A mean that is low rather than right

Electronegativity equalisation solved in closed form gives a weighted mean of the free atoms’ values, with each atom weighted by the reciprocal of its hardness. Sanderson’s geometric mean is close to that answer and not equal to it, and that the disagreement is largest where the hardnesses are furthest apart. The hardness is the quantity no scale prints and the whole difference between the two answers lives in it.

That leaves an obvious question, and it is the one this essay is about. Sanderson’s rule is a geometric mean specifically. It has survived sixty years of use in that form, and no derivation has ever been offered for the choice. So: is the geometric mean doing something, or would any unweighted average have done as well?

The answer is that it is doing something, that the something is not what it looks like, and that no rule of its kind can be right — which is provable rather than merely measurable, because the family of means it belongs to is bounded and the exact answer is not.

The family the rule belongs to #

Every candidate rule proposed for combining electronegativities is a member of one family. The power mean of order $p$ over a set of positive numbers is the $p$-th root of the mean of their $p$-th powers, and it gives the harmonic mean at $p = -1$, the geometric mean in the limit at $p = 0$, the arithmetic mean at $p = 1$ and the root mean square at $p = 2$.

The family has a property that decides this essay: it is increasing in $p$. For any set of positive numbers that are not all equal, a power mean of lower order is strictly smaller than one of higher order. The geometric mean is below the arithmetic mean, always, for every molecule, with no exceptions and no conditions.

That is not an empirical observation about the twelve molecules here. It is an inequality, and it holds for any set of numbers whatever.

The exact answer is not in this family at all. Setting every atom’s chemical potential equal under a quadratic energy gives

$$\chi_{\text{mol}} = \frac{\sum_i \chi_i / \eta_i}{\sum_i 1 / \eta_i}$$

which is an arithmetic mean carrying weights, and the weights are the reciprocal hardnesses. When the hardnesses are all equal the weights are all equal and this is the plain arithmetic mean exactly — checked here to ten decimal places. When they are not, the answer moves towards the softest atom, and it can move either up or down depending on which atom that is.

What the twelve molecules say #

Measured against the exact answer across the same twelve molecules, the mean absolute errors are:

So the rule’s reputation is earned on the first column: the geometric mean is better than the arithmetic mean by a factor of 2.4 on average, and better than the root mean square by a factor of 3.7.

The second and third columns say something else. The geometric mean is the closest rule on only four of the twelve, and the harmonic mean beats it on five. A rule that is best on a third of the cases and best on average is a rule that is well placed rather than well founded, and the difference matters because it says where it will fail. That is the same complaint a population analysis attracts: a number that agrees with another number is not thereby a measurement of anything.

Why it is well placed #

The exact answer is a mean weighted by reciprocal hardness, so it is pulled towards the softest atom. For most of chemistry the softest atom in a molecule is also the least electronegative one, because hardness and electronegativity are correlated across the periodic table — both rise towards fluorine and both fall towards caesium.

Where that correlation holds, the weighting drags the answer downwards, below the plain average. And the geometric mean is also below the plain average, unconditionally, by the inequality above. Two quantities that both sit below a third will tend to sit near each other, and that is the whole of Sanderson’s accuracy.

The size of the drag is set by how far apart the hardnesses are. Measured as a rank correlation between the spread of a molecule’s hardnesses — the largest divided by the smallest — and the size of Sanderson’s error, the answer is 0.622 across the twelve. The two molecules with the widest spread are silane at 1.90 and boron trifluoride at 1.75, and they carry the two largest errors: 0.212 and 0.184 eV. Substituent series are where a spread like that is built deliberately, and the fluoromethanes are the standing example.

Where it cannot be #

The correlation between hardness and electronegativity is strong and it is not exact, and the exceptions are where the rule breaks in a way no adjustment can repair.

Chlorine is more electronegative than hydrogen and softer than it. Chlorine’s values are 8.290 eV and 4.678; hydrogen’s are 7.176 and 6.422. So in hydrogen chloride the reciprocal-hardness weighting favours the more electronegative atom, and the exact answer is pulled above the plain arithmetic mean rather than below it: 7.8205 against 7.7330 eV. That reversal is the same one a difference does not make a transfer found in the charges: which way the charge goes is decided by two quantities and not by one.

The geometric mean for that molecule is 7.7129 — below the arithmetic mean, as it must be, and therefore on the wrong side of the answer by 0.108 eV. There is no order $p$ below one at which a power mean reaches 7.8205, because every one of them is bounded above by 7.7330. The rule does not merely miss here; it is excluded.

Water is the same case at a tenth of the size. Oxygen at 7.5395 eV and 6.0785 is more electronegative and softer than hydrogen at 7.176 and 6.422, and the exact answer sits 0.0045 eV above the arithmetic mean. Two of twelve is not a rare pathology, and both are molecules a chemist meets before anything else.

This is the same shape of result as a stabilisation measured from somewhere one field over: a quantity that looks like a property of the object turns out to be a property of a construction, and the construction has a range it cannot leave.

Two constants and a curve #

The clearest demonstration that no unweighted rule can be the right one takes the electronegativities out of the argument entirely.

Fix two atoms at carbon’s 6.2615 eV and chlorine’s 8.290 eV, and then sweep the ratio of their hardnesses over a factor of sixty-four, from one-eighth to eight. Every unweighted rule takes the two electronegativities as its only input, so all four answers are constants — 7.134, 7.205, 7.276 and 7.346, spanning 0.212 eV between the extremes of the family.

The exact answer runs from 8.065 to 6.487, a range of 1.578 eV. It leaves the family at the top end and again at the bottom, and it crosses every member of it on the way. A constant cannot track a curve, and the curve here is seven and a half times as long as the interval the constants occupy.

That last observation is the answer to the question. The four rules agree with each other, and with the truth, precisely when the hardnesses are equal. The reason no one has ever had to choose between them is that in most molecules the hardnesses are within a factor of two, and over that range the whole family spans a fifth of an electronvolt — which is smaller than the disagreement between two published electronegativity scales for the same element.

So the rule’s long survival is a fact about how rarely the hardnesses differ. That was a guess; it is now a measurement with a coefficient in it. The coefficient is what makes it usable: given the hardnesses of two atoms, it says in advance how far any unweighted mean will sit from the equalised value, without solving for the molecule at all.

What is quoted, and what is computed #

The ionisation energies and electron affinities are quoted, and they are measurements. Everything built from them is computed: the Mulliken electronegativity as their mean, the hardness as half their difference, the equalised molecular value from the closed form, all four power means, every error, the rank correlation, and the sweep.

No electronegativity table is used anywhere in this essay. The Pauling and Allen numbers appear where four scales are compared and they cannot appear here, because the equalisation argument needs a hardness and only the Mulliken construction supplies one.

The quadratic energy is a model and is the one assumption the whole thing rests on. An atom’s energy expanded to second order in its electron count is what makes the chemical potential linear in the charge, which is what makes the fixed point a closed form rather than an iteration — and the frame that was allowed to relax is the case where a similar feedback is not linear and has to be iterated instead.

What this cannot say #

There is no geometry. A molecule here is a formula, so two isomers get the same answer and a charge that depends on which atom is next to which is beyond it. Nor is there an atom with an inside, which is where zero dipole is not no interaction starts.

Three affinities are negative — beryllium, nitrogen and magnesium do not bind an extra electron — so their Mulliken values are extrapolations rather than measurements. None of the three appears in the twelve molecules, and the catalogue of them says which they are.

The power-mean family is not every rule. A rule that used the hardnesses would not be in it, and would not be subject to the inequality that excludes the family here. The claim is about unweighted rules, which is what every published one is.

And twelve molecules is twelve molecules. The rank correlation of 0.622 is computed over that set and is quoted as a description of it rather than as a coefficient to carry elsewhere. The inequality is not: that one holds for any molecule at all.

What a chemist should take from it #

Sanderson’s rule is a good approximation with a stated range. It is accurate to about 0.07 eV where the hardnesses are within a factor of two, which is most molecules, and its error grows with the hardness spread at a rank correlation of 0.62.

Its errors have a sign, and the sign is predictable. It comes out below the exact answer whenever the most electronegative atom is also the hardest one, which is the usual case, and above it otherwise. Hydrogen chloride and water are the two exceptions here, and both are exceptions because a halogen or an oxygen is softer than a hydrogen.

And the exact calculation is not harder than the rule. It is one weighted mean and one subtraction — no iteration, no convergence, no starting guess — so there is no computational reason to use an approximation to it. A linear relation makes a hard problem easy here in the same way it does for a set of non-orthogonal hybrids, and it is worth noticing which conveniences are worth what. The rule is a convenience from an era before the hardnesses were tabulated, and they have been tabulated since 1983.

Which quantity predicts an atom’s charge #

The weights have been established for the molecule’s summary value. What they do to the individual charges follows in one line from the same solution, and the answer is that neither tabulated quantity predicts a charge on its own.

At the fixed point every atom sits at the molecule’s chemical potential $\mu$, so its charge is whatever it took to get there:

$$q_A = \frac{\mu - \chi_A}{2\eta_A}.$$

The numerator is how far the atom’s own electronegativity is from the molecule’s; the denominator is how strongly it resists moving. The electronegativity decides the sign and the hardness decides the size.

That is a sharper statement than either half of the received account. An atom well above the molecular mean does not necessarily carry a large negative charge — if it is hard, the same displacement in chemical potential buys it very little charge. An atom barely displaced from the mean can carry a large one if it is soft.

So the two quantities are not competing predictors to be ranked against each other. They are numerator and denominator of one ratio, and predicting a charge from either alone fails in a direction that can be stated in advance: using electronegativity alone over-predicts the charges on hard atoms and under-predicts them on soft ones, by exactly the factor their hardnesses differ from the set’s.

Which is the same finding as the equalisation result, arriving in a different place. There, the difference in electronegativity was found not to determine the transfer because the hardness sum was missing. Here, an atom’s own electronegativity is found not to determine its charge because its own hardness is missing. Both are the same missing quantity, and the fact that it is missing from the tables rather than from the theory is the awkward part: every number needed to form the ratio has been measured for every element in the set.

What was checked #

Every power mean of order below one is at or below the arithmetic mean, for all twelve molecules — the inequality the argument rests on, checked rather than cited.

At least one exact answer is above the arithmetic mean, which is what makes the family insufficient rather than merely inaccurate. Two are, and for each the check additionally requires that the molecule’s most electronegative atom is not its hardest — so the reason is checked and not only the fact.

The geometric mean is the better of the two commonest rules over the twelve, or the essay’s account of its survival would be wrong.

Sanderson’s error tracks the hardness spread at a rank correlation above 0.3; the measured value is 0.622.

The exact answer moves by more than an electronvolt when only the hardnesses are swept, while each unweighted rule returns a number that does not move at all.

At equal hardnesses the exact answer is the plain arithmetic mean, to ten decimal places — which is the check that the weighted formula reduces correctly.

And the refusal is a molecule of one element. With nothing to average, every rule and the exact answer must return that element’s own value; a comparison that showed a difference there would be reporting on its own arithmetic rather than on chemistry.

Still open: testing the charge ratio #

The obvious open question is the ratio derived above, tested. It says an atom’s charge is a displacement in chemical potential divided by that atom’s hardness, which is a prediction with two measurable inputs and no fitted content — and it can be checked against charges obtained another way, from a fitted electrostatic potential or from a population analysis, across a set chosen so that the two inputs disagree. What the closed form cannot supply is the check itself, because both sides of it would come from the same three tabulated numbers.

The nearer question is about what happens when the quadratic is not enough. The expansion is in the electron count, and it is a good description for a fractional charge and a poor one past about half an electron. Every charge computed in these essays is well below that, but a strongly ionic molecule is not — and the fixed point of a cubic energy is a quadratic equation rather than a linear one, so it has two roots and something has to choose between them. Whether the second root is spurious or is the ionic solution is a question with a definite answer, and it would say where the closed form stops being a closed form.