Six electrons in a ring that is not all carbon

Put six π electrons into a five-membered ring of carbons and the answer is forced by symmetry: five equivalent atoms, five equal shares, $1.200$ electrons on each. There is nothing to compute beyond the total.

Replace one carbon with a nitrogen and the symmetry is gone. The ring still has five atoms and six π electrons, the shell still closes, and the electrons are no longer shared equally — by an amount that is computable from a single change to the matrix.

What the heteroatom is, in the matrix #

A Hückel matrix has one row and column per π centre. For a hydrocarbon every diagonal entry is α and every entry between bonded neighbours is β, so the matrix is the adjacency matrix of the ring and nothing else.

A heteroatom is entered as two modifications, both of them in the standard parameterisation Hückel with a heteroatom introduces:

A diagonal entry $\alpha + h\beta$. For a nitrogen contributing two π electrons, $h = 1.5$: the level is pushed down, because nitrogen holds an electron more tightly than carbon.

Off-diagonal entries $k\beta$ to its neighbours. For the same nitrogen, $k = 0.8$: the C–N π interaction is weaker than a C–C one.

That is the whole model. Two numbers, both quoted from the literature rather than computed here, and everything that follows is arithmetic on the resulting matrix.

The numbers are fitted, and it is worth saying so plainly. They are chosen to make computed quantities agree with measured ones across many molecules, so a result that depends sensitively on their exact values is a result about the fit. What this essay uses them for is a comparison — nitrogen against oxygen, heteroatom ring against all-carbon ring — where the direction of the effect is robust and the third decimal place is not.

Where the electrons actually sit #

Diagonalising and filling six electrons into pyrrole gives π charge densities of

$$1.720, \quad 1.035, \quad 1.106, \quad 1.106, \quad 1.035$$

with the first entry the nitrogen. It brought two π electrons and it keeps $1.720$ of them, so $0.280$ of an electron has been donated to the ring — a bit over a quarter of one electron, distributed unevenly over four carbons with the two furthest from the nitrogen taking the most.

Furan, with an oxygen at $h = 2$ and $k = 0.8$, gives

$$1.791, \quad 1.015, \quad 1.090, \quad 1.090, \quad 1.015$$

so oxygen donates $0.209$: three quarters of what nitrogen donates.

That ordering is the whole point. The more electronegative heteroatom shares less. It is the same statement as the diagonal entry being more negative — the level is deeper, so the orbital coefficients concentrate on it — and it is the computed form of the observation that furan behaves far less like an aromatic ring than pyrrole does.

The comparison with the all-carbon case makes the size of the effect clear. The cyclopentadienyl anion puts $1.200$ on every atom: a fifth of an electron of excess charge per carbon, spread with perfect uniformity because symmetry demands it. Pyrrole’s carbons carry between $1.035$ and $1.106$, and its nitrogen carries $1.720$.

The bonds tell the same story #

Charge is one output of the eigenvectors and bond order — computed as bond order from the eigenvectors computes it — is the other, and here they agree.

Pyrrole’s π bond orders are $0.440$ for each C–N bond, $0.790$ for the two bonds next to them, and $0.553$ for the bond across the back. Furan’s are $0.385$, $0.811$ and $0.537$.

Two things are visible in those numbers.

The bond to the heteroatom is the weakest π bond in the ring, and it is weaker in furan than in pyrrole. Part of that is the $k = 0.8$ entered by hand; the rest is the charge distribution, since a bond order is a product of coefficients on the two atoms and the coefficients on the ring carbons are smaller when the heteroatom holds more.

The ring is not uniform. In the all-carbon anion every bond order is the same; here they run from $0.385$ to $0.811$, a factor of more than two. A picture of furan with a circle in the middle of the ring — the standard notation for a delocalised π system — is drawing something the calculation does not support, in the same way that a double bond is not two single bonds shows a doubled line drawing something the overlaps do not support.

How much of that is the parameter #

A result computed from two numbers somebody else fitted deserves to be tested against those numbers, so the diagonal entry is swept while everything else is held.

Two things are worth reading off it.

The row at $h = 2$ is furan, to four decimal places, and it was reached here by moving pyrrole’s diagonal entry rather than by asking for furan. So the difference between the two molecules in this model is the diagonal entry — the off-diagonal $k = 0.8$ is the same for both — and the essay’s comparison is a comparison of one number.

The dependence is smooth and monotone but not steep. Moving $h$ by a third, from $1.5$ to $2.0$, changes the donation by a quarter and the bond order by an eighth. So the ordering nitrogen-donates-more-than-oxygen survives any plausible re-fitting of these parameters, and a claim that pyrrole donates exactly $0.280$ of an electron does not: that figure is the parameterisation speaking.

Sweeping the off-diagonal entry instead gives the opposite dependence — $k$ from $0.6$ to $1.0$ takes the donation from $0.184$ to $0.371$ — which is the expected direction, since a stronger interaction with the ring is what lets density leave the heteroatom in the first place.

The pairing theorem falls over #

Something else changed when the heteroatom went in, and it is worth pulling out because it is easy to attribute to the wrong cause.

For an alternant hydrocarbon — one whose π centres can be two-coloured so that no two neighbours share a colour — the eigenvalues come in pairs symmetric about α. That is the pairing theorem, and it is checked in both directions: alternant systems must satisfy it and the three non-alternant systems here must fail it. A check made in one direction only would pass on a diagonaliser that had stopped working, which is why the same ring, three charges checks its shell closures against a predicate rather than against a table.

Pyrrole’s levels are $2.320$, $1.189$, $0.618$, $-1.008$, $-1.618$. Not paired, and the reason is that an odd-membered ring is not bipartite — a five-cycle cannot be two-coloured.

But pyridine’s carbon skeleton is bipartite, being a six-ring, and its levels are $2.107$, $1.167$, $1.000$, $-0.841$, $-1.000$, $-1.934$. Also not paired.

The theorem is a property of the matrix, not of the drawing. It requires the matrix to have zero diagonal in the α-shifted basis, and a heteroatom’s $h\beta$ destroys that regardless of what the graph looks like. A Hückel calculation should therefore refuse to apply the pairing theorem to any system carrying diagonal entries, and say so.

Two ways a nitrogen can enter a ring #

Pyridine and pyrrole both have a nitrogen in an aromatic ring and the nitrogens are doing different jobs, which the parameterisation records.

Pyridine’s nitrogen contributes one π electron, like each of the carbons, and keeps its lone pair in an sp² hybrid in the ring plane. Its diagonal entry is a mild $h = 0.5$, and its π charge comes out at $1.195$ — it draws a little extra density in, being electronegative, but it contributes nothing extra.

Pyrrole’s nitrogen contributes two, its lone pair being part of the π system, which is what makes a five-membered ring with four carbons reach six π electrons. Its diagonal entry is the much larger $h = 1.5$, because an atom donating two electrons to a π system is doing so from an orbital it holds tightly.

The chemical consequences are opposite and familiar: pyridine is a base at nitrogen, pyrrole is not, and the reason is that pyrrole’s lone pair is spent. Basicity is not computed here, but the electron bookkeeping above is where the explanation starts, and it is the same bookkeeping as the s-character budget in the angle does not fix the hybridisation: a pair used for one thing is unavailable for another.

Furan’s five levels take its six electrons with a gap of 1.566β to the next, so by the shell-closure test furan is aromatic. That is true and is not the whole story: the test asks only whether the highest occupied shell is full, and every ring in this essay passes it.

What a delocalisation energy cannot be quoted for #

A conspicuous absence in this essay is any delocalisation energy for pyrrole or furan.

The reason is the one delocalisation sets out: a delocalisation energy is a difference against a reference state of isolated double bonds, and the reference has to be stated rather than assumed. For a hydrocarbon the reference is unambiguous — so many ethene units. For a ring with a nitrogen or an oxygen in it, the isolated reference would have to include a localised C=N or C–O π unit whose energy in this parameterisation is another fitted quantity, and the difference of two fitted numbers is not a measurement of anything.

So no delocalisation energy is quoted for these rings at all. The comparisons that survive are the ones made here — charge distributions, bond orders and level orderings against an all-carbon ring — because those are ratios within one parameterisation rather than differences across two.

Pyrrole’s lowest π orbital has its largest coefficient on the nitrogen, which is the eigenvector’s version of the charge above: the deepest level is the one most concentrated on the lowest-lying atom. The charges and the coefficients are the same six numbers read two ways.

The same model on an open chain #

The heteroatom parameters are not specific to rings, and running them on a chain shows the effect without any aromaticity in the way.

Acrolein is the open-chain member of the same family, and its charges — 0.771, 1.034, 0.667 and 1.529 — put the oxygen half an electron above what it brought and the third carbon furthest below. That depleted position is the one conjugate addition attacks, out of a four-by-four matrix.

The oxygen ends up with $1.529$ π electrons against the one it contributed, and the carbon two bonds away from it carries $0.667$. That is a large polarisation for one diagonal entry, and it is the computed content of the statement that a carbonyl group withdraws π density along a conjugated chain.

What the calculation does not supply is anything about reactivity, which is a rate and belongs to a different subject — the same boundary conjugation, and its limits keeps when it stops at a ground-state property. The charge distribution is a ground-state property and stops there.

The Hückel diagonal parameters run in the same order as every published electronegativity scale — carbon, then nitrogen, then oxygen — which is reassuring and is not evidence. The parameters were fitted to spectra, and agreeing with a scale they were not fitted to is a consistency check rather than a derivation.

The third member of the family, and why it is absent #

The conventional ranking of the five-membered heterocycles has three members and only two are computed here. Thiophene, with a sulfur in place of the nitrogen or oxygen, is usually placed at the aromatic end of the series — more benzene-like than pyrrole — and it is not in this collection’s parameter set.

The reason is that the standard account of why sulfur behaves that way invokes its 3d orbitals, and what that invocation is worth has already been computed. Hypervalency without d orbitals shows the bonding in the classic hypervalent sulfur compounds arriving from three-centre four-electron arrangements with no d participation required, and six bonds and four orbitals computes the symmetry side of the same question. Adding a thiophene parameterisation whose only content is a fitted diagonal entry would produce a number without settling anything, and the honest position is that a π model with one parameter per atom cannot distinguish sulfur is a poor π donor from sulfur has extra orbitals available, because both would be entered as the same adjustment.

That is a boundary rather than an omission, and it is the same boundary the ring parameters run into elsewhere: the model has exactly as many degrees of freedom as it has fitted numbers, and any explanation that needs a mechanism the model does not contain will be absorbed into a parameter without complaint.

What the missing third member would need #

The family this essay computes has two members and the experimental ranking has three, with the absent one at the top. That absence is worth more than a note, because putting sulfur in would test the account rather than extend it — and the test is one the account could fail.

The story so far is a story about one parameter. Nitrogen and oxygen differ in $h$ — 1.5 against 2.0 — and share a $k$ of 0.8, so the whole difference between pyrrole and furan is the diagonal entry, and the more electronegative atom holds its electrons more tightly and donates less. One number moves, one ordering follows, and the ordering matches the measurements.

Sulfur breaks that arrangement, because both parameters move and they move in opposite directions.

Its $h$ is smaller — around 1.0 against nitrogen’s 1.5 — because a sulfur 3p orbital sits higher in energy than a nitrogen 2p, closer to carbon’s, so sulfur is the readier donor of the three. On the one-parameter story that alone would make thiophene the most aromatic, which is what the measurements say.

Its $k$ is smaller too — around 0.7 against 0.8 — and that runs the other way. A carbon–sulfur bond is 1.71 ångström where a carbon–nitrogen bond is 1.37, and the sulfur orbital is a diffuse third-shell function overlapping a compact second-shell one. Poorer overlap means a weaker interaction across those two bonds, and a weaker interaction is less delocalisation, not more.

So the two parameters disagree about thiophene, and which of them wins is a quantitative question rather than a trend to be read off electronegativity. That is exactly the situation in which computing is worth more than arguing, and it is why the third member is the interesting one: pyrrole against furan tests nothing, because only one parameter moves and the answer is contained in the sign of its change.

Two cautions belong with any such calculation, and both are about the parameters rather than the arithmetic.

Sulfur’s values are the least transferable in the table. The nitrogen and oxygen entries were fitted across many molecules; sulfur’s were fitted across fewer, and different compilations disagree about them by more than the difference between pyrrole and furan.

And thiophene carried an extra argument for decades. Its unusual stability was long attributed to sulfur’s empty 3d orbitals joining the π system — an explanation that would sit entirely outside this model, since it needs an orbital the matrix has no row for. That account is now regarded as largely mistaken, and the modern reading is closer to the one above: a good energy match, a poor overlap, and the first winning.

Which leaves a clean statement of what the missing calculation would be for. It would not be a third data point on a trend. It would be the one case in the family where the model’s two parameters pull against each other, so the answer is a genuine output rather than a restatement of an input — and the experimental ranking is already known, so the calculation could come out wrong.

Who found it, and when #

The heteroatom parameterisation is essentially Adolf Streitwieser’s, laid out in Molecular Orbital Theory for Organic Chemists (1961), which collected values that had been in circulation since the 1940s. The scheme is deliberately crude — one number for the atom, one for each bond to it — and it survived because it is transferable enough to be useful across a large family of molecules.

The relative aromaticity of the five-membered heterocycles was an experimental result long before it was a calculated one. Pyrrole, furan and thiophene are conventionally ranked thiophene ≳ pyrrole > furan on the strength of how much each behaves like benzene and how much like a diene, and the ordering follows the heteroatom’s electronegativity downward. That the crudest possible model reproduces the ordering from one parameter per atom is the sort of result that made Hückel theory worth keeping long after better methods existed.

Still open: where a model’s central number stops being defined #

The argument about delocalisation started with a hydrocarbon question and a stated reference state, and a heteroatom breaks the symmetry that made the hydrocarbon cases easy. What it gains is a quantity that varies — the charge distribution — and what it loses is the delocalisation energy, which cannot be quoted once the reference stops being a count of ethenes.

The trade is worth noticing because it recurs. Every model of this kind has a regime where its central number is well defined and a neighbouring regime where the number becomes a difference of two fitted quantities. Recognising which side of that line a calculation sits on is most of what separates a computed claim from a plausible one.