Five coordinates for six vibrations

A formula that predicts minus eleven vibrations took an old explanation — one totally symmetric vibration per orbit of internal coordinates, less one per redundancy — and turned it into an identity with two corrections. The first was that an orbit contributes nothing when an operation that fixes its coordinate also reverses it, and it was demonstrated on a single kind of signed coordinate, the out-of-plane displacement of a planar centre. It named the sharper test, a torsion, and proposed a ring with a twofold axis through two of its bonds to supply one.

Adding torsions runs that test, and the rule holds on every molecule. The proposed ring could not have run it, for a reason one sentence settles. And the calculation found something the test was not looking for: when hydrogen peroxide’s torsion was added, its count of totally symmetric vibrations went from three to four — which a property of a molecule is not allowed to do.

A torsion’s sign is a handedness #

A torsion about the bond b–c is the angle between the plane of the atoms a, b, c and the plane of b, c, d, taken with a sign. The sign says which way the four atoms twist: looking down the bond from b, whether d is turned clockwise or anticlockwise from a. It is a handedness, the same property chirality is a statement about, carried by four atoms rather than by a whole molecule.

That settles how every operation acts on a torsion before anything is computed. A proper operation — a rotation, or leaving everything where it is — moves a twisted arrangement without changing which way it twists, so the dihedral keeps its value wherever the operation puts it. An improper one — a mirror, an inversion, a rotation followed by a reflection — turns every handedness into its opposite and reverses every dihedral. Naming the four atoms backwards gives the same angle, so a torsion carried onto itself end for end has still been carried onto itself.

The orbit rule’s condition therefore becomes exact and short for torsions. An orbit of torsions contributes one totally symmetric combination unless some improper operation carries its torsion onto itself; if one does, it contributes nothing.

Where that happens #

Four of the molecules here have torsions — a bond with further neighbours at both ends: ethene, benzene, both ferrocenes and hydrogen peroxide.

In a planar molecule every torsion lies in the plane of the molecule, which is a mirror. It fixes every atom and reverses every dihedral, so every torsion orbit of ethene and of benzene is reversed. Most are reversed by a second operation as well, and which one depends on where the torsion’s bond sits. A torsion at 0° is also reversed by the mirror perpendicular to its bond through the midpoint, which swaps the two ends. Ethene’s trans torsion at 180° is also reversed by the centre of inversion, which sits at the middle of its C=C bond, swaps the two carbons and sends each hydrogen to the one opposite — the anti arrangement reversed by an inversion, found inside a molecule that did not need adding to show it. Benzene’s mixed torsions are at 180° too, but about a ring bond whose midpoint is not an inversion centre, and the plane alone reverses them.

Ferrocene is not planar and still has reversed torsions. Each of its rings has five vertical mirrors, and each passes through one carbon and bisects the C–C bond opposite. The ring torsion and the H–C–C–H torsion about that bond are carried onto themselves backwards by that mirror and reversed. The mixed H–C–C–C torsions are not: the same mirror carries each of them to the mixed torsion at the other end of the bond, which is a different coordinate, and nothing but the identity fixes one. So twenty mixed torsions form one symmetric orbit while ten of each other kind are reversed. The eclipsed and staggered ferrocenes give identical rows.

Hydrogen peroxide’s one torsion is fixed by its twofold axis, turned end for end, and left with its sign.

Across all of them no proper operation reverses a torsion it fixes, which is the sign rule stated as what it forbids, and the identity holds with torsions on every one of the fifteen molecules. That is a sharper check than it sounds. The torsions’ action enters the coordinates’ representation and their redundancies, and a wrong sign for a torsion would have broken the identity’s agreement with a count computed without them — which is what the rest of this essay is about.

The twofold axis cannot do it #

Benzene’s ring torsion is fixed by four of the twenty-four operations of its group. Two keep its sign — the identity, and a twofold axis through the midpoint of the bond, which turns the torsion end for end. Two reverse it — the mirror bisecting the bond and the plane of the molecule. The twofold axis is doing exactly what the proposed test wanted it to do, carrying the torsion onto itself, and because it is a rotation it cannot touch the sign.

That is why a ring with a twofold axis through two bonds would never have served. A cyclohexane chair is the natural example: its twofold axes pass through the midpoints of opposite C–C bonds and fix the ring torsions about them, and they are proper. Its improper operations — three vertical mirrors through opposite carbons, a centre of inversion, two rotation-reflections — each carry a C–C bond to a different bond, so none fixes a ring torsion. Every orbit of torsions in a chair would come out symmetric and the test would have passed without asking anything. The chair is not among the molecules here, so that is argued from its group rather than computed; benzene’s cards show the part that carries the argument. It is the same reason hydrogen peroxide’s twofold axis did not serve, and the sentence that ruled out peroxide should have ruled out the chair with it.

What does serve is any improper operation that fixes a bond: a molecular plane, a mirror bisecting the bond, or an inversion centre at its midpoint. Those are found in planar molecules and in molecules like ferrocene whose mirrors cut across a ring, and the four molecules here supply all three.

A count that moved #

Adding a torsion to hydrogen peroxide changed its count of totally symmetric vibrations from three to four.

A count of vibrations is a property of the molecule — how many times the totally symmetric species appears in the representation its nuclear displacements carry — and a choice of coordinates can change how that number is computed and never what it is. So one of the two numbers was not a count of the molecule’s vibrations, and there is a way to find out which that shares nothing with internal coordinates at all.

The Cartesian displacements of hydrogen peroxide’s four atoms carry a representation of their own; take away the three translations and three rotations, as every vibrational count has to, and what is left is 4a ⊕ 2b — four totally symmetric vibrations. The coordinates as built are three stretches and two bends. Their rank is five against six vibrations, so they are not a basis for the molecule’s motion: turning one O–H bond about the O–O bond while the other stays put changes no bond length and no angle, and no combination of these five coordinates can see it. The representation they carry is 3a ⊕ 2b. What had been reported as the vibrational representation was the coordinates’ representation less their redundancies, and that is the vibrational representation only when the coordinates span every vibration. These span five of six, and the one they miss is totally symmetric.

With the torsion the rank is six of six, the coordinates carry 4a ⊕ 2b, and the count is four — the Cartesian answer, from a route with nothing in common with the first.

Three of fifteen #

Once the question is asked of one molecule it has to be asked of all of them, and it is a comparison of two integers: the rank of a coordinate set against the number of vibrations.

Ten of the fifteen coordinate sets span every vibration as built. Five do not: ethene’s thirteen coordinates have rank eleven against twelve vibrations, benzene’s thirty-six rank twenty-seven against thirty, hydrogen peroxide’s five rank five against six, and both ferrocenes’ sixty rank forty-four against fifty-seven.

Two of the five were right about the count anyway, by luck of symmetry. Ethene’s missing vibration is the twist about its double bond and benzene’s three are out-of-plane ring motions, and none of them is totally symmetric, so the sets that miss them still count the totally symmetric ones correctly. The other three were wrong by one: hydrogen peroxide and both ferrocenes, each counted at three where the molecule has four.

Those three counts had been reported as the molecules’ own in the essay that found the identity, and the comparison with the Cartesian displacements that would have caught them was described there and not made. Made, it corrects that essay’s tally — the old formula is right for seven molecules rather than eight, because hydrogen peroxide’s three was matched against a count that was also three and also wrong — and it leaves the identity itself untouched, which is true of any coordinate set and a count of vibrations only for one that spans them.

An iron with no bonds #

Torsions complete ethene, benzene and hydrogen peroxide. They take ferrocene’s rank from forty-four to forty-eight, and leave it nine vibrations short.

The missing nine belong to the iron. Bonds are listed out to 1.85 ångström, which is a length for bonds between light atoms, and ferrocene’s iron is 2.064 ångström from each of its ten carbons. So no coordinate involves the iron at all: the rings are two separate pieces as far as the coordinates know, and every motion of the rings against the metal — including the totally symmetric stretch of the metal–ring distance — is invisible to them.

Counting bonds out to 2.1 ångström takes in exactly the ten iron–carbon distances and nothing else, since the next atoms are the hydrogens at 2.838. That gives 135 coordinates and 78 redundancies, rank fifty-seven of fifty-seven, and four totally symmetric vibrations, which is the molecule’s count.

It is the second time a fixed bond length has decided what a molecule’s coordinates are. The earlier essay found hydrogen peroxide carrying two extra bonds because two O···H distances of 1.82 ångström fell inside the cutoff; here ten real bonds fall outside it. A single length is a rule for one kind of bond, and it is wrong in both directions as soon as a molecule contains another kind.

The formula, completed #

Completing the five short sets gives the old formula one more chance, and it does worse.

Benzene’s formula goes from −4 to −22 and each ferrocene’s from −11 to −66, against true counts of two and four. Ethene, which the formula had right, goes from three to two and is wrong. Hydrogen peroxide goes from three to four and is right, now for the right reason. The mechanism is the one the earlier essay identified, pushed further: completing a set adds orbits slowly and redundancies quickly. Benzene’s twenty-four torsions add three orbits and twenty-one redundancies; ferrocene’s seventy-five extra coordinates add seven orbits and sixty-two redundancies, most of them among the angles at the iron.

The corrected count — symmetric orbits less symmetric redundancies — taken on the completed sets is the molecule’s own in every row. A formula that improves as a coordinate set is completed and one that degrades are distinguishable by exactly this test, and it is not a test either can make of itself.

What was computed #

Torsions are generated from the bond list: one for each bond and each choice of a further neighbour at both ends, skipping any trio of atoms in a straight line, where a dihedral has no value. Each is a signed angle, and its derivatives are taken on the circle, because anti and planar dihedrals sit at ±180° where the angle wraps and a straight-line difference there would be off by a full turn. An operation carries a torsion to another with a sign equal to the determinant of the operation, which is the handedness argument above; the operations themselves are the ones closing each molecule’s candidate symmetries, and the kind of each is read from its matrix.

The molecule’s own count comes from the displacements of its atoms reduced in its group — the route every vibrational representation here is built on — and has nothing to do with internal coordinates.

The checks are these. Adding torsions drops no molecule and the identity holds for all fifteen. No proper operation reverses a torsion it fixes. Every torsion orbit of ethene and benzene is reversed by the plane of the molecule; ferrocene’s ring and H–C–C–H torsions are reversed by the mirror bisecting their bond and its mixed torsions are fixed by the identity alone; hydrogen peroxide’s is fixed by its twofold axis and not reversed. Ten sets are complete as built, torsions complete three more, and only bonds to the iron complete the ferrocenes, torsions alone leaving them at forty-eight of fifty-seven. And the refusal: for every molecule, every convention that spans the vibrations gives the same count, and it is the molecule’s.

What four molecules with torsions cannot settle #

Four molecules is enough to find all three ways a torsion can be reversed and not enough to say how often each happens. They are small, rigid and symmetric, and each torsion here is a small displacement from a fixed structure. A molecule whose torsion is a large motion over a low barrier — ethane is the standard case — is not described by a point group at all, which is the subject of the group of a molecule that will not hold still.

The 2.1 ångström that completes ferrocene is a second fixed length, and it works here only because nothing lies between 2.064 and 2.838. A bond list that knows about metals would need a length for each pair of elements; it is not attempted, and the claim here is only that one cutoff miscounted three molecules.

And the chair is argued rather than computed, because it is not among the structures here.

A count that depends on a convention #

The transferable point is about when a number computed in a basis is a number about the thing the basis describes.

The identity was never wrong: symmetric orbits less symmetric redundancies is, for any set of coordinates, the count of totally symmetric vibrations those coordinates describe. It was read as the molecule’s count, and for three molecules the coordinates described too little. Nothing inside the identity could say so, because an identity is true of whatever it is given — which is the same observation the earlier essay made when a wrong bond list left its arithmetic intact.

What exposed it was a property that moved when a convention changed. That is the warning worth keeping: a quantity that is supposed to belong to an object and shifts when only the description shifts is evidence that the description was incomplete. The check it points to — rank against dimension — is the cheapest in the subject, and it has to come before any count read from a basis is quoted as a count of anything.

Where the pieces come from #

Internal coordinates, torsions among them, are the valence coordinates of Wilson, Decius and Cross’s treatment of molecular vibrations, and counting a representation’s totally symmetric part by orbits is Frobenius reciprocity. The sign rule for torsions, the completeness census and the three corrected counts are computed here.

The earlier essay deserves the credit for proposing torsions as the next test and for saying, of hydrogen peroxide’s extra bonds, that the identity holds for a wrong coordinate set exactly as for a right one. That sentence contained this finding; it needed a count to move before it could be seen.

Still open: the projection, and a bond rule that knows about metals #

The obvious open question has stood since the coarse species test was first applied: where a species appears more than once, only a projection onto normal coordinates says which vibration a distortion goes into, and that needs a force field, which exists here for six molecules. With complete coordinate sets now checked, the coarse test and the projection can be run side by side on those six with every count they rest on verified.

The nearer question is the bond list. One length miscounted three molecules, in both directions, and a rule built from per-element bonding radii would decide ferrocene’s iron and peroxide’s hydrogens without a special case. Which of the fifteen sets would change under such a rule — and whether any complete set would stop being complete — is a census over bond lists rather than over coordinates, and nothing here has run it.