Molecular Geometry & Bonding
what the shape decides
The collection

Every essay — page 39

Page 39 of 39, continuing through the fields in the same order.

Orbitals Where the atoms go Bonding models What symmetry decides Beyond the octet What a spectrum settles When the molecule does not stop What the shape is for What is taught wrongly Series Named objects Orbitals Refutations Search

What is taught wrongly

The explanations that are confident, memorable and false — stated fairly and then tested against a calculation rather than an opinion.

Letting disputed pairs through narrows the rule and barely moves the panel. Over 61,353 straight boundaries in the plane of difference and dispute, the fewest pairs any boundary flags for each number of disputed pairs it lets through (bars), and the number of tables that flagged count needs at the measured correlation of 0.325 (dots). Missing none, nineteen is the least — the published rule. Each miss saves one or two flags and two to five tables; at eight misses, half the disputed pairs, the rule flags 8 and still needs 46 tables.

The rule is not the lever

One exception among nineteen flagged bond pairs would need seventy-three electronegativity tables to mean anything, and a rule that flagged fewer pairs looked like the way to need fewer. Searched over sixty-one thousand boundaries, nineteen is already the fewest that misses no disputed pair; each missed pair buys a table or five; and a rule flagging a single pair would still need fourteen. The requirement lives in the correlation between the tables, which moves it nine times as far as any rule can.

6 figures Six removal lines, each smooth, and the contrast is whichever two sit at the cut. The weight of each of the six strongest removal lines of the half-filled chain of six, followed from U = 16 upward by continuity in energy and labelled by the energy it tends to. Every line is monotone from U = 19. The contrast is the third strongest over the fourth, so it changes whenever two lines exchange those ranks: at U = 28.92 the line tending to +0.45 overtakes the one tending to −1.80 and the two weights are equal, and at U = 157 the fourth and fifth exchange. Lines at ±E, drawn in one colour, converge to one weight.

The limit of one is a parity

A half-filled chain of six has no degenerate removal lines, and its intensity contrast still goes to one — after dipping near U = 32 and rising again to U = 128. Followed line by line, every removal line is smooth and monotone; the dip is an exact tie between two lines at U = 28.916, and the rise ends where two satellites change places. At large repulsion the lines pair up at ±E with equal weights, so the contrast goes to one exactly when the cut falls inside a pair. On a chain of four it does not, and the limit is 1.3125.

6 figures Eighty systems, and only sixteen of them on an arm. The composite's error at every combination of repulsion and staggered site energy for a four-site ring, with the correction transferred from one reference — the point at the top left, where it is exact by construction. The area of each mark is the error. Every system the diagnostics have previously been tested on lies along the top edge or the middle column; everything else is new, and it is most of the square.

The arms were the kindest part of the square

Seven cheap quantities have been tried as diagnostics for when a transferred correlation correction will fail, and every one of them was tested on systems lying along one of two arms — a repulsion swept at zero site energy, or a site energy swept at one repulsion. Filling in the square makes every one of them worse, by up to a factor of twelve, so the diagonal is not a region where they might work but the region where they fail hardest. And a grid asks a question a cross cannot: whether a candidate even sorts the systems the way the error does. One of them does not.

6 figures A ratio is not a score until the gap it was taken at is beside it. Each candidate placed by how far its nearest comparison on the other axis is — the median gap in diagnostic value — and by the worst error ratio it produces. The bottom left is where a working diagnostic sits: close comparisons and ratios near one, which is where the control sits. The two candidates the banded rule could not score at all sit far to the right, so they were blank because they separate the two families rather than because they are good.

A blank is not a pass

Two of the seven cheap diagnostics for a transferred correlation correction could not be scored at all, because a scoring rule that only compares systems whose diagnostic values agree to fifteen per cent finds no such pair for them. The census reported them as blanks, which is correct and reads as the best entry in the table. Dropping the band and reporting the gap beside the ratio scores both — and the two turn out to be unscoreable for opposite reasons, one because it separates the two families as a monotone function of something that already failed, and one because it is unrelated to anything.

6 figures The conventional start misses the lowest mean field on one band, and nowhere else. A four-site ring at every combination of repulsion and staggered site energy on a 12 by 12 grid. A small dot is a system where the conventional broken-symmetry start reaches the lowest mean-field solution; a circle is one where it lands above it, with area proportional to how far. There are 9, every one with a site energy one or two below the repulsion, and the largest miss is 1.124 at a repulsion of six and a site energy of four.

The worst system in the square was the solver's

Every score in the transferability square rests on an unrestricted mean field, and every one was computed from a single conventional start. On a band of nine systems, each with a site energy one or two below the repulsion, that start lands above the lowest solution the method has — by 1.12 at a repulsion of six and a site energy of four, which was the largest error in the whole square. Recomputed on the lowest field, the factor of forty-six that first showed the polarisation to be no rule falls to 2.61 and passes; the grid still fails every candidate, by a factor of fifteen rather than forty-eight; and the typical comparison, which no worst case could show, is far better than chance for every candidate but one.

8 figures