The line was holding the answer up

One basis size where it is worth doing ended by naming its own weakest step, and it named it accurately.

Its finding was a three-regime picture, built on the calculation that separates a basis-set superposition error from a real interaction. At small basis sizes the three-body counterpoise correction is large enough to matter but the pairwise assembly of it is wrong by more than a kilocalorie a mole. At large basis sizes the correction itself has fallen below a kilocalorie and there is nothing worth assembling. In between there is exactly one basis size where the correction matters and the pairwise assembly reproduces it — one cell in a grid of six basis sizes and four separations.

Every one of those verdicts is a comparison against a kilocalorie a mole, and the margin at the one usable cell is under one per cent. So the essay wrote that the whole picture is more sensitive to that convention than to anything computed, and that sweeping the line would say which parts of the finding are about the chemistry and which are about the convention.

The answer is: mostly the convention.

What the line is #

A kilocalorie a mole is chemical accuracy — the conventional figure for how close a computed energy has to be before a chemist treats the difference as unimportant, and the same order as the half of the correction that cannot be computed at all. It is not derived from anything. It is a round number in a unit that is itself a convention, adopted because it is roughly the precision at which computed and measured thermochemistry started to agree.

Nothing about the calculation cares about it. The correction being compared to it is the difference between two ways of counting the same basis functions, which the assembly that counts one share twice established has no natural scale of its own. The correction has a size, the assembly has an error, and the line decides which of those count as large. Moving it does not change a single computed number; it changes only which comparisons come out true.

Four answers and a staircase #

Over two decades of line — from a tenth of a kilocalorie a mole to ten — four different basis sizes are the answer, and they arrive in order.

Basis 4 is the answer from 0.100 to 0.117. Basis 3 from 0.271 to 0.464. Basis 2 from 0.993 to 1.722. Basis 1 from 3.98 to 5.41.

The order is the check that this is structure rather than noise. If the comparison were marginal in a way that made the outcome arbitrary, the answers would scatter — a size would recur, or the sequence would jump. It does neither: the usable size falls monotonically as the line loosens, no size appears twice, and each stretch occupies a comparable width on a logarithmic axis.

That regularity has a reason. Both quantities being compared to the line — the size of the three-body correction, and the error the pairwise assembly makes — fall roughly geometrically with basis size. A threshold sweeping across a set of geometrically-spaced quantities crosses them one at a time and at regular multiplicative intervals, which is exactly a staircase in the logarithm. So the pattern is what the underlying numbers must produce, and finding it is a check on the sweep rather than a discovery about basis sets.

And more of the range has no answer #

Between the four answers are four stretches — 0.117 to 0.271, 0.464 to 0.993, 1.72 to 3.98, and 5.41 upward — where no basis size is usable at every separation.

Those gaps are wider than the answers. For most choices of line, the statement “there is exactly one basis size where this is worth doing” is not merely differently answered; it has no answer, because either the correction fails to clear the line anywhere the assembly can reproduce it, or the assembly error clears the line everywhere the correction still matters.

This is the part that no single choice of line could ever have revealed. Running the calculation once at a kilocalorie a mole produces a clean result and gives no indication that the result is unusual.

Seven tenths of a per cent #

The published answer holds for lines from 0.9926 to 1.7224 times a kilocalorie a mole. The standard line is 1.000.

So the finding is robust upward by 72 per cent and fragile downward by 0.7 per cent. A convention of 0.99 kcal/mol instead of 1.00 would have produced no usable basis size at all and no three-regime picture to report.

Both edges are found by bisecting on the answer rather than read off the sweep’s grid, so the 0.7 per cent is the real distance to the edge and not the spacing between lines tried.

What happens at that edge is a single cell. One basis-and-separation pair whose assembly error was just under the line goes just over it, and because “usable everywhere” requires all four separations, one cell is enough to remove the whole answer.

A finding that rests on a single cell has no redundancy in it, and the right way to report it is with the width of the window rather than with the value that happened to be chosen.

The asymmetry is not an accident #

The window is 0.7 per cent wide below and 72 per cent wide above, and it is worth asking why the standard line landed so near one edge rather than in the middle.

The two edges are made by different things. The lower edge is where the pairwise assembly’s error stops being under the line: tighten the line and the assembly is no longer adequate. The upper edge is where the three-body correction itself stops being over the line: loosen it and there is nothing worth computing. At the published cell those two quantities are not equally far from a kilocalorie a mole — the assembly error is just under it and the correction is comfortably over it — so the window has to be lopsided, and the lopsidedness is a direct readout of where the cell sits between them.

That says something the single-count report could not. That report was that one cell satisfies both conditions; the sweep can say that it satisfies one of them by a factor of nearly two and the other by seven parts in a thousand. Those are different kinds of satisfaction, and averaging them into “usable” hides which is doing the work.

It also says which way an improvement would have to go. Making the assembly better — a better fragmentation, a correction for the term being dropped — moves the lower edge down and widens the window from the side where it is narrow. Computing at a larger basis does not: it moves the correction down towards the line and closes the window from above. So the useful direction of work is the one the assembly’s own double-counting points at, and the sweep says so quantitatively rather than by preference.

What survives #

Not everything here is convention, and it is worth separating what is.

The staircase itself is a property of the calculation. That the usable basis size falls as the line loosens, in order and at regular multiplicative intervals, follows from how both compared quantities scale, and it holds at every line. Somebody using a different convention would find a different answer and the same structure.

The direction of the trade survives too. There is always a tension between the correction being big enough to matter and small enough to assemble, and it always resolves at a particular basis size or at none. The underlying finding — that the correction gets harder to assemble as it gets smaller — is what makes the window narrow at every line, and nothing in this sweep touches it.

What does not survive is the specific claim: one basis size, and that size being two. That is what the convention decided, and the honest form of the result is a range of lines with a range of answers, quoted together.

What was computed, and how #

Nothing was recomputed. The three-body counterpoise corrections and the pairwise assembly errors are the ones already computed for six basis sizes at four separations of three centres on a line, and the entire sweep is 121 comparisons of those stored numbers against 121 different thresholds.

The edges of the published window are found by bisection, and the bisection is on the answer — “does this line give the same usable set as the standard line” — rather than on any continuous quantity. That is the right thing to bisect here because the answer is what the essay is about, and it means the reported edge is where the reported conclusion changes rather than where some proxy for it does. Both directions are bracketed before any midpoint is taken, and a direction with no bracket inside the swept range returns no edge rather than a number pinned to the end of the sweep.

Six checks. One reproduces the single-count answer at the standard line. One says the answer changes with the line, which is the finding. One says the sizes step down in order without repeating, which distinguishes structure from noise. One says there are stretches with no answer, which is the part a single run cannot see. One says both edges of the published window are inside the range swept, so the margin quoted is a measurement rather than a bound. And the last says the standard line is within a sixth of the lower edge — a deliberately loose threshold on a check, because the interesting thing is that it is close at all, and a check tuned to 0.7 per cent would fail on any change that moved it to one per cent while making exactly the same point.

Where the model stops #

Three centres on a line with one electron each is the model used throughout, and the corrections are small absolute energies where a fractional change in a basis produces a large fractional change in a difference. So the numbers being compared to the line are the ones most sensitive to everything, which is part of why the window is narrow.

Six basis sizes is a coarse grid in the quantity the answer is expressed in, and the sizes themselves are an even-tempered construction rather than any published series. The staircase’s steps are one basis size wide because that is the resolution available, and the true boundaries in a continuous measure of basis quality would fall somewhere inside them.

And “usable everywhere” means at all four separations, which is a strict reading and is the one originally chosen. A weaker reading — usable at some separation — gives a broader answer at every line and would narrow the gaps; the strict reading is the one that supports a transferable recommendation, which is why it is kept, but the fragility reported here is partly the strictness.

The generalisation #

The transferable point is about which conventions are worth sweeping, and the answer is not “the ones that look arbitrary”.

A kilocalorie a mole looks arbitrary and is, and that is why it was flagged. But plenty of arbitrary conventions are harmless — the choice of unit, the number of decimal places reported, which of two equivalent definitions of an error is used. What made this one dangerous is a different property: the finding was a count, and the count was one. The same shape appears where one basis size was the whole of an answer and the interesting question turned out to be how wide the conditions for it were.

A result of the form “there is exactly one X” has no margin by construction. Any perturbation that removes the single case takes the whole statement with it, and any that adds a case changes the number. Compare a result of the form “the correction falls by two orders of magnitude across this range”, which would survive a line moving by a factor of two without being restated at all.

So the rule worth carrying is that a convention needs sweeping when the finding is a threshold count rather than a magnitude — and that the sweep should report the window rather than the value, because the window is the part that transfers.

How the result should have been quoted #

It is worth writing out what the single-count result should have reported, because the correction is small and the shape of it is the transferable part.

Not “basis 2 is the one size where the three-body correction is worth computing”. That sentence is true, is what the calculation showed, and carries a hidden conditional that nothing in it announces.

Not “basis 2, at a kilocalorie a mole” either, which is better and still invites the reading that the line is a unit rather than a lever. A reader who prefers half a kilocalorie will substitute it and expect the answer to survive, and it does not.

The form that transfers is the window: “the usable size falls from 4 to 1 as the accuracy line loosens from a tenth of a kilocalorie to five; at a kilocalorie the answer is basis 2, and that answer holds only between 0.99 and 1.72.” It is longer, it says which parts move, and it lets a reader with a different convention read off their own answer instead of inheriting this one’s.

That is more than a style preference. The single-count version is the kind of claim that gets carried into a later argument as an established fact, and there is a specific mechanism for that — one argument quotes another and the conditional does not survive the quotation. Writing the window in makes the conditional part of the sentence rather than part of the method section, which is the only place it reliably travels.

Who found it, and when #

The counterpoise correction is Boys and Bernardi’s, the many-body expansion is standard, and chemical accuracy as a kilocalorie a mole is a piece of professional custom with no single author. The three-regime picture and this sweep are new here.

The single-count result deserves the credit for the finding, and specifically for the sentence that named the line as the thing its picture was most sensitive to. That sentence is what made this a follow-up rather than a correction, and it was written when the result looked clean.

Still open: a fourth centre, and a weaker test #

The obvious open question is the fourth centre, which has been named twice and never run. With four fragments the pairwise assembly leaves out four three-body terms as well as the four-body one, and the sweep says what to measure: not whether there is a usable basis size at a kilocalorie a mole, but over what range of lines there is one, and how wide that window is compared with this one. A four-centre window wider than the three-centre one would say the assembly gets more forgiving as fragments are added, which nobody expects; a narrower one would say the method’s usable region shrinks, and would be worth knowing before anybody assembles a many-fragment correction. The calculation generalises directly and the extra cost is a larger linear solve, so what has kept it unrun is only that it was named as an afterthought — the same reason the two-variable boundary waited.

The nearer question is the strictness. “Usable at every separation” is what makes a single cell load-bearing, and the natural weaker reading — usable at the separations a real calculation would encounter — is a different and more useful question the existing numbers can answer. Whether the window widens under it, and by how much, would say whether the fragility found here is a fact about the method or about the strictness of the test applied to it. That is the same distinction just drawn about the line, applied one level up.