Molecular Geometry & Bonding
what the shape decides
Concept

Empirical scale — where it appears

A set of numbers fitted to reproduce observations rather than derived from a theory, useful for ordering and unreliable for arithmetic. Two such scales for the same property can rank things identically and still disagree about any difference computed from them.

Named by 7 essays across 2 fields — each of them below, with the objects they name alongside it.

One number separates the disputed bonds from the agreed ones. Every bond in the collection, ordered by how big its electronegativity difference is as a fraction of each table's range. The six the tables disagree about are all below 0.0507; the 25 they agree about are all above 0.0584. Nothing lies between, and which elements a bond joins does not enter — a dispute is what happens when the difference is small enough that the tables' own disagreement about it is larger.

A dispute is a small difference

Four electronegativity tables disagree about the direction of a quarter of a set of hydrogen bonds, and the interhalogens look like the natural test: five compounds with no hydrogen in them, to say whether the disagreement is about one awkward element or about the whole idea. All four tables agree about every interhalogen. They agree because the differences are large, and that is not a fact about halogens.

shape · Dipole The gap closed when the other hundred and twenty-two pairs arrived. Every pair sorted by the size of its normalised electronegativity difference: the 31 common bonds on the left, all 153 pairs the four tables cover on the right. Filled marks are pairs the four tables put on different sides of zero. On the small set they are the 6 smallest with nothing in between; on the full set 27 uncontested pairs sit below the largest contested one.

A gap that was a choice of bonds

One number separates every disputed bond from every agreed one with nothing in between — on thirty-one bonds, which is enough to see a gap and not enough to know it is real. It is not real. The line is in the same place on all one hundred and fifty-three pairs the four tables cover, and twenty-seven agreed pairs now sit below the largest disputed one.

shape · Dipole 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.

wrong · Electronegativity 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.

wrong · Electronegativity Correlated tables stop cancelling each other's luck. The number of the 19 flagged pairs expected to show all tables agreeing by chance, against the number of tables, for four correlations between them. Independent tables halve it with every table added, and it falls below the 0.0513 at which one agreement would be significant at ten tables. Correlated tables share part of their luck, and the expectation falls only as a power of the panel size: at a correlation of a half it is still 0.095 with four hundred.

No panel of this kind can find an exception

Four electronegativity tables leave three exceptions to a rule for which bond polarities they dispute, and four independent coins would produce two and a half. How many tables before one exception would mean something? Ten, if tables were coins. Near the boundary they are not: they correlate at 0.32, two of the four are nearly one table, and the requirement becomes seventy-three — with a plausible range running past any panel that could exist.

wrong · Dipole 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.

wrong · Electronegativity Letting disputed pairs through narrows the rule and barely moves the panel. Over 61,353 straight boundaries in the plane of difference and dispute, the fewest pairs any boundary flags for each number of disputed pairs it lets through (bars), and the number of tables that flagged count needs at the measured correlation of 0.325 (dots). Missing none, nineteen is the least — the published rule. Each miss saves one or two flags and two to five tables; at eight misses, half the disputed pairs, the rule flags 8 and still needs 46 tables.

The rule is not the lever

One exception among nineteen flagged bond pairs would need seventy-three electronegativity tables to mean anything, and a rule that flagged fewer pairs looked like the way to need fewer. Searched over sixty-one thousand boundaries, nineteen is already the fewest that misses no disputed pair; each missed pair buys a table or five; and a rule flagging a single pair would still need fourteen. The requirement lives in the correlation between the tables, which moves it nine times as far as any rule can.

wrong · Dipole

Named alongside it

The objects these essays reach for when they reach for this one.

ElectronegativityConventionModel limitRank correlationBond dipoleChemical hardnessElectron affinityIonisation energyPeriodic trendPolarityClosed formCorrelation

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