Molecular Geometry & Bonding
what the shape decides
Concept

Counterpoise — where it appears

The standard repair for basis-set superposition error. Each fragment is recomputed in the whole system's basis functions with the other nuclei's charges switched off, and whatever it gained is subtracted. It is defined for a pair and is not simply additive over more.

Named by 6 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 The correction collapses and the error made assembling it grows. At a separation of 2 bohr, two quantities against the number of Gaussians a centre. The trimer's own counterpoise correction falls by a factor of 2717 from one function to six — a bigger basis has less to borrow. The fraction by which summing the pairwise corrections overshoots it rises from 5.3 per cent to 38.0. Improving the calculation makes the assembly proportionally worse.

The correction that gets harder to assemble

Summed across a trimer, pairwise counterpoise corrections come to fifteen per cent more than the trimer's own. Three Gaussians a centre is a small basis, so the natural question was whether the fraction shrinks with a better one or stays put. It does neither. The correction falls by three orders of magnitude and the fraction grows fivefold.

orbitals · Basis Both quantities, and the line they have to be read against. At the closest separation, the counterpoise correction itself and the error a pairwise assembly of it makes, against the number of Gaussians a centre. Both fall — the correction by a factor of 3631, the error by 793 — and the fraction rises by exactly the ratio of those two. The dashed line is a kilocalorie a mole. The only basis where the correction is above it and the error below it is two.

One basis size where it is worth doing

The pairwise assembly's error grows with the basis — 5.25 per cent at one Gaussian a centre, 37.98 at six — while the correction itself falls by a factor of three thousand. Which of the two should a practitioner care about? Drawing a line at a kilocalorie a mole answers it: there is exactly one basis size at which the correction is worth computing and its pairwise assembly is accurate enough to use.

orbitals · Basis Which basis size is usable, against where the line is drawn. The basis size usable at every separation — big enough that the three-body correction matters, small enough that the pairwise assembly reproduces it — against the accuracy line, over two decades. It is not one answer. Four different sizes are the answer over this range, and for much of it there is no answer at all. The standard kilocalorie a mole gives basis 2, and it sits 0.7 per cent above the edge where that answer stops.

The line was holding the answer up

There is exactly one basis size where the three-body correction is worth computing and the pairwise assembly reproduces it, and the accuracy line is the choice that whole picture is most sensitive to. Sweeping the line over two decades gives four different answers, long stretches with no answer at all, and a published window whose lower edge sits seven tenths of a per cent below the standard kilocalorie a mole.

orbitals · Basis Every usable window, with three centres and with four. For three pairs of arrangements on a line — every centre alike, the heavy centres inside, the heavy centres outside — the stretch of accuracy line over which each basis size is usable at every separation, across three decades of line. Adding a centre takes the covered share from 44% to 58%, from 27% to 54%, and from 67% down to 60%. The four-centre uniform chain is the one arrangement with no usable basis size at a kilocalorie a mole.

The overshoot was one arrangement

A fourth fragment leaves four three-body terms out of a pairwise counterpoise assembly as well as the four-body one, so the window in which the assembly is usable was expected to narrow. Asked of three arrangements that each gain one centre, it widens twice and narrows once, the uniform chain loses its answer at a kilocalorie a mole altogether, and the overshoot every earlier calculation reported turns out to belong to the arrangement with the heavy centre inside.

orbitals · Basis The usable stretches, assembled from pairs and from triples. For each four-centre arrangement, the stretches of accuracy line with a usable basis size when each fragment's correction is assembled from pairs, and when the three-body increments are added to it, labelled with the sizes usable there. The covered share rises from 58% to 67% (1-1-1-1), 54% to 75% (1-2-2-1), 60% to 80% (2-1-1-2), and stretches where more than one basis size is usable appear where there were none.

A repair that costs more than the whole

Four fragments are the first system in which a counterpoise correction can be assembled from something between pairs and the whole. Adding the three-body increments leaves what is still missing below a kilocalorie a mole on every cell, widens the usable range for every arrangement and turns a single usable basis size into two — and under a cubic model of cost it is dearer than the full calculation it stands in for until the cluster has eleven fragments. Keeping only the consecutive triples, which pays from five, works for two arrangements and does worse than pairs for the third.

orbitals · Basis

Named alongside it

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

Basis set superposition errorMany-body expansionModel limitFragment methodApproximationBasis setConventionGaussian basisVariationalBasisBasis set superpositionBond length

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