A correction computed at one length

Computed properly, the basis-set superposition error — the amount each atom’s energy falls when the other atom’s functions are put beside it, with no nucleus attached to them — is 0.64 microhartree against an incompleteness error of 11.85 millihartree at four Gaussians a centre. It also varies by a large factor across the range where a chemical bond sits, which is easy to notice and not follow up.

That variation is the subject here, because of what is usually done with it. Recomputing the correction at every point of a potential surface means three calculations instead of one at every geometry, which for a surface of any size is not done. What is done instead is to evaluate it once, at a reference geometry, and subtract that number everywhere.

The cost of that shortcut turns out to be exactly computable, and it is not what the shortcut’s users would guess. It does not degrade the bond length; it removes the correction to the bond length entirely, at any reference geometry, in any basis, by an argument with no arithmetic in it.

The system, and why it is small #

Everything here is the hydrogen molecular ion in a basis of $s$-type Gaussians, two per centre, with every integral in closed form and every energy from a two-by-two generalised eigenvalue problem. There is one electron, so there is no repulsion and no correlation, and the exact answer is known: a binding of 0.102634 hartree at 1.9972 bohr.

That smallness is deliberate. The counterpoise argument is about a basis and not about a method, so a system where the method is exact isolates it. The same choice was made where the two errors were first separated, and for the same reason: a variational improvement that moves the answer the wrong way is only visible where the right answer is known. Every number below is an artefact of the functions and of nothing else, and the exact answer is available to check against — which it is not in the calculations where counterpoise is actually used.

The correction itself is the difference between two calculations of the same atom: one in its own functions, and one in its own functions plus the other centre’s, with the other centre’s nucleus removed. The ghost is what makes it a measurement rather than a bookkeeping exercise, and its whole content is that a function centred somewhere else is still a function that can be borrowed.

The correction is not a constant, and its shape is the problem #

Across the range from 1.5 to 5.5 bohr, which is where a well and its outer wall sit, the artefact runs from 10.1 microhartree at 1.75 bohr to 236.0 microhartree at 4.25 — a factor of 23.4.

The shape is more informative than the range. It is not monotone. At very small separation the two atoms’ function sets are nearly coincident, so there is little to borrow that is not already there; at very large separation they do not reach each other and there is nothing to borrow at all. In between there is a maximum, and near the equilibrium separation there is a minimum.

That last is the awkward fact. The reference geometry a practitioner picks is almost always the equilibrium one, because that is where the calculation was being done anyway — and that is the point on the curve where the correction is smallest. Freezing it there under-corrects everywhere else on the surface, and under-corrects most at the separations a dissociation curve spends most of its length at. It is the same trap as a correction transferred between systems, one variable over: there the quantity was assumed to be the same for two molecules, here for one molecule at two geometries, and both assumptions are made because recomputing is expensive rather than because anything says they hold.

Why the bond length comes back #

Here is the argument the essay is built on, and it takes one sentence.

A frozen correction subtracts the same number from the binding at every separation. Subtracting a constant from a function does not change where its maximum is. So the frozen-corrected binding curve has its maximum at exactly the same separation as the uncorrected one, and reports exactly the uncorrected bond length.

The properly corrected curve does not, because the correction it subtracts varies with the separation, and a varying subtraction tilts the curve. That tilt is the whole of what counterpoise does to a structure. A correction that changes an energy and a correction that changes a geometry are different objects, and the distinction appears elsewhere — a bond length that depends on the isotope is a length that moves because the shape of a curve is sampled differently, not because its depth changed.

Measured here: the uncorrected minimum is at 2.228058 bohr, the properly corrected one at 2.223602, and the frozen one at 2.228058 — agreeing with the uncorrected number to nine decimal places, at every one of seven reference geometries from 1.6 to 6 bohr. The bond length error a frozen correction leaves is 4.456 millibohr, and it is the same 4.456 millibohr whatever reference is chosen, because it is not a function of the reference at all.

What the choice of reference does decide #

Everything else, and by a great deal.

The well depth. The error left in it is the difference between the artefact at the reference and the artefact at the minimum, and across seven references it runs from −77 to +135 microhartree, changing sign twice on the way. Freezing at 1.6 bohr over-corrects the depth; freezing at 4 bohr under-corrects it by nearly three times as much in the other direction.

The asymptote. This one is worse and is rarely mentioned. At infinite separation the true correction is exactly zero — there is nothing to borrow — so the corrected binding curve goes to zero there. A frozen correction subtracts a non-zero constant at infinite separation as well, so the frozen curve has a spurious offset at dissociation equal to the whole of whatever was frozen: from 18 microhartree at a reference of 1.6 bohr to 230 at 4.

That offset does not cancel in a dissociation energy computed as the difference between the asymptote and the minimum — it cancels only if the two ends of the subtraction carry the same frozen constant, which they do, so the dissociation energy is safe. It is a barrier height, an interaction energy quoted against a computed asymptote, or any comparison between two frozen surfaces with different references that is not. Which quantities survive a shared error and which do not is the question two wrong numbers and a right difference is entirely about, and the answer here has the same shape: what cancels is what the error is common to.

The two errors, and which one this is #

It is worth restating what the correction is for, because the relative size of the two errors makes the whole exercise more delicate than it looks.

A finite basis makes two mistakes of opposite sign. It describes the molecule incompletely, which makes the binding too small; and it lets each atom borrow the other’s functions, which makes the binding too large. Counterpoise removes the second, and the first measures eighteen thousand times the size of the second at four Gaussians a centre — so correcting moves the answer away from the exact one at every basis size in that calculation.

That does not make counterpoise pointless, and the reason it does not is the one this essay is about. The incompleteness error varies with geometry too, and the two variations are not the same shape — so a correction that removes one of them changes the shape of the surface even where it worsens its absolute position. A calculation is often after a shape rather than an energy, and a shape is exactly what a frozen correction cannot touch. That is also why a basis fitted to an energy fails on a property it was not fitted to: energies and shapes are different things to be right about, and being right about one is no evidence at all about the other.

What a practitioner should take from it #

A frozen correction is an energy correction and nothing else. It cannot change an optimised geometry, a vibrational frequency, or any quantity that is a derivative of the surface, because a constant has no derivatives. Reporting a “counterpoise-corrected geometry” from a frozen correction is reporting the uncorrected geometry.

Three points cost almost nothing. The correction here is smooth and has one interior maximum, so a quadratic through three computed values reproduces it across the whole range at a fraction of the cost of recomputing it everywhere — and it restores the derivative, which is the part that matters. The decision is the same one a contraction is: a cheap approximation made once and then carried, whose cost is invisible unless somebody computes what was given up.

And the reference should not be the equilibrium geometry. It is the minimum of the artefact, so it is the least representative point on the curve. A reference near the artefact’s own maximum would be nearer the mean of what it takes over the surface, which is a strange-sounding rule with a simple justification: the reference is being asked to stand in for an average, not for a typical geometry.

The bond length is not the only casualty, and the rule covers them all #

The argument that recovers the uncorrected bond length used nothing about bond lengths. It used only that a constant has zero derivative, so it applies unchanged to every quantity read off the shape of a potential curve rather than off its value.

The force constant is the second derivative at the minimum, and a constant does not change it. The harmonic frequency is that force constant divided by a reduced mass, so it does not change either. Nor does the anharmonicity, which is the third and fourth derivatives; nor the rotational constant, which is fixed by the bond length already shown to be untouched; nor the zero-point energy, which is built from the frequencies; nor the whole vibrational–rotational spectrum that follows from them.

So the rule has a clean statement. A frozen counterpoise correction moves the depth and the asymptote of a potential curve and nothing else whatever. It is exactly right for a binding energy computed at one fixed geometry, and it is precisely worthless for anything spectroscopic — not inaccurate, worthless, in the sense that it changes no digit of the answer.

That is a more comfortable position than it first sounds, and it should be read as a licence as well as a warning. A calculation of a vibrational spectrum in a basis with a superposition error does not need to worry about whether the correction was frozen, because a frozen one could not have helped; what it needs to worry about is whether the correction varies over the range the vibration samples, which is the quantity this essay measures at a factor of 23.4 and which no single number can carry.

The two failures are therefore separable and should be reported separately: an energy that is wrong by a constant, and a curvature that is wrong because a varying artefact was replaced by its value at one point.

What is quoted, and what is computed #

The exact H₂⁺ binding and separation are quoted — 0.102634 hartree at 1.9972 bohr — and they are the only quoted numbers.

Everything else is computed from closed-form Gaussian integrals: the overlaps, the kinetic energies, the nuclear attractions through the Boys function, the generalised eigenvalue problem at every separation, and the three separate calculations that go into every value of the artefact. The minima are found by fitting a parabola through the best three computed points rather than by any assumed curve shape, and the identity between the frozen and uncorrected minima holds at that resolution because both are fitted the same way through the same points.

The integrals themselves are checked against their one-centre forms at zero separation and the Boys function against a direct quadrature, both in the Gaussian integrals used throughout.

What this cannot say #

One electron. There is no correlation here, and the superposition error in a correlated calculation is much larger and has a different geometry dependence — the correlation energy borrows functions more greedily than the mean field does. The identity about the bond length survives that unchanged, because it is about constants and not about magnitudes.

Only $s$ functions. A real basis has $p$ and $d$ shells whose borrowing has a different reach, and supplying them directly might reduce the borrowing rather than increase it. Which way that goes is still not known here.

And the geometry is fixed while the electrons move. Every point on the curve is a clamped-nuclei calculation, which is the approximation the atoms are not at the points is about — so the measured bond length a frozen correction would be compared against is not the minimum of any of these three curves in the first place.

One coordinate. A diatomic has one separation, and a surface has many. On a surface the frozen correction is still a constant and still cannot move a stationary point, but the variation it discards is now a function of several variables and could be much larger than a single curve suggests.

And the numbers are small. Two hundred microhartree is half a millihartree, which is below chemical accuracy, because this is a one-electron system in a tiny basis. The ratios are the transferable part: a factor of twenty-three of variation, an interior minimum at equilibrium, and a structural correction that a constant cannot carry.

What was checked #

The artefact varies by more than a factor of ten across the range where a well sits. Measured: 23.4.

And it is smallest in the middle of that range rather than at an end, which is what makes the usual reference geometry the worst available one.

A correction frozen at a given separation returns the uncorrected bond length exactly, to nine decimal places, at seven reference geometries — and that is not the bond length a full correction gives, which is checked separately so that the identity cannot be satisfied by a correction that does nothing.

The bond length error does not depend on the reference at all, across seven of them, to better than a nanobohr.

While the depth error depends on it by more than a factor of five, which is the contrast the essay is about.

And the control is a reference at forty bohr. There the artefact is essentially zero, so the frozen correction is no correction at all and the frozen curve must coincide with the uncorrected one to better than a microhartree at every separation — a check that the subtraction removes what it says it removes.

Still open: how many points an interpolation needs #

The obvious open question is the interpolation recommended here and not tested. Three computed values of the artefact and a quadratic through them would restore the derivative that freezing destroys, and the question is how many points are actually needed — whether the artefact’s one interior maximum makes it a two-parameter object over the range that matters, or whether its behaviour at short separation, where it turns over sharply, needs more. The answer is a number, and it would turn “recompute it everywhere” and “freeze it” into a stated cost per unit of accuracy rather than a choice between two extremes.

The nearer question is about what happens when the two atoms are not alike. Everything here is symmetric, so the two halves of the correction are equal and the reference geometry is one number. In a heteronuclear pair the smaller atom borrows more from the larger one than the reverse, which means the correction is not shared equally and the two halves peak at different separations — so a single frozen number is standing in for two functions with different shapes. Whether that makes the error larger or merely more confusing is a computation that needs only one exponent changed, and it is the case every real application is.