The same calculation, two bindings

One of the figures on a basis is not a thing: The same electrons written two ways with the density unchanged, and four electronegativity scales that disagree.

Eleven essays draw this figure, each at the values its own argument needs rather than at the setting shown above. What each one uses it to show is below, in the words of its own caption.

The basis the other atom lent

One correction, two halves, three orders of magnitude apart. The two atoms of an unequal pair do not borrow equally: the one with the poorer basis borrows far more, and the correction that is quoted as a single number is a sum of two quantities that differ by a factor of a thousand.

The binding of H₂⁺ in two Gaussians a centre, measured from the atom in its own functions and again from the atom with the ghost. The second reference is lower, so the binding measured from it is smaller. The two minima are not at the same separation, which makes the artefact a structural matter rather than only an energetic one.

What the ghost is worth, against how far away it is. Nothing at 1.4 bohr, a maximum of 235.7 microhartree at 4.19, and nothing again by 9. A quantity that vanishes at both ends of a range has a maximum in the middle of it, and a maximum in the middle is what moves a bond length rather than merely deepening a well.

A correction computed at one length

Three binding curves for the hydrogen molecular ion: uncorrected, corrected at every separation, and corrected once at three bohr with that one number carried across. The third curve is the first one moved down the page, and moving a curve down the page does not move where its minimum is.

The artefact against separation. It has an interior minimum — near where the bond is — and rises in both directions, reaching more than twenty times its smallest value before falling away at large separation where the two sets of functions no longer overlap.

Seven reference geometries. The depth error and the asymptote error move by hundreds of microhartree and change sign; the bond length error is the same number in every row. A shortcut whose cost does not depend on how carefully it is applied is a shortcut that has removed something rather than approximated it.

A correction that is two functions

What fraction of the correction belongs to the lighter of two unlike atoms, against their separation. The dashed line at a half is the symmetric case, where it does not move.

The two halves themselves, on a logarithmic scale, with the symmetric pair’s coincident halves drawn for comparison. Three orders of magnitude separate the two curves where a bond would be.

The binding with and without the correction, across the separation. Everything above is that gap, taken apart.

The half that cannot be computed

The two halves against separation. One falls smoothly over two decades; the other scatters over more than one between neighbouring points.

The second difference of each half as a fraction of its own value, point by point.

The two differences, and how many figures of cancellation each asks for. Both are computed the same way and only one of them survives it.

The assembly that counts one share twice

The counterpoise correction of a three-fragment system computed directly, against the sum of its three pairwise corrections. The sum is the larger everywhere the difference is above the solver’s noise.

The three geometries, with each fragment’s nuclear charge and the correction the whole trimer’s basis gives it.

How far the assembled correction exceeds the trimer’s own, for the total and for each fragment separately.

The correction that gets harder to assemble

At a separation of two bohr: the trimer’s own correction against the basis size, and the fraction by which assembling it from pairs overshoots. The two go in opposite directions.

The overshoot against the basis size at four separations. Every curve rises, and the one that starts smallest rises fastest.

Each fragment’s own overshoot across the sweep. The heavy centre carries all of it.

One basis size where it is worth doing

The counterpoise correction and the error in assembling it pairwise, both against the number of Gaussians a centre, with a line at one kilocalorie a mole.

At the closest separation: whether the correction is worth computing, whether its pairwise assembly is accurate enough, and therefore whether the procedure is worth doing at all.

The basis sizes at which the correction is above the line and its pairwise assembly below it, at each of the four separations.

The line was holding the answer up

The basis size usable at every separation, against where the line is drawn. Shaded stretches are lines at which nothing is usable.

The stretches over which each basis size is the answer, logarithmically. One size down for roughly every fourfold loosening.

The stretches at which no basis size is usable at every separation. They cover more of the range than the answers do.

The overshoot was one arrangement

Each bar is the stretch of accuracy line over which one basis size is usable at every separation, for three centres and for four.

The covered share of the line with three centres and with four, for each arrangement.

Each basis size’s window width as a ratio of its top to its floor, with three centres and with four.

A repair that costs more than the whole

For every cell, the size of the pairwise error and of the remainder after triples, on a logarithmic scale against a kilocalorie a mole.

Each basis size’s window assembled from pairs (thin) and with triples added (thick). The tops coincide; the floors move.

The stretches of accuracy line with a usable basis size, assembled from pairs and with triples, labelled with the sizes usable in each.

Length did not rescue the consecutive triples

The share of three decades of accuracy line with a usable basis size, against the number of fragments, for three arrangements and three assemblies.

Each fragment’s assembly error on the uniform chain of eight at three Gaussians a centre and two bohr, for the three assemblies.

On the uniform chain, the median share of each cell’s three-body increment carried by triples that are not consecutive, by size and with its sign, against the number of fragments.