Molecular Geometry & Bonding
what the shape decides
Concept

Hardness — where it appears

Half the difference between an atom's ionisation energy and its electron affinity, which is the second derivative of its energy with respect to electron count. It is the resistance to charge flowing, and it appears in no electronegativity table although it varies by a factor of nearly four across the main group.

Named by 6 essays across 2 fields — each of them below, with the objects they name alongside it.

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.

bonding · Electronegativity 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 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.

bonding · Electronegativity 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 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.

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

Named alongside it

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

ElectronegativityClosed formConventionElectron affinityIonisation energyCharge transferModel limitPartial chargeMulliken scalePolarityExpectation valuePauling scale

All concepts