Molecular Geometry & Bonding
what the shape decides
Concept

Relaxation — where it appears

Letting a structure or a wavefunction respond to a change instead of holding it fixed. A relaxed answer is always lower in energy than a rigid one, so the difference between them is a quantity in its own right — and it is the part of a response that a single-point calculation leaves out.

Named by 7 essays across 3 fields — each of them below, with the objects they name alongside it.

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.

bonding · Delocalisation 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.

bonding · Delocalisation Half the series stops on its own; the other half stops where it is told. How far from the chain's end the local decay rate can still be read, against how much of the chain the reading is allowed to cover. The dashed line is the cap itself. A curve that flattens below it has ended on the noise floor and the window never mattered; a curve that tracks the cap is being cut, and has more to say. The soft chains do the first and the stiff chains do the second.

The window that was not a plateau

The near and far ends of this fit both sit on plateaus, and it is tempting to expect the same of every window. The third choice — that the local decay rate is read from the first quarter of the chain — is not a plateau. It never bound the soft half of the series and it was setting the answer for the stiff half, where opening it moves an exponent by a seventh, always downward. And a profile allowed to end on its own always runs 13.26 of its own decay lengths, which is the ruler that shows three cases are still cut at the midpoint — four, once a stiff profile is counted only to the bond where its local rate turns back.

solids · Peierls distortion On an opened profile the local rate stops falling, and turns. The local decay rate at every bond, read to half the chain, for three stiffnesses. The soft chain's rate falls all the way to the last bond that produces one, where its excess meets the noise floor. The two stiff chains' rates reach a minimum and climb, because the excess is measured against a bulk value read from the chain's middle and the middle still carries both ends' healing. Every bond past the minimum is reading that reference, not the tail.

The rate that turned back

Opening the Peierls fit's window to half the chain added forty to eighty bonds to every stiff profile, and those bonds were read as tail. Most of them are not. The local decay rate on an opened stiff profile falls to a minimum and climbs again, because the bulk it is measured against is read from the chain's middle, which still carries both ends' healing — and a model of two ends and that reference, built from the quarter window alone, puts the minimum within two bonds of where it is. Read only to the turn, the fall of a seventh in the stiff exponents disappears, and one of the two cases the wider window was credited with resolving is not resolved.

solids · Peierls distortion Three bulks, and only one is the chain's. For each of the ten stiffnesses, how far three candidate bulk values sit from the bulk at the chain's own shift, on a logarithmic axis. A free infinite chain and a free ring of 320 sites are off by 10⁻⁵ to 10⁻³ at every stiffness — up to thirty thousand times the relaxation's floor of 10⁻⁸ — because neither has the shift the chain's fixed length imposes. The chain's own middle agrees with the shifted bulk to 10⁻¹¹ at every stiffness to 2.0 and then rises above it, by the two ends' tails, to 10⁻⁴ at the stiffest.

A ring is not the bulk

The local decay rate on a stiff Peierls chain turned back because its bulk was read from the chain's own middle. The remedy proposed was a ring of the same length, with no ends to heal from. A ring gets the soft chains wrong by thirty thousand times the floor — because the chain's relaxation holds its length fixed, and the ends' extra alternation is paid for by a shift in every bond that goes as one over the length. With that shift in the bulk, the soft middles agree to 10⁻¹¹, no stiff profile turns back, two tails describe every profile to its middle, and the stiff exponents keep rising where the published ones had turned down.

solids · Peierls distortion 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.

beyond · Delocalisation Freeing the chain's length leaves every tail exponent where it was. The tail exponent p of a two-ended fit A·b^(−p)·e^(−b/λ) from bond 6, at ten stiffnesses, for the chain whose total length is held fixed and for the chain at zero tension. The fixed chain's middle is compressed by an amount that goes as one over its length; the free chain's is not compressed at all. The exponents differ by at most 0.0009 — the constraint was never in them.

The chain that pays for nothing

Held to a fixed length, a Peierls chain pays for its ends' extra alternation with a compression of every bond. Let the length go and nothing is paid: the middle is the free infinite chain to twelve figures, the ends adjust locally — and every tail exponent comes out where the fixed chain put it, to under a thousandth. The constraint was never in the exponents. Nor is the drift in them a missing 1/b term: fitting one makes the exponent wander three times further.

solids · Peierls distortion

Named alongside it

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

Model limitDelocalisationApproximationFinite-size effectPeierls distortionBond orderHückel theoryLeast-squaresReference stateBond alternationConvergenceThermodynamic limit

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