Four bonds, or one a₁ and three t₂

One of the figures on a basis is not a thing: The same electrons written two ways with the density unchanged, and four electronegativity scales that disagree.

Six 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.

Hybrids are a basis

The two coefficient matrices side by side. The left-hand orbitals each sit on one hydrogen. The right-hand ones have the totally symmetric combination in the first column, with equal coefficients on all four hydrogens and no carbon 2p at all, and three more that each carry a single 2p — which nobody imposed. The number at the foot is how much the two densities differ by.

The localisation transformation, demonstrated

Methane’s four occupied bonding orbitals, written twice. On the left each sits on one hydrogen; on the right one is shared equally over all four and three follow the Cartesian directions. The transformation between them is a four-by-four orthogonal matrix, and the number at the foot is how much the electron density differs.

The same transformation at a longer bond. Every coefficient moves and the two descriptions still give the same density, because the invariance is a property of the transformation rather than of the geometry it is applied at — which is what makes it a theorem and not a numerical coincidence at one distance.

And at half the hydrogen admixture, which is the free parameter nothing in the construction determines. The coefficient matrices are quite different and the density is the same to the last bit a double holds, so the invariance survives the one quantity the calculation cannot fix.

One scale, from two centres to a cage

Eight systems, with the two counts drawn as the ends of a line. Where the sharing is even they coincide; where one atom holds most of the pair they differ by more than a whole centre. The tetramer’s four localised pairs sit on four atoms apiece and have a participation number of 2.540, and that gap of 1.460 is what electron deficiency looks like from the inside.

Ten answers, and none of them is another one turned round. That is what makes the count a count rather than an artefact of where the search started: the descriptions are compared after being brought to a canonical orientation, so two that differ only by a relabelling are one answer and not two.

Canonical orbitals on one side, localised on the other, and a density that does not move. Everything in this essay is that operation on a series of systems, with the number of centres read out of the right-hand side — so the number reported is a property of a chosen basis, and the choosing is the whole of what a localisation criterion does.

How many descriptions a cage has

What a localisation produces: the same occupied space written in a basis that makes each orbital sit on as few atoms as it can. Nothing observable changes — the density is identical to 10⁻¹⁶ — and everything a chemist would call a bond appears only in this description.

How many distinct answers had been found after each batch of twelve starts. Seven after the first batch, ten after the second, and nothing at all in the three batches after that. Three dry batches is what turns a number of answers into a count of them.

Every answer the search found, placed by the value of the functional it maximises, with the two controls above. The ten span 0.0236 and the two closest differ by 9.89 × 10⁻⁶ — seven orders of magnitude above the 10⁻¹³ the sweep converges to, so they are different maxima rather than one maximum reached to different precision.

The answer a search is most likely to give

Each of the cage’s localised descriptions with the number of starting points that reached it, in order of how often. The basins are uneven and the unevenness is not the kind that would make a search reproducible — and the description marked as the best is not the one at the top.

The descriptions themselves: each a genuine maximum, separated by far more than the sweep’s tolerance, with the controls beside them. Everything about that picture survives; what is added here is how often a search finds each one.

How the count of distinct answers grows with the number of starts. A curve that has not flattened is a search that has not finished, and the extra descriptions found here are the ones the earlier stopping rule was on the wrong side of.

Where the count stops being an effort

The discovery curve out to four thousand starts, with the closed form the measured basin sizes predict. The last new description arrives at 124.

The start at which each description first appeared. Ten within sixteen starts, four more by 124, and then nothing.