Molecular Geometry & Bonding
what the shape decides
Bonding models

Overlap decides

Two orbitals interact in proportion to how much they overlap, and the sign of the overlap decides which of the two combinations is the lower in energy. It is one integral, and almost everything about bonding follows from it.
10 min read 6 figures Counted, not quotedExactly zero

An orbital is a one-electron wavefunction. Bring two atoms together and their orbitals do not stay separate. The states of the pair are combinations of the atomic orbitals, and how strongly they combine is set by a single number.

S=∫ψA ψB dVS = \int \psi_A\,\psi_B\,dV

The overlap integral: the product of the two wavefunctions, integrated over all space.

1s with 1s at 2.8 bohrThe two orbitals in the plane containing both nuclei, with the regions where their product is positive and negative shown faintly. The overlap integral is the signed volume of that product, and where symmetry makes the two regions mirror images it comes out exactly zero.1s · 1sS = 0.38997sigma interactionseparation 2.8 bohrcontours at 50% of each densitythe signed product integrated over all spaceone electron
Fig. 1 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.

What it measures

SS is large when the two functions are substantial in the same region of space and small when they are not. It falls off as the atoms are separated, because the region where both are appreciable shrinks.

For two 1s orbitals in atomic units it has a closed form,

S=e−R(1+R+R23),S = e^{-R}\left(1 + R + \tfrac{R^2}{3}\right),

which is 0.75 at 1.4 bohr, 0.59 at 2, and 0.19 at 4. That closed form is worth having for a reason beyond elegance: it lets the numerical integrator used everywhere else on this site be checked rather than trusted.

Overlap against separationHow the overlap integral falls as two atoms are pulled apart, for several pairs of orbitals. Where a closed form exists it is drawn over the computed curve, so the integrator is checked rather than trusted.1234561s-1ssigma1s-2pzsigma2px-2pxpidots: the closed formoverlap Sseparation / bohrintegrated numerically, checked against the exact resultone electron
Fig. 2 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.

Why the sign matters

Two orbitals combine in two ways: added and subtracted. Both combinations exist, and they differ.

Where the two functions have the same sign, adding them makes the wavefunction larger between the nuclei, so density builds there. An electron in that region is attracted to both nuclei at once, and the combination is lower in energy than either atomic orbital. That is a bonding combination.

Where they have opposite signs, the sum cancels between the nuclei and a node appears there. Density is expelled from exactly the region where it would have been most useful, and the combination is higher in energy. That is antibonding.

So the sign of the wavefunction, which is not itself observable, decides which combination is which — and that is why every lobe on this site is coloured by sign rather than drawn as a bare shape.

Splitting goes with overlapFor each pair, the atomic levels on the outside and the combinations they form in the middle, with the splitting drawn in proportion to the computed overlap. A pair that symmetry forbids does not split at all, because its overlap is exactly zero.1s-1sσ overlapS = 0.39002pz-2pzσ overlapS = 0.45852px-2pxπ overlapS = 0.75291s-2pxsymmetry forbids itS = 0 exactlyno splittingoverlaps computed at 2.8 bohr, levels in proportionone electron
Fig. 3 Splitting drawn in proportion to the computed overlap. The two combinations move apart by an amount that goes with S, and a pair whose overlap vanishes does not split at all.

Sigma and pi

The two thread systems of bonding, named for how they look along the internuclear axis.

A sigma interaction is cylindrically symmetric about the axis: two s orbitals, or two p orbitals pointing at each other end-on. Looking down the bond, nothing changes as the view rotates.

A pi interaction has a node containing the axis: two p orbitals side by side, overlapping above and below. Looking down the bond, the sign alternates.

2pz with 2pz at 2.8 bohrThe two orbitals in the plane containing both nuclei, with the regions where their product is positive and negative shown faintly. The overlap integral is the signed volume of that product, and where symmetry makes the two regions mirror images it comes out exactly zero.2pz · 2pzS = 0.45852sigma interactionseparation 2.8 bohrcontours at 50% of each densitythe signed product integrated over all spaceone electron
Fig. 4 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.
2px with 2px at 2.8 bohrThe two orbitals in the plane containing both nuclei, with the regions where their product is positive and negative shown faintly. The overlap integral is the signed volume of that product, and where symmetry makes the two regions mirror images it comes out exactly zero.2px · 2pxS = 0.75286pi interactionseparation 2.8 bohrcontours at 50% of each densitythe signed product integrated over all spaceone electron
Fig. 5 A pi interaction between two p orbitals side by side. The overlap is split between the region above the axis and the region below, and the axis itself lies in a node.

The names come from the Greek letters matching s and p, because the symmetry about the axis is the same as that of an s or p orbital about a nucleus.

Which is stronger

The usual claim is that sigma bonds are stronger than pi bonds, and it is true at real bond lengths for a reason worth understanding rather than memorising.

A sigma interaction has the orbitals pointing directly at each other, so the region of maximum density of each is where the other is largest. A pi interaction has them side by side, so the maxima are displaced from one another and the useful overlap is smaller.

But that is a statement about the geometry at a particular separation, not a law. At very short separations the ordering can reverse, because the sigma lobes have passed through each other while the pi lobes are still approaching. Real molecules sit where the total energy is minimised, and there sigma wins.

Overlap and bond strength

A caution, because the connection is looser than it is usually presented.

Larger overlap does mean stronger interaction in the sense of a larger splitting between the two combinations. It does not straightforwardly mean a stronger bond, because a bond’s strength depends on which combinations are occupied, on the energies of the parent orbitals, and on the nuclear repulsion that grows as the atoms approach.

In particular, overlap keeps growing as two atoms are pushed together, and bonds do not keep getting stronger. What stops them is that the antibonding combination rises faster than the bonding one falls, and that the nuclei repel. Overlap is one term in a balance, not the answer.

Energy matching, the second condition

Overlap is necessary and not sufficient. Two orbitals also have to be close in energy to interact strongly.

The reason is that the mixing coefficient goes roughly as the overlap divided by the energy difference. Two orbitals with large overlap and very different energies barely mix: the low one stays low, the high one stays high, and the combinations are nearly the parents.

That is why a hydrogen 1s interacts strongly with a carbon 2p and hardly at all with a carbon 1s, despite the carbon 1s being right there. The core orbital is far too low in energy to be perturbed.

The third condition is where symmetry can forbid an interaction outright. So a strong interaction needs three things: appreciable overlap, comparable energy, and the right symmetry. The third is absolute — where symmetry forbids it, no amount of the other two helps.

How the integral is computed here

Every overlap on this site is evaluated numerically on a Gauss–Legendre product grid, and validated where a closed form exists.

The check is worth describing because it is the kind that catches real errors. The numerical result for two 1s orbitals is compared against the exact expression at five separations from 0.8 to 4 bohr, and the worst difference is about six parts in a hundred thousand. That establishes the quadrature is adequate before it is used on the cases with no closed form.

The alternative — trusting a numerical integral because it looks reasonable — is exactly how a plausible wrong number gets into a figure. An overlap of 0.7 and an overlap of 0.75 look equally sensible on a page.

Where the model stops

Three limits, and the first is the standing one.

These are hydrogenic orbitals, and a many-electron atom has none that are exact. Real atoms’ valence orbitals are contracted by the nuclear charge and distorted by the other electrons, so the overlaps here have the right form and are not the numbers for any particular element.

Overlap is a one-electron quantity. It says how two basis functions relate; it does not include electron–electron repulsion, which is a large part of what determines a real bond energy.

Two orbitals is a simplification. A real molecule mixes many orbitals at once, and the two-orbital picture is the first term of something larger. It is a very good first term.

Where the idea came from

The overlap integral in this role is Mulliken’s and Hund’s, in the late 1920s, as molecular orbital theory took shape. The competing valence bond approach of Heitler, London and Pauling used overlap too, in a different formal role, and the two descriptions turn out to be closely related.

Pauling’s The Nature of the Chemical Bond (1939) made overlap a household idea among chemists, including the principle of maximum overlap: bonds form in the directions that maximise it. That principle is a good heuristic and it is not a theorem, for the reasons in the section above — overlap is one term in a balance.

Reading a curve

The overlap-against-separation plot repays a closer look, because three separate facts are visible in it.

Overlap against separationHow the overlap integral falls as two atoms are pulled apart, for several pairs of orbitals. Where a closed form exists it is drawn over the computed curve, so the integrator is checked rather than trusted.1234561s-1ssigma2pz-2pzsigmadots: the closed formoverlap Sseparation / bohrintegrated numerically, checked against the exact resultone electron
Fig. 6 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.

Overlap falls monotonically with distance, roughly exponentially, because both wavefunctions decay exponentially and their product decays about twice as fast.

Compact orbitals give short-range interactions. A 1s orbital of hydrogen has an appreciable overlap only within a few bohr; a 3d orbital reaches much further. That is the geometric reason core orbitals do not participate in bonding — quite apart from their energies, they are too small to reach a neighbouring atom.

The pi curve crosses the sigma one. At very small separations the side-by-side arrangement can overlap more than the end-on one, because the end-on lobes have already passed through each other. At real bond distances the ordering is the familiar one, and the crossing is a reminder that “sigma is stronger than pi” is a statement about a range rather than a law.

None of that required a calculation of a molecule. It is geometry of the atomic functions, and it is the part of bonding that can be understood before any energy is evaluated.

Where the ladder goes next

The case where the integral vanishes for a reason is exactly zero, which is the sharpest statement in this field.

The two frameworks that use overlap differently are molecular orbital and valence bond theory.

What the pictures here cannot show. The overlap figures draw a plane through both nuclei, and the integral is over all space — so what is shown is a slice of what is being computed. The number printed is the full three-dimensional integral, which no two-dimensional drawing can display.

Where this goes next

1s · 2pxS = 0 exactlycomputed -9.6e-17 — arithmetic noiseseparation 2.8 bohrcontours at 50% of each densitythe contributions cancel in pairs, so the integral vanishesone electron Exactly zero Where symmetry forbids an interaction the overlap is not small. It is zero — and computing it and finding arithmetic noise is a different kind of statement from computing it and finding a small number. 1s-1sσ overlapS = 0.39002pz-2pzσ overlapS = 0.45852px-2pxπ overlapS = 0.75291s-2pxsymmetry forbids itS = 0 exactlyno splittingoverlaps computed at 2.8 bohr, levels in proportionone electron Molecular orbital and valence bond Two frameworks, taught as rivals, describing the same molecules. One starts from delocalised orbitals and localises; the other starts from localised bonds and delocalises. Pushed far enough they meet. 2pzencloses 90% of the densitycontour at |ψ| = 9.48e-30 radial nodes1 angular nodecontour solved for by integrating the density1 shell · one electron What an orbital is Not a region the electron occupies, not a path it follows, and for any atom but hydrogen not an exact anything. An orbital is a one-electron wavefunction, and almost every difficulty in this subject comes from forgetting that.