Molecular Geometry & Bonding
what the shape decides
Concept

Threshold — where it appears

A value at which a verdict changes, chosen rather than discovered. Point group assignment, activity and closure all depend on one, and the honest form of such a decision states the continuous quantity underneath it.

Named by 7 essays across 4 fields — each of them below, with the objects they name alongside it.

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 The twist the ring forces on the bond. For each ring size, the largest torsion about the double bond that the ring will close on — 180° being a flat trans arrangement and 90° being a π bond broken outright. The six-ring will not close at any torsion tested. The eight-ring reaches 139.3°, against 136° measured in trans-cyclooctene by diffraction: a model with bond lengths and bond angles in it and no energy anywhere agrees with the crystal to a few degrees.

The double bond a ring cannot hold

A trans double bond needs a ring of nine carbons to sit flat, and the eight-ring will hold one twisted by 40.7 degrees — against 136 degrees measured in trans-cyclooctene. The same eight-ring threshold, applied by counting ring sizes, sorts nine bridgehead alkenes correctly with no bridgehead anywhere in the argument.

shape · Strain How far benzene is from its own group, against how far it has been pushed. The measure — the distance to the nearest structure with the ideal group — against the size of the distortion, for two ways of distorting the same molecule. Each curve is smooth and quadratic; the markers on it are where the point-group search changes its verdict, which happens at one step and says nothing about the steps either side of it.

How much symmetry is left

A point group is a verdict and every real structure fails it. Two distortions of benzene that are two per cent apart in how far they sit from D6h need tolerances 1.88 times apart before the search will call either of them D6h — and one tolerance, applied to six molecules, admits amounts of asymmetry differing by a factor of fifty-six.

symmetry · Point group 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 The coupling at the closest pair, and at the typical one. For runs of four to eight low sites at half concentration: the splitting between two runs one site apart, which is the largest coupling any chain can produce, and the splitting at the separation two runs typically have. Both on a logarithmic axis spanning 173 decades. A band width taken from the second is not a small number; it is not a number.

A band that is a hundred and seventy decades of nothing

The chain length at which a defect's levels become a band can be located by asking when the coupling between the closest pair exceeds the level spacing. A band also has a width, and a width is set by the typical coupling rather than the closest one. For runs of eight at half concentration those two numbers differ by a hundred and seventy-three orders of magnitude.

solids · Defect 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.

Model limitBands in a solidLocalisationBand edgeDefectDopingLevel spacingParticipation ratioTight-binding modelsClosed formDefect stateDistortion

All concepts