Molecular Geometry & Bonding
what the shape decides
Concept

Correlation — where it appears

The part of a many-electron problem that no single arrangement of electrons in orbitals describes. It is defined as a subtraction, so its size depends on what is subtracted, and it is not one thing but at least three.

Named by 5 essays across 2 fields — each of them below, with the objects they name alongside it.

Orbitals at the 90 per cent contour. Several orbitals drawn at the same enclosed fraction and, unless stated otherwise, at the same scale — so the sizes on the page are the sizes. Each contour was solved for separately by integrating that orbital's own density. Contours drawn: 1s at 90% of its density, |ψ| = 3.94e-2; 2s at 90% of its density, |ψ| = 7.40e-3; 2pz at 90% of its density, |ψ| = 9.48e-3.

Orbitals are not where the electron is

A many-electron atom has no exact orbitals at all. The orbital picture is a basis for an approximation — an extremely good one — and treating it as a description of reality is the source of most of the confusion in this subject.

wrong · Approximation 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 The capacity against the quantity it was supposed to be. The chemical capacity of 14 atoms against their second ionisation energy, with the capacity on a logarithmic axis because it spans nearly three orders of magnitude. The suspected relation is not there: N and K are 2.02 electronvolts apart in the second ionisation energy and differ by a factor of 193 in capacity, which no function of one variable can produce.

A correlation is not an account

A quantity called the chemical capacity is ordered against the thing it was supposed to predict and correlates instead with the second ionisation energy, which invites asking how much of it that accounts for — on the reasoning that a quantity which is ninety per cent of one input has a simpler name than the one it carries. It is three per cent of it. Nitrogen and potassium sit two electronvolts apart in the second ionisation energy and differ two hundredfold in capacity.

wrong · Electronegativity The ranking is fragile and the headline is not. The correlation required to disturb two different things. Reordering the easiest neighbouring pair needs ρ = 0.154, which is weak enough to expect. Displacing the collection's most fragile claim outright needs ρ = 0.954, which is near-perfect correlation. So a per-predictor structure scrambles the middle of the ordering and leaves the top of it alone.

The ranking moved and the headline did not

One correlation applied across a collection multiplies every price by the same number, so it slides the fragility ordering rigidly and cannot scramble it. Give each predictor its own correlation and the ordering does move — but not at the top. The two most fragile claims come from the same predictor, share a factor, and are locked in order at every correlation whatever; the easiest swap anywhere below them needs only ρ = 0.15.

bonding · Models One ranking, three chains. The collection's priced claims in order of fragility, with each claim placed in its own predictor's column. Two claims sharing a predictor share its error correlation, so their prices carry the same factor and their relative order cannot be changed by any correlation structure whatever — a column is rigid. Claims in different columns can be reordered at a price. So the ranking is not one ordering but three chains interleaved, and only the within-column statements need no assumption about anybody's errors.

Three chains and ninety orderings ruled out

Two claims priced against the same predictor share its error correlation, so no correlation structure can reorder them. That makes the collection's fragility ranking three chains rather than one list — and thirty of the hundred and twenty orderings of five claims are reachable, with the other ninety forbidden before a single measured difference is looked at.

bonding · Models

Named alongside it

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

Closed formElectronegativityError propagationMeasurement uncertaintyPredictorUnderdeterminationApproximationBasisCharge transferChemical hardnessDensity matrixEigenvalue

All concepts