The sites are not the same size

Every arrangement drawn in this collection has been found the same way: place $n$ points on a sphere, minimise the repulsion between them, and read off the angles. The tetrahedral angle is not written down anywhere here — it comes out of the minimisation — and neither is the trigonal bipyramid’s 90 and 120.

On a sphere is doing something in that sentence, and it has never been examined. It says every site is at the same distance from the centre; that is, every bond is the same length. It is exactly true for methane — where symmetry makes every bond the same — and exactly false for phosphorus pentafluoride, whose axial bonds are 1.577 Å and whose equatorial bonds are 1.534. It is wildly false for a molecule with two different substituents, which is most molecules.

Where the assumption bites #

Five-coordination is where it matters, because five is the smallest number for which the sites of the minimum are not all equivalent. Two kinds of site: an equatorial position with two neighbours at 90° and two at 120°, and an axial position with three neighbours at 90°. Which one a particular substituent takes is a real chemical question with a real answer, and every textbook gives the same reason for it — the bulkier group goes equatorial, where there is more room.

The model behind that sentence has no bulk in it. It has a repulsion between point charges at fixed distance from a centre, and the only thing that can distinguish one site from another is the weight — how strongly a domain repels — which is what a lone pair’s extra demand has been represented by since the first figure here, and which is also how water’s bent shape is obtained. Bulk, in the sense of a substituent taking up space, is not a quantity the model possesses.

Giving it one is a change of two lines: let each site have its own radius, and measure the distance between two sites as the distance between the points rather than between the directions.

What happens when a bond gets longer #

The seeded minimisation is run twice for each length — once starting from the odd substituent axial, once from it equatorial — and each is relaxed into its own basin. The difference between the two energies is the preference.

The model puts the long bond axial. This is the opposite of the rule.

It is not a numerical accident near the crossing point: the margin grows steadily from 0.0028 at a ratio of 1.1 to 0.0156 at 1.6, and the ordering is monotone throughout. Nor is it a defect of the relaxation, since the two structures are checked to have settled in the basins they were started in — an axial substituent still has three neighbours near 90° and an equatorial one still has two.

The reason is arithmetic and it is worth following, because it says which term is really deciding. Push a site further from the centre and every interaction it has weakens; in the limit of a very long bond it stops mattering altogether. What is left is the arrangement of the four sites behind it, and those four have a choice: three equatorial and one axial, or two and two. The first is worth 3.853 in these units and the second 3.906. Three-and-one wins, so the long bond goes axial and leaves the short bonds in the better arrangement.

That is a real effect and it is not the effect the rule names. The word bulky in the textbook sentence is not doing the work it appears to be doing.

The arithmetic behind the sign #

The frame argument deserves the numbers, because it is short and because it explains why the effect is inescapable rather than a property of a particular potential.

Split the total repulsion into two parts: the interactions the odd site has with the other four, and the interactions the other four have among themselves. The second part does not involve the odd site at all, and it takes one of exactly two values.

If the odd site is axial, the four left behind are three equatorial and one axial. Their mutual energy is three pairs at 90° — distance $\sqrt{2}$ — and three pairs at 120°, distance $\sqrt{3}$: $3/\sqrt{2} + 3/\sqrt{3} = 3.853$.

If the odd site is equatorial, the four left behind are two axial and two equatorial. That is one pair at 180° at distance 2, four pairs at 90°, and one pair at 120°: $0.5 + 4/\sqrt{2} + 1/\sqrt{3} = 3.906$.

The difference, 0.052, is fixed. It does not depend on the odd site’s length at all, and it always favours the axial placement. The odd site’s own terms fall towards zero as its bond lengthens, so at long range the fixed difference is the whole answer — and its sign was decided before the substituent was mentioned.

At equal lengths the odd site’s terms make up exactly the same 0.052 in the other direction, which is why the two placements are one structure there. Everything between is a competition between a term that shrinks and a term that does not.

The consequence is that the sign of the long-bond preference is not a property of the Coulomb exponent, or of the particular minimiser, or of how carefully the structures were relaxed. It is a property of which four-site arrangement is lower, and that is a two-line calculation with no free parameters in it.

The knob that does reproduce the rule #

The other knob does give the observed answer. A site demanding more room — a lone pair, or the bond to a less electronegative partner, which holds its pair closer to the central atom — prefers equatorial, at equal bond length, by a clear margin. That is the model’s version of Bent’s rule and it is what this collection’s weighted arrangements have said all along.

So the two effects are opposed. Length says axial; demand says equatorial. Real substituents that are said to be bulky are usually both larger and less electronegative than the atoms they replace, so the two are confounded in almost every example, and the observed preference has been attributed to the one that does not produce it.

Why a window and not a threshold #

The first attempt to draw this asked for one boundary length per demand and bisected between 0.4 and 2.6. It was refused, because at low demand the two ends of that range have the same sign: the axial placement wins everywhere.

At equal demand there are two crossings, not one. The equatorial placement wins only between 0.551 and 1.000 times the others’ length. Above the upper edge the frame argument above takes over. Below the lower edge something else does: a very short bond pulls the substituent so far in that it stops being the crowded one — its neighbours are then further away than they were, and the crowding it was avoiding by going equatorial has gone.

The upper edge is pinned at exactly equality by symmetry, and that is the fixed point the whole map hangs on. The lower edge is a real length, and the window widens with demand — 0.234 wide at a demand of 0.95, 0.449 at 1.00, 1.026 at 1.30 — so a substituent that wants more room can afford a longer bond before length wins. The demand axis is the same axis the substituent series is measured along, which is what makes the two halves of the map commensurable at all.

The critical demand at which the window opens at all is 0.933, and nothing put it there: it is where the largest value of the odd site’s own energy difference, over all lengths, comes to equal the fixed difference between the two arrangements of the four sites behind it.

The angles, which an experiment reports #

The second thing the length knob buys is a measurable. At equal lengths, symmetry fixes the axial angle at exactly 180° and the axial-to-equatorial angles at exactly 90°. With one bond of a different length neither holds.

The first version of this measurement required the angle to fall monotonically with length and was refused at 176.8 → 178.0 → 180.0 → 177.5, which is the shape of a maximum rather than of a defect. Exact equality is a maximum of the axial angle, and the model is symmetric about it in the sense that any departure bends the frame — for opposite reasons on the two sides. A short bond brings the substituent closer and pushes the axials away from it; a long one is further off and lets them relax past it.

That is a testable statement about real five-coordinate structures, and the direction is right: molecules with one long equatorial bond have axial angles several degrees short of 180°. A model whose sites all sit on one sphere cannot produce that at all — symmetry forbids it — so the bend has never been available as evidence for or against the repulsion picture.

Reading a real molecule #

Three cases, and each shows the confounding directly.

Phosphorus pentafluoride has one substituent and two bond lengths — 1.577 Å axial, 1.534 equatorial, a ratio of 1.028. The molecule has full D₃ₕ symmetry with those two lengths, because nothing relates an axial position to an equatorial one. This is the reference case: the difference in length is a consequence of the two site types rather than an input, and the model above, which takes lengths as given, has nothing to say about it.

Chlorotetrafluorophosphorane replaces one fluorine with chlorine. The chlorine sits equatorial, and its bond is 2.005 Å against about 1.55 for the fluorines — a ratio of about 1.3. On the map above, a ratio of 1.3 requires a demand of at least about 1.2 before the equatorial placement wins. Chlorine is less electronegative than fluorine, so it does hold its bonding pair closer to the phosphorus and does demand more room; the observation is consistent, and it is consistent because of the demand, with the length pulling the other way the whole time.

Sulfur tetrafluoride has a lone pair — the case the weighted arrangement was built for — and the lone pair goes equatorial with a demand far above one — which is the case the model has always got right and the case where length is irrelevant, since a lone pair has no bond length to be long.

The pattern across the three is the same. Where the observed preference is strong, the demand is large; where the demand is small, the preference is weak and the assignments are often disputed in the literature. That is what a two-variable answer looks like when it is being reported as though it had one variable.

What the model still cannot supply #

The radii have to come from somewhere, and the model has no way to produce them. That is the honest statement of its limit and it is a sharper limit than the usual one.

A repulsion model determines angles and says nothing whatever about lengths. Handed a set of lengths it now produces angles that depend on them, which is more than it did before; but the lengths are experimental input, exactly as the demand weight is. So the map above has two axes and the model supplies neither coordinate — it supplies the surface over them.

Nor can it settle a case where the two effects genuinely conflict, because their relative scale is set by the weight, and the weight is fitted. What it can do is say that a conflict is possible, where the boundary between the two regimes lies at any given weight, and that an argument naming only bulk is incomplete rather than merely imprecise.

The five- and six-site arrangements minimised under three different repulsion exponents barely move, and the six-site one does not move at all — which is the distinction between what symmetry fixes and what the model chooses, and the reason the six-site case is immune to everything in this essay.

The reason nobody names, computed from the count of neighbours #

The last paragraph above says the received rule is right for a reason this model does not contain, and leaves the reason unstated. It is worth stating, because the leading part of it needs no force field and no contact radius — only a count.

Put five substituent atoms at the ends of five bonds of length $L$ in a trigonal bipyramid and ask what each one’s neighbours are. The distances follow from the angles alone: two sites at 90° are $L\sqrt{2}$ apart, two at 120° are $L\sqrt{3}$, and the two axial sites at 180° are $2L$.

An axial substituent has three neighbours at $1.414L$ — all three equatorial ones — and one at $2L$.

An equatorial substituent has two at $1.414L$ — the two axial ones — and two at $1.732L$.

Every site has four neighbours; what differs is how many of them are at the closest distance, and the axial site has three against the equatorial site’s two.

Summing any decreasing function of the distance therefore favours the equatorial site, and by an amount that depends entirely on how sharply the function falls:

At a Coulomb law the preference is two per cent, which is nothing. At the short-range repulsion that actually decides contacts it is 39%, and it is in the direction the received rule claims.

So the two halves of the situation separate cleanly. The domains at the centre put the long bond axial, which is what the sections above compute and which is the opposite of what is taught. The atoms at the ends put the bulky substituent equatorial, because the axial position has one more close neighbour than the equatorial one does. The received rule is about the second and is invariably justified by the first.

And the second half is not a fitted claim. It needs no radius, because the count of neighbours at each distance is fixed by the geometry, and no potential, because every decreasing function gives the same sign. What a contact radius would supply is the size — whether the effect is worth two per cent or forty — and that is exactly the quantity a force field is for and this model is not.

What five-coordination does next #

Two directions, and both are already visible in the numbers.

The first is that the two placements are close in energy over the whole range — a few thousandths, against a total of about six and a half — which is why five-coordinate molecules exchange their axial and equatorial substituents so freely. That exchange has a name and a mechanism, and it is the one case where a molecule’s ground-state structure is the wrong thing to describe it with: what a spectrum sees is an average over a rearrangement fast enough that a rigid point group is not the right group at all.

The second is that the length knob applies wherever the sites are inequivalent, which is not only five. Seven-coordinate arrangements have three kinds of site and much flatter minima; six-coordination has one kind of site and is therefore immune to the whole argument, which is a good part of why octahedral geometry is so reliably octahedral.

There is a third direction, and the section above takes its leading term. A substituent’s actual bulk is a property of the atoms hanging off it, and the count of close neighbours at each site settles the sign with no radius needed. What is still missing is the size, which needs a contact radius for every substituent — a force field rather than a symmetry argument, and building one here would replace a model whose limits are stated with one whose parameters are not.

The claim to carry away is small and awkward. A repulsion model with one radius is a model of angles, and everything it says about which substituent goes where has been said with a knob it did not have. Given the knob, it says the opposite of the received rule — and the received rule is nevertheless right, for the other reason, which the same model gets correct and which nobody names.