The width of a band is a bond length

A photoelectron spectrum is usually read as a list of energies: a band here, another there, and each one assigned to an orbital. Two things about those energies are settled elsewhere: what they are and are not — an ionisation energy is a difference between two states, not the energy of an orbital — and how many there are, since symmetry decides how many bands there can be.

Every band also has a shape, and the shape is a second measurement made at the same time and usually discarded.

Nitrogen’s spectrum is the standard example because its three bands could not look more different. The first is a single sharp line. The second is a long progression of evenly spaced lines. The third is a line with one shoulder.

The reason is a bond length, and the whole of it can be computed from measured constants.

Three states of one ion #

Removing an electron from a nitrogen molecule leaves an ion that is not the same shape.

From the 3σg orbital, weakly bonding: the bond lengthens from 1.0977 to 1.1164 ångström, by 18.7 thousandths.

From the 1πu orbital, strongly bonding: it lengthens to 1.1749, by 77.2 thousandths.

From the 2σu orbital, weakly antibonding: it shortens to 1.0740, by 23.7 thousandths — because the electron that left was holding the atoms apart.

Each of those numbers is measured, and each comes with a vibrational frequency for the state it belongs to: 2207, 1904 and 2420 wavenumbers, against the neutral molecule’s 2359. The bonding orbital’s removal softens the bond; the antibonding one’s stiffens it.

What the ion is left in #

The electron leaves in about an attosecond and the nuclei do not move in that time. So the molecule arrives in the ion’s potential well still at the neutral’s bond length, in whatever superposition of the ion’s vibrational states that position corresponds to.

The intensity of each vibrational line is therefore the square of an overlap: the neutral molecule’s lowest vibrational wavefunction against each of the ion’s.

Both potentials here are Morse curves built from the measured vibrational constants, solved on the same grid by finite differences, and the overlaps are sums over grid points. Nothing is fitted.

The three bands, computed #

That figure is the essay. Three bands whose shapes are entirely different, computed from measurements that say nothing about spectra — two bond lengths and two frequencies per state — and matching what the spectrum shows.

The parameter that organises it has a name. For two harmonic wells of the same frequency displaced by $\Delta r$, the intensities are a Poisson distribution $e^{-S}S^v/v!$, with

$$S = \frac{\mu \omega \, \Delta r^2}{2\hbar}$$

so a band’s shape is set by one number, quadratic in the geometry change. For the three states $S$ is 0.083, 1.319 and 0.139.

The brightest line is the one nearest $S$: the ground state for the first and third bands, the first excited state for the second.

The width in quanta is $\sqrt{S}$: 0.29, 1.15 and 0.37.

And the gap between the vertical and adiabatic ionisation energies is $S\hbar\omega$: 0.024, 0.349 and 0.041 electronvolts. The middle band’s peak is a third of an electronvolt above its onset, and reading its maximum as the ionisation energy is an error of that size.

The closed form and the calculation #

The Poisson expression is harmonic and the wells are not, so the two disagree — and where they disagree is informative.

The pattern is the usual one for anharmonicity: it is invisible where the motion is small and grows with the excursion, so the broad band — the one that reaches high vibrational states — is the one where the harmonic form fails.

The check that makes both believable is that the intensities of a band sum to one, over the ion’s bound states, in every case. An electron that leaves puts the ion in some vibrational state, and a calculation whose intensities did not add up would be losing amplitude off the end of its grid.

Two ionisation energies, and the one that is quoted #

A band with structure in it has two energies that could be called the ionisation energy, and they differ by more for the broad band than the resolution of any modern instrument.

The adiabatic ionisation energy is the onset: the transition from the neutral molecule’s lowest vibrational state to the ion’s lowest. It is the true energy difference between the two ground states.

The vertical one is the maximum: the transition to whichever vibrational state has the greatest intensity, which is the ion’s energy at the neutral’s geometry.

For nitrogen’s first band the two differ by 0.024 electronvolts, for the third by 0.041, and for the middle one by 0.349. A table of ionisation energies that does not say which convention it uses is carrying a third of an electronvolt of ambiguity on exactly the bands where the chemistry is most interesting.

This is a shape that recurs throughout chemistry: a quantity that is unambiguous where the effect is small and needs a convention exactly where it matters. It is also the reason Koopmans’ theorem’s comparisons have to say which energy they are approximating, since a frozen-orbital calculation is by construction a vertical quantity.

What this makes the spectrum say #

A photoelectron spectrum with resolved vibrational structure is therefore a measurement of the ion’s geometry, and by difference of what each orbital was doing in the neutral molecule.

A sharp band means the orbital was nearly non-bonding. That is a statement about the molecule, arrived at without any calculation of it — and it is the kind of statement an absent band also makes, from the other direction. Removing the electron changed nothing about the bond, so the ion is born at its own equilibrium length.

A broad progression means the orbital was strongly bonding or strongly antibonding, and the vibrational spacing says which: a spacing smaller than the neutral molecule’s frequency means the bond got weaker, so the electron was bonding; larger means it got stronger, so the electron was antibonding.

And the spacing itself is a frequency of the ion, measurable to a wavenumber, which is a considerably more precise number than a band position.

That is a lot of chemistry to read off a shape, and it is the reason vibrationally resolved photoelectron spectroscopy was worth the effort of achieving. It also settles assignments that band positions cannot: two orbitals close in energy but different in bonding character produce bands of quite different shape, and the shape is what tells them apart.

The assignment this settles #

Nitrogen’s spectrum has a further wrinkle that makes it the textbook case, and the shapes are what resolved it.

The 3σg orbital is nominally the σ bonding combination of the two 2p orbitals and should be strongly bonding; the 1πu orbitals are π and should be less so. The observed shapes say the opposite: the σ band is sharp and the π band is broad.

The reason is mixing. The 3σg orbital is not a pure 2p σ combination — it mixes with the 2s σ combination, which is antibonding at that separation, and the mixing largely cancels its bonding character. That mixing is also why nitrogen’s 3σg lies above its 1πu, which is the famous ordering anomaly of the second-row diatomics.

So the band shapes and the level ordering have the same cause, and either one confirms the other. A spectrum read only for its positions gives the ordering and no explanation; read for its shapes as well, it gives both.

The counting half of the same subject predicts a spectrum’s number of bands from symmetry alone, before any energy is mentioned. Positions, counts and shapes are three separate questions about one spectrum, and this essay is about the third — which is the only one of the three that reports a distance.

What the same argument does elsewhere #

The mechanism here is not a fact about photoelectron spectra. It is what happens whenever a molecule changes electronic state faster than its nuclei can move, and it turns up under three different names.

In electronic absorption, the same overlaps give the vibrational structure of an absorption band, and the same $S$ decides whether a band is a sharp line or a broad hump. A dye with a large geometry change between its ground and excited states has a broad absorption, and the Stokes shift between absorption and emission is $2S\hbar\omega$.

In electron transfer, $S\hbar\omega$ is the reorganisation energy — the energy the surroundings and the molecule need in order to arrive at the geometry the product wants — and it is the central quantity of Marcus theory.

And in a solid, the same parameter is the Huang–Rhys factor, which decides how strongly a defect’s optical transition couples to the lattice and how broad its emission is.

Three fields, three names, one integral. What makes the photoelectron case the clearest is that the two states are a molecule and its own ion, both of whose geometries are measured independently, so the calculation can be checked rather than fitted.

Where the model stops #

Two potential curves, one coordinate. A diatomic has one vibration and every band is a progression in it. A polyatomic has 3N − 6, the ion’s geometry can change along several at once, and the band is a product of progressions — which is why polyatomic photoelectron bands are usually a smooth hump rather than a comb.

Only totally symmetric modes appear. A geometry change that lowers the symmetry cannot appear at first order, so the modes a band progresses in are a subset chosen by symmetry, which is the same rule that governs which vibrations a transition can excite.

The electronic transition moment is taken as constant. The intensities here are vibrational overlaps alone; the assumption that the electronic part does not vary over the range of the vibration is the Franck–Condon approximation itself, and it fails where a band overlaps another electronic state.

The two potentials are Morse curves. A Morse potential has the right shape near the minimum and the right dissociation behaviour, and it is a two-parameter fit rather than the real curve — the isotope-dependent bond lengths of one such curve come from the same kind of fit.

And the temperature is zero. The neutral molecule is taken to be in its lowest vibrational state; at room temperature a small population sits in the first excited state and produces lines on the low-energy side of the band, which is a real feature of measured spectra and is not here.

What the count of lines is not #

One caution belongs here, because the essay has been counting lines and a count is easy to over-read.

The number of resolvable lines in a band is not a property of the molecule alone: it depends on the instrument’s resolution, on the vibrational spacing, and on where the threshold for “resolvable” is put. The counts quoted above use a threshold of a tenth of the strongest line’s intensity, which is a choice.

What is not a choice is the distribution, and every claim in this essay that matters is about it: which line is brightest, what fraction the first line holds, how far the intensity extends. Those are computed numbers with no threshold in them, and the counts are a convenient summary of them for prose.

That distinction — a robust quantity and a convenient summary of it — is the same one this collection insists on for a tolerance that decides a point group and for a moment read off a fitted law. Where a summary needs a threshold, the threshold is part of the claim.

The same arithmetic run backwards is a structure determination #

Everything here goes from a known geometry change to a predicted band shape, and the calculation is used in practice in the other direction — because for the species it applies to there is often no other way to get a bond length at all.

The objects in question are molecular ions. A nitrogen cation in its second excited state is not a substance: it cannot be put in a cell, it does not survive long enough for a rotational spectrum, and there is no crystal of it to diffract. Its geometry is nevertheless a well-defined quantity, and the photoelectron band that produces it carries the information.

The route is the calculation here, inverted. Measure the intensities of the vibrational lines within a band; fit the geometry change that reproduces them; and read off how much the bond lengthened or shortened when the electron left. A single sharp line means the geometry barely moved; a long progression means it moved a great deal; and the ratio of the first few lines fixes the amount.

The precision is better than the crudeness of the idea suggests. The intensities depend on the square of the displacement, so a progression is a sensitive measure of a small change, and geometry changes of a few thousandths of an ångström are recoverable — which is the scale the numbers here sit at.

That gives the vibrational structure of a photoelectron band a standing that a first reading would not suggest. It is not decoration on the band; it is a structure determination on a species no other technique can reach, and it is why the states of small molecular ions have known bond lengths at all.

Two conditions limit it, and both are visible in the three bands here. The mode has to be one the filter passes, so only the totally symmetric coordinates are measured this way. And the progression has to be resolved, which requires the band to be sharp enough to show lines — so a strongly bonding ionisation, which produces the longest progression and therefore the most information, is also the one most likely to be a broad unresolved hump.

That second condition has a consolation, and it is one the numbers here already contain. Even an unresolved band carries the information in its width: a progression whose lines cannot be separated is still a feature whose total extent is set by how far the geometry moved, so the envelope’s width is a cruder version of the same measurement. A band that is a single sharp line and a band that is a broad hump differ by a geometry change of tens of thousandths of an ångström, and that much is readable from a spectrum with no resolution at all — which is why the qualitative rule sharp band, non-bonding electron is taught long before the arithmetic behind it.

What is quoted, and what is computed #

Quoted: the vibrational constants and equilibrium bond lengths of nitrogen and of the three states of its cation, from the standard compilations; the three ionisation energies; the assignment of each band to an orbital.

Computed: the Morse potentials from those constants; the vibrational wavefunctions by diagonalisation on a grid; every overlap and every intensity; the displacement parameter and its consequences; and the Poisson distribution the calculation is compared against.

What the calculation requires #

The intensities of each band sum to one over the ion’s bound states, to two per cent. Without this the grid is not holding the wavefunctions.

The computed 0–0 intensity agrees with the Poisson form to within what anharmonicity can explain, in every band.

The middle band is the broad one, by a count of resolvable lines, and it is the band of the orbital a bonding analysis calls strongly bonding.

The antibonding electron’s removal shortens the bond and the bonding one’s lengthens it, and the displacement parameters differ by more than a factor of five between them.

And the refusal: an ionisation that changes no bond length puts everything in one line, to a part in a thousand. A calculation that spread intensity there would be measuring its own numerics.

Still open: which modes a polyatomic band shows #

The obvious open question is the polyatomic case, and the interesting part of it is not the extra coordinates but which ones appear.

A band’s progression is in the modes along which the geometry changed, and those are the totally symmetric ones — so a photoelectron band is a filter that selects a few of a molecule’s vibrations and reports their frequencies in the ion. Which modes appear is a symmetry statement that follows from the coordinates, and how strongly each appears is a displacement along that mode. Together they would turn a photoelectron band into a partial structure determination of an ion that lasts microseconds and cannot be crystallised.