The assembly that counts one share twice

The half that cannot be computed established that a counterpoise correction is not one number but two, and that the two are very unequally divided: for an unequal pair the heavier centre’s half is a hundred times the lighter one’s over the range where a bond would be, and which half is the larger changes with the separation.

It then named the first place that has a consequence for a number anybody quotes. An interaction energy between three fragments is assembled from pairwise counterpoise corrections, each of which is now known to be unequally divided — so the assembly double-counts one fragment’s share and undercounts another’s by amounts that do not cancel.

It does, and both halves of the prediction are right. The sizes are 17.0 per cent and −2.5 per cent.

Two quantities, both exact #

Three centres on a line, each with its own Gaussian basis fitted to its own nuclear charge, with the middle one twice the charge of the other two.

The three-body correction is what a counterpoise correction is, done properly for three fragments: each fragment’s energy in its own basis alone, less its energy in the whole trimer’s basis with the other two nuclei switched off. Summed over the three, that is the artefact the trimer’s binding carries.

The assembled correction is what practice does: for each of the three pairs, compute the pairwise counterpoise correction, and add them up.

If a second neighbour’s basis helped a fragment exactly as much as the first neighbour’s had, the two quantities would agree. They do not, because the help saturates: a fragment already improved by one set of extra functions is improved less by a second set that partly spans the same space.

The assembly overshoots #

At 1.6 bohr the assembled correction is 30.4 per cent larger than the trimer’s own; at two bohr, 15.7; at 2.5, 4.3; at three, 0.58. By five the two agree to a part in thirty thousand.

The absolute size at two bohr is $1.89 \times 10^{-4}$ hartree, which is half a kilojoule a mole. That is negligible against a bond and it is not negligible against the interaction energies counterpoise corrections are applied to — which is the whole point of applying them.

And it is wrong in both directions #

The interesting part is not the total.

At two bohr the heavy central fragment’s assembled correction exceeds its true one by 16.98 per cent, and each light outer fragment’s falls short by 2.54 per cent. The signs are opposite and the magnitudes are nothing like each other.

They do not cancel, and the reason they do not is the unequal division itself. The heavy centre’s correction is $1.13 \times 10^{-3}$ hartree and a light end’s is $4.0 \times 10^{-5}$ — a factor of twenty-eight — so the total’s sign and most of its size are the heavy fragment’s, and the light fragments’ under-correction contributes almost nothing to the net.

That is a prediction confirmed with a mechanism. The unequal division predicts a non-cancelling assembly error without computing one; the computation gives the same answer and says which fragment is which.

Why the light ends go the other way #

The signs deserve an account, because “the assembly overshoots” and “one fragment undershoots” are not obviously compatible.

What a fragment gains from a neighbour’s basis is the improvement in its own energy. For the heavy centre, both neighbours are the same kind of thing — a light atom’s diffuse functions — and they span much the same space, so the second one adds little and the pairwise sum counts the same improvement twice. That is the over-correction, and it is large because the heavy centre’s own basis is the tightest and has the most room to be improved.

For a light end, the two neighbours are different: one is the heavy centre nearby and the other is the far light end. The heavy centre’s tight functions are the useful ones, and the far end’s contribute almost nothing on their own — but in the trimer’s full basis they contribute through the heavy centre’s, because the three together span more than any two of them do. That extra is missing from every pairwise calculation, and it is why the light ends come out under-corrected.

So the two signs come from the same fact seen from two ends: the three bases overlap. Where two of them overlap heavily the sum double-counts, and where a third adds something no pair contains the sum misses it. An assembly of pairs can only be wrong in one direction if the bases are either redundant or complementary, and here they are both.

The error is not the near pair’s #

The obvious repair would be to estimate the error from the closest pair, since that is where the bases overlap most.

Holding the first separation at two bohr and moving only the second, the overshoot falls from 15.69 per cent to 0.45 — five sixths of it gone, with the near pair unchanged. So the error is a genuine three-body quantity: it is about the two neighbours’ bases overlapping each other, not about either of them overlapping the fragment they are correcting.

That also settles what kind of correction would fix it. A pairwise scheme with a pairwise patch cannot, because the quantity being missed is not attached to any pair.

What it costs a practitioner #

The numbers have a plain reading for anybody who applies counterpoise corrections to a cluster.

A pairwise-assembled correction over-corrects, so the binding it reports is too weak. At two bohr the excess correction is half a kilojoule a mole on a total correction of three; on a cluster of many fragments the pairs outnumber the fragments and the excess grows faster than the correction it belongs to. The direction is the same in every geometry tested here, so it is a bias rather than a scatter — and a bias in the direction of under-binding is the one an interaction-energy calculation is least able to tolerate, since it is already fighting a superposition error that biases the other way.

The other reading is about which fragment. If a cluster’s interaction is being decomposed — this much from the metal, this much from each ligand — then the fragment with the tightest basis takes the whole of the assembly’s error, and it takes it as an over-correction. A contraction is a decision made once and it decides which fragment that is: the one whose basis was contracted hardest is the one whose share is most misreported.

The control #

The two outer fragments of a symmetric trimer are related by a reflection, so their corrections must be identical. They are, to the last digit the solver carries, by both routes — which is what says the asymmetry above is the physics rather than an indexing error in the assembly.

There is a corollary for the divided correction itself. The two halves of a pairwise correction change places at a definite separation — near a bond the heavy centre has essentially all of it, and by nine bohr two thirds belongs to the lighter one. In a trimer that crossover has no meaning at all, because there are no halves to divide: a fragment’s correction in the full basis is one number and the two neighbours do not have separable shares of it. A frozen correction taken at one separation was one hazard; a divided correction taken at any separation is the new one.

One more thing the geometry sweep says #

The overshoot’s fall with separation is worth one number rather than a shape. Between 1.6 and 3 bohr — a factor of under two in distance — it falls from 30.4 per cent to 0.58, a factor of fifty-two. That is much faster than the correction itself falls over the same range, which is only a factor of 1.955.

So the assembly’s error lives at short range even by the standards of a quantity that is already a short-range artefact. It is a product of two basis overlaps rather than one, and each of those falls exponentially, so the error falls with roughly the square of what the correction does. A correction computed at one length is a hazard already recorded; the assembly’s error has a shorter range than the correction, so freezing it at one length is a worse approximation than freezing the correction was.

That has a consequence for the practice the assembly exists to serve. Pairwise assembly is used because a full counterpoise on a large cluster is expensive: it needs one calculation per fragment in the whole cluster basis, and the number of those grows with the cluster while the number of pairs grows faster still. The trade is supposed to be accuracy for cost. What the range dependence says is that the trade is not uniform — the assembly is nearly exact wherever the fragments are loosely packed and worst wherever they are tightly packed, and a cluster is interesting precisely where it is tightly packed. The approximation is cheapest to justify in the regime where it was not needed.

It also says where to spend a correction if only some triples can be afforded. The overshoot falls with roughly the square of the correction, so ranking triples by their closest contact ranks them by their assembly error much more sharply than ranking them by their own counterpoise size would. A cluster with one close trio and twenty distant ones has essentially all of its assembly error in the one, and computing a genuine three-body counterpoise for that trio alone recovers most of what the full treatment would give.

What was computed, and how #

Every energy is the lowest eigenvalue of a generalised eigenproblem in a basis of s-type Gaussians placed on the line, with the overlap, kinetic and nuclear-attraction integrals in closed form. A “ghost” centre is one whose functions are kept and whose nuclear charge is set to zero, which is what counterpoise means and is implemented as exactly that rather than as anything approximate.

Each centre’s basis is fitted to its own nuclear charge by a variational optimiser and then held fixed, so nothing in the comparison is refitted between geometries. That matters here: a basis refitted at each separation would make every difference a difference of two different calculations rather than of two ways of counting one.

The refusal is separation. At six bohr the two quantities must agree to the solver’s noise, and they do — the difference is $-0.003$ per cent, which is the last digit of a subtraction of two numbers agreeing to eight figures. An arithmetic error in the assembly would not go away when the fragments are pulled apart.

Where the model stops #

One electron per fragment, s functions only, and three centres on a line. So the numbers here are the size of an effect in a caricature, not an estimate of what a real trimer’s assembly error is. What is being established is that the effect exists, has a sign, and is a three-body quantity — none of which depends on the size of the basis.

The fragments have no electrons to correlate and no exchange between them, so the interaction being corrected is not a real intermolecular interaction. What a basis-set superposition error actually is is a variational artefact of an incomplete basis, and that is present here in full; what is absent is everything else about an interaction.

The basis the other atom lent is what this whole argument is about, and this is one more thing to say about it: what one atom lends is well defined, and what two atoms lend is not the sum of what each lends.

And a real three-fragment calculation would not use the site-site scheme tested here. The literature’s three-body counterpoise schemes are more careful — the whole point of naming them is that the naive assembly is known to be wrong — and what is added here is a measurement of how wrong, resolved by fragment.

A last practical note, because it decides whether any of this is worth acting on. The over-correction is 30 per cent of a quantity that is itself an artefact, so what it costs a binding energy is 30 per cent of an artefact rather than 30 per cent of a binding. In this model at 1.6 bohr that is $4.2\times10^{-4}$ hartree — about one kilojoule a mole — against a binding of a different order entirely. Whether that matters is a question about the calculation being done, and the honest answer is that it matters exactly when the counterpoise correction itself mattered, since it is a fixed fraction of it.

The generalisation #

An extensive quantity assembled from pairwise pieces double-counts whatever the pieces share, and what a counterpoise correction’s pieces share is basis-function space, which is exactly the thing they are made of.

That statement is not specific to basis sets. It is the same arithmetic that makes a sum of pairwise interaction energies wrong for three bodies, and the same arithmetic that makes an inclusion–exclusion correction necessary whenever overlapping contributions are counted separately. What is unusual here is that the overlap being double-counted is an overlap of descriptions rather than of physical interactions, so nothing about the geometry suggests it needs correcting.

The same shape turns up twice more, and both times it is a convention that cancels within one system and not between two. A reference bond order common to one molecule put a floor under a decay when two molecules were differenced; an electronegativity table’s multiplier cancelled inside one molecule and not across a series. Here it is a basis: what a fragment gains from a neighbour is well defined for one neighbour and is not additive over two.

Who found it, and when #

The counterpoise correction is Boys and Bernardi’s, from 1970. That the naive extension to many fragments over-corrects has been known since the eighties and is the reason the literature distinguishes the site–site function counterpoise scheme from the pairwise sum: the former uses the whole cluster’s basis for every fragment, which is what the “three-body” column here is.

Which of the two is right has been argued about, and the argument is not settled by anything here — the site–site scheme is the one that follows from the definition of the artefact, and objections to it are about whether the full cluster basis is the right reference rather than about the arithmetic.

What does not seem to be standard is the per-fragment resolution: that the assembly’s error has opposite signs on different fragments, and that the total’s sign is decided by whichever fragment has the largest correction rather than by anything about the geometry.

Still open: a fourth fragment, and the basis size #

The obvious open question is the fourth fragment. With three centres there are three pairs and one three-body term; with four there are six pairs, four triples and one four-body term, and the assembled error should grow faster than the number of fragments. Whether it grows as the number of pairs or as something slower is a question the same calculation can answer directly, and the answer decides whether a pairwise-assembled correction is usable on a cluster of ten.

The nearer question is the basis size. Everything here uses three Gaussians a centre, which is a small basis and therefore one with a large superposition error to begin with. A larger basis has a smaller correction, and whether the fraction the assembly overshoots by shrinks with it or stays put is what decides whether this is a small-basis artefact or a permanent feature — and the basis series already runs from one function to six, so the sweep is one loop.