Overlap against separation

One of the figures on overlap: Two functions on two centres, integrated — including the integrals symmetry requires to vanish, which come out at arithmetic noise.

Seven 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.

Overlap decides

How the overlap falls with separation, for three pairs of orbitals. The dots on the 1s–1s curve are the closed form, drawn over the computed one — so the integrator is validated at every separation rather than at one.

Two sigma interactions compared. The 1s–1s curve falls off fast, because 1s orbitals are compact; the 2p–2p curve is broader, because 2p orbitals extend further. The dots are the closed form for the first, drawn over the computed curve.

A double bond is not two single bonds

The σ and π overlaps of two carbon 2p orbitals against separation, computed at the nuclear charge a carbon 2p electron actually feels. The two curves have different shapes and different signs, cross nothing in common, and are not related by any factor. There is no arithmetic in which one is half or twice the other.

Overlap is not interaction

The factor the rule of thumb tracks, computed. Overlap against separation for three pairs, checked against the closed form where one exists. Every curve here is between identical orbitals, so the gap is zero throughout and the overlap really is the whole story — which is why this figure is the one the rule of thumb was learned from.

Closer is not more overlap

Where the overlap peaks and where the bonding peaks, which are not the same separation. Every extra radial node gives the product another region of the opposite sign, so the number of ways an overlap can turn over grows with the principal quantum number — and the separation at which it is largest drifts away from the separation at which the bond is strongest.

The overlap of a 1s with a 2s, against the separation. It is nearly zero at contact, grows in magnitude to 0.2820 at 4.17 bohr, and falls away beyond. The extremum is found by bisection on the integral itself rather than read off the sampled curve.

Two 2p orbitals pointing at each other along the axis between them. The overlap is +0.973 at one bohr, passes through zero at 5.06 bohr, and is −0.319 at eight — larger in magnitude than it was at four. The root is found by bisection to six figures.

A bond is not two atoms overlapping

The overlap itself against separation, which is the quantity underneath everything above. Where it is large the two combinations differ sharply; where it has fallen to a per cent they are almost the same orbital drawn twice, and the picture’s division into two surfaces is the visual statement of that.

The same overlap, a different bond

The stabilisation of three pairs against separation, each scaled to its own largest. Two 1s orbitals fall away monotonically, as everybody expects. The two pairs with a radial node in them turn over: a 1s with a 2s at 4.168 bohr and a 2s with a 2p at 7.583 — separations at which two atoms are further apart than any bond and are bonded more strongly than at contact.

Where the overlap peaks and where the bonding peaks, for two pairs with a radial node, both refined off the grid. They agree to about a hundred-thousandth of a bohr, which is the refinement’s own precision. The grid spacing is a tenth of a bohr, so a comparison that read the maxima off the samples could not have said anything either way.

The σ overlap of two 2p orbitals against separation, which does not fall monotonically. It rises, turns over and changes sign, so the same numerical value of the overlap occurs at three different separations — and the bond those three describe is not the same bond. That is the essay’s claim in the one curve where it cannot be argued away.

A regime that belongs to the neighbours

The overlap integral, for its simplest pair. Nothing here says the integral misbehaves — only that a bond is not a function of it alone.