Flat rings: the angles, the torsions and what is measured

One of the figures on where the atoms go: Repulsion minimised on a sphere, the angles that fall out of it, and the arrangements that are not all alike.

Eleven essays draw this figure, each at the values its own argument needs rather than at the setting shown above. What each one uses it to show is below, in the words of its own caption.

The strain that is not in the angles

Every ring from three to eight, held flat, with the angle strain its geometry forces and the torsional strain of having every bond eclipsed. The fifth row is the one to read: 0.4 kilojoules of angle strain against a measured twenty-six.

Where the term starts existing at all, which is the other half of the account. Below a certain ring size there is no pair of bonds far enough apart round the ring for the torsional term to have anything to sum over, so the strain in the small rings is angular by default — and above it the torsional term appears and grows, which is the crossover the rest of this essay measures.

A six-ring built at a series of interior angles, with its dihedrals read off the coordinates rather than assumed. The angle term is a parabola about the tetrahedral value; the torsional term is whatever the closure leaves; and the minimum of the sum sits where the two trade off rather than at either one’s own minimum.

The atoms that meet across a ring

The number of pairs of ring atoms four or more bonds apart, against ring size, with each ring’s measured strain beneath it. Below eight there are none: not few, none. From eight upwards the count grows quickly — four pairs in an eight-ring, nine in a nine-ring, fifteen in a ten, thirty in a twelve.

Twelve closed conformers of a ten-membered ring at a bond angle of 111 degrees, placed by torsional energy and by the closest approach of two atoms four or more bonds apart. An account built from angles and torsions reads the horizontal axis and has nothing to say about the vertical one. The circled point is the conformer that account prefers.

The two-term account itself: angle strain and torsional strain for the rings it covers, against the measured strain of each. It works up to seven and the disagreement grows after that, which is the pattern this essay explains.

The explanation with the wrong sign

The six-ring in the conformation the torsional term prefers, with what a closure search actually returns beside it. The alternating form comes out of the search rather than being imposed, and its bond angles are within a degree of the tetrahedral preference — which is why the angular account has almost nothing to explain here and the torsional one has everything.

The five-ring family: the same angles, the same torsions, and not the same molecule. Substituting a carbon changes which arrangements are cheap without changing any bond angle to speak of, so the relief a substituted carbon gives is torsional — and it is a help below five and a hindrance from five up, which is the sign the received account has backwards.

The factor the closable population gains as the penalty on each extended rotation rises. A penalty no larger than a gauche interaction — 3.8 kJ/mol, the measured value for butane — is already worth a factor of 10.8, with no bond angle changed anywhere.

A ceiling that rises where the measurements fall

The ceiling against the ring being closed, with the two measured gem-dimethyl accelerations on the same logarithmic axis. The curve rises by a factor of eleven from a five-ring to a seven; the measurements fall by a factor of twenty-five from a five-ring to a six. One measurement is above its own ceiling and the other is far below it.

The ceiling itself, plotted against the cost it is built from. A chain with three internal rotations closes only from an arrangement in which every rotation is gauche and all of one sign, so making the extended arrangements expensive makes the chain spend more of its time closable — and the factor that buys is bounded above however expensive they are made. The measured acceleration is above the bound at a gauche cost of 3.8 kJ/mol, which is the number the account itself uses.

The angular problem the five-membered ring has and the six-membered one does not: the largest bond angle a closed equilateral ring of each size can hold, against the tetrahedral angle. Five is below it and six is not, which is the asymmetry the missing term would have to exploit.

An estimate that can be wrong by two

The ceiling against the rotor count, with both measurements drawn across it. The question is not where each closure sits but how far it would have to move.

What taking rotors out of the count does to each closure. The bars shorten in both cases; only one of the two verdicts is anywhere near the line.

How the ceiling is approached at the six-membered closure’s own count: the acceleration against the substituent’s penalty, with the closed form drawn across. The measurement sits far below it, which is what a comfortable verdict looks like.

The curve between two rows

What one rotation is worth, from free to locked. The table’s two rows are the two ends of this curve; the ester it was arguing about is off to the right of everything interesting.

The same factor tabulated. An ester is past every energy at which the interpolation makes a difference.

Both ceilings against hindrance with both measurements drawn across. One verdict has hindrance pushing it further into safety and the other has it pushed towards the line.

A verdict inside its own error bar

The computed ceiling with the band one quoted input puts on it, against the measured rate ratio with an assumed uncertainty. The two overlap over most of their length.

The six-membered closure’s ceiling against the butane gauche energy, with the quoted value marked, its uncertainty shaded, and the measured tenfold crossed.

The derivative by a central difference and by that expression, for each number of free rotors. The two agree to a part in a million.

One number decides which way it breaks

The rotamer ceiling against temperature, with the measured tenfold drawn flat and as a rate ratio with three different activation-energy differences.

The temperature at which the two sides meet, against the measurement’s own activation-energy difference, with the gap where they never meet.

The margin between ceiling and measurement across the range, for five assumed activation-energy differences.

The count that was never written down

The three defensible counts against ring size. Two rise with the ring and one falls, so no constant relates them.

The ceiling against the rotor count, logarithmically. Each rotor multiplies it by 3.32, so the conventions are a factor of eleven apart.

Three measured accelerations against three conventions. Nine ceilings, and the verdict column is constant within every row.

The lever that was supposed to be smaller

The rotamer ceiling against the gauche energy, for the rotor counts each convention gives a five-membered and a six-membered closure, with the three measured accelerations drawn across. The shaded band is the quoted error bar.

Each of the nine combinations, with a bar drawn where it calls the rotamer account sufficient. Two of them change during the sweep and one changes inside the quoted error bar.

The three rotor counts against ring size. Which convention a case is read under decides not only its margin but, now, whether its verdict moves at all when the gauche energy does.

Two sweeps and one lever

Five matched pairs: the ceiling reached by moving the gauche energy, against the ceiling reached by leaving it alone and moving the temperature to keep g/RT fixed. The two agree bitwise.

The two sweeps measured in the one variable the ceiling has. The temperature range is a strict subset of the gauche range.

The logarithm of every ceiling in these sweeps, divided by its own rotor count, against g/RT. Thirty-six combinations from two sweeps and three rotor counts, on one curve.