A band is a filter on the modes

The vibrational structure of a photoelectron band is simplest when a molecule has one vibration to structure it: nitrogen’s three bands, their intensities coming out at 0.917, 0.263 and 0.880 in their first lines, from bond-length changes of +18.7, +77.2 and −23.7 thousandths of an ångström. A diatomic has one coordinate, so a band is a progression in it and the only question is how long.

It ended by naming what the polyatomic case adds, and the interesting part is not the extra coordinates but which of them 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.

Computing the filter shows it to be narrower than the phrase suggests.

The projection, which is one line #

Ionising a molecule changes its equilibrium geometry, and the vibrational structure of the band is that change projected onto the neutral molecule’s normal modes. The projection is exact and needs no fitting: the modes are mass-weighted orthonormal, so the amount of mode $k$ in a Cartesian displacement $\mathbf{d}$ is

$$q_k = \sum_i m_i \,\mathbf{d}_i \cdot \mathbf{s}_{ki}.$$

Divide by that mode’s own zero-point spread and the dimensionless displacement follows; half its square is the Huang–Rhys factor, which is the mean number of quanta the band puts into that mode and the parameter of the Poisson progression it shows.

The zeros are not small numbers. Methane’s eight non-symmetric modes come back between $1.5 \times 10^{-35}$ and $7.5 \times 10^{-29}$; boron trifluoride’s out-of-plane mode comes back at exactly zero, because no change in any bond length displaces a planar molecule out of its plane at all.

Why they are zero #

A symmetric change of geometry is a displacement that the group leaves alone. A non-symmetric mode is a displacement that the group does not. The overlap between them is an integral of something odd about an operation the molecule has, and it vanishes for the same reason every other selection rule in this collection does — one theorem, applied to a projection rather than to a transition.

So the filter is set before any calculation. How many progressions a band can show is the number of totally symmetric vibrations the molecule has, which is read off the character table.

Methane has one, so one progression. Boron trifluoride has one. Water and ammonia have two each, sulfur dioxide two. Benzene has two of its thirty vibrations totally symmetric, so a band that could in principle report thirty frequencies reports two.

The control #

Zeros are only evidence if something could have been non-zero, and a projection that returned zero for everything would pass the first half of the claim on its own. So the whole calculation is run again with a change of geometry that breaks the symmetry — one bond stretched rather than all of them.

Every molecule’s non-symmetric modes wake up. Water’s antisymmetric stretch goes from $7 \times 10^{-30}$ to 0.137; methane’s t₂ stretches go from $10^{-29}$ to 0.109; boron trifluoride’s e′ pair from $10^{-19}$ to 0.150. So the modes are perfectly capable of carrying intensity, and the symmetry is what stops them.

The one that stays at exactly zero under both is boron trifluoride’s out-of-plane bend, and its reason is different and worth separating: no change in any bond length has a component out of the plane, whatever its symmetry. That mode is invisible to this kind of geometry change rather than to this symmetry of it, and it would appear if the ion were pyramidal.

What a band therefore reports #

The practical form of the result is worth stating, because it inverts what a spectrum looks like it is offering.

A photoelectron band that shows vibrational structure is showing the frequencies of the totally symmetric vibrations of an ion that exists for microseconds and cannot be isolated. That is a genuinely remarkable measurement — a partial structure determination of a species nobody can put in a bottle. The same point is sharper still for a diatomic, because there the one frequency reported is the only frequency the ion has: nitrogen’s three photoelectron bands give the three ion states’ vibrational constants outright, and the bond lengths follow. A polyatomic gives a fraction of them, and the fraction is the group’s business rather than the experiment’s.

And it is partial in a way the spectrum does not advertise. The frequencies it does not report are not weak; they are absent, by a symmetry argument, and no improvement in resolution brings them out. A reader looking at a clean progression in one frequency is looking at a molecule that may have eight other vibrations, all of them silent for a reason that has nothing to do with how strongly they are excited.

The comparison with an infrared spectrum is instructive because the filters are different. Infrared activity needs a changing dipole, which for methane admits the t₂ modes and excludes the a₁ one — exactly the reverse of the filter here. So the two spectroscopies of one molecule report disjoint sets of frequencies, and neither of them reports the e modes at all.

That last point is the useful qualification. Symmetry says which modes can appear; the geometry change says which of those do. Water’s bend is allowed and does not appear, because the particular change of shape tested here has almost no component along it. A real ionisation whose ion is more bent would light it up.

The intensities within a progression #

Each active mode carries a Poisson progression: the intensity of the line with $v$ quanta is $e^{-S}S^v/v!$, so the brightest line is at $v \approx S$ and the number of resolvable lines goes as $\sqrt{S}$.

That gives the shapes their names. Methane’s $S = 0.905$ puts most of the intensity in the first two lines, which is a band with a shoulder. Boron trifluoride’s $S = 3.75$ puts the brightest line fourth, which is a long progression with a weak origin. Sulfur dioxide has one of each, superimposed.

So the shape of a band and the number of progressions in it are two independent pieces of information, and the second is the group’s. A band with two obviously superimposed progressions is a molecule with at least two totally symmetric vibrations, which for a small molecule narrows the possible point groups considerably — the same kind of structural inference that mutual exclusion offers and does not quite deliver.

Which frequencies a band actually delivers #

It is worth setting out what a spectrum of each of these molecules would report, because the counting is the essay’s practical content.

Methane is the sharpest case. It has nine vibrations at four distinct frequencies, and a photoelectron band from a non-degenerate ionisation reports one of them. Three quarters of what a molecule can do vibrationally is unavailable to the measurement, and the unavailability is exact.

That is a different kind of limitation from a weak signal. A band whose lines are weak can be measured harder; a band whose lines are forbidden cannot, and the distinction is the same one that separates an absent band from an unobserved one elsewhere in this field.

Where the intensity comes from, and where it does not #

One thing the calculation makes precise and a picture does not. The Huang–Rhys factor is the square of a dimensionless displacement — the geometry change measured in units of that mode’s own zero-point spread — so a low-frequency mode, whose spread is large, needs a bigger change of geometry to reach the same factor.

Sulfur dioxide shows it. Its bend at 518 cm⁻¹ and its symmetric stretch at 1168 both respond to the same change of shape, and the stretch gets fourteen times the factor. Part of that is that the change was a bond lengthening and the stretch is a bond lengthening; part is that the higher-frequency mode has the smaller zero-point spread, so the same absolute displacement counts for more.

The same projection, run on nitrogen #

For nitrogen it is worth checking that the polyatomic projection reduces to the diatomic calculation, because the two look nothing alike.

A diatomic has one vibration and it is totally symmetric, so the filter passes everything and the projection is trivial: the whole geometry change is along the one mode. The Huang–Rhys factor then reduces to $\mu\omega\,\Delta r^2/2$, which is the diatomic closed form, and the Poisson progression it produces is the band.

That reduction is worth checking rather than assuming, because the two constructions share no code: the diatomic one solves two Morse potentials on a grid and sums a product over grid points, and this one projects a Cartesian displacement onto a set of eigenvectors. They agree because both are computing the same overlap between two harmonic ground states, one of them displaced.

So the polyatomic construction contains the diatomic one as the case where the group has nothing to say. What it adds is a projection that can return zero, and the reason a diatomic band is always a clean progression is that its projection cannot.

That is the general shape of what symmetry does to a spectrum, and it is worth naming: a group never adds anything to a spectrum. It removes, and the removals are exact where everything that is left is a model’s.

It is worth saying what the filter costs a molecule that has other spectroscopies. Methane’s infrared-active modes are its t₂ set, and a photoelectron band of the same molecule reports the a₁ mode and nothing else — so the two measurements name disjoint sets of frequencies, and between them they still leave the e modes unreported by either.

The exception, and what it reveals when it happens #

The filter passes only the totally symmetric modes, and the derivation says why: the geometry change on ionisation preserves the molecule’s symmetry, so only displacements that preserve it can be excited. That premise is worth stating separately, because when it fails the filter opens and the opening is informative.

The premise fails when the ion has a lower symmetry than the neutral. If removing an electron leaves a degenerate electronic state, the ion cannot keep the parent’s geometry — a degenerate state distorts — and the distortion is along a coordinate that is not totally symmetric in the parent’s group. That coordinate then appears in the band, with a progression of its own, in violation of the rule.

So an unexpected progression is not a failure of the analysis. It is evidence that the ion has distorted, and the mode it appears in names the distortion.

Methane is the standing example and it is one of the molecules computed here. Removing an electron from its threefold-degenerate level leaves a degenerate ion, which cannot stay tetrahedral; the cation distorts, and the first photoelectron band is correspondingly broad and complicated rather than the clean single progression the filter would predict for an undistorted ion.

That makes the rule and its exception a pair of diagnostics rather than a rule with a caveat.

A band showing one progression in a totally symmetric mode is a band whose ion kept the parent geometry, and the progression measures how much the bond lengths changed.

A band showing structure in other modes is a band whose ion distorted, and the modes appearing name the coordinate it distorted along — which is a symmetry statement about the ion obtained from the parent’s spectrum.

The second is the more valuable of the two, because the ion in question is usually a species nothing else can be measured on, and because the alternative route to the same conclusion is a calculation of a distorted open-shell ion, which is among the harder things to compute reliably.

There is a third case between the two, and it is the commonest of all in molecules with no symmetry. A filter defined by a point group has nothing to filter when the group is trivial: every mode of a molecule with no symmetry elements is totally symmetric, every one of them is permitted, and the band’s structure is a superposition of progressions in as many modes as the ionisation happens to displace.

That is why photoelectron bands of small symmetric molecules show clean progressions and those of ordinary organic molecules show broad structureless humps. The difference is not resolution or sample quality. A symmetric molecule has a filter and an unsymmetric one does not, so the first concentrates its intensity into a few lines and the second spreads it over a great many — and the same measurement is a structure determination in one case and an envelope in the other.

Which sets the scope of everything here. The filter is a gift of symmetry, it is what makes a vibrational progression readable, and it is available for exactly the molecules whose ions were interesting enough to study in the first place.

What this cannot say #

The ion’s geometry is put in. This calculation projects a stated change of shape onto the modes; it does not compute what removing an electron does to a molecule. That would need the ion’s potential energy surface, which is the standing gap in this whole field. The diatomic calculation has the same limitation and takes its bond-length changes from measured spectroscopic constants.

The modes are the neutral’s. A proper treatment uses the ion’s normal modes, and the two sets differ by more than a little when the geometry change is large — and by a rotation, in general, which is the Duschinsky effect and is not here.

Harmonic, and one dimension at a time. Each mode’s progression is an independent Poisson distribution, which is exact for a harmonic surface displaced along that mode and is not exact for anything else.

Koopmans’ approximation is nowhere in this. Which orbital an electron came from, and what its binding energy is, are questions this essay does not touch — and the approximation that connects them is a separate matter with its own failures. What is computed here is the structure within a band, given that the band exists.

And the ionisation is assumed non-degenerate. Removing an electron from a degenerate orbital gives a degenerate ion state, which distorts — the whole of the first-order Jahn–Teller argument — and then non-symmetric modes appear in the band for exactly the reason this essay says they cannot. Methane’s own first photoelectron band is such a case, and it is not the case computed here.

What was checked #

Every totally symmetric mode active and every other mode below the tolerance, for five molecules and twenty-nine modes.

The tolerance taken from the coordinates rather than from the arithmetic, because a stored geometry rounded in the fifth decimal cannot support a claim in the twelfth — ammonia’s is, and it is the one molecule whose zeros are at $10^{-9}$ rather than $10^{-30}$.

The control on every molecule: an unsymmetric change of the same size activating at least one mode the symmetric change could not.

And the count. The number of progressions found equals the number of totally symmetric vibrations the character table gives, molecule by molecule, which is the check that ties the computed projection to the group rather than merely observing that both are small.

Still open: the degenerate ionisation #

The natural open question is the degenerate ionisation, which is the case excluded here and the case most first bands actually are. A degenerate ion state distorts, the distortion is along a non-symmetric coordinate, and the band then shows a progression in a mode that a symmetric analysis forbids — so the presence of a forbidden progression is a diagnostic for a Jahn–Teller ion, and its length measures the distortion. Both halves of that computation are already here.

The nearer question is what the band’s silence costs. A photoelectron band reports two frequencies of an ion with thirty vibrations, and the ion’s geometry has as many parameters as the neutral’s. So the structure determination the band offers is underdetermined in exactly the way three rotational constants underdetermine a structure — and by much more, since two numbers is fewer than three. How large the family of ion geometries consistent with a measured band is, is a question the projection could answer by running backwards. It would belong to a recurring thread: a measurement that leaves a family standing rather than a single answer.