The s character a bond angle requires

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

Three 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.

The angle does not fix the hybridisation

The s fraction two equivalent hybrids must have in order to meet at a given angle, with four measured hydride angles marked and the budget each implies beside them. The curve runs from pure p at 90° to half s at 180°, and the three labelled points on it are the angles at which the arithmetic closes on a whole number.

Coulson’s relation itself: the angle between two equivalent hybrids against the s character they share. It is exact, and it is a relation between two quantities rather than a determination of either — which is the whole difficulty this essay is about. The tetrahedral angle is nowhere in it as an input; it is where the curve crosses a quarter.

The same relation over the whole range an angle can take. At 90 degrees the hybrids are pure p and at 180 they are half s, and every angle between requires a stated composition — but the composition is required of a set of equivalent hybrids, and a molecule whose bonds and lone pairs are not equivalent has no single number to be required of it.

Three bent bonds, and the same hybrid

Coulson’s relation, with four cases marked. A quarter gives the tetrahedral angle exactly; a third gives 120 degrees; a half gives 180. A sixth gives 101.53696 degrees, and it is marked once for two different bonds, because a double bond’s bent components and a triple bond’s have the same s character and therefore the same angle.

The s character of a set of equivalent hybrids against the angle between them, with measured angles of real molecules marked. Every bent bond in this essay is a point on this curve at one sixth, and the two that coincide there come from opposite ends of the framework axis — an sp² carbon and an sp one.

The hybrids that point outside the bonds

The relation the whole essay is one application of: the angle two equivalent hybrids can make, against how much s character they share. Everything above is that curve read backwards.