One basis size where it is worth doing

The correction that gets harder to assemble asked whether the pairwise assembly of a counterpoise correction — computing the three-body correction as a sum of two-body ones — gets better as the basis improves. It does not. From one Gaussian a centre to six, at a separation of two bohr, the assembly overshoots the true correction by 5.25, 7.10, 15.69, 23.16, 32.27 and 37.98 per cent, and it grows at all four separations tested.

It also noted, and left, an awkward fact. The absolute difference between the assembled correction and the true one is falling across the same sweep even as the fraction rises. Which of the two a reader should care about depends on whether the correction is being compared against a chemical accuracy target or against itself, and the two answers point in opposite directions.

That is a question with an answer, and the answer needs one number from outside: what counts as an energy small enough to ignore. A kilocalorie a mole is the convention, which is 1.594 × 10⁻³ hartree. Drawing it turns two opposed readings into a table with three regimes in it.

Three regimes #

At one Gaussian a centre, at a separation of 1.6 bohr, the correction is 5.42 × 10⁻² hartree — thirty-four times chemical accuracy, so it is emphatically worth computing. The pairwise assembly of it is wrong by 6.27 × 10⁻³ hartree, which is four times chemical accuracy. So the correction matters and the shortcut is not good enough.

At two, the correction is 7.87 × 10⁻³ hartree, still five times the line. The assembly is wrong by 1.58 × 10⁻³ hartree, just under it. So the correction matters and the shortcut will do.

At three, the correction is 1.39 × 10⁻³ hartree — below the line. There is nothing to assemble. The 30 per cent overshoot a fraction would report is thirty per cent of an energy no calculation cares about.

At four, five and six it is further below, and at six it is 107 times below. That is the whole of the sweep and it is the whole of the answer.

Two of those three regimes are not regimes in any useful sense. The first has one member and the third has four, and the third is the one a modern calculation is actually in: a basis good enough that the other atom’s functions are barely worth borrowing is a basis in which the whole superposition error has dropped below anything a chemist would act on. That is not news — it is the reason basis-set superposition error is discussed less than it was — but it is worth having as a number rather than as a sentiment, and the number here is a factor of a hundred.

Where the window is, and which way it opens #

The three regimes hold at the closest separation. At the others the boundaries move, and the way they move is informative.

Two Gaussians a centre is usable at all four separations and is the only basis that is. At 2.5 and 3 bohr, one Gaussian joins it — because the assembly’s error falls with distance faster than the correction does, so at large separation even the crudest basis assembles well enough.

The window therefore opens downwards with distance rather than upwards. Nothing is gained at the top end by pulling the fragments apart: three functions a centre is below the line at every separation tested, so the correction is never worth computing there whatever the geometry.

The asymmetry has a plain reading. What the pairwise assembly leaves out is the genuinely three-centre part of the borrowing — the amount by which the third fragment’s basis changes how much the first two lend each other — and that requires all three to be close at once. The two-body corrections it does include decay more slowly, because they need only two. So pulling the fragments apart kills the omitted term faster than it kills the terms that are kept, and the fraction improves. The numbers say so directly: at one Gaussian a centre the overshoot runs 11.56, 5.25, 2.28 and 0.74 per cent across the four separations, falling by a factor of sixteen while the correction itself falls only sixfold.

That is the reverse of the basis-size behaviour and it is the reverse for the same reason, read the other way. Improving the basis kills the correction faster than the omitted term; separating the fragments kills the omitted term faster than the correction. The fraction is a race between two decays and it says which is winning, not whether anybody should care.

Which is a more useful statement than either the fraction or the absolute error alone. The pairwise assembly of a counterpoise correction is a technique with a range, and its range is narrow and at the bottom.

The same picture also says which cells are not close calls. At one Gaussian a centre and 1.6 bohr the assembly’s error is four times the line, which no reasonable redrawing of the line rescues; at six functions and 3 bohr the correction is 144 times below it, which no reasonable redrawing brings back. The fragile verdicts are the two nearest cells at two functions, and they are fragile in the same direction.

Two boundaries that happen to coincide #

The window is one basis size wide, and it is worth asking why it is not wider or empty, because nothing arranged it.

There are two boundaries and they are set by different things. The necessity boundary is where the correction itself drops below the line, and at the closest separation it falls between two Gaussians a centre and three. The adequacy boundary is where the assembly’s error drops below the line, and it falls between one and two. They are consecutive, so exactly one basis size sits between them.

Had the adequacy boundary been one place later — had the assembly’s error at two functions been 1.7 × 10⁻³ rather than 1.58 × 10⁻³ — the two boundaries would have coincided and the window would have been empty. There would then be no basis size at which the pairwise assembly of a counterpoise correction was worth doing at 1.6 bohr: below it the shortcut is too crude, above it the correction is negligible, and nothing in between.

That is not a remote possibility. The margin is 0.74 per cent. So the correct statement of the finding is that the window exists and is one size wide and that it very nearly does not exist, and the second half is the part that would be lost by quoting the first alone.

Why both readings are true #

The two opposed readings are not a paradox and they are not a matter of emphasis. They are one piece of arithmetic seen twice.

Across the sweep at 1.6 bohr the correction falls by a factor of 3,631 and the error in assembling it falls by a factor of 793. The fraction is the second divided by the first, so the fraction grows by 3,631/793 = 4.579 — and the measured growth is 11.56 per cent to 52.92 per cent, which is 4.579.

That identity holds to a part in 10⁹ and is checked, because if the three numbers were computed by three routes that did not agree, one of them would be wrong. It is the same discipline a correction computed at one length needed: check that the aggregate and the parts are one calculation before drawing a conclusion from the difference between them. It also settles the question in general: a fraction rises whenever its denominator falls faster than its numerator, and a correction being computed better is exactly a denominator falling fast. The rising percentage was never evidence that anything was getting worse.

The percentage on top of nothing #

The strongest version of that point is at the far end of the sweep.

At six functions a centre the correction is between 107 and 144 times below chemical accuracy at the four separations, and the pairwise assembly overshoots it by between 9.5 and 52.9 per cent. Both of those are correct. Only one of them is a number anybody should act on, and it is the first.

That is the honest reading of the headline number. “The overshoot grows to 38 per cent with a good basis” is true and it is a fact about a quantity that has become negligible. The correction falls by orders of magnitude and the fraction gets reported anyway, because the fraction is what comparisons of assembly schemes conventionally report; this is a correction to that presentation rather than to the arithmetic.

What was computed, and how #

The system is three one-electron centres on a line, each carrying a Gaussian basis fitted to it, at four separations from 1.6 to 3 bohr. The counterpoise correction is the difference between each fragment’s energy in its own basis and in the full basis of the trimer; the true three-body correction is the version with all three fragments, and the pairwise assembly is the sum of the three two-body ones.

Both quantities are in hartree throughout, and the line is one kilocalorie a mole, 4.184/2625.4996 = 1.5936 × 10⁻³ hartree. That is a convention rather than a measurement, and it is the one number here that did not come out of a calculation.

It is also worth saying what “the assembly’s error” is not. It is not the error in the interaction energy; it is the error in the correction to the interaction energy. A calculation that skipped the correction entirely would be wrong by the whole of it, and a calculation that assembled it pairwise is wrong by the difference — so at two Gaussians a centre the shortcut converts an error of 7.87 × 10⁻³ hartree into one of 1.58 × 10⁻³, a factor of five. The comparison being made throughout is between the shortcut and the full correction, not between the shortcut and doing nothing, and the shortcut wins that second comparison everywhere in the sweep.

The check requires seven things: that the fraction’s growth equals the ratio of the two shrinks; that the correction falls by more than three orders; that the error falls too and more slowly; that exactly one basis size is usable at every separation and that it is two; that the two closest separations admit only that one; that the two furthest admit one function a centre as well; and that the one usable cell clears the line by a margin small enough to depend on where the line is drawn.

And the refusal: that at the largest basis the overshoot is above 40 per cent and the correction is more than fifty times below the line, at every separation. A case where a large fraction sits on a negligible energy has to exist in the output, or nobody reading the fractions has been shown what they are fractions of.

Where the model stops #

The margin at the one usable cell is 0.74 per cent. The assembly’s error there is 1.582 × 10⁻³ hartree against a line at 1.594 × 10⁻³, so a convention one per cent tighter — or a system slightly closer together — would close the window at the two nearest separations entirely, leaving pairwise assembly usable only far apart, where it is least needed. That is stated rather than smoothed over because the verdict genuinely turns on where a conventional line is drawn.

The choice of a single line is also a simplification of how anybody works. A practitioner comparing conformers wants relative energies good to a fraction of a kilocalorie and a practitioner computing an atomisation energy will accept several; the window found here is the window for one target and it would be a different window for another. The sweep in the last figure carries the raw energies precisely so a reader with a different target can redraw it.

One electron a centre is also not a molecule, and the fragments here are single Gaussians rather than the contracted sets a real calculation uses. What transfers is the shape of the argument — a correction and its assembly error both falling, at different rates, past a fixed threshold — rather than the basis size at which the window sits. The half that cannot be computed applies here as well: the counterpoise correction is one half of the basis-set superposition problem and is not the whole.

And “three-body” here means three centres, not three atoms. A cluster of ten fragments has 120 three-body terms and a great many four-body ones, and nothing above says how the window moves with the number of fragments. That is the other open question, and the assembly that counts one share twice is where the double-counting the expansion suffers from was first isolated.

The generalisation #

Two things travel, and the first is small and mechanical.

A fraction is not a size, and the same shape appears elsewhere — a property that gets worse as a basis improves is a property whose error is falling more slowly than the quantity it is an error in. When a quantity and its error are both shrinking, the ratio between them says which is shrinking faster and says nothing about whether either matters. The only way to find out whether either matters is to put both on the same axis as something fixed, and the fixed thing has to come from outside the calculation. Here it is a kilocalorie a mole; elsewhere it is an instrumental resolution, or a thermal energy, or a bond length one could measure.

And an approximation has a window, not a direction. The original question was framed as “does it get better or worse”, which presupposes a monotone answer. The answer is that it gets proportionally worse and absolutely better and stops being needed, which is three statements that only assemble into advice once a threshold is named. The useful output of a study like this is a range of conditions rather than a trend.

That reframing is available in most places where an approximation is being assessed by how its error scales. The scaling exponent is a fact about the method; the range is a fact about the method and the problem together, and it is the one somebody deciding whether to use the method actually needs.

Who found it, and when #

Counterpoise correction is Boys and Bernardi, 1970, and the many-body expansion of interaction energies is older still. The chemical-accuracy convention of a kilocalorie a mole is folklore with no single origin. Everything computed above is original arithmetic on a three-centre model, done to settle a question that had been left as a choice between two readings.

The number worth carrying is not the window’s position, which belongs to this model. It is that the two readings differ by exactly the ratio of two shrink factors, which is arithmetic and belongs to everybody.

Still open: a fourth centre, and where the line is drawn #

The obvious open question is the fourth centre, which is not touched here. With four fragments the pairwise assembly leaves out four three-body terms as well as the four-body one, and the question is whether the window found here — one basis size wide, at the bottom of the range — narrows further or moves. The calculation generalises directly and the cost is a larger linear solve.

The nearer question is the line itself. Every verdict above is a comparison against a kilocalorie a mole, and the margin at the one usable cell is under one per cent, so the whole three-regime picture is more sensitive to that convention than to anything computed. Sweeping the line from a tenth of a kilocalorie to ten and reporting how the window moves would say which parts of the finding are about the chemistry and which are about the convention — and the answer is worth having before anyone quotes “two Gaussians a centre” as though it were measured.