The two coordinates a square plane was missing

Every internal coordinate, redundancy and totally symmetric count in this collection is built on a bond list, and the bond list comes from a length cutoff. That essay rebuilt the cutoff on measured radii and reported what changed. Its closing paragraph named the gap that remained: a square-planar centre’s coordinate set spans seven of its nine vibrations, and the two it misses are the out-of-plane ones.

The candidate it named is the pair of pyramidalisation coordinates built from opposite ligand pairs — the distance of the centre from the line through each pair, signed. This builds them.

What is missing, and why an angle cannot see it #

A square-planar XY₄ has nine vibrations. Four of them are in the plane and involve the bond lengths; three more are in the plane and involve the angles. Two are not in the plane at all.

In the first the centre leaves the plane of its four ligands. Every Y–X–Y angle in the plane is a function of where the atoms are, and to first order in the displacement not one of them changes — a small perpendicular displacement of the apex of a shallow pyramid changes its base angles at second order. In the second the ligands move alternately up and down with the centre fixed; again every in-plane angle is unchanged to first order, because the displacement is perpendicular to every bond.

So a coordinate set made of stretches and angles is blind to both, and the rank of its B matrix is seven. That is not a subtlety of the construction — it is the same fact that makes a planar three-coordinate centre need an out-of-plane coordinate, which these essays have had since the beginning, applied to a case with two such motions instead of one.

The consequence of leaving it out is not a small error. A force field written in an incomplete set has directions it cannot resist, so the molecule has vibrations at zero frequency and the count of genuine zero modes — six for a non-linear molecule, and the check the whole force-field apparatus rests on — comes out too high by the number of missing species. For a square plane it would come out at eight rather than six, which is the kind of failure that would be noticed at once by anyone who fitted a field in that set. Nobody had, because nobody could: the molecules are not in the table.

One coordinate cannot fix it. A single coordinate spans at most one species, and two are missing.

Which two they are is fixed by the group and can be read off without any geometry. In D4h the nine vibrations are a1g + b1g + b2g + a2u + b2u + two eu pairs, and the two with a u subscript and no partner — a2u and b2u — are the out-of-plane ones. A set of stretches and in-plane bends spans the g species and the eu pairs and nothing else, which is seven. So the shortfall is two, it is the same two for every square-planar molecule, and a coordinate set that spans them is a coordinate set that spans everything.

The pair, and why it is a pair #

The four ligands form two pairs of trans partners, and each pair defines an axis through the centre. The coordinate is the signed distance of the centre from that axis, measured along the normal the two axes build between them, and there are two of them because there are two pairs.

Both are zero when the five atoms are coplanar, which is what makes them coordinates of the reference geometry rather than constants.

And they span exactly the two missing species, which is the whole reason for choosing them:

• the centre moving out of the plane takes it away from both axes by the same amount, so both coordinates rise together;
• the ligands puckering — one trans pair up, the other down — moves one axis up and the other down, so one coordinate rises and the other falls.

The symmetric combination is the first motion and the antisymmetric combination is the second. A pair of coordinates that both measured the same thing would have spanned one species twice and left the rank at eight.

The measurement says so. Xenon tetrafluoride goes from ten coordinates at rank seven to twelve at rank nine, and tetrachloridoplatinate the same. The rank rises by exactly the number of coordinates added, so neither of the two is redundant against the existing set or against the other.

Nineteen molecules are untouched #

The census’s other nineteen molecules get no new coordinate at all, and that is worth checking rather than assuming, because a construction keyed on “a four-coordinate centre” could reach further than intended.

The condition it is keyed on is coplanarity: four ligands whose two trans axes define a normal, with the centre on the plane those axes span. Methane’s four hydrogens are four ligands and their centre is nowhere near the plane of any pair of axes, so it raises nothing. Tetrachloridonickelate is the same tetrahedral case with heavier atoms, and it was complete already at rank nine of nine.

That leaves exactly two molecules, both square-planar, and both are completed.

The two are worth naming as chemistry rather than as rows. Xenon tetrafluoride is square-planar because xenon has two lone pairs above and below, which this collection derives rather than looks up; tetrachloridoplatinate is square-planar for the electronic reason the sixteen-electron count exists for. They are the two kinds of square plane there are, and both were absent.

The three molecules that remain short in this figure — ethene, benzene and hydrogen peroxide — are short for the reason an earlier essay established: they need a torsion, which is a different construction and is off in this comparison so that one thing changes at a time.

The accounting survives #

The identity an earlier essay repaired is that a molecule’s count of totally symmetric vibrations equals its number of symmetric orbits of internal coordinates, less its number of symmetric redundancies. Adding coordinates is exactly the operation that could break it.

It does not. The two new coordinates form one new orbit — they are exchanged by the quarter-turn about the axis perpendicular to the plane — and they add no redundancy, so the orbit count goes from three to four and the redundancy count stays at three. The count of totally symmetric vibrations stays at one, which is the molecule’s own.

The new orbit is therefore not a symmetric one, and the reason is the mechanism the identity is built on: an orbit contributes a totally symmetric combination only when its coordinate survives its own stabiliser, and the operations that fix one of these coordinates include reflections that reverse its sign. A quantity defined along a normal is reversed by anything that reverses the normal.

That is the satisfying part of the check. The identity does not merely survive; it survives because the new orbit falls on the side of it that the identity was repaired to account for.

Why nobody had measured it #

The more useful finding in this essay is not about coordinates.

A xenon–fluorine bond is 1.94 ångström and a platinum–chlorine bond is 2.32. The census’s default cutoff is 1.85. So at the default neither molecule has a single bond, therefore no stretches, no bends, no coordinates at all — and the census refuses, the refusal is caught, and the molecule is dropped from the table.

Sixteen molecules were reported and two were absent. The absence is silent by construction: a loop that tries each molecule and keeps the ones that work produces a table whose rows are all correct and whose missing rows are invisible.

That is the same shape of defect as the one the whole essay below was about, arriving one level up. That essay found five molecules given no bonds by a length cutoff and rebuilt the rule; it did not ask what the census had done with the molecules the old rule refused, and the answer is that it had quietly shortened itself.

So the incompleteness of a square-planar coordinate set was not a subtlety anybody missed. It was a row that was not printed.

What the construction assumes #

Coplanarity to within a twentieth of an ångström. The condition for adding the coordinates is that the centre lies on the plane the two trans axes span, tested with a tolerance. A genuinely pyramidal five-atom centre — a square pyramid — does not get them, and should not: its out-of-plane motions are already visible to its angles.

Trans pairs found by angle. The pairing is by largest angle at the centre, which for a square plane is a hundred and eighty degrees and is unambiguous. For a distorted four-coordinate centre the pairing could in principle be ambiguous, and nothing here handles that: the construction is for a square plane and is applied only where the coplanarity test passes. The same ambiguity was met before in deciding which atoms are bonded at all, and the resolution was the same: a rule with a stated tolerance, and a refusal rather than a guess when the tolerance is not met.

The normal is built from the geometry, not fixed in advance. Both coordinates are measured along the cross product of the two pair axes, which is a function of the current positions. That is what makes the coordinate a smooth function of the geometry rather than a projection onto a frame, and it is why a reflection that reverses the normal reverses the coordinate — which is the sign behaviour the symmetry analysis needs.

And the cutoff is now 2.5 ångström throughout this essay, so that the comparison is between coordinate conventions and not between bond lists. That is a larger cutoff than is used anywhere else here, and it is chosen to admit the two molecules rather than because 2.5 is right.

That is an uncomfortable choice and it is worth being explicit about the alternative. The essay below replaced the cutoff with a rule based on measured radii, which is the right repair and which this essay does not use — because a comparison between two coordinate conventions has to hold the bond list fixed, and the radius rule and the length cutoff give different bond lists on some of the nineteen untouched molecules. Using the radius rule here would have mixed two changes. The honest order is: fix the bond rule, then compare the conventions, and this essay is the second of those with the first one’s cutoff frozen at a value that admits everybody.

What the two coordinates are worth to a force field #

The point of a complete coordinate set is that a force field can be written in it, and it is worth saying what these two coordinates would carry.

Each is a length, like a stretch and like the out-of-plane coordinate of a three-coordinate centre, so its diagonal force constant has the same unit as a stretching constant and needs no conversion factor. That is deliberate in the construction: the convention here is that a bend is multiplied by the geometric mean of its two bond lengths so that every entry of a force-constant matrix is comparable, and a coordinate that is already a length sidesteps it.

The two would also have a coupling between them, and the coupling is the interesting entry. Their symmetric and antisymmetric combinations are the two out-of-plane species, which have different frequencies, so the two diagonal constants and the one coupling constant are three numbers determined by two observed frequencies plus one relation — the symmetry requires the two diagonals to be equal, since the quarter-turn exchanges the coordinates. So two frequencies fix two constants exactly, with nothing left over.

That is the tightest determination anywhere in the force fields. Most blocks have more constants than frequencies and an earlier essay measured the flat directions that leaves; this block has exactly as many of each. Whether it is worth fitting is a different question, since neither molecule’s out-of-plane frequencies are quoted here, and that is the shortfall this essay records.

A loop that skips is a table that lies #

The habit: a census built by trying each case and keeping the ones that work reports a complete table of an incomplete set.

Every number in the rows that were printed was correct. What was wrong was the count of rows, and nothing in the output said so — the census had no line reading two molecules raised no coordinates and were dropped, because a catch that continues has nowhere to put that. The repair is not a better cutoff; it is that a skip has to be reported as loudly as a failure.

The corollary is about where to look for this. It is an omission rather than an error, and every check in the apparatus reads what exists and asks whether it is right. A census that drops a molecule has nothing wrong in it to find, which is why three essays of increasingly careful work on coordinate sets went past it, and why the thing that finally exposed it was going looking for a molecule that ought to have been there.

There is a cheap repair and it is the one to take from this: make the count of cases attempted part of the output. A census that printed twenty-one molecules attempted, nineteen reported, two refused for want of bonds would have carried its own defect in its first line, and the cost of that is one number.

Who counted what, and when #

Internal coordinates, the B matrix and its rank are Wilson’s, from the 1940s, and the requirement that a coordinate set span every vibration is the condition for a force field written in it to be complete. The out-of-plane coordinate of a planar three-coordinate centre is standard; the pair used here for a four-coordinate one is the natural extension and is not, as far as the sources go, a named construction. The species of a square-planar XY₄’s vibrations are textbook.

What is computed here is the two coordinates, the rank they add, the census over every molecule with and without them, the identity checked under both, and the cutoff that had removed the two relevant molecules from the table.

The numbers worth carrying are seven and nine — a coordinate set’s rank before and after two coordinates, on the two molecules that were not in the census.

Still open: the projection, and a square pyramid #

The obvious open question is the one every coarse species test has deferred: where a symmetry species appears more than once, only a projection onto the normal coordinates says which vibration a distortion goes into, and a projection needs a force field. With every coordinate set now either complete or incomplete for a stated reason, the coarse test and the projection can be compared on the molecules that have force fields, and nothing in the counting underneath either of them is still in doubt. That comparison is the natural next step and it needs no new construction — the projections exist and the coarse test exists, and they have never been run against each other.

The nearer question is the square pyramid. A five-coordinate centre with four ligands in a plane and one above has both of this essay’s motions available and its angles see them, because the axial ligand breaks the symmetry that made the angles blind. Whether its coordinate set is complete without any pyramidalisation is a rank calculation on a molecule the census already holds — and if it is, that is the sharpest statement of what the two new coordinates are for: they are needed exactly when a symmetry makes the angles blind, and not merely when a centre is four-coordinate.