Molecular Geometry & Bonding
what the shape decides
Concept

Unitary transformation — where it appears

A change of basis that preserves every inner product and therefore every observable. Localising orbitals is one, which is why the resulting picture is a description rather than a discovery.

Named by 19 essays across 7 fields — each of them below, with the objects they name alongside it.

methane — Td. The molecule with the point group found from its coordinates rather than looked up: every candidate operation was applied and kept when it permuted the atoms among themselves.

Hybridisation does not explain

Methane's photoelectron spectrum has two bands, not one. Four equivalent sp³ bonding orbitals cannot produce that, and the resolution is that hybridisation was never a claim about what a measurement would find.

wrong · Hybrids sp3 hybrids. The directions the hybrids point, with the angle between them computed from the coefficients rather than quoted. The set is an orthogonal transformation of the atomic orbitals, so it describes the same space in different coordinates.

Hybrids are a basis

An sp³ hybrid set is an orthogonal matrix applied to the atomic orbitals. Rotating a basis changes no observable, so asking whether the electrons are really in hybrids is asking which coordinate system nature prefers.

bonding · Hybrids Orbitals at the 90 per cent contour. Several orbitals drawn at the same enclosed fraction and, unless stated otherwise, at the same scale — so the sizes on the page are the sizes. Each contour was solved for separately by integrating that orbital's own density. Contours drawn: 2pz at 90% of its density, |ψ| = 9.48e-3; 2px at 90% of its density, |ψ| = 9.49e-3; 2py at 90% of its density, |ψ| = 9.49e-3.

Complex harmonics against real ones

The p orbitals every chemist draws are not eigenfunctions of anything. They are real combinations of the complex solutions, chosen because they point along axes — and the choice is invisible until a magnetic field makes it matter.

orbitals · Orbital 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 localisation transformation, demonstrated

Four equivalent bonds or one a₁ and three t₂ — two descriptions of methane's bonding electrons that disagree about everything except the electron density, which they agree about to the last bit a double can hold.

bonding · Hybrids Two answers from one projector: E1g. The E1g projection operator of benzene, applied to the pz function on one atom and then to the one on its neighbour. Both results belong to the same two-dimensional representation and span the same subspace; neither is more correct than the other; and they are different pictures, overlapping by 0.500. The circle areas are the coefficients and the two colours are their signs.

The projector is unique, the basis is not

A reduction says how many times each representation appears. It cannot say which combinations of orbitals they are — that takes a projection operator, and for a degenerate representation the operator returns a different pair of orbitals depending on which function it is handed first. Both pairs are correct, they span the same space, and every energy computed from either is identical.

symmetry · Representation 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 One double bond, two descriptions. A carbon–carbon double bond drawn twice in the plane perpendicular to the molecule: as a σ orbital along the axis with a π orbital above and below it, and as two equivalent bent bonds tilted 50.8 degrees either side of the axis. The two descriptions are related by a rotation and have the same density everywhere.

Two bent bonds, or a σ and a π

A carbon–carbon double bond is drawn two ways and the two look like rival claims about what is there. They are one occupied space in two bases, related by a rotation of exactly forty-five degrees: the density is identical to the last bit a double holds, the bent pair are sp⁵ hybrids at 50.77° to the axis, and their charge sits 0.235 Å off the plane of the molecule where neither canonical orbital's does.

bonding · Hybrids 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 Three bent bonds, at a hundred and one degrees to each other. A carbon–carbon triple bond in its localised description: three equivalent bent bonds, spaced by 101.54 degrees, each tilted 63.43 degrees off the axis and each carrying 0.17 of an s orbital — an sp⁵ hybrid. Its charge sits 0.32 ångström off the axis, where every canonical orbital's sits on it. The mixing that makes the three equivalent is a rotation in the three-dimensional occupied space, so the density is untouched.

Three bent bonds, and the same hybrid

A triple bond localises into three bent bonds at 101.537 degrees to one another, each an sp⁵ hybrid with exactly one sixth s character. A double bond's two bent components are sp⁵ hybrids at 101.537 degrees. The two are the same hybrid, built out of different frameworks and in different numbers, and the reason is that a third of a half is a half of a third.

shape · Hybrids 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 The metal's charge is a coordinate, and the count is not. The metal's charge in an octahedral d6 complex, against how much of each shared pair the ligand is given. Half each is Mulliken's rule and the whole to the ligand is the assumption an oxidation state makes; the answer runs over 2.06 electrons between them. The oxidation state itself is 0, which is off the end of the range, and the electron count is the same number at every point on it.

An integer nobody measured

The oxidation state of chromium in the hexacarbonyl is zero. Its charge, computed from the same wavefunction, is anywhere between −3.04 and −0.98 depending on how the shared electrons are divided — and the integer sits outside that whole range. The electron count, meanwhile, is eighteen at every point on it.

applied · Electron count 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 The hybrids do not point at the atoms. For each cycloalkane, the angle between its two ring hybrids — fixed by orthogonality once the measured H–C–H angle has said how much s character the hydrogens take — against the angle between its carbons. Cyclopropane's differ by 45.5°, so each hybrid points 22.75° outside the bond it makes; cyclohexane's agree to 0.035°, which is the control.

The hybrids that point outside the bonds

Coulson's relation says two equivalent hybrids sharing an s orbital are orthogonal only between ninety and a hundred and eighty degrees. Cyclopropane's carbons make sixty, so its ring hybrids cannot point at the atoms they bond to — and the same relation says by how much they miss: 22.75 degrees each, falling to 0.03 in cyclohexane.

bonding · Hybrids A functional with no interior maximum. The Boys functional against the mixing angle for a carbonyl, evaluated directly at a hundred and eighty-one angles. It is a quadratic in cos²θ − ½ with no linear term, so it is symmetric about forty-five degrees and its maximum is at the middle or at the ends and nowhere else. Here the coefficient is positive, so the best is at 0° — the canonical σ and π. There is no angle to search for, however unsymmetrical the molecule is.

The angle that does not have to be searched for

A carbonyl's two bent components have no symmetry making them equivalent, so the mixing that best localises them looks like something to search for and their s characters look like two different numbers. Neither happens. The localisation functional is a quadratic with no linear term, so its maximum is at forty-five degrees or at the ends — bent bonds or canonical ones, decided by one inequality, with nothing in between.

bonding · Hybrids 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 What a rotation group buys over a single angle. The Boys functional at the canonical orbitals, at the best mixing of the two bonds with the lone pair left alone, and at the maximum over the whole three-dimensional rotation group. The first step is 11.70 per cent and the second, which no two-orbital treatment can reach, is a further 2.50.

An interior maximum a third orbital allows

Two orbitals related by a mirror plane give a Boys functional with no linear term, so it has no interior maximum and no angle is ever searched for. Add the oxygen lone pair and the search over a three-dimensional rotation group finds one — eight and a half degrees of lone pair mixed into two bent bonds, worth a further two and a half per cent that no pair of orbitals could reach.

bonding · Hybrids The criterion is the reciprocal of the p amplitude. The two lone pairs' mixing criterion — twice the dipole between them over the separation of their centroids — against the p fraction of the outward hybrid. The curve is 1/√(p fraction), written down rather than fitted; the marks are the numerical integrals. They agree to 15 parts in a million at every hybrid, and the criterion never falls to one, so the mixed description wins at every s character there is.

The five figures were an identity

Two lone pairs satisfy the two-orbital mixing criterion by a margin of 1.732128, against √3 = 1.732051 — agreement to five figures on a number assembled from three integrals over a numerical grid. It is exact. The criterion is the reciprocal of the p amplitude of the outward hybrid, and everything else in the molecule cancels out of it.

bonding · Hybrids 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.

LocalisationCanonical orbitalsModel limitBasisClosed formConventionOrthogonalityDegeneracyLocal minimumMulticentre bondingOverlap integralElectron-deficient bonding

All concepts