.17 electronvolts of difference moves 0.421 of an electron.

And hydrogen fluoride, the textbook polar bond, comes tenth by difference and fifteenth by transfer, at 0.120 — less than the 0.123 of an Si–H bond whose electronegativity difference is a quarter smaller. Hydrogen and fluorine are the two hardest atoms in the table after nitrogen, and their sum, 13.43 electronvolts, is the second largest denominator in the whole list.

Why the tripwire matters here #

An essay whose finding is a reordering has an obvious failure mode: the two rankings might differ because of how ties were broken, or because of an error in one of the two sorts.

So the same computation is run with every element given the same hardness. The denominator is then a constant, the two rankings become the same ranking multiplied by a number, and the discordant count must be exactly zero. It is. Whatever the reordering above is, it is not an artefact of the sorting.

The second half of the tripwire is the spread: the hardnesses must differ by more than a factor of two across the table, or the reordering would be a curiosity at the edges rather than a fact about ordinary chemistry. They differ by 3.8.

The measurement that would settle it #

A transferred charge is not observable. What is observable, for a diatomic, is a dipole moment — and the temptation is to read one off the other by multiplying the charge by the bond length.

The reading does not work, and the dipole is not a sum of bonds explains why: a molecular dipole is a property of the whole charge distribution and is not a sum of bond contributions, the lone pairs contribute as much as the bonds do, and two quite different structures can produce the same moment. Even for a diatomic, where there is only one bond and no lone-pair geometry to argue about, the moment includes the polarisation of every filled shell and not only the transfer between the two valence orbitals.

So the transfer computed above is not a prediction of anything measurable on its own. What it is good for is the comparison between bonds, which is exactly what the ranking uses it for — and the comparison is what an electronegativity difference is used for too.

Where the two halves come apart #

The pattern in the movements is not random, and it has a name in the literature that is older than the arithmetic here.

Soft atoms — large, polarisable, with a small gap between taking an electron and losing one — have small hardness and transfer easily. Hard atoms resist, whatever the difference across the bond. The bonds that move up when ranked by transfer are the soft–soft ones, and the bonds that move down are the hard–hard ones. This is the hard-and-soft classification, and the arithmetic above is one route to why it is a classification at all: it is the denominator of a fraction whose numerator is the only thing anybody tabulates.

Even the numerator is not agreed: a bond can have its polarity direction disputed between two tables that correlate at better than 0.99. So the missing second quantity is not the only difficulty — it is the one that survives after the first is fixed.

What a two-level model says about the same thing #

There is a second way to see why a difference cannot be the whole answer, and it needs no hardness at all — only the algebra of two orbitals at different energies.

Two orbitals separated in energy and mixed by a resonance integral share unequally, with the lower level dominated by the deeper orbital — and how unequally depends on the coupling as well as on the separation. That is the orbital version of the same missing quantity.

The mixing between two orbitals goes as the interaction divided by the energy gap. Increase the gap — which is what increasing the electronegativity difference does — and the mixing falls: the electrons localise on the deeper atom and the bond becomes ionic, but the further transfer produced by a further increase gets smaller and smaller. That saturation is the same physics the hardness expresses, arrived at from the orbital side, and it is why a linear reading of the difference overestimates the polarity of the most polar bonds.

For real orbitals the same statement holds: two 1s functions on identical centres share equally, and a 1s with a 2p on centres of different energy does not. Every quantity in such a picture is computed from an overlap and two site energies, and the second of the two is what no electronegativity scale prints.

Where the model stops #

Three limits are worth stating plainly, because the expression is easy to over-read.

It is second order. The energy of an atom is expanded to two terms in the electron count, and for transfers approaching a whole electron there is no reason for two terms to be enough. Every transfer in the table above is below half an electron, which is where the expansion is defensible and is also where real covalent bonds sit.

It has no geometry. Two atoms at three ångström and two atoms bonded at one and a half get the same answer, which is plainly wrong. What the expression describes is the limit in which the two atoms are in contact and nothing else about the contact matters.

Hardness itself is an atomic property being used in a molecule. The second derivative measured on a free atom is not the second derivative that atom has in a bond, for the same reason that a free atom’s orbitals are not the orbitals it brings to a bond, and for the same reason that an orbital energy is not an ionisation energy once the other electrons are allowed to notice.

None of the three affects the conclusion, because the conclusion is a reordering rather than a set of values. Any correction that scales all the transfers by a similar factor leaves the crossings in the figure where they are.

Inside a molecule the same arithmetic gives the charge each site ends with, and it depends on the parameter’s size rather than only on its sign. A scale that prints one number per element cannot supply that dependence, and a table of hardnesses is exactly the second column it is missing.

What is quoted, and what is computed #

The ionisation energies and electron affinities are measurements and are quoted; a site that pretended to compute them would be worse rather than better. The four electronegativity scales are quoted for the same reason.

The chemical potentials and hardnesses are computed from those measurements, and the reconstruction figure is the check that the computation agrees with the tabulation where it should.

The transfer expression is quoted — it is Parr and Pearson’s — and everything downstream of it, the thirty-five transfers, the two rankings, the twenty-nine discordant pairs and the flat-hardness control, is computed here.

What was checked #

The reconstruction holds where the anion is bound and fails where it is not. Both halves; a table that agreed everywhere would have been patched.

Ranking by transfer is not ranking by difference — twenty-nine of five hundred and ninety-five pairs come out reversed, and at least one reversed pair is separated by more than an electronvolt of electronegativity difference, so it cannot be dismissed as two bonds that were tied.

With one hardness for every element the two rankings coincide exactly. Zero discordant pairs, which is what makes the reordering a fact about the elements.

The hardnesses are not nearly constant: a factor of 3.8 from potassium to nitrogen.

The missing half costs nothing to supply #

There is one feature of this omission that makes it harder to excuse than most, and it is arithmetic rather than physics.

The Mulliken electronegativity is half the sum of an atom’s ionisation energy and its electron affinity. The hardness is half their difference. So the two quantities are built from the same pair of measurements, and any laboratory that determined the first has, by that act, determined the second.

A table that prints the sum and not the difference is therefore not omitting something expensive or hard to obtain. It is discarding half of what it already has — one subtraction where it performed an addition — and the discarded half is the one that decides how much charge actually moves.

That is a different situation from most incomplete descriptions in this collection. A delocalisation energy needs a reference somebody has to construct; a partial charge needs a partition somebody has to choose; a bond dipole needs a quantity nobody can measure. Here the missing quantity exists, is measured, and is thrown away by the format of the table.

What would fix it is one more column, and the column that would actually be used is neither of the two. What governs the transfer is the difference divided by the sum of the two hardnesses — a dimensionless number that says how far apart the atoms are relative to how strongly each resists giving or taking. That number can be computed for any pair from the same two measurements per atom, it needs no fit and no convention, and printing it would replace a subtraction that is misleading with a ratio that is not.

What a careful statement looks like #

The practice that follows is small and is already standard in the parts of the literature that think about this.

Quote the pair, not the difference. “A hard–hard bond of difference 3.2” and “a soft–soft bond of difference 3.2” are different claims about polarity, and only one number is usually given.

Do not convert a difference into a partial charge. A partial charge is in any case a quantity with a convention inside it, so the conversion has an arbitrary quantity on both sides of it. The conversions in circulation — ionic character as a function of Δχ — were fitted to a set of bonds whose hardness sum happened to lie in a narrow range, and they inherit that range silently.

Say which scale. The numerator is not agreed either, which is the four-scales finding and has not gone away.

And treat the reordering as the result. The transfers themselves inherit every limitation of a second-order expansion with no geometry in it; what survives those limitations is the order, in the same way that a tolerance sweep’s ordering survives a choice of tolerance that its individual answers do not.

Still open: electronegativity equalised in a molecule #

The obvious open question is what happens when electronegativity stops being treated as a property of the atom at all. 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: the atom that gained charge has become less electronegative and the one that lost it more so, until the two chemical potentials are equal and no further transfer lowers anything.

That is the equalisation principle, and it makes the electronegativity of an atom in a molecule a computed quantity rather than a tabulated one — different for the same element in two compounds, and different again for two positions in one compound. The tables would then be the starting values of an iteration rather than the answer, which is a very different object from a column of numbers used by subtraction.