Molecular Geometry & Bonding
what the shape decides
Concept

Numerical precision — where it appears

The resolution the arithmetic itself has, below which a computed difference is the gap between two numbers that are equal rather than a small quantity. It bounds what a calculation can measure as firmly as the model does, and it does so without appearing in the model.

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

The exponent against how far the tail can be seen. The fitted power against the reduced reach — how many coherence lengths of the decay survive above the floor before the excess is numerical noise. The six cases with a reach past six give a power between 0.41 and 0.57 and are drawn solid; the rest are hollow and their fitted powers are off this scale in the negative direction. The reach is not a choice — it falls as the gap closes, because the excess the tail starts from falls with it.

The exponent was the floor

Fitting the local decay rate against the reciprocal distance reads a power off the slope. It runs from 0.41 to 0.66 across ten stiffnesses and appears to settle near two thirds. It is not settling. The tail is dropping below the arithmetic's own floor sooner at every step, so each case's power is taken over a shorter piece of the curve than the last.

solids · Peierls distortion The exponent against where the fit is allowed to start. For each stiffness, the fitted tail exponent as the near end of the fitting window is moved outward from two bonds to thirty. The standard start is six, by a rule of thumb — three coherence lengths — and the question was whether that choice is doing any work. Below six the exponent rises steeply; from six outward it is nearly flat. The rule of thumb sits on a plateau.

The rule of thumb was on the flat part

Fitting the Peierls tail discards the first few bonds of every profile, on a rule of thumb — three coherence lengths. Does that unexamined choice hide a second exponent? It does not. From six bonds outward the fitted power moves by half a per cent to nine; below six it moves seven times as much, and starting at two would have halved the very trend the fit reports.

solids · Peierls distortion The far end stops mattering, abruptly. The fitted tail exponent as the far end of the window is opened from twenty bonds to a hundred and twenty. Each curve is flat past a stiffness-dependent point and exactly flat past it — because beyond a profile's own reach there are no more local rates to add, so a larger window is the same fit. The standard sixty bonds is inside the flat part for every case.

The other window was a plateau too

Sweeping where the fit begins finds a plateau. The far end is the other window and nobody had swept it: inside each profile's own reach the exponent moves by at most 5.3 per cent, and past that reach every larger window returns exactly the same fit — because there are no more points to add. What the reach is depends on the stiffness, and for half the series it is an arbitrary rule rather than the physics.

solids · Peierls distortion The mass is worth a factor of 1.6, and the other two 1e+4 and 9e+4. Ammonia's umbrella well, with each of phosphine's three differences substituted into it one at a time and then all together. The reduced mass is 11 per cent larger and costs a factor of 1.61. The pyramid is 2.01 times taller and costs 9.6e+3; the barrier is 6.1 times higher and costs 9.4e+4. Phosphine's own splitting is below what the arithmetic resolves, so it is drawn at that bound.

It was never the mass

Phosphine does not invert, and the reason given is that phosphorus is heavier than nitrogen. Three things about phosphine differ from ammonia. Substituting each into ammonia's own well one at a time, the reduced mass costs a factor of 1.61, the pyramid height costs 9,600 and the barrier 94,000 — and the mass is the smallest of the three by four orders of magnitude.

shape · Inversion One sentence, three displacements, three answers. For each symmetry-distinct internal coordinate of the five molecules, how concentrated the projection is across the occurrences of the species the label names — under the B row the published measurement used, under the displacement that changes the coordinate at the least kinetic cost, and under the one that changes this coordinate and no other. Ammonia's bond angle goes from 0.516, a coin, to 0.883 and 0.926. Boron trifluoride's angles are the worst case under one clean displacement and its bonds under the other, both at 0.573.

Three ways to stretch a bond

A symmetry label was measured at half the distance from a coin to a decision about which vibration a distortion goes into. The distortion was a B-matrix row, which moves a central atom as far as a ligand — and in the metric where the vibrations are orthogonal, three quarters of water's bond-angle distortion turns out to be the molecule moving rather than deforming. Two displacements that realise the same sentence and carry no rigid motion at all double what the label decides, from eleven coordinates of twenty-nine to twenty-two, and move the worst case from ammonia to boron trifluoride. Between them sits an exact identity that found a rounding error in a stored molecule.

symmetry · Point group Eighteen paths, two hundred and seventy geometries, one gap. Every one-parameter path between two arrangements of the same ligand count, each sampled at 15 geometries, with each geometry coloured by whether its group is named and tabulated, refused by the finder's tolerance, or a finite group with no table. The one gap is a group of order 10 at the pentagonal-pyramidal end of two paths — C5v, which no table here reaches. The path with a linear end is excluded, since a continuous group is declined deliberately rather than missing.

The gap found on purpose

A twist between two coordination arrangements passes through a point group with no character table among the twenty in use, and that was found by accident. Running every path between the arrangements — eighteen of them, two hundred and seventy geometries — finds exactly one more gap, at a group of order ten. It also finds something the search was not looking for: a group without a table had been reported as an infinite group.

beyond · Hypervalency Two pyramids, and only one of them has ever shown a line. What ammonia and phosphine each bring to the same one-dimensional model. The barrier is six times larger and the apex sits twice as far out, so the action under the barrier is seven times larger and the splitting is twenty-four orders of magnitude smaller. What is not six times anything is the transverse zero-point difference between a molecule and its deuterated twin: the modes that carry it are the ligands' own, the ligands are the same three hydrogens, and the two numbers are within thirty per cent of each other.

The same fraction at six times the barrier

Ammonia's two fitted barriers leave a gap that an isotope-dependent term would have to fill, and the term was sized at five per cent of the transverse zero-point difference. Phosphine can be asked the same question and cannot answer it the same way: its splitting is twenty-four orders of magnitude below anything the solver can represent, and the solver does not fail cleanly — it reports values that rise as the barrier rises. Calibrated where a splitting is still visible, the action carries the question across, and the same fraction moves phosphine's isotope ratio by sixteen per cent against ammonia's twenty-one.

shape · Inversion

Named alongside it

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

Model limitBond alternationCoherence lengthExtrapolationPeierls distortionTight-binding modelsApproximationConventionInversion splittingReduced massTunnellingBand gap

All concepts