Molecular Geometry & Bonding
what the shape decides
Series

Delocalisation — the series

15 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. benzene — D6h. The molecule with the point group found from its coordinates rather than looked up: every candidate operation was applied and kept when it permuted the atoms among themselves.

    Delocalisation

    Benzene does not alternate between two structures. It has one structure, and the two Kekulé forms are basis functions in a description of it — which is a different and much less exciting claim than the one usually made.

    part 1 · beyond
  2. Hückel levels of benzene. The orbital energies of the pi system, computed as the eigenvalues of the molecule's adjacency matrix. Degenerate levels share a line. The electrons are placed by aufbau with Hund's rule, so a half-filled degenerate shell shows as two unpaired spins.

    Hückel theory and what it gets right

    A conjugated system's orbital energies are the eigenvalues of a matrix of ones and zeroes. Nothing about carbon enters, no geometry enters, and the results that survive are exactly the ones that depend on neither.

    part 2 · bonding
  3. π bond orders in naphthalene. Each bond drawn with a thickness set by its computed pi bond order, and the number printed beside it. The orders come from the eigenvectors and cost nothing extra once the matrix is diagonalised.

    Bond order from the eigenvectors

    Having diagonalised the matrix, the coefficients are already there. Two sums over them give every bond's order and every atom's charge, and naphthalene's three kinds of bond come out in the order the measurements find them.

    part 3 · bonding
  4. Hückel levels of hexatriene. The orbital energies of the pi system, computed as the eigenvalues of the molecule's adjacency matrix. Degenerate levels share a line. The electrons are placed by aufbau with Hund's rule, so a half-filled degenerate shell shows as two unpaired spins.

    Conjugation, and its limits

    Every extra double bond in a chain lowers the gap between the highest occupied and lowest empty orbital, which is why long conjugated molecules are coloured. The trend has a limit, and the limit is not where the arithmetic says it should be.

    part 4 · beyond
  5. pyridine against benzene. The Hückel levels of pyridine beside those of benzene, which is the same graph with one diagonal entry and the bonds touching it changed. The parent's levels are symmetric about α because its matrix has nothing on the diagonal; the substituted system's are not, and the asymmetry is the size of the fitted parameter rather than a result.

    Hückel with a heteroatom

    A hydrocarbon's Hückel matrix contains no fitted number at all. Put one nitrogen in the ring and two arrive at once, the levels stop being symmetric about α, and the delocalisation energy stops being quotable — all from changing three entries in a matrix.

    part 5 · bonding
  6. pyrrole against cyclopentadienyl anion. The Hückel levels of pyrrole beside those of cyclopentadienyl anion, which is the same graph with one diagonal entry and the bonds touching it changed. The parent's levels are symmetric about α because its matrix has nothing on the diagonal; the substituted system's are not, and the asymmetry is the size of the fitted parameter rather than a result.

    Six electrons in a ring that is not all carbon

    Pyrrole and the cyclopentadienyl anion hold the same six π electrons in the same five-membered ring. The anion spreads them perfectly evenly; pyrrole's nitrogen keeps 1.720 of them and furan's oxygen keeps 1.791 — the more electronegative atom donates less, and the bond orders to it fall with it.

    part 6 · beyond
  7. benzene: what alternation costs and gains. The π energy of benzene as its bonds are alternated by β(1 ± δ), the elastic cost of alternating them at a stated stiffness, and the sum. The π curve falls quadratically in δ, which is measured here and is the whole difference between a molecule that distorts and one that does not. The stiffness is a stated parameter, so where the minimum falls is not a claim; which power of δ each curve goes as is.

    The hexagon is the frame's doing

    Benzene's delocalisation energy is quoted as 2β and read as the reason its bonds are equal. Let the bonds alternate and the π energy goes down, not up — at every ring size, for 4n+2 as much as for 4n. The π electrons are not what keeps the hexagon regular; the σ frame is, and the margin between them is uncomfortably thin.

    part 7 · beyond
  8. The two figures, per electron. For each ring, the delocalisation energy against isolated double bonds and against the same ring with one bond deleted, divided by the number of pi electrons. The upper bar of each pair is the usual reference; the lower one is the cycle's own contribution, and the two are nowhere proportional.

    A stabilisation is measured from somewhere

    Benzene's delocalisation energy is 2β against three isolated double bonds and 1.0121β against the same six carbons with one bond deleted. Cyclobutadiene's is exactly zero against the first reference and −0.4721β against the second, so one of the two references records the most famously destabilised ring in the subject as neutral.

    part 8 · beyond
  9. Which numbers survive a change of frame, and which are coordinates. Every quantity this field quotes, against the four things that can be changed without changing the molecule: the zero of energy, the unit, the reference state a stabilisation is measured from, and what a "per" quantity is divided by. A filled mark is a quantity that moves. three of the 9 survive all four, and every one of them is a property of the eigenvectors rather than of the energies. The first two changes are exact symmetries of the model, so a quantity that moves under either is a coordinate and not a quantity at all.

    Which numbers carry a frame

    A Hückel calculation is written in two numbers nobody computes and quoted against reference states nobody measures, so every quantity it prints is a quantity in a frame. Nine of them, tested against four changes of frame: three survive all four, and the one that survives both exact symmetries and looks safest — a dimensionless ratio between two molecules — is the most fragile of all, because against one reference its denominator is exactly zero.

    part 9 · bonding
  10. One pi energy, two delocalisation energies. For each of six rings: the computed pi energy, the energy of the same atoms with one bond deleted, and the delocalisation energy that follows from each of the two reference states. Every entry is an eigenvalue sum; the two right-hand columns differ only in what was subtracted from the second column.

    The frame that was allowed to relax

    Four changes of frame were tested on nine quantities and none of them could move a bond order, because a bond order is a property of the eigenvectors and every change left the eigenvectors alone. Letting the geometry answer back does move them — butadiene's central bond falls from 0.4472 to 0.3676 — and it moves naphthalene's the other way, because its weakest bond is the one two rings share and relaxation strengthens it.

    part 10 · bonding
  11. The shared bond's response falls and reverses, at every coupling. How far the relaxation moves the most interior cross bond of an acene, against the number of rings, at three couplings. Every series falls monotonically and every one changes sign — at λ = 0.2 between 3 and 4 rings, at λ = 0.4 between 3 and 4 rings, at λ = 0.6 between 4 and 5 rings. Where it crosses is a property of the coupling; that it crosses is not. The natural reading — that the effect grows with the number of rings feeding a bond — is refused by every one of these curves.

    An anomaly that is not the first of a series

    Naphthalene's shared bond behaves differently when the geometry is allowed to answer back, and the obvious reading is that two rings feeding one bond is what does it. Run it up the acenes and the effect falls at every step and changes sign: +0.02036 for two rings, +0.00678 for three, −0.00029 for four, −0.00603 for five. Two rings feeding a bond is not the beginning of anything.

    part 11 · bonding
  12. One heteroatom, and how far the relaxation carries it. The change every ring's bonds undergo when one carbon of the end ring is given a site energy, on a chain of 12 fused hexagons with its gap held open. The upper curve measures each molecule's couplings from its own mean bond order, which is what this collection's relaxation has always done; it stops falling at 4.48e-5 and stays there. The lower curve measures both from the same mean and keeps falling to 7.89e-9. Nothing about the molecule differs between them.

    The floor was in the bookkeeping

    A heteroatom in one ring of a fused chain is a perturbation with a place, and the relaxation carries it along the molecule. Measured directly, the response stops falling after five rings and sits at four hundredths of a millionth for ever. That floor is not the molecule. It is the relaxation measuring each system's couplings from its own mean, and removing it recovers nine orders of magnitude.

    part 12 · bonding
  13. A heteroatom on ring 5, and the profile in both directions. The response of each ring to a heteroatom placed on ring 5 of 12, on a logarithmic scale. Two rings share the peak, because an interior ring's outermost carbon belongs to two rings at once, and the profile falls away at the same rate on both sides — which is the answer wanted: the reach is the molecule's and not the end's. Every point is a mean over the six bonds of its ring.

    The reach is the molecule's

    Every measurement of the relaxation's reach so far puts the heteroatom on the end ring, because that gives the longest run to measure a decay over — and the response of a molecule to a perturbation at its end is not the response to one in its middle. Moved inward, the decay is the same in both directions and the same as the end's. The amplitude is not: it halves, and splits across two rings.

    part 13 · bonding
  14. One kink, and the heteroatom's own ring notices only at the kink. The response on the heteroatom's own ring, on a straight chain of twelve and on one with a kink — two adjacent angular fusions — in the middle, as the heteroatom is moved along. The two agree to within one and a half per cent except on the kink's two rings: on the first the bent chain reads 18.9 per cent lower and on the second 3.1 per cent lower. An end halves the amplitude.

    A bend is not an end

    Moving a heteroatom from the end of a straight chain of twelve rings to its middle leaves the decay unchanged and halves the amplitude, which raises the question of whether a bend in the chain is a boundary of the same kind. It is not a boundary of the same kind: the kink — two adjacent angular fusions — leaves the fitted decay length within two per cent and costs the heteroatom's own ring a fifth only on the kink's first angular ring. It does multiply down the response on the rings beyond it, which a decay length does not see.

    part 14 · bonding
  15. Angular rings leave the heteroatom's own ring alone and scale the rest. Each ring's response as a fraction of the straight chain's, with the heteroatom on ring 0 and the first one, two, four or ten interior rings angularly fused, on a logarithmic axis. The heteroatom's ring stays within one and a half per cent in every case. The rings beyond the angular fusions fall, to under half by the third ring once two are angular, and to between an eighth and a third along the fully angular chain, alternating from ring to ring.

    An angular ring rescales what lies beyond it

    A heteroatom's influence along a chain of fused rings decays with a length near 0.7 rings, and one bend was found to change almost nothing. Counted properly, that bend was two angular rings, and counting angular rings one at a time shows what they do: the heteroatom's own ring never moves by more than one and a half per cent, the rings beyond angular fusions fall to between an eighth and a half of their response — or rise by up to two fifths — and the decay length changes by a tenth.

    part 15 · beyond

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