Molecular Geometry & Bonding
what the shape decides
Figure

The sp3 transformation

Each row is one hybrid, written in the basis of the atomic orbitals it is made from. The rows are orthonormal, so the matrix is a rotation — and a rotation of a basis changes no observable quantity whatever.
The sp3 transformation. Each row is one hybrid, written in the basis of the atomic orbitals it is made from. The rows are orthonormal, so the matrix is a rotation — and a rotation of a basis changes no observable quantity whatever.

One of the figures on bonding models: Valence bond, molecular orbital and hybrids, drawn as descriptions of one thing rather than as competing pictures.

Six essays draw this figure, each at the values its own argument needs rather than at the setting shown above. What each one uses it to show is below, in the words of its own caption.

In the essays

Hybrids are a basis

The same thing written as a matrix, one row per hybrid. Every row is normalised and every pair of rows is orthogonal, so the matrix is orthogonal — and an orthogonal matrix is a rotation.

Bent's rule, against the substituent series

The sp³ matrix written out, orthonormal to better than one part in 10¹⁵. Bent’s rule is a statement about what happens when the four rows stop being identical — the matrix stays orthogonal, the compositions differ, and the directions move apart.

An interior maximum a third orbital allows

The Boys functional at the canonical orbitals, at the best mixing of the two bonds alone, and at the maximum over the whole rotation group of three.

The overlap matrix of the three orbitals as built. One entry is not zero, and a Boys functional over a set that is not orthonormal is not a Boys functional.

The dipole matrices of the three orthogonalised orbitals, across the bond and along it. The component perpendicular to both is zero throughout, by the mirror plane.

The five figures were an identity

The mixing criterion against the p fraction of the outward hybrid, with the closed form drawn through it and the rival form beside it.

The criterion at three effective charges and three bond lengths, where the two quantities it compares move together and it does not.

The departure from the square root of three as the quadrature is refined from forty points a dimension to a hundred.

The criterion that has never said no

The criterion’s margin for every orbital pair available. Three were asked about and all three clear the line; of the two that were not, one fails it and one lies outside the test.

The two margins side by side. The tempting contrast runs the wrong way.

The σ–π margin against the oxygen’s s fraction. Its floor is 2.412, and it never approaches the boundary.

A verdict that flips and changes nothing

The criterion’s margin for each pair as the heteroatom runs from carbon to fluorine. σ–π falls and never reaches one; π–n rises and crosses it between oxygen and fluorine.

The π–n margin between H₂C=O and H₂C=F⁺. It equals one at a Slater exponent of 4.772 and an electronegativity of 8.52 eV.

The oxygen and fluorine members’ π–n margins as the bond length and the heteroatom’s s fraction are varied. The crossing survives every bond length and most of the polarisation range.

Every figure · Every orbital, by what it encloses · All essays