Molecular Geometry & Bonding
what the shape decides
The collection

Every essay — page 27

Page 27 of 42, continuing through the fields in the same order.

Orbitals Where the atoms go Bonding models What symmetry decides Beyond the octet What a spectrum settles When the molecule does not stop What the shape is for What is taught wrongly Series Named objects Orbitals Refutations Search

When the molecule does not stop

A chain of two hundred atoms is a molecule and behaves like a solid. Bands, gaps, metals, defects and surfaces, every one of them out of a finite matrix — and a clear account of what that route cannot reach.

Seven nets, and the one column that sorts them. Every wrapped net here, with its dimension, coordination, third moment and band shape, beside the exponent the coupled-band measurement returns for it. The 4 nets whose third moment vanishes all give an exponent within 0.03 of −2 and a quotient constant to a fiftieth of a per cent; the 3 that do not all give one near −1 and no constant at all. Dimension does not sort them and neither does coordination — each takes values in both groups.

Seven points that looked like a switch

A constant belongs to a square net and not to a triangular one, which points at the coupling graph. Seven wrapped nets say which property of it: the third moment, and neither the dimension nor the coordination. They also say it is a switch — and made continuous, it is a crossover that every one of the seven sits twenty-five times past.

6 figures Nine combinations, and the column that sorts them is not the bands'. Two bands and a coupling, varied separately. The composite graph's third moment splits into triangles that lie inside a band and triangles that use two coupling bonds, and only the second sorts the table: every row with no gap-crossing triangle gives an exponent near −2 and a nearly constant quotient, whatever its bands are made of. Triangular bands carrying an intra-band moment of 7.296 behave exactly like square ones when the coupling is a matching.

The triangles that were never in the bands

A switch in how a gap scales is usually attributed to a band's third moment, and the attribution cannot be tested while the coupling runs along one of the bands. Separated, the bands turn out to decide nothing. Two triangular bands coupled along a matching — which cannot close a triangle across the gap — behave exactly like square ones.

6 figures The count, for five run lengths. The fraction of runs whose nearest neighbour of the same length is coupled more strongly than a threshold, against how many decades below the closest possible coupling that threshold sits. Every curve is a straight line over this range, because the fraction is small and the geometric tail is linear in the separation there. The slope is what the next figure is about: it is the whole content of the distribution, and it is a product of two numbers.

A count rather than an average

Two couplings quoted from one distribution sit a hundred and seventy decades apart. Neither is a summary of it. The quantity that decides how much of a spectrum near a box level is resonant pairs is a count of pairs above a threshold — and it has a closed form, which is a density times a reach times the logarithm of ten.

6 figures How often a random start reaches the best of them. The share of random starts that reach the best arrangement found, against the fraction of sites raised, for two nets at two contrasts. Every curve dips in the middle of its left half and recovers: the hard compositions are between a quarter and a third, and the half-filled one — the rightmost point of each curve — is among the easiest. The hardest points are 5 of 16, 6 of 16, 9 of 36, 12 of 36.

The composition that is hard is not the full one

A landscape of arrangements measured at one composition on each of two nets raises the question of where the hardest one sits — and the natural guess is the half-filled one, where there is most to arrange. It is the easiest. On thirty-six sites at a contrast of one, half filling has one local optimum and nine billion arrangements, and a quarter filling has six optima and a hundredth as many.

6 figures The exponent against how far the tail can be seen. The fitted power against the reduced reach — how many coherence lengths of the decay survive above the floor before the excess is numerical noise. The six cases with a reach past six give a power between 0.41 and 0.57 and are drawn solid; the rest are hollow and their fitted powers are off this scale in the negative direction. The reach is not a choice — it falls as the gap closes, because the excess the tail starts from falls with it.

The exponent was the floor

Fitting the local decay rate against the reciprocal distance reads a power off the slope. It runs from 0.41 to 0.66 across ten stiffnesses and appears to settle near two thirds. It is not settling. The tail is dropping below the arithmetic's own floor sooner at every step, so each case's power is taken over a shorter piece of the curve than the last.

6 figures The same collapse, with the cases made comparable. The alternation divided by its own cold value against the reduced temperature, for the published five cases and for five chosen so that every ring is the same size in its own alternations. The published set agrees to 3.41 per cent and the matched set to 1.62 — so the residual left was finite size, as suspected.

Five rings that were five different sizes

Five warmed rings have scaled alternation curves that lie on one another to 3.41 per cent, and the departure from the bulk amplitude turns out to be a function of the ring measured in its own alternations. The five cases span a factor of seven in that quantity. Choosing sizes that make them comparable halves the residual — and runs into a floor the lattice itself imposes.

6 figures The exponent against where the fit is allowed to start. For each stiffness, the fitted tail exponent as the near end of the fitting window is moved outward from two bonds to thirty. The standard start is six, by a rule of thumb — three coherence lengths — and the question was whether that choice is doing any work. Below six the exponent rises steeply; from six outward it is nearly flat. The rule of thumb sits on a plateau.

The rule of thumb was on the flat part

Fitting the Peierls tail discards the first few bonds of every profile, on a rule of thumb — three coherence lengths. Does that unexamined choice hide a second exponent? It does not. From six bonds outward the fitted power moves by half a per cent to nine; below six it moves seven times as much, and starting at two would have halved the very trend the fit reports.

6 figures 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.

6 figures Three targets, and the residual keeps falling. The worst spread across the five scaled curves, at three values of the matched product n·δ∞. It falls from 2.51 per cent at 5 to 1.62 at 9.6, monotonically. The unmatched cases sit at 3.41 per cent throughout, because they are the same five rings whatever target is being aimed at — which is what makes the comparison a comparison.

Three points, and they all go down

Matching five rings at one value of n·δ∞ tightens the temperature collapse from 3.41 per cent to 1.62, and what is left might be the even-site rounding rather than anything physical. At three targets the residual falls monotonically — and at the smallest one it is a third of what the rounding leaves, which the rounding cannot explain.

6 figures The far end stops mattering, abruptly. The fitted tail exponent as the far end of the window is opened from twenty bonds to a hundred and twenty. Each curve is flat past a stiffness-dependent point and exactly flat past it — because beyond a profile's own reach there are no more local rates to add, so a larger window is the same fit. The standard sixty bonds is inside the flat part for every case.

The other window was a plateau too

Sweeping where the fit begins finds a plateau. The far end is the other window and nobody had swept it: inside each profile's own reach the exponent moves by at most 5.3 per cent, and past that reach every larger window returns exactly the same fit — because there are no more points to add. What the reach is depends on the stiffness, and for half the series it is an arbitrary rule rather than the physics.

6 figures One net has a two-colouring and the other cannot. Sixteen sites wrapped into a square net and into a triangular one, with the wrapping bonds left undrawn. The square net is bipartite: its sites split into two classes with every bond running between them. The triangular net is not, and the obstruction is a triangle — three sites in a cycle of odd length cannot be two-coloured. It has thirty-two of them, two per site, and forty-eight bonds against the square net's thirty-two.

A net with no two-colouring

Half filling is the easiest composition to search and the reason given was the two-colouring: on a bipartite net it is the unique arrangement with every bond unlike, so the optimum has nothing competing with it. A triangular net has no two-colouring, its half-filled composition is reached by every one of two hundred starts, and its best arrangement is two bonds short of what counting allows.

7 figures On the frustrated net the widest gap wins everywhere but just past a change of winner. Every basin found at half filling on the triangular net, at eleven contrasts, placed by the gap it opens at the Fermi level; the winning basin is filled and the rest open, sized by how many of two hundred starts reach them. Up to a contrast of 2.5 there is one basin. From 3 a second appears, and the winner is the one with the wider gap — except at 4, where the winner has a gap of 4.947 and a runner-up has 5.088. Squares mark basins with thirty-two unlike bonds and circles thirty.

The gap follows the winner late

On a triangular net at half filling, the arrangement that binds best was expected to be the one that opens the widest gap at the Fermi level. Across twenty cases it is, eighteen times. The two exceptions are not noise: the frustrated net's winner changes at a contrast of 3.790, from an arrangement with thirty unlike bonds to one with thirty-two, and the gaps of the two do not cross until 4.849. For a whole unit of contrast the better binder has the narrower gap.

6 figures