How much of a band is a bond stretch

A vibrational assignment is a table of bands with a description beside each one, and the description is usually a percentage: 1454 cm⁻¹, 87 per cent B–F stretch, 13 per cent F–B–F bend. The numbers look like measurements of the mode. They are not.

They are the result of sharing one mode out among a set of coordinates somebody chose, by a rule somebody chose, with the angles converted into lengths by a factor somebody chose. Change any of the three and the number changes.

The three choices #

What is being shared out. Displace the molecule along a normal mode and every internal coordinate changes. Squaring those changes and normalising shares out what moves. Weighting by the force constants instead shares out where the potential energy is. Those are different questions, and a mode answers them differently whenever a coordinate moves a lot in a soft direction or a little in a stiff one.

Whether the couplings are counted. The potential energy of a distorted molecule is $\sum_{ij} F_{ij}\Delta R_i \Delta R_j$, and the off-diagonal terms belong to no single coordinate. The standard potential-energy distribution gives each coordinate its own diagonal term plus its half of every cross term; the sum is then exactly the energy, and individual entries can fall outside nought and one. Dropping the cross terms and renormalising keeps every entry a genuine fraction and no longer adds up to the energy.

How an angle is made comparable with a length. A stretch is measured in ångström and a bend in radians, and the two cannot be squared and added until one is converted. Wilson’s convention multiplies an angle by the geometric mean of the two bond lengths it lies between; a bare radian is also in use, and so is a fixed one ångström. For boron trifluoride the difference between the first two moves the stretch content from 36.7 to 49.9 per cent.

None of these is a mistake. Each is a reasonable answer to how shall this be shared out? — and the trouble is only that the answer is almost never stated beside the number it produced.

Where they agree #

The conventions are not simply noise, and the case where they agree is the case that shows what they are for.

A mode that is one coordinate is one coordinate under every convention. Boron trifluoride’s 888 cm⁻¹ band comes out at 100.0 per cent stretch four times over; water’s two O–H stretches likewise; carbon dioxide’s 2396 cm⁻¹ band likewise. Nothing is at stake for a localised mode, which is why the numbers in an assignment table so often look uncontroversial.

The disagreement is entirely in the mixed modes. Which is exactly where the percentage is doing work — where somebody is deciding whether a band can be called a C=O stretch, whether a group frequency has survived into a new molecule, whether an isotope shift is where it should be.

Why the two disagree so violently on that one mode #

The mechanism is worth having, because it is not a numerical accident and it says which convention to prefer for which purpose.

The 1454 cm⁻¹ mode moves the angles a great deal and the bonds rather little. But the bonds are stiff — boron–fluorine is one of the strongest single bonds there is — and the angles are soft. So a small stretch stores a lot of energy and a large bend stores little, and the two ways of counting come out on opposite sides.

Both statements are true of the mode. It is mostly a bending motion and mostly a stretching energy. Which of those a caption should report depends on what the caption is for: a description of the atomic motion wants the first, and an analysis of where the restoring force comes from wants the second.

What a caption may not do is print one of them as though the other did not exist.

The number that is outside nought and one #

The potential-energy distribution has a known embarrassment and it is worth demonstrating rather than describing.

Because the cross terms are shared out, an entry can come out negative — a coordinate with a negative percentage of the energy — or above one hundred. On the small molecules here the excursion is small and it is real: water’s bend at 1649 cm⁻¹ comes out at 100.1 per cent of the energy, with each of the two O–H stretches at −0.03 per cent.

A negative share is not a fault in the arithmetic. It is the arithmetic saying that the coordinate’s own motion and its coupling to another have opposite signs, and that no division of the energy into per-coordinate parts exists. On a larger molecule with strong couplings the numbers get much worse, and published tables occasionally carry entries like −15 per cent with no comment at all.

The convention chosen without saying so #

Mode compositions are routinely quoted under one of the four conventions above without saying which. A common choice is squared displacements with Wilson’s angle scaling — the first column — with the reassurance that the force-constant-weighted version “agrees wherever the force field is nearly diagonal and the difference is not what any conclusion turns on”.

The first half of that is right. The second half is now measured, and for boron trifluoride it is wrong: the difference is sixty-one percentage points on the mode that matters, on an ordinary molecule.

The repair is not to switch conventions. It is that the earlier statement was a claim about a quantity nobody had computed, made in the direction that required no work — which is the shape of error this site’s whole apparatus exists to catch, met here in its own library rather than in a textbook.

What the earlier essays actually said survives intact, because none of them turned on a mixed mode’s percentage. Normal modes are not bond stretches argues that a mode is not a bond stretch at all when the shares are near a half, and near a half is where every convention agrees that no coordinate owns the mode. When a mode becomes a bond stretch reports HOD at 99.5 and 99.7 per cent localised, which is the regime where all four conventions give the same answer to a tenth of a per cent.

So the claims stand and the number they were made with now has a stated provenance, which it did not have before.

Where the conventions came from #

The potential-energy distribution is Morino and Kuchitsu’s, from 1952, and it was invented for exactly the purpose it is still used for: to decide which of several molecules’ bands correspond to one another when the modes are mixed. Wilson’s angle scaling is older and comes out of the FG matrix formalism, where the internal coordinates have to be given commensurate units before a matrix can be written down at all.

Both were designed as working tools for people who understood what was in them, and both were adopted as reported quantities by people who did not. That transition is visible in the literature: papers of the 1950s state the definition beside the table, papers of the 1990s state the number alone.

The situation is the same one electronegativity is in, and the same one a resonance energy is in. A quantity defined by a construction gets quoted as though it were measured; the construction is dropped somewhere along the way; and the number survives, comparable with nothing.

The remedy in every case is the same and it is small: say which construction produced the number. For a mode composition that is one clause — by the potential-energy distribution, or by squared displacements with Wilson scaling — and it costs a line in a caption.

Across five molecules #

The size of the disagreement varies, and it varies in a way that says what to be suspicious of.

Water and methane are nearly safe — their modes are either clean stretches or clean bends, so almost any convention gives almost the same table. Ammonia’s umbrella mode at 1020 cm⁻¹ runs from 9.1 to 42.2 per cent stretch. Sulfur dioxide’s 1168 runs from 72.5 to 98.5. Boron trifluoride’s 1454 runs from 36.7 to 97.9.

The pattern is that the disagreement is largest where a soft coordinate and a stiff coordinate move together, and that is the same condition under which a group frequency stops being transferable — which the transferability of group frequencies shows from the frequencies alone. The two diagnostics agree, and they are computed from different things: one from how the frequency moves when a substituent changes, the other from how the mode divides among coordinates at one geometry.

What a mixed mode does to a group frequency #

There is a practical consequence and it is the reason vibrational spectroscopists care about these numbers at all.

A group frequency is the claim that a particular band belongs to a particular bond and appears at nearly the same wavenumber in every molecule containing it. The claim holds when the mode is localised on that bond and fails when the mode mixes — and the percentage is how the mixing is reported.

So a table that says 87 per cent C=O stretch is being read as this band is safely a carbonyl band. Under a different convention the same mode might be 55 per cent, and the reader’s confidence in the assignment would be different. The band has not moved; the sentence about it has.

It is worth being clear about what is not in dispute. The spectrum itself — every band at its computed position — is untouched by anything in this essay. Nothing here moves a line. What is in dispute is the words written under them, and only for the pair at 1454.

The honest way to use the numbers, and the one the literature settled on where it was careful, is comparative rather than absolute: compute the composition the same way for a whole series and watch it change. A share falling from 95 to 60 per cent across a substituent series says something about the series whatever convention produced both numbers, because the convention is held fixed. A single number quoted alone says almost nothing.

What was computed, and what is fitted #

The eigenvectors are the same in every column: one Hessian, one diagonalisation, one set of modes. Only the sharing-out differs.

The force constants are fitted to the observed frequencies, which is the standing caution of this whole field and matters twice over here. The energy-weighted conventions use those constants directly, so their percentages inherit the fit; the displacement-weighted ones use only the eigenvectors, which the fit also produced but less directly. So the two families are not equally exposed to the same uncertainty, and a comparison between them is a comparison between one quantity that depends on the fit strongly and another that depends on it weakly.

Three things are checked in the computation.

Every convention’s shares sum to one, to 10⁻⁹, for every mode of every molecule here. A convention whose shares did not would not be a sharing-out at all.

A mode every convention calls localised is called the same coordinate by all of them. That is the agreement case made mechanical, and it is what stops the essay’s claim from being the numbers are arbitrary.

Some entry of the potential-energy distribution lies outside nought and one somewhere in these molecules, reported with its size. A check that a known embarrassment never happens would be worth nothing; this one requires it to happen and prints how big it is.

Nor is the species assignment in dispute. Which species each mode falls into is fixed by the group, with no force constant anywhere in the computation, before any of this essay’s conventions are chosen — and none of them can move a mode from one species to another. What a convention decides is how a mode inside a species is described, which is a weaker thing to be deciding and is exactly why the disagreement survives.

What none of this touches #

The frequencies. Every convention above shares out the same modes at the same wavenumbers. Nothing here changes a computed or measured frequency, and a spectrum is unaffected.

The symmetry species. Which species a mode belongs to is a group-theoretic fact and is computed from the coordinates with no force constant in it. The percentages cannot move it.

The isotope shifts. They follow from the masses and the force field, and are arithmetic. A convention for describing a mode does not enter.

So what is at stake is exactly one thing: the words and numbers written beside a band in a table. That is a smaller claim than it sounds, because those words are how a spectrum is turned into a statement about a molecule, and they are copied forward into every later paper that cites the assignment.

Summing to one is free #

All four are defensible, all four sum to one is offered as evidence that none can be dismissed, and the second half of that sentence deserves to be taken away from all four of them — because summing to one is a property of the constructions rather than a property of the molecule.

Each of the four is built to be a distribution: the shares are defined so that their total is the mode’s, and the total is then divided out. Nothing about the molecule enforces it, no measurement checks it, and a fifth convention invented this afternoon would sum to one as well, because that is what normalising means.

So the agreement everybody notices first is the one piece of information the four do not carry. What they disagree about — 36.7 against 97.9 per cent on one band — is where all the content is, and the disagreement is not a small tail on a shared answer but the whole range from a minor contribution to essentially the entire mode.

There is a further awkwardness that the normalisation conceals and that is worth stating, because it is the reason the conventions differ at all. The internal coordinates of a molecule are not orthogonal: a bond stretch and an adjacent angle bend move some of the same atoms, so a mode’s energy contains cross terms between them, and those cross terms have to be assigned to somebody. The four conventions are four ways of assigning them — split evenly, folded into the diagonal, discarded, or kept — and a cross term can be negative, which is how a single coordinate’s share can exceed the whole mode or fall below zero before the normalisation tidies it away.

Two practical consequences follow.

Quote the convention. A potential energy distribution without the rule that produced it is a number a reader cannot reproduce, and the spread here is sixty-one points.

And read the label rather than the percentage. Which coordinate contributes most is stable across the four conventions far more often than how much it contributes, so predominantly a B–F stretch survives where 36.7 per cent B–F stretch does not. That is the same retreat that works elsewhere — from a magnitude to an ordering — and it is available here for the same reason: the ordering does not depend on where the cross terms were sent.

There is one case where even the label fails, and it is worth knowing how to recognise. When two coordinates are strongly coupled and close in frequency, the cross term between them is comparable to either diagonal term, and which of the two comes out largest can change between conventions. A band whose assignment flips between two coordinates as the convention is changed is a band with no dominant coordinate at all — and reporting it as either is reporting a choice. The honest description there is the one the mode actually has: a mixture, named by both coordinates, with no percentage attached.

Still open: redundant coordinates, and what is free of convention #

The obvious open question is the redundant coordinate set. A molecule with a ring has more internal coordinates than it has degrees of freedom, so the coordinates are not independent and the sharing-out has a null space in it — a set of shares can be changed by adding something that describes no motion at all. Benzene’s assignment tables are built on such a set, and the standard repair is a projection whose choice is another convention.

The nearer question is whether any convention-free statement is available. There is one: the eigenvector itself, in Cartesian displacements, which is what the picture above draws and what nothing about this essay disturbs. Every percentage in every assignment table is a lossy summary of that object, and the argument here is not that the summaries are useless but that a summary is not the thing summarised.