Molecular Geometry & Bonding
what the shape decides
Concept

Rank correlation — where it appears

A measure of how far two orderings of the same set agree, computed from ranks rather than values. It is what shows that two electronegativity scales disagree about the ordering itself and not merely about the numbers.

Named by 14 essays across 5 fields — each of them below, with the objects they name alongside it.

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.

wrong · Electronegativity A 6-ring at 111°: the twist-boat. A closed ring of 6 equal bonds meeting at 111°, drawn from the coordinates the closure conditions produce. Its torsions are 16.9°, -63.6°, 44.2° and repeat; its puckering amplitude is 0.508 bond lengths at a phase of 344°, with q₃ exactly zero, so it lies on the equator. Turning the ring changes none of those numbers, which is what makes them a description of the shape rather than of the view.

The ring that cannot hold still

Cyclohexane's chair is rigid and its boat is not, and that is a statement about the rank of a matrix rather than about strain. Hold every bond length and every bond angle fixed and count what is left: the chair has nothing, and the boat sits on a continuous loop of shapes with the same bonds and the same angles.

shape · Strain 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.

bonding · Electronegativity What boron trifluoride's bands are strong in, and what they move. Every infrared-active mode of boron trifluoride, with its band strength and the root-mean-square displacement of its atoms in the zero point, each scaled to its own largest. The two do not order the modes the same way — the rank correlation between them is 0.2 — and the strongest band belongs to the mode at 719 cm⁻¹, in which 89.93 per cent of the motion is the lightest atom's. Every mode moves the same weighted amount of mass, exactly, so that is not what separates them either.

The mode that moves least radiates most

Boron trifluoride's strongest infrared band is the one in which the fluorines barely move: ninety per cent of the motion belongs to the boron, which is a fifth of the molecule's mass. The mode that moves the most mass is nine and a half times weaker. Across five molecules the rank correlation between band strength and how far the atoms actually go runs from +1 to −0.66, and every normal mode carries exactly the same weighted motion by construction.

shape · Dipole The splitting against the square of one computed overlap. five chromium(III) complexes: the measured ligand-field splitting against the square of the metal–ligand σ overlap, computed from Slater-type functions at the measured bond lengths. The angular overlap model says the splitting is proportional to that square and to no other power, and the line drawn through the origin is that proportionality with nothing fitted but its slope. Across a series in which the splitting doubles, the ratio varies by 30.75 per cent.

The splitting against something structural

The angular overlap model says a ligand field splitting is proportional to the square of one overlap integral and to no other power. Computing that integral from Slater functions at the measured bond lengths, for five chromium complexes whose splittings run from 13,600 to 26,700 wavenumbers, the ratio varies by thirty-one per cent with nothing fitted. And a power law on the donor's effective charge, at an exponent nobody predicted, does slightly better.

applied · Ligand field 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.

wrong · Electronegativity A ninety per cent surface with a neighbour beside it. The contour enclosing 90 per cent of a one-electron ion's density at an effective charge of 1.6, drawn with no field as a circle and in a field of 0.05 atomic units as the closed curve. The surface moves out by 47.3 millibohr on the side the field pulls the density towards and in by the same amount on the far side — 2.84 per cent of its own radius. What the sphere encloses does not change to first order; only where the surface is does.

The surface a neighbour moves

An ion in a crystal sits in the field of the ion next to it, and that field moves its contour. The displacement has a closed form, it is checked against the polarisability it implies, and it turns out to be almost perfectly anti-correlated with the discrepancy it was proposed to explain — the pairs that need the most correction get the least.

orbitals · Contour Four tables, one answer, and a reason it could not be otherwise. The intensity–motion correlation — infrared intensity against how far the atoms move — computed with charges from four published electronegativity tables. Every molecule gives the same number on all four, to machine precision, because each is made of two elements: its charges are one number times a fixed pattern, a change of table changes only that number, and a rank correlation does not notice a rescaling. The dipole moments beside them do notice, which is the check that the tables are genuinely different.

The table that could not have mattered

Charges taken from one of four electronegativity tables invite a worry, because the tables disagree with each other. They do disagree — hydrogen cyanide's dipole runs over a factor of eleven between them — and for the molecules in question the worry could not have applied, because a molecule of two elements has charges that are one number times a fixed pattern and a rank correlation does not notice a rescaling.

shape · Dipole A cheap number that predicts an expensive failure. Thirty-two systems. Along the bottom, how far the mean field's own symmetry breaking has moved between the reference and the target — a quantity available before any exact calculation. Up the side, how wrong the transferred correction turns out to be. They rank together at 0.902, and the open marks are the systems whose broken solution has collapsed entirely, which is where the diagnostic stops being a scale and becomes a warning.

The warning a cheap calculation gives

A correlation correction computed on one system and carried to another works until it does not, and nothing in the scheme says in advance which. The mean field's own symmetry breaking says: it collapses at a definite field, and the transfer fails where it goes. Across thirty-two systems the two rank together at 0.90.

wrong · Approximation Twelve hydrogen bonds, four tables. The bond dipole of hydrogen against each partner, in debye, on each of the four tables after all four are anchored to the same hydrogen–fluorine separation. Positive is hydrogen at the positive end. A bond whose marks straddle the axis is one the tables disagree about the direction of, and there are 3 of them.

Four tables and one molecule to disagree about

A molecule of two elements is provably safe from the choice of electronegativity table, and a series down a group should land where the tables do disagree. It does. All four agree that hydrogen iodide is the exception — and they disagree about what its dipole is by 1.245 debye, which is two and a half times the 0.448 that was measured.

shape · Dipole 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 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

Named alongside it

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

Model limitElectronegativityConventionPartial chargeDipole momentMulliken scalePolarityEffective nuclear chargePauling scaleEmpirical scaleBond dipoleBond length

All concepts