Molecular Geometry & Bonding
what the shape decides
Concept

Chemical potential — where it appears

The derivative of a system's energy with respect to its electron number, equal to the negative of its electronegativity. A curve fitted smoothly through the integers gives one value at each count; the piecewise-linear curve the exact theory requires gives two, one for each side of every integer.

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

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 Straight segments, and the curve fitted through their ends. Cl's energy against the charge it carries. The exact theory says the energy is straight between integers, with a kink at each one: the slope below the neutral atom is the electron affinity and the slope above it is the first ionisation energy, and the two are different numbers. Every quantity in this argument comes from the smooth curve fitted through those points instead — and the curve's second derivative, which is the hardness, is a property the segments do not have at all.

Four quantities go and one question stays

The exact theory says an atom's energy against electron count is straight segments between integers, and that model gets the alkali metals right for free — which is exactly where the fitted curves give lithium a negative electronegativity. What it costs is the hardness, the capacity, the electronegativity and equalisation, and what it leaves is a question whose answer is an integer.

wrong · Electronegativity

Named alongside it

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

Electron affinityElectronegativityIonisation energyModel limitPartial chargeApproximationCharge transferChemical hardnessClosed formConventionDipole momentHardness

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