Molecular Geometry & Bonding
what the shape decides
Concept

Mean-field approximation — where it appears

A description in which each electron moves in the averaged field of all the others, so a many-electron problem becomes a set of one-electron ones. It is exact when the true state is a single determinant, and what it misses is the correlation energy, defined as the difference from the exact answer.

Named by 8 essays across 2 fields — each of them below, with the objects they name alongside it.

The slope goes to a half, and a window fit stops short of it. The local slope of the alternation against the reduced temperature, between each neighbouring pair of points, on a ring of 40 at K = 1.6. It rises monotonically from 0.4115 to 0.5053 as the transition is approached, crossing a half at about a part in a thousand of the reduced temperature. The fitted 0.44 is the average of the left-hand end of this curve; the exponent is one half, which is what a free energy analytic in one order parameter is obliged to give.

The exponent was the window's

A fit over the last decade before a distortion vanishes gives an exponent of 0.44, and running it on larger rings should say whether the number belongs to the transition or to a forty-site ring. It belongs to neither. The local slope runs to 0.5020 as the transition is approached, and 0.44 is what a fit over that particular decade returns — on every ring size and every stiffness, because the whole curve is one curve.

wrong · Metal The second number is the first one, rearranged. The composite's error against the change in the correlation energy, at every point on both axes of the square. They lie on the diagonal because they are the same quantity: the composite is the target's mean field plus the reference's correlation energy, so its error is the reference's correlation energy minus the target's. The largest departure across 20 points is 2.2e-16, which is the arithmetic's own precision and not a measurement.

The second number is the error, rearranged

A cheap diagnostic for a composite method leaves a scatter it cannot explain, and the number that ought to close it is the change in the correlation energy, already computed at every point, so the test is arithmetic rather than a calculation. It is arithmetic, and the arithmetic is the answer. The composite's error is that change with a sign on it.

wrong · Approximation Five candidates, five failures, five different places. Every point of the square, with each cheap diagnostic's failing pair joined by a line. The five tested candidates fail on five different pairs involving 10 different points — no line shares an end with another. Had they all failed on one corner the lines would have converged on it, and the honest conclusion would have been that composites are safe away from that corner.

Five failures in five different places

Seven quantities a mean field produces for nothing have been tried as diagnostics, and none of them is usable. The question left is whether that is one finding or seven — whether the same awkward corner of the square breaks every candidate, or each is broken somewhere else. Each is broken somewhere else. Five candidates, five failing pairs, ten systems, and not one of them appearing twice.

wrong · Approximation 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.

wrong · Approximation 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.

wrong · Approximation 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.

wrong · Approximation At a fixed number of coherence lengths, the curve still depends on the stiffness. The scaled alternation of twenty-five rings at reduced temperature 0.03, against each stiffness's bulk amplitude δ∞: five stiffnesses from 1.2 to 2.0, each at five sizes around seven coherence lengths, rings of 4m + 2 sites filled and rings of 4m sites in the warning colour. Size barely moves a point; stiffness moves it along a straight line, 1.63% from the softest to the stiffest, and rings of 4m sit 0.15% lower at every stiffness. The line runs back to 0.29646 at δ∞ = 0, against BCS's weak-coupling 0.29709 (open ring).

The residual was the stiffness

A matched Peierls collapse leaves 1.6 to 2.1 per cent between five rings that hold the same number of coherence lengths, and the question was how much of it the even-site rounding explains. Twenty-five rings answer it: a ring's curve moves by a few hundredths of a per cent per coherence length, so the rounding explains a hundredth of the residual. The rest is the stiffness itself — a term linear in the bulk amplitude, which no choice of sizes can match away — and its intercept at zero amplitude is BCS's gap curve to a fifth of a per cent.

solids · Metal 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.

wrong · Approximation

Named alongside it

The objects these essays reach for when they reach for this one.

Composite methodCorrelation energyTransferabilityReference stateModel limitSymmetry breakingError cancellationLeast-squaresLocal minimumPeierls distortionBand gapBond alternation

All concepts