Length did not rescue the consecutive triples

A counterpoise correction for a cluster can be assembled from pieces, and assembling it has its own ways of going wrong that the correction itself does not. The pairs give each fragment its correction from every other fragment’s functions one at a time; the three-body increments add what two sets of ghost functions do together that neither does alone. On four fragments adding every triple repaired the pairwise assembly in all three arrangements and cost more than the whole calculation it stood in for. Keeping only the consecutive triples — the ones anybody would actually run, whose number grows with a chain’s length rather than as its cube — worked with the heavy centres inside or outside, and did worse than pairs alone when every centre was alike.

That last result came with an obvious reservation. On four fragments a triple’s gap can be at most one separation wide, and an end fragment belongs to one consecutive triple and two with a gap. A chain that short is mostly ends. Whether the uniform chain’s failure is a property of assembling from consecutive triples or a property of having four fragments is a question with a definite answer, and it costs nothing to ask: the same calculation, on longer chains.

The uniform chain’s failure is general #

The answer for the uniform chain is unambiguous. At every length from four to eight the consecutive assembly covers less of the line than pairs alone, and the gap between them widens. Pairs cover 57.6 per cent at four fragments and 67.9 at eight; the consecutive assembly covers 52.4 at four, falls to 45.6 at five and then settles, at 47.1, 48.8 and 48.1. The shortfall is 5.2 points at four and 19.8 at eight. Every triple together keeps improving throughout, from 66.6 to 81.3 per cent.

So the reservation is answered in the direction nobody wanted. The consecutive assembly does not fail on a uniform chain because the chain is short; it fails in general, and its failure is sharper the longer the chain. That is the difference between a method with a warning attached and a method with an exception, and for a chain of like fragments this is the second.

The covered share is the right first measure because it is the one that decides whether any basis size is worth using at all at a given line, and it is built from the worst cell at each basis size. That makes it deliberately pessimistic, and the question of whether the whole grid moves or only its worst cells is worth checking before trusting it. It is the whole grid. On the uniform chain of eight, at three Gaussians a centre and two bohr, the consecutive assembly over-corrects every one of the eight fragments, and it is the worst of the three assemblies on six of them.

The shape along the chain is informative. Pairs alone err in both directions — under-correcting the third and sixth fragments by 0.022 of a kilocalorie a mole and over-correcting the ends slightly. Adding the consecutive triples pushes every fragment upward by a similar amount, which cures the under-correction in the middle and turns the ends and their neighbours into the worst-corrected fragments, at 0.021 and 0.023. Adding every triple instead brings the errors back to within about the pairwise ones, with both signs. The consecutive triples add a correction in one direction everywhere, and the triples with a gap are what takes it back.

The gap triples do not fall away #

The lead that suggested this calculation offered a specific test: on a longer chain a triple’s gap can be two or three separations wide, and if the share the gap triples carry falls with the gap, the consecutive assembly’s failure should shrink with length. It does not shrink, and the reason is that the share does not fall.

By size, the triples with a gap carry a median of 0.42 of each cell’s three-body increment at four fragments — the 42 per cent the four-fragment calculation reported. At five it is 0.82. From six it is 1.13, 1.14 and 1.14: the triples with a gap carry more than the whole increment, which is only possible because the consecutive triples carry more than the whole increment in the opposite direction and the two nearly cancel. With its sign the median moves from +0.01 to −0.27, so the gap triples are, on balance, taking back.

The share stops growing at six, and that is the point at which it is a property of the chain rather than of its length. A seventh or eighth fragment adds triples whose members are too far apart to matter, and the share stays where it is. If the gap triples were a short-chain effect, this curve would turn down; it flattens instead.

Where the increment actually lives #

A median share over all the triples with a gap hides which ones they are, and the difference between them is the most useful thing in the calculation.

Each triple can be described from the point of view of the fragment whose correction it contributes to: either both ghosts are on the same side of it or one is on each side, and each is some number of separations away. Filed that way, one shape carries 142 per cent of the increment: both ghosts on the same side, one and two separations away — the consecutive triple that runs outward from a fragment. The shape a chemist would think of as the consecutive triple, one neighbour on each side, carries very little, because a fragment with a neighbour on each side is already surrounded by functions in both directions and a second set on the far side of one of them adds almost nothing.

The shapes that take the excess back are also same-side ones. Two ghosts one and three separations away carry +20.8 per cent; two and three away, −11.0; three and four away, −16.5. These are not small, and they do not follow the gap. A two-gap shape carries more than a one-gap shape, and they have opposite signs. What they have in common is that they are all seen from one side, where a fragment’s own functions face an empty half-line and ghost functions there fill it.

What does fall away is distance. Nothing whose nearer ghost is more than three separations from the fragment carries more than about a per cent; the largest such share is 0.9. That refines the explanation offered for the four-fragment result, which was that light centres have diffuse functions whose reach does not respect the chain’s spacing — the same reason most of an overlap can lie outside the surfaces drawn for it. The reach is finite, three or four separations; the failure is that within that reach the corrections from a fragment’s empty side do not add by triples, and truncating them at the consecutive ones keeps the overshoot and drops the correction to it.

A complete order that is worse #

The three arrangements are the ones the four-centre comparison chose so that each question is asked three times rather than once, and they now separate further. With the heavy centres inside, length changes nothing about the comparison: the consecutive triples cover exactly what every triple covers, to the last digit, at every length from three to eight, and both climb slowly from 74.8 to 77.5 per cent. Heavy centres in the interior have tight functions, the chain’s ends are light and short-reaching, and the gap triples carry nothing that matters.

With the heavy centres outside, the result turns over. From five fragments, the consecutive triples alone cover more than every triple together: 77.4 against 70.8 at five, 80.9 against 56.2 at six, and 87.9 against 62.3 at eight. Adding the triples a practical assembly would drop makes it worse by up to twenty-five points of the line.

The reason is what every triple leaves out. At four fragments the four-body remainder was below a kilocalorie a mole on every cell of every arrangement, which is what made the three-body order look adequate. On longer chains it does not stay there. With the heavy centres outside it reaches 1.05 kilocalories a mole at five fragments and 2.70 at eight, on the cell with one Gaussian a centre at 1.6 bohr; the uniform chain crosses the line at eight, at 1.19; with the heavy centres inside it never exceeds 0.32.

So the many-body expansion of this correction is not a series that improves term by term. Adding the complete three-body order moves the assembly towards the full correction by a large step and leaves a remainder that grows with the chain; dropping some of the triples can land closer, by accident of cancellation, than keeping all of them. The same shape appeared at four fragments on individual cells, where the next order made nine cells worse. On a chain of six with the heavy centres outside it is no longer a matter of cells. The whole range is worse.

That result has a familiar relative. A better energy is not a better answer for a property the energy was not built to get right, and the same logic applies one level up: a more complete assembly is closer to the correction in a specific sense — it includes more of the terms — and has no obligation to be closer in the sense that decides usability, which is the size of the worst error.

What the line sees #

The covered share is a summary over three decades of line. At the standard line itself — a kilocalorie a mole — what matters is which basis sizes are usable, and here length changes the answer in a way the covered share does not show.

Pairs alone give no answer at the line for the uniform chain at four or five fragments, and none for the heavy-outside chain from five fragments on. The consecutive assembly gives an answer in every arrangement from six fragments — two Gaussians a centre for the uniform and heavy-outside chains, up to three for the heavy-inside one. And for the heavy-outside chain, the assembly that covers a quarter less of the range gives the same answer at the line: two Gaussians a centre, under both.

So at the one line most calculations are judged against, the consecutive assembly’s failure on the uniform chain is invisible from six fragments on, and the heavy-outside chain’s preference for fewer triples is invisible everywhere. That is the same situation the accuracy line itself was found to be in: a single answer at a single threshold, standing on a structure that changes sharply a short distance away.

What was solved #

Each fragment carries one electron and an optimised set of one to six s Gaussians on a line — six sizes because a pairwise assembly’s error does not simply shrink with a better basis and every conclusion has to be read across them — with charge one for a light centre and two for a heavy one, and the fragments are equally spaced at 1.6, 2, 2.5 and 3 bohr. A fragment’s correction from a set of ghost functions is how far its energy falls when those functions are added with their charges set to zero, and every energy is the lowest root of the one-electron generalised eigenvalue problem, solved by symmetric orthogonalisation — the construction the basis the other atom lent set up.

For a chain of $N$ fragments each fragment needs its correction from all $N-1$ others, from each one alone and from each pair of them, so a chain of eight costs 29 solves a fragment and 232 a cell, across 24 cells and three arrangements. Every three-body increment is the correction from a pair of ghosts less the two single-ghost corrections. A triple is consecutive when its three members span two separations; its shape is read from the fragment being corrected, as the side each ghost is on and the separations to each.

A basis size is usable at a line when every cell’s correction exceeds the line and every cell’s assembly error, by size, is below it; that defines one window per basis size, and the covered share is the fraction of three decades of line, logarithmically, lying inside at least one window.

The checks: at four fragments the general chain reproduces the four-centre calculation’s full correction, consecutive increment and whole three-body increment to a part in 10¹², for all three arrangements at three basis sizes and two separations; at three fragments every assembly with triples is exact; the uniform chain’s consecutive assembly covers less than pairs at every length from four, by more at eight than at four; the heavy-inside chain’s consecutive and complete assemblies cover the same share at every length; the heavy-outside chain’s consecutive assembly covers more than the complete one at six, seven and eight, where the four-body remainder exceeds the line; and on the uniform chain of six the one-gap shapes seen from one side carry large shares of opposite sign. The refusal is the remainder: below the line for every arrangement at four fragments, and above it for at least one at eight, so the three-body order is not uniformly adequate.

The limits of a line of fragments #

One dimension and one electron a fragment. A line is the only geometry in which “consecutive” has a single meaning; in a three-dimensional cluster the obvious truncation is by distance, and a fragment surrounded on every side has no empty half-line for ghost functions to fill. The same-side shapes that carry the increment here are an end effect of a chain, which may be exactly why they would be weaker in a compact cluster — or stronger at its surface. Nothing here decides which.

Even-tempered s Gaussians with charges one and two. Real fragments have angular functions, and polarisation functions are the ones counterpoise corrections are most sensitive to. The arrangement dependence found here is a dependence on how diffuse a centre’s functions are, and a real basis has several degrees of diffuseness on one centre.

And the cost model is a model. It charges a cubic in the number of functions per solve and nothing for integrals. Under it the consecutive assembly costs 0.435 of the full calculation at six fragments and 0.228 at eight, and every triple costs 1.44 and 1.22 times it; the conclusion that the cheap assembly is the better one for the heavy-outside chain does not depend on the model at all.

A truncation is a claim about what it drops #

The general lesson is about what truncating an expansion claims. Keeping the consecutive triples claims that the triples dropped are small. They are not small; they are large and opposite in sign to part of what is kept. In the heavy-inside chain that part happens to be negligible and the truncation is harmless. In the uniform chain the part kept overshoots and the part dropped would have corrected it. In the heavy-outside chain the complete order overshoots in its turn, and the truncation removes an error the complete order introduces.

Whether a truncation is safe is a property of the cancellation between what is kept and what is dropped, and not of the size of either. That is checkable only by computing what is dropped at least once, on the geometry that matters. Four fragments could not show it, because at four the gap shapes and the end effects are the same triples; six can.

Still open: a distance truncation, and the heavy-outside remainder #

The obvious open question is the truncation a real calculation would use. A consecutive rule is a topological one; the practical one is geometric — keep every triple whose members are all within some distance of each other — and on a line that rule keeps the shapes one and three separations away before it keeps shapes two and three away, which is exactly the pair found here to carry opposite signs. A distance cutoff swept from two to four separations would say whether there is a cutoff at which the uniform chain’s covered share passes pairs, and whether that cutoff is the same for all three arrangements.

The nearer question is what makes the heavy-outside chain’s four-body remainder grow. It is largest with one Gaussian a centre at the shortest separation, where the heavy ends’ single function is tight and the light interior’s is diffuse, and it doubles between five fragments and eight. Four-body increments are computable here at the price of every quadruple, and reading them by shape, as the triples were read, would say whether the remainder lives at the ends of the chain, where the heavy centres are, or in its light interior.