Overlap decides

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.

$$S = \int \psi_A\,\psi_B\,dV$$

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

What it measures #

$S$ 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,

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

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 a numerical integrator be checked rather than trusted.

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 a lobe should be coloured by sign rather than drawn as a bare shape.

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.

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 here 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.

What it costs #

More than any other quantity this simple-looking, and the figure is worth stating because it is what makes the check above possible.

Each overlap is a three-dimensional Gauss–Legendre product rule with ninety points along each axis: seven hundred and twenty-nine thousand evaluations of a pair of wavefunctions, each of which is a Laguerre polynomial times an exponential times a spherical harmonic. On a modern machine that is a fraction of a second, and a comparison across many pairs and separations needs enough of them that storing each result by orbital pair and separation pays for itself.

Two choices inside that deserve naming, because both were made deliberately and either could have been made badly.

Gauss–Legendre rather than a uniform grid. These functions decay exponentially, and a product Gauss rule converges several orders of magnitude faster on such an integrand than an evenly spaced sum with the same number of points. That difference is what makes a grid fine enough to reach $10^{-17}$ on a forbidden case affordable in advance.

Cartesian rather than spherical. A spherical grid is the natural choice for a single orbital and the wrong one here, because the integrand is centred on two different points and no single spherical grid suits both. Choosing the coordinate system that suits neither centre particularly well, but treats them alike, is what keeps the cancellation in a forbidden overlap exact — a grid aligned to one centre would break the symmetry the cancellation depends on and return a small number instead of a vanishing one.

That last point is the one that pays. A forbidden overlap computed this way comes out at around $10^{-17}$, which is arithmetic noise, and it does so because the quadrature respects the symmetry that makes the integrand odd. Had the grid been chosen for convenience the same integral would return something like $10^{-6}$ — small, plausible, and a completely different kind of claim.

Where the overlap criterion stops being the answer #

The heuristic that bonds form where overlap is greatest is used constantly and it is worth knowing where it breaks, because it breaks in a specific and predictable place.

Overlap is one factor in a balance and the other is the energy match between the two orbitals. Two orbitals overlapping strongly but lying far apart in energy interact weakly, because the interaction goes roughly as the overlap squared over the energy gap. Two orbitals well matched in energy interact strongly on a modest overlap.

The consequence is that maximising overlap alone gives wrong answers whenever the partners are badly matched, and the standard case is the d-orbital account of hypervalency: sulfur’s 3d orbitals have a substantial spatial overlap with fluorine’s 2p, which is why that account was believable for fifty years. They lie about ten electronvolts too high to be usefully mixed, and the overlap does not rescue them.

There is a second, quieter failure. Overlap between two orbitals is defined for a chosen pair of orientations, and the maximum-overlap principle silently assumes the orientations can be chosen freely. In a molecule they cannot: the directions are fixed by everything else attached to the atom, and a hybrid that would maximise the overlap in one bond is not available if it is already committed elsewhere. The principle is a statement about an isolated pair, applied to a system where nothing is isolated.

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 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 a computed overlap turns into a level diagram #

The reason to have the number rather than the intuition is that the number places things.

A bonding and an antibonding combination are split apart by an amount that goes with the overlap, so a molecular orbital diagram drawn from computed overlaps has its levels placed rather than sketched — and the case that matters is the one where the overlap is zero, because then the level does not move at all.

Three things follow from reading a diagram that way. A non-bonding level is not “weakly interacting” — it is at the atomic energy exactly. A level’s displacement is a measurable consequence of a computed integral rather than an artistic choice, so two diagrams drawn from the same integrals are comparable. And the asymmetry that real diagrams show, with the antibonding level pushed up further than the bonding one is pushed down, is a consequence of the overlap appearing in the normalisation as well as the interaction — which is a fact about the arithmetic and is why an electron in an antibonding orbital costs more than one in a bonding orbital gains.

That last asymmetry is the reason helium has no molecule. Two electrons in σ and two in σ* is not a wash; it is a net destabilisation, and the size of the effect goes with the same overlap integral that would have made the bond.

It is also, incidentally, why the overlap criterion and the repulsion-minimising account of shape so often agree without either being evidence for the other. Both are dominated by the same geometric fact — that two things cannot occupy the same direction — and a shape that maximises overlap in every bond is generally a shape that keeps the bonds apart. The agreement is over-determined, which is worth remembering whenever two independent-looking arguments reach the same answer about a molecule.

One integral doing three jobs #

The overlap is introduced here as the quantity that decides how strongly two orbitals interact, and it is worth noticing that the same integral appears three times in the machinery of this collection, under three names, with only one of the three connections being a theorem.

It is the non-orthogonality. Two atomic functions on different centres are not orthogonal, and their overlap is exactly the amount by which they fail to be. That is a definition and carries no assumption.

It is the denominator of the secular problem. Solving for the levels of two interacting orbitals requires the overlap matrix, and the asymmetry between the bonding and antibonding levels comes from it — which deserves an argument of its own. That too is exact.

And it is assumed proportional to the coupling. The resonance integral, the quantity that actually sets how far the levels move, is taken to be proportional to the overlap in essentially every semi-empirical treatment in the subject. That one is neither a definition nor a theorem: it is an approximation with a name and a fitted constant, and it is the reason an overlap can be used as a proxy for an interaction at all.

Keeping the three separate matters because only the third can fail, and it fails in a specific way. The overlap is a purely geometric quantity — two functions and a distance — while the coupling involves the potential as well, so the proportionality holds well between similar pairs and less well between pairs on different elements or at very different separations.

Which is why arguments from overlap are built on comparisons within a family rather than on absolute values. Comparing two orientations of the same pair of orbitals at the same distance is a comparison in which the fitted constant cancels; comparing an overlap on carbon with one on sulfur is a comparison in which it does not, and the proportionality is doing work it was never fitted for.

Where to read on #

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.

The last caution is about the word itself. Overlap names an integral, and an integral has a sign; the habit of speaking of “more overlap” as though it were unambiguously good discards exactly the information that separates a bonding combination from an antibonding one.

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.