Molecular Geometry & Bonding
what the shape decides
Figure

Two is the ordinary answer, and one is a different kind of correlation

The local exponent of the correlation energy in the repulsion, against the repulsion, for three systems at half filling. The two closed-shell systems tend to two as the repulsion vanishes, which is ordinary perturbation theory. The ring of four tends to one, at repulsions fifty times smaller than the hopping.
Two is the ordinary answer, and one is a different kind of correlation. The local exponent of the correlation energy in the repulsion, against the repulsion, for three systems at half filling. The two closed-shell systems tend to two as the repulsion vanishes, which is ordinary perturbation theory. The ring of four tends to one, at repulsions fifty times smaller than the hopping.

One of the figures on when repulsion is in the model: Hubbard systems small enough to diagonalise exactly: the singlet a one-electron model cannot find, the coupling between two spins, and a gap where band theory says there is none.

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.

In the essays

A third kind of correlation

The exponent beside the natural occupations for three systems at the same repulsion. For two centres the exponent and the largest occupation are the same number to five figures. For four in a chain the exponent is above every occupation. For four in a ring it is nowhere near any of them.

The exponent against the repulsion for three half-filled systems. The two closed-shell ones tend to two as the repulsion vanishes, which is ordinary perturbation theory. The ring of four is at 1.008 where the repulsion is a fiftieth of the hopping, and never approaches two at any strength.

Every natural occupation of all three systems, against the repulsion, rather than the three rows of the table above at one value of it. The two-centre pair separates smoothly from two and zero and the chain’s four do the same more slowly. The ring’s middle two lines are pinned at exactly one and nothing moves them — which is what makes “already maximally multireference” a statement about the whole range rather than about one point on it.

The warning a cheap calculation gives

Two kinds of correlation, told apart by how they scale. The diagnostic in this essay is a cheap way of asking which kind a system has, and the answer decides whether a number computed elsewhere is worth carrying.

Where the electrons are, without subtracting anything

The two kinds of correlation told apart by how they scale. The distributions above cannot separate them, because a lattice hole is one bond wide whichever kind it is — which is the limitation this essay ends on.

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