Molecular Geometry & Bonding
what the shape decides
Concept

Hybridisation — where it appears

Mixing an atom's own orbitals into directed combinations pointing at its neighbours. It is a change of basis and therefore explains nothing on its own: the same molecule described with or without it has the same density.

Named by 15 essays across 4 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 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 Four scales, four orderings. Each column ranks the elements by one electronegativity scale, most electronegative at the top, with a line joining each element across the columns. Every crossing is a pair of elements that two scales order differently.

Electronegativity is not one quantity

Four scales, four definitions, four sets of units, and a defence — "they correlate well" — that answers a question nobody asked. Two scales can correlate at 0.99 and still put hydrogen on the wrong side of carbon.

wrong · Electronegativity 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 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 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 largest angle each ring size can have. The ceiling on a bond angle in a closed ring of equal bonds, 180°(n−2)/n, which is the interior angle of the regular planar polygon and follows from a closed curve having to turn through a full circle. The line at 109.47° is the tetrahedral angle: rings of 3, 4, 5 atoms cannot reach it at any geometry whatever, and every larger ring can, by leaving the plane.

The angle a ring cannot have

A closed ring of equal bonds has to turn through a full circle, so its bond angles cannot average more than 180°(n−2)/n. Three, four and five atoms are below the tetrahedral angle at every geometry whatever; six is above it, and reaches it only by leaving the plane.

shape · Strain 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 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 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: π–n fails it, and σ–n lies outside it, because its dipole runs along the separation and its best rotation is a few degrees. 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 A p function fills the polar bond's minimum, and a pair of them fills it nine times over. Carbon monoxide's σ profile along the bond at equal coefficients, for three bases: one s function on each atom, a quarter p mixed into carbon's, and a quarter p on both atoms in the sense that maximises their overlap. The minimum at π/R is 1.35% of the peak with s alone, 3.63% with p on carbon and 12.53% with p on both. The p function proposed to reduce the floor raises it, and the pair raises it most.

The third function fills the floor

A polar σ bond built from one s function per atom has a momentum profile whose minimum at π/R never reaches zero — for carbon monoxide, 1.35 per cent of the peak. The natural hope was that a p function mixed into the diffuse atom would reshape it towards the compact one and lower that floor. It cannot. At π/R an s function on one atom and a p function on the other are exactly out of phase, so a p function on one atom cancels nothing and only adds itself. With p on both atoms in the sense that makes the bond, the pair adds fifteen times as strongly as the s pair cancels: a quarter p takes the minimum to 12.5 per cent. That is eight times the whole range through which polarity moves the s-only bond.

orbitals · Orbital Carbon monoxide's three σ orbitals, and what each does at π/R. The momentum profile along the bond of each occupied σ orbital of carbon monoxide, with weights from an extended Hückel calculation over carbon and oxygen 2s and 2pσ, each scaled to its own peak, against momentum in units of π/R. At π/R the lowest, 3σ, reads 2.52% of its peak; the middle, 4σ, 8.40%; and the highest, 5σ, 85.3% — its profile peaks at 0.82 π/R and has no minimum near π/R at all.

The lone pair sits where the floor was

A polar σ bond's momentum profile has a minimum at π/R whose floor was proposed as a measure of polarity, and the essay that swept the p weight could not say whether carbon monoxide's own σ bond sat near the s-only floor of 1.35 per cent or the quarter-p value of 12.5. With the weights computed — carbon and oxygen 2s and 2pσ, extended Hückel — the answer is that carbon monoxide has no such bond. Its lowest σ orbital reads 2.5 per cent, its middle one 8.4 per cent of its own peak, and its highest, the orbital with the lone pair's sense, has no minimum at π/R at all: it peaks there. The six σ electrons together have no minimum anywhere, and the polar molecule reads lower at π/R than nitrogen does.

orbitals · Orbital Along the series one pair crosses the boundary, and it is the one nobody asks about. The two-orbital criterion's margin for each pair of the carbonyl-type trio as the heteroatom runs through carbon, nitrogen, oxygen and fluorine at one bond length. Above one the mixed description wins. The σ–π margin falls from far off the scale to 1.179 and never reaches the boundary; the π–n margin rises from 0.470 to 1.350 and crosses it between oxygen and fluorine. The σ–n margin is drawn faint because the criterion does not apply to that pair at all.

A verdict that flips and changes nothing

Run the two-orbital mixing criterion along H₂C=CH⁻, H₂C=NH, H₂C=O and H₂C=F⁺ and one pair finally crosses its boundary — the π and the lone pair, a third of the way from oxygen to fluorine. It is the first real test the criterion has had, and the localised orbitals on the far side of it are the ones on the near side: two bent bonds and a lone pair that stays itself.

bonding · Hybrids

Named alongside it

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

Lone pairBasisBond angleLocalisationModel limitElectronegativitys charactersp³ hybridsMolecular orbitalOrthogonalityOverlap integralTetrahedral angle

All concepts