The ligand the rule was waiting for

The gap that only a tetrahedron closes found that a square-planar complex’s gap above eight d electrons is 2eσ exactly, independent of the π strength — because d(z²) points along the axis perpendicular to the plane and a square-planar ligand π set contains nothing of that symmetry. Then it bent all four ligands towards a tetrahedron and found the exactness gone at the first degree of bend, while the gap itself fell only slowly.

It closed by naming the distortion a sixteen-electron complex actually undergoes. It is not a bend. A sixteen-electron complex is called reactive because it can add a ligand, and the ligand arrives along the axis d(z²) points down.

The gap closes, and it closes exactly #

Bring an axial donor in at a fraction $f$ of the equatorial ligands’ σ strength and the gap is $(2-f)e_\sigma$. Bring two in and it is $(2-2f)e_\sigma$. Not approximately: the measured gap and that expression agree to twelve figures at every point tested.

So a full square pyramid — five equal ligands — has a gap of 1eσ, half the plane’s. And an octahedron has exactly zero.

The mechanism is visible in the levels and is entirely one-sided. Three levels sit at the foot and stay there. d(x²−y²) does not move at all, because an axial ligand lies in its nodal cone and has no σ overlap with it whatever. All the motion is d(z²), which the incoming donor points straight at, rising by $e_\sigma$ per unit of axial strength until it catches d(x²−y²) and the two become the eg pair of an octahedron.

That is why the closure is linear rather than merely monotone: it is one level moving at a fixed rate towards another that is standing still. Both halves of that are exact statements about the model rather than approximations in it — the angular overlap contribution of a ligand to a d orbital is the square of an angular factor, and for d(x²−y²) against a ligand on the z axis that factor is identically zero.

The endpoint is worth stating plainly because it settles something about the rule. An octahedron has no sixteen-electron gap at all — d(z²) and d(x²−y²) are degenerate there by symmetry, and no filling can find a gap between two levels at the same energy. Whatever a count of eighteen electrons is about in a six-coordinate complex, it is not this gap, because this gap does not exist there. The two rules are not the same rule at different coordinations.

What the approach does not do #

The finding about exactness was the sharper half of it, and the natural expectation is that adding a ligand destroys it faster than bending does — a new ligand is a bigger change to the molecule than a few degrees of bend.

It does not destroy it at all. The derivative of the gap with respect to $e_\pi$ is zero at every point of both axial approaches, to $10^{-9}$ — including at the octahedron, where the gap itself is zero.

The reason is the same reason the plane had it, and the approach does not touch it. The fourfold axis survives: at every $f$ the complex is D₄ₕ, d(z²) belongs to a₁g and d(x²−y²) to b₁g, and no π combination the ligands can form belongs to both. A quantity that is a difference between two levels in different representations cannot be moved by a perturbation that respects the symmetry relating them.

So the exactness is not a property of the plane. It is a property of the axis, and an approach along the axis is the one distortion that keeps it. That is a sharper statement than the bending path could support, because there the only path available broke the axis and the two facts — plane, and exactness — could not be separated.

The same distinction appears for a level placed exactly by a symmetry, where breaking the symmetry moved it at once and at half the perturbation. The difference here is that the perturbation is large and the symmetry is untouched, so nothing moves.

There is a second reading of that zero which is worth separating from the first, because it says what the gap is rather than only what it survives.

A quantity protected by a symmetry is not protected because the perturbation is weak. It is protected because the perturbation has no matrix element to work through: d(z²) is a₁g and d(x²−y²) is b₁g, so a π combination belonging to eg or b₂g cannot connect them at any strength whatever. The gap is therefore a difference between two labels as much as between two numbers, and it stays exact while the labels stay meaningful.

The moment they stop being meaningful — the moment the fourfold axis goes — the gap becomes a difference between two states with no names, and the exactness has nothing to be a statement about. That is what the fold does, and it is why a two-ligand fold finds the two frontier orbitals mixing rather than merely moving.

The gap and the binding go opposite ways #

There is a reading of the sixteen-electron rule in which the gap is the reason the count stops: a large gap above the eighth electron is what makes sixteen a stable number, so closing the gap should destabilise the complex.

The stabilisation rises the whole way. With two axial donors it runs 2.0, 2.4, 2.8, 3.4, 4.0, 5.0, 6.0 eσ as $f$ goes from nothing to full — trebling — while the gap falls from 2 to 0.

That is not a paradox and it is not subtle. An approaching σ donor lowers nothing; it raises the levels it overlaps, and the filled levels here are the three at the foot plus d(z²). Adding a ligand adds σ interaction, and the sixteen electrons are more stabilised relative to the free ion the more ligands there are. Meanwhile the gap above them closes because the level being raised is the highest filled one.

So a reader using the gap as a proxy for stability along this path gets the sign wrong at every step. The gap is a statement about what the next two electrons would cost, and the stabilisation is a statement about the sixteen that are there. An integer nobody measured is the standing warning about reading a count as a measured quantity, and this is the same warning applied to the gap beside it. Nothing requires them to move together, and along the one path a sixteen-electron complex actually takes, they do not.

What the linearity is worth #

A closed form is worth more than a curve when it can be inverted, and this one can.

The gap is $(2 - kf)e_\sigma$, so a measured gap and a known equatorial $e_\sigma$ give $kf$ — the total axial σ donation — without any assumption about how far away the axial ligand is or what it is. That is a quantity a d–d spectrum reports and a structure does not: two complexes with the same axial bond length and different axial ligands have different $f$, and the gap distinguishes them where a bond length cannot.

It also says what a weakly interacting axial contact does. A solvent molecule loosely associated at $f = 0.1$ takes the gap from 2.00 to 1.80 eσ — a tenth of it — which is a shift a spectrum can see and which nothing about the four-coordinate description would predict. The bare plane is the $f = 0$ limit of a family rather than a case on its own, and most real “square-planar” complexes in solution are somewhere in that family.

The two-axial case is the one where the arithmetic is sharpest, because the endpoint is fixed by symmetry rather than by the model. At $f = 1$ the gap must be exactly zero — d(z²) and d(x²−y²) are the eg pair of an octahedron — so the line has both its intercept and its zero pinned, and the only content of the model is that it is straight in between. It is.

What was computed, and how #

Everything is the angular overlap model: each ligand contributes $e_\sigma$ along its own direction and $e_\pi$ perpendicular to it, the contributions are rotated into the d-orbital frame and summed, and the five-by-five matrix is diagonalised exactly. The axial ligands carry their own strengths, so the approach is a scaling of one ligand’s parameters rather than a change of geometry — which is the right way round for what is being asked, since a donor at a longer distance is a weaker donor and not a differently placed one.

The gap is the difference between the fifth and fourth levels, which is what a sixteen-electron count leaves open. The stabilisation is twice the sum of the four lowest levels, taken against the barycentre the model’s own sum rule fixes.

The π derivative is a symmetric finite difference at $\pm 0.05e_\sigma$, and it is checked to be zero rather than small: $10^{-9}$ on a quantity of order one. That is the check the whole reading rests on, and it is the one that would fail first if the geometry were entered wrongly — a mistyped direction would break the fourfold axis and the derivative would come back non-zero immediately.

Three things are checked and all three could have failed. The gap must equal $(2-kf)e_\sigma$ to twelve figures, at both axial counts and at every fraction. The octahedron must give exactly zero rather than something small. And the derivative must vanish at every point of both paths.

The comparison path is the two-ligand fold, computed the same way, and it is what says the zero derivative is a result rather than a property of the method: the same finite difference on the same calculation returns 0.74 there.

The two rules are not one rule #

The octahedral endpoint settles something electron counting has circled for a long time, and it settles it by arithmetic rather than by argument.

A count of sixteen and a count of eighteen are usually presented as the same idea at two coordinations — a closed shell of metal and ligand electrons, with sixteen the four-coordinate version because one orbital is left out of the bonding. On that reading the gap above sixteen in a square plane and the gap above eighteen in an octahedron are the same object seen twice.

They are not, and this path shows why: the gap above the eighth d electron falls continuously from 2eσ to exactly zero as the coordination goes from four to six. There is no gap there at all to be the octahedral version of it. What an octahedron has instead is a gap above the tenth d electron — the eg set to the σ-antibonding levels above — which is a different pair of levels, involves a different set of orbitals, and closes for entirely different reasons.

So the continuity the two-rule picture suggests does not exist. The sixteen-electron gap is a four-coordinate phenomenon that vanishes on the way to six, and the eighteen-electron gap is not its continuation. Eighteen is a count made the case that the eighteen-electron rule is bookkeeping rather than a gap; this adds that even where a gap is available at sixteen, it is not the same gap.

Where the model stops #

The angular overlap model has no electron–electron repulsion in it, so “the gap above eight electrons” is a one-electron gap and not an excitation energy. A real sixteen-electron complex’s reluctance to add a ligand involves the pairing energy and the reorganisation of the whole d shell, neither of which is here.

More specifically, the approach is modelled by turning a strength up rather than by moving a ligand in. Those are the same thing only if $e_\sigma$ falls smoothly with distance and nothing else changes — and the equatorial ligands do move when an axial one arrives, since a real square-planar complex becomes a square pyramid with the equatorial set bent slightly away. That bend is exactly the distortion shown here to cost the exactness, so a real approach loses the protection to a second-order effect the model has left out.

The strengths are also independent of each other here. In a real complex an axial donor competes with the equatorial ones for the same metal orbitals, so $e_\sigma$ for the plane falls as the axial ligand arrives — the trans influence, which is an overlap argument of its own and which is not in this sweep. Nor is the change in the equatorial count that an added ligand brings, which the count that cannot be broken by strength is about.

And $f$ is not a distance. It runs from 0 to 1 and the physical scale is set by how $e_\sigma$ falls with distance, which the model does not contain.

The generalisation #

An exactness protected by a symmetry survives any perturbation that respects it, however large. The axial approach is not a small change — it takes a four-coordinate complex to a six-coordinate one and closes the gap from 2eσ to nothing — and the derivative stays at zero throughout. The size of a perturbation and whether it breaks a symmetry are independent, and only the second decides whether an exact statement survives.

That is the complement of what a symmetry holds or it does not finds about a level placed exactly: there an arbitrarily small symmetry-breaking moved a level at once. Together the two make the rule: ask which symmetry produces an exactness, then ask only whether a perturbation breaks it. Never ask how big the perturbation is.

And two quantities computed from the same levels need not move together. The gap and the stabilisation are both differences of the same five numbers, and along this path one falls by everything it has while the other trebles. A rule stated in terms of one of them is not a rule about the other, and the sixteen-electron rule is stated in terms of a gap while the intuition behind it is about stability.

Who found it, and when #

That a square-planar d⁸ complex has a large gap above its filled levels, and that adding an axial ligand raises d(z²) into it, is the standard picture in every account of why sixteen-electron complexes are the ones that undergo oxidative addition and associative substitution. The angular overlap model is Schäffer and Jørgensen’s, and the linearity of a level in an axial ligand’s σ strength is immediate in it.

What is done here is to put a number on the two halves separately and find them going opposite ways: the gap linear and exactly closing, its exactness untouched, and the stabilisation rising through all of it. The zero derivative is the part that could have come out otherwise, and it is the part the bending result made worth checking.

Still open: the pyramidal bend, and a two-ligand distortion #

The obvious open question is the distortion that accompanies a real approach. A square-planar complex becomes square pyramidal with the equatorial ligands bent away from the incoming donor by a few degrees, and that bend breaks the fourfold axis — so the protection found intact here would be lost to a second-order effect while the gap’s size is barely touched. Combining the two moves in one path is one more set of directions, and it would say whether a real approach keeps the exactness in practice or only in the idealisation. Sixteen is also a count argues the number is a count rather than a gap; this is the first measurement for which the two readings would give different answers.

The nearer question is the other four-coordinate distortion, which moves two ligands rather than four and which the comparison above has already used without exploring. In C₂ᵥ d(z²) and d(x²−y²) share a representation and may mix, which neither the plane nor the tetrahedral path allows — so the gap there is not a gap between two named orbitals, and what it does is not predictable from either path measured so far.