Molecular Geometry & Bonding
what the shape decides
Figure

Three answers to one question

The energy to remove an electron from a half-filled four-site system, computed three ways against the repulsion: exactly, by solving a self-consistent field twice — once for the molecule and once for the ion — and by reading the highest occupied orbital energy straight off the molecule, which is Koopmans' theorem. All three agree exactly at zero repulsion. The theorem always sits above the two-calculation answer, because letting the ion relax can only lower it; the exact answer sits above both, because the molecule is more correlated than its ion. The two errors have opposite signs and do not cancel: the residue grows to 6.03.
Three answers to one question. The energy to remove an electron from a half-filled four-site system, computed three ways against the repulsion: exactly, by solving a self-consistent field twice — once for the molecule and once for the ion — and by reading the highest occupied orbital energy straight off the molecule, which is Koopmans' theorem. All three agree exactly at zero repulsion. The theorem always sits above the two-calculation answer, because letting the ion relax can only lower it; the exact answer sits above both, because the molecule is more correlated than its ion. The two errors have opposite signs and do not cancel: the residue grows to 6.03.

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.

Six 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

Two kinds of correlation, and only one is small

Where a mean field gains by releasing its spin symmetry, against the exact energy. Above the instability the broken solution is lower in energy and has a spin density the exact singlet ground state does not have — a better number attached to a worse picture, which is the same trade the occupations describe.

A method that is not additive

The same dimer, once, twice and three times over. The exact energies are exactly additive. The truncated ones are not: exact for one unit, 0.19 short for two, 0.49 short for three — and the error per unit grows rather than staying fixed.

Koopmans' theorem is exact for nothing

The three answers against the repulsion, for a half-filled chain of four sites. They start together and separate immediately. The theorem’s answer is always the highest, the two-calculation answer is in the middle, and the exact one is above both — so the two errors have opposite signs and neither is small.

Three answers to one question, on the same system at the same repulsion: the orbital energy, the energy difference between the neutral molecule and the ion computed with the neutral’s orbitals frozen, and the same difference with the ion allowed to relax. Koopmans’ theorem is the claim that the first equals the third. It equals the second exactly and by construction, and the gap between the second and the third is the relaxation the theorem drops.

The energy of the same four-site system three ways: exact, in a mean field constrained to keep the spins alike, and in one released from that constraint. Below the critical repulsion the released search returns to the constrained answer on its own, which is the check that the instability belongs to the model rather than to the starting guess. Above it the two part company and neither ever falls below the exact answer.

A mean field cannot get out of the way

The same four-site system three ways: exact, in a mean field constrained to keep the spins alike, and in one released from that constraint. At U = 16t the constrained field misses 12.11 and the released one misses 0.21, on the identical system with the identical repulsion.

A hundred lines and no way to sort them

Why the fundamentals move as well as splitting: the energy needed to take an electron out is not the level it came from, and the gap grows with the repulsion. The bands whose range the satellites invade are themselves in the wrong place.

The boundary belongs to the gap

What a removal spectrum is a spectrum of, and where the fundamentals come from: the ionisation energies against the repulsion. A satellite is one of these plus an excitation, and the excitation costs the gap.

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