A σ contour at 90 per cent, and the two atomic ones

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.

Three 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.

A bond is not two atoms overlapping

The section through both nuclei of the surface enclosing ninety per cent of the bonding combination’s density at two bohr, with the two atomic contours of the same stated fraction drawn as circles behind it. The molecular surface has no waist between the nuclei; the pair of atomic ones has a pronounced one, and reaches further out along the axis at each end.

Where the picture divides, against the enclosed fraction chosen. With one centre a smaller fraction gives a smaller sphere and nothing else changes; with two, there is a fraction below which the surface is one connected object and above which it is two, so a caption saying only “the bonding orbital” is not distinguishing between pictures of different shapes. The fraction at which it happens is computed here rather than eyeballed.

The same section at four bohr rather than two. The surface is still one object and its waist has deepened; the two atomic circles have separated entirely, so the picture usually drawn has become two spheres that do not touch while the real contour has not yet divided.

The tenth that is not drawn

The separation at which the drawn picture stops being one surface and becomes two, against the fraction the contour is drawn at. A half contour divides at 2.67 bohr, a ninety per cent one at 5.32, a larger one later still — one orbital, one pair of atoms, and a separation-of-parting that moves by a factor of two because a caption asked for a different number. Nothing happens to the orbital at any of them.

The share of the overlap that falls outside both drawn surfaces, against separation, with the overlap itself listed underneath. At a bond length the picture contains almost all of it; by a van der Waals contact it is missing more than a third; at seven bohr, more than half.

That concentration, measured: the share of each combination’s density lying between the two nuclei, at four separations, for the bonding combination and the antibonding one. The bonding combination holds more at every separation, which is what a bonding orbital is; and the two converge as the atoms part, to within two per cent of each other at eight bohr, because by then there is nothing between the nuclei for either of them to differ about. The region that makes a contour work at a bond length has emptied out by the separations the rest of this essay is about.

A bond with nothing in the middle

The combination the model fills at three bohr, drawn as contours of the wavefunction with the two signs in the two colours. The dashed line is a nodal plane through the midpoint of the bond.

The overlap against separation, through the zero. Neither orbital changes anything at the crossing.

Both combinations along the axis, at a separation below the crossing and one above. The heavier curve is the filled one.