One coordinate, three point groups

A structure determination fixes some coordinates tightly and others hardly at all. Hydrogen peroxide is the smallest molecule where the difference matters: its O–O bond is $1.475$ Å, its O–H bonds $0.950$, and the O–O–H angle $94.8°$ — all three stiff, all three known to three or four figures. The fourth coordinate, the dihedral angle between the two O–H bonds seen along the O–O axis, is soft, and the molecule spends its time swinging through a wide range of it.

That one coordinate decides the molecule’s point group, and the group decides two properties outright.

The search, not a lookup #

The structures are built rather than tabulated. Two oxygens on the y axis at $\pm 0.7375$ Å, and each hydrogen placed at $94.8°$ to the O–O direction at an azimuth of $\pm\phi/2$ about that axis, so that a twofold rotation perpendicular to the O–O bond carries one onto the other for every $\phi$.

The group is then found from the coordinates alone: candidate axes are taken from the atoms themselves, every candidate operation is applied, and one is kept when it permutes the atoms among themselves to within a tolerance. Nothing is looked up, and the search has no idea what answer it is expected to produce.

What it produces is three groups and no others.

The eclipsed structure is planar, and the plane it lies in is a mirror; a second mirror bisects the H–O–O–H unit perpendicular to the first. The anti structure is planar too, and has an inversion centre at the midpoint of the O–O bond, which the eclipsed one does not. Everywhere between, the twofold rotation survives and nothing else does.

What the two ends decide differently #

The two achiral structures are achiral for different reasons, and only one of them is forbidden a dipole.

Chirality is forbidden by any improper operation whatever — a mirror, an inversion centre, or a rotoreflection. Both ends have one, so both ends are achiral. That is chirality is a symmetry statement’s rule applied to a family instead of to a list of molecules, and it is why the chirality rail in the figure is a single unbroken band with two holes punched in it.

A dipole is forbidden by any operation that moves every direction — which for these groups means an inversion centre or a rotation axis with something perpendicular to it. C₂ᵥ has neither: its twofold axis lies in both mirrors, so a vector along that axis is left alone by every operation, and a dipole may exist. C₂ₕ has an inversion centre, and an inversion reverses every vector, so a dipole must vanish exactly.

So the polarity rail has exactly one hole in it, at $180°$, and the chirality rail has two, at $0°$ and $180°$. One coordinate, two different patterns of vanishing, both exact.

The bond-vector argument, and what it misses #

The familiar way to reach the same conclusion adds up bond dipoles. Two O–H bond moments, each at a fixed angle to the O–O axis, projected onto the axis and onto the perpendicular plane; at $180°$ their perpendicular components are opposite and cancel, and their axial components are opposite too, so the total vanishes.

That argument gets the right answer here, and it is worth being precise about why it is nevertheless the weaker one — the case the dipole is not a sum of bonds makes.

It needs an input the symmetry argument does not. A bond moment is a number somebody measured or estimated, and the O–O bond’s moment — between two identical atoms in an unsymmetrical environment — is not zero and is not known.

It gives an approximate zero. The cancellation is exact only if the two bond moments are exactly equal and exactly placed. Symmetry says the total is zero because an operation of the molecule reverses it, which is a statement about the exact Hamiltonian rather than about a model of bond polarity, and it holds at whatever level of theory anybody cares to use.

It cannot see the other property at all. Nothing in a sum of bond vectors says whether the molecule is chiral. The same group-theoretic argument delivers both, from the same operations, at no extra cost.

There is a fourth objection and it is the one that generalises. A bond moment is a fiction of a particular partition of the electron density into bonds, and electronegativity is not one quantity measures how badly that partition is agreed on — four published scales disagree about the direction of six ordinary bonds, C–H among them. An argument built on a quantity whose sign is disputed cannot be the exact one, whatever answer it happens to give. The symmetry argument never asks which atom is more electronegative, and would give the same answer if the roles were reversed.

The group is exact and the search is not #

A search for symmetry operations has to decide whether an operation maps the structure onto itself, and no structure built from floating-point arithmetic maps onto itself exactly. So the search carries a tolerance — six hundredths of an ångström here — and near the two planar arrangements that tolerance is visible.

Sweeping in one-degree steps from the eclipsed end, the search reports C₂ᵥ at $0$, $0.5$, $1$, $2$ and $3$ degrees, and C₂ from $5$ degrees on. From the anti end it reports C₂ₕ down to $178°$ and C₂ at $175$.

The width of that band is arithmetic and can be predicted. A hydrogen sits $0.947$ Å from the O–O axis, and the mirror the eclipsed structure has maps the azimuth $+\phi/2$ onto $-\phi/2$, so it misses by $2 \times 0.947 \sin(\phi/2)$. Setting that equal to the tolerance gives $\phi = 3.6°$, which is where the reported symbol changes.

Two things follow and they point in opposite directions.

The band is not physics. A structure at $2°$ has no mirror plane; it has an almost-mirror, and the group of the exact structure is C₂. Anything read off the reported symbol in that band is read off a rounding.

The band is not a defect either, provided its width is known. A tolerance loose enough to accept a published structure — whose coordinates are quoted to four decimals and whose atoms therefore sit some tens of micro-ångströms off perfect — is a tolerance that will call a nearly-planar molecule planar. The expansion phase found the same tolerance being right for deciding whether an operation exists and wrong for using one, and the repair there was to treat the approximate matrix as a question about which atom goes where and rebuild the operation exactly from the answer.

The figure samples at fifteen-degree intervals, well outside the band, so nothing drawn here sits in the ambiguous region. That is a choice made because the region exists, which is the useful form of knowing about it.

What decides the angle, and why it is not here #

The molecule adopts about $111.5°$ in the gas phase. Nothing in this essay predicts that, and symmetry cannot.

The dihedral is decided by a balance between the repulsion of the two lone pairs on each oxygen, which favours a twisted arrangement, and whatever weak attraction favours planarity — a total-energy question requiring electron-electron repulsion, which is a calculation beyond anything here.

What is measured, and can be quoted: the barrier to passing through the anti arrangement is about $4.6$ kJ/mol, and through the eclipsed one about $29.5$. Both are small enough that at room temperature the molecule crosses them freely, which is why hydrogen peroxide cannot be resolved into two hands even though every instantaneous structure has one.

That last sentence is the one worth being careful with. The molecule is chiral and its enantiomers are not separable, and those are different statements about different things: the first is a property of a structure and is exact, the second is a property of a barrier and a temperature. Collapsing them makes chirality depend on how cold the sample is.

Where the group appears in another measurement #

The dihedral shows up in a second place, and it is one that can be computed from the geometry alone: the moments of inertia.

At $111.5°$ the three principal moments are $1.678$, $19.378$ and $19.979$ u Ų, which makes the molecule an asymmetric top with an asymmetry parameter of $-0.994$ — very near the prolate limit, since two of the three moments are close and the third is small. That is a spectroscopic quantity: it sets where the rotational lines fall.

More useful is the inertial defect, $I_c - I_a - I_b$. For a planar rigid structure it is exactly zero, as the moment that is the sum of the other two sets out, and for hydrogen peroxide it is:

• $0$ at a dihedral of $0°$,
• $-0.84$ u Ų at $60°$,
• $-1.08$ at $111.5°$,
• $0$ at $180°$.

So the defect is a direct measure of the coordinate this essay is about — zero at both planar arrangements and largest in between. A rotational spectrum measures the moments; the moments give the defect; the defect gives the twist. That is how the angle was determined in the first place, and it is a good example of a symmetry-adjacent quantity doing quantitative work.

A group is a property of a structure #

The general point is worth stating plainly, because the language of the subject works against it.

Chemists say methane is Td and water is C₂ᵥ, and treat the group as a property of the substance. It is a property of a structure — a specific arrangement of nuclei — and substances whose structure is rigid can get away with the shorthand.

The moment a molecule has a soft coordinate, the shorthand breaks. Ethane at any twist between staggered and eclipsed is D₃d, D₃h or D₃ depending on where it is; biphenyl is planar or twisted depending on substituents; a protein has no single point group in any useful sense. Hydrogen peroxide is the smallest place the failure can be seen completely, because the whole family of structures fits in one figure.

This matters for what the group is allowed to be used for. Symmetry forbids a dipole and point groups from coordinates both derive properties from a group and both are exact — for the structure that group belongs to. A measurement made on a molecule that is moving between structures sees an average, and an average of exact zeros and non-zeros is not zero.

A sentence of reasoning, computed #

Chirality is a symmetry statement already contains the sentence “rotate about the O–O bond to make it planar and it becomes C₂ᵥ or C₂ₕ, both achiral”. That was written as a piece of reasoning about what the operations would be, and it was right.

This essay is that sentence turned into thirteen structure builds and thirteen searches, with the two properties read out of the recovered symbols rather than out of the reasoning. The value of doing it that way is not that the earlier sentence was in doubt. It is that the computation can fail: an error in the geometry construction, in the axis search or in the polarity rule would show up as a group that does not match the pattern, and the checks demand the pattern in the direction that could break — chiral strictly between the ends, achiral at them, polar except at the anti arrangement.

A check that has never rejected anything proves nothing, so the same search is asked for the arrangement at exactly $180°$ and required to report a molecule that cannot be polar. It does.

A chirality that lasts a picosecond #

The group says the molecule is chiral at every angle but two, and that is exactly true of a structure. It is worth asking what it means for the molecule, because the answer is almost nothing, and the reason is the barrier this essay declines to compute.

A twisted hydrogen peroxide has a mirror image which is a different structure — the same dihedral with the opposite sign. Getting from one to the other means passing through a planar arrangement, and there are two ways to do it: through the anti form at 180°, or through the eclipsed form at 0°. Both are measured, and they are not the same size. The anti barrier is about 390 wavenumbers and the eclipsed one about 2,460, so the cheap route is the anti one, over a barrier of a few kilojoules a mole.

That is low enough that the molecule does not have to climb it. The two twisted forms are separated by a barrier thin enough to tunnel through, and the tunnelling shows up directly in the spectrum as a splitting of every torsional level — the ground state’s is about 11.4 wavenumbers, which is large enough that it was one of the first things the microwave spectrum revealed.

A splitting of that size corresponds to an interconversion time of

$$\frac{1}{2 \times 11.4\ \text{cm}^{-1} \times c} \approx 1.5\ \text{picoseconds},$$

so the two enantiomeric structures exchange about a million million times a second, at any temperature, with no collision required.

A molecule that changes hands in a picosecond is not a chiral substance. Its true stationary states are symmetric and antisymmetric combinations of the two twisted forms, each of which is achiral; no sample of it rotates polarised light; and nothing about it could be resolved into two bottles. The point group of the structure is $C_2$ and the symmetry of the molecule is higher.

That is the general condition chirality has, made unusually visible by a molecule small enough to tunnel. Every resolvable chiral compound is one whose interconversion barrier is high enough that the tunnelling is negligible and the thermal crossing is slow on the timescale of the experiment — which for an ordinary carbon stereocentre means a barrier of hundreds of kilojoules a mole rather than a few.

So the two properties are not on the same footing after all. The dipole is a property the group settles absolutely, since it is forbidden at 180° by a centre of inversion whatever the molecule is doing. The chirality is a property the group settles for a structure, and whether the molecule has it depends on a barrier — which is exactly the quantity the essay says is not computed here.

Who found it, and when #

The twisted structure of hydrogen peroxide was settled by Paul-Antoine Giguère in the 1950s from infrared and Raman spectra, after two decades in which planar structures — both the eclipsed and the anti — had been proposed and defended. The decisive evidence was a band count: a planar molecule of either kind has a centre or a plane that makes certain vibrations inactive, and the observed spectrum showed bands that neither planar structure permits. That is what an absence proves run in reverse, using a presence to rule out two symmetries.

The dihedral itself came later and from the rotational spectrum, through exactly the inertial-defect route above. It is a case where the awkward, soft coordinate turned out to be the one carrying the interesting physics, and where three decades of argument were settled by counting lines rather than by any calculation.

Still open: how big an allowed dipole is #

The argument about dipoles began by taking a vector sum apart. This essay shows the same quantity being switched off and on by one coordinate, and shows the symmetry argument delivering a second property — chirality — that the vector sum has no access to at all.

What neither touches is the size of a dipole that is allowed to exist. That is not a symmetry question and it is not computed here: it needs the charge distribution, which needs the electron-electron repulsion. The pattern is one that keeps recurring — symmetry says exactly which quantities vanish and never says how big the others are — and every field here has its own version of it. Exactly zero computes a forbidden overlap and gets arithmetic noise rather than a small number; selection rules are one theorem gets the same kind of zero for a transition moment; and in each case the allowed quantity beside it is a number the argument declines to supply.