Molecular Geometry & Bonding
what the shape decides
Concept

Fragment method — where it appears

Any scheme that computes a large system by cutting it into pieces, treating the pieces, and adding the results with corrections for what the cut removed. Its accuracy rests on the neglected terms being small, which is a claim about the system rather than about the method.

Named by 5 essays across one field — each of them below, with the objects they name alongside it.

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 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 The consecutive triples fail for a uniform chain at every length. The share of three decades of accuracy line over which some basis size is usable, against the number of fragments from three to eight, for three arrangements and three ways of assembling the correction. On the uniform chain the consecutive-triple assembly covers less than pairs alone at every length from four, and the shortfall grows. On the heavy-inside chain it covers exactly what every triple covers. On the heavy-outside chain it covers more than every triple from five fragments on.

Length did not rescue the consecutive triples

On four fragments, keeping only the consecutive triples of a counterpoise assembly worked for two arrangements and did worse than pairs for the uniform chain, and a short chain was the obvious excuse. Carried to eight fragments the excuse fails: the uniform chain's consecutive assembly settles at 48 per cent of the line against 68 for pairs. And the heavy-outside chain turns the lesson over — from five fragments every triple together covers less than the consecutive ones alone.

orbitals · Basis

Named alongside it

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

Basis set superposition errorMany-body expansionCounterpoiseModel limitGaussian basisBasis setVariationalApproximationConventionFinite-size effect

All concepts