Molecular Geometry & Bonding
what the shape decides
Concept

Graph — where it appears

A set of vertices with edges between them, which is all a Hückel calculation knows about a molecule. Level patterns, shell closures and defect states follow from the graph alone, with no chemistry entering anywhere.

Named by 18 essays across 6 fields — each of them below, with the objects they name alongside it.

⟨x²⟩ is the average number of neighbours. For each structure, the mean coordination counted off the edge list beside the mean of x² measured off the eigenvalues. They are equal by an identity about graphs, not by a limit — the full band width beside them obeys no such rule.

The width of a band is a count of neighbours

The mean of the squared level energies equals the average coordination — exactly, for any structure, with no limit taken and no periodicity assumed. It is the one statement in this field that is arithmetic rather than physics, and the usual textbook formula for band width is a special case of something weaker.

solids · Bands in a solid Chains of 4, 16, 64, 160: the levels crowd, the edges do not move. Every level of a chain of 4, 16, 64, 160 sites, drawn at its computed energy on one axis. Adding sites adds levels inside a fixed interval rather than widening it, which is what makes the large system a band rather than a wider molecule.

A band with no structure in it

Everything in this field is computed from a finite matrix with no periodicity assumed, which is a real method and a real limitation. It produces a band and cannot produce a band structure — and the difference between those two words is worth an essay, because it is the boundary of what a finite matrix can honestly say.

solids · Bands in a solid Hückel levels of benzene. The orbital energies of the pi system, computed as the eigenvalues of the molecule's adjacency matrix. Degenerate levels share a line. The electrons are placed by aufbau with Hund's rule, so a half-filled degenerate shell shows as two unpaired spins.

Hückel theory and what it gets right

A conjugated system's orbital energies are the eigenvalues of a matrix of ones and zeroes. Nothing about carbon enters, no geometry enters, and the results that survive are exactly the ones that depend on neither.

bonding · Delocalisation The radial set of a 6-vertex cage. The energies of the 6 radial orbitals of a deltahedron — one per vertex, pointing at the centre — as the eigenvalues of the cage's own adjacency matrix. The top level is nodeless and is the only one that is, which is Perron's theorem about a connected graph rather than an observation. Degenerate levels share a line. The electrons are placed by aufbau with Hund's rule, so a half-filled degenerate shell shows as two unpaired spins.

A cage needs one pair more than it has corners

Every closed borane holds n+1 skeletal electron pairs for n vertices, and the extra one is a theorem about connected graphs rather than an observation about boron. A cage's radial orbitals have exactly one nodeless combination, always, whatever its shape.

beyond · Multicentre One carbon, three lithiums, and no way to make it two. The four localised orbitals of the tetramer, with the share of each pair on each of the eight framework atoms. Each sits on exactly four: one carbon and the three lithiums of the face that carbon caps, which was not put in anywhere. The four shares are not equal — about 59 per cent of the pair is on the carbon — so the participation number comes out near 2.5 rather than at four. A bond over four centres is not a bond divided into four.

Four centres, and the pair that will not localise

The occupied orbitals of a molecule can be mixed freely without changing anything observable, and the freedom is usually spent on making them as local as possible. For the methyllithium tetramer the answer is four centres — one carbon and the three lithiums of the face it caps — and no mixing of the four pairs reduces it.

beyond · Multicentre The twist the ring forces on the bond. For each ring size, the largest torsion about the double bond that the ring will close on — 180° being a flat trans arrangement and 90° being a π bond broken outright. The six-ring will not close at any torsion tested. The eight-ring reaches 139.3°, against 136° measured in trans-cyclooctene by diffraction: a model with bond lengths and bond angles in it and no energy anywhere agrees with the crystal to a few degrees.

The double bond a ring cannot hold

A trans double bond needs a ring of nine carbons to sit flat, and the eight-ring will hold one twisted by 40.7 degrees — against 136 degrees measured in trans-cyclooctene. The same eight-ring threshold, applied by counting ring sizes, sorts nine bridgehead alkenes correctly with no bridgehead anywhere in the argument.

shape · Strain ⟨x²⟩ is the average number of neighbours. For each structure, the mean coordination counted off the edge list beside the mean of x² measured off the eigenvalues. They are equal by an identity about graphs, not by a limit — and the binding each bond supplies, in the last column, obeys no such rule and falls as neighbours are added.

The bond that weakens as neighbours multiply

Wrapped structures with two, four and six neighbours per site give a mean of x² of exactly 2, 4 and 6. Their binding per site goes 1.272, 1.611, 1.979 — and their binding per bond falls from 0.636 to 0.330, which is why a metal atom with twelve neighbours has weak bonds and a great many of them.

solids · Cohesion The same neighbours, and a fifth of the binding between them. five structures in which every site has 4 neighbours. Their second moments are identical — 4 for every one, which is the coordination and is what the band width is read from. Their bindings per site are not: they run from 1.28 to 1.64, and the least bound is the one with the most four-step walks.

Two structures with the same neighbours

Five structures in which every atom has exactly four neighbours. Their second moments are 4.000 to nine decimal places, because that identity is the coordination and nothing else. Their bindings per site run from 1.2756 to 1.6363 — a spread of twenty-two per cent — and two of them that agree on the second, third and fourth moments together still differ in the third decimal place.

solids · Cohesion A band becomes a bell curve, and the dimension is the sample size. The normalised fourth moment of a wrapped hypercubic structure's density of states against its dimension, with the closed form 3 − 3/2d drawn through it. They agree to eight decimal places at every dimension from one to 6, and the reason is a limit theorem: a hypercubic spectrum is the sum of d independent one-dimensional ones, so its cumulants fall as powers of d exactly as a sum of independent samples does. The Gaussian value of three is approached and never reached.

A band becomes a bell curve

The normalised fourth moment of a hypercubic structure's density of states is 3 − 3/2d exactly, from one dimension to six, to eight decimal places — because the spectrum is a sum of d independent one-dimensional ones and the central limit theorem is what it is obeying. Reach further in one dimension instead and the shape overshoots a Gaussian rather than approaching it: a chain touching its third neighbour has a cubic structure's coordination and a fourth moment of 3.389 against 2.500.

solids · Bands in a solid 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 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 Two bands, square below, triangular above. The 128 levels of a structure carrying two orbitals on every one of its 64 sites, where the lower orbitals hop on a square net and the upper ones on a triangular net, separated by 12 and coupled at 0.06. The two bands have different widths and different shapes before anything mixes, which is the situation a single net cannot construct. Every level here is also reproduced by a two-by-two problem, one per mode, to 3e-13.

The constant that belonged to one net

Divide one second-order departure by another and the coupling cancels, leaving one constant times the gap — which reads as arithmetic. It was arithmetic about a square net. On a triangular one — the same graph for both bands, nothing else changed — the quotient drifts by a hundred and twenty-eight per cent.

solids · Bands in a solid 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 Nine clusters, two candidate rules, and one of them survives. Every cluster's exponent, with whether it is frustrated and whether its count is even. Frustration is decided by whether the coupling graph is bipartite, since an antiferromagnet can satisfy every bond exactly when a two-colouring exists. All four decisive cases come out positive — 0.2136, 0.2084, 0.1924, 0.2048 — so the parity rule survives and the frustration account does not. The gap between the lowest positive exponent and the highest negative one is 1.0497.

The frustrated cluster with an even count

A parity account of the exponent's sign replaced a frustration account and left the two still confounded: every case tested had frustration and odd parity aligned. A tetrahedron of four spins is frustrated and even. Its exponent is +0.2084, and so are those of three more clusters the two accounts disagree about — but the mechanism proposed with the parity rule is refuted along with the account it replaced.

applied · Magnetism 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 Six pairs off the diagonal, and every one of them on an axis. Each cage-and-filling pair's Pipek–Mezey spread against its Boys spread, both logarithmic, with the two thresholds drawn. Agreements sit in the two opposite corners: spreads that are zero in both, or large in both. The six disagreements do not sit between them — they sit on the axes, with one coordinate at the floor. A criterion-dependent cage is not one the two criteria half-agree about; it is one where the difference between its descriptions is invisible to one of them entirely.

The cage is on both sides

Two localisation criteria classify six cage-and-filling pairs differently, and a symmetry explanation for the six is the natural first guess. The icosahedron's graph has a hundred and twenty automorphisms, the most in the family, and supplies three of the six disagreements and nine of the agreements. What the six do have in common is sharper than a symmetry: in every one, one criterion's spread is not small but zero.

beyond · Multicentre

Named alongside it

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

Tight-binding modelsModel limitCohesionBands in a solidDensity of statesEigenvalueEnergy per siteBand widthSecond momentAdjacency matrixClosed formCoordination

All concepts