The direction the gap cannot see

A square-planar sixteen-electron complex has a gap between its fourth and fifth d levels, and the whole of the rule of sixteen rests on that gap being large enough to keep the fifth empty. The distortion that opens the gap asked what distortions do to it and found two that do opposite things: folding two trans ligands out of the plane opens the gap, and bending four of them closes it.

It then made an observation rather than a measurement. If one distortion opens the gap and another closes it, then a family of distortions mixing the two must contain a direction along which the gap does not move at all to first order — and a distortion the gap is blind to is the one along which a complex is softest without paying for it. Finding it is a two-parameter scan of a calculation already set up.

Two folds rather than a fold and a bend #

The natural two-parameter family is not the fold and the bend, which move different numbers of ligands, but two folds: one pair of trans ligands lifted out of the plane by an angle p and the other pair by q. At q = 0 it is the two-ligand fold. Along p = q it is the symmetric fold that carries a square plane towards a tetrahedron.

And the two known effects are the two axes of that plane. Folding one pair by ten degrees raises the gap from 2.000000 to 2.007682. Folding both by ten degrees lowers it to 1.994545. Opposite signs, on the same coordinate space, from the same starting point.

So the blind direction has to exist and the question is only where it points.

It is the antisymmetric fold, and it is exact #

On the symmetric line the two derivatives are identical — at ten degrees and ten degrees both are −1.078 × 10⁻³ — so the direction perpendicular to the gradient is at forty-five degrees to both axes. That is the antisymmetric fold: one pair folded further and the other unfolded by the same amount.

Which could have been a numerical coincidence and is not.

Start from both pairs at ten degrees and fold one to thirteen while unfolding the other to seven. Now do the opposite: seven and thirteen. Those are the same arrangement with the two pairs exchanged, and exchanging them is a symmetry of the whole family — so the two gaps must be identical, and the arithmetic returns them identical to every bit it carries.

That makes the gap an even function of the antisymmetric coordinate about every point of the symmetric line, and an even function has a vanishing first derivative at its centre. The blindness is a theorem about the coordinate, not a root somebody found by searching, and it holds at every amplitude along the symmetric line rather than at one point of it.

It is worth pausing on why the two axes pull opposite ways, because it is not obvious and it is what makes the whole question well posed.

Folding one pair out of the plane removes the fourfold axis and lowers d(x²−y²) — the orbital that was pointing straight at four ligands now points at two — while d(z²) is pushed up as the folded pair approaches the axis. Those two moves separate the fourth and fifth levels and the gap opens.

Folding both pairs equally keeps a fourfold axis, so nothing separates on symmetry grounds; what happens instead is that every ligand moves away from the xy plane and towards the z axis, which is the path to a tetrahedron — and a tetrahedron is where the gap closes entirely. The gap falls because the arrangement is on its way to the one that has none.

So the two axes are not two versions of the same distortion with different amplitudes. They are a symmetry-lowering and a symmetry-preserving move, and the sign of the gap’s response is set by which of those it is.

Not at the square plane, and not fixed #

Two things spoil the tidiest version of the statement, and both are worth having.

The first is the square plane itself, where the gradient is not merely small but zero in both directions: 4.9 × 10⁻⁸ against 2.9 × 10⁻³ ten degrees out. Every direction is flat there to first order, so no direction is distinguished and a null direction computed there is undefined. The calculation reports that rather than returning an angle, which it would otherwise do — a perpendicular to a vanishing vector is whatever the rounding says.

The second is that off the symmetric line the blind direction moves. At ten degrees and zero it is at ninety degrees — the gap is blind to folding the unfolded pair, which makes sense because that pair is still in the plane and moving it out is second order. At fifteen and five it is at 59.52°, and at twenty and ten at −72.76°.

So the blind direction is a direction and not a subspace, and it is a property of where the molecule sits rather than a coordinate of the problem. Only on the symmetric line is it fixed, and there it is fixed by a symmetry.

That second point is the more useful one for a chemist and it is worth stating without the geometry. Fold one pair and the gap opens; the arrangement is then no longer symmetric between the pairs, and what the gap has become insensitive to is the pair that has not moved. Moving it out of the plane changes the gap only in second order, because the first-order effect of lifting a ligand out of a plane depends on the plane still being there — and it is not, for the pair already lifted.

So the blind direction near a partly folded arrangement points at the ligands that have not yet moved, and near a symmetrically folded one it points at the difference between the pairs. Those are two different physical statements and the plane holds both.

Blind to first order only #

A vanishing first derivative is not a flat function. Three degrees along the blind direction from a five-degree symmetric fold moves the gap by 0.15 per cent; from ten degrees, 0.55; from fifteen, 1.04. And the movement is upward every time, so the symmetric line is a valley floor in the gap and any departure from it costs.

That is the honest form of “a distortion the gap cannot see”: the gap does not respond linearly, it responds quadratically, and the quadratic response is small but not zero. A complex vibrating along this coordinate modulates its own gap at twice the vibrational frequency and by a per cent, which is a real if minor statement about how a sixteen-electron complex’s frontier gap couples to its own motion.

What this adds to a rule of sixteen #

The subject here is a counting rule, so it is worth saying what a blind direction has to do with counting.

Sixteen is a count only because one d orbital is pushed far enough above the other four that filling it is not worth the energy. The count is therefore a statement about a gap, and a gap is a statement about a geometry — so a rule quoted as a property of an electron configuration is really a property of a configuration and a shape together.

What the two-fold plane adds is a map of how much shape the rule can tolerate. Along the symmetric fold the gap falls: 2.000, 1.9997, 1.9945, 1.9731 at zero, five, ten and fifteen degrees — slow, and the rule survives. Along one fold it rises, which strengthens the rule. Along the blind direction it does neither to first order.

So of the three directions in this plane, one weakens the rule, one strengthens it, and one leaves it alone — and a real complex vibrating in this space samples all three. The count is not fragile here; what is established here is the geometry of how it is not fragile, which is a thing to know before quoting a gap from an idealised square plane.

What was computed, and how #

Four ligands at unit σ strength and no π interaction, at directions parameterised by two fold angles, with the d levels from the angular overlap model and the gap taken between the fourth and fifth. The gradient is a central difference in each angle at a quarter of a degree; the blind direction is the perpendicular to it.

The check requires seven things: that the square plane gives the model’s own sixteen-electron gap of two; that its gradient vanishes there, so no direction is defined; that one fold opens the gap and two close it, so something between them must be flat; that on the symmetric line the blind direction is the antisymmetric fold at every amplitude; that the two arrangements either side agree to the last bit, which is the exactness; that away from that line the direction is something else, so it is not a fixed coordinate; and that the gap does move quadratically along it, so the blindness is first order only.

Why the square plane has no gradient #

The stationary point at the square plane deserves its own explanation, because it is the reason the question had to be asked at a distorted geometry rather than at the undistorted one.

Both fold coordinates change sign under a reflection through the square plane, and that reflection is a symmetry of the undistorted arrangement. Folding a pair up and folding it down give the same arrangement, so the gap is an even function of each coordinate separately about the origin — and both first derivatives vanish for the same parity reason the antisymmetric direction’s does further out.

That is why the gradient there is 4.9 × 10⁻⁸ rather than merely small: it is zero, and the number is the central difference’s own arithmetic. It also means the square plane is not a useful place to look for a blind direction, because every direction is blind there and the distinction the question is about does not exist.

The general shape is worth naming: a parity argument that makes a derivative vanish is powerful and it applies wherever the symmetry does, which at a high-symmetry point can be everywhere. A blind direction is only informative where some directions are not blind.

Where the model stops #

The angular overlap model has no π interaction here and no repulsion between ligands at all, so nothing in this calculation says which of these arrangements a complex would adopt — only what its gap would be if it were there. The fold that opens the gap makes the same separation and it matters more here, because a valley floor in the gap is not a valley floor in the energy and only the first is computed here.

The distortions are also all rigid: bond lengths are fixed and only directions change. A real fold shortens or lengthens bonds as it goes, which the model has no term for, and the gap is sensitive to length through eσ — the same limitation the ligand the rule was waiting for has to work around.

And the family is two-dimensional out of the many more a four-coordinate complex has. Two pairs folding independently is the family the two known effects live in, so it is the right place to look for a direction joining them; a fold that moved three ligands rather than four, which is also worth naming, is a different family and might have a different blind direction or none.

One more limit, and it is about the word blind rather than about the model. A direction along which the gap does not change to first order is not a direction along which it does not change. The finding here is that the gradient has a null direction, which is a statement about the linear term; the gap still moves along it quadratically, and how far one must go before that movement is measurable is a question not asked here. A direction that is flat to first order and steep to second is blind only in a neighbourhood, and the size of that neighbourhood is what would decide whether the blindness is a curiosity of the derivative or something a spectrum would actually fail to see. That is one more sweep along the null direction at increasing amplitude, and it is the obvious thing to run before quoting the result as a property of real complexes rather than of the model’s gradient.

The generalisation #

The transferable point is that a blind direction can be exact or found, and the difference is worth establishing before the direction is used.

A direction located by searching for where a gradient vanishes is blind to the precision of the search, and it moves when anything about the problem moves. A direction fixed by a symmetry — here, that the two pairs are exchangeable — is blind identically, at every amplitude, and stays blind under any change that preserves the symmetry. The first is a fact about a calculation and the second is a fact about the system.

The test is the one used above and it is cheap: find the operation that would map the distortion to its negative, check that it is a symmetry of the arrangement, and then check that the two arrangements agree bitwise rather than nearly. If they do, the vanishing derivative is a parity argument and needs no numerics at all.

The corollary is that the interesting blind directions are the ones a symmetry does not fix, because those carry information about the particular system. The 59.52° at fifteen and five degrees is such a direction, and it is the one there is least to say about here. That asymmetry — the symmetric cases being the ones an argument settles and the general ones being the ones a computation must — recurs throughout symmetry arguments, and it is worth expecting rather than discovering each time. The count that cannot be broken by strength is the same shape in electron counting: what a symmetry fixes is exact, and what it does not fix is where the work is.

Who found it, and when #

The angular overlap model is Schäffer and Jørgensen’s, and the square-planar sixteen-electron rule is standard organometallic chemistry — the gap only a tetrahedron closes is where the shape of that gap is established. The two-fold family, the gradient, the blind direction and the parity argument are new here, and they answer a question the fold calculation raised.

Still open: the energy along the blind direction, and the triplet #

The obvious open question is the energy. This essay finds a direction along which the gap is stationary and says nothing about whether it costs anything to move along — and the two questions have different answers, since the ligand–ligand repulsion the model omits is precisely what would resist an antisymmetric fold. Computing the repulsion along the same family, on the unit-charge repulsion model, would say whether the direction the gap cannot see is also cheap, which is what “softest without paying for it” would actually require.

The nearer question is the triplet, which neither calculation has run. Along the symmetric fold the gap falls — 2.000, 1.9997, 1.9945, 1.9731 at zero, five, ten and fifteen degrees — and somewhere it becomes small enough that the two frontier electrons would rather occupy both orbitals than pair in the lower one. Pairing energies can be computed. Putting the two on one axis along this coordinate would say at what fold angle a sixteen-electron complex stops being a singlet, which is the point at which the rule of sixteen stops applying at all.