Four scales, four orderings

One of the figures on a basis is not a thing: The same electrons written two ways with the density unchanged, and four electronegativity scales that disagree.

Twelve essays draw this figure, each at the values its own argument needs rather than at the setting shown above. What each one uses it to show is below, in the words of its own caption.

The dipole is not a sum of bonds

Four scales, four orderings. Lines that stay parallel are agreement; every crossing is a pair of elements that two adjacent scales rank the opposite way round. Spearman’s coefficient between neighbouring columns is printed at the foot — high, and not one.

Electronegativity is not one quantity

Four scales, four rank columns, with a line joining each element across them. Lines that stay parallel are agreement; every crossing is a pair of elements two adjacent scales order the opposite way round. Spearman’s coefficient between neighbouring columns is printed at the foot.

The same four scales restricted to a set of elements spread widely apart. Across a range this large the lines are nearly parallel and the orderings agree: fluorine is at the top on every scale, potassium at the bottom on every scale, and nothing in between crosses.

Carbon’s charge across the compounds it appears in, which is the sharper form of the same objection. An element’s electronegativity is a single number and the charge it actually carries is a different number in every molecule — so what a scale reports is a starting point rather than a property, and two scales that disagree about the starting point disagree about nothing measurable at all.

A ranking is not a difference

Twelve elements on four scales, with the six that carry most of the disagreement marked. Hydrogen’s position is the one that moves most between scales, which is why every badly disputed bond in the table above has a hydrogen in it.

The value that only exists in the bond

Carbon’s charge across the four fluoromethanes, with hydrogen’s marked where it appears. Carbon runs from +0.0692 to +0.3074 as fluorines are added — a quarter of an electron — and hydrogen changes sign between the first two.

Fluorine’s charge across the same four molecules, with hydrogen’s overlaid again. It runs from −0.2143 in fluoromethane to −0.1608 in tetrafluoromethane — so each fluorine takes less charge as fluorines are added, because the four of them are competing for it and the carbon they are taking it from has already been drained. The element every account calls the most electronegative there is does not have a charge either; it has a charge per compound, and the spread is a third of what it holds in the first one.

Each molecule’s equalised electronegativity beside the geometric mean of its atoms’ values, with the difference and the spread of hardnesses in the same row. The two agree where the hardnesses do and disagree where they do not — silane by 0.21 eV, boron trifluoride by 0.18.

A mean that is low rather than right

The starting picture: each molecule’s exact equalised value against Sanderson’s geometric mean. The two are close, and the closeness is the thing this essay is asking about rather than the thing it is reporting.

Four tables and one molecule to disagree about

The bond dipole of hydrogen against twelve partners, on each of the four tables after all four are anchored to the same hydrogen–fluorine separation. A bond whose four marks straddle the axis is one the tables disagree about the direction of.

Each table’s multiplier, the charge it then gives hydrogen in hydrogen fluoride, and the charge it gives hydrogen in hydrogen iodide. The first two columns agree by construction. The third does not.

A mark where the table puts hydrogen at the negative end of the bond. Four rows are agreed on by every table; three are Mulliken’s alone.

Where a closed form stops being one

The linear model’s answer, which is what all of this was proposed as a correction to: charges from a hardness-weighted mean, computed for a dozen molecules. The cubic changes water’s sign and refuses two alkali metals; it does not improve any of these.

A dispute is a small difference

Every bond in the collection, ordered by the size of its electronegativity difference as a fraction of each table’s own range. The disputed ones are all at the top.

Each bond’s difference against how far the four tables spread on it. A vertical line separates the two groups; no horizontal line does.

Each interhalogen pair on each of the four tables. Every mark for every pair is on the same side of zero.

A gap that was a choice of bonds

The collection’s thirty-one bonds and all one hundred and fifty-three pairs, sorted by the size of the difference, with the disputed ones marked.

How many pairs a single threshold gets wrong, as the threshold is moved, with the minimum at six rather than zero.

The six pairs the best threshold misclassifies, with the spread across tables that explains each.

Two numbers caught what one could not

Every pair the four tables cover, by difference and by dispute, with the disputed ones filled and the best boundary drawn.

The two rules by how many pairs each gets wrong, and in which direction.

The sixteen disputed pairs against both rules. The one-variable rule misses the ones with large differences and large disputes.

No panel of this kind can find an exception

The correlation between the four tables over the pairs with the smallest differences, from their values and from their signs, as the window widens.

The correlation between each pair of tables over the nineteen pairs with the smallest differences.

How often all four signs agree, observed against the prediction from the pairwise agreement, for each window.

The rule is not the lever

For each number of disputed pairs a boundary lets through, the fewest pairs any boundary flags, and the tables that count needs.

The pairs in the plane of dispute and difference, with the narrowest boundaries that miss none, four and eight disputed pairs.

Tables needed before agreement on one flagged pair is significant, against the correlation between tables.