Bond order from the eigenvectors

A Kekulé structure gives every bond in benzene an order of one and a half, and gives every bond in naphthalene either one or two depending on which structure is drawn. Neither answer is right, and the second is not even self-consistent.

The eigenvectors settle it, and they cost nothing extra: the matrix has already been diagonalised for the energies.

What the quantity is #

For occupied orbital $k$ with coefficient $c_{ki}$ on atom $i$, the pi bond order between atoms $i$ and $j$ is

$$p_{ij} = \sum_k n_k\, c_{ki}\, c_{kj},$$

summed over occupied orbitals with $n_k$ the occupation. The charge density on atom $i$ is the same sum with both coefficients on the same atom:

$$q_i = \sum_k n_k\, c_{ki}^2.$$

Both are one line of arithmetic over numbers the eigensolver has already produced.

The interpretation is worth stating carefully. $p_{ij}$ is not a count of electron pairs; it is a measure of how much the occupied orbitals reinforce between two atoms, and a full pi bond in ethene gives exactly one. The total bond order in a conjugated system is conventionally quoted as $1 + p_{ij}$, the one being the sigma bond that the treatment never looked at.

Benzene, where the answer is uninformative #

Every one of benzene’s six bonds comes out at exactly two thirds.

That is the right answer and it is weak evidence for anything, because it could not have come out otherwise. Benzene’s sixfold symmetry carries every bond onto every other, so any quantity defined on a bond has to take one value. A theory that got this wrong would be broken in a way visible without the theory.

It is worth noticing the exact figure, though. Two thirds, not one half — the Kekulé averaging that gives “one and a half bonds” corresponds to $p = 0.5$, and the computed value is a third larger. The two accounts differ, the difference is not small, and only one of them is derived.

Naphthalene, where the answer is a test #

Naphthalene has three symmetry-distinct kinds of carbon–carbon bond, and they come out at 0.725, 0.603 and 0.518.

Read those against the structure. The largest is the bond between the two carbons adjacent to a ring fusion — the 1,2 bond in the usual numbering. The middle value belongs to the 2,3 bond across the top of a ring. The smallest belongs to the bond shared between the two rings.

The measured bond lengths run in the opposite order, as they must: 1.371, 1.412 and 1.420 ångström. Higher bond order, shorter bond. The correlation is monotonic and it was not built in — nothing in the calculation knows what a bond length is.

Now compare the naive account. Naphthalene has three Kekulé structures. Averaging them with equal weight gives the 1,2 bond a double-bond character in two of the three, the 2,3 bond in one of three, and the shared bond in one of three. That predicts the 2,3 and the shared bond to be identical, and they are not — by 0.085 in computed order and by 0.008 ångström in measured length.

So the eigenvector calculation and the structure-counting calculation make different predictions about a specific pair of bonds, and the measurement agrees with the eigenvectors. That is what a test looks like.

Alternation, in a chain that has no reason to alternate #

Butadiene is the smallest system in which the calculation says something the structural formula does not.

The drawn structure implies orders of 1, 0 and 1 for the pi contributions. The computation gives 0.894, 0.447 and 0.894. Both outer bonds are weaker than a full double bond and the central one is very far from a single bond — half of a pi bond’s worth of order sits where the formula puts none.

That is conjugation stated as a number. The measured lengths follow: 1.34 ångström for the outer bonds against 1.33 in ethene, and 1.47 for the central bond against 1.54 in ethane. Both are shifted toward each other, in the directions and roughly in the proportions the orders give.

Extending the chain further, the alternation continues to narrow, and the limit of an infinite chain of equally spaced carbons would have every bond at the same order. Real polyacetylene does not reach that limit: it alternates, for a reason the theory cannot supply, because escaping a uniform chain requires moving the atoms and the geometry was discarded at the first step.

Charge density, and the theorem that makes it dull #

The same sum with both indices on one atom gives the charge density, and for an alternant hydrocarbon the answer is always one.

That is Coulson and Rushbrooke’s theorem, and it is a consequence of the pairing property: in a graph whose atoms can be two-coloured with no edge joining two of the same colour, the eigenvalues come in plus-and-minus pairs and the coefficients are related in a way that forces every charge density to unity. Benzene, naphthalene, butadiene, anthracene — all uniform, all trivially.

So charge density is uninformative for exactly the molecules bond order is most interesting in, which is a useful thing to know before spending time on it.

Where it becomes informative is in the ions and the odd systems.

The reason is one eigenvector. Allyl’s middle orbital lies at exactly α — non-bonding — and its coefficients are $(1/\sqrt2,\, 0,\, -1/\sqrt2)$, with zero amplitude on the central carbon. The cation, the radical and the anion differ only in how many electrons occupy that orbital, and an orbital with no amplitude in the middle cannot put or remove charge there.

So the arrow-pushing result — that allyl’s charge sits on the termini — arrives as arithmetic, from a single zero in an eigenvector. The radical, with that orbital singly occupied, comes out uniform at 1.0 everywhere, which is the odd-alternant case of the Coulson–Rushbrooke theorem and is why the radical is the least informative of the three.

What was computed, and how #

Every number on this page comes out of a cyclic Jacobi diagonalisation of an adjacency matrix, and the eigenvectors it returns need one piece of care that the eigenvalues do not.

Degenerate eigenvectors are arbitrary within their subspace. Benzene’s two degenerate pairs can be rotated into one another freely, and the individual coefficients printed for one of them are not meaningful on their own. What is meaningful is any quantity summed over a fully occupied degenerate set, because such a sum is invariant to the rotation — which is exactly what bond order and charge density are.

That invariance is why those two quantities are safe to quote for benzene and why an orbital-by-orbital reading of the coefficients is not. It is the same distinction the localisation transformation turns on: the individual orbitals are a choice, and the sums over an occupied set are not.

Two assertions guard the calculation. Every eigenvector is required to be normalised and every pair orthogonal, checked directly against the matrix rather than assumed from the solver. And a figure claiming uniform bond orders is required to find them uniform to $10^{-9}$ — which is fed naphthalene deliberately in the site’s gate, and must refuse it.

The surprise: a Kekulé average is not a bond order #

The result worth carrying away is negative and it is precise.

Averaging resonance structures with equal weights is a procedure, it produces numbers, and the numbers are not the bond orders. For benzene it gives 0.5 where the calculation gives 0.667. For naphthalene it makes two inequivalent bonds equal.

The reason is structural rather than numerical. Resonance structures are basis functions, and a basis is not a thing; combining them with equal weights is an assumption about the coefficients of an expansion, and there is no reason for the coefficients of a valence-bond expansion to be equal. A proper valence-bond calculation weights them unequally and reproduces the molecular-orbital answer, at which point the two frameworks agree as they always do.

What the equal-weight average is, is a mnemonic. It gets the qualitative picture right — all six benzene bonds alike, naphthalene’s bonds unequal — and it gets the numbers wrong, and the numbers are the part that can be checked.

What it costs #

Two nested sums over the occupied orbitals: for naphthalene, ten atoms and five occupied orbitals, so a few hundred multiplications. Against the diagonalisation that produced the eigenvectors, it is free.

The cost that is real is interpretive, and it is worth pricing because bond order is a quantity people reach for casually.

It is not observable. A bond order is defined within a scheme, and different schemes give different numbers for the same molecule. The Hückel value, a Mulliken population, a Mayer bond order and a natural bond orbital analysis will not agree. Comparing bond orders across methods is comparing quantities that share a name.

It is not a bond length. The correlation between the two is empirical, monotonic and useful, and it is a correlation. A bond order of 0.518 does not convert to an ångström without a fitted relationship.

And it says nothing about strength. Bond dissociation energy is a difference between a molecule and two fragments, and nothing in a ground-state calculation of the molecule sees the fragments.

Three things a number is often taken to mean and does not, which is a fair summary of what a derived quantity costs to use.

Where the model stops #

Three limits.

Pi only. Every number here is the pi contribution. The sigma framework was assumed separable and then ignored, so a “total bond order” of 1.518 for naphthalene’s shared bond is one plus a computed number, and the one is an assumption.

No repulsion, as everywhere in this treatment. The coefficients come from a one-electron model with no electron–electron term at all.

The correlation with length is borrowed. That higher bond order means shorter bond is an empirical regularity supported here by three data points from one molecule. It holds broadly and it is not derived anywhere on this page, and an essay that claimed otherwise would be pretending a correlation was a theorem.

What a bond order does not commit anybody to #

One further caution, because the quantity is unusually easy to over-read and the over-readings are all plausible.

It does not imply a Lewis structure. Benzene’s 0.667 corresponds to no drawable arrangement of double bonds, and that is not a failure of the calculation; it is the point. A bond order is a continuous quantity computed from a wavefunction, and a Lewis structure is a discrete labelling. Asking which structure a bond order of 0.667 corresponds to is asking for an integer where the theory produced a real number.

It does not divide the electrons up. Summing every bond order in benzene gives four, and there are six pi electrons. The sums do not have to match and there is no partition of the density into bonds that would make them — the same objection that defeats the bond-dipole method applies here, and for the same reason: the density is a single function and does not arrive partitioned.

It is not comparable between molecules of different size. Naphthalene’s largest bond order is 0.725 and benzene’s is 0.667, and it would be wrong to conclude that naphthalene has stronger bonds. The two numbers were computed with different numbers of orbitals in the sum.

Each of those is a case of the same thing: a number derived within a scheme being read as though it were a property of the molecule. The general version of that mistake is the subject of a whole field on this site.

Who found it, and when #

Coulson introduced the bond order in this form in 1939, and it was among the first quantities molecular orbital theory produced that chemists could use directly — a number attached to a bond, comparable across molecules, correlating with something measurable.

The Coulson–Rushbrooke theorem on uniform charge densities in alternant hydrocarbons followed in 1940, and it is a good example of a result whose value is in what it forbids: it says that a whole class of calculations will return the answer one, so nobody need do them.

The bond-order-to-bond-length correlation was worked out empirically over the following decade, largely against the growing body of X-ray structures, and naphthalene was one of the molecules that established it. That a theory with no lengths in it predicts the ordering of three lengths correctly remains the cleanest demonstration of what a connectivity determines.

Where the ladder goes next #

The theory the eigenvectors come from is Hückel theory.

The claim about benzene that these numbers make precise is delocalisation.

The count that decides whether a ring is worth having at all is aromaticity as a shell closure.

And the general point about quantities that depend on a description rather than on the molecule is hybrids are a basis.

The habit worth taking from all this is to ask, whenever a bond order is quoted, two questions that are rarely asked together: by what scheme, and against what other bond. Neither answer is usually printed, and without both the number is a decoration. With both it is one of the more useful derived quantities the subject has — a single figure that ranks three chemically distinct bonds in a molecule and gets the ranking right.

What the pictures here cannot show. These figures draw a graph, and the thickness of a bond in them encodes a computed number rather than a distance. Nothing on this page is drawn to scale in any physical sense: the positions of the carbons are a layout convenience, and the measured bond lengths quoted in the text appear nowhere in the drawings because no length was ever computed.