The overshoot was one arrangement

The line was holding the answer up ended on a calculation it had named twice and never run. Three fragments on a line had a pairwise counterpoise assembly that was usable at exactly one basis size, inside a window of accuracy lines seven tenths of a per cent from vanishing, and the obvious question was what a fourth fragment does to that window. With four fragments the pairwise assembly leaves out four three-body terms as well as the four-body one. The expectation, stated in advance, was that the window would narrow: that the region in which the method can be trusted shrinks as fragments are added, which would be worth knowing before anybody assembles a correction for a cluster of ten.

It does not narrow as a rule. It widens for two arrangements of four centres and narrows for the third, and the reason it can go either way undoes the premise that every earlier calculation on this question rested on — that a pairwise assembly over-corrects.

One arrangement is not a method #

The three-centre calculation used one arrangement: a centre of nuclear charge two between two of charge one. That was the right choice for the question it was built to answer, which was how a correction divided unequally between centres gets assembled, and that question needs centres that differ. But “what does a fourth fragment do”, asked of that arrangement alone, is really “what does a fourth fragment do to this arrangement”. There is not even a single way to add a centre to it. A second heavy centre can go inside, the light centres can move outside, or the symmetry can be broken.

So the question is asked three times, each time of a pair of arrangements that differ by exactly one added centre and in nothing else:

• every centre alike, 1-1-1 against 1-1-1-1;
• the heavy centres inside, 1-2-1 against 1-2-2-1;
• the heavy centres outside, 2-1-2 against 2-1-1-2.

Everything else stays where the three-centre sweep put it. Each fragment carries one electron and its own even-tempered s Gaussians, optimised on the free atom; there are one to six functions a centre; the centres sit equally spaced at four separations from 1.6 to 3 bohr. A basis size is usable at a given accuracy line when, at every separation, the full counterpoise correction is above the line and the pairwise assembly’s error is below it.

The first thing to establish is that the four-centre calculation is the three-centre one when there are three centres, because otherwise the comparison is between two different questions. It is, cell for cell, to a part in 10¹². Asked for the published window, it returns 0.9926 to 1.7224 kilocalories a mole — the edges the three-centre sweep had to find by bisection, obtained here directly for a reason the next section but one explains.

Two wider and one narrower #

Read along three decades of accuracy line, from three hundredths of a kilocalorie a mole to thirty, the share of the range at which some basis size is usable changes in all three comparisons, and not in one direction.

Every centre alike, the covered share goes from 44 per cent to 58. With the heavy centres inside it goes from 27 to 54, nearly doubling. With the heavy centres outside it falls, from 67 to 60. A fourth fragment is not a penalty the assembly pays, and it is not a gift either; it is a change whose sign depends on where the fourth fragment goes.

The arrangement the question was first asked about is the one most changed, and changed in the direction nobody expected. The published answer for it — two Gaussians a centre at a kilocalorie a mole — sat 0.7 per cent above the lower edge of its window, which was the whole reason that result was called fragile. With a second heavy centre beside the first, the same answer holds from 0.764 to 2.026. The standard line is now 24 per cent clear of the edge rather than seven tenths of one. For this arrangement, a fourth fragment made the pairwise method a good deal less fragile than it was with three.

That is not a reason to think the method gets better as clusters grow. It is a reason to think that the three-centre verdict, fragile or robust, was a verdict about three particular centres, and that a verdict about the method needs more than one arrangement behind it.

A window is two numbers #

Why it can go either way comes out of one observation about what a window is, and the same observation is what lets its edges be read off rather than searched for.

A basis size is usable at a line when every cell’s correction is above the line and every cell’s assembly error is below it. Both conditions are about extremes. The first holds for every line below the smallest correction at any separation; the second holds for every line above the largest assembly error. So each basis size has a window with a floor and a top — the worst error and the smallest correction — and the window exists only when the top is above the floor. On a logarithmic axis its width is the logarithm of their ratio. Nothing else about the twenty-four cells matters to it.

That turns “what does a fourth fragment do to the window” into two separate questions. The top is the smallest correction — the whole of what each fragment borrows, which is not one quantity shared evenly even between two centres — and a fourth centre raises it at every basis size in every arrangement: there are more functions to borrow from, so every fragment’s correction grows. At two Gaussians a centre it goes from 1.722 to 2.026 with the heavy centres inside and from 1.669 to 2.097 with them outside. The top only ever moves the window’s way.

The floor is the worst error, and it can move in either direction. With the heavy centres inside it falls, from 0.993 to 0.764 at two Gaussians. With them outside it rises, from 0.422 to 0.707. So whether a window widens is a race between a top that always rises and a floor that sometimes rises faster.

The race comes out the same way at every basis size for one arrangement and not for the others. The heavy-inside window widens at all six sizes, the uniform chain’s at five of six, and the heavy-outside window at one of six. The covered shares in the previous figure are these ratios added up in the logarithm, which is why they move in the directions they do. And the question of why the floor moves the way it does is a question about the error’s sign, which is where the premise gives way.

The overshoot belonged to the heavy centre inside #

Four earlier calculations on this question reported that the pairwise assembly over-corrects. The assembly that counts one share twice found it overshooting the three-fragment correction by 15.7 per cent at two bohr and gave the mechanism: a fragment already helped by one neighbour’s functions is helped less by a second, because the improvement saturates, so counting the two neighbours separately counts their overlap twice. The correction that gets harder to assemble found the overshoot’s fraction growing with the basis. Both findings were right, and both used the heavy centre inside.

With the heavy centres inside, the assembly over-corrects on all twenty-four cells, with three centres and with four. Nowhere else does it. Every centre alike, the three-centre chain is under-corrected on seventeen cells of twenty-four and the four-centre chain on thirteen. With the heavy centres outside it is sixteen and eighteen.

So the overshoot was one arrangement. The saturation mechanism is real and is still the right account of the cells where it wins, but there is plainly a second contribution of the opposite sign — help a fragment receives only when two sets of ghost functions are present together, which pairs cannot see — and the arrangement decides which of the two dominates. Nothing here identifies that second contribution. What is established is that it exists, that it is not small, and that a statement of the form “pairwise assembly over-corrects” is a statement about where the heavy centres were put.

Opposite signs on one line #

Taking the error apart fragment by fragment shows how a total can be small for two quite different reasons, and why a small total says less than it seems to.

In the three-centre chain with the heavy centre inside, nearly all of the error is on the heavy fragment: it is over-corrected by 0.999 kilocalories a mole, 21 per cent of its own correction, while each light fragment is under-corrected by three thousandths. Add a second heavy centre and each heavy fragment’s over-correction falls to 0.392, or 11 per cent, while the light fragments’ under-corrections grow to nine thousandths. The total falls from 0.993 to 0.764. That is the floor falling, seen from inside.

The uniform chain shows the thing no position-based account survives. Its end fragment is under-corrected by 43 per cent of its own correction with three centres. Add a fourth centre at the other end and the same end fragment — same charge, same neighbour, same place in the chain — is over-corrected by 18 per cent. Its sign changed because of a centre two separations away from it. The sign of a fragment’s error is not a property of the fragment.

And the four-centre uniform chain’s total, −0.094 kilocalories a mole, is eleven per cent of its whole correction, assembled from fragment errors of 18 and 40 per cent in opposite directions. It is the pattern two wrong numbers and a right difference found in a composite energy, where the cancellation improves and the thing being cancelled does not: a pairwise assembly with a small total error has not necessarily corrected any fragment well. For a total energy that is enough. For anything that needs a correction on one fragment — a property localised on one site, a partitioned interaction energy — it is not.

The size of an error, not its sign #

The three-centre test compared the signed error with the line: the assembly was adequate when the difference between the assembled correction and the true one was below a kilocalorie a mole. For the heavy-inside chain that is harmless, because the difference is positive on every cell, and every result reported on that chain stands unchanged.

It is not harmless in general. With the heavy centres outside, one Gaussian a centre under-corrects by 5.17 kilocalories a mole — five times the line — and a test on the signed difference would call that assembly adequate. Every window in this essay is built from the error’s size. The point is worth making as a point rather than as a detail of method, because it is exactly the kind of condition that an extension inherits silently: a test written for a case where one sign never occurs keeps working until the other sign does. It is a cousin of the failure a bisection that found two different events ran into, where a test reported a root and a pole in identical words: in both, the check was sound for the case it was written against and said nothing to announce that the case had changed.

Where the line falls through the uniform chain #

The uniform chain is the arrangement with nothing chosen about it, and at the standard line it is the one that loses its answer.

With three centres, one Gaussian a centre is usable from 0.471 to 2.433 kilocalories a mole, comfortably around the line. With four, that window’s floor — the worst assembly error at one Gaussian — rises to 1.063, six per cent above the line, while two Gaussians’ window tops out at 0.696. The standard line falls in the gap between them. The four-centre uniform chain has no usable basis size at a kilocalorie a mole.

That changes what the single-answer sentence has to carry. The line was holding the answer up concluded that “at a kilocalorie a mole the answer is two Gaussians” needed its line quoted with it. It also needs its arrangement quoted, and the number of fragments, because each of the three can move the answer or remove it.

What was computed, and from what #

Each fragment’s energy was computed alone, and then in the functions of every subset of the other centres present as ghosts — charges removed, functions kept — which for four centres is eight solves a fragment and thirty-two a cell, over six basis sizes and four separations for each of six arrangements. Every solve is the same one-electron generalised eigenvalue problem in s Gaussians on a line that the basis the other atom lent introduced, with integrals in closed form.

Each window is read exactly from its two numbers rather than swept. That is a claim about a rule, so the rule is checked against the definition it replaces: at six hundred and one lines across the three decades, for every arrangement, the basis sizes the floor-and-top rule calls usable are exactly the ones found by testing every cell.

Beyond that, the checks are these. The general calculation reproduces the three-centre one to a part in 10¹² and returns its published window. The heavy-inside chain over-corrects on every cell and the uniform three-centre chain under-corrects on most. Adding a centre widens two covered shares and narrows the third; every window’s top rises with it; the sizes whose windows widen number five, six and one. The uniform four-centre chain has no answer at the standard line. One four-centre cell has fragments of both signs. Some arrangement under-corrects by more than the line, which is what requires the error’s size rather than its sign. And at twelve bohr the four-centre correction is numerically nothing, which says the effects above belong to overlapping functions and not to the arithmetic.

What a line of s functions cannot settle #

One electron a fragment removes every electron–electron term, so this measures a basis effect in its purest form and says nothing about how it combines with correlation — which has a many-body structure of its own, and one that a truncated method fails to make additive for reasons unrelated to any basis. The functions are s Gaussians on a line, which have no angular freedom and no geometry beyond a spacing; a real cluster’s ghosts sit in three dimensions and carry p and d functions whose reach is directional.

Six arrangements are three comparisons, not a survey. They are enough to show that the answer depends on arrangement, which is the finding, and not enough to say which arrangements widen and which narrow in general. The charges are one and two only.

The covered shares depend on the three decades chosen to read them over; the per-basis ratios do not, and the directions of change at each basis size are the more robust statement. “Usable at every separation” is kept from the three-centre work, deliberately, so that the comparison is like for like — and it is still a strict reading whose effect on all of this has not been measured.

And the second contribution to the error, the one of opposite sign to saturation, is inferred from the signs and not isolated. The uniform chain’s end fragment, whose sign flips when a far centre is added, is where isolating it would start.

A property of the method, or of the example #

The general point is about how a method gets characterised, and it has nothing to do with basis sets.

Every earlier result on this assembly was obtained on one arrangement, and every one was correct. The overshoot was reproducible, it had a mechanism that made sense, and it grew with the basis in a way that invited explanation — all of which made it read as a property of pairwise counterpoise assembly. It was a property of a heavy centre in the middle. Nothing in any of those calculations could have shown that, because a property of the example and a property of the method look identical from inside one example.

The design that separates them is cheap, and it is the one used here: ask the question of pairs of cases that differ in exactly the thing being asked about, and ask it more than once with everything else varied. One comparison would have returned “wider” or “narrower”, and either would have been quoted. Three returned both, which is the answer.

Where the pieces come from #

The counterpoise correction is Boys and Bernardi’s. Extending it to clusters of many fragments, with a hierarchy of corrections at increasing orders, was set out by Valiron and Mayer. The one-electron chains, the arrangement comparison and the windows are new here.

The three-centre calculations deserve the credit for naming the fourth centre twice and for writing down what to measure when it was run: not whether a usable basis size exists, but over what range of lines, compared with the three-centre one. That was the right quantity, and it is the one that exposed the arrangement.

Still open: the three-body order, and what sets a fragment’s sign #

The obvious open question is the order of the expansion. Four fragments are the first system in which a correction can be assembled from something between pairs and the whole: each fragment has three three-fragment ghost sets between its pairs and its full basis, and restoring the three-body increments is a real option rather than a relabelling of the full calculation. A window’s top is the correction itself and cannot move under any assembly, so the three-body order can only lower floors — and whether it lowers them enough to make the arrangement stop mattering, and at what cost against the calculation it approximates, is the next thing to measure.

The nearer question is the sign. Saturation explains over-correction and something else explains under-correction, and the cleanest place to separate them is the uniform chain’s end fragment, which changes sign when a centre two separations away is added. Its three-body increment splits into a part from its near neighbour and its far one taken together, and the sign of that increment against the separation of the far centre would say whether the second contribution is help relayed through the functions in between — which is the obvious candidate and has not been tested.