Molecular Geometry & Bonding
what the shape decides
Concept

Multireference — where it appears

The condition of a state that no single determinant describes, so that a method built on one has no good starting point. It is measured by the weight of the leading determinant or by the occupations of the natural orbitals, and it is where error cancellation between two calculations stops being available.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

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.

A third kind of correlation

The two-site model gave an exact identity — the power of the correlation energy in the repulsion equals the occupation of the bonding natural orbital — and asked whether anything like it survives with more orbitals. It does not, and the way it fails is better than the identity was: a ring of four gives a power of one where every closed-shell system gives two, at repulsions fifty times weaker than the hopping, because its reference was never a single state.

beyond · Correlation One exact state, two correlation energies. The energy each of two mean fields misses, against the repulsion, for one system whose exact energy is a single smooth curve. Below U = 2 the unrestricted search returns the restricted answer and the two definitions agree to the last bit. Above it they part: at U = 32 the restricted reference reports -27.82 and the unrestricted one -0.11, a factor of 259.43, and one is growing while the other falls.

The reference decides the correlation

The correlation energy of one exact state, measured against two references that are both called Hartree–Fock, is −27.82 and −0.107 at the same repulsion — a factor of 259, on a system whose exact energy is a single smooth curve. Below the instability at U = 2 the two agree to the last bit; above it one grows without limit while the other falls, and the reference that reports almost no correlation has ⟨S²⟩ = 1.99 where a singlet is zero.

beyond · Correlation Nearly all of the error cancels, and the answer gets worse. For each repulsion: the error a spin-paired mean field makes in the total energy of one four-site system and of two two-site ones with the same number of electrons, and the error left in the difference between them. The cancellation improves from 83 to 97 per cent along the axis. The residue as a share of the quantity being computed goes the other way, from 1 to 423 per cent, because the reaction energy shrinks faster than what survives.

Two wrong numbers and a right difference

A mean field gets the total energy of a four-site system wrong by 12.11 and of two two-site systems wrong by 12.49, and 96.9 per cent of that error cancels out of the difference between them. The residue is 0.38 — and the reaction energy it is a residue of is 0.09, so the cancellation improves and the answer gets worse at the same time. Change the pair being compared to a singlet and a triplet and nothing cancels at all: the sign goes.

wrong · Approximation

Named alongside it

The objects these essays reach for when they reach for this one.

Correlation energyExact diagonalisationHubbard modelOpen-shell configurationsHartree–FockModel limitReference stateSpin stateApproximationConventionConvergenceCrossover

All concepts