Only one force law brings the orbit back to where it started

One of the figures on radial functions: The one-dimensional half of a wavefunction: where the density is, what screening does to it, and how far out an orbital reaches.

Seven 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.

The symmetry that is not a rotation

The classical version. An orbit under an inverse-square force closes on itself; the same particle under any other power precesses, and the long axis creeps round by a measured amount each turn. A closed orbit means the direction of the long axis is conserved, and a conserved direction is a conserved vector.

The residue that is two numbers

How far each model’s predicted separation is from the measured one, pair by pair, for four ways of assigning radii.

The eight separations as a grid of cations against anions. They fall into two complete blocks that share no ion.

The first block’s radii shifted by a stated amount, added to every cation and taken from every anion, with the predicted separations and residuals recomputed each time.

The residue is below its own noise

The residue whose sign is wanted, beside the accuracy of the numbers it is a difference of.

For each ion pair with a measured separation, how far the model’s equilibrium distance is from it.

The model’s residue for each of the twenty-one blocks it can build, sorted, with the two measured residues drawn across.

The error was the row, not the charge

Where the six pairs sit in the grid of charge against shell count. Both charges appear at more than one shell count.

The model’s relative error on each separation, against how many of the pair have a third-row valence shell. Three groups, no overlap.

The same six errors against the charge. The two groups overlap almost completely.

The sum of the exponents, not the softer ion

The six errors against the exponent of the more diffuse ion in each pair.

The six errors against the sum of the two ions’ exponents, with the line through all six.

The count and seven continuous measures: the size of each rank correlation, its exact p-value, and whether it orders the middle three.

One contraction for two conditions

The shift in each pair’s equilibrium when its repulsion is steepened about five bohr, for the model’s repulsion and for one exponential that is the same for every pair.

Each pair’s error as built and with every p exponent contracted by 1.1.

The mean error of each group of pairs as the third-row p exponents alone are contracted.

Three contractions for one shell

Each middle pair’s error as its one third-row ion is contracted, against the band the two second-row pairs span. Three ions with the same shell reach the second-row mean at three different factors.

The two second-row pairs as built, and three pairs at contractions fixed elsewhere: potassium chloride at its own ions’ factors, and each cation given the other cation’s factor.

Each ion’s factor from the pair it is the only third-row ion in, and potassium’s found a second time from potassium chloride with chloride’s factor held.