Molecular Geometry & Bonding
what the shape decides
Concept

Band gap — where it appears

The energy between the highest occupied level of a solid and the lowest empty one, when that energy does not shrink as the structure is made larger. It is the cost of promoting one electron with the atoms held still, and is not convertible into a bond energy.

Named by 24 essays across 5 fields — each of them below, with the objects they name alongside it.

Where two bands lie, as their centres are pulled apart. The σ band and the π band of a two-orbital chain, drawn as the intervals they occupy, against the difference in site energy between the two orbitals. Below a difference of 3 the two intervals overlap and the filled-band count stops deciding anything.

A full band is not an insulator

Two electrons per atom, two orbitals per atom, and the lower set exactly full: the count says insulator, and magnesium is a metal. The count is not wrong about the count. What it assumes is that the two sets of levels occupy separate ranges of energy, and whether they do is a comparison of four numbers that has nothing to do with how many electrons there are.

wrong · Metal 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 Two kinds of answer to a flux, and only one of them is a curve. The π binding of three rings against the magnetic flux through them, in units of beta and of the flux quantum, measured from each ring's own value at zero flux. Beta is negative, so a binding that FALLS is an energy that rises: benzene's does, which is what a diamagnetic ring current is. Cyclobutadiene's rises in both directions from a corner — its energy has no second derivative at zero field at all, and the two one-sided slopes differ by 12.57.

Two rules that share no arithmetic

Hückel's rule is a statement about a gap. Put a magnetic flux through the same rings instead and ask which of them push the field out, and the answer is the same set — eighty cases, no exceptions — although the second calculation counts nothing and has no shells in it. And the rings the rule excludes turn out to have no magnetic susceptibility at all: their energy has a corner at zero field.

symmetry · Aromaticity How many times more a metal carries, and when. The ratio of the carriers in a uniform ring to those in an alternating one of the same size, at four sizes and six temperatures. Down the left-hand column the two are indistinguishable, because a ring of 42 at kT = 0.002 has a level spacing larger than the temperature and is no more a metal than the gapped one is. Along the bottom row they are indistinguishable again, because the temperature is larger than the gap. The word only means anything in the middle.

The metal a thermometer cannot find

A metal is a system with excitations of arbitrarily small energy, which is a claim about a sequence of finite systems rather than about any one of them. Put a temperature on it and the claim needs a second limit, and the two do not commute: a uniform ring of forty-two at kT = 0.002 carries exactly as much as an alternating one, and at kT = 0.1 a ring of three hundred and twenty-two carries only four times as much.

wrong · Metal How anisotropic a structure has to be. The fourth moment of a layered band, divided by the square of its second so that the width drops out, against the coupling between layers. The curve is the closed form that follows from adding cumulants — 3 − 3(2 + λ⁴)/(2(2 + λ²)²) — with nothing fitted; the points are eigenvalues. At λ = 1 it is the three-dimensional value 2.5 and at λ = 0 the two-dimensional 2.25, and it is halfway between them at λ = 0.47 — so the question of when a band is effectively two-dimensional has a number rather than an opinion.

Two bands, and the shape of each

A band's shape has a closed form when the structure it belongs to factorises. Put a second band beside it and the form survives exactly while the two do not talk, and fails as the square of the coupling once they do. Make one direction weaker instead and the form survives everywhere — 3 − 3(2 + λ⁴)/(2(2 + λ²)²), agreeing with eigenvalues to nine decimal places — so how anisotropic a structure has to be before its band is two-dimensional is a number: λ = 0.46518.

solids · Bands in a solid A mixture is not the average of its ends. The binding per site of a square net whose sites are of two kinds, against how many of each. The straight line is arithmetic rather than a fit: a structure of one kind only has every level shifted by ±δ, so the two ends and the line between them are known before anything is diagonalised. Every mixture lies above it — more bound — by as much as 0.43 per site at the middle, and that departure is the whole of what makes an ordered compound worth forming.

A mixture is not the average of its ends

Half of a structure's sites raised and half lowered, and the line between the two pure ends is arithmetic — no diagonalisation needed. Every mixture lies below it, by 0.42987 per site for the ordered arrangement and 0.23690 for the segregated one at the same composition, so composition fixes neither the binding nor the gap. And the departure is not quadratic in the contrast: it grows as its 1.79 power in a chain and its 1.28 power on a square net.

solids · Cohesion 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 Where the two halves of an alloy come apart. The gap at the centre of a binary alloy's band against the contrast between its two components, for a chain of 2048 sites arranged three ways. The ordered arrangement's gap is exactly twice the contrast and opens at once; the segregated one's is exactly the contrast less the band width and opens at 2; and the random one, which has no closed form, opens a little before the segregated one and stays a little wider. Below the openings the curves sit at one level spacing rather than at zero, which is what a finite chain has instead of a gap.

Two bands, if the chain is short enough

Take a chain, raise half its sites and lower the other half, and ask when the band comes apart into two. The ordered arrangement splits at once, the segregated one at a contrast equal to the band width, and the random one splits earlier than either — and then closes again as the chain is made longer, because a long chain contains a long run of like atoms and a long run is a narrow sub-band.

solids · Defect 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 slope goes to a half, and a window fit stops short of it. The local slope of the alternation against the reduced temperature, between each neighbouring pair of points, on a ring of 40 at K = 1.6. It rises monotonically from 0.4115 to 0.5053 as the transition is approached, crossing a half at about a part in a thousand of the reduced temperature. The fitted 0.44 is the average of the left-hand end of this curve; the exponent is one half, which is what a free energy analytic in one order parameter is obliged to give.

The exponent was the window's

A fit over the last decade before a distortion vanishes gives an exponent of 0.44, and running it on larger rings should say whether the number belongs to the transition or to a forty-site ring. It belongs to neither. The local slope runs to 0.5020 as the transition is approached, and 0.44 is what a fit over that particular decade returns — on every ring size and every stiffness, because the whole curve is one curve.

wrong · Metal Two curves that cross, and two that do not. Every pair of parameters that reproduces one measured number, for three numbers, with the true system marked at Δ = 12 and t⊥ = 0.06. The lower band's shape is a function of the ratio of the two, so its curve is a straight line through the origin; the excess gap is a function of the coupling squared over the separation, so its curve bends. The two cross at one point. The upper band's shape draws a line almost on top of the first, because it is a function of the same ratio — a second measurement lying along the first fixes nothing the first had not already fixed.

Two ways of being second order

A band's shape and a band's gap are both second order in the coupling that mixes two bands, which sounds like a reason to measure only one of them. They are second order in different ways — one goes as the square of the ratio and the other as the square over the separation — and that single difference of one power is what turns a curve of possible answers into a point.

solids · Bands in a solid A state in the gap is a particle in a box the alloy happened to make. The participation ratio of the 40 levels nearest the gap centre, over 5 chains of 400 sites, against the length of the run of like sites each one sits on. The line is 2(L+1)/3, the participation ratio of the ground state of an isolated chain of L sites, with nothing fitted. 38 of 40 lie on it to within 3.1 per cent. The ones above it are states shared between two runs close enough to talk, which is a defect band beginning.

A particle in a box the alloy made

A random alloy's gap is set by the longest run of like sites. What sits at the edge of that gap turns out to be the simplest state in quantum mechanics: a particle in a box of L sites, occupying 2(L+1)/3 of them, to within three per cent and with nothing fitted — except for the two states in forty that found a second run to share.

solids · Defect A length that keeps growing, and one that stops. The fitted decay length of the ring-current response, against the number of rings. The bare acene's runs 1.397, 1.669, 1.896, 2.104, 2.302 — up by a factor of 1.65 and still climbing — while its own gap falls from 0.590 to 0.1102. With a gap held open the same measurement gives 0.678, 0.642, 0.636, 0.636, 0.639, which has stopped moving by the third molecule. There is a magnetic reach, and an acene is too nearly gapless to have one.

A reach that has no length

A ring current's response to a neighbouring ring falls with distance, which invites asking for the length. Every acene computed gives a longer one — 1.397, 1.669, 1.896, 2.104, 2.302 rings — because the gap that would set the length is closing at the same time. Give the same molecule a gap that stays open and the number settles at 0.636 by the third one and does not move.

symmetry · Aromaticity Two lengths off the same chains, and only one of them has an exponent. The length an end's influence reaches into a chain of 320, against the gap the bulk has opened, over ten elastic constants and a factor of twenty in the gap. Fitted over the first twelve bonds — as a short-window fit does — the exponent is -0.476. Taken from the local decay rate extrapolated to a bond infinitely far from the end, it is -1.029, and every neighbouring pair of points gives between -1.06 and -0.91. The argument says −1. The two lines are the same ten profiles read two ways.

A decay that keeps slowing down

A healing length fitted over the first twelve bonds of each relaxed chain goes as the gap to the power −0.60, against an argument that says −1. A fitted exponent can belong to its window, and this one does — but not because the window is too short for the chain. The profile is not an exponential at all, so there is no single length for an exponent to be of.

solids · Peierls distortion Where the coupling overtakes the spacing. Two quantities that both depend on the chain length, for runs of 6. The coupling is the splitting of the closest pair of runs, which rises as the chain grows because a longer chain brings some pair closer together. The spacing is the mean separation of the gap levels in energy, which falls as the chain grows because there are more of them. They cross at 8,675 sites, and a set of levels coupled more strongly than they are spaced is a band.

The length at which levels become a band

Two runs of low sites share a state when they are close enough, and the splitting falls exponentially over two sites. A longer chain brings some pair closer while spreading its levels thinner, so the two quantities run against each other and cross — at about a thousand sites for runs of four and eighty thousand for runs of eight. Chains of four hundred sites are far below that, which is why every gap state on them is a box.

solids · Defect 6 rings fused two ways. Two catacondensed chains of 6 hexagons: the linear one, where every fusion continues the line, and the angular one, where the fusions alternate. They have the same formula and the same number of bonds and they are not the same graph. Ring centres are numbered; in the linear molecule two rings k steps apart have centres 1.7321k units apart and in the angular one they do not, which is the whole reason this pair can be asked the question.

Neither of the two separations

A linear acene cannot pose the question, because the number of fusions between two rings and the distance between their centres are the same variable there. Bending the molecule pulls them apart — and the response follows neither. Two pairs of rings the same distance apart differ by two thirds, and the larger one is at the greater distance.

symmetry · Aromaticity 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 One heteroatom, and how far the relaxation carries it. The change every ring's bonds undergo when one carbon of the end ring is given a site energy, on a chain of 12 fused hexagons with its gap held open. The upper curve measures each molecule's couplings from its own mean bond order, which is what this collection's relaxation has always done; it stops falling at 4.48e-5 and stays there. The lower curve measures both from the same mean and keeps falling to 7.89e-9. Nothing about the molecule differs between them.

The floor was in the bookkeeping

A heteroatom in one ring of a fused chain is a perturbation with a place, and the relaxation carries it along the molecule. Measured directly, the response stops falling after five rings and sits at four hundredths of a millionth for ever. That floor is not the molecule. It is the relaxation measuring each system's couplings from its own mean, and removing it recovers nine orders of magnitude.

bonding · Delocalisation 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.

solids · Bands in a solid 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 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.

solids · Bands in a solid 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.

solids · Peierls distortion 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.

solids · Metal

Named alongside it

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

Tight-binding modelsModel limitClosed formThermodynamic limitBands in a solidBond alternationConventionDegeneracyLeast-squaresPeierls distortionEigenvalueExact diagonalisation

All concepts