The valence-bond and molecular-orbital directions in one plane

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 kinds of correlation, and only one is small

The exact ground state of the dimer in the plane spanned by the molecular orbital and valence bond descriptions, at a series of repulsions. It never leaves the plane, and its position in it is the same information the occupations carry — with one extra coordinate, the sign, that no scalar diagnostic reports.

Two pictures, one plane

The two functions as directions, drawn at the angle their overlap requires: they overlap by 0.7071, so they sit 45° apart. The exact ground state’s direction is marked at eight repulsions, swinging from one towards the other and reaching neither. The plane is the whole of the space the exact answer occupies — the projection of the exact state onto it is 1 to twelve decimal places at every U.

The ionic weight against the repulsion, with the two pictures’ fixed values as horizontal lines. The exact curve leaves one at U = 0 and approaches the other asymptotically. Every point on it is a molecule that neither picture describes.

The same plane traced over a wider range of repulsion than the figure above uses. At zero the exact state sits exactly on the molecular-orbital direction; by sixteen it has moved most of the way to the valence-bond one and has still not left the plane. Two descriptions and one two-dimensional space is the whole geometry, and every repulsion the model can be given lands inside it.

A difference does not make a transfer

The same state read a third way. A molecular orbital description and a valence bond one are two vectors in a plane, and the exact state lies somewhere between them; the repulsion is what moves it. A polarity read off the first of the two is a polarity read off a vector the state has left.

A weight that depends on how it is weighed

The two pictures as two vectors in a plane, with the exact ground state of a two-site model lying between them at every repulsion. The model’s sites are orthogonal by construction, so the weight of the ionic configurations in the exact state is unambiguous: 0.5000 with no repulsion, 0.2764 at U = 2t, 0.1464 at U = 4t and 0.0149 at U = 16t.

One spectrum, a line of models

The other tradition’s version of the same freedom: a molecular orbital wavefunction and a valence bond one are two coordinates on one plane of states, and a mixing parameter slides between them. A model’s parameters and a model’s basis are two different things to be underdetermined about, and both have now been found.