Concept

Basis set superposition — where it appears

The spurious part of a computed interaction energy that comes from each fragment borrowing the other's basis functions as they approach. It shrinks as the basis grows and never changes sign, so an interaction always looks stronger than it is; the counterpoise correction estimates it by giving each fragment the whole basis on its own.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

$The two halves change places. What fraction of the counterpoise correction belongs to the lighter of two unlike atoms, against their separation. At a bonding distance it is 1.26 per cent — essentially the whole correction is the heavier atom's — and by 9.0 bohr it is 67. The two change places at 6.29 bohr. A symmetric pair's share is exactly a half everywhere, which is what makes one frozen number a complete description there and nowhere else.$

A correction that is two functions

A symmetric pair's counterpoise correction splits exactly in half, at every separation, to the last digit — which is why one frozen number describes it. Give the two atoms different charges and the split runs from 0.03 per cent to 67, changing places at 6.29 bohr: the correction a single number was standing in for is two functions of different shapes.

orbitals · Basis

One half is a curve and the other is not. The two halves of a counterpoise correction for an unequal pair, against separation, on a logarithmic axis. The heavier centre's falls smoothly over two decades; the lighter centre's scatters over more than one decade between neighbouring points. It is not a rough function — it is a difference of two energies of order a hartree whose difference is a millionth, and the solver does not have seven figures to spare.

The half that cannot be computed

How many points does each half of a counterpoise correction need to interpolate? The natural expectation is two different numbers. The answer is that the question is not yet askable: the lighter centre's half is a difference of two energies agreeing to six figures, its second differences are seven per cent of its own value, and no interpolation of it means anything. The third thing worth checking — the symmetric-pair check — works perfectly.

orbitals · Basis

The trimer's correction, and the sum of its pairs. The counterpoise correction of a three-fragment system computed directly — each fragment's energy alone less its energy in the whole trimer's basis — against the sum of the three pairwise corrections, on a logarithmic axis. The sum is the larger everywhere the difference is above the solver's noise: 30.4 per cent at 1.6 bohr and nothing by six.

The assembly that counts one share twice

A counterpoise correction is divided unequally between its two centres, and the first place that matters is a three-fragment system, where the pairwise corrections are added up and the assembly must double-count one share and undercount another. It does: the heavy centre is over-corrected by seventeen per cent and the light ones under-corrected by two and a half, and the two do not cancel.

orbitals · Basis

Named alongside it

The objects these essays reach for when they reach for this one.

Approximation Basis Closed form Model limit Reference state Variational principle Convergence Normalisation Bond length Convention Counterpoise Intermolecular force

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