The distortion that opens the gap

Two distortions of a square-planar complex have been followed for what they do to the gap above its eighth d electron. Bending all four ligands towards a tetrahedron closes it, slowly at first and to zero at the end, and costs its exactness at the first degree. Bringing in an axial donor closes it exactly linearly and costs the exactness nothing at all.

Both close it. That is the whole of the evidence, and it makes closing look like what distortion does.

It gets bigger first #

Fold two of the four ligands to the same side — the distortion towards a see-saw, which is what a real square-planar complex does when it distorts without adding anything — and the gap rises: 2.000013 at two degrees, 2.000511 at five, 2.007682 at ten, 2.034853 at fifteen, and 2.093826 at twenty.

It falls after that, through 1.772 at twenty-five and 1.460 at thirty, reaching 1.000 at forty-five. But there is a whole range of ordinary distortion — up to about twenty-two degrees — in which bending the complex makes its sixteen-electron gap larger than the plane’s.

The levels say where it comes from. Two stay at the foot; the third leaves it. The top two — which start as d(z²) at 1 and d(x²−y²) at 3 — do not simply drift: they repel. In C₂ᵥ both belong to the a₁ representation, so they may mix, and two levels that may mix push apart when anything couples them. That repulsion is worth more than the σ redistribution is worth in the other direction, and the net effect is a larger gap.

So the direction the gap moves is not a property of “bending”. It is a property of which ligands bend, and specifically of what the bend does to the symmetry relating the two frontier orbitals. The four-ligand path keeps them in different representations (D₂d has d(z²) in a₁ and d(x²−y²) in b₁) and they cannot repel; the two-ligand path puts them in the same one and they must.

There is a way to see that the repulsion is the whole of it, and it is arithmetic rather than a picture. The sum of the five levels is fixed at $4e_\sigma$ by the model’s own trace rule, whatever the geometry — a distortion redistributes a total it cannot change. So a gap that grows must be paid for somewhere below, and it is: the third level leaves the foot and climbs, from 0 at the plane to 0.045 at five degrees and 0.375 at fifteen. The gap at the top opens because the levels beneath it are compressing, and both are the same redistribution seen at two ends.

That also explains why the growth stops. Past about twenty degrees the third level has climbed far enough to start pushing on the fourth from below, and the fourth is one of the pair whose separation is being measured. The turnover at twenty-two degrees is where the two effects balance, and it is not a feature of either one alone.

The exactness goes while the gap is growing #

At two degrees the derivative with respect to $e_\pi$ is −0.00490 per unit. At ten it is −0.137. At twenty — where the gap is at its largest, five per cent above the plane’s — it is −0.639.

So every point at which this distortion has made the gap bigger is a point at which the gap has stopped being exact. The two properties identified in the plane — that the gap is 2eσ, and that it is 2eσ whatever the π strength is — come apart here in the most awkward possible way: the first improves while the second is destroyed.

That is the reason to keep them apart as separate claims rather than folding them into “the plane has a good gap”. A gap of 2.09eσ that depends on a parameter nobody has measured is worth less than a gap of 2.00eσ that does not, and no measurement of the gap alone can tell them apart. That is the same trade found in a level’s position, where what replaced an exactness was a formula rather than nothing; here what replaces it is a number with an unmeasured parameter in it.

It stops being a gap between two orbitals #

The mixing is not a small admixture. The purity of the less pure frontier orbital — the fraction of it that is a single d function — falls to 0.9974 at ten degrees, 0.9678 at twenty, 0.8949 at thirty and 0.7500 at forty-five.

At forty-five degrees a quarter of the upper frontier orbital is the other one. Calling the gap “d(z²) to d(x²−y²)” there is not a rough description; it is a description of two objects that are not present. The states are a₁ combinations of both, and which of them is “the” d(z²) is a question with no answer.

On the four-ligand path this never happens. The purities stay at 1.000 to the last figure all the way to the tetrahedron, because D₂d forbids the mixing — so that path can close the gap to zero and still have two named orbitals at every point. The orbital a ligand cannot reach is where that naming was first shown to be doing real work rather than being a convenience. The two paths differ in what they do to the identity of the levels, not only in what they do to the numbers.

Four columns, and they do not agree about whether the distortion is good for the complex. The gap improves to twenty degrees; the stabilisation of the sixteen electrons improves the whole way, from 2.000 to 3.000 eσ; the exactness is gone from two degrees; and the purity falls throughout. A reader watching any one of them would form a different view. The stabilisation column is the one a total-energy calculation would report, and it is the only one of the four that is monotone — which is why the effect above is not in the literature: the quantity everybody computes does not show it.

What three paths make of one plane #

That makes three distortions of one complex, measured the same way, and it is worth putting them beside each other because no two agree about anything.

Three columns and three different answers in each. The only generalisation that survives all three is the one the axial path made available: the exactness depends on whether the fourfold axis lives, and nothing else — the gap’s size is not a proxy for it, its direction is not a proxy for it, and the purity of the orbitals is a fourth thing again.

That is a modest conclusion for three calculations, and it is the honest one. A reader wanting to know what a distortion will do to a sixteen-electron complex has to be told which distortion, and the answer changes sign between two of them.

The one that is not a distortion #

There is a second way to move two ligands out of the plane and it is worth measuring precisely because it looks the same and is not.

Tilt the pair rigidly, so that the two stay opposite each other, and every level is unchanged at every angle — 0, 0, 0, 1, 3 at zero degrees and at ninety and everywhere between, to nine figures.

That is not a discovery about d orbitals; it is a fact about geometry, and stating it is what makes the fold’s result mean anything. Two perpendicular linear pairs always span a plane, so tilting one pair rigidly while the other stays gives four ligands in a different plane — the square plane in a new orientation, not a distorted square plane. A calculation that reported a change there would be reporting its own choice of axes.

The distinction is sharp and easy to lose: both moves take two ligands out of the xy plane by the same angle, and only one of them changes the molecule. The fold breaks the trans relationship and is a distortion; the rock preserves it and is a rotation.

What a real complex would show #

The measurement is idealised, and the question of whether any of it is visible has a clean answer at each of the three columns above.

The gap is what a d–d spectrum reports, and a five per cent change is inside what a band maximum resolves — so a series of complexes distorted by increasing amounts should show the transition energy rising and then falling, with a maximum near twenty degrees. That is a prediction with a shape rather than a value, and a shape is what a series can test even when every complex in it has a different $e_\sigma$.

The exactness is not directly visible at all, and that is the point of quoting it. Nothing measures $\partial(\text{gap})/\partial e_\pi$; what it changes is whether the gap can be predicted from the σ framework alone. In the plane it can, so a π-donor and a π-acceptor with the same σ strength give the same gap. Two degrees off the plane they do not, and the difference between two such complexes is then a measurement of the fold rather than of the ligands — which is a confound in every attempt to order ligands by their π character from four-coordinate data. Where a d–d band falls is assembled from exactly that kind of comparison.

The mixing is visible in intensities rather than in energies. A transition between two pure d orbitals is Laporte-forbidden in a centrosymmetric complex and weakly allowed otherwise; between two mixed a₁ states the selection rules are different again, and the fold removes the centre of symmetry the plane has. So the same distortion that opens the gap also brightens the band — two effects with one cause, which is the kind of coincidence that makes a series hard to read.

What was computed, and how #

The angular overlap model throughout: each ligand contributes $e_\sigma$ along its own direction and $e_\pi$ perpendicular to it, the contributions are rotated into the d frame and summed, and the five-by-five matrix is diagonalised exactly. The fold takes the two ligands on the x axis to $(\cos\beta, 0, \sin\beta)$ and $(-\cos\beta, 0, \sin\beta)$ — the same side — and leaves the y pair alone. The rock takes them to $(\cos\beta, 0, \sin\beta)$ and $(-\cos\beta, 0, -\sin\beta)$, which keeps them trans.

The purity of a level is the largest squared coefficient of its eigenvector in the d basis, so a level that is a single d orbital has purity 1 and an even mixture of two has 0.5. It is reported rather than assumed, and on the four-ligand path it is what says the eigenvectors really are the orbitals the argument names them by.

The π derivative is a symmetric finite difference at $\pm 0.05e_\sigma$, the same one used on the axial path, so the −0.639 here and the zero there are the same measurement on the same calculation.

Three checks carry the reading. The gap at a small fold must exceed the plane’s, which is the finding and would fail if the fold were entered as a rock. The derivative must be non-zero at the first few degrees. And the frontier purity must fall below one, since a version of this in which the two orbitals stayed pure would mean the geometry had not broken the symmetry it was supposed to.

The refusal is the rock, and it is checked level by level rather than in summary: every one of the five must be where it was, at every angle, to $10^{-9}$.

Where the model stops #

There is no electron–electron repulsion here, so “the gap above eight electrons” is a one-electron gap. Where the two frontier orbitals mix strongly the many-electron problem is not a filled-shell problem at all — a small gap between two mixed a₁ states is exactly the situation in which a triplet may lie lowest — and nothing in this model can see that.

The fold is a bare angular change with the ligand strengths held fixed. A real fold moves the two ligands relative to the metal as well as relative to each other, so $e_\sigma$ would change too, and the direction it changes is not obvious: the folded ligands are closer to each other and no further from the metal.

Forty-five degrees is where this family is taken to, and past it the two folded ligands approach each other closely enough that the model’s assumption of independent contributions is doing more work than it should. At ninety they coincide. Nothing above uses the region past forty-five.

And the whole comparison is at $e_\pi = 0$ except where the derivative is taken. A complex with a genuinely large π interaction is not near this family’s zero, and the derivative says only how fast it leaves.

The generalisation #

Two quantities can both be called “the gap” and behave oppositely, and the distinguishing thing is a symmetry rather than a size. The four-ligand path and the two-ligand fold are both bends of the same complex through comparable angles, and one closes the gap while the other opens it. What separates them is whether the frontier pair stays in different representations. Anyone predicting the effect of a distortion from its magnitude would get the sign wrong here, and the magnitude is the only thing a structure reports.

And a level’s name is a claim that can fail. Calling a state d(z²) is a claim that it is one, and the claim has a number — its purity — that is almost never quoted. On this path it falls to three quarters, and every sentence of the form “d(z²) rises” becomes a sentence about something that is no longer d(z²). The same warning appears in orbitals that are a basis rather than a thing, and the purity is the cheapest possible instrument for it: one number per level, free from the diagonalisation that was going to happen anyway.

Who found it, and when #

That square-planar d⁸ complexes distort towards a tetrahedron, and that the barrier to doing so is what makes them stereochemically rigid, is standard; so is the observation that palladium and platinum resist it while nickel does not. The two-ligand see-saw distortion is the one associated with an incipient fifth coordination rather than with a flattening, and the angular overlap model for all of it is Schäffer and Jørgensen’s.

What is added here is the sign. A gap that grows under distortion is not what any of the standard accounts would lead a reader to expect, and it is not a subtle effect — five per cent at twenty degrees, which is well inside the range real complexes sample. Why a d–d band is weak is the account of the intensities the same distortion changes. The reason it is not remarked on is probably that the quantity usually looked at is the total energy, which behaves monotonically, rather than the frontier gap, which does not.

Still open: the triplet, and where the two paths cross #

The obvious open question is the triplet. Where the two frontier orbitals mix and the gap between them is small, a one-electron gap is the wrong instrument entirely: the question is whether the two remaining electrons pair in the lower state or occupy both, and that is a competition between the gap and the pairing energy. Pairing energies are computable, so the two can be put on one axis, in the way the pairing energy decides the moment already does for an octahedron — and the interesting output is whether the fold’s larger gap at twenty degrees is enough to keep a singlet where the four-ligand path’s smaller one is not.

The nearer question is where the two paths cross. The fold opens the gap and the four-ligand bend closes it, so a distortion that mixes the two — three ligands moving, or two moving by different amounts — must pass through a direction in which the gap does not move at all to first order. Finding that direction is a two-parameter scan of a calculation already set up, and a distortion the gap is blind to is the one along which a complex is softest without paying for it.