The mode that moves least radiates most

Two questions about an infrared spectrum can be separated, and the division between them is not where a reader expects. Which bands exist is settled by the point group, exactly, with no charges and no force field; how strong each band is needs a model and is where that model is worst. The ordering that leaves is exact, good, poor — existence, position, strength.

There is also a loose observation. Boron trifluoride’s strongest band by that model is the out-of-plane bend, which is also the mode in which the heavy atoms barely move. This essay takes that observation and asks what a band’s strength is a strength of, since it is plainly not a strength of the motion.

Three quantities that get confused with each other #

For every mode there are at least three numbers that could be meant by “how big it is”.

The band strength is the square of the dipole derivative — how much the molecule’s dipole changes per unit of normal coordinate — and it is what the area under a band measures.

The amplitude is how far the atoms actually go: the root-mean-square Cartesian displacement in the zero point of that mode, which is a length in ångström and is the thing a picture of a vibration draws.

The reduced mass is what is moving: one over the sum of the squared displacements, which is the effective mass of the coordinate.

A fourth quantity that is the same for every mode #

There is a fourth number that sounds as though it ought to distinguish the modes and does not, and noticing that it does not is worth a paragraph.

A normal coordinate is normalised so that $\sum_i m_i |\mathbf{d}_i|^2 = 1$, where the sum runs over atoms. That is not a convention that could have been otherwise: it is what makes the mass-weighted Hessian’s eigenvectors orthonormal, which is what makes them normal modes — the property that makes a mode a mode rather than a description of one. So every mode of every molecule moves exactly the same weighted amount of mass, to the precision of the eigensolver, and the phrase “this mode moves the heavy atoms” cannot mean that quantity.

What it can mean is the reduced mass, which is the same sum without the weighting inverted, and that does vary — from 1.03 for one of ammonia’s bends to 20.3 for sulfur dioxide’s antisymmetric stretch, a factor of twenty across the molecules here. It is the same quantity that makes an isotope shift arithmetic, which is the one place in this collection where it does the work it is usually credited with.

Five molecules, and two of them run backwards #

The single case is suggestive and is not evidence. Across five molecules the rank correlation between band strength and amplitude is +1.00 for methane, +0.50 for water, +0.20 for boron trifluoride, −0.50 for sulfur dioxide and −0.66 for ammonia.

Methane’s +1.00 is worth discounting rather than counting. Its nine modes fall into two active sets, t₂ at 1363 and t₂ at 3173, so the correlation is a comparison of two numbers and could only have been ±1. It is the control that shows the measure works rather than a result.

Ammonia’s −0.66 is the sharpest of the real ones. Its umbrella mode at 1020 cm⁻¹ has by far the largest amplitude of any of its six and is the third strongest band; its degenerate stretch at 3567 has the smallest amplitude and is the strongest.

Why they need not agree #

The reason is not subtle once it is stated, and it is worth stating because the received impression survives it.

An amplitude is set by the frequency and by nothing else: the zero-point spread of a normal coordinate is $\sqrt{\hbar/2\omega}$, so a high-frequency mode is a small one, always. That relation is what a zero-point spread is, and it contains nothing but the frequency. That is why every stretch in the table above has a smaller amplitude than every bend.

A band strength is set by how much the dipole changes per unit of that coordinate, which is a question about where the charge is and which way it moves — and it has no frequency in it at all. The two quantities are therefore independent by construction, and a molecule in which they happened to correlate would be a molecule in which the high-frequency modes also happened to be the ones that move the charge most.

That independence is the same one a force constant has from a bond strength, one step further along: there, a frequency was shown not to measure a bond because it carries a mass; here, an intensity is shown not to measure a motion because it carries a charge distribution. Both are cases of a spectrum’s two axes reporting on different halves of a molecule, and the same shape turns up in what a dipole predicts about a spectrum and in what a mode’s composition says about its coordinates.

The reduced mass ranks it backwards #

The reduced mass is the more interesting of the two comparisons, because for one molecule it is not merely uncorrelated with the intensity but perfectly anti-correlated.

Boron trifluoride’s rank correlation between band strength and reduced mass is exactly −1.00. Its three modes with light reduced masses — 11.8 and 12.3, the ones in which the boron moves — are the three strongest bands; the pair at 17.9, in which the fluorines move, are the weakest. So on this molecule “how much mass moves” predicts the intensity exactly, and with the wrong sign.

That is a coincidence of one molecule and it is a useful one, because it makes the general point unmissable. The quantity is not a bad predictor with noise in it. It is a good predictor of the wrong thing.

The same three quantities on a molecule where they agree #

Water is the useful contrast, because on water the disagreement is mild and it is mild for a reason.

Its three modes are a bend at 1649, a symmetric stretch at 3833 and an antisymmetric stretch at 3943. The amplitudes fall in that order, as they must, since the amplitudes are fixed by the frequencies. The intensities are 1.000, 0.525 and 0.995, so the strongest band is the one with the largest amplitude and the second strongest is the one with the smallest — a correlation of +0.50, which is neither agreement nor disagreement.

The middle mode is where the whole argument sits. Water’s symmetric stretch moves the two hydrogens outward together, and the two bond dipole changes have components that cancel along the molecular axis; the antisymmetric stretch moves one out and one in, and they add. Two modes at almost the same frequency, therefore almost the same amplitude, differ by a factor of two in strength for a reason that has nothing to do with either.

What actually decides it #

The dipole derivative in a fixed-charge model is $\sum_i q_i \mathbf{d}_i$: the charges are constants and the displacements are the mode. So the intensity is large when the charged atoms move and in the same direction, and small when they move oppositely or when the atoms that move are the ones carrying little charge.

Boron trifluoride is the case where those two criteria come apart most sharply. Its charges are large and its boron carries the largest of them; its fluorines carry three smaller ones of the opposite sign, arranged at 120°, so any motion of the fluorines alone that is not totally symmetric cancels a great deal against itself. Moving the boron alone cancels against nothing.

That is a statement about geometry rather than about mass — the same kind of cancellation that makes carbon dioxide’s symmetric stretch silent while its antisymmetric one is strong, and it is why the mass correlation runs backwards on this molecule: the atom that is light is also the one that is alone.

Where this leaves the ordering #

The ordering was exact, good, poor: existence from the group, position from a fitted force field, strength from a model of the charge. This essay adds a fourth item and it sits at the top.

The amplitude is exact, in the sense that it follows from the frequency by a relation with no model in it — given the frequency, the zero-point spread of that coordinate is fixed. So the quantity that looks like the most physical thing about a vibration, and the quantity a picture of a mode actually draws, is the one that carries the least information: it is the frequency, rewritten.

Which is why comparing it against the intensity is worth doing. The comparison is between a quantity that is a rewriting of the position and a quantity that is a property of the charge distribution, and their disagreement is not a failure of either. It is the statement that a spectrum’s two axes measure different things about a molecule, which sounds obvious written down and is not what a picture of a strong band suggests.

Drawn as a spectrum — band positions from the force field, heights from the charge model — the result is a picture in which the tallest line belongs to the mode with the least mass in motion. Nothing about that is visible in a table of frequencies, which is why the intensity has to be computed rather than inferred from the modes.

The column that is missing, which is the measured one #

Four quantities have been ranked against one another and all four come out of the same model. The fifth column — what a spectrometer actually reports — has not been added, and it is worth being explicit about what happens when it is, because the answer separates this essay’s argument from its example.

Boron trifluoride’s infrared spectrum is well measured. Its most intense band is the antisymmetric B–F stretch near 1,450 wavenumbers, and the out-of-plane bend near 720 is substantially weaker. So the ordering the fixed-charge model produces for this molecule is not the measured ordering, and the headline case — the strongest band being the one where the heavy atoms barely move — is a property of the model rather than of the molecule.

The reason is the term named as the next step. A fixed-charge model has the dipole change only because the charges move; a real molecule also has the charges change as it distorts, and for a bond stretch that flux is large, because stretching a polar bond is exactly the motion that redistributes electron density along it. A bending mode moves charge around without stretching anything, so it gains much less from the flux. Adding the missing term therefore promotes the stretches relative to the bends, which is the direction the measurement requires.

What survives is the argument, and it survives untouched. The four rank correlations computed here are between four quantities all defined within one model, and their disagreement with each other is exact: the weighted motion really is identical for every mode by construction, boron trifluoride’s out-of-plane bend really does put 89.9 per cent of its motion on the lightest atom, and the mode that moves the most mass really is weaker in that model by a factor of 9.6. Intensity, amplitude, reduced mass and frequency are four different quantities, and nothing about which of them a better model would reorder makes them the same quantity.

What does not survive is using boron trifluoride as the demonstration that a real spectrum behaves this way. It demonstrates that a fixed-charge model behaves this way, which is a weaker claim and is the one the arithmetic supports.

That is worth stating plainly rather than leaving in a caveat, because the distinction is the whole point. A quantity computed from a stated model is a result about the model; it becomes a result about a molecule when a measurement agrees, and here one does not. The honest form of the finding is therefore about the independence of the four quantities rather than about which mode of which molecule comes top — and the independence is what the rank correlations running from $+1$ to $-0.66$ across five molecules actually establish.

What this cannot say #

The charges are a model. Every intensity here comes from a fixed-charge model with electronegativity-derived charges, and the omitted term — the charge that flows along the bonds as they stretch — is comparable in size to the one that is kept. That is the standing caution about infrared intensities and it applies unchanged.

The rank correlations are over small sets. Three modes for water and sulfur dioxide, five and six for the others. A rank correlation over three items takes one of a few values, and the numbers should be read as the direction of a comparison rather than as a statistic.

Degenerate modes are counted once each. Ammonia’s two e pairs contribute two entries apiece with identical values, which ties two ranks and is handled by averaging them. Removing the duplicates changes the correlations by a little and none of the signs.

Degenerate partners are not independent measurements. Boron trifluoride’s two e′ pairs and ammonia’s two e pairs each contribute two identical rows, so the effective number of distinct comparisons is smaller than the number of modes, and the correlations are computed over the modes rather than over the distinct frequencies. Doing it the other way changes the numbers and none of the signs.

And a zero-point amplitude is not what a spectrometer sees. The amplitude quoted is the ground state’s spread. What an absorption measures is a transition between two states, and its strength involves the derivative and not the spread — which is exactly the distinction this essay is about, arriving one more time. The same separation between a quantity and its derivative is what makes a force constant not a bond strength.

There is a practical consequence worth stating, because it is the form in which the mistake usually gets made. A picture of a molecule’s vibrations, with the strongest bands drawn as the largest arrows, is a picture of two unrelated quantities superimposed. Nobody draws it that way deliberately; it happens because a figure of a mode has to choose an arrow length and the obvious thing to scale by is how important the band is. On boron trifluoride that produces a drawing in which the fluorines swing furthest in the mode where they barely move.

What was checked #

The control, which is that intensity ranks with itself perfectly, on every molecule. Without it a measure returning small numbers for everything would pass the main claim.

That the weighted motion is one for every mode, to $10^{-9}$, so that the comparison is against the amplitude and the reduced mass rather than against a quantity that cannot vary.

That at least one molecule has a correlation below one half, and at least one has a negative correlation.

And the case first noticed, as a number rather than an impression: boron trifluoride’s strongest band has 89.9 per cent of its motion on the lightest atom, and the mode that moves the most mass is 9.6 times weaker.

Still open: the charge flux, and a series with different central atoms #

The natural open question is the quantity that is missing from all of this, which is the charge flux — the derivative of the electron density with respect to a nuclear displacement. It is a standard output of a real calculation and it is the term that would turn the qualitative statement above into a quantitative one, since it can be comparable to the fixed-charge term and can have the opposite sign.

The nearer question is one the numbers here raise and do not settle. Every molecule whose correlation runs backwards has its charge concentrated on the central atom, and every molecule whose correlation is positive does not. That is four data points and a plausible mechanism, which is not enough for a claim — but it is a testable one, because the charges are computed here rather than assumed, and a series of molecules with the same shape and different central atoms would separate a fact about geometry from a fact about polarity. The charges themselves come from a scale that four sources disagree about, so the series would need to be run on all four before anything was claimed.