Molecular Geometry & Bonding
what the shape decides
Concept

Lone pair — where it appears

A pair of valence electrons not shared with another atom. It occupies space and pushes bonds together, and it is a feature of a localised description rather than an object a photoelectron spectrum resolves as one band.

Named by 19 essays across 4 fields — each of them below, with the objects they name alongside it.

4 sites, minimised. The arrangement of 4 points on a sphere that minimises their mutual repulsion. The angles printed were measured off the result rather than quoted, and the shape was not assumed.

VSEPR, computed

The tetrahedral angle is not 109.5 degrees because a textbook says so. It is arccos(−1/3), and it falls out of minimising the repulsion of four points on a sphere without ever being written down.

shape · VSEPR What the group settles. For each molecule, the point group found from its coordinates and the two properties that follow from the group alone. Neither column required knowing anything about the bonds.

The dipole is not a sum of bonds

Adding bond dipoles as vectors gets the easy cases right and rests on a quantity with several incompatible definitions. The symmetry argument is exact, needs no electronegativities, and says when the answer must be zero.

wrong · Dipole water — C2v. 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.

Why water is bent

The standard answer is lone pair repulsion, it predicts the right direction, and it cannot predict the magnitude. A better rule can, and the heavier hydrides show where both accounts run out.

shape · VSEPR 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.

Bent's rule, against the substituent series

More electronegative substituents get more p character, so the angle between them shrinks. The rule predicts a trend, the trend is observed, and the quantity it depends on turns out not to be one quantity.

shape · VSEPR Bond angle against lone-pair weight, 2 lone and 2 bonding. The bond angle a weighted repulsion minimisation gives for 2 lone pairs and 2 bonding pairs, as the lone-pair weight runs from one to 3.2. The marked molecules are water and hydrogen sulfide, each placed at the weight that reproduces its measured angle.

What a lone pair is worth

A lone pair repels more than a bonding pair, says VSEPR, without saying how much more. Put a number on it and fit that number to water, and the same number is wrong for hydrogen sulfide by a factor of two — which means it was never a property of a lone pair.

shape · VSEPR The s character a bond angle requires. The fraction of the one s orbital each of two equivalent hybrids must carry in order to meet at a given angle, from pure p at 90 degrees to half s at 180. The marks are measured bond angles — H₂O, NH₃, H₂S, PH₃ — and the column beside them is the budget each implies: what one bond hybrid takes, and what is left for the lone pairs to share. The relation is orthogonality, not a model of bonding, and nothing here is fitted.

The angle does not fix the hybridisation

Two equivalent hybrids are orthogonal only at one relation between their s character and the angle between them. Run it backwards on measured bond angles and water's bonds come out sp³·⁹⁹, hydrogen sulfide's sp²⁷, and phosphine's lone pair takes eighty-four per cent of the one s orbital.

shape · Hybrids The moment that does not depend on where the origin is. For each molecule, the dipole and the largest quadrupole component about the centre, and the same two about an origin moved 1.6 bohr away. The lowest non-vanishing moment is unchanged in every row and the one above it moves in every row. The 4 non-polar molecules here have a quadrupole that is a property of the molecule; the polar ones have one that is a property of a choice.

What a dipole cannot tell apart

A dipole moment is three numbers extracted from a whole charge distribution, and enormously many distributions give the same three. The moment above it is not even a property of the molecule unless the one below it vanishes — which is why a quadrupole is quoted with an origin and a dipole is not.

shape · Dipole How far two hybrids are from orthogonal. The overlap between two equivalent s–p hybrids of a stated label, against the angle between them. Each curve crosses zero at exactly one angle — sp3 at 109.47°, sp2 at 120.00°, sp at 180.00° — and a molecule whose measured angle is not that angle has hybrids that overlap. The largest here is cyclopropane at 0.63.

Hybrids that were never orthogonal

Two sp³ hybrids are orthogonal at 109.47° and nowhere else. Water's are drawn at 104.5° and overlap by 0.062; cyclopropane's are drawn at 60° and overlap by 0.625, which is not a small correction to a basis but a description that has stopped being one.

shape · Hybrids water: 4 valence bands. The measured valence photoelectron bands of water, each labelled with the symmetry species of the orbital it comes from, and beside them the species the valence basis spans — the central atom's s and p functions and one s on each ligand, reduced in the molecule's own group. A band carrying a species the reduction does not produce would stop this figure being drawn.

Water's lone pairs are not a pair

Every course draws two equivalent lone pairs on water, pointing away from the hydrogens like a pair of ears. Its photoelectron spectrum shows the two bands they would produce at 12.6 and 14.7 electronvolts, two point one apart, in different symmetry species.

bonding · Photoelectron The bond sum, the measurement, and what is left over. For each pyramid: the vector sum of three bond moments estimated from the electronegativity difference and the measured geometry, the measured dipole, and the difference between them. Positive is towards the lone pair. Ammonia's bonds point that way and nitrogen trifluoride's point the other, which is why the trifluoride's far more polar bonds give it a dipole six times smaller. The leftover is 0.58 D for ammonia and at least 1.48 D for the trifluoride, so it is not one lone pair's property.

The lone pair is not the missing term

Ammonia's dipole is 1.47 debye and nitrogen trifluoride's is 0.235, although the N–F bonds are far more polar than the N–H ones. The bond sums explain the reversal exactly and point in opposite directions — and the lone pair that is supposed to make up the difference has to be worth 0.58 debye in one molecule and at least 1.48 in the other.

shape · Dipole 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 Where the orphan pair actually sits. A σ-only Hückel model of each molecule, built from its own coordinates: the central atom's four valence orbitals, one σ orbital on each ligand, and the coupling between them the direction cosine of that ligand. Lone pairs need no special handling — they come out of the diagonalisation as the central orbitals no ligand combination transforms like. The charge is measured rather than assigned, and PF₅ comes out with two kinds of fluorine at 3 at -0.13 and 2 at -0.3, the more charged pair being the axial one that carries the orphan.

The count is the population

The census counted how many ligand combinations have no partner on the central atom and called the count n + L − 4. A σ-only model built from each molecule's own coordinates says what that count is worth: with no electronegativity difference anywhere, the mean charge on a ligand is minus the orphan count divided by the ligand count, exactly, in all ten cases. And the prediction the census made — that the charge grows with the orphan count — is refused by the divisor.

beyond · Hypervalency The even sharer is the one the repulsion likes least. trigonal bipyramid: 2 kinds of ligand, spread 0.1137, repulsion 6.4747; square pyramid: 2 kinds of ligand, spread 0.1658, repulsion 6.4844; pentagonal planar: 1 kind of ligand, spread 0.0000, repulsion 6.8819. The planar arrangement gives all five ligands exactly the same charge and costs 6.3 per cent more in repulsion than the bipyramid, which is the arrangement chemistry actually adopts — so the two models disagree about which arrangement is preferred, and about how much charge is moved.

Two models that disagree about the shape

The identity says how much charge a hypervalent molecule's ligands must share and nothing about how. Working out which arrangement shares it most evenly puts the σ model and the repulsion model on one axis for the first time: the even sharer is the pentagonal plane, which is the arrangement the repulsion likes least — and along the interchange chemistry actually uses, one model sees 0.15 per cent of a change and the other sees 54.

beyond · Hypervalency Ten molecules, twenty-six arrangements, three disagreements. Every arrangement of every one of the ten molecules, with the point group recovered from the arrangement's own coordinates and the ligand σ set reduced in it. 27 of them can be worked in a tabulated group; the formula n + L − 4 is right for 23 and wrong for 4. Every failure is a planar arrangement of four or more ligands, and every one of them is a molecule that does not adopt that arrangement.

The square that wastes an orbital

The orphan count that prices hypervalency was treated as a property of a molecule's composition — ligands plus lone pairs minus four. Run on twenty-six arrangements of the same ten molecules it is right for twenty-three and wrong for three, and all three are flat. A planar arrangement gives a main-group centre three usable orbitals rather than four, so the count is a property of the shape.

beyond · Hypervalency The repulsion against the count, and they do not sort together. Every arrangement of the census by how many ligand σ combinations are left without a central partner and by how far its ligand repulsion sits above the best arrangement of that many points. The three that break the formula are marked. They are not the expensive ones: they sit at 4.2, 6.3, 9.8 per cent while the arrangements the formula gets right run to 26.5.

Expensive is not the same as unadopted

A counting formula right for twenty-three arrangements and wrong for three invites a reading: the failures are the arrangements nothing adopts, so the formula is reliable because chemistry stays away from where it breaks. Put the repulsion energy on the same axis and the reading fails — the most expensive arrangement in the census is one the formula gets right.

beyond · Hypervalency 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 Five pairs, one boundary, and every case anybody asked about on one side. The mixing criterion's margin for every pair of orbitals available from the carbonyl and from ethene. Above one the mixed description wins. Three pairs were put to the criterion and all three clear the line; two more come from the same three orbitals and the same dipole matrices, and both fail it. The instrument discriminates — the selection did not.

The criterion that has never said no

The two-orbital mixing criterion is a tautology on a carbonyl's lone pairs, and it is tempting to contrast that with the double bond, where the criterion seems to discriminate. Both halves are wrong. The double bond's margin is wider, ethene's is infinite because symmetry puts both centroids at the same point, and the criterion has never returned a negative on any case it is usually put to — though the same three orbitals supply two pairs it refuses outright.

bonding · Hybrids Folding the ring off its plane never gives the orbital back. For rings of four to eight ligands round a centre with no lone pair, the orphan count minus the formula n + L − 4, at polar angles from 60° to 120°. The upper dot in each row is the bare ring and the lower the same ring with a ligand on the axis. The bare ring is one over the formula at every angle, including 90°, where the ring is flat and its group is D₄ₕ, D₅ₕ or D₆ₕ; there is no table here for D₇ₕ or D₈ₕ. The capped ring agrees with the formula at every angle, 90° included, where the ring is exactly flat and the apex alone keeps its group C₄ᵥ to C₈ᵥ.

Folding the ring does not give the orbital back

Three arrangements break the orphan count n + L − 4, all flat, and the reason given was flatness: the p orbital perpendicular to the ring has no ligand combination of its species. Fold the ring into an umbrella at any angle and that orbital becomes totally symmetric — and the count stays wrong by exactly one, for rings of four to eight. The ring offers one symmetric combination to a centre with two symmetric orbitals. Writing C₅ᵥ to see it also counts the pentagonal pyramid at last, and the formula holds there.

beyond · Hypervalency

Named alongside it

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

ElectronegativityModel limitBond angleHybridisationHypervalencyReduction formulaConventionDipole momentIrreducible representationsLocalisationPartial chargePoint group

All concepts