What a photoelectron spectrum measures

A photoelectron spectrum is a histogram of kinetic energies. A photon of known energy is absorbed, an electron leaves, and the difference between what went in and what came out is the energy it took to remove that electron. What the experiment produces is a list of ionisation energies, and every one of them is a difference between two total energies: the molecule’s, and one state of the ion’s.

Those are perfectly good observables. The thing they are almost always called — orbital energies — is not.

Koopmans’ approximation, and the two errors in it #

The identification of an ionisation energy with minus an orbital energy is Koopmans’ approximation, and its derivation makes an assumption that is visibly false and then neglects a quantity that is definitely not zero.

The assumption. Remove one electron from a Hartree–Fock wavefunction and leave every remaining orbital exactly where it was. Then the total energy changes by exactly minus the orbital energy of the one removed. That is a theorem, and its content is entirely in the words “leave every remaining orbital exactly where it was” — the frozen orbital approximation.

Real ions relax. Losing an electron reduces the screening felt by the rest, they contract, and the ion’s energy comes out lower than the frozen calculation says. So Koopmans overestimates the ionisation energy, typically by one to two electronvolts for a valence orbital.

The neglect. Hartree–Fock has no electron correlation. The molecule, with more electrons, has more correlation energy to lose than the ion does, so the true ionisation energy is lower than the correlated calculation would put it — an effect of the same order and the opposite sign.

The two errors are each about an electronvolt and they partly cancel, which is why Koopmans works as well as it does for valence ionisations of closed-shell molecules — typically to within an electronvolt or so. It is a cancellation between two things that were not computed, not a theorem about orbitals, and the cancellation fails where either error is unusual: for inner-shell ionisations, where relaxation is large, and for open-shell systems, where the frozen-orbital picture is wrong from the start.

Neither error is computed here. That needs a self-consistent field, and an unchecked one is worse than none, for the reason orbitals are not where the electron is gives about the whole subject: a wrong calculation that produces plausible numbers is exactly the failure worth preventing.

Why an orbital energy is not an observable at all #

The deeper problem is not the size of the errors. It is that the quantity being approximated does not have a unique value.

An orbital is a one-electron function in a chosen basis, and the choice is free. Hybrids are a basis makes this concrete: applying an orthogonal matrix to a set of occupied orbitals changes every orbital and changes no observable, because the determinant built from them is unchanged up to a factor of modulus one. The localisation transformation, demonstrated computes the case: methane’s canonical orbitals and its four equivalent localised bond orbitals give the same electron density to 10⁻¹⁶, at four different values of the free mixing parameter.

The two descriptions and the density they share are drawn there: one a₁ and three t₂ orbitals on one side, four equivalent bond orbitals on the other, related by an orthogonal transformation that changes every orbital and no observable, with the densities agreeing to the last bit a double can hold.

Those two sets have different orbital energies. The canonical ones are eigenvalues of the Fock operator and are distinct; the localised ones are not eigenvalues of anything and are all equal. Since no measurement distinguishes the two descriptions, no measurement is a measurement of an orbital energy.

What survives is narrower and is genuinely there. The ion’s states are physical: they have energies, and they have symmetry species, because the ion is a molecule with a point group like any other. A photoelectron spectrum resolves those states. Assigning each band to an orbital is a convenience that works because the canonical orbitals are the ones whose symmetry matches the ion states — which is exactly the reason the canonical set is privileged for this purpose and no other.

What symmetry does determine #

Here symmetry has something exact to say, and it is the part of the assignment that does not depend on any energy calculation at all.

The molecule’s valence orbitals span a representation that follows from the coordinates. Reduce it, and the number of distinct symmetry species is the number of distinct ion states available; the dimension of each species is the degeneracy, and therefore the relative intensity.

Methane’s spectrum shows exactly that: two bands, at 12.7 and 23.0 electronvolts, with the lower one about three times as intense. Hybridisation does not explain is the essay about what that refutes, and the argument there is entirely a counting argument.

The energies are measured and the assignment is computed, and keeping those apart is the whole of what can honestly be claimed here about a photoelectron spectrum.

What the bands do tell a chemist #

Setting aside the identification with orbital energies, the experiment is informative, and the reasons are worth listing because they survive the criticism above.

Band positions are ionisation energies, which are real thermodynamic quantities and enter real arguments — about acidity, about electron transfer, about where a molecule sits relative to a metal’s work function.

Band structure carries vibrational information. A band with vibrational structure whose spacing differs from the neutral molecule’s frequency says the ion has a different geometry, and therefore that the electron removed was doing something structural. That is a genuinely orbital-like conclusion drawn from something other than an energy.

Band counts and ratios carry symmetry information, exactly as computed above, and this is the part that is exact.

Comparisons across a series are robust in a way absolute assignments are not. The relaxation and correlation errors are similar for similar molecules, so a trend in ionisation energies across a series of related compounds is meaningful even when each individual identification with an orbital energy is not.

Everything in the figures above — which species exist, which are degenerate, what may be assigned to what — comes from a table whose columns were counted on the molecule before the table was opened. Why a character table stops where it stops is why those columns cannot be otherwise: the number of species is the number of classes and the squares of their dimensions sum to the order, so Td’s largest degeneracy is three and C₂ᵥ’s is one, and no arrangement of electrons can change either.

Where Koopmans fails visibly #

The cancellation holds well enough for valence ionisations of closed-shell molecules that its failures are informative, and there are three standard ones.

Inner-shell ionisation. Removing a core electron leaves a hole the remaining electrons relax around very strongly — the relaxation energy for a carbon 1s hole is of order ten electronvolts — and there is no correlation error of comparable size to cancel it. Koopmans overestimates core ionisation energies badly, and core-level spectroscopy is interpreted with computed relaxation rather than with orbital energies.

Open-shell molecules. The frozen-orbital derivation assumes a closed-shell determinant; for an open shell there are several ion states from removing one electron and the identification with a single orbital energy is not even well posed.

Where two ion states are close. The measured quantity is a state of the ion, and two states of the same symmetry can mix. Then there is no orbital that either corresponds to, and the spectrum shows two bands where an orbital picture predicts one — a breakdown of the whole picture rather than an error in it.

None of those is a small correction to a good approximation. They are cases where the quantity being approximated stops existing, and the pattern matches the third refutation at the head of this essay: the approximation is not close or far from the truth, it is of the wrong kind.

Water is the molecule where the count is largest and the check is tightest. Its four valence bands sit at 12.6, 14.8, 18.6 and 32.2 electronvolts and carry the labels b₁, a₁, b₂ and a₁ — four bands for a molecule with two bonds, and water’s lone pairs are not a pair draws them with the assignment checked against the reduction as the figure is built. The energies are measured; the number of bands and their labels are computed, and they are the part that could have come out differently.

What can and cannot be said here, exactly #

It is worth setting out the division, because here the boundary of a one-electron treatment is closest to the subject.

Computed here: which symmetry species the valence orbitals span, in the molecule’s own group, generated from the coordinates; how many distinct ion states that permits; the degeneracy of each and therefore the intensity ratio; and whether a proposed assignment is even possible — a band labelled with a species the reduction does not produce is ruled out.

Quoted here: every band position, in electronvolts, as a measurement.

Not available here: the relaxation energy, the correlation energy, the absolute ionisation energy, and any statement about which orbital is “at” a given energy in an absolute sense.

The last of those is the one worth being firm about. Even with a self-consistent field in hand, the computed orbital energy would be a property of a chosen set of orbitals rather than an observable — so the missing calculation would improve the approximation to the measurement and would not turn an orbital energy into a measurable thing. The deferral costs accuracy; the argument does not depend on it.

The transformation the whole argument turns on is a matrix, and hybrids are a basis writes it out: an orthogonal matrix applied to a set of occupied orbitals produces a different set describing the same many-electron state, so orbital energies change and nothing observable does. That is the reason an orbital energy cannot be measured, and it is arithmetic rather than philosophy.

The same shape of error, elsewhere #

The mistake this essay is about has a form that recurs: a quantity internal to a model being treated as a measurement because a measurement happens to agree with it.

The aufbau order is not a property of the atom is the same shape. The filling order is a property of a sequence of many-electron systems, not a list of one-electron energies, and every one-electron estimate puts 3d below 4s at every element while 4s fills first. Reading the periodic table as a list of orbital energies is reading a model quantity as a measurement.

Electronegativity is not one quantity is a third instance. Four scales, four definitions, and a defence — they correlate well — that answers a question nobody asked, because the disagreements are where the quantity is being used.

The general repair is always the same: name the model that produced the number, and say what would have to be true for it to be a measurement. For Koopmans that is a frozen ion with no correlation, and neither condition holds.

What a band’s shape adds #

Band positions are the part of a photoelectron spectrum most often quoted and the least informative about structure. The shapes carry something the positions do not, and it is worth setting out because it is a genuinely orbital-like conclusion drawn without any orbital energy.

A photoelectron band is not a line. It is a progression, because the ion can be left in any of its vibrational states, and the pattern of that progression says how much the geometry changed on ionisation.

A sharp band with one strong peak means the ion has nearly the same geometry as the molecule, so the electron removed was doing little structurally — a non-bonding orbital.

A broad band with extended vibrational structure means the geometry changed, so the electron was bonding or antibonding in the coordinate that changed. The spacing of the structure is the ion’s vibrational frequency in that mode, which is a direct measurement of how the bonding changed.

Water’s spectrum shows both: its 1b₁ band is sharp and its 3a₁ band is extended, which says the first orbital is non-bonding and the second is angle-determining. Water’s lone pairs are not a pair leans on that difference, and it is evidence of a kind Koopmans’ theorem has nothing to say about.

The modes whose quanta appear in that structure are the molecule’s own three, counted in a spectrum counts environments, not atoms, and it is the bending mode at 1,595 wavenumbers whose progression appears on the 3a₁ band: the ion is bent differently from the molecule, so ionising into it excites the bend. The frequency in the ion differs from the neutral molecule’s, and that difference is the measurement.

What is deferred, and why #

A proper treatment would compute the ionisation energies directly — the molecule’s energy and the ion’s, separately, each self-consistently — and would then be able to say how large the relaxation is rather than quoting it. That needs two-electron integrals and a self-consistent field, neither of which is used here.

The reason for leaving it out is specific. A minimal Gaussian-basis calculation that produced plausible numbers without being checkable would be worse than no calculation: it would look like the computed claims around it and would not be one.

What can be done without it is what is done above: the counting, the species, the degeneracies and the ratios, all exact, plus a clear statement of what the measured numbers are measurements of.

The molecule where the cancellation gets the order wrong #

Two errors of about an electronvolt with opposite signs is a comfortable position while the two are the same size for every orbital. They are not, and the clearest demonstration is a molecule where the residue is large enough to reverse two bands.

Nitrogen’s photoelectron spectrum has its first band at 15.58 electronvolts, and the state it produces is the one reached by removing an electron from the σ orbital along the bond. The next band, at 16.98, comes from the π pair.

Hartree–Fock orbital energies for the same molecule put them the other way round. The π level comes out slightly above the σ — by a couple of tenths of an electronvolt, small but unambiguous — so reading the spectrum off the orbital energies predicts the first ionisation from π, and the molecule ionises from σ.

The theorem gets the ordering wrong for one of the most-studied molecules in chemistry.

The reason is the part of the cancellation that does not cancel. Removing an electron from a σ orbital concentrated along the bond leaves a hole in a compact region, and the remaining electrons relax strongly into it; removing one from a π orbital spread around the axis leaves a more diffuse hole and less relaxation. So the correction is larger for the σ, it lowers that ionisation energy more, and the two swap.

Nothing about that is exotic. It is the general statement that relaxation depends on which orbital the hole is in, so the correction is orbital-specific and a cancellation calibrated on one orbital does not hold for another.

That sets the scope of the identification more sharply than a stated error bar would. Koopmans’ theorem is reliable when the orbitals being compared are of similar spatial extent, which for the valence orbitals of an ordinary closed-shell organic molecule they usually are — and it is unreliable exactly when two bands are close together and the orbitals differ in character, which is precisely when the ordering is being asked for.

So the theorem is best at what it is least often needed for. Where two bands are far apart, the assignment was never in doubt; where they are within an electronvolt of each other, the correction is the same size as the gap being resolved.

There is a repair and it costs a calculation rather than an argument. Instead of reading the ionisation energy off an orbital energy, compute the neutral and the ion separately and subtract — which includes the relaxation exactly, because the ion is optimised in its own right, and leaves only the correlation difference. That is more expensive by a factor of two and it gets nitrogen’s ordering right.

What it gives up is the thing the theorem was valued for. A subtraction of two total energies produces one ionisation energy and no orbitals at all, so the picture of a spectrum as a ladder of levels — which is how most spectra are read — is a consequence of the approximation rather than of the measurement. The honest position is that that picture is a useful fiction whose failures are locatable, and nitrogen is where to look for one.

Who Koopmans was #

Tjalling Koopmans published the theorem in 1934, as a physicist, in a paper about the ordering of orbital energies in the Hartree–Fock scheme. He left physics for economics shortly afterwards and won the Nobel Prize in economics in 1975 for work on the allocation of resources; the theorem that carries his name in chemistry was an early paper in a career he did not continue.

The experimental technique arrived thirty years later — Turner’s ultraviolet photoelectron spectrometer in the 1960s — and it is worth noting how the two met. Koopmans’ theorem was a statement inside a computational scheme with no experiment to compare it against; when the experiment arrived, the agreement to about an electronvolt was taken as a vindication of both. The cancellation that produces that agreement was understood later.

One molecule, one picture, and the spectrum #

The sharpest test takes one molecule and one picture and lets the spectrum settle it. Almost everybody is taught that water has two equivalent lone pairs pointing away from the hydrogens; its photoelectron spectrum shows four distinct valence bands, two of which are the ones that picture says should be identical, and they are 2.1 electronvolts apart. That is water’s lone pairs are not a pair.