Five ligands, and no ideal to aim at

A σ-only bond energy sees the directions of its ligands through two numbers and nothing else: their summed direction and their second-moment tensor. At fourth order in the coupling, where geometry first enters, it charges for each. A net direction costs $\kappa_\text{dip}|\sum_i\mathbf u_i|^2$; a frame excess — a departure from treating the three directions of space equally — costs $\kappa_\text{frame}\bigl(\sum_{ij}(\mathbf u_i\cdot\mathbf u_j)^2 - n^2/3\bigr)$. Both terms are non-negative, and an arrangement that makes both zero pays neither. That arrangement is a spherical 2-design, a set of points that averages every polynomial of degree two exactly as the sphere does. The tetrahedron is one and the octahedron is one, and each is the most stable arrangement of its count at every coupling.

That essay closed on a count it had not examined: five. The trigonal bipyramid, it noted, is not a 2-design, but comes closest of the five-ligand arrangements in its census. Both halves of that sentence were true for the model’s own heights. Neither says what happens to a count with no ideal to come close to.

Five points on a sphere cannot form a 2-design. Four can, and every count from six upward can; five is the gap. So at five the σ energy cannot have both of the things it prefers, and it has to trade.

A count with no ideal #

The gap can be seen without any theorem. With the two terms weighted equally, a search over arrangements from twelve seeded starts finds a cost of zero for four, six, seven and eight ligands — to fourteen decimal places, which is rounding — and for five a least cost of exactly one sixth, reached by the trigonal bipyramid. That is the geometric fact Mimura proved in 1990 after the definition of spherical designs by Delsarte, Goethals and Seidel: 2-designs on the ordinary sphere exist for four points and for every count from six, and for no other.

The reason the bipyramid falls short is visible in its second moment. Its axial pair contributes two to the axial direction and nothing across it; its equatorial triangle contributes one and a half across and nothing along. The moment tensor has eigenvalues 1.5, 1.5 and 2 where a tight frame of five would have five thirds in every direction. The bipyramid is perfectly balanced — its net direction is zero — and it pays for its balance with anisotropy. Every arrangement of five pays for one virtue with the other; the only question is at what rate the energy converts one into the other.

One number sets the exchange rate #

Divide the fourth-order cost by $\kappa_\text{frame}$ and one parameter is left:

$$q = \frac{\kappa_\text{dip}}{\kappa_\text{frame}} = \left(\frac{S_s}{S_p}\right)^2 r(1 + r),\qquad r = \frac{\Delta_p}{\Delta_s}.$$

Here $S_s$ and $S_p$ are the overlaps of a ligand function with the central s and p orbitals, and $\Delta_s$ and $\Delta_p$ are how far those orbitals sit above the ligand level. A large q means a net direction is expensive and anisotropy cheap; a small q the reverse. The model used throughout the σ calculations, with p one and a half times as high as s and equal overlaps, has $q = 3.75$.

The bipyramid’s cost does not depend on q at all: its net direction is zero, so it pays one sixth of $\kappa_\text{frame}$ whatever the rate. A square pyramid’s does. With its apex on an axis and its four basal ligands at an angle θ from it, the net direction is $(1 + 4\cos\theta)^2$ and the frame excess $24\cos^4\theta - 8\cos^2\theta + \tfrac23$, so for each q the best pyramid is the minimum of one quartic in $\cos\theta$. As q falls the best pyramid leans further — gives up more balance for more isotropy — and its cost falls with it.

The two branches cross at q = 0.862, where the best pyramid has its base 108.7° from the apex, a net direction of 0.081 and a frame excess of 0.097. Below that q the square pyramid is the fourth-order floor for five ligands; above it the bipyramid is. At q approaching zero the pyramid tends to the one square pyramid that is a tight frame, with its base at 114.1° and a frame excess of zero, and costs nothing but its lean.

A search over arbitrary arrangements finds nothing else. Well above the crossing, sixteen seeded starts all end at the bipyramid; below it they end at the square pyramid of the closed-form branch, with the same angle to the third decimal. The floor at five is made of these two shapes and no third one.

The two shapes the floor is made of #

Drawn side by side, the trade is easy to see. The bipyramid points nowhere on average, and its excess of one sixth is the whole of its cost. The pyramid at q = 0.3 leans — its net direction squared is 0.219 — and in exchange comes within 0.024 of a tight frame. It is a shape that averages the second moment almost as a sphere does while pointing somewhere, which the bipyramid can never do without ceasing to be a bipyramid.

The lead the earlier essay left was that the bipyramid, not being a 2-design, “has no reason to be the only shape with its moments”, and that five ligands should have a continuum of shapes the σ model cannot tell apart. The count is right for a generic shape. Up to rotation an arrangement of five directions has seven degrees of freedom, and the energy depends on five rotational invariants of the two moments, so an irregular arrangement has a two-parameter family of twins — the rank of the five invariants’ derivatives at a seeded irregular shape is five, leaving exactly two. The bipyramid has none. Wherever it is the floor it is the only minimum: its curvature matrix has exactly three zero eigenvalues, which are the three rotations, and its softest genuine bend has a curvature of 0.58 at q = 1. A twin of the bipyramid would have its balance and its excess, so it would be a second minimum of the same cost, and none of the searches that land on the bipyramid ever finds one. The σ model can tell the bipyramid apart from every other shape; what it cannot tell apart is most of the shapes nobody has ever proposed.

Berry’s path, both ways #

The bipyramid and the square pyramid are not unrelated shapes. Berry proposed in 1960 that a trigonal bipyramid exchanges its axial and equatorial ligands through a square pyramid: one equatorial ligand stays put as a pivot, the axial pair bends towards the opposite side, the other two equatorial ligands open to meet them, and at the halfway point the four form the base of a square pyramid with the pivot as its apex. Continuing takes the molecule to a second bipyramid with its axes exchanged. It is the standard explanation of why the five fluorines of phosphorus pentafluoride look equivalent on the timescale of an NMR measurement, although the bipyramid’s axial and equatorial sites are not alike in any single structure. Like the planar geometry through which ammonia turns inside out, the pyramid is visited rather than occupied — and, as with ammonia, what a spectrum measures is not the barrier itself but its consequences.

In that account the square pyramid is the transition state — the top of the path — and the bipyramid its minimum. The fourth-order cost along the same path says the account is a statement about q.

At the model’s own $q = 3.75$ the cost rises all the way from the bipyramid to the pyramid, by 0.068 of $\kappa_\text{frame}$, and the pyramid is a first-order saddle: exactly one of its curvatures is negative, and the direction it points along is Berry’s. That is pseudorotation as it is taught, recovered from nothing but a σ bond energy. At q = 0.9 both ends are minima and a barrier of 0.008 separates them, with the bipyramid 0.003 lower. At q = 0.8 the barrier is 0.004 and the pyramid is now 0.005 lower. At q = 0.5 the path runs downhill the whole way to the pyramid, which lies 0.040 below.

Below q = 2/3 the roles are exchanged outright. The bipyramid is then the saddle, and Berry’s path read backwards is a pseudorotation of square pyramids through a bipyramidal transition state.

Where each shape stops being a minimum #

The two changes of character happen at different values of q, and neither is where the floor changes hands. The bipyramid’s softest bend — a doubly degenerate one, as it has to be in a threefold-symmetric shape — has positive curvature above q = 2/3 and negative below it, and two thirds is where it crosses to the seventh decimal the second derivatives can carry. The square pyramid’s softest bend changes sign at q = 0.955. Between them both are minima, and inside that interval, at 0.862, the lower of the two changes ends.

That ordering is the familiar shape of a first-order transition, and it has the familiar consequence: in the window from two thirds to 0.955 which shape a molecule sits in depends on where it started. The σ model’s five-coordinate chemistry, for a centre whose heights put q in that window, would have two shapes and a barrier between them, and nothing in a single energy minimisation would reveal the other.

The exact energy keeps the pyramid in a corner #

Every number so far is the fourth-order cost, which is the energy’s leading geometric term at weak coupling. The exact σ energy can be minimised over the same two shapes at any coupling.

At weak coupling it agrees with the series: with equal overlaps the pyramid wins when p sits below 0.554 of s’s height, the ratio that gives q = 0.862. As the coupling grows the window closes. The pyramid’s advantage never survives a coupling of 0.3 in units of s’s height — at best 0.290, near a ratio of one sixth — and near the critical ratio it is gone by a coupling of a few hundredths. At the model’s own heights the bipyramid wins at every coupling, and the square pyramid sits above it on Berry’s path by 0.0010 at a coupling of a half, against a bond energy of 2.88, and by 0.015 at a coupling of 2, against 19.6 — parts in a thousand of the bond, which is how a pseudorotation barrier ought to look beside one.

The window’s location says what the square pyramid needs in this model: p orbitals lower than s — nearer the ligands by nearly half — and weak coupling. No main-group centre has its p orbitals below its s. For every one of them r is above one, q is above two, and the σ-only model places the bipyramid as the only minimum and the square pyramid as the top of the pseudorotation path.

Averages, again #

This is the third time averaging has decided an argument in this collection. The icosahedron’s indifference to the orientation of its half-filled shell came from its twelve vertices forming a spherical 5-design. The octahedron’s victory at six ligands came from its forming a 2-design, and so did the tilted prism that shares its energy. Here the same object appears by its absence. Five is the one count at which the ideal the energy wants does not exist, and the consequence is not a small correction but a change in kind: a trade, a parameter that sets it, and two shapes that exchange the roles of minimum and transition state across it.

It is also a small instance of a principle that runs through the count of orbitals the ligands can reach. A symmetry label says which arrangements are special. What decides which of them a model prefers is a quantity the label does not mention — here, a ratio of orbital heights and overlaps — and that quantity can move a shape from minimum to saddle without any symmetry changing.

What the σ model cannot say about five #

Real five-coordinate shapes are not set by σ bonding alone. Pentaphenylantimony is a square pyramid in the crystal and its tolyl analogue a bipyramid; the difference is packing and ligand–ligand contact, not the central atom’s orbital heights, and the σ model has no term for either. Every molecule with a lone pair on the centre — BrF₅, XeF₅⁻ — is outside the model’s premise that every ligand level is filled and no central level is.

Ligand repulsion is absent. The repulsion minimum and the bonding minimum are different questions, and the two models do not always agree about a shape. For five ligands the repulsion minimum VSEPR computes is the bipyramid on its own. A model with both would add a term that pushes q’s effective value upward.

There are no d orbitals and no π. A $d^0$ transition-metal centre with five ligands bonds through d orbitals as well as s and p, and pentamethyltantalum, for instance, is a square pyramid; the domain picture does not reach such a centre and nothing here speaks to it except by analogy, since a d orbital’s second moment is a different object. π donors add rows perpendicular to each bond, with a net direction and a frame excess of their own.

The window is a statement about weak coupling. The four regimes are exact for the fourth-order cost and the exact energy confirms the window’s edges, but at couplings where the series fails the saddle and minimum structure has to be recomputed rather than read off q.

How it was computed #

The fourth-order cost is the series term of the σ-only model with every ligand level filled, in units of the frame coefficient. The least cost for each ligand count is found by pattern search over the ligands’ polar angles from twelve seeded starts; the five-ligand floor at each q from sixteen. The square-pyramid branch is minimised in closed form over the cosine of its basal angle, and the crossing q by bisection on the difference with one sixth. Curvatures are second differences of the cost in each ligand’s tangent plane, with a step of 10⁻⁴, and the q at which the softest one changes sign is bisected for each shape. The exact energy is twice the sum of the five lowest eigenvalues of the nine-by-nine model matrix.

The checks, made wherever these figures are drawn: zero least cost for four, six, seven and eight ligands and one sixth for five; the bipyramid as the floor from every start well above the crossing and the closed-form pyramid below it; the crossing between 0.860 and 0.863; the bipyramid’s change of character at two thirds to within 10⁻⁴ and the pyramid’s between 0.95 and 0.96; the bipyramid rigid with exactly three zero curvatures where it is the floor; the pyramid above the bipyramid and a saddle on Berry’s path in the exact energy at the model’s heights, at four couplings; a two-parameter twin family at an irregular shape; and a window edge below 0.3 at every ratio under the critical one. The refusal is the octahedron, which as a 2-design must cost exactly nothing at every q — so the one sixth at five is the count’s and not the search’s.

Who found what #

Berry described pseudorotation in 1960. Spherical designs were defined by Delsarte, Goethals and Seidel in 1977; that 2-designs on the sphere exist for four points and every count from six, and not for five, is Mimura’s. The electron-rich three-centre σ model is Rundle’s and Pimentel’s, the account in which six bonds share four orbitals, and the fourth-order series with its two geometric terms is the one derived in the previous essay of this argument.

What is computed here is the five-ligand floor of that series at every exchange rate, its two branches and the rate at which they cross, the curvature of each branch along Berry’s path and the two rates at which each changes character, the window in the exact energy, and the count of twins at the bipyramid and at an irregular shape.

Still open: π rows, and a repulsion term #

The obvious open question is still π. A π-donor ligand adds two functions perpendicular to its bond, and with them a second net direction and a second frame excess, weighted by π overlaps and heights. Whether π moves a main-group centre’s effective q down towards the window — and so whether a strongly π-donating set of five ligands could make the square pyramid a minimum in a model that still has no repulsion — is a computation of exactly the shape of this one, with more terms.

The nearer question is the repulsion. The window here is for bonding alone, and a Coulomb or contact repulsion between ligands has its own preference at five, which is the bipyramid. Adding it with one weight turns the four regimes into a two-parameter map, and it would say whether any combination of the two makes the pyramid the floor for a centre with p above s — or whether in a model with both terms the square pyramid is always the top of Berry’s path for such a centre.