Molecular Geometry & Bonding
what the shape decides
Figure

2 pairs the VSEPR angle cannot separate

VSEPR assigns an arrangement from two integers, so two molecules with the same numbers of bonds and lone pairs are given the same answer. Every pair it assigns the same value, with what was actually measured for each. The largest gap is 14.5 °, which is 92.4 per cent of the 15.7 ° the whole set spans — a floor on the error of every model of this form, established with nothing fitted.
2 pairs the VSEPR angle cannot separate. VSEPR assigns an arrangement from two integers, so two molecules with the same numbers of bonds and lone pairs are given the same answer. Every pair it assigns the same value, with what was actually measured for each. The largest gap is 14.5 °, which is 92.4 per cent of the 15.7 ° the whole set spans — a floor on the error of every model of this form, established with nothing fitted.

One of the figures on what a measurement can fix: The other half of every model: which of its parameters a measurement can recover, which combinations no measurement of that kind can see, and how much a second number of a different shape is worth.

Eight essays draw this figure, each at the values its own argument needs rather than at the setting shown above. What each one uses it to show is below, in the words of its own caption.

In the essays

Two systems a model cannot tell apart

Four predictors built elsewhere in these essays, each audited by its own ties. The bar is the share of the variation that no model of that form can account for.

The two pairs VSEPR assigns the same arrangement, with the gap between what was measured for each. Both pairs are a hydride against a heavier hydride of the same group, which is the axis the counts do not carry.

Four ties among nine ions, ordered by how far apart the measurements are. Every pair fails in the same direction, which is a fact about what is missing rather than about noise.

The pair that is not a tie

Every pair of six rings, plotted as the separation the predictor gives against the separation the measurement gives. A model with slope at most L lies below the line of slope L, and the steepest point sets the floor.

The ratio for each numeric predictor. The control is the one that fails hardest, and it is the one the exact instrument had nothing to say about.

The six rings at their predicted and measured values, with a line drawn between every pair the two orderings disagree about.

The error bar that would be needed

Every discordant pair in the four sets, priced by the measurement error that would reverse it, as a fraction of that predictor’s own measured range.

The three discordant pairs, drawn on the set they come from.

The one refutation a small error would undo, with bars of exactly that size drawn on the two points.

The other end of the bracket is not a number

The multiplying factor against the correlation, at five ratios of the two measurements’ precisions.

Each of the five priced claims against the assumed correlation, at equal precision.

How much correlation each claim would need before its price reached a tenth of its predictor’s range.

The ranking moved and the headline did not

The price multiplier against correlation. Applied to everything it changes no ordering; what reorders claims is the ratio of two of these at different values.

The claims grouped by the predictor they depend on. Claims in one column share a factor whatever it is.

The two most fragile claims as their shared predictor’s correlation rises. The gap scales; it never closes.

Three chains and ninety orderings ruled out

The collection’s priced claims in order of fragility, with each placed in its own predictor’s column.

The number of orderings five claims admit, the number the chain structure permits, and the number it forbids.

Every pair of priced claims, with the correlation one predictor’s errors would need before the pair changed order; a dash marks a pair sharing a predictor.

A denominator needs three currencies

For each slope floor, what it costs to halve it by halving its measured rise and by doubling its computed run.

What each predictor’s own difference is made of, and whether anything defensible can move it.

The Hückel run drawn against the predictor’s own range and against the measured range.

A floor on models written in one scale

For each predictor, what survives thirteen strictly increasing reparameterisations of its own scale.

The Hückel eigenvalue predictor’s slope floor as the overlap between bonded atoms is restored, from zero to three tenths.

Each predictor’s slope floor against the exponent of the power reparameterisation, with the identity member marked.

Every figure · Every orbital, by what it encloses · All essays