Molecular Geometry & Bonding
what the shape decides
Concept

Local minimum — where it appears

A point where the energy is lower than everywhere nearby but not necessarily lowest overall. A minimisation returns one, so a search that starts in the wrong place converges neatly on the wrong answer.

Named by 24 essays across 4 fields — each of them below, with the objects they name alongside it.

Eight points: the cube loses. The cube and the minimised arrangement of eight points on a sphere, with the repulsion energy of each computed. The minimum is a square antiprism — the cube twisted by forty-five degrees on one face — and the margin is about one part in three hundred.

The shapes above six coordination

Eight points on a sphere do not arrange themselves in a cube. They twist one face by forty-five degrees, and above six the arrangements stop being the ones anybody would name and start being the ones a minimisation finds.

shape · VSEPR Angles at exponents 1, 2, 3, 6, 12. The distinct angles of the minimised arrangement of 4, 5, 6, 7 points, under a repulsion going as one over r to the power 1, 2, 3, 6, 12. Where the arrangement is the maximally symmetric one the angles do not move at all; where it is not, both the angles and how many of them there are depend on the law assumed.

Which angles are symmetry and which are the model

VSEPR says electron pairs repel and never says by what law. For four, five and six domains it makes no difference whatever — change the exponent by a factor of twelve and not one angle moves. For seven it decides the answer.

shape · VSEPR A 6-ring at 111°: the twist-boat. A closed ring of 6 equal bonds meeting at 111°, drawn from the coordinates the closure conditions produce. Its torsions are 16.9°, -63.6°, 44.2° and repeat; its puckering amplitude is 0.508 bond lengths at a phase of 344°, with q₃ exactly zero, so it lies on the equator. Turning the ring changes none of those numbers, which is what makes them a description of the shape rather than of the view.

The ring that cannot hold still

Cyclohexane's chair is rigid and its boat is not, and that is a statement about the rank of a matrix rather than about strain. Hold every bond length and every bond angle fixed and count what is left: the chair has nothing, and the boat sits on a continuous loop of shapes with the same bonds and the same angles.

shape · Strain 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 How many centres, by two measures that do not agree. For each of eight systems, the number of atoms one localised pair has real amplitude on, and its participation number — which weights those atoms by how much of the pair each holds. Where the sharing is even the two coincide; where it is not they differ by more than a whole centre, and that gap is what electron deficiency looks like from the inside. one of the systems has more than one localisation, so for it neither number is an answer.

One scale, from two centres to a cage

How many centres a pair of electrons holds together is two different numbers, and they separate exactly where the bonding is most deficient: the methyllithium tetramer's pairs sit on four atoms each and have a participation number of 2.54. Run the same measurement up the scale and the twelve-vertex borane refuses it — its localisation has at least ten maxima differing by six parts in a thousand, so the number of centres is not an output for it at all.

beyond · Multicentre Ten answers, and none of them is another one turned round. Every localised description the search found for a twelve-vertex cage, placed by the value of the functional it maximises. There are 10 of them, spanning 0.02, and the two closest differ by 0 — far more than the 10⁻¹³ the sweep converges to, so they are different maxima rather than one maximum reached to different precision. Each has its own multiset of participation numbers, and a symmetry of the cage permutes sites without changing that multiset, so no two of these are related by one. The controls above find one answer each.

How many descriptions a cage has

A localisation is a maximisation, and running it once reports the maximum it reached rather than the maximum there is. Run to exhaustion on a twelve-vertex borane it finds ten answers and then three batches of twelve starts in a row that find nothing new — and none of the ten is another one seen from a different side, because a symmetry of the cage cannot change a multiset of participation numbers and all ten multisets differ.

beyond · Multicentre 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 shallowest slope, and the steepest. For three molecules, the softest and stiffest vibrations, the species each belongs to, and what a unit distortion along each costs. The cost is not taken from the frequency: it is computed by resolving the distortion onto the normal coordinates in the mass-weighted metric and adding up ω²q². That it comes back as the frequency is the identity the whole comparison rests on — the species decomposition uses only the coordinates and the group, and the cost uses only the masses and the force constants, and the two have to agree on this case before they can be asked to disagree on any other.

The coordinate it was already soft along

A distorted structure can be resolved into the symmetry species of its ideal group, and that resolution cannot say which coordinate a molecule fell down by itself and which one something outside pushed it along. The force field answers that, and the two halves agree on the case where they must — a unit distortion along a normal mode costs exactly that mode's frequency, to a part in a million, computed from masses and force constants by one side and from coordinates and characters by the other.

symmetry · Point group 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 Four bonds, or one a₁ and three t₂. The same four occupied orbitals written in two bases. On the left each orbital sits on one bond; on the right one is shared over all four hydrogens and three follow the Cartesian directions. The transformation between them is orthogonal, so the density is unchanged.

The answer a search is most likely to give

If a cage has ten equally good localised descriptions, why does the literature agree about its picture? The hoped-for answer was that one basin is very large. Counting four hundred and eighty starting points says it is not: two independent searches agree one time in ten, and the description they most often return is not the best one.

beyond · Multicentre Every arrangement, and the winner is not the one with the most unlike bonds. All 1820 ways of raising 4 of 16 sites on a wrapped square net, at a contrast of 4, each placed by its count of unlike bonds against the binding it gives. The best arrangement has 12 unlike bonds where 16 is available, and it binds at 1.103953 against 1.080031 for the best of those that do have the most. The count and the spectrum are two different orderings.

The arrangement a count cannot pick

Three arrangements of one composition came out ordered by their count of unlike bonds, which looked like a rule. Enumerating every arrangement instead of three shows it is not one: it holds at every composition on a square net at a small contrast, fails at four of them at a large contrast, and fails on a triangular net at any contrast at all.

solids · Cohesion 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 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 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 The count is a curve, and the curve flattens. How many distinct localised descriptions of a twelve-vertex cage had been found after each batch, out of 4,000 random starts. The last new one appears at start 124; the remaining 3,876 add nothing. The dashed curve is what the basin sizes measured here predict — the chance of having hit each description at least once — and it is a closed form rather than a fit to the points.

Where the count stops being an effort

A cage's localised descriptions were counted by a search that stopped when it stopped finding new ones, which makes the count a property of the stopping rule. Run to four thousand starts the count is fourteen and the last new description appears at start 124 — after which three thousand eight hundred and seventy-six starts add nothing. The four rarest are found six times in a thousand, which is what says nothing rarer is hiding.

beyond · Multicentre Three basin populations from one search. The share of 4000 random starts landing on each localised description, in rank order, on a logarithmic axis, for three cages and fillings. a twelve-vertex cage at twelve electrons gives 14 descriptions and two groups with a gap; a nine-vertex cage at eight electrons gives 63 descriptions and a smooth tail; a twelve-vertex cage at twenty electrons gives 15 descriptions and one basin and dust. Only the first is the twelve-vertex shape, and the rule it suggests is a rule about that shape.

Three shapes from one search

A cage's fourteen localised descriptions fell into two groups with a factor of five between them and nothing in the gap, which made counting the big basins look like a stopping rule. Two more cases from the same family give a smooth tail over sixty-three descriptions and a single basin holding ninety-eight per cent. One search, one family, three shapes.

beyond · Multicentre How many places a local search can stop. The number of distinct arrangements a steepest-ascent search settles at, for each net and contrast, with the fraction of starts reaching the best of them written beside it. 16 sites, contrast 1: 2 from 200 starts, best reached 67 per cent of the time; 16 sites, contrast 4: 2 from 200 starts, best reached 59 per cent of the time; 36 sites, contrast 1: 4 from 20 starts, best reached 80 per cent of the time; 36 sites, contrast 4: 12 from 20 starts, best reached 10 per cent of the time. The larger net at the larger contrast is a different kind of landscape.

Twelve basins where there were two

Sixteen sites can be searched exhaustively and thirty-six cannot, so the only instrument available past about twenty sites is a local search — and how much weaker it is has never been measured against the answer on a net where both can be run. On sixteen it is barely weaker at all. On thirty-six it fails nine starts in ten, and the arrangement it is beaten by has fewer unlike bonds than the one it starts from.

solids · Cohesion Three cages, three answers, and none of them reassuring. For each cage: how many descriptions the search finds, what share the commonest takes, how far the whole set spreads in the functional, whether the commonest is the best, and the verdict. A search that always agrees with itself is agreeing about a choice that does not matter; a search whose descriptions genuinely differ does not return the best one.

Fifty descriptions of one molecule

A search whose largest basin takes ninety-eight per cent of its starts will report one description however long it is run, and nobody runs four thousand starts when the first fifty agree. Whether the rare ones are worse descriptions or merely rarer is one number per description, already computed and never looked at. On two of three cages they are not worse — they are the same answer, to parts per million.

beyond · Multicentre How many descriptions, against how much they differ. Every cage-and-filling pair, by the number of distinct descriptions its localisation finds and by how far apart they are in the functional. If the count measured ambiguity the points would rise from left to right. The case with the most descriptions — 44 of them — has a spread of two parts in a hundred thousand, and sits at the bottom right.

Counting was right except where it mattered

A cage whose localisation gives dozens of descriptions that are all the same answer raises a worry: if degeneracy is common across the family, counting descriptions is the wrong measure of ambiguity. Across forty-eight cage-and-filling pairs it is the right measure on eleven of the thirteen that have anything to count — and it fails on the one leaned on hardest.

beyond · Multicentre 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.

solids · Cohesion 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.

solids · Cohesion 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.

solids · Cohesion

Named alongside it

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

Model limitApproximationBond angleConvergenceDegeneracyElastic energyMinimisationUnderdeterminationClosed formLocalisationTight-binding modelsCanonical orbitals

All concepts