Molecular Geometry & Bonding
what the shape decides
Figure

Which site a long bond takes, and it is not the one the rule says

The energy of the axial placement minus the energy of the equatorial one, against the odd bond's length. Above the line the equatorial placement wins and below it the axial one does. The textbook sentence is that a bulkier group goes equatorial where there is more room; given a longer bond and nothing else, the repulsion model puts it AXIAL, at every length, by a margin that grows. A site pushed further out interacts with everything less, so what is left to decide is the arrangement of the four sites left behind — and three equatorial with one axial is the better set of four.
Which site a long bond takes, and it is not the one the rule says. The energy of the axial placement minus the energy of the equatorial one, against the odd bond's length. Above the line the equatorial placement wins and below it the axial one does. The textbook sentence is that a bulkier group goes equatorial where there is more room; given a longer bond and nothing else, the repulsion model puts it AXIAL, at every length, by a margin that grows. A site pushed further out interacts with everything less, so what is left to decide is the arrangement of the four sites left behind — and three equatorial with one axial is the better set of four.

One of the figures on where the atoms go: Repulsion minimised on a sphere, the angles that fall out of it, and the arrangements that are not all alike.

Eight essays draw this figure, each at the values its own argument needs rather than at the setting shown above. What each one uses it to show is below, in the words of its own caption.

In the essays

The sites are not the same size

Where each placement wins, in the plane of the two knobs, at a larger site weight than the map above uses. The boundary moves and the shape of the region does not — so which site a long bond takes is decided by the ratio of the two effects rather than by either of them, and no single number settles it.

The energy of the axial placement minus the energy of the equatorial one, against the odd bond’s length. At equal lengths the two are one structure and the difference is exactly zero. As the bond lengthens, the axial placement wins, and the margin grows monotonically.

For each electron demand, the range of bond lengths over which the equatorial placement wins. It is a window rather than a threshold, and below a demand of 0.933 it closes altogether — the axial placement then wins at every length.

The long bond goes to the crowded site

The energy of the odd bond placed axially minus placed equatorially, against its length relative to the others, at five and at seven sites.

The angles from each kind of site to its neighbours, with the sums of 1/r and 1/r¹² over them, at five and at seven sites.

The axial and equatorial bond lengths of phosphorus pentafluoride and iodine heptafluoride, from gas-phase electron diffraction.

The ring held flat by its long bonds

IF₇’s composition placed all four ways, each relaxed and tested with small random kicks, with its energy above the lowest in units of one short bond’s equatorial penalty.

Every internal curvature of the flat structure, on a logarithmic axis, with equal bonds and with IF₇’s. The degenerate pair at the far left is the ring’s pucker.

The curvature along the pucker against the ring-to-axis bond ratio, under three repulsion laws. Below zero the flat ring is not a minimum.

The fluorines' range decides the ring

The pucker’s curvature with every repelling point placed at a fraction of its bond, rescaled by that fraction to the law’s power and divided by the value at the bond’s end. Two power laws give exactly one at every fraction; the exponential does not, and changes sign at 0.723 of the bond.

For each contact law, the share of the Coulomb energy the contact term must carry before IF₇’s flat ring stops being a minimum, filled, and the same with equal bond lengths, open. Three laws hold the ring flatter at any weight.

Each law’s curvature along the pucker split into its four kinds of contact, scaled to the ring-neighbour term. Only the five ring-neighbour pairs favour the pucker; the ten ring-to-axis pairs resist it.

The lone pair goes through a face

For each candidate structure at a lone-pair weight of 1.5 with no contact term, the angles from the lone pair to the six fluorines.

The capped and edge-capped octahedra’s energies above the pentagonal pyramid for lone-pair weights from 1.05 to 3, with no contact term.

At a lone-pair weight of 1.5, the capped and edge-capped octahedra’s energies relative to the pentagonal pyramid against the fluorines’ Born–Mayer contact share, with the lowest structure found from any nudged start.

A lone pair weaker than a bond

The spread of the twelve cis F–Xe–F angles in the capped octahedron, against the lone pair’s weight, with no contact between the fluorines.

With the domains alone, the capped and edge-capped octahedra’s energies relative to the pentagonal pyramid, for weights from 0.1 to 1.5.

The range of lone-pair weights over which each symmetric structure is a true minimum, with the domains alone.

The model lengthens the wrong three

The energy of the capped octahedron with its long three round the opened face, minus with them opposite, against the lone pair’s weight, with the domains alone and with a one-per-cent contact.

The outward push on the three bonds round the opened face minus the push on the three opposite, split into what the lone pair contributes and what the fluorines contribute, and their sum enlarged below.

The lowest equal-length structure over lone-pair weights from 0.1 to 3 and contact shares from 0 to 6 per cent, with a dot wherever the capped octahedron prefers its long bonds round the opened face.

The lone pair has no bond to stretch

The lone pair’s distance from the xenon, as a fraction of the mean bond, under each reading at one weight and stiffness.

Near three over far three, minus one, against the lone-pair weight at stiffness 20. Kept among the bonds the long three are opposite below 0.725; left where it was or on a spring, they are round the face over that range.

The split under the lone pair kept among the bonds, at six stiffnesses, each scaled by its own largest value. Every curve crosses zero at 0.7253.

Every figure · Every orbital, by what it encloses · All essays