Molecular Geometry & Bonding
what the shape decides
Concept

Bent bond — where it appears

A bond whose density is greatest off the line between the two nuclei, so that the orbitals meet at an angle rather than head on. It is what a small ring forces on its bonds, and it is why such a ring's strain does not follow from its internal angles alone.

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

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 inequality, decided twice. The quantity the localisation criterion compares — the centroid separation over twice the off-axis dipole — computed with hydrogenic radial functions and with Slater ones at the same effective charges. Below one the localised description is a pair of bent components; above it, σ and π. The hydrogenic answer is 1.8090 and the Slater answer is 0.3937, and they are on opposite sides.

The node that decided a picture

A hydrogenic calculation cannot say whether formaldehyde's localised description is two bent components or σ and π, because the integral that decides it came out at 0.00718 where it should be the largest in the molecule. Replacing the hydrogenic radial functions with nodeless Slater ones moves that overlap by a factor of ninety-seven, leaves the π overlap identical to the last bit, and flips the answer: the description is bent.

bonding · Hybrids 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

Named alongside it

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

LocalisationModel limitApproximationClosed formConventionDipole momentHybrid orbitalMolecular orbitalOverlap integralUnitary transformationBasisCanonical orbitals

All concepts