Molecular Geometry & Bonding
what the shape decides
Concept

Participation ratio — where it appears

A measure of how many sites a state is spread over, computed from its coefficients. It distinguishes a state confined near a defect from one spread through the whole structure without needing a picture.

Named by 11 essays across 3 fields — each of them below, with the objects they name alongside it.

The state on a defect at site 31 of 60. The amplitude of one eigenvector at each site of a 60-site chain whose site 31 has its energy raised by 1.6β. The two colours are the two signs. A state belonging to the whole chain would reach across it; this one does not.

A defect is a level in the gap

Change one site's energy in a chain of a hundred and sixty and a level leaves the band, carrying a state that lives on a handful of atoms. In one dimension it happens for any change however small — the threshold measured on chains of forty, eighty and a hundred and sixty halves with every doubling, so there is no threshold at all.

solids · Defect The state on the end of a chain of 60. The amplitude of one eigenvector at each site of a 60-site chain whose site 1 has its energy raised by 0.6β. The two colours are the two signs. A state belonging to the whole chain would reach across it; this one does not.

The end is the hardest place to bind

A site in the middle of a chain traps a state for any energy difference however small. The site at the end demands a whole β before it traps anything — measured at 1.025, 1.013 and 1.006 on chains of forty, eighty and a hundred and sixty, converging on exactly one. The intuition runs the other way and is wrong.

solids · Defect Two impurities at 2 to 14 sites apart. The splitting between the two levels a pair of impurities of strength -2β pulls out of a chain of 61, against how far apart they are, on a logarithmic scale. It falls by a constant factor per site of separation, and that factor is the decay of the isolated bound state computed from its energy alone. The two levels close on the single impurity's level as the pair separates.

Two defects, and the level between them

One impurity pulls a level out of a band at exactly −√(h²+4). Two of them pull out a pair, split by an amount that falls by a factor of 0.41421 for every site of separation — which is √2 − 1, predicted from the isolated level's energy alone and measured to six figures.

solids · Defect One carbon, three lithiums, and no way to make it two. The four localised orbitals of the tetramer, with the share of each pair on each of the eight framework atoms. Each sits on exactly four: one carbon and the three lithiums of the face that carbon caps, which was not put in anywhere. The four shares are not equal — about 59 per cent of the pair is on the carbon — so the participation number comes out near 2.5 rather than at four. A bond over four centres is not a bond divided into four.

Four centres, and the pair that will not localise

The occupied orbitals of a molecule can be mixed freely without changing anything observable, and the freedom is usually spent on making them as local as possible. For the methyllithium tetramer the answer is four centres — one carbon and the three lithiums of the face it caps — and no mixing of the four pairs reduces it.

beyond · Multicentre One impurity is a level; many are a band. The impurity levels of a ring of 160 with sites of depth -3, drawn as a bar from the lowest to the highest, against the fraction of sites that are impurities. At the lowest concentration every level is at the same energy and the bar has no height at all. By 30 per cent the levels span 2.44 and have closed to within 0.25 of the host band, which is shaded.

One defect is a level, many are a band

A single deepened site in a chain pulls one state out of the band to −√(h² + 4), exactly, and holds it on 1.42 sites. Put in more and the levels spread: at one site in ten they span 1.45 in the same units and have closed to within 0.69 of the host band, and above one site in eight the count of levels stops matching the count of defects, because two defects on neighbouring sites push one of their pair back into the band.

solids · Defect How many sites a state occupies, and whether that depends on the ring. The participation ratio of the states at the middle of the band — the number of sites a state occupies — against the width of the disorder, for rings of 50, 100, 200 sites. With no disorder the three curves are three different numbers, each two thirds of its own ring. At the right they have converged: 7.47 sites on a ring of 50 and 9.82 on a ring 4 times larger.

The third way to be an insulator

A ring of two hundred sites with a half-filled band has its levels crowding together as 1/n, which is the usual electronic-structure criterion for a metal, and it goes on holding at every disorder tested. Meanwhile the states at the middle of the band go from occupying 127 sites to occupying 10 — and at that disorder the number stops depending on how large the ring is at all.

wrong · Metal How many centres, by two measures that do not agree. For each of eight systems, the number of atoms one localised pair has real amplitude on, and its participation number — which weights those atoms by how much of the pair each holds. Where the sharing is even the two coincide; where it is not they differ by more than a whole centre, and that gap is what electron deficiency looks like from the inside. one of the systems has more than one localisation, so for it neither number is an answer.

One scale, from two centres to a cage

How many centres a pair of electrons holds together is two different numbers, and they separate exactly where the bonding is most deficient: the methyllithium tetramer's pairs sit on four atoms each and have a participation number of 2.54. Run the same measurement up the scale and the twelve-vertex borane refuses it — its localisation has at least ten maxima differing by six parts in a thousand, so the number of centres is not an output for it at all.

beyond · Multicentre Ten answers, and none of them is another one turned round. Every localised description the search found for a twelve-vertex cage, placed by the value of the functional it maximises. There are 10 of them, spanning 0.02, and the two closest differ by 0 — far more than the 10⁻¹³ the sweep converges to, so they are different maxima rather than one maximum reached to different precision. Each has its own multiset of participation numbers, and a symmetry of the cage permutes sites without changing that multiset, so no two of these are related by one. The controls above find one answer each.

How many descriptions a cage has

A localisation is a maximisation, and running it once reports the maximum it reached rather than the maximum there is. Run to exhaustion on a twelve-vertex borane it finds ten answers and then three batches of twelve starts in a row that find nothing new — and none of the ten is another one seen from a different side, because a symmetry of the cage cannot change a multiset of participation numbers and all ten multisets differ.

beyond · Multicentre Four bonds, or one a₁ and three t₂. The same four occupied orbitals written in two bases. On the left each orbital sits on one bond; on the right one is shared over all four hydrogens and three follow the Cartesian directions. The transformation between them is orthogonal, so the density is unchanged.

The answer a search is most likely to give

If a cage has ten equally good localised descriptions, why does the literature agree about its picture? The hoped-for answer was that one basin is very large. Counting four hundred and eighty starting points says it is not: two independent searches agree one time in ten, and the description they most often return is not the best one.

beyond · Multicentre Where the two halves of an alloy come apart. The gap at the centre of a binary alloy's band against the contrast between its two components, for a chain of 2048 sites arranged three ways. The ordered arrangement's gap is exactly twice the contrast and opens at once; the segregated one's is exactly the contrast less the band width and opens at 2; and the random one, which has no closed form, opens a little before the segregated one and stays a little wider. Below the openings the curves sit at one level spacing rather than at zero, which is what a finite chain has instead of a gap.

Two bands, if the chain is short enough

Take a chain, raise half its sites and lower the other half, and ask when the band comes apart into two. The ordered arrangement splits at once, the segregated one at a contrast equal to the band width, and the random one splits earlier than either — and then closes again as the chain is made longer, because a long chain contains a long run of like atoms and a long run is a narrow sub-band.

solids · Defect A state in the gap is a particle in a box the alloy happened to make. The participation ratio of the 40 levels nearest the gap centre, over 5 chains of 400 sites, against the length of the run of like sites each one sits on. The line is 2(L+1)/3, the participation ratio of the ground state of an isolated chain of L sites, with nothing fitted. 38 of 40 lie on it to within 3.1 per cent. The ones above it are states shared between two runs close enough to talk, which is a defect band beginning.

A particle in a box the alloy made

A random alloy's gap is set by the longest run of like sites. What sits at the edge of that gap turns out to be the simplest state in quantum mechanics: a particle in a box of L sites, occupying 2(L+1)/3 of them, to within three per cent and with nothing fitted — except for the two states in forty that found a second run to share.

solids · Defect

Named alongside it

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

LocalisationBands in a solidBand edgeModel limitTight-binding modelsCanonical orbitalsElectron-deficient bondingLevel spacingMulticentre bondingUnitary transformationClusterDefect

All concepts