Water's lone pairs are not a pair

The picture is universal. Oxygen, two bonds to hydrogen, and two lone pairs drawn as a matched pair pointing back and away — the same shape, the same size, at the tetrahedral angle to each other. It explains water’s bent shape, it explains hydrogen bonding, it explains why ice has the structure it has, and every one of those explanations works.

The two lone pairs are not equivalent, and the spectrum that says so has been available since 1970.

What the group permits before any measurement #

Water’s point group is C₂ᵥ, and character tables and reduction computes its four irreducible representations by generating the group from the coordinates: A₁, A₂, B₁ and B₂. All four are one-dimensional. The group has no degenerate representation at all, which by degeneracy is a group theorem means water can have no degenerate orbital, ever, whatever the bonding.

That single fact already settles the question. Two equivalent lone pairs would have to be exchangeable by some operation of the group, and two orbitals exchanged by an operation belong to a degenerate pair. Water has no degenerate pair available, so no two of its orbitals can be equivalent — not the lone pairs, not the bonds, none of them.

Water’s group is generated from three atomic positions and sorted into classes before any table is opened, and it holds four classes and therefore four representations, every one of them one-dimensional — why a character table stops where it stops is the arithmetic that forces that. A group with no degenerate representation cannot have two degenerate orbitals, and that single sentence is most of this essay.

The valence basis then decides how many bands there can be. Oxygen’s 2s spans a₁; its three 2p functions span a₁ ⊕ b₁ ⊕ b₂; the two hydrogen 1s functions span a₁ ⊕ b₂. Together that is 3a₁ ⊕ b₁ ⊕ 2b₂ — six orbitals, of which the four lowest are occupied by water’s eight valence electrons.

What the spectrum shows #

The measured valence bands of water sit at about 12.6, 14.7, 18.5 and 32.2 electronvolts, and they are assigned 1b₁, 3a₁, 1b₂ and 2a₁.

The two highest — 1b₁ and 3a₁ — are the ones the equivalent-lone-pair picture describes. They are 2.1 electronvolts apart, which is not a small discrepancy in a spectrum whose whole valence range is twenty, and they belong to different symmetry species, which is not a matter of degree at all.

They also behave differently in ways beyond their positions. The 1b₁ band is sharp; the 3a₁ band shows extended vibrational structure, because removing an electron from 3a₁ changes the bond angle substantially while removing one from 1b₁ hardly does. That is a structural conclusion drawn from band shape, and it is exactly what the symmetry labels predict: 1b₁ is the out-of-plane p orbital, which is non-bonding and has nothing to do with the angle; 3a₁ lies in the plane and is heavily involved in it.

What is refuted, precisely #

It matters to be exact about what falls, because localised descriptions are perfectly legitimate and nothing here says otherwise.

Refuted: that water has two equivalent lone pairs. Two orbitals are equivalent when a symmetry operation carries one onto the other, and water’s group has no operation that does. The two are of different symmetry, different energy, different shape and different bonding character.

Refuted: that the two would appear as one band. The equivalent-pair picture, taken as a claim about what a measurement will find, predicts a single ionisation from a doubly degenerate pair. Water’s spectrum has no degenerate band anywhere and cannot have one.

Not refuted: the use of localised orbitals. Hybrids are a basis and the localisation transformation, demonstrated settle this. An orthogonal transformation of the occupied orbitals leaves the electron density unchanged — demonstrated to 10⁻¹⁶ for methane, at four different values of the mixing parameter — so a description in terms of two equivalent localised lone pairs is a perfectly valid alternative basis for the same many-electron state.

The transformation the defence rests on is drawn for methane in the localisation transformation: four equivalent localised orbitals on one side, one a₁ and three t₂ on the other, and a density that does not move between them. Two descriptions disagreeing about everything except every observable is exactly the situation, and it is why the localised picture is a basis rather than a claim.

The distinction is between a basis and a state. A basis may be chosen freely. The ion’s states cannot: they are eigenstates of a Hamiltonian that commutes with the group, so each carries a symmetry label, and a photoelectron spectrum resolves them. Localised orbitals are not eigenfunctions of anything, so there is no ion state corresponding to “remove an electron from lone pair number one” — that operation produces a superposition of the 1b₁ and 3a₁ ion states rather than any state of the ion.

That is why the picture fails as a prediction while succeeding as a description. It predicts the wrong thing about ionisation and the right thing about geometry, hydrogen bonding and the direction of the dipole, because those depend on the density and the density is what the transformation preserves.

The four orbitals, one at a time #

Naming what each band is makes the argument concrete, and every label below is a species the reduction produced rather than a claim about shape.

1b₁, at 12.6 eV. The oxygen 2p orbital perpendicular to the molecular plane. It is in a symmetry species of its own — b₁ is spanned by nothing the hydrogens carry — so it cannot mix with them at all and is a pure, non-bonding oxygen orbital. This is the closest thing water has to a textbook lone pair, and there is exactly one of it.

3a₁, at 14.7 eV. In the plane, and a mixture: oxygen 2s and 2pz with the symmetric combination of the two hydrogen 1s functions, all of which are a₁. It is partly non-bonding and partly bonding, which is why removing an electron from it changes the bond angle and gives the band its extended vibrational structure.

1b₂, at 18.5 eV. The antisymmetric combination of the hydrogen 1s functions with the oxygen 2py — genuinely bonding, and the reason water is held together at all.

2a₁, at 32.2 eV. Mostly oxygen 2s, deep and largely non-bonding.

Two of those four are commonly called lone pairs and they could hardly be more different: one is a pure p orbital forbidden by symmetry from mixing with anything, the other a three-way mixture that determines the bond angle. They are not a pair in any sense except that both are occupied and neither is a bond.

The prediction the picture actually makes #

It is worth being fair to the equivalent-pair picture by writing down what it predicts, because the prediction is specific and is what gets refuted.

Two equivalent orbitals related by a symmetry operation belong to a degenerate pair, so the picture predicts one band of double intensity where the spectrum shows two bands of single intensity 2.1 electronvolts apart. It also predicts that the two would respond identically to anything done to the molecule, where in fact one is angle-determining and the other is not.

And it predicts something about a related molecule that is also wrong. If lone pairs were equivalent sp³ hybrids, ammonia — with one lone pair and three bonds — and water — with two and two — would show closely analogous spectra with the lone-pair bands in corresponding places. Ammonia’s valence spectrum shows three bands, in species a₁, e and a₁, and its highest is the nitrogen lone pair in a₁ rather than in anything resembling water’s b₁.

Why the picture is worth keeping anyway #

Three things the equivalent-lone-pair picture gets right, all of them consequences of the density rather than of the orbitals:

The shape. Four electron domains around oxygen, two bonding and two not, arranged to minimise repulsion, giving an angle below tetrahedral. Why water is bent computes the arrangement and finds that the lone-pair correction is a fitted number rather than a derived one — but the direction is right and the reason is the density’s shape, which the localised picture describes well.

Hydrogen bonding. Ice’s structure has each oxygen accepting two hydrogen bonds in roughly tetrahedral directions, which the two-equivalent-pairs picture predicts and which is essentially right — the density in those directions is what an approaching proton sees, and it is the same density in either description.

The dipole. Its magnitude and direction come from the charge distribution, which is again basis-independent. The dipole is not a sum of bonds is about a different error in the same area.

So the correct statement is narrower than “the rabbit-ear picture is wrong”. It is: the picture is a basis, not a set of states; it makes correct predictions about anything determined by the density, and incorrect predictions about anything that resolves states. Photoelectron spectroscopy resolves states, which is why it is the experiment that catches it.

The same argument, on methane #

The identical situation arose for methane in the essay that counted its hybrids, and it is worth putting the two side by side because the numbers differ and the structure of the argument does not.

Methane is the cleaner case because its bonds genuinely are equivalent — all four sit in one orbit under a group of twenty-four operations — and the spectrum still shows two bands, because equivalence of bonds does not mean equivalence of the orbitals describing them. Water is the harder case and the more common teaching error, because there the two objects in question are not equivalent by any measure at all.

Why the wrong picture is taught #

It is worth asking why a picture refuted in 1970 is still drawn in every introductory course, because the answer is not stupidity and the pattern recurs wherever a memorable picture is taught first.

It gets the geometry right. Four domains around oxygen, two bonding and two not, arranged to minimise repulsion — that is the VSEPR argument why water is bent computes, and the equivalent-pair picture is a perfectly good way of holding it in mind.

It gets hydrogen bonding right. Two acceptor directions, roughly tetrahedral, which is what the structure of ice shows.

It is memorable, and its competitor is not. “Two lone pairs, like ears” is a picture; “3a₁ is an in-plane mixture of oxygen 2s and 2pz with the symmetric hydrogen combination, and 1b₁ is a pure out-of-plane 2p” is a paragraph.

And the measurement that refutes it needs tools a first course has not built. Photoelectron spectroscopy needs the idea of an ion state, symmetry species, and the distinction between a basis and a state — none of which is available in week three.

So the picture survives because it is right about what a first course tests and wrong about what a first course cannot measure. The honest teaching repair is not to remove it but to say what kind of object it is: a way of holding the density in mind, not a claim about what a spectrum will show.

C₃ᵥ does have a two-dimensional representation, which is what makes ammonia the instructive comparison: its three N–H bonding orbitals can be partly degenerate, a₁ ⊕ e, where water’s group has nowhere to put a degeneracy at all. So the two molecules are not two versions of one argument — one of them permits the textbook picture in part and the other forbids it outright.

The general form of the correction is worth carrying beyond water. A picture that is right about the density can be wrong about the states, and only an experiment that resolves states will notice. Geometry, dipole moments, hydrogen bonding and reactivity depend on the density; ionisation, excitation and magnetic resonance resolve states. A model tested only against the first kind will survive indefinitely.

What this essay does not do #

No energy is computed. The band positions are measured and are quoted as measurements; the species are computed. No self-consistent field is run, and the deferral is deliberate — what a photoelectron spectrum measures sets out why, and adds the further caution that even a computed orbital energy would not be the measured quantity.

No claim is made about which basis is better. The canonical orbitals are the right basis for discussing ionisation because they carry the ion states’ symmetry; localised orbitals are the right basis for discussing local chemistry because they are local. Asking which is real is asking which coordinate system nature prefers.

Ice and hydrogen bonding are gestured at rather than computed. The claim that two hydrogen bonds are accepted in roughly tetrahedral directions is quoted from the structure of ice, not derived here.

The symmetry all of this rests on is visible without any spectrum. Water’s two hydrogens are one orbit and its oxygen is another — two environments — and a spectrum counts environments, not atoms sorts them. Two environments and four one-dimensional representations is the whole of the input; everything else in this essay is arithmetic on those two facts.

The picture the spectrum refuses is the one the structures need #

The spectrum settles what a photoelectron measurement sees, and it does not settle the argument, because the equivalent-lone-pair picture is defended on quite different evidence — and that evidence is worth putting beside the spectrum rather than leaving out.

Water accepts hydrogen bonds, and where it accepts them is measurable. In ice, and in the local arrangement of liquid water, each oxygen has four neighbours in a roughly tetrahedral arrangement: two hydrogens it donates to and two directions it accepts along. Those two acceptor directions are, to a good approximation, where a pair of equivalent lone pairs would point.

The canonical orbitals do not point that way. One of the two is in the plane of the molecule and the other is perpendicular to it, they have different energies and different shapes, and nothing about them suggests two equivalent tetrahedral directions.

So one picture matches the spectrum and the other matches the structures, and each is used by the community that measures the thing it matches.

The resolution is the one that keeps recurring and it is worth stating in full here, because this is the case where it does the most work. Neither picture is a claim about the molecule; both are bases for the same occupied space. The total density is identical in the two, and so is everything computed from it — including where an approaching proton finds the density it is attracted to.

That means the acceptor directions are not evidence for the localised picture. They are a property of the density, the density is basis-independent, and the localised orbitals reproduce it exactly as the canonical ones do. What the localised picture supplies is a convenient way of talking about the density’s two maxima; what it does not supply is a second physical fact.

And the same is true in reverse. The spectrum’s two bands are not evidence against the localised picture; they are evidence that ionisation couples to the canonical orbitals, which it does because those are the eigenfunctions of the one-electron Hamiltonian, and which is a fact about the experiment rather than about the molecule.

So both bodies of evidence stand, neither refutes the other, and the appearance of a dispute comes entirely from treating a choice of basis as a claim. What the spectrum does refute is the specific statement that water has two ionisation energies from equivalent lone pairs — which is a claim about a measurement, is made routinely, and is wrong by 2.1 electronvolts.

The tetrahedral acceptor geometry deserves one further note, because attributing it to the lone pairs at all is a step worth examining. An approaching hydrogen-bond donor is a small positive charge looking for negative density, and the density around an oxygen is not two lobes: it is a single distribution with a broad region of excess on the side away from the hydrogens. Whether that region has two maxima or one is a question about a computed density, the answer depends on where the contour is drawn, and calculations disagree.

So the structural evidence is weaker than it is usually presented as being. Ice is tetrahedral, and a tetrahedral arrangement of four neighbours around an oxygen is also what packing four things around a small centre gives. The geometry is consistent with two lone pairs pointing where they are drawn, and it does not require them — which leaves the localised picture where the spectrum leaves it, as a convenient basis rather than a claim.

Who measured it #

Water’s valence photoelectron spectrum was recorded by Turner and co-workers around 1970, in the burst of work that followed the development of the ultraviolet photoelectron spectrometer, and the four-band structure was clear immediately. The assignment followed from symmetry — exactly the argument above — supported by the vibrational structure on the 3a₁ band.

The conclusion has been in the literature for over fifty years and the two-equivalent-lone-pairs picture is still drawn in almost every introductory course. That is a familiar pattern, and the reason is usually the same: the picture is memorable, it is right about the things a first course tests, and the measurement that refutes it needs tools a first course has not built.

Where symmetry takes over #

Past the photoelectron spectrum the next questions are about symmetry rather than spectroscopy: an infinite group, worked in a finite one deals with the molecules whose groups are infinite, and the vibration that lowers the symmetry computes a case where a molecule escapes a degeneracy by distorting — which is what a molecule does when it finds itself in the state water cannot have.