Molecular Geometry & Bonding
what the shape decides
Bonding models

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.
12 min read 6 figures Exactly zeroSymmetry settles itCounted, not quoted

Overlap decides how strongly two orbitals interact. Put a 1s orbital on one atom and a p orbital on its neighbour, oriented perpendicular to the line joining them. Compute the overlap.

The answer comes out at about 10−1710^{-17} — a number with no significant digits in it.

1s 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.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
Fig. 1 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.

That is not a small number. It is zero, expressed in double-precision floating point, and the difference between those two readings is the whole of this essay.

Why it vanishes

The s orbital is symmetric about the plane containing the internuclear axis. The p orbital is antisymmetric about it: positive on one side, negative on the other, and identical in magnitude.

Their product is therefore antisymmetric about that plane. Every volume element above it is matched by one below with the same magnitude and the opposite sign, and the integral of an odd function over a symmetric region is zero.

Not approximately zero. Zero, in the same way that ∫−11x dx\int_{-1}^{1} x\,dx is zero — because the contributions cancel in pairs, term by term, with nothing left over.

What “non-bonding” means

Chemistry has a word for this situation and the word is often used loosely.

A non-bonding interaction is one where the overlap vanishes by symmetry. The two orbitals do not mix at all: neither combination is stabilised, neither is destabilised, and the resulting molecular orbital is unchanged from the atomic one.

That is a stronger statement than “weakly bonding”. A weak interaction has a small non-zero overlap, mixes a little, and shifts the levels slightly. A non-bonding one does nothing whatever, and pushing the atoms closer does not help — the cancellation is exact at every separation.

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. 2 Splitting drawn in proportion to overlap. The pairs that overlap split; the pair that symmetry forbids sits exactly where it started, which is what non-bonding looks like on an energy diagram.

Why “exactly” is worth insisting on

A reader might reasonably ask what turns on the distinction, since a very small interaction and no interaction have similar consequences.

Three things.

It does not depend on the numbers. A small overlap depends on the orbital exponents, the separation and the element. A symmetry-forbidden one is zero for every element at every separation, because it follows from the shapes rather than from the sizes.

It cannot be tuned away. Making the atoms closer, the orbitals more diffuse, or the environment different changes a small overlap and leaves a zero one at zero. The only way to make such an interaction happen is to break the symmetry that forbids it.

It licenses a different kind of argument. From “S is small” one can conclude “the effect is probably minor”. From “S is zero by symmetry” one can conclude “this term is absent”, and absent terms are what make selection rules and orbital-symmetry arguments possible.

The general rule

The specific case generalises, and the general form is the foundation of a large part of chemistry.

An integral over all space vanishes unless its integrand is totally symmetric under every operation of the relevant symmetry group. If any operation changes the sign of the integrand, the integral is zero, because the operation maps the region onto itself and pairs each contribution with its negative.

That single statement gives:

  • Which orbitals may combine, which is this essay.
  • Which spectroscopic transitions are allowed, since a transition moment is an integral of the same kind.
  • Which molecules may be polar, since a dipole moment is an integral of a vector quantity — and that has its own essay.
  • Which vibrational modes appear in an infrared or Raman spectrum.

All four are the same theorem wearing different clothes, and none of them requires computing anything.

The forbidden cases, listed

For two atoms on the z axis, with the standard orientations, the vanishing pairs are worth having explicitly.

2pz 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.2pz · 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
Fig. 3 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.

An s orbital with a pxp_x or pyp_y: forbidden. A pzp_z with a pxp_x or pyp_y: forbidden. A pxp_x with a pyp_y: forbidden. In every case the two functions have different symmetry about a plane containing the axis, and the product is odd about it.

By contrast s with s, s with pzp_z, pzp_z with pzp_z and pxp_x with pxp_x are all allowed, and all give non-zero integrals that fall smoothly with separation.

Reading a computed zero

The figures on this site print S=0S = 0 exactly for the forbidden cases, and report the computed value alongside — a number of order 10−1710^{-17}, whose digits are quadrature noise and change with the grid.

That number is worth explaining rather than hiding. Double-precision arithmetic carries about sixteen significant digits, so a quantity that should be zero and is built from sums of terms of order one comes out at around 10−1610^{-16} or 10−1710^{-17}. Getting that is the correct outcome; getting 10−610^{-6} would indicate a real error somewhere, and getting exactly 0.00.0 would be suspicious, since it would suggest the terms were never computed.

The site’s gate asserts the magnitude is below 10−1210^{-12}, which is far below anything physical and far above the noise floor. And it asserts the converse: an allowed overlap declared forbidden must fail, so the check cannot pass by being permissive.

The same theorem in four places

The rule this essay is an instance of is worth writing out in its general form, because seeing four apparently different chemical rules collapse into one is the sort of thing that makes a subject smaller.

The rule. An integral over all space vanishes unless the integrand is unchanged by every operation of the symmetry group. If any operation reverses its sign, the operation maps the integration region onto itself and pairs each contribution with its negative, so the total is zero.

Four consequences:

Orbital mixing. The integrand is ψAψB\psi_A \psi_B. Different symmetry means the product changes sign, so the overlap vanishes and the orbitals cannot mix — this essay.

Dipole moments. The integrand is the charge density weighted by position. Position is a vector, so any operation reversing a direction kills that component — and only a few groups leave any direction alone.

Electronic transitions. The integrand is the initial state, the dipole operator and the final state. That is the selection rule for absorption and emission, and it is why some transitions are strong and others invisible.

Vibrational activity. Whether a mode appears in an infrared or Raman spectrum is the same integral with a different operator.

What the group settlesFor each molecule, the point group found from its coordinates and the two properties that follow from the group alone. Neither column required knowing anything about the bonds.moleculegroupmay be polarmay be chiralwaterC2vyesno2 σcarbon dioxideD∞hnonohas iammoniaC3vyesno3 σmethaneTdnono6 σboron trifluorideD3hnono4 σhydrogen peroxideC2yesyes—bromochlorofluoromethaneC1yesyes—both columns derived from the symbol, not from the bonds
Fig. 4 One of the four, tabulated. The polarity column is the same theorem as the vanishing overlaps in this essay, applied to a different integrand — and neither required computing anything about the bonds.

Four rules that look unrelated in a first course, and one argument underneath all of them.

What non-zero does not guarantee

The converse of the rule deserves stating because it is weaker than people assume.

Symmetry can say an integral must vanish. It cannot say an integral must not. An interaction that symmetry permits may still be negligible — if the overlap is tiny, or the orbitals are far apart in energy, or the geometry is unfavourable.

So “symmetry allowed” means “not forbidden”, which is a much weaker statement than “happens”. The energy-matching condition is what decides whether an allowed interaction is actually important, and it is a quantitative matter rather than a symmetry one.

That asymmetry — clean exclusion, weak inclusion — is characteristic of symmetry arguments everywhere. It is why they are so useful for ruling things out and so unhelpful for predicting magnitudes, and why a chemist reaching for symmetry is usually trying to eliminate possibilities rather than to choose among them.

Breaking the symmetry

The rule has an obvious escape and it is chemically important.

Bend the molecule, substitute an atom, apply a field — anything that removes the symmetry plane — and the integral is no longer forced to vanish. The interaction turns on, usually weakly at first and growing with the distortion.

That is the mechanism behind a large family of effects. Vibrations that lower the symmetry make forbidden electronic transitions weakly allowed, which is why some formally forbidden absorptions are seen faintly. Substituting a hydrogen for a methyl group in an aromatic ring turns on couplings that were zero in the parent. And a whole class of reaction pathways proceeds by distorting a molecule until an orbital interaction that was forbidden becomes allowed.

So “forbidden” means forbidden in that symmetry, and much of chemistry is about what happens when the symmetry goes.

Seeing the cancellation

The figures on this page draw the plane containing both nuclei, with the regions where the product of the two wavefunctions is positive and negative shown faintly. For a forbidden pair those regions are mirror images.

2s 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.2s · 2pxS = 0 exactlycomputed 9.7e-17 — arithmetic noiseseparation 2.8 bohrcontours at 50% of each densitythe contributions cancel in pairs, so the integral vanishesone electron
Fig. 5 An s orbital of a different shell against a perpendicular p: forbidden for exactly the same reason as the 1s case. The radial part is quite different and the symmetry argument does not care, which is the point — the vanishing depends on the shapes, not the sizes.
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. 6 An allowed interaction for contrast: two p orbitals pointing at each other along the axis. Nothing cancels, the integral is substantial, and the value printed is the number the same routine produced.

The comparison is worth making deliberately. The two figures were produced by the same code, integrating the same way over the same grid. One returns 0.78 and one returns 10−1710^{-17}, and the difference is entirely in the symmetry of the integrand.

That is the strongest form of the argument available without a proof: the same machinery, applied to two cases, giving a substantial number and arithmetic noise. If the vanishing were an artefact of the method, the method would not produce sensible numbers for the allowed cases — and the closed-form check establishes separately that it does.

Where the model stops

Two limits.

Overlap is not the whole interaction. Zero overlap means the two basis functions do not mix directly. In a real many-electron molecule there are indirect routes — mixing through a third orbital, or electron correlation — and “non-bonding” in a one-electron picture does not always mean “no interaction at all”.

The symmetry has to be exact, and a real molecule’s symmetry holds only to a tolerance. A molecule that is nearly symmetric has nearly-zero integrals, not zero ones, and how nearly matters. Real molecules vibrate, so a symmetry that holds at the equilibrium geometry is broken instantaneously most of the time.

Where the ladder goes next

The quantity itself is overlap, and the two frameworks that use it are molecular orbital and valence bond theory.

The same theorem applied to a different integral gives why symmetry forbids a dipole, which is the cleanest example of a property decided without any calculation of the bonding at all.

What the pictures here cannot show. These figures draw a plane through both nuclei, and the cancellation being described is between the half-space above that plane and the half-space below it. A single slice cannot show two half-spaces cancelling; the number printed is the three-dimensional integral, and the drawing is an aid to seeing why it comes out as it does.

Where this goes next

1s · 1sS = 0.38997sigma interactionseparation 2.8 bohrcontours at 50% of each densitythe signed product integrated over all spaceone electron 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. moleculegroupmay be polarmay be chiralwaterC2vyesno2 σcarbon dioxideD∞hnonohas iammoniaC3vyesno3 σmethaneTdnono6 σboron trifluorideD3hnono4 σhydrogen peroxideC2yesyes—bromochlorofluoromethaneC1yesyes—both columns derived from the symbol, not from the bonds Symmetry forbids a dipole Whether a molecule can have a dipole moment follows from its point group alone. The usual argument — adding up bond vectors — gets the right answer for easy cases by a route that does not generalise. 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.