Molecular Geometry & Bonding
what the shape decides
Concept

Open-shell configurations — where it appears

A configuration with a partly filled shell, so that more than one arrangement of the electrons is available at the same energy. It brings degeneracy, a choice of spin state and, for a mean-field description, the possibility that the symmetric solution is not the lowest one.

Named by 9 essays across 4 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 How many times more a metal carries, and when. The ratio of the carriers in a uniform ring to those in an alternating one of the same size, at four sizes and six temperatures. Down the left-hand column the two are indistinguishable, because a ring of 42 at kT = 0.002 has a level spacing larger than the temperature and is no more a metal than the gapped one is. Along the bottom row they are indistinguishable again, because the temperature is larger than the gap. The word only means anything in the middle.

The metal a thermometer cannot find

A metal is a system with excitations of arbitrarily small energy, which is a claim about a sequence of finite systems rather than about any one of them. Put a temperature on it and the claim needs a second limit, and the two do not commute: a uniform ring of forty-two at kT = 0.002 carries exactly as much as an alternating one, and at kT = 0.1 a ring of three hundred and twenty-two carries only four times as much.

wrong · Metal Three lines, then a hundred. The exact removal spectrum of a 6-site Hubbard ring at half filling: every final state of the ion, at the energy it costs to reach and with the intensity the matrix element gives it. With no repulsion there are 3 lines and they are the occupied orbital energies. At U = 8 there are 100, on a molecule with 6 orbitals — so the spectrum cannot be read as a list of orbital energies, because there are more bands in it than there are orbitals to name.

More bands than there are orbitals

A photoelectron spectrum is read as a list of orbital energies, one band per occupied orbital. Computed exactly for a six-orbital ring, it has three bands with no repulsion and a hundred with eight — and by then fifty-three per cent of the intensity is in lines that no orbital corresponds to. The total intensity is three at every repulsion, exactly, because that is a sum rule and not a fit.

spectra · Photoelectron The whole difference lives at the two ends. The energy difference between the two dimerisations of an open chain, held at the same distortion, multiplied by the number of sites. It settles on a constant — 1.09 in units of the hopping — so the difference per site falls as one over the length, with a fitted exponent of -1. An end is a bond that is not there, and it is worth the same amount whatever it is attached to.

The distortion the ends decide

A chain of an even number of sites has an odd number of bonds, so its two dimerisations are different molecules rather than one molecule translated. Held at the same distortion they differ by 1.08715 in units of the hopping, whatever the length — a fixed amount of energy living at the two ends, with the per-site difference falling as one over the length at a fitted exponent of −0.99986. And below a hundred and twenty-eight sites the second dimerisation does not exist at all.

applied · Peierls distortion A cheap number that predicts an expensive failure. Thirty-two systems. Along the bottom, how far the mean field's own symmetry breaking has moved between the reference and the target — a quantity available before any exact calculation. Up the side, how wrong the transferred correction turns out to be. They rank together at 0.902, and the open marks are the systems whose broken solution has collapsed entirely, which is where the diagnostic stops being a scale and becomes a warning.

The warning a cheap calculation gives

A correlation correction computed on one system and carried to another works until it does not, and nothing in the scheme says in advance which. The mean field's own symmetry breaking says: it collapses at a definite field, and the transfer fails where it goes. Across thirty-two systems the two rank together at 0.90.

wrong · Approximation The carriers a distortion was hiding. A half-filled ring of 40 — one of the 4m rings that carry exactly one pair of carriers at every temperature — allowed to distort. Cold, it alternates by 0.1232, opens a gap of 0.4927 and carries 3.6e-15 carriers rather than one pair. The alternation is undone continuously at kT = 0.1358, and the carrier count comes back as it goes.

The carriers a distortion was hiding

A half-filled ring of 4m carries exactly one pair of thermal carriers at every temperature, which is true only of a ring held rigid. Allowed to move, it does not carry them: it alternates, opens a gap of 0.4927, and carries none at all until a temperature that undoes the distortion.

wrong · Metal Where the broken solution appears. The mean field's spin polarisation against the on-site repulsion, at half filling and no site-energy modulation, for three systems. Two of them are symmetric below a threshold and polarised above it — 1.672 for a chain of four and 2.355 for a ring of six. The third is polarised at every repulsion tested, because its half-filled shell is degenerate and the symmetric solution is unstable however small the repulsion is.

The half of the square a ring of four cannot show

There is a warning that says in advance whether a transferred correction will hold: the mean field's own symmetry breaking, which collapses at a definite site-energy modulation and takes the transfer with it. The other axis of the same square has a threshold too — on the other side — and the ring of four it was all measured on is the one system with no threshold to find.

wrong · Approximation

Named alongside it

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

Exact diagonalisationHubbard modelCorrelation energyDegeneracyModel limitHartree–FockOn-site repulsionReference stateSymmetry breakingBand gapElectron correlationMultireference

All concepts