Molecular Geometry & Bonding
what the shape decides
Concept

Ring strain — where it appears

The excess energy a cyclic molecule carries over an open chain of the same composition. It is defined against a reference that has to be chosen, and the choice decides how much of it is attributed to angles rather than to torsions or to non-bonded contacts.

Named by 10 essays across 2 fields — each of them below, with the objects they name alongside it.

The two rows of a table, and the curve between them. The rotamer ceiling of a closure freezing 4 rotations, against how hard each of a stated number of them is to turn. The top curve is the free-rotor row and the right-hand end of the bottom curve is the absent-rotor row; everything else is what a real hindered tether does. The measured acceleration of 10-fold is drawn across it. It falls below that line when 3 of the 4 are hindered by 4.09 kJ/mol, or when 4 of the 4 are hindered by 2.70 kJ/mol.

The curve between two rows

A rotation a ring closure has to freeze was treated as free or as absent, and the two answers sat in adjacent rows of a table. A real hindered rotor is neither. The factor one rotor contributes runs from 3.316 to one along a curve nobody had drawn, and where it matters is between one and eight kilojoules a mole — which is a torsional barrier rather than a conformational preference.

shape · Strain The two sides of a verdict, with error bars. The computed ceiling for the six-membered closure — 10.9945 — with the band a gauche energy of 3.8 ± 0.4 kJ/mol puts on it, against the measured tenfold rate ratio with an assumed 20 per cent uncertainty. The two bands overlap over most of their length, and the nine per cent margin the verdict was decided by sits inside both of them.

A verdict inside its own error bar

The tightest comparison in the rotamer argument is a computed 10.99 against a measured 10 — a nine per cent margin, offered as a verdict. The computed side is built on one quoted energy known to ±0.4 kJ/mol, and that alone puts a band of twenty-three per cent on it. The verdict turns on four tenths of one standard deviation of a number nobody had put an error bar on.

shape · Strain 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.

bonding · Models The measurement is not a horizontal line. The rotamer ceiling against temperature, with the measured tenfold drawn four ways: flat, and as a rate ratio whose two activation energies differ by 3, 5.31 and 8 kJ/mol. Flat, it is crossed at 312.1 K, which is the flat-rotor answer. At 5.31 kJ/mol the two curves are parallel and never meet. At 8 they meet on the other side, and it is cooling rather than heating that refutes the account.

One number decides which way it breaks

The six-membered verdict gives way at 312.1 K, and that is a statement about the ceiling rather than a prediction about the ratio — because a real rate ratio has a temperature dependence the model has no term for. Putting that term in moves the crossing, and at 5.308 kJ/mol it removes it entirely.

shape · Strain The five-membered ring would need more rotors than its chain has bonds. For each measured acceleration, the least rotor count whose ceiling reaches it, against the most rotors any convention gives that ring and against the number of bonds the chain has at all. The five-membered cases need five and eight rotors from a chain with four bonds, so they are refused by supply rather than by a margin — a refutation that needs no convention to be chosen, because every convention is below the requirement.

The count that was never written down

The rotamer ceiling for a ring closure needs a count of frozen rotations, and the counts in use disagree in direction: one gives the six-membered ring more frozen rotations than the five, another fewer. Every measured verdict survives all three defensible conventions unchanged — and the margins move by a factor of eleven, which is why a verdict decided by ten per cent was worth being nervous about.

shape · Strain The ceiling moves by 8.7-fold across the gauche energy's own reported range. The rotamer ceiling against butane's gauche energy, for the rotor counts each convention assigns to a five-membered and a six-membered closure, with the three measured accelerations drawn as horizontal lines. Across the reported range the ceiling moves by up to 8.70-fold, and two of the nine verdicts cross a measurement during the sweep. The natural prediction is that this input would be the smaller lever of the two; it is the larger.

The lever that was supposed to be smaller

Sweeping the rotor conventions leaves every verdict unmoved, and suggests that the other quoted input will be a smaller lever. It is a larger one. Sweeping butane's gauche energy over its reported range moves the ceiling by 8.7-fold, flips the six-membered verdict at 3.630 kilojoules a mole — inside the quoted error bar — and takes the five-membered refutation with it at 4.422.

shape · Strain Two sweeps, and one of them covers 28 per cent less ground than the other. What each of the two sweeps reaches, measured in the one variable the rotamer ceiling has: g/RT. The temperature sweep lies entirely inside the gauche sweep, so it explored no arrangement the other does not. Their widths are 0.581 and 0.807, their union is 0.807, and treating them as independent would have credited them with 1.387.

Two sweeps and one lever

The rotamer ceiling depends on the gauche energy and on the temperature only through their ratio, so a sweep of either traces the same curve. Measured in that one variable, the temperature sweep lies entirely inside the gauche sweep — it covered no ground the other does not, and the whole reported spread in one energy is worth a temperature swing from 196 to 353 kelvin.

shape · Strain 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.

bonding · Models 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.

bonding · Models 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.

bonding · Models

Named alongside it

The objects these essays reach for when they reach for this one.

Model limitApproximationConformerConventionRotamerUnderdeterminationClosed formError propagationReference stateBoltzmann distributionMeasurement uncertaintyPredictor

All concepts