Molecular Geometry & Bonding
what the shape decides
Concept

Elastic energy — where it appears

The cost of displacing atoms from their positions, quadratic in the displacement for small ones. It is what a distortion has to overcome, and its being quadratic is why a linear electronic gain always wins near the origin.

Named by 13 essays across 5 fields — each of them below, with the objects they name alongside it.

Four fillings, four periods. four fillings of a ring of 120, and for each of them what a distortion of every available period is worth. Every period pays the same elastic cost, so the bars compare what the electrons give back and nothing else. The winner is one over the filling in every case.

The distortion the filling chooses

A half-filled chain of equal bonds is unstable and alternates — long, short, long, short. That is the case everyone is shown, and it is one case. Fill the chain a third of the way instead and the alternation is worthless: what wins is a pattern that repeats every three bonds, and the period is one over the filling at every filling tried.

solids · Peierls distortion The total energy against the distortion, at several gaps. The elastic cost plus the second-order lowering, for five gaps between the ground state and the excited state it mixes with. The critical gap is 1.00: above it the symmetric structure is the minimum, below it the minimum has moved off zero, and nothing about the molecule is degenerate in either case.

A distortion needs two states

A degenerate electronic state cannot survive — that is the Jahn–Teller theorem, and it can be computed. A closed shell can fail to survive too, and the condition is a number: the symmetric structure holds only while the nearest excited state of the right symmetry lies above 2λ²/k, and one of ten symmetry species in an octahedron is the right one.

applied · Peierls distortion Flat rings: the angles, the torsions and what is measured. For each ring from three to eight, held flat: its interior angle, how far that is from tetrahedral, the angle strain that follows, the torsional strain of having every bond eclipsed, and the measured strain energy. The five-ring is the row the essay is about — its angles are almost ideal and it is strained.

The strain that is not in the angles

Cyclopentane's flat bond angles are 108°, a degree and a half from tetrahedral, and its angle strain computed from a standard bending constant is 0.4 kJ mol⁻¹. Its measured strain is twenty-six. The missing sixty kilojoules are torsional — one ethane barrier for every bond in the ring, which no account built on bond angles mentions.

shape · Strain The same angles, the same torsions, and not the same molecule. twelve closed conformers of a ring of 10 at a bond angle of 111.5 degrees. Every one has exactly the same bond angles, so an account built from angles and torsions places them all on the horizontal axis alone. The vertical axis is the closest approach of two atoms four or more bonds apart, which no term in that account mentions: two of these differ by 0.08 kilojoules in torsional energy and by 0.75 ångström in how close they come.

The atoms that meet across a ring

Twelve closed conformers of a ten-membered ring at one bond angle, so every one has identical angle strain by construction. Two of them differ by 0.026 kilojoules a mole in torsional energy and by 0.80 ångström in how close two atoms on opposite sides of the ring come — 2.331 against 3.131, where two carbons are in contact at about 3.4. An account built from angles and torsions calls those two structures the same.

shape · Strain An effect that cannot distort a molecule, deciding how far it distorts. The energy along one distortion coordinate, three times. With only the first-order term the minimum is at 0.6; with only the second-order term there is no minimum away from zero at all, because the gap of 1.5 is above the critical 1 for a closed shell on its own. With both, the molecule distorts to 1.06 — well past the first-order answer — and gains 0.09 more than the two separate stabilisations add up to.

Two distortions in one coordinate

A second-order Jahn–Teller effect that cannot distort a molecule by itself — its gap is half again above the critical value — nearly doubles the distortion when a first-order effect is already acting. The molecule goes to 1.075 instead of 0.600 and gains twice the energy, and it does it while the gap the second-order term divides by is opening rather than closing.

shape · Peierls distortion The whole difference lives at the two ends. The energy difference between the two dimerisations of an open chain, held at the same distortion, multiplied by the number of sites. It settles on a constant — 1.09 in units of the hopping — so the difference per site falls as one over the length, with a fitted exponent of -1. An end is a bond that is not there, and it is worth the same amount whatever it is attached to.

The distortion the ends decide

A chain of an even number of sites has an odd number of bonds, so its two dimerisations are different molecules rather than one molecule translated. Held at the same distortion they differ by 1.08715 in units of the hopping, whatever the length — a fixed amount of energy living at the two ends, with the per-site difference falling as one over the length at a fitted exponent of −0.99986. And below a hundred and twenty-eight sites the second dimerisation does not exist at all.

applied · Peierls distortion The 6-ring at 111.5°: the alternating form and what a search finds. The alternating ring — every atom displaced above or below the plane in turn — with its dihedral angles and its two strain terms, beside the lowest-torsion member a search constrained only by bond angles and closure returns. Both satisfy every geometric constraint; only one of them is staggered.

The explanation with the wrong sign

Two methyl groups on one carbon make a ring easier to close, by up to eleven thousand-fold, and the textbook reason is that they compress the ring's internal angle. Computed from a standard bending term, that compression helps a three-ring and a four-ring and hinders every ring from five up — predicting a slowing of 0.754-fold for the five-ring measured to speed up 250-fold. The account with the right sign is about rotations rather than angles, and it has a ceiling of 36.5 that the measurement is already above.

shape · Strain The ceiling rises and the measurements fall, so they cross. The largest acceleration the rotamer account can produce, against the ring being closed, with the measured gem-dimethyl accelerations on the same axis. Closing a bigger ring means freezing more rotations, so the ceiling rises steeply; the measurements go the other way. The five-membered ring's 250-fold acceleration is above its own ceiling of 36.5 and the six-membered ring's tenfold one is far below its 121 — so the account is refused at one size and sufficient at the next.

A ceiling that rises where the measurements fall

The rotamer account of the gem-dimethyl effect has a largest possible acceleration, which looks like a limitation. It is a prediction: the ceiling is a closed form in the number of rotations a closure freezes, it rises steeply with ring size, and the measurements fall — so the account is refuted for the five-membered ring and more than sufficient for the six.

shape · Strain The hybrids do not point at the atoms. For each cycloalkane, the angle between its two ring hybrids — fixed by orthogonality once the measured H–C–H angle has said how much s character the hydrogens take — against the angle between its carbons. Cyclopropane's differ by 45.5°, so each hybrid points 22.75° outside the bond it makes; cyclohexane's agree to 0.035°, which is the control.

The hybrids that point outside the bonds

Coulson's relation says two equivalent hybrids sharing an s orbital are orthogonal only between ninety and a hundred and eighty degrees. Cyclopropane's carbons make sixty, so its ring hybrids cannot point at the atoms they bond to — and the same relation says by how much they miss: 22.75 degrees each, falling to 0.03 in cyclohexane.

bonding · Hybrids How far the rotor count would have to be wrong. The ceiling against the number of rotations a closure freezes, with the two measured accelerations drawn across it. The five-membered closure freezes three and its ceiling is 36.46; the ceiling does not reach the measured 250 until 5 rotors, so the count would have to be wrong by 2 on a ring that has three rotations to freeze. The six-membered closure freezes four at a ceiling of 120.88, and stays above its measured 10 down to 2 — so the refusal is airtight and the sufficiency is comfortable.

An estimate that can be wrong by two

The ceiling on the gem-dimethyl effect is exponential in the number of rotations a closure freezes, and that number was taken as n − 2 without counting — which left the refutation at five rings probable rather than airtight. It is airtight. The ceiling does not reach the measured 250 until five rotors, on a ring that has three, and no hindering of the tether can raise it.

shape · Strain The carriers a distortion was hiding. A half-filled ring of 40 — one of the 4m rings that carry exactly one pair of carriers at every temperature — allowed to distort. Cold, it alternates by 0.1232, opens a gap of 0.4927 and carries 3.6e-15 carriers rather than one pair. The alternation is undone continuously at kT = 0.1358, and the carrier count comes back as it goes.

The carriers a distortion was hiding

A half-filled ring of 4m carries exactly one pair of thermal carriers at every temperature, which is true only of a ring held rigid. Allowed to move, it does not carry them: it alternates, opens a gap of 0.4927, and carries none at all until a temperature that undoes the distortion.

wrong · Metal The chain distorts hardest where it stops. The alternation of each bond along a relaxed chain of 64, at four elastic constants, with the bulk value of each drawn as a dashed line. Every chain alternates more at its end than in its middle — by 1.21 times at the stiffest and 3.15 at the softest — and the excess dies away over a handful of bonds. The uniform alternation usually assumed is the flat part of these curves.

The chain distorts hardest where it stops

Holding the alternation uniform is what made the end energy a clean constant, and it is the one assumption the end-energy calculation had to make. Letting every bond find its own value shows the distortion is largest at the end and decays inwards over a measurable length — one and a half bonds in a strongly dimerised chain, five in a weak one.

solids · Peierls distortion The alternation a spring buys, three ways. The alternation against the elastic constant, on a logarithmic axis. The middle line solves (2/π)(K − E)/(1 − δ²) = K for the complete elliptic integrals; the lower one is the exponential form every account of a Peierls distortion quotes, which is its own asymptote and is 6.5 per cent low at K = 1.2; the upper one is a ring of 40, which leaves the infinite chain as the spring stiffens because a smaller alternation is a longer coherence length.

The amplitude the collapse left behind

Five Peierls curves became one curve when each was divided by the alternation its ring settles at cold, so that amplitude is the whole of what distinguished them — and it was five golden-section searches over diagonalisations with no formula anywhere. It has one, exactly, as a sum of square roots; and writing it down says that one of the five rings was never measuring a long chain.

wrong · Metal

Named alongside it

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

Model limitLocal minimumBond angleThermodynamic limitBand gapClosed formBond alternationDegeneracyConformationDistortionInternal coordinateJahn–Teller distortion

All concepts