Electronegativity — the series · Molecular Geometry & Bonding

Four scales, four orderings. Each column ranks the elements by one electronegativity scale, most electronegative at the top, with a line joining each element across the columns. Every crossing is a pair of elements that two scales order differently.

Electronegativity is not one quantity

Four scales, four definitions, four sets of units, and a defence — "they correlate well" — that answers a question nobody asked. Two scales can correlate at 0.99 and still put hydrogen on the wrong side of carbon.

part 1 · wrong

Bonds the scales disagree about. For each bond, which atom each scale calls the more electronegative. Every row is a bond whose polarity would be drawn in opposite directions depending on which of these four tables in common use was consulted: Pauling, Mulliken, Allred–Rochow, Allen.

A ranking is not a difference

Four electronegativity scales agree about order to a Spearman coefficient of 0.99 and disagree about size by a factor of sixty-two. Put each on its own range and the O–H bond is 39 per cent of the way across on three scales and 4.5 on the fourth.

part 2 · wrong

The same difference, at five repulsions. The charge transferred to the more electronegative of two atoms against the difference in their orbital energies, at five strengths of the repulsion between the two electrons. Only the topmost curve is the two-level result; every other one moves far less charge at the same difference.

A difference does not make a transfer

Every electronegativity scale reports one thing: how far apart two atoms are in their appetite for electrons. Put that difference into a model with repulsion in it and the charge it actually moves is not determined at all — the same difference of one moves 0.447 of an electron with no repulsion and 0.0019 with a strong one, a factor of two hundred and forty-two, and nothing on any scale distinguishes the two cases.

part 3 · bonding

Two derivatives of the same energy, and only one is tabulated. The 18 elements of the electronegativity tables, placed by their chemical potential — half the sum of the ionisation energy and the electron affinity, which is the Mulliken electronegativity — against their hardness, half the difference of the same two numbers. Hardness runs from 1.92 to 7.3 electronvolts, a factor of 3.8, and does not follow the horizontal axis. Ringed points are the three elements whose anion is not bound, so whose affinity is not a measurement.

The quantity no scale prints

Every electronegativity table is half of a calculation. The other half is the hardness — half the difference of the same two measurements the Mulliken scale is half the sum of — and it varies by a factor of 3.8 across eighteen elements. Put both halves in and B–F, with an electronegativity difference of 6.12 electronvolts, moves less charge than lithium iodide, whose difference is 3.75.

part 4 · bonding

Carbon is not one number. The charge equalisation puts on carbon in the four fluoromethanes, and on hydrogen in the three that have one. Carbon's runs from 0.07 to 0.31, and hydrogen's changes sign between methane and fluoromethane, so the same C–H bond is polarised one way in one molecule and the other way in the next. A table with one number for carbon is printing the value an iteration starts from.

The value that only exists in the bond

A difference in electronegativity moves charge, and moving charge closes the difference — until every atom in the molecule has the same chemical potential. Solving that gives one number per molecule and a charge per atom, and carbon's runs from +0.0692 in methane to +0.3074 in tetrafluoromethane. Hydrogen's changes sign between the first and the second, so the same C–H bond is polarised one way in one compound and the other way in the next.

part 5 · bonding

Four unweighted rules against the answer the hardnesses give. Twelve molecules, each drawn with the exact equalised electronegativity as a filled mark and the four unweighted means of the free atoms' values as open ones. The geometric mean — Sanderson's rule — is the closest on average, at 0.0706 eV against the arithmetic mean's 0.1666, and it is the closest on only 4 of the twelve. Every rule below the arithmetic mean is constrained to sit below it, and 2 of the exact answers do not.

A mean that is low rather than right

Sanderson's rule takes the geometric mean of the atoms' electronegativities, and it works better than the plain average. Computed against the exact equalised answer for twelve molecules it is better by erring downwards — and two of the twelve have exact answers above the plain average, where no geometric mean can go at any parameter.

part 6 · wrong

A fourth data point, and 3 negative electronegativities. Each element's electronegativity from three points on its energy curve, the cubic coefficient a fourth point adds, and what the fourth point leaves. The cubic coefficient is one sixth of a second difference of the ionisation series, so it is largest where that series has a kink — and the alkali metals, whose second electron comes out of a closed shell, are pushed to Li -7.91, Na -3.42, K -1.49 eV. The last column is where the two roots of the fixed-point equation collide, beyond which the atom has no solution at all.

Where a closed form stops being one

What happens when the quadratic energy is not enough? A cubic makes the equalisation condition a quadratic with two roots, and something has to choose between them. The choice is easy and the finding is somewhere else — a cubic fitted through the dication gives lithium an electronegativity of −7.907 eV, a capacity of 0.073 of an electron, and a molecule of two alkali metals no solution at all.

part 7 · wrong

The capacity, against whether the anion exists. Each atom at its electron affinity and the capacity the cubic model gives it — the largest amount of electron the model says it can accept. A negative affinity is an anion that is not bound, which is the statement that the true capacity at the integer is zero. The three atoms with negative affinities are beryllium, magnesium and nitrogen, and they have the largest capacities in the set: two of them infinite and the third 31.5. The three smallest capacities all belong to atoms whose anions are bound.

A capacity that is largest where there is none

A cubic through four charge states implies a largest amount of electron an atom can accept, and it is natural to ask whether the idea survives its own model. It does not. The three atoms whose anions are not bound — beryllium, magnesium and nitrogen — have the largest capacities in the set, two of them infinite; the three smallest capacities all belong to atoms whose anions are bound. And among the fourteen where the test cannot bite, the ordering is sensible.

part 8 · wrong

The capacity against the quantity it was supposed to be. The chemical capacity of 14 atoms against their second ionisation energy, with the capacity on a logarithmic axis because it spans nearly three orders of magnitude. The suspected relation is not there: N and K are 2.02 electronvolts apart in the second ionisation energy and differ by a factor of 193 in capacity, which no function of one variable can produce.

A correlation is not an account

A quantity called the chemical capacity is ordered against the thing it was supposed to predict and correlates instead with the second ionisation energy, which invites asking how much of it that accounts for — on the reasoning that a quantity which is ninety per cent of one input has a simpler name than the one it carries. It is three per cent of it. Nitrogen and potassium sit two electronvolts apart in the second ionisation energy and differ two hundredfold in capacity.

part 9 · wrong

The input the question named is the worst of the six. For each candidate input, the largest capacity ratio between two atoms that are neighbours in that input. A quantity the capacity were a function of would have a small bar. The dashed line is the floor — neighbours in the capacity itself still differ by ×5.0, because fourteen atoms spread over a factor of four hundred cannot do better. The second ionisation energy is ×193, which is 39 times that floor.

The worst of the six was the one we asked about

The capacity's correlation with the second ionisation energy explains 2.9 per cent, and the natural next step is the same pair test against every other candidate input. Every input fails it — but the second ionisation energy fails it by a factor of twenty-four more than the best, and the test itself had to be repaired first, because the version the question implied reports the capacity failing to be a function of itself.

part 10 · wrong

Six of fifteen change verdict

The chemical capacity is finite or infinite according to the sign of a coefficient that is one sixth of a second difference of three measurements, and the pair test asked how much of its reported ordering is the quantity and how much is the fit's resolution. Adding a fifth point changes the verdict on six of fifteen atoms — and the five it unbounds are the five it had largest.

part 11 · wrong

One size scale, computed for every atom at once. Each atom's valence shell at the effective charge Slater's rules give it, with the mean radius of a hydrogenic orbital of that shell and that charge, and its root-mean-square radius. Both are closed forms in the principal and angular quantum numbers and the charge, so both are on one scale for every atom by construction — which is the difficulty a tabulated radius has and this does not. Neither quantity is anywhere in the cubic, which knows three energies and no length.

A size the fit was not made from

Every input tested against the chemical capacity so far has been inside the cubic that produced it, so the search was constrained to fail. A size is not: a hydrogenic orbital at the effective charge Slater's rules give an atom's valence shell has an exact radius, on one scale for every atom, and the cubic knows three energies and no length at all. It fails the test by sixteen times the floor.

part 12 · wrong

Straight segments, and the curve fitted through their ends. Cl's energy against the charge it carries. The exact theory says the energy is straight between integers, with a kink at each one: the slope below the neutral atom is the electron affinity and the slope above it is the first ionisation energy, and the two are different numbers. Every quantity in this argument comes from the smooth curve fitted through those points instead — and the curve's second derivative, which is the hardness, is a property the segments do not have at all.

Four quantities go and one question stays

The exact theory says an atom's energy against electron count is straight segments between integers, and that model gets the alkali metals right for free — which is exactly where the fitted curves give lithium a negative electronegativity. What it costs is the hardness, the capacity, the electronegativity and equalisation, and what it leaves is a question whose answer is an integer.

part 13 · wrong