Molecular Geometry & Bonding
what the shape decides
Concept

Ionic radius — where it appears

A length assigned to each ion so that the sum of two of them reproduces a measured separation. Dividing one measured distance between two ions is a convention rather than a measurement, which is why two tabulations can differ by tenths of an ångström and both reproduce the same crystals.

Named by 8 essays across one field — each of them below, with the objects they name alongside it.

A control that ranked better than the mechanism. Rank correlations against the additivity shortfall, over eight ion pairs. The overlap of the two closed shells ranks at 0.8571 — but the cation's formal charge, which cannot be a mechanism, ranks at 0.9524, so the set is confounded: its eight pairs split four and four by charge and everything else rises with it. Held fixed within a charge group the overlap still ranks at 0.80 — and so does the softness, at -1.00. Four pairs cannot separate two candidates.

A control that outranked the mechanism

Ruling polarisation out left one candidate, and the closed-shell overlap ranks at 0.857 against the additivity shortfall — which looked like the answer until the control was read. The cation's formal charge, which cannot be a mechanism, ranks at 0.9524. Eight pairs split four and four by charge cannot separate anything, and within a charge group two candidates both rank perfectly.

orbitals · Contour Eight wells, and where each one puts its pair. The total energy of each pair against separation, with the measured distance marked on every curve. The wells are deep and their minima are in the right region — tenths of an ångström from the measurements — which is what makes the comparison worth making. What they are not is closer to the measurements than the sum of two tabulated radii, and that is the result.

A size a confound cannot supply

A rank correlation of 0.857 was beaten by a control that cannot be a mechanism, so a size is the next thing to ask for: does a closed-shell repulsion of the computed magnitude displace two ions by the tenths of an ångström the additive radii are wrong by. It does not. The balance of a Madelung attraction against six computed repulsions predicts six separations to 0.242 ångström where adding two tabulated radii predicts them to 0.183, and the displacement it produces ranks at 0.14 against the shortfall it was proposed to explain.

orbitals · Contour Four ways of predicting the same six separations. How far each model's predicted separation is from the measured one, pair by pair. The best additive model misses by 0.0098 ångström on average, the tabulated radii by 0.0237, radii read off the ions' own densities by 0.1828, and the balance of a Madelung attraction against a computed repulsion by 0.2421.

The residue that is two numbers

Is a shortfall between a sum of radii and a measured separation a property of the pairs, or of the compromise a universal table makes? The eight separations are two complete two-by-two blocks, so what no assignment of radii can reproduce is not a residual at all — it is an alternating sum, computed by subtraction, and it comes to three hundredths of an ångström and six.

orbitals · Contour The residue is four times below the model's own error. The quantity whose sign is wanted, beside the accuracy of the numbers it is a difference of. The measured residues are 0.031 and 0.056 ångström; the model gets a single separation right to 0.242 on average. A fourfold alternating difference of quantities known that badly cannot resolve something that small, and that arithmetic was available before any of this was computed.

The residue is below its own noise

Two interaction terms, both negative, leave the sign a coin toss — and a larger block would settle it. There is a larger block: a model that needs no measured separations supplies twenty-one. It cannot settle anything, because a fourfold alternating difference of distances known to a quarter of an ångström cannot resolve three hundredths of one.

orbitals · Contour The error runs with the row of the periodic table. The model's relative error on each measured separation, against how many of the two ions have a third-row outermost shell. The three groups do not overlap and they run in order: two second-row ions and the model is about fourteen per cent short, one of each and it is within seven per cent long, two third-row ions and it is eighteen per cent long. The narrowest gap between groups is 11.2 percentage points.

The error was the row, not the charge

An ionic model checked against six measured separations has a wildly uneven error — under one per cent on two pairs, thirteen to eighteen on three others — and the pattern is not obviously size or charge. It is the row of the periodic table. Counting how many of a pair's two ions have a third-row valence shell separates the errors completely, with an eleven-point gap; counting the charge separates nothing.

orbitals · Contour The error falls along the sum of the two exponents. The ionic model's relative error on each of the six checkable separations, against the sum of the two ions' Slater exponents, with the least-squares line through all six. The rank correlation is −0.986 and only 4 of the 720 possible orderings of six points do as well, where the count of third-row ions it replaces is matched by 24. The three pairs with one third-row ion, which the count could not tell apart, fall in the order the line runs.

The sum of the exponents, not the softer ion

The row of the periodic table sorted an ionic model's errors into three groups, and a count that takes three values can say nothing inside a group. Made continuous, the variable the proposed mechanism names — how diffuse the softer ion is — carries no information: an oxide's 2p and a chloride's 3p have the same exponent to a hundredth. The sum of the two exponents carries nearly all of it, orders the middle group, and, asked about that group without having seen it, predicts its spread at twice the size.

orbitals · Contour Two conditions, and one factor that nearly meets both. The mean error of each group of pairs as the third-row p exponents alone are contracted, with the second-row pairs untouched by construction. One factor has to bring both other groups onto them. The pairs with one third-row ion arrive at 1.376 and potassium chloride at 1.346, 2.2 per cent apart — a test that could have produced two factors nowhere near each other, and did not.

One contraction for two conditions

The ionic model's errors run with the row of the periodic table, and the test proposed for that — stiffen the repulsion and watch for second-row pairs moving out and third-row pairs moving in — produces exactly that pattern from a repulsion that knows nothing about shells. The test that can fail contracts the third-row shells alone and asks one factor to bring two different groups of pairs onto the second-row ones. The two factors needed are 1.35 and 1.38.

orbitals · Contour Three ions with the same shell, and three different contractions. Each pair with one third-row ion, its error plotted against a contraction of that ion's p exponents alone. Sodium chloride's error falls to the second-row pairs' mean when chloride is contracted by 1.302, potassium fluoride's when potassium is contracted by 1.387, and calcium oxide's when calcium is contracted by 1.441. The single factor that contracts every third-row shell at once, 1.376, is drawn faint: it sits between the three, and the three span 10.7 per cent.

Three contractions for one shell

One contraction of the third-row p shells removed the row pattern from an ionic model's errors, with two conditions met by factors 2.2 per cent apart, and left the order of three pairs untouched. Taken ion by ion, chloride needs 1.302, potassium 1.387 and calcium 1.441 — the pairs' own order — and potassium chloride, fitted on nothing, lands among the second-row pairs. The single factor's two conditions agreed because both were averages of these three.

orbitals · Contour

Named alongside it

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

Model limitOverlap integralEffective nuclear chargeApproximationClosed-shell configurationsConventionLeast-squaresProbability densityReference stateScreeningClosed formRepulsion

All concepts