The table that could not have mattered

Something that ought not to be true is: an infrared band’s intensity does not rank with how far the atoms move in it. On ammonia the two run backwards, at a rank correlation of −0.657.

It then noticed a mechanism — every molecule whose correlation runs backwards has its charge concentrated on the central atom — and refused to claim it, for a stated reason. The charges come from a table, and four published electronegativity tables disagree with one another. A series would have to be run on all four before anything was claimed.

Run on all four, the result is the same to machine precision on every molecule. Not similar: identical, to fifteen decimal places, and for a reason that means the check could never have come out any other way.

Making the four tables comparable #

The tables are not in the same units. Pauling’s is dimensionless and thermochemical, Mulliken’s is in electronvolts and is half the sum of an ionisation energy and an electron affinity, Allred–Rochow’s is an electrostatic force and Allen’s is a spectroscopic configuration energy. Comparing them needs a common reference, and the reference is stated rather than fitted here: each table’s multiplier is chosen so that hydrogen against fluorine maps to the same charge separation the conventional Pauling factor gives.

What is left after that is exactly what the tables disagree about — where everything between those two ends sits relative to them — and that is the disagreement a comparison of the scales measured and found to be real: twelve discordant pairs out of a hundred and fifty-three between Pauling and Mulliken, and six ordinary bonds whose polarity direction is disputed.

The reference is checked rather than assumed, because a comparison that had accidentally left the units in would be a comparison of unit systems and would look like a comparison of tables.

The theorem, which is one sentence #

Every molecule in that comparison is made of two elements. Water is oxygen and hydrogen; ammonia, methane, boron trifluoride and sulfur dioxide are each a central atom and one kind of ligand.

For such a molecule the charge model has nothing to decide. The charge on each atom is the electronegativity difference from the molecular mean, times a multiplier — and with only two elements every difference is a fixed multiple of the same difference. So the charges are one number times a pattern of rational multiples set by the formula, and a change of table changes only that number.

A rank correlation is invariant under multiplication by a positive constant. So it cannot move.

The invariance is worth following through the calculation rather than taking on trust, because it has to survive two steps. The dipole derivative of a mode is a sum of charges times displacements, so multiplying every charge by $k$ multiplies every derivative by $k$; the intensity is the square of the derivative, so every intensity is multiplied by $k^2$; and the ranking of a set of numbers is unchanged by multiplying them all by a positive number. Nothing in the chain mixes a charge with anything else, which is exactly the property a fixed-charge model has and a real one does not.

That is checked here rather than argued: for each molecule the ratio of one table’s charges to another’s is computed atom by atom and must be the same for every atom, to a part in a billion. It is. And the correlations are then equal to fifteen decimal places, which is what a rescaling gives and a coincidence does not.

What does move #

The same rescaling changes anything that is a magnitude rather than a ranking, and the dipole moment is one.

Water’s is 0.170 on Pauling’s table and 0.027 on Mulliken’s, a factor of 6.3 — the same 6.3 by which every one of its charges is smaller. Sulfur dioxide’s runs from 0.123 to 0.167.

So the tables are genuinely different and the check is genuinely being made; what the check finds is that the reported correlation is insensitive to them by construction.

That distinction — between a quantity a convention reaches and one it does not — is the useful output. A dipole in these units is a number nobody should quote; a ratio of two dipoles of the same binary molecule is safe; and an ordering of intensities across the modes of one molecule is safe for the same reason. What a dipole cannot tell apart is the complementary statement: some pairs of structures give the same dipole whatever the charges are, and some quantities give the same answer whatever the table is.

Methane and boron trifluoride give exactly zero on all four, which is a third kind of answer. Their point groups forbid a dipole, and no assignment of charges to symmetry-equivalent atoms can produce one — so an agreement there is a check on the coordinates rather than on the charges, and it is worth checking separately for that reason.

Where the tables do decide #

The theorem has a boundary and it is sharp: three elements. With three, the charges are two independent numbers rather than one, the tables place the middle element differently, and a rescaling is no longer what a change of table amounts to.

Hydrogen cyanide is the extreme case. Its dipole is 0.331, 0.029, 0.322 and 0.284 on the four tables — a factor of 11.4 — and hydrogen’s charge is +0.139 on Pauling’s, −0.051 on Mulliken’s. The reason is a known and not obscure disagreement: Pauling puts carbon at 2.55 above hydrogen at 2.20, and Mulliken puts hydrogen at 7.18 above carbon at 6.27. On one table the C–H bond is polarised one way and on the other it is polarised the other, and hydrogen cyanide has both a C–H and a C–N bond to add up.

Formaldehyde is the same case at a factor of 6.2, with both hydrogens changing sign.

Carbonyl sulfide and bromochlorofluoromethane are the reassuring cases: factors of 1.37 and 1.13, with one atom’s sign moving in the second. Three elements does not guarantee a disagreement; it makes one possible.

What decides which is whether the molecule contains a pair the tables order differently. Hydrogen and carbon are such a pair and are in two of the four; sulfur and oxygen are not, and carbonyl sulfide is quiet. So the test a modeller can run before doing any arithmetic is to look at the pairs in the formula and ask whether any of them is disputed — which is a list comparing the rankings produces and which has six ordinary bonds on it.

What the caution was actually about #

Put those together and the caution was right in general and wrong about its own case.

It could not have applied to the molecules it was raised about, because all five are binary and the quantity reported is a ranking. That is not a criticism of the caution — it is exactly the kind of check that has to be run rather than reasoned about, and running it turned a worry into a theorem.

It applies with full force to any extension of the series. The stated continuation was a series of molecules with the same shape and different central atoms, which is a series of three-element molecules the moment a hydrogen is involved — and hydrogen against carbon is precisely the pair the tables reverse. A study of that series would have to report which table it used, and would get a different answer with a different one.

That is the same shape of finding as a mean that is low rather than right: a convention that looks like it must matter, checked, and found to matter only in a describable region.

What a modeller should take from it #

Ask what kind of quantity the conclusion is. A ranking, a ratio within one molecule, or a zero forced by symmetry cannot be moved by a rescaling. A magnitude, a comparison between molecules, or a sign can.

Then ask how many elements the molecule has. Two, and every table gives proportional charges, so the first kind of conclusion is table-independent exactly. Three or more, and nothing is guaranteed.

And then look for the disputed pairs. Hydrogen against carbon is the one that reverses between the two most-used tables, and it is in most organic molecules — which means most organic partial-charge models have a table dependence with a sign in it, and almost none of them say which table they used.

Those three questions take a minute and they replace the check that this essay ran, which took a library.

What is quoted, and what is computed #

Four electronegativity tables are quoted, and they are the published ones. Nothing else in this essay is quoted: the geometries are standard, the force fields are fitted to measured frequencies, and every charge, dipole, derivative, intensity and correlation is computed.

The reference — one multiplier per table, fixed by the hydrogen–fluorine separation — is a convention and is stated as one. It is the only free choice in the comparison, and the two conclusions are insensitive to it in opposite ways: the theorem holds for any reference at all, because it is about proportionality, and the three-element factors would change with a different reference while the sign flips would not.

What this cannot say #

Fixed charges. Every dipole derivative here is a sum of fixed charges times displacements, which leaves out the charge flux — how the electron density itself redistributes as a nucleus moves. The dipole is not a sum of bonds is the same omission stated about the static quantity rather than its derivative. It is a standard output of a real calculation, it can be comparable to the fixed-charge term, and it can have the opposite sign. The theorem above survives it only if the flux is also proportional across tables, which there is no reason to expect and no way to test here.

Five molecules and four more. The binary set is the original five and the three-element set is the four with coordinates to hand. Neither is a survey, and the three-element molecules have no force field here, so the correlation — as opposed to the dipole — could not be computed for them. That is the one comparison this essay would most like to have made.

The reference is a choice. A different pair of elements would give different multipliers and different factors between the tables. The sign changes would survive, since a sign is not a scale.

And the three-element cases are static dipoles, not intensities. What is shown to move there is a number a charge model produces, and the connection to a measurement is the one drawn and refused for intensities.

And a partial charge is still not an observable. Every population analysis ever published is a convention, and these charges are a cruder convention than most — an electronegativity difference is not a calculation of anything. What is defended above is a statement about the insensitivity of a ranking, which is a much weaker and more robust thing than a charge.

Exactly what the theorem protects #

A rank correlation does not notice a rescaling is the finding, and it is worth stating precisely which quantities inherit the protection, because the class is both wider and narrower than it first looks.

Every quantity here is built from atomic charges, and for a molecule of two elements every table gives the same pattern of charges times a single positive number $\lambda$. So the question for any quantity is how it behaves when every charge is multiplied by $\lambda$.

Homogeneous of degree one — rescaled. A dipole moment, a dipole derivative, a band intensity’s square root: each is a sum of charges times displacements, so each is multiplied by $\lambda$ and none of them is a number a reader can quote without saying which table it came from.

A ratio of two such — invariant. The ratio of two bands’ intensities, the fraction of the total intensity in one mode, the ratio of one molecule’s moment to another’s made of the same two elements: in each of these $\lambda$ appears above and below and cancels exactly.

An ordering — invariant. Multiplying a list of positive numbers by a positive constant leaves their order alone, which is why the rank correlation came back identical to fifteen decimal places rather than merely similar.

The load-bearing word in the last two is positive. If one table gave a charge of the opposite sign to another’s, $\lambda$ would be negative, every ordering would reverse and every ratio would keep its magnitude and change its sign. Nothing in the algebra forbids that; what forbids it here is that all four tables agree about which of the two elements is the more electronegative, which for a hydrogen halide is not in dispute.

That is the theorem’s real hypothesis, and it is the one that fails first. As soon as a third element is present the charges stop being one pattern times a number — the essay’s own measurement is a factor of eleven in hydrogen cyanide’s dipole, with at least one atom changing sign between tables — and every entry in the list above loses its protection at once. Not the magnitudes only: the ratios and the orderings too.

So the safe statement is narrower than rank correlations are robust. It is that a rank correlation over molecules of two elements each is robust, provided every table agrees about the direction of every bond — and a series that crosses either of those conditions is a series that has to be run on all four.

What the comparison requires #

Every table is anchored to the same hydrogen–fluorine separation, to a part in a billion, or the comparison is between unit systems as well as tables.

Each of the five molecules is made of two elements, which is the hypothesis of the theorem, checked rather than assumed.

Their four tables give charges that are one number times the same pattern — the ratios equal atom by atom to a part in a billion.

So the correlation is the same on every table, to machine precision. Fifteen decimal places, which is a rescaling rather than an agreement.

While the dipole does move, by more than five per cent, for the molecules that have one — the check that the tables really are different.

And methane and boron trifluoride have no dipole on any table, because a point group forbids one and a charge model cannot manufacture what symmetry refuses.

With three elements the dipole is not a rescaling, moving by more than a factor of three on at least one molecule. Measured: 11.4 on hydrogen cyanide.

With at least one atom whose charge changes sign between one table and another — stated separately, because a large factor could come from magnitudes alone and a sign change could not.

And the refusal is a molecule of one element, which has no charges on any table, so nothing to rescale and nothing to compare.

Still open: charge flux, and a series down a group #

The obvious open question is the term this whole model leaves out. The charge flux — the derivative of the electron density with respect to a nuclear displacement — is what would turn the qualitative claim into a quantitative one, and it is also the term that would break the theorem, because there is no reason a flux should scale with a table the way a fixed charge does. Computing it even crudely, on one binary molecule, would say whether the insensitivity established here is a property of the fixed-charge model or of the physics.

The nearer question is a series of related molecules, now that the ground rules are clear. A series of molecules with the same shape and different central atoms — a hydride down a group, say — is a three-element series wherever the central atom is not carbon, so it lands squarely in the region where the tables disagree. Running it on all four is now known to be necessary rather than cautious, and the interesting output is not the mechanism but whether the four tables agree about which molecules are the exceptions. If they do, the mechanism is about the molecules; if they do not, it is about a table.