One double bond, two descriptions

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

Five 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.

Two bent bonds, or a σ and a π

The two diagonal entries of the density matrix as the mixing runs through ninety degrees, with the off-diagonal entry beneath them. Both stay at one and zero to the last bit a double holds. Forty-five degrees is marked, and there is nothing special about it in this figure — the density does not know that the two members have become equivalent there.

The two descriptions drawn in the plane perpendicular to the molecule. On the left the σ along the axis and the π above and below it; on the right the two bent bonds, tilted 50.77 degrees either side. The pictures look like different physics and are one occupied space seen in two bases.

The arithmetic of the bent-bond description, computed rather than quoted: the s character, the hybrid index, the tilt, the angle between the two, the centre of charge, and the two overlaps.

Three bent bonds, and the same hybrid

The three bent components of a carbon–carbon triple bond, at 101.537 degrees to one another, each tilted 63.435 degrees off the internuclear axis. The charge of each sits 0.31519 ångström away from the axis — the same distance for all three, which the threefold axis requires and which is computed from the wavefunctions rather than from the mixing.

The bent-bond description of a single, a double and a triple carbon–carbon bond. A single bond’s component is an sp³ hybrid; a double bond’s two and a triple bond’s three are the same sp⁵ hybrid, at the same angle to one another, although one comes from a trigonal framework and the other from a linear one and there are different numbers of them.

Eight quantities describing the bent-bond account of the carbon–carbon bond, every one computed rather than quoted: the s character of the hybrid each component uses, the angle it makes with the internuclear axis, and where its charge sits relative to that axis. The s character is the number that comes out equal for the double bond and the triple one; the off-axis distance is one of the two that do not.

The hybrids that point outside the bonds

For each cycloalkane, the angle between its two ring hybrids against the angle between its carbons. The shaded bar is the gap the bond has to bridge, and half of it is how far each hybrid misses by.

The band two orthogonal hybrids can occupy, with the angle each planar ring needs. Five and up are inside it, four is exactly on the boundary, and three is outside.

The double-bond series: how much s character a carbon puts into a double or triple bond, and how many equivalent hybrids it divides that between. The bending measured there is a property of the bond order; the bending measured here is a property of the ring.

The angle that does not have to be searched for

The localisation functional against the mixing angle for a carbonyl, evaluated directly at a hundred and eighty-one angles. It is symmetric about forty-five degrees, and its best is at an end.

The two quantities the inequality compares, against how much of the σ orbital sits on oxygen. Bent components are the localised description inside the shaded band and not outside it.

The same pair at sixty degrees rather than at forty-five, which is where Coulson’s relation between the angle and the shared s character would put a set of equivalent hybrids. The relation is exact and it fixes the composition once the angle is chosen; what it does not do is choose the angle, which is the whole of what this essay has to supply.

The node that decided a picture

The quantity the criterion compares, computed twice at the same effective charges. Below one the description is bent components; above it, σ and π.

The two radial functions at the same effective charge, with the node and the other nucleus marked.

The two overlaps in the two bases. The two π bars are the same number.