Molecular Geometry & Bonding
what the shape decides
Figure

Two molecules, one eigenvalue, and 1.27 eV between them

Six alternant hydrocarbons placed by the Hückel eigenvalue of their highest occupied level — computed by diagonalising each molecule's own adjacency matrix — against the measured first π ionisation energy. Ethene and benzene share an eigenvalue of exactly 1 and their measurements differ by 1.27 eV; butadiene and naphthalene share 0.618 and differ by 0.94. A model that reads only the eigenvalue is a function of it, so it must give each pair one answer, and the two vertical pairs are the whole of its error.
Two molecules, one eigenvalue, and 1.27 eV between them. Six alternant hydrocarbons placed by the Hückel eigenvalue of their highest occupied level — computed by diagonalising each molecule's own adjacency matrix — against the measured first π ionisation energy. Ethene and benzene share an eigenvalue of exactly 1 and their measurements differ by 1.27 eV; butadiene and naphthalene share 0.618 and differ by 0.94. A model that reads only the eigenvalue is a function of it, so it must give each pair one answer, and the two vertical pairs are the whole of its error.

One of the figures on hückel systems: Adjacency matrices diagonalised: levels, coefficients, bond orders, and the shell closures that decide which rings are stable.

Two 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

A parameter that never finds a value

Six alternant hydrocarbons, placed by the computed eigenvalue of their highest occupied level against the measured energy needed to take an electron out of it. The two vertical bars are pairs of molecules that share an eigenvalue exactly, and they are the whole of what follows.

The scan restricted to the values an overlap between two neighbouring p orbitals can actually take. Every point is an exact least-squares fit of α and β at that overlap. The curve falls from 0.5093 eV to 0.5047 and is still falling where the range ends, so there is no best overlap anywhere inside the parameter’s own meaning — which is a stronger statement than the fit being poor.

The same scan taken past the point where the parameter still means anything, from −0.5 to 1. The curve falls the whole way, is still falling at the edge of the shaded region, and is still falling at the end. A parameter with a meaning has a best value; this one has a direction.

Two systems a model cannot tell apart

The two ionisation ties, drawn against the fitted curve they bound. A model of this form cannot pass between two points on one vertical line, however it is parameterised.

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