Molecular Geometry & Bonding
what the shape decides
Concept

s character — where it appears

The fraction of s orbital in a hybrid, which follows from the angle between hybrids under one convention. Different routes to it — an angle, a coupling constant, a localised orbital — give different numbers for the same bond.

Named by 9 essays across 3 fields — each of them below, with the objects they name alongside it.

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 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 Bonds the scales disagree about. For each bond, which atom each scale calls the more electronegative. Every row is a bond whose polarity would be drawn in opposite directions depending on which of these four tables in common use was consulted: Pauling, Mulliken, Allred–Rochow, Allen.

A ranking is not a difference

Four electronegativity scales agree about order to a Spearman coefficient of 0.99 and disagree about size by a factor of sixty-two. Put each on its own range and the O–H bond is 39 per cent of the way across on three scales and 4.5 on the fourth.

wrong · Electronegativity 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 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 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 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

Named alongside it

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

Bond angleHybridisationLone pairBent's ruleElectronegativityLocalisationOrthogonalityUnitary transformationBasisModel limitsp³ hybridsTetrahedral angle

All concepts