The collection

Every essay — page 43

Page 43 of 43, 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.

The chain of four's strongest line is a hole's standing wave read against the singlet's ends. For the half-filled chain of four at infinite repulsion, each removal line's weight is a sum over the four sites of two factors. The first is the hole's standing wave squared on each site: the inner levels at ±0.62 put more of the hole on the end sites, the outer levels at ±1.62 more on the middle ones. The second belongs to the spin chain alone: removing an up electron at an end site leaves the stronger spin doublet with weight (2 + √3)/8 = 0.4665, at a middle site 1/4. Their sums are 0.4067 and 0.3098, which are 1/4 + √3/16 ± √15/80, and the limit of the contrast is their ratio, 1.312498.

The limit was two chains at once

At infinite repulsion every removal line of a half-filled chain is a hole's standing wave read against the spin chain the hole moves through, and every hole level carries exactly half an electron. On the chain of four that makes the contrast's limit (20 + 5√3 + √15)/(20 + 5√3 − √15) = 1.3124978 — not 21/16, which it misses in the sixth figure. On chains of eight and twelve, which no exact spectrum had reached, the limit is 1.0458 and 1.0195, and an energy that vanishes in the limit decides it.

6 figures

Where the correction comes from moves the worst case more than the start did. The polarisation difference's worst error ratio on the square's grid, with the correlation correction carried from eight references: the published one, three beside the band with no site energy, and the four systems where the two mean-field starts disagree. On the lowest mean field it runs from 9.7 to 213.2; the start alone moved the published reference's from 48.0 to 14.7. Rings are the conventional start; the dashed line is the census threshold of three.

The worst case measured the reference

Every score on the transferability square carries one correlation correction, fitted at a repulsion of eight and no site energy. Carried instead from seven other systems — three beside the band where the mean field's start misses, four on it — the polarisation's worst error ratio on the lowest mean field runs from 9.7 to 213, where fixing the start had moved it only from 48 to 14.7. Its median stays between 1.21 and 1.55. A worst ratio divides by the smallest comparable error it can find, and the reference decides where that is. And a reference fitted to the wrong solution passes the census at 2.05 while every system's error is above a half.

7 figures

A constant ties the polarisation

Every score on the transferability square moved with where the borrowed correction was fitted, and an absolute score was the natural repair. It repairs nothing, because a change of reference slides every error along one straight line. The only thing a diagnostic can be asked to predict without a reference is the correlation energy itself, and asked that, the polarisation fails more plainly than any ratio showed: twenty-seven systems read exactly zero, including both the system with the least correlation on the grid and the one with the most. No function of it can beat a single constant, and a second reference does not help.

7 figures