Molecular Geometry & Bonding
what the shape decides
Series

Models — the series

14 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Splitting goes with overlap. For each pair, the atomic levels on the outside and the combinations they form in the middle, with the splitting drawn in proportion to the computed overlap. A pair that symmetry forbids does not split at all, because its overlap is exactly zero.

    Molecular orbital and valence bond

    Two frameworks, taught as rivals, describing the same molecules. One starts from delocalised orbitals and localises; the other starts from localised bonds and delocalises. Pushed far enough they meet.

    part 1 · bonding
  2. How often two electrons are in the same place. Double occupancy per site in the exact ground state of a chain of 2, against the repulsion. It starts at the independent-electron quarter and falls to nearly nothing. The second curve is what the state actually pays in repulsion, which peaks and then falls because the electrons have stopped meeting.

    Where molecular orbital theory dissociates

    The molecular orbital description of a two-electron bond puts both electrons on the same atom half the time — at every bond length, including infinite. The exact answer falls from a half to 0.0039 as the atoms separate, and the point where the two standard models are equally wrong is exactly U = 4t.

    part 2 · bonding
  3. The valence-bond and molecular-orbital directions in one plane. The valence-bond function and the molecular-orbital function as two directions, at the angle their overlap requires — 45.0°, since they overlap by 0.7071. The exact ground state lies in the plane they span at every repulsion, to twelve decimal places, and swings from one to the other as the repulsion grows without ever arriving. Neither picture is a special case of the other and the answer is not either of them.

    Two pictures, one plane

    Molecular orbital theory and valence bond theory are taught as rival descriptions of a two-electron bond. In a model small enough to solve exactly they are two vectors in a two-dimensional space, the exact answer lies in the plane they span at every repulsion, and it is neither of them at any repulsion but two.

    part 3 · bonding
  4. The ionic weight against how much ionic structure is in the wavefunction. The percentage each convention calls ionic, at a fixed structure overlap, as the amount of ionic structure in the wavefunction is raised from none to the molecular orbital value. They meet at both ends of the sweep and disagree everywhere between. Every curve is a weight and every set sums to one.

    A weight that depends on how it is weighed

    The ionic character of a two-electron bond is quoted as a percentage. For one wavefunction at hydrogen's bond length, three conventions in the literature give 18.73, 34.74 and 5.88 per cent — a factor of six — and on a wavefunction with no ionic structure in it at all, one of them still reports a quarter.

    part 4 · bonding
  5. seven sets of parameters, one spectrum. Benzene's π levels from seven sets of the three Hückel parameters, each fitted to reproduce the two measured ionisation energies exactly. The two occupied levels sit at the same energy in every column, because that is what was fitted. The empty level moves from -3.15 to 4.69 electronvolts across the family, and the resonance integral from -3.05 to -7.66.

    One spectrum, a line of models

    Hückel theory has three parameters and benzene's photoelectron spectrum supplies two numbers, so the fit has a curve of solutions rather than a point. Along it the resonance integral runs from −3.05 to −7.66 electronvolts, the empty level moves by eight, the terminal-to-central bond order ratio in butadiene goes from 2.000 to 2.671 — and the delocalisation energy is 6.100 electronvolts in every member.

    part 5 · bonding
  6. Two molecules, one eigenvalue, and 1.27 eV between them. Six alternant hydrocarbons placed by the Hückel eigenvalue of their highest occupied level — computed by diagonalising each molecule's own adjacency matrix — against the measured first π ionisation energy. Ethene and benzene share an eigenvalue of exactly 1 and their measurements differ by 1.27 eV; butadiene and naphthalene share 0.618 and differ by 0.94. A model that reads only the eigenvalue is a function of it, so it must give each pair one answer, and the two vertical pairs are the whole of its error.

    A parameter that never finds a value

    Show a two-parameter model six measurements instead of two and it stops being underdetermined and starts being wrong. Adding the third parameter improves the fit by one part in eighty, moves the resonance integral by a factor of four, and never finds a best value at all — because nine tenths of the error is a term the model does not have.

    part 6 · bonding
  7. What each of these predictors cannot see. Four predictors built elsewhere in this collection, each audited by its ties: the share of the variation in what was measured that the predictor demonstrably cannot account for, because two systems it assigns the same number to were measured to differ by that much. the spin-only moment leaves 23 per cent; the VSEPR angle leaves 92 per cent; the highest occupied Hückel eigenvalue leaves 41 per cent. The control has no ties at all and the instrument returns nothing for it, which is what it must do. No fitting anywhere: a tie is a claim a model of that form cannot escape.

    Two systems a model cannot tell apart

    Two molecules that share a Hückel eigenvalue but were measured to differ bound a whole family of models at 0.456 eV. That is a reusable instrument, and chemistry has plenty of predictors of exactly the same shape. Turned on themselves: VSEPR cannot account for 92 per cent of the variation in the four angles it predicts, and no fitting is involved anywhere.

    part 7 · bonding
  8. Two instruments, and they do not agree. Each of this collection's four predictors under both tests. The exact-tie instrument reports the share of the variation a predictor demonstrably cannot account for, and it ranks VSEPR worst and has nothing at all to say about the ring-strain control. The near-tie instrument reports how much steeper any model must be somewhere than the set is overall, and it ranks the control worst and cannot speak about VSEPR, whose predictor is a label rather than a number. Neither instrument is the general one, and a predictor that passes one has not been audited.

    The pair that is not a tie

    An exact tie bounds every model of a form with no fitting, and it is available only for predictors that read integers. A near tie bounds the model's derivative instead, and a pair the predictor orders the wrong way round refuses monotonicity outright. Run on four standard predictors, the new instrument ranks them in a different order from the old one — and its worst case is the control the old one could not see.

    part 8 · bonding
  9. What it would cost to destroy each refutation. For every discordant pair in the collection, the measurement error that would be needed to reverse it, as a fraction of that predictor's own spread of measurements. Nothing is quoted: the question is not what the uncertainties are but how large they would have to be. The cheapest to destroy needs 3.4 per cent of the range and the dearest needs 51.6.

    The error bar that would be needed

    Two things a pair of measurements can say without any model both treat their numbers as exact. The quoted measurements carry no uncertainties, and inventing some would be worse than having none — so the question is asked the other way round. Not what the errors are, but how large they would have to be. The three refutations that seemed most worrying turn out to be the sturdiest of the lot.

    part 9 · bonding
  10. The bracket has no upper end. Every price is multiplied by √2 over √(1 + r² − 2ρr), where ρ is the correlation between the two measurements' errors and r is the ratio of their sizes. For equal precision the factor is 1/√(1 − ρ), which is one at independence and unbounded at perfect correlation. The other extreme the question asked for is not a number.

    The other end of the bracket is not a number

    Every claim can be priced by the measurement error it would take to overturn it, assuming the two errors independent, and the correlated extreme ought to turn each price into a bracket. At perfect correlation between equally precise measurements the difference is exact and the price is infinite — and every price is multiplied by the same factor, so the ordering of prices cannot change.

    part 10 · bonding
  11. The ranking is fragile and the headline is not. The correlation required to disturb two different things. Reordering the easiest neighbouring pair needs ρ = 0.154, which is weak enough to expect. Displacing the collection's most fragile claim outright needs ρ = 0.954, which is near-perfect correlation. So a per-predictor structure scrambles the middle of the ordering and leaves the top of it alone.

    The ranking moved and the headline did not

    One correlation applied across a collection multiplies every price by the same number, so it slides the fragility ordering rigidly and cannot scramble it. Give each predictor its own correlation and the ordering does move — but not at the top. The two most fragile claims come from the same predictor, share a factor, and are locked in order at every correlation whatever; the easiest swap anywhere below them needs only ρ = 0.15.

    part 11 · bonding
  12. One ranking, three chains. The collection's priced claims in order of fragility, with each claim placed in its own predictor's column. Two claims sharing a predictor share its error correlation, so their prices carry the same factor and their relative order cannot be changed by any correlation structure whatever — a column is rigid. Claims in different columns can be reordered at a price. So the ranking is not one ordering but three chains interleaved, and only the within-column statements need no assumption about anybody's errors.

    Three chains and ninety orderings ruled out

    Two claims priced against the same predictor share its error correlation, so no correlation structure can reorder them. That makes the collection's fragility ranking three chains rather than one list — and thirty of the hundred and twenty orderings of five claims are reachable, with the other ninety forbidden before a single measured difference is looked at.

    part 12 · bonding
  13. Two prices, and which is cheaper is a property of the claim. For each slope floor, what it costs to halve it two ways. The measurement price is a standard error on the measured rise, as a fraction of the measured range — the currency every price in this argument has been quoted in. The model price is the change in the computed run, as a fraction of the predictor's own range. They are different currencies and their ordering differs between claims: the angle strain's floor is three times cheaper to move through its model, and the spin-only moment's cannot be moved through its model at any price at all.

    A denominator needs three currencies

    Every price in this argument has been a price on a measured rise. A slope floor is a rise over a run, the run is computed rather than measured, and pricing it turns out to need three currencies rather than one — because one predictor's run is a graph eigenvalue, one is a convention, and one is a difference of square roots of integers that nothing defensible moves.

    part 13 · bonding
  14. The verdicts hold and the number does not. For each predictor, what survives thirteen strictly increasing reparameterisations of its own scale. The count of exact ties, the share of the variation those ties leave unexplained, and the count of discordant pairs are identical under every one of them — to the last bit, because each asks only about the order of the predicted values and an increasing map preserves order. The slope floor asks for a ratio of differences, and an increasing map does not preserve differences.

    A floor on models written in one scale

    A tie asks whether two predicted values are equal and a discordance asks whether two differences have the same sign. Both are questions about order, and a strictly increasing change of scale preserves order — so both are exactly invariant under thirteen reparameterisations of all three predictors. The slope floor asks for a ratio of differences, and it moves by factors of nineteen, ten and eleven thousand.

    part 14 · bonding

All series