1s with 1s at 2.8 bohr

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.

Eight 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

Two 1s orbitals close enough to interact, with the regions where their product is positive shown faintly. The integral of that product is the overlap, and it is positive throughout because both functions are positive everywhere.

A pair with different quantum numbers on the two centres, at the same separation as the three above. Nothing about the integral changes kind: the product is positive over a region and negative over none, and the value comes out at 0.51 — larger than the 1s–1s pair at the same distance, from functions that look far less alike.

A sigma interaction between two p orbitals pointing along the internuclear axis. The lobes that meet in the middle dominate the integral, and the result is cylindrically symmetric about the line joining the nuclei.

What an orbital is

Two 1s orbitals close enough to interact, with the regions where their product is positive shown faintly. That product, integrated, is the overlap — and its sign is what decides which of the two combinations is the lower in energy.

Exactly zero

An s orbital and a p orbital perpendicular to the internuclear axis. The positive contribution above the axis and the negative one below are mirror images, so they cancel exactly — and the computed integral is arithmetic noise rather than a small physical quantity.

The other pair that survives, for contrast with the three above that do not. Two s functions on different centres have no plane about which the integrand is odd, so nothing cancels and the integral is a number of ordinary size — 0.44 here. The distinction the essay is about is not between large and small overlaps; it is between an integral with a number in it and one with a cancellation in it.

A p orbital along the axis with a p orbital perpendicular to it: forbidden. The same cancellation as before, with different shapes on either side of the plane.

A double bond is not two single bonds

The σ interaction: two 2p orbitals pointing along the axis between the nuclei, at carbon’s screened charge and the C=C bond length. The overlap region sits between the nuclei, which is what makes this component strong and what makes it resist stretching.

The π interaction between the same two atoms at the same separation: two perpendicular 2p orbitals overlapping side by side. The overlap is in two regions, above and below the axis, and there is a nodal plane containing both nuclei. Nothing about this picture is half of the previous one.

The third overlap that is available at the same separation and is not part of either bond as usually drawn: an s function on one atom against a p function on the other. It is not small. A picture of a double bond as one σ and one π is a choice of which two of several available interactions to name, and the naming is what makes the two look like independent objects to be added.

The antibonding level goes up more

Two 1s orbitals at hydrogen’s bond length, with the region where their product is positive shaded. The overlap integral is the volume of that product, computed here at 0.7529 against the closed form’s 0.7529. Every contour drawn states the fraction of its own density it encloses, so the two pictures are comparable with each other.

The same two orbitals at four times the separation. The overlap has fallen to 0.063 and the two contours barely meet; at this distance the symmetric diagram is nearly right, and there is nearly nothing to be right about. Both figures draw their contours at half of each orbital’s density, so the comparison between them is a comparison of the same thing.

Closer is not more overlap

Two 2s orbitals at 2.8 bohr with the sign of their product shaded. Both functions change sign at the same radius, so the regions where the product is negative are small and symmetric, and the integral is dominated by the positive ones.

A full band is not an insulator

The vanishing this model rests on. A p orbital along the axis and a p orbital across it, with the regions where their product is positive and negative shaded. The two regions are mirror images and the integral over them is 4×10⁻¹⁷.

The tenth that is not drawn

The two atomic contours at exactly the separation where they lose contact, with the overlap integral computed and stated. The picture says the atoms have parted; the number says the interaction is a tenth of what it is at bonding distance and larger than a hydrogen bond’s.