Every essay — page 23
Orbitals Where the atoms go Bonding models What symmetry decides Beyond the octet What a spectrum settles When the molecule does not stop What the shape is for What is taught wrongly Series Named objects Orbitals Refutations Search
Where the atoms go
VSEPR as a repulsion minimisation rather than a table, the angles that fall out of it, and the case where five sites are not all alike.
Deuterium cannot tell the masses apart
A reduced mass built by holding ammonia's bonds rigid predicts a deuterated molecule differently from any constant mass, and ND₃'s splitting is measured. Run as a test, it cannot choose. Every well and every mass needs a barrier for ND₃ several per cent lower than for NH₃, every prediction of the isotope ratio from a well fitted to NH₃ overshoots by eighteen to twenty-nine per cent, and the two masses differ by less than either misses — in opposite directions in the two wells.
The two that are not on the line
Ammonia and its fully deuterated twin are two points, and two points cannot show a curve. Putting the partly deuterated molecules between them needs a term neither symmetric one has — three ligands of unequal mass move their own centre of mass sideways as the apex descends — and it moves the prediction by half a per cent, which is what a whole change of mass construction was worth.
The ordering a manifold picks
A position-dependent mass leaves the kinetic energy with no unique quantum form, and an earlier sweep of the five orderings in use found half a per cent between them. A one-dimensional reduction is a one-dimensional manifold, a manifold has a distinguished Laplacian, and carrying it to the flat measure lands on exactly one of those five — not the one with no extra potential, and not the one anybody reaches for.
The long bond goes to the crowded site
Given one bond longer than the others, the repulsion model puts it axial in a trigonal bipyramid — against the rule it is usually cited for. The reason is a crowding count, and at seven sites the count reverses: the pentagonal bipyramid's crowded site is equatorial, so a long bond goes equatorial and a short one axial. PF₅'s long bonds are axial and IF₇'s are equatorial. And at seven the site a bond avoids is not even a minimum.
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.
The band that cannot supply what it changes
A missing transverse zero-point term was sized at five per cent of the difference between two molecules' frozen frequencies, as though the five frozen modes were one thing. They are three bands carrying 26, 50 and 24 per cent of that difference, so the requirement is not one fraction but three, spanning a factor of two — and the band a flattening pyramid changes most is the one that would have to move furthest. What survives the choice is the barrier: every version implies the same bare electronic value to one and a third per cent, because every band of a pyramidal hydride has the same isotope frequency ratio.
The steps were a square root of a mass
Replacing ammonia's hydrogens one at a time does not lower its splitting in equal steps, and the obvious readings are the potential's shape or the tunnelling arithmetic. Both lines — the ground doublet and the first excited one, which nothing in this argument has ever been fitted to — come out straight in the square root of the reduced mass and bent against the count, by a factor of eight, so the unequal steps are the shape of a square root drawn against the wrong variable. And the excited line is the one test the missing correction cannot pass by construction: it improves it threefold and over-shoots it.
The ring held flat by its long bonds
Given iodine heptafluoride's own composition — five long bonds and two short — the repulsion model puts the short pair on the axis, as the molecule does, and keeps the equatorial ring flat. But the flatness is the softest thing in the arrangement by a factor of a hundred. With equal bonds, a pucker of the ring costs nothing at all to second order under an inverse-square law and lowers the energy under any steeper one. What holds IF₇'s ring flat is the four per cent by which its ring bonds are longer, and that holds only under laws softer than about the 4.4th power.
The fluorines' range decides the ring
IF₇'s flat ring was held by its bond lengths under every repulsion softer than about 1/r^4.4, with the fluorines left out. Put them back as a second set of repelling points and under the same law they change nothing at all — they sit on the same rays, and the model cannot tell where along a ray a point sits. They matter only as a contact repulsion of a different shape. Then a steep power puckers the ring with a few thousandths of the energy. An exponential contact puckers it only if its range is under 0.346 ångström, and a longer one holds it flatter. Fluorine's own orbital gives a range near 0.16. And the ring that puckers does not settle: its envelope and its twist differ by a ten-thousandth of the pucker's depth, so the pucker runs round the ring.
The lone pair goes through a face
Xenon hexafluoride is the other famously floppy seven-domain molecule, and its seventh domain is a lone pair: a domain with no atom at its end, which a contact repulsion between fluorines cannot see. With the domains alone the model puts the lone pair on the axis of a pentagonal bipyramid at every lone-pair weight — a pentagonal pyramid of fluorines. Add the six fluorines' contact repulsion and, past a share of the energy that rises with the lone pair's weight, the lowest structure becomes a distorted octahedron with the lone pair through one face. Every contact law tried does this. On the way the lone pair leaves the axis continuously, and past it the molecule's floppiness is the lone pair hopping from face to face over an edge — not a ring's pucker moving round a ring.
A lone pair weaker than a bond
The repulsion model put xenon hexafluoride's lone pair on an axis at every weight VSEPR allows, and only a contact between the fluorines could move it into a face. Below the weight VSEPR allows — a lone pair that repels less than a bond, which is what an inactive pair is — it goes into a face on its own. As the weight falls to zero the distorted octahedron becomes a regular one continuously, at 58.5° of angle spread per unit weight; and between the face and the axis there is a region nobody had looked at, where the lone pair sits over an edge.
The model lengthens the wrong three
Xenon hexafluoride's capped octahedron has its three longer bonds round the face the lone pair opens. Give the repulsion model three long bonds and three short and ask where it puts them: wherever the face is its lowest structure — at every lone-pair weight and every contact share mapped — it puts the long three on the opposite side. The lone pair's push on its neighbours and the fluorines' push on each other nearly cancel, and the lone pair wins only above a weight of 0.727, where the face has already given way to the edge.