Molecular Geometry & Bonding
what the shape decides
Concept

Crossover — where it appears

A point at which two competing quantities become equal and the system's preferred state changes, without anything discontinuous happening to the system itself. A spin crossover at Δ = P and a symmetry-breaking instability are both of this kind, and each is a property of a competition rather than of a substance.

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

A moment that is not an integer's worth of anything. The effective magnetic moment against temperature for an iron(II) complex whose two spin states lie close together, with the moments belonging to whole numbers of unpaired electrons drawn across. The curve spends its time between them and settles on neither.

A moment between two integers

A magnetic moment is celebrated as one of the few chemical measurements that returns an integer: count the unpaired electrons, feed the count into a formula, and nine first-row ions come out right. That works when one state lies far below the others. Sit a complex at its own crossover and the same measurement returns 0.30 at 80 K and 3.61 at 400 K — a quantity that counts nothing and is a temperature in disguise.

applied · Magnetism 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

Named alongside it

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

Spin stateBoltzmannClosed formConventionCorrelation energyDegeneracyExact diagonalisationHartree–FockHubbard modelLigand fieldMagnetic momentModel limit

All concepts