Priced against the whole bond

A contact between five ligands makes a main-group square pyramid only if it is steeper than the fifteenth power of their separation, because five points repelling each other change their own best arrangement from the bipyramid to the square pyramid at s = 15.05, and every softer law sides with the bipyramid the bonds already prefer. That was the condition. The price was a share: the contact energy of the bipyramid at the switch divided by the bipyramid’s σ cost, and it ran from about a hundred at the thirtieth power to two thousand at the sixteenth.

The σ cost in that share was the fourth-order term of an expansion in the coupling between the ligands and the central orbitals — the term the essay on five ligands and no ideal used to find that five points cannot be both balanced and isotropic. The same essay found that the exact σ energy behaves differently from its expansion once the coupling is not small. So the question left was whether the exact energy makes the contact’s job easier or harder, and whether a hundred to two thousand is even the right order.

It is the right order only at weak coupling, and its units are the wrong ones anywhere else. In the units the bonds actually supply — the σ bond energy — the price settles at a fraction set by the contact’s steepness, and the dependence on the centre that the fourth-order share was built around disappears.

The same bisection, with the energy exact #

The model is unchanged. Five ligand σ functions at a common energy couple to a central s orbital one unit above them and three p orbitals r units above, with couplings β in units of the s level’s height and equal overlaps; the σ parameter of the earlier essays is q = r(1 + r), so a main-group centre with q = 2 has its p level at the same height as its s level — r = 1 — and q = 5 puts it at 1.79 times the s height. The exact σ bond energy is twice the sum of the five lowest levels of the resulting nine-by-nine matrix, which is a diagonalisation rather than a series.

To that is added the contact, a law 1/rˢ of the distance between each pair of ligand positions on the unit sphere, and the switch is found as before: for each weight, the best square pyramid over its apex angle against the bipyramid, and a bisection in the weight for the point where they are equal. The share keeps the earlier normalisation, the bipyramid’s contact energy over $\beta^4\kappa/6$ — the bipyramid’s fourth-order σ cost in absolute units — so that the two columns compare directly.

Ten couplings, from β = 0.02 to 1.5; four σ parameters, q = 1, 2, 3 and 5; three steep laws, the sixteenth, twentieth and thirtieth powers. That is 120 switches, each a bisection over diagonalisations, which together take under half a minute.

Weak coupling returns the fourth-order answer #

The first check is the one that says the exact calculation is pricing the same thing. At β = 0.02, where the expansion should be good, every one of the twelve exact shares is within two per cent of its fourth-order value — 1,117 against 1,123 at q = 2 under the sixteenth power, 100.0 against 100.6 under the thirtieth. That is the refusal built into the comparison: an exact energy that did not reduce to its own expansion would be answering a different question.

It is worth being exact about what small means here, because it is smaller than it looks. At β = 0.05 the exact shares are already 3 to 6 per cent below the fourth-order ones, and at β = 0.1 between 10 and 20 per cent. The expansion is in $\beta^2$ over the square of a level spacing, and at q = 1 the p level sits only 0.62 units above the ligands; β = 0.1 is not a weak coupling for that centre.

The bipyramid’s advantage stops growing as the fourth power #

What moves the price is the thing the contact has to overcome: how much the bonds prefer the bipyramid. With no contact the exact σ energy prefers the bipyramid at every coupling and every main-group q, as the fourth-order term does — the shape does not change. What changes is by how much.

At fourth order the preference grows as $\beta^4$. The exact preference does too while β is small — a local slope of 3.97 on logarithmic axes at β = 0.05 — and then stops. The slope falls to 3.45 by β = 0.2, 2.05 by 0.5, 1.22 by 1 and 1.06 by 1.5: at strong coupling the bipyramid’s advantage grows nearly in proportion to β, like the bond energy itself. By β = 1.5 the exact advantage at q = 2 is 0.65 per cent of what the fourth-order term says it is.

The reason is the ordinary one for a perturbation series pushed past its radius. At strong coupling the ligand levels are pushed down by an amount that grows as β rather than $\beta^2$, because each bonding combination is a two-level problem whose splitting is $\sqrt{\Delta^2 + 4\beta^2\lambda}$ rather than $\Delta + 2\beta^2\lambda/\Delta$; every geometric preference inherits that, and a preference that was a fourth-order difference of two second-order quantities becomes a first-order difference of two first-order ones. The expansion was never going to be right there. The point of computing it is to see how wrong, in what direction, and whether it matters for the shape.

So the share falls #

The share is the contact needed over a fixed fourth-order unit, so it falls exactly as the exact preference falls below the fourth-order one. Under the sixteenth power at q = 2 it is 1,123 at fourth order, 720 at β = 0.2, 222 at 0.5, 46 at β = 1 and 15 at 1.5. Under the twentieth power the same column runs 233 to 3.1, and under the thirtieth 101 to 1.3. At q = 1 and β = 1.5 the thirtieth-power share is 0.3: the contact needed is a third of the bipyramid’s fourth-order σ cost, where the earlier essay needed seventy times it.

Read in the earlier essay’s units, then, the exact energy makes the contact’s job easier — by a factor of 11 to 70 at β = 1 and 28 to about 230 at β = 1.5, most at q = 1 and least at q = 5. The shares of a hundred to two thousand were the right order only for β below about 0.1.

Two things do not move, and they are the two the earlier essay built its argument on.

The condition. Under the soft laws — Coulomb, the sixth and the twelfth powers — no share up to ten million switches any main-group q at β = 0.2, 1 or 1.5. The exact σ energy prefers the bipyramid at every coupling, and a contact softer than the fifteenth power prefers it too, so there is nothing for any weight to trade. The threshold s = 15.05 belongs to five points on a sphere and no σ energy can touch it.

The shape. The apex angle of the pyramid at the switch is 97.8° under the sixteenth power, 97.3° under the twentieth and 96.1° under the thirtieth, and at every coupling it agrees with the fourth-order switch’s to within half a degree. At the switch the contact has chosen the pyramid, and the σ energy only decides how much contact that took.

A seeded search over every arrangement of five directions, with the exact energy and the contact, confirms at three points either side of a switch that the lowest arrangement is one of the two shapes, to 10⁻⁹. The bipyramid and the square pyramid are still the only candidates.

The units were the wrong ones #

A share that falls three-hundredfold because its denominator was computed in the wrong approximation is not yet a price. The denominator — $\beta^4\kappa/6$ — is a number the exact model never produces; it is the fourth-order term evaluated where the fourth-order term has stopped describing anything. The quantity a chemist would compare a contact against is the bonding it competes with.

So the second normalisation is the contact energy at the switch divided by the bipyramid’s exact σ bond energy. At weak coupling it is small — 0.07 to 0.75 per cent at β = 0.02 — because the σ preference the contact has to overcome is a fourth-order quantity and the bond energy a second-order one. As the coupling grows both become first order, and the fraction levels off:

• under the thirtieth power, at 14.8 per cent of the σ bond energy by β = 1.5;
• under the twentieth, at 35.3 per cent;
• under the sixteenth, at 173 per cent — the contact energy of the bipyramid has to exceed its entire σ bonding before the square pyramid wins.

The fraction is flat from about β = 1 onward, within one and a half per cent of its value at 1.5. So in the regime where the exact energy differs most from the expansion, the price has a stable meaning after all: a contact must carry a fixed fraction of the bond, set by its steepness.

The centre drops out #

The earlier essay’s shares rose with q at every steepness — 823, 1,123, 1,422 and 2,020 at q = 1, 2, 3 and 5 under the sixteenth power — and that rise was a finding: a centre whose p level sits further above its ligands prefers the bipyramid more, so it needs a stiffer push. At fourth order the push grows 2.5-fold under the sixteenth power and 2.8-fold under the thirtieth across that range.

At β = 1.5, measured against the bond, it barely grows at all. Under the sixteenth power the fraction is 173, 173, 171 and 162 per cent at q = 1, 2, 3 and 5; under the twentieth 35, 35, 35 and 33; under the thirtieth 15, 15, 15 and 14. Whatever the centre, a contact of a given steepness must carry the same share of the bond.

That is the surprising connection, and it runs against the chemistry the fourth-order picture suggested. There, q was a property of the central atom — how far its s and p levels are apart relative to the ligands — and the contact’s job depended on it. In the exact model at strong coupling q still sets the bond energy and still sets the σ preference, but it sets them in proportion, so their ratio no longer knows which atom is at the centre. A main-group square pyramid needs a contact of a certain steepness carrying a certain fraction of the bond, and whether the centre is phosphorus, arsenic or antimony enters only through whether the ligands can actually supply that contact.

What the fraction says about pentaphenylantimony #

The fraction is large, and that is the part worth sitting with. A contact carrying a seventh of the σ bond energy is not a small steric correction to the bonding, and one carrying more than all of it is not a correction at all. The reason the price is so high is visible in the earlier essay’s own numbers: near its threshold a steep law distinguishes the two shapes by only a few per cent of itself — 0.41 per cent at the sixteenth power, 2.3 at the twentieth, 6.7 at the thirtieth — so to produce even a small difference between the shapes it has to be large in both.

Pentaphenylantimony is a square pyramid in one crystal and a bipyramid in a cyclohexane solvate, which the earlier essay read as a molecule so close to its switch that packing decides. In this model, close to the switch at strong coupling means a contact energy within a narrow band around a large fixed fraction of the bond — and a contact that large is something a crystal environment could shift by the few per cent that would move the molecule across. The model cannot say whether phenyl rings supply a thirtieth-power contact worth a seventh of the Sb–C bonding; it says that anything less steep or less large does not do it.

Why the share had to fall as β cubed #

The two normalisations are related by one line of arithmetic, and it is worth doing because it says the fall in the share is not a detail of this model. The share divides the contact energy at the switch by $\beta^4\kappa/6$. The contact energy at the switch is whatever it takes to cancel the σ preference, and at strong coupling that preference grows as $\beta^{1.1}$ or less. So the share must fall roughly as $\beta^4$ over β — as $\beta^3$ — and it does: under the twentieth power at q = 2 it goes from 9.3 at β = 1 to 3.1 at 1.5, a factor of 3.0 where $\beta^3$ gives 3.4.

Any quantity quoted in units of a fourth-order cost will do this once the coupling is strong, and so will any argument that compares a steric term with the fourth-order bonding term. The earlier essay’s shares were correct statements about the expansion. They were not statements about how bulky a ligand has to be, and a reader carrying a hundred to two thousand times the σ cost into a real molecule would have carried a number whose size depended on an approximation that real bonds are far outside.

What the three contact results now share #

A contact between ligand atoms has now decided a shape three times in these essays, and with the exact energy in hand they read more alike than they did.

In iodine heptafluoride the range of an exponential contact between fluorines decided whether its flat ring puckered, with a threshold at 0.346 Å. In xenon hexafluoride every contact tried moved the lone pair into a face, and the range set only the price. Here the steepness decides whether a main-group five-coordinate centre can be a square pyramid at all, and — the new part — the price, measured against the bonding, depends on the steepness and not on the centre.

What the three share is that the contact acts on a choice the bonds have nearly made already, and its effect is set by which pairs of ligands are closest in each competing shape. Where the shapes differ in their closest pairs only by a slight shortening — five-coordination’s six contacts at 90° against the square pyramid’s four at 89° and four at 97° — only a very steep law can tell them apart, and it has to be large to do it. VSEPR’s account of five-coordination treats domains as repelling without saying how steeply, and the three distinct angles five sites carry are exactly why the steepness is the whole question.

Every switch #

The table is the whole measurement. Every exact share is below its fourth-order value and falls with the coupling, at every law and every q; the refusal at β = 0.02 holds to two per cent throughout; the apex angles agree with the fourth-order ones to half a degree; and no soft law switches anything.

Where the model stops #

The coupling is a dial, not a measurement. β is in units of the s level’s height above the ligands, and nothing here fixes it for any real molecule. The conclusions are stated as functions of β for that reason, and the one that does not depend on it — the fraction flat from β ≈ 1 — is the one to carry.

Equal bonds and point ligands, as before. A bipyramid’s axial bonds are longer than its equatorial ones, and a phenyl ring is a flat object with an orientation; the contact here is between points at one distance, the limit of a round ligand at a fixed bond length.

σ only. The two π functions each ligand carries add rows the model does not have. Whether they move a main-group centre’s effective q is the other half of the question the earlier essay left, and it is untouched here; the finding that q drops out of the contact’s fraction at strong coupling would, if it held with π rows, make that question less important than it looked.

The contact is added, not computed. A 1/rˢ law is a stand-in for ligand–ligand repulsion whose steepness is the only thing the argument uses. The Born–Mayer range that failed to switch anything fails here too, because whether a contact prefers the pyramid is a property of the contact alone; the exact σ energy changes how much preference is needed, never which contacts can supply it.

Who worked out which part #

The five-point Riesz problem and its switch near the fifteenth power belong to the mathematics of point energies on a sphere. The σ-only molecular-orbital picture of five-coordination, with its s–p balance, is Rundle’s and Pimentel’s three-centre bonding extended to five ligands, as the count of reachable orbitals set it out here. That a perturbation series for a bond energy fails once the coupling is comparable to the level spacing is textbook.

What is computed is the specific comparison: the switch shares under the exact energy, the growth exponent of the bipyramid’s advantage, and the fraction of the bond a contact must carry, with its independence of q.

Still open: the π rows, and bonds of two lengths #

The obvious open question is still π donation. Two π rows per ligand would add a second set of couplings with their own geometry — a π function has a direction perpendicular to its bond, so balance and isotropy mean something different for it — and whether they help the square pyramid or the bipyramid is a diagonalisation of a nineteen-by-nineteen matrix instead of nine-by-nine. The finding here sharpens it: if π rows also scale with β at strong coupling, the contact’s fraction would stay independent of the centre and the π question would be about the fraction’s size, not about which atom sits in the middle.

The nearer question is the bond lengths. The model puts every ligand at one distance, and a real bipyramid’s axial bonds are several per cent longer than its equatorial ones. Letting the axial pair take a second radius, as the sites of a five-coordinate molecule already do for other purposes, changes the bipyramid’s closest pairs — the six axial–equatorial contacts that a steep law is dominated by — and so changes the few-per-cent difference the whole price rests on. It is one more parameter in the same bisection.