Molecular Geometry & Bonding
what the shape decides
Concept

Electron-deficient bonding — where it appears

Having fewer electron pairs than there are connections to hold together. Such molecules are held by multicentre bonding rather than by two-centre bonds, which is the general case rather than an anomaly.

Named by 12 essays across 3 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 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 F s on sulfur hexafluoride: a₁g ⊕ eg ⊕ t₁u. The character of the basis under each class of operations, which is a count of what did not move, and the multiplicities that come out of the reduction formula. The multiplicities must be whole numbers, and that is the check.

Six bonds and four orbitals

The six fluorine σ functions of sulfur hexafluoride span a₁g ⊕ eg ⊕ t₁u. Sulfur's 3s and 3p supply a₁g and t₁u and nothing else, so four bonding orbitals hold twelve electrons across six bonds — a bond order of two thirds, computed from characters with no energy anywhere in it.

symmetry · Hypervalency 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 Three orbitals in a line, four electrons in them. The three levels of a linear three-centre system, with the two lowest filled. The lower one is bonding across all three centres; the second has exactly zero amplitude on the middle atom, by symmetry rather than by arithmetic, so the two electrons in it sit entirely on the ends. That is where the charges come from: -0.5 on each end and +1 in the middle. Each of the two bonds has an order of 0.7071, which is 1/√2 and not the half the electron count suggests.

Hypervalency is about the ligands

The three-centre four-electron bond is offered as the reason sulfur hexafluoride needs no d orbitals. Read it forwards instead of backwards and it is a requirement rather than a permission — the arrangement puts half an electron onto each ligand before any electronegativity difference is applied, which is why the hypervalent compounds are fluorides and SH₆ is not a compound.

beyond · Hypervalency 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 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 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

Named alongside it

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

Multicentre bondingClusterThree-centre bondingBond orderCanonical orbitalsHypervalencyLocalisationParticipation ratioUnitary transformationEigenvectorLocal minimumModel limit

All concepts