A double bond is not two single bonds

Bond energies for carbon, in kilojoules per mole: single 348, double 614, triple 839.

If a double bond were two single bonds it would be 696, and if a triple were three it would be 1044. Neither number appears. The ratios are 1 : 1.76 : 2.41, and the deficit grows as the bonds are added.

The two integrals #

A double bond between two carbons is made of two different interactions between different orbitals.

The σ component is the head-on overlap of two 2p orbitals pointing at each other along the internuclear axis. Their lobes meet directly, and the region of overlap is on the line between the nuclei.

The π component is the side-on overlap of two 2p orbitals perpendicular to that axis. Their lobes are parallel, the overlap is in two regions above and below the axis, and nothing of it lies on the internuclear line at all.

Those are different integrals over different functions in different regions of space. At carbon’s screened charge and the C=C bond length of 1.34 ångström — 2.53 bohr — they come out at

$$|S_\sigma| = 0.325, \qquad S_\pi = 0.270.$$

Comparable, not equal, and neither is twice the other. There is no reason they should be: the only thing they have in common is the pair of atoms.

Why they fall off differently #

The distance dependence is the more useful difference, because it is what makes the two components behave differently in every situation a chemist cares about.

The σ overlap is large at short range and falls quickly as the atoms separate, and it changes sign: pointed lobes that overlap constructively at contact overlap destructively once the far lobes dominate. The π overlap is smaller at short range, falls more slowly, and does not change sign in the range drawn.

Two consequences follow directly.

Shorter bonds favour σ disproportionately. Compressing a bond raises the σ overlap faster than the π, which is part of why a triple bond is shorter than a double and a double shorter than a single, and why the shortening is not proportional to the number of components.

Longer conjugated systems are held together by π at long range. The overlap that survives out to larger separations is the side-on one, which is why π conjugation runs along a chain of atoms that are individually held together by σ bonds.

Why the energies cannot add #

Even if the two overlaps were equal, the energies would not add, and the reason is the antibonding level goes up more.

A bond’s energy is not proportional to its overlap. It comes out of a secular determinant in which the overlap appears in a denominator, so the stabilisation of a bonding level goes as $\beta/(1+S)$ — sub-linear in the interaction and saturating as $S$ grows. Two interactions of the same size do not give twice the stabilisation of one, because the second one is competing for the same electron density in the same region.

There is a second, larger reason. The bonds being compared are not the same length. A C–C single bond is 1.54 ångström and a C=C double bond is 1.34. Subtracting 348 from 614 to get “the π bond energy” compares a σ bond at 1.34 with a σ bond at 1.54 and attributes the whole difference to π, which it is not: a σ bond compressed by 0.2 ångström is stronger than one that is not, and some of the 266 is that compression.

So the familiar figure of about 266 kJ/mol for a π bond is a difference between two molecules rather than a property of anything. It is useful as a bookkeeping quantity, in the same way an average bond enthalpy is useful, and it is not a measurement of a π bond.

The same failure, three bonds along #

The triple bond makes the point harder to avoid, because the deficit grows.

Three components — one σ and two π — give 839 kJ/mol against 1044 for three single bonds, so the shortfall is 205 where the double bond’s was 82. The third contribution is worth less than the second, which was worth less than the first.

That ordering is what a saturating interaction produces. Each added component acts on the same pair of atoms, in the same region, competing for the same density; the first has the region to itself and the third does not. Nothing in the geometry changes between them — the two π components of a triple bond are perpendicular to each other and to the axis, and their overlaps are identical by symmetry — so the falling increments cannot be a geometric effect.

There is a second effect running the same way. Each added component shortens the bond, and shortening raises the repulsion between the two nuclei and between the core electrons. Some of the shortfall is that cost, and none of it is visible in an overlap integral.

What the model gets right #

The picture does earn several predictions, and they are worth separating from the one it does not support.

Rotation about a double bond is restricted and about a single bond is not. The σ overlap is unchanged by twisting, because it is cylindrically symmetric about the axis; the π overlap goes as the cosine of the twist angle and vanishes at ninety degrees. So a double bond has a rotational barrier and a single bond has essentially none, and the barrier is roughly the π contribution.

A twisted double bond is a diradical. At ninety degrees the two p orbitals are orthogonal and the π interaction is exactly zero — the same kind of exact zero as exactly zero computes for a symmetry-forbidden overlap. The two electrons then sit in two degenerate non-interacting orbitals, which is what makes the transition state of a cis–trans isomerisation what it is.

Conjugation is a π phenomenon. Because the π overlap survives at longer range and is unaffected by rotation about the σ framework, a planar chain of alternating bonds has a continuous π system while its σ bonds remain local. Every result in Hückel theory and what it gets right depends on that separation being clean enough to treat the two independently.

The rotational barrier is the one place a π energy can be measured. Twisting an alkene by ninety degrees switches off the π interaction and leaves the σ framework, so the barrier is a much cleaner estimate of what the π component is worth than any subtraction of bond enthalpies. For ethene it is about 270 kJ/mol, which happens to land close to the 266 the subtraction gives — a coincidence worth noticing rather than a confirmation, since the twisted geometry is not the σ bond of ethane either.

Bond order counts, and does not weigh. The computed π bond order in benzene is exactly 2/3 on every bond, and adding the σ framework gives a total of 1.67 — which correctly predicts that benzene’s bonds are between single and double in length, at 1.397 ångström. What it does not predict is the energy, and it is not trying to.

What the additive scheme is actually for #

None of this makes tabulated bond enthalpies useless, and it is worth being clear about what they are.

They are a fitting scheme. A large set of measured heats of formation is reduced to a small set of numbers by assuming additivity, and the numbers are whatever makes the sum work best across the set. Used to estimate an unmeasured heat of formation they are accurate to perhaps twenty kilojoules per mole, which is often enough to answer the question being asked.

What they are not is a set of measurements of individual bonds. No experiment isolates a C–C bond and weighs it. The number 348 is a regression coefficient, and asking what a π bond is worth by subtracting two of them is asking a regression to answer a question about a mechanism.

The distinction matters most exactly where the additive scheme fails hardest, which is where the interesting chemistry is: aromatic systems, strained rings, conjugated chains. Benzene’s famous stabilisation is defined as the failure of the additive scheme, and the range of values quoted for it — 120 to 180 kJ/mol — is the range of reference states people have chosen, as delocalisation sets out. A quantity whose value depends on what it is compared with is a comparison, not a property.

What was computed, and how #

Quoted: the bond energies 348, 614 and 839 kJ/mol, and the lengths 1.54, 1.34 and 1.20 ångström. Those are measurements, and this site quotes measurements rather than pretending to compute them.

Computed: every overlap integral, by Gauss product quadrature over a fourteen-bohr box at ninetieth order, cached between builds and re-verified at fortieth order on every restore. The integrator is validated against the closed form for two 1s orbitals to six parts in a hundred thousand, and against the requirement that a symmetry-forbidden overlap come out at arithmetic noise rather than merely small.

The screened charge matters here and is stated. The bare hydrogenic 2p orbital has a mean radius of five bohr, which is twice the C=C bond length: two of them overlap almost completely at any separation drawn, and the comparison would say nothing about carbon. Drawn at the charge Slater’s rules give a carbon 2p electron — 3.25, as computed in what an electron actually feels — the orbital is a bond-length-sized object and the comparison means something. Every figure states the charge it was drawn at, and the orbital index lists both.

Nothing here is a bond energy. The overlaps are pure numbers and the level splittings are in units of an unfitted parameter. Converting either into kilojoules would require a calibration not made here.

Where the model stops #

Two orbitals at a time. A real double bond involves the whole valence shell of both atoms, and the σ component in particular is not a bare 2p–2p interaction: it includes s character, which is what makes it directional in the first place.

The orbitals are hydrogenic with a screened charge, which is a caricature of an atomic orbital in a molecule. The comparison between σ and π is meaningful because both are caricatured the same way; the absolute values are not quotable.

Nothing here says anything about the σ–π separation being exact. It is not. The separation is exact only for a planar molecule with a well-defined mirror plane, where σ and π belong to different symmetry species and cannot mix. Twist the molecule and they mix, which is what makes the rotational barrier finite rather than infinite.

The σ and π components are not independent even in a planar molecule. They belong to different symmetry species and so do not mix, which makes the separation exact in the sense that matters for a symmetry argument. It does not make them energetically independent: both are built from the same electrons on the same atoms, and the total energy is not a sum of two terms one of which can be varied while the other is held.

And the additivity that fails here fails for other reasons too. Bond enthalpies are averages over many molecules and are not properties of a bond at all; the standard tables carry uncertainties of several kilojoules and disagree with each other. Building an argument on the difference of two of them is building on a difference of two averages.

The generalisation #

The failure here belongs to a family: a quantity is defined for a simple case, given a name, and then used as though it were additive.

Bond energies are the case in this essay. Electronegativity is not one quantity is the same failure applied to a scale. The dipole is not a sum of bonds is the same failure applied to a vector, and it fails for the same underlying reason: the parts being added were never independent.

The test is cheap and worth applying everywhere. Take the quantity, compute it two ways, and see whether the answers agree to the precision the argument needs. Here two ways of getting a double bond’s strength — twice a single, or a single plus a π — give 696 and 614, and the measurement is 614 because it was defined that way.

What survives is what was computed rather than assembled: an overlap integral is a definite number about two definite functions, and every claim in this essay that rests on one is safe. Every claim that rests on adding two of them is not.

The same arithmetic one row down, where it fails #

The π component being a different integral with a different distance dependence is a claim that can be tested by changing the distance — and the periodic table supplies the test, because the same two bonds exist for silicon at a longer separation.

Carbon’s numbers are the ones above: a single bond at 348 kilojoules a mole and a double at 614, so the second component is worth 266.

Silicon’s single bond is about 226 and its double about 315, so its second component is worth roughly 90.

The single bonds differ by a third and the π components differ by a factor of three. That is the essay’s claim in its strongest form: the two halves of a double bond are not two of anything, and they respond quite differently to the same change.

The reason is geometric and is exactly the distance dependence. A σ interaction is head-on, so lengthening the bond weakens it steadily; a π interaction is side-on, between two orbitals whose lobes are already offset from the internuclear line, and stretching the bond pulls the lobes apart much faster. A silicon–silicon bond is 2.35 ångström against carbon’s 1.54, and the sideways overlap has fallen far more than the head-on one.

The chemical consequence is well known and was for a long time stated as a rule rather than explained. Silicon forms no stable double bonds under ordinary conditions — nor do phosphorus, sulfur or the heavier elements generally — and the first compound with a genuine silicon–silicon double bond was not isolated until 1981, protected by substituents bulky enough to stop it polymerising.

Carbon’s chemistry is built on double and triple bonds; silicon’s, with the same valence and the same four bonds, is built almost entirely on single ones. The difference is one integral evaluated eight tenths of an ångström further out.

The comparison also says why the rule was ever stated as a rule about the second row. Nothing forbids a heavier element from making a π bond — the disilene exists, and so do compounds with phosphorus–phosphorus and germanium–germanium double bonds — and what the arithmetic says is that such a bond is weak, not that it is impossible. A weak bond in a reactive molecule is a bond that does not survive meeting another molecule, which is why the compounds needed bulky protecting groups rather than a new kind of bonding, and why they were made once somebody thought to shield them rather than once somebody solved a theoretical problem.

Who found it, and when #

The σ and π classification is Mulliken’s, from the early 1930s, and the labels are his — Greek letters chosen to match the s and p of the atomic orbitals they most resemble in symmetry about the axis.

That double bonds are not twice single bonds was known long before any of the orbital theory: thermochemical measurements through the nineteenth century established the additivity scheme and its failures together, and Pauling’s discussion of bond energies in The Nature of the Chemical Bond is explicit that the additive scheme is an approximation with known deviations.

The restricted rotation about a double bond has an older and more visible history still: the existence of cis and trans isomers of but-2-ene was the phenomenon van 't Hoff’s 1874 tetrahedral carbon was partly proposed to explain, half a century before there was an orbital to attribute it to.

Still open: what overlap cannot answer #

Four essays on overlap start from one integral and end here, at what happens when two of them are added by mistake. The overlap has now been computed, checked, screened, and taken to the limit where it vanishes exactly — which leaves the questions overlap cannot answer, and those need a model in which the electrons stop being one at a time.