The basis the other atom lent

A basis set is attached to atoms. That is the whole convenience of it — functions centred where the electrons are — and it has a consequence that took the field two decades to name.

Compute a molecule and each atom sits in a basis that includes its own functions and its partner’s. Compute the separated atoms and each sits in its own functions alone. The molecule has therefore been given a better description than the pieces it is being compared with, and the difference between the two descriptions is added straight to the binding energy. Nothing has gone wrong with any calculation: both are correct answers to the questions they were asked, and the questions were not the same.

This is the basis set superposition error, and this essay computes it in the smallest system that can carry one.

The system, and why it is one electron #

H₂⁺ — two protons and one electron — is where this collection’s contraction argument already lives, and it has the property that matters here: the exact answer is known. The one-electron two-centre problem separates in prolate spheroidal coordinates and has been solved to many figures since Burrau, giving a binding of 0.102634 hartree at 1.9972 bohr. So a basis can be told not only how it compares with another basis but how far short it falls.

The basis is n s-type Gaussians on each proton, with the exponents found by descent on the free atom rather than taken from a table, exactly as the fitted basis for a single atom was found. Everything needed for two centres is one integral more than the one-centre problem already needs: the product of two Gaussians on different centres is one Gaussian on the line between them, and the nuclear attraction to a third point brings in the Boys function.

Three energies where a binding needs two #

The calculation is run three times at every separation.

The dimer is the electron in all 2n functions, attracted to both protons. The atom alone is the electron in the n functions on one proton, attracted to that proton — which is what a calculation on the separated fragment gives. The atom with a ghost is the electron in all 2n functions, attracted to one proton only: the far centre carries functions and no charge.

The third is the counterpoise construction, and it is not a trick. Adding functions to a variational calculation can only lower the energy, so the atom with a ghost is at or below the atom alone, and the gap between them is the whole of what one atom borrows from the other’s basis.

Two checks say the construction is what it claims to be. Put the ghost fifty bohr away and the atom’s energy returns to the atom-alone value to better than a nanohartree — so the effect is the reach of the far functions and not a bookkeeping error. And the artefact is positive at every separation and every basis size, which it must be, because it is a variational lowering of one side of a subtraction.

It is largest where the atoms are not touching #

The first thing the calculation says is not what a reader would guess.

At contact the two atom-centred sets are so nearly the same set that the far one has little to add — the functions have collapsed onto each other, and adding a nearly linearly dependent function is a function that is already there, where a duplicate buys almost nothing. Far out there is nothing within reach. In between there is room for the far centre’s diffuse functions to describe the tail of the near atom’s density better than the near centre’s own can, and that is where the borrowing happens.

The consequence is the shift in the minimum. Because the artefact grows with separation over the range where the well is, the uncorrected curve is being tilted outward: for two Gaussians a centre the minimum moves from 2.2271 bohr uncorrected to 2.2225 corrected, and for one Gaussian a centre it moves by 0.015. A correction that only deepened a well would not change a structure; this one does.

The other error, and its sign #

Now the part that decides what the correction is worth.

A finite basis makes two mistakes, and they have opposite signs.

Superposition makes the binding too large, by treating the molecule better than the fragments. That is the one counterpoise removes.

Incompleteness makes the binding too small, because the molecule itself is described imperfectly — and the molecule is harder to describe than the atom, so the imperfection costs the molecule more.

Both are computable here, because the exact binding is known. The incompleteness error is what is left of the shortfall after the superposition part is taken out.

The ratio is the result. At four Gaussians a centre, the error the correction removes is one part in eighteen thousand of the error it does not. Correcting it moves the answer from 0.090785 to 0.090784 hartree, against an exact 0.102634 — a change in the fifth decimal place of a number wrong in the second.

And because the two errors have opposite signs, removing the smaller one makes the total worse. The uncorrected binding is nearer the exact value at every basis size in the table. That is not an argument against counterpoise in general, and the next section says why; it is a statement about what the correction is and is not for.

Where the incompleteness stops falling #

There is a second thing in that table and it is the more useful of the two.

The atomic energy converges nicely: the error in the free hydrogen atom falls from 75.6 millihartree at one Gaussian to 0.054 at six, three orders of magnitude, exactly as a variational sequence should. The incompleteness error in the binding does not. It falls from 27.3 millihartree to 11.85 and then stops: 11.85, 11.94, 12.08 at four, five and six functions.

What is missing is not a count. It is a shape: these are s functions only, and an electron in a molecule is pulled towards the other nucleus, which needs a p function on each centre to describe. No number of s functions supplies one. So the residual twelve millihartree is a polarisation error, and adding functions of the kind already present cannot touch it — which is the property that gets worse arriving from a different direction: a basis improves at what it was built to do and need not improve at anything else.

That also explains the sign of the failure. The superposition error is the mechanism by which an s-only basis does get some polarisation — the far centre’s functions sit on the side the electron is being pulled towards — so it is partly repairing the incompleteness rather than merely inflating the answer. Removing it removes a repair.

What the correction is actually for #

None of this says counterpoise is wrong. It says what it is a correction of, and the distinction is worth stating plainly, because the practice is nearly universal and the reason for it is often given wrongly.

Counterpoise removes an artefact that is not variational. Everything else about a finite basis is: adding functions lowers the energy, and the sequence converges from above. Superposition breaks that, because it lowers one side of a subtraction and not the other, so the binding does not converge from any consistent direction as the basis grows. A sequence of uncorrected bindings can rise, fall, or wander.

What it does not do is make the number right. Here the incompleteness error dominates by four orders of magnitude, so the corrected number is worse; in a calculation of a weak intermolecular complex with a large basis it is the other way round, because the binding is small, the incompleteness error is small with it, and the superposition error is not. The ratio in the table is the quantity that decides, and it is a property of the system rather than of the method.

Two things that were checked rather than assumed #

The two-centre integrals reduce to the one-centre ones at zero separation, for overlap, kinetic energy and nuclear attraction alike, to the last bits a double holds. That is the only check available on them that does not use the same formulae twice.

And the Boys function — the one new special function a second centre needs — is computed here from a series and an asymptote rather than through an error function, and is checked against a direct quadrature of its own defining integral at five arguments. A special function taken on trust is exactly the sort of thing that produces a smooth, plausible, wrong potential energy surface.

The arithmetic of a borrowed function #

It is worth saying in one place what is actually being lent, because the phrase basis set superposition suggests something more mysterious than it is.

An atom’s electron is described by a sum of Gaussians centred on its own nucleus. Those Gaussians are all spherical about that point, so the density they can build is spherical about that point. Put a second centre four bohr away and give the electron access to its functions too, and the sum can now be lopsided: a small admixture of a diffuse function on the far centre pulls density towards it. That is not a mysterious lowering. It is one more variational parameter, doing what variational parameters do.

The reason it counts as an error is entirely about the comparison. A binding energy is a difference, and the two halves of a difference have to be computed the same way. The same discipline governs a resonance energy, where the reference is a construction rather than a molecule, and it governs the frames a Hückel number does and does not survive: a quantity built by subtraction inherits every asymmetry between the two things subtracted.

The two-centre problem underneath all of this is the ordinary one: two 1s functions, a bonding combination and an antibonding one. Everything above is an attempt to build that picture out of a basis that is not complete on either centre, and the artefact is what the incompleteness costs.

The objection this system is immune to #

The counterpoise correction has been argued about for forty years, and the argument is not about whether the artefact exists. It is about whether removing it the standard way removes too much. The system here happens to be the one in which that objection cannot be raised, which is worth saying because it is what the one-electron restriction buys rather than what it costs.

The objection runs like this. A ghost calculation gives an atom the whole of its partner’s basis to use. In the real molecule the partner’s functions are not entirely available: the partner’s own electrons are sitting in them, and antisymmetry forbids a second electron from occupying an orbital already occupied. So the ghost atom is described better than the atom in the molecule is, the correction computed from it is larger than the artefact actually present, and subtracting it overshoots.

That is a real effect and the reason the dispute lasted. It also requires two electrons.

There is one electron here. Nothing is occupying the far centre’s functions, so nothing blocks the near electron from using them, and the ghost calculation gives that electron exactly the freedom it has in the molecule — no more. The counterpoise correction computed in this essay is therefore not an estimate of the artefact; it is the artefact, and the usual caveat has nowhere to attach.

Two things follow, and they run in opposite directions.

The measurement here is cleaner than any that could be made on a real closed-shell pair, which is the point of choosing it. A factor of eighteen thousand between the superposition error and the incompleteness error is a statement about the two errors and not about a correction scheme’s over-enthusiasm.

And it is correspondingly less transferable. In a two-electron system the same arithmetic would return a correction somewhat too large, so the ratio measured here is a lower bound on how favourably counterpoise behaves rather than a typical value. The literature’s compromises — applying half the correction, or applying it only to the fraction of the partner’s basis that is genuinely vacant — are responses to a difficulty that is invisible from inside this calculation.

What survives the difference is the ordering, which is what the essay’s refusal is about. The occupancy objection makes the correction too large; it cannot make the artefact bigger than it is. So an argument that the artefact is small beside the incompleteness error is not weakened by admitting the objection — a smaller true artefact only widens the gap. The one-electron restriction limits what the number means and leaves the comparison it was computed for intact.

What this cannot say #

One electron, so no correlation. Real superposition errors in correlated calculations are much larger than in Hartree–Fock ones, because the correlation energy is far more basis-sensitive than the orbital energy. Nothing here can show that: there is one electron and no correlation energy to be sensitive.

s functions only. The residual twelve millihartree is a polarisation error and adding p functions would remove most of it, which would change every ratio in the table. What would not change is the structure of the argument — two errors of opposite sign, one variational and one not — and the ratio would still be the quantity that decides.

The exponents are the free atom’s. A molecule-optimised basis would have a smaller incompleteness error and a smaller superposition error, and which fell faster is a question not asked here.

And the eight per cent shortfall is not a scandal. Six s Gaussians a centre giving 0.0906 against 0.1026 hartree is a perfectly ordinary result for a minimal basis on a molecule; it is quoted here to give the artefact something to be measured against, not as a criticism of the basis.

What was checked #

The two-centre integrals reduce to the one-centre ones, at three exponent pairs, for all three kinds of integral — the check that the two-centre integrals are the one-centre ones with a separation in them.

The Boys function agrees with a direct quadrature at five arguments, to three parts in a million.

A ghost at fifty bohr lends nothing, to a nanohartree, which identifies the effect as an overlap of functions.

The artefact is positive at every basis size and every separation, because it is a variational lowering; a negative value would mean the eigensolver had not returned the lowest state.

It shrinks as the basis grows, by a factor of more than three between two functions and six, which is what makes it an artefact rather than a physical term.

It peaks in the middle of the range rather than at either end, which is what lets it move a minimum.

And the refusal: at every basis size the incompleteness error is the larger of the two, and the uncorrected binding is no further from the exact one than the corrected. Both halves are checked together, because either alone would be a weaker and less interesting claim than the pair.

Still open: p functions, and whether counterpoise transfers #

The obvious open question is the one the model cannot reach: p functions. Adding one shell of them on each centre would remove most of the residual twelve millihartree, and it would do something to the superposition error that is not obvious in advance — a p function is exactly the thing the far centre’s s functions were standing in for, so supplying it directly might reduce the borrowing rather than increase it. Which way that goes is a question with an answer and it is one integral more than is computed here.

The nearer question is about the practice rather than the model. Counterpoise is applied by taking the difference of two calculations, and the difference of two calculations is the shape of thing where error cancellation has been measured before. What has not been asked is whether the correction is transferable: the artefact is computed at one geometry and often applied across a whole potential surface, and the curve above says it varies by a factor of thirty across the range where a well sits. Measuring what that assumption costs would need only the calculation here and a second geometry.