The frequency is not the bond strength

“The C=O stretch is at 1,700, so it is a strong bond.” The inference is made constantly, it is usually harmless, and it is wrong in two separate ways at once — one about mass and one about what a mode is.

A frequency is a ratio. The harmonic oscillator gives $\tilde\nu \propto \sqrt{k/\mu}$, and only the numerator is a property of the bonding. Change the denominator and the frequency moves with no bond having changed at all, which is exactly what the isotope shift is arithmetic demonstrates by changing nothing but the masses. So a frequency is a measurement of two things and a spectrum reports the product.

Ordering by force constant is not ordering by frequency #

The six fitted fields give bond stretching constants running from 5.47 mdyn per ångström for C–H in methane to 16.0 for C=O in carbon dioxide. Their stretching frequencies run from 1,168 wavenumbers for sulfur dioxide to 3,943 for water. Set the two lists beside each other and they disagree in an obvious place.

Sulfur dioxide: k = 10.15, bands at 1,168 and 1,382. Water: k = 8.46, bands at 3,833 and 3,943. The S–O bond has the larger force constant by a fifth and the smaller frequency by a factor of three. Nothing subtle is happening: oxygen is sixteen times heavier than hydrogen, the reduced mass for S–O is 10.66 against 0.948 for O–H, and the square root of eleven is 3.35.

This is the whole of the first error, and it is worth being blunt about how large it is. The mass factor between an X–H bond and a bond between two heavy atoms is a factor of three in frequency, which swamps every difference in force constant that ordinary chemistry produces. Comparing an X–H frequency with a heavy-atom frequency says almost nothing about the bonds and almost everything about hydrogen.

Where the rule does work is within a series. Comparing C–H at 5.47 with N–H at 6.21 and O–H at 8.46 — all bonds to hydrogen, all reduced masses within 2 per cent of one another — the frequencies run 3,026, 3,490 and 3,833 in the same order. The mass is being held constant, so the frequency is reporting the force constant, and that is when the inference is sound.

Carbon dioxide, where the same bond gives two frequencies #

The second error survives even when the masses are held fixed, and carbon dioxide is the cleanest case of it.

A thousand wavenumbers separate two modes of the same molecule, drawn from the same bonds, with the same force constant. Neither frequency is the frequency of a C=O bond, because neither mode is a motion of a C=O bond: as normal modes are not bond stretches computes, both are exactly half in each of the two bonds.

What separates the two stretches is two things and neither is a force constant of a bond. The first is kinematics: in the symmetric stretch the carbon stays still and the two oxygens move outward together, while in the antisymmetric stretch the carbon moves against both, so the effective mass differs. The second is the interaction constant — 1.25 mdyn per ångström in the fitted field, coupling one bond’s stretch to the other’s — which adds to the restoring force in one combination and subtracts in the other.

Put a number on the naive estimate. The diatomic formula with k = 16.015 and μ = 6.856 gives 1,991 wavenumbers. The molecule’s actual stretches are at 1,354 and 2,396: one is 32 per cent below the estimate and the other 20 per cent above. The estimate is not a bad approximation to either frequency; it is an approximation to something the molecule does not have.

Where the diatomic formula is worth using #

It is worth stating clearly when the formula does earn its keep, since this essay is not an argument against ever using it.

For a genuine diatomic it is exact within the harmonic model, and the force constant read off is the force constant. For a bond to a heavy fragment where the light atom does almost all the moving — the isolated O–H of HOD, a C–H in a large molecule, a metal–hydride stretch — it is accurate to a few per cent, and the reason is the composition: those modes really are one coordinate, so the two-body reduced mass really is the mass involved.

The failure cases are exactly the cases where the mode is not one coordinate, and there is a test for that which does not require chemical judgement. Compute the mode’s composition; if one internal coordinate holds nine tenths of the motion, treat it as an oscillator and the formula will work. If the largest share is a half, as it is for every stretching mode of water, carbon dioxide and methane, the formula is describing a fiction. Group frequencies, and where they stop takes that criterion and applies it across a set of molecules.

Sulfur dioxide has the second-largest bond force constant of the six and its stretching bands sit below 1,400 wavenumbers — lower than every X–H bend here. The mass in motion is what puts it there, and no ordering by frequency can recover the ordering by constant when the masses differ by a factor of thirty.

The three quantities a band position confuses #

Pulling the argument apart gives three distinct things, all of which get called “bond strength” at some point, and none of which is any of the others.

The force constant is the curvature of the electronic energy at the bottom of the well. It has units of force per length, it is what a vibrational spectrum is sensitive to, and it says how hard the bond resists a small displacement.

The dissociation energy is the depth of the well. It has units of energy, it is what a thermochemical measurement gives, and it says how much work it takes to separate the atoms completely.

The bond order is a count, or a matrix element standing in for one. Bond order from the eigenvectors computes it for conjugated systems directly from the coefficients, and it is a property of a description rather than a measurement at all.

The three correlate within a family and diverge between families, which is the worst possible behaviour for a quantity people want to compare across a table. A steep well can be shallow — a bond that resists small displacements strongly and comes apart at moderate ones — and a deep well can be flat near its minimum. Nothing forbids either.

The comparison that does work #

Having ruled out the loose inference, it is worth stating carefully the tight one, because it is used constantly and correctly.

Hold the masses and the mode composition fixed, and a frequency shift is a force-constant shift. Comparing the C=O band of one ketone with the C=O band of another; comparing an O–H stretch in a free molecule with the same stretch in a hydrogen-bonded one; comparing a metal–carbonyl stretch across a series of complexes. In every case the two modes have nearly the same composition and exactly the same masses, so the only thing that can have moved is the curvature.

The size of the effect is worth knowing. A shift of 30 wavenumbers on a band at 1,700 is 1.8 per cent in frequency and therefore about 3.6 per cent in force constant, which is a small change in bonding and a large, easily measured change in a spectrum. Vibrational spectroscopy is an exquisitely sensitive detector of small changes and a poor absolute measure, and both halves of that follow from the same square root.

The failure mode to watch for is a comparison in which the composition quietly changes. Substituting a heavy atom next to the bond of interest changes the mode’s composition as well as its masses, and a shift then reports both at once with no way of separating them. That is exactly the situation group frequencies, and where they stop computes a criterion for.

What the force constant does measure #

Having taken the frequency away as a measure of bond strength, it is fair to ask what the force constant is a measure of, and the answer is narrower than it sounds.

A force constant is the curvature of the energy at the bottom of the well. A dissociation energy is the depth of the well. Those are different quantities and they are not related by any general rule: a shallow well can be steep near its minimum and a deep one can be flat. For bonds between the same pair of elements they correlate well, which is why the confusion survives, and across different elements they correlate poorly.

A double bond is not two single bonds makes the neighbouring point about dissociation energies: carbon’s single bond is 348 kJ per mole and its double is 614, which is not twice anything, because the two halves are different integrals over different orbitals. Force constants show the same non-additivity for the same reason, and neither quantity is a count of bonds.

The most useful thing a force constant does is compare the same bond in different environments. The C=O constant in carbon dioxide against the C=O constant in a ketone, the C–H constant across a series of substituted methanes: those comparisons hold the masses fixed and the mode composition roughly fixed, and the differences that survive are differences in the bonding. That is a real technique and it is the honest form of the inference this essay is refusing.

The mass effect, isolated #

The cleanest demonstration that the mass half is real is to change only the mass, which isotopic substitution does exactly.

Substituting the oxygens of sulfur dioxide moves every frequency and no force constant, which is the cleanest demonstration available that the two are independent quantities. The shift is arithmetic in the masses and the constants are what the arithmetic is applied to.

The shifts here are small because the mass change is small: oxygen-18 is 12 per cent heavier than oxygen-16, against deuterium’s 100 per cent over hydrogen. They are also entirely a mass effect, with no accompanying change in anything else, which is what makes isotopic substitution the standard tool for confirming a vibrational assignment.

The one number a spectrum gives directly #

There is a bond property that a vibrational spectrum does determine well, and it is not the strength: it is the symmetry of the environment. How many bands there are, which species they belong to and which are active in what are all fixed by the shape, exactly, before any force constant is chosen. That is what selection rules are one theorem establishes and what two structures, two spectra uses to settle a geometry.

So the reliable inference runs the other way round from the usual one. A spectrum is excellent evidence about shape and poor evidence about strength, and the reason is the same in both cases: the positions depend on a force field that a spectrum cannot determine, while the count and the activity depend on a group that a spectrum determines outright.

What a spectrum is good evidence about #

Having taken a great deal away, it is worth ending on what a vibrational spectrum determines well, because the list is short and sharp.

Shape. How many bands there can be, in what species, active in which experiment, is fixed exactly by the point group — before any force constant is chosen. Two structures, two spectra settles a geometry that way, by counting.

Change. A shift in a band whose composition has not changed is a shift in a force constant, and the sensitivity is excellent: 30 wavenumbers on a band at 1,700 is a 3.6 per cent change in curvature, easily measured.

Identity. The pattern of a spectrum is characteristic of a molecule in the way a fingerprint is characteristic of a person, and for the same reason — it is a property of the whole rather than a list of parts.

What it is poor evidence about is any absolute statement about a bond, and the reason has been the whole of this essay: the observable is √(k/μ) with a composition in front of it, and a single number cannot report three.

How many vibrations each molecule has and how many species they fall into is a count the group settles, and it is the count the fitting problem is against. It matters here only as a reminder that the constants above are fitted quantities rather than measured ones.

A last way of putting it, since the confusion is so persistent. A frequency is a rate; a force constant is a stiffness; a dissociation energy is a depth. The first depends on the second and on a mass, and the second and third are related by nothing general at all.

The mass in the formula is not a mass of two atoms #

The usual repair for the essay’s complaint is to divide out the mass — take the frequency, multiply by the reduced mass of the two bonded atoms, and call the result a force constant. That repair does not work, and the reason is worth stating because it is the same reason the frequencies were delocalised in the first place.

A normal mode’s frequency involves the mode’s own reduced mass, which is a property of the whole pattern of motion rather than of any pair of atoms. Water’s antisymmetric stretch moves both hydrogens and the oxygen; its effective mass is a combination of all three, weighted by how far each moves, and it is not the reduced mass of an O–H pair. For a mode with contributions from several coordinates there is no pair to take a reduced mass of at all.

So converting a frequency to a force constant requires knowing the mode’s composition — which requires the force field — which is what the conversion was supposed to supply. The inference runs in a circle, and the middle step is the underdetermined one a fitted force field leaves open.

That said, the practice is not worthless, and the condition under which it works is stateable. A comparison is safe when the effective mass is the same across the series being compared.

A carbonyl stretch is the standard case. In ketone after ketone the mode is dominated by the same two atoms moving against each other, with much the same participation from the rest of the molecule, so the effective mass barely changes from one compound to the next — and the frequency shifts that remain are shifts in the force constant. That is why a carbonyl frequency is a genuinely useful measure of how much double-bond character a C=O has, and why a shift of twenty wavenumbers between two ketones is read confidently as an electronic effect.

The same comparison across different bonds is what fails. Sulfur dioxide against water is a comparison in which both the constant and the mass have changed, and the frequency reports their combination — which is the essay’s finding, and it is not repaired by dividing by anything a table supplies.

The division between the two cases is therefore one that keeps turning up from different directions. A difference within a series where one thing is varying is a measurement of that thing. A difference across a comparison where two things vary is a measurement of neither, and no amount of arithmetic on the two numbers separates them, because the information required to separate them was never in the two numbers.

Who this is due to #

The distinction between the curvature and the depth of a potential well is as old as the Morse potential — Philip Morse, 1929 — which gives both quantities from one functional form and makes plain that they are independent parameters of it. Badger’s rule, from 1934, is the best-known attempt to tie force constants to bond lengths, and it works within a row of the periodic table and fails between rows, which is about what a purely empirical correlation of this kind should be expected to do.

The mode-composition argument is Wilson’s again, from the same 1955 apparatus that normal modes are not bond stretches is built on. The habit of quoting a band position as a bond strength long predates all of it and shows no sign of stopping, which is why this essay exists.

Where to read on #

The practical version of the same question is this: given that a frequency is not a bond’s property, why does the carbonyl band sit at 1,700 wavenumbers in every ketone anybody looks at? That regularity is real and needs an account rather than a dismissal — group frequencies, and where they stop computes when a mode is localised enough for a group frequency to exist, and finds the boundary in the arithmetic rather than in a table.