Hückel with a heteroatom

The remarkable thing about Hückel theory applied to a hydrocarbon is what it does not contain. The matrix is ones and zeroes; α and β never receive numbers; and every result — benzene’s delocalisation energy of exactly 2β, the 4n+2 closures, the uniform bond order of two thirds — is a property of a graph.

Put a nitrogen into the ring and that ends. Two fitted numbers arrive with the atom, and this essay is about what they cost.

The two numbers #

A heteroatom differs from a carbon in two ways that the model can represent. Its own orbital sits at a different energy, and its bonds to carbon have a different strength. Both are written as multiples of the parameters already in play:

$$\alpha_X = \alpha + h_X\beta, \qquad \beta_{CX} = k_{CX}\beta.$$

So a nitrogen becomes a number $h$ on the diagonal of the adjacency matrix and a number $k$ on the two edges that touch it. Everything downstream is unchanged: the matrix is still real symmetric, the same eigensolver applies, and the same bond orders and charges come out of the same sums over coefficients.

The values used here are Streitwieser’s, from the 1960s, fitted to spectra and charge distributions:

The first thing to notice is that nitrogen appears twice with quite different numbers. These are not element parameters. A nitrogen that contributes one electron to the π system, as in pyridine, is a different object from one that contributes two, as in pyrrole, and the distinction is the one most often lost when the tables are used from memory.

Pyridine, computed #

Substituting one nitrogen into benzene’s ring and diagonalising gives levels at

$$2.107,\; 1.167,\; 1.000,\; -0.841,\; -1.000,\; -1.934$$

against benzene’s 2, 1, 1, −1, −1, −2.

Three things have happened. The lowest level has dropped further, because the nitrogen’s own orbital was lower to begin with. The degeneracy of the first excited pair has broken — 1.167 and 1.000 where benzene has two levels at exactly 1 — because the sixfold symmetry is gone. And the whole spectrum has slid: the sum of the eigenvalues is no longer zero but 0.5, which is the trace of the matrix, which is $h$.

The charges are the more chemically useful output. Pyridine’s come out at

$$1.195,\; 0.923,\; 1.004,\; 0.950,\; 1.004,\; 0.923$$

with the nitrogen first and the ring positions following round. The nitrogen has drawn about a fifth of an electron out of the ring, and the positions that lose the most are the two ortho and the one para — 0.923, 0.923 and 0.950. Those are exactly the positions an electrophile avoids in pyridine and the positions a nucleophile attacks, which is a real prediction from a matrix that knows nothing about chemistry beyond one number on a diagonal.

Both readings come off the same eigenvectors. The lowest π orbital has its largest coefficient on the nitrogen, which is what “the nitrogen is more electronegative” means once it has been turned into arithmetic — the low-lying orbital puts most of its amplitude on the low-lying atom — and the charges above are the sum of those squared coefficients over the occupied set. Neither is a new calculation; both are the same six numbers read down a column instead of along a row.

The theorem that stops holding #

Benzene’s levels are symmetric about α: for every level at $+x$ there is one at $-x$. That is the pairing theorem, and it is a genuine result — it holds for any alternant system, meaning any graph whose atoms can be two-coloured with no two same-coloured atoms bonded.

Pyridine’s graph passes that test. It is a six-membered ring, it is bipartite, and the colouring succeeds. And its levels are not paired.

That is not a contradiction. It is the theorem being stated more precisely than usual. The proof of pairing works by flipping the sign of every coefficient on one colour, which turns an eigenvector of eigenvalue $x$ into one of eigenvalue $-x$ — and that transformation preserves the matrix only if every diagonal entry is the same. Put 0.5 on one diagonal entry and the transformation no longer maps the matrix to itself.

Pairing is a property of the matrix, not of the graph. The check that matters runs in the direction that can fail: a substituted system is required to come out unpaired, so arithmetic that had quietly dropped the diagonal entry would be caught rather than produce a symmetric spectrum that looked reassuring.

What the delocalisation energy becomes #

Benzene’s delocalisation energy is exactly 2β against a reference state of three isolated double bonds, and the reference is written into the system’s own definition rather than left implied. That is the repair delocalisation makes to the usual vague figure of “about 150 kJ/mol, depending what one compares it to”.

Pyridine does not get one, and the reason is worth being exact about. The reference state would have to be two isolated C=C double bonds and one isolated C=N — and the energy of an isolated C=N in this model is $2\sqrt{1 + h^2/4}$-ish rather than 2, because the heteroatom’s own α shift enters it. So the delocalisation energy would be a difference between a computed number and a number containing the same fitted parameter, and its value would move with $h$ for reasons that have nothing to do with delocalisation.

The systems here therefore declare no reference and report no delocalisation energy. That is a deliberate refusal rather than an omission: a number that moves with a fitted parameter is not a measurement of anything, and quoting it would undo the discipline that makes benzene’s 2β worth having.

The other three #

Pyrrole and furan are the interesting pair, because they run the opposite way to pyridine. Their heteroatoms donate into the ring: pyrrole’s nitrogen carries 1.72 electrons where a carbon would carry 1, and every carbon carries more than one — 1.03, 1.11, 1.11, 1.03. The ring is electron-rich, which is why pyrrole is attacked by electrophiles far faster than benzene and pyridine is attacked far slower.

Furan pushes it further still, with oxygen’s $h = 2.0$ giving a charge of 1.79 on the oxygen and a slightly less activated ring. The ordering pyrrole > furan for electrophilic substitution comes out of the coefficients, and it is right.

Acrolein — a butadiene with the last carbon replaced by a carbonyl oxygen — is the case where the polarisation is largest: charges of 0.77, 1.03, 0.67 and 1.53, so the oxygen has taken half an electron and the carbon two bonds away from it is the most depleted position in the molecule. That position is the one a nucleophile attacks, which is the whole of conjugate addition drawn out of a four-by-four matrix.

How much of the answer is the parameter #

A fitted model deserves the same test what a lone pair is worth applies to VSEPR’s lone-pair clause: how much does the answer move when the parameter does?

For pyridine’s charges, not very much. Halving $h$ from 0.5 to 0.25 moves the nitrogen’s charge from 1.195 to about 1.10, and doubling it to 1.0 takes it to roughly 1.37 — so the magnitude of the polarisation tracks $h$ almost proportionally. What does not move is the pattern: the ortho and para positions are depleted and the meta positions are not, at every value of $h$ that has been tried, because that pattern comes from the ring’s symmetry and from which coefficients are large in which orbital.

That is the useful split, and it is the same one that keeps turning up. The qualitative prediction — where an electrophile goes — is a consequence of the graph and survives the parameter. The quantitative prediction — how much charge moves — is the parameter, restated.

An account that reports only the second gives the impression of a calculation. An account that reports only the first gives up a real prediction. Reporting both, and saying which is which, is the whole of what this essay is for.

What was computed, and how #

Quoted: the four $h$ and $k$ values in the table above, from Streitwieser’s tabulation. Different tables differ by enough to move a level by a tenth of a β.

Computed: everything else. The matrices are built by substituting into the parent hydrocarbon’s own definition, so pyridine is benzene-with-one-atom-changed in the calculation as well as in the argument, and the three entries that differ are the three entries the essay is about.

The checks that a hydrocarbon system passes are all re-run, and two of them had to be generalised to survive the change. The trace relations were written as “the eigenvalues sum to zero” and “their squares sum to twice the bond count”, which are the values those traces take for a matrix of ones and zeroes; they are now read off the matrix itself, so they still say something when the diagonal is occupied and the bonds are not all one. That generalisation is worth recording because the un-generalised version would have passed for every hydrocarbon and failed for every heteroatom system — a check that stops applying exactly when a new feature is introduced is worse than no check.

The pairing check is the other one, and it branches: an alternant hydrocarbon must pair, a non-alternant one must not, and a heteroatom system must not. All three directions are checked, so each of the three claims can fail.

Whether the parameters could be computed instead #

Two fitted numbers arriving with an atom is a cost, and the natural question is whether it has to be paid. The parameter $h$ is supposed to say that a heteroatom’s orbital sits at a different energy from carbon’s, and orbital energies of atoms are measured. So the obvious repair is to source $h$ from an atomic measurement rather than from a fit to molecular spectra.

The measurement to use is the energy of the relevant valence orbital of the atom — the quantity a photoelectron measurement returns for the electron that will be donated to the π system. The standard values are about $-11.4$ electronvolts for carbon’s 2p, $-13.4$ for nitrogen’s and $-14.8$ for oxygen’s. So a nitrogen’s orbital sits 2.0 electronvolts below carbon’s and an oxygen’s 3.4.

If $h$ were that difference divided by $\beta$, the ratio of the two heteroatoms’ parameters would be fixed at $3.4/2.0 = 1.70$, whatever $\beta$ turned out to be. The table gives $1.0/0.5 = 2.00$ for the one-electron cases and $2.0/1.5 = 1.33$ for the two-electron ones. One atomic ratio, two tabulated ratios, and the atomic one lies between them.

Running it the other way is more damaging. Each row of the table implies a value of $\beta$, by dividing the atomic energy difference by the tabulated $h$:

Four values spanning a factor of three, from a table whose whole purpose is to be used with one $\beta$.

So the parameters cannot be sourced from atomic energies, and the arithmetic says why with more force than the observation that nitrogen appears twice. A quantity that would be an atomic property if the model were what it claims to be comes out different by a factor of three depending on which molecule it was fitted to. What $h$ actually absorbs is everything the model left out — the σ framework’s response, the formal charge a two-electron donor carries, the fact that a nitrogen donating a lone pair is a nitrogen with a positive formal charge and therefore a much lower orbital energy than a neutral one.

There is a family of methods that takes the atomic route anyway, and what it pays is instructive. Extended Hückel theory puts measured valence orbital energies on the diagonal directly, which removes $h$ entirely and forces $\beta$ to become a real number in electronvolts rather than a symbol. The off-diagonal elements then need a formula relating them to the overlap between the two orbitals, and that formula carries a fitted constant of its own.

Which is the ordinary outcome. The parameter does not disappear; it moves, from a table of per-atom values to a single constant in an expression, and the trade is a smaller number of fitted quantities against the loss of every result that was previously a property of a graph. Plain Hückel theory gets benzene’s degeneracies exactly and its delocalisation energy exactly in units of an unfitted $\beta$; a method with real energies on its diagonal gets none of those exactly and gets a number for everything.

The choice between them is therefore not a choice between a crude method and a better one. It is a choice about which kind of statement is wanted. A graph with no numbers in it can say that two levels are exactly degenerate, that a stabilisation is exactly twice a parameter, and that a shell closes at a particular count — statements that are either right or wrong and cannot be nearly right. A method with measured energies on its diagonal says that a level is at $-11.4$ electronvolts, which is a statement that can be nearly right and can never be exactly right. Plain Hückel theory makes only the first kind, and a heteroatom is where the second kind arrives, two numbers at a time.

Where the model stops #

Two fitted numbers are two more than none. The value of the whole hydrocarbon exercise was that no quantity in it had been tuned. That is gone here, and every result in this essay is contingent on a table published sixty years ago.

The parameters are not transferable in the way the table implies. Pyridine’s nitrogen and pyrrole’s nitrogen already need different numbers; a nitrogen in a five-membered ring next to a carbonyl arguably needs a third. The literature contains several inconsistent tabulations, and the standard advice — use whichever set the comparison being made was built on — is an admission.

Charges here are π charges only. The σ framework is untouched, and a real molecule’s charge distribution has both. Pyridine’s dipole moment is dominated by the nitrogen lone pair in the σ plane, which this model does not contain at all, so the π charges above should not be added up and called a dipole — the mistake the dipole is not a sum of bonds is about.

The parameters were fitted to observables this model then predicts. Streitwieser’s values came in part from ultraviolet spectra, which are transition energies between the very levels computed here. Reproducing those spectra is therefore not evidence; it is the fit closing on itself. What counts as evidence is a prediction of a different kind of quantity — the substitution pattern, say — and those are the predictions this essay leans on.

No geometry enters. The nitrogen’s shorter bonds are not represented; only a different $k$ on them is, and $k$ is a parameter rather than a length.

The generalisation #

The pattern worth carrying is what happens to a model’s status when a parameter is admitted.

Before the substitution, Hückel theory made statements that were true of a graph and could not be adjusted: the 4n+2 closures come out of the filling and would come out of it whatever numbers α and β took. After the substitution, it makes statements that depend on a fit, and the honest report of those statements has to name the fit.

The same boundary runs through this whole site. Symmetry conclusions are exact and are stated without hedging. Results that follow from a graph — degeneracies, closures, pairings — are exact given the graph. Results that follow from a fitted number are as good as the fit, and the essay that introduces the fit is the essay that has to say so.

What is striking is how much survives on the graph side. Everything in Hückel theory and what it gets right, everything in aromaticity as a computed shell closure, and the bond orders in bond order from the eigenvectors are parameter-free. The heteroatom is where the free lunch ends.

Who found it, and when #

Hückel’s own papers of 1931 and 1932 are about hydrocarbons. The extension to heteroatoms was worked out through the 1940s and 1950s by several groups, with the notation $\alpha + h\beta$ becoming standard through Coulson and Longuet-Higgins’s series of papers on the molecular orbital theory of conjugated systems.

Streitwieser’s 1961 book collected the parameter values and is the source most later tabulations trace back to. His own discussion is careful about their provenance and about their limits; the tables are what got copied.

The pairing theorem is Coulson and Rushbrooke’s, from 1940, and their statement of it is explicit that the diagonal elements must be equal — the condition that later summaries tend to compress into “for alternant hydrocarbons”, which is true and is not the same statement.

Still open: when the graph grows without limit #

Plain Hückel theory is a matrix of ones and zeroes and what a graph alone can decide. A heteroatom lets two fitted numbers in, and this essay has watched what they cost. The next question is what happens as the graph grows without limit — where the levels stop being a list and become a band.