Molecular Geometry & Bonding
what the shape decides
Concept

Multicentre bonding — where it appears

Bonding by orbitals spread over three or more atoms, holding them together with fewer pairs than connections. It is the general case, and two-centre bonding is the special one in which a pair happens to be localised.

Named by 19 essays across 2 fields — each of them below, with the objects they name alongside it.

Which rings close a shell. Each ring is filled with its own number of pi electrons and asked whether the highest occupied shell came out full. Of the rings drawn here, C6 and C10 close — at 6 and 10 electrons — which is Hückel's 4n+2, produced here rather than recalled.

Where two-centre bonding stops

A bond between two atoms is a special case, not the general one. Rings, clusters and metals are held together by orbitals spread over many centres, and the arithmetic that describes them is the arithmetic already used for benzene.

beyond · Multicentre Rings of 6, 10, 20, 60 and the band at 2000. The Hückel levels of rings of 6, 10, 20, 60 atoms, all of them inside the same interval from −2 to +2, beside the density of states of a ring of 2000. The histogram is the computed levels; the line through it is the closed-form density, which diverges at both band edges.

The band limit

Every level of a ring of n atoms lies between −2β and +2β, however large n gets. The levels do not spread out as the molecule grows; they crowd into a fixed interval — and that crowding, computed, is a band with its density of states diverging at both edges.

beyond · Multicentre One metal orbital, two ligands competing for it. Metal–ligand bond orders in a three-orbital model as the left-hand ligand's interaction is turned up. Its own bond order rises and the bond order to the ligand opposite falls, from 0.62 at equal strengths to 0.42 at the strongest. Nothing else in the model can carry the effect: switch the second bond off and it vanishes exactly.

The trans influence is an overlap argument

Two ligands on opposite sides of a metal both bond through the same metal orbital, and there is only one of it. Strengthen one and the bond order to the other falls — computed exactly on three levels, and measured as a bond length that grows by a tenth of an ångström.

applied · Overlap Hückel levels of three-centre four-electron. 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.

Three-centre bonding, computed

Three orbitals in a line and four electrons: a bonding level, a level with exactly zero amplitude on the central atom, and an empty antibonding one. The middle atom never exceeds an octet, and the ligands carry the charge — which is why every molecule that needs this arrangement has electronegative ligands.

beyond · Hypervalency π bond orders in benzene. Each bond drawn with a thickness set by its computed pi bond order, and the number printed beside it. The orders come from the eigenvectors and cost nothing extra once the matrix is diagonalised.

What one pair can hold together

Put a single electron pair into a ring of any size and it supplies a total bond order of exactly two and a π energy of exactly 4β — three atoms, eight atoms or six hundred. Spreading a pair over more centres divides the bonding among them; it neither creates nor destroys any.

beyond · Multicentre 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 The bonds are what is left over. Thirteen carbonyls and clusters, with the total valence electron count, the number of metal–metal bonds that leaves over from eighteen per metal, and the number the crystal structure has. They agree for every cluster up to five metals. At six they disagree by one, in both entries tested — and the skeletal count in the last column, which is the rule boranes are analysed with, comes out at n + 1 for both, meaning a closed deltahedron, which an octahedron is.

The bonds are what is left over

Every metal wants eighteen electrons and a metal–metal bond gives one to each of its partners, so the number of bonds in a cluster is what is left over after the counting. It works for every carbonyl cluster up to five metals and fails at six by exactly one bond — where the other counting rule, the one boranes are analysed with, is right.

applied · Electron count 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 Strong outside, weak inside. The bond orders along each chain. A three-centre system has two equal bonds of 0.707 — not the one half the electron count suggests — and every longer chain alternates, strong at the ends and weak in the middle. The spread grows with the chain: 3:0.000, 5:0.211, 7:0.271, 9:0.296. Nothing here is about iodine.

Hypervalency does not stop at three centres

The three-centre four-electron bond is written up everywhere as an arrangement peculiar to hypervalent molecules. It is the first member of a family — five centres and six electrons, seven and eight — and the family predicts alternating bond strengths that the polyiodide crystal structures have.

beyond · Hypervalency Four, however many ligands there are. For each molecule, the number of ligand σ combinations that find a partner among the central atom's four s and p orbitals, against the number of ligands and lone pairs it has. The matched count rises along the diagonal and then stops at four, because there are four orbitals; everything above the ceiling is a pair with nowhere on the central atom to go, and that is what the word hypervalent names.

Four is all that s and p can match

Reduce the ligand σ set of ten molecules in each one's own point group and ask how many of its components transform as one of the central atom's four valence orbitals. The answer is never more than four — not by arrangement, in every geometry from linear to octahedral — and what is left over is n + L − 4, with exactly twice that many electrons in excess of an octet.

beyond · Hypervalency 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 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 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 Both criteria leave a gap, and Boys leaves a chasm. Every non-zero relative spread under each criterion, on one logarithmic axis, with the largest empty stretch shaded. Pipek–Mezey's runs a factor of 206; Boys's runs 2.8e+7 — seven decades, from numerical zero to a real spread with nothing between. So the bimodality belongs to the cages rather than to the functional, and under the second criterion the threshold matters even less.

A second criterion left a gap too

The localisation spreads are bimodal — an empty factor of two hundred around the degeneracy threshold — and the gap might belong to the criterion rather than to the cages. Boys localisation leaves a gap of seven decades on the same forty-eight pairs, so it belongs to the cages. But the two criteria disagree about six of them, in both directions, and the family's most ambiguous cage under one is exactly degenerate under the other.

beyond · Multicentre 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 Forty-eight rows, thirty-three calculations. Every cage-and-filling pair the family survey covers, one cell per pair, with cells that hand the localisation the same set of orbitals joined. A localisation here selects its occupied orbitals by asking which have any occupation at all, and Hund's rule puts one electron into each member of a degenerate shell before pairing any of them — so adding two electrons to a half-filled shell pairs a spin and changes nothing the search can see. Fifteen of the forty-eight rows repeat an input already in the table.

Six disagreements and three calculations

Two localisation criteria classified six of the family's forty-eight cage-and-filling pairs differently, and three of the six were consecutive fillings of one icosahedron with spreads identical to three figures. They are not three coincidences and not one degeneracy being filled: they are one calculation, because the survey's forty-eight rows are thirty-three distinct questions.

beyond · Multicentre The boundary falls inside a shell, and the shell has no inside. The 12-vertex cage at 10 electrons: its Hückel levels, with each degenerate shell drawn as its members and the occupied ones filled. The occupied set takes 2 of the 5 members of one shell — and the members of a degenerate shell are not distinguishable. Whichever combinations the eigenvalue routine happened to return are the ones that get occupied, so the density being localised is a choice made by a diagonaliser rather than a property of the cage.

The basis a diagonaliser happened to return

Six of the cage family's thirty-three inputs have their occupied set cut through the middle of a degenerate shell, and the members of a degenerate shell are interchangeable. Re-orienting the shell changes nothing about the cage, the filling or the criterion — and it moves the best localisation functional by up to twenty-one per cent, moves the count of descriptions from ten to fourteen, and flips the degeneracy label on three of the six.

beyond · Multicentre

Named alongside it

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

Electron-deficient bondingLocalisationDegeneracyModel limitClusterThree-centre bondingUnderdeterminationBond orderElectron countLocal minimumUnitary transformationApproximation

All concepts