Normal modes are not bond stretches

A water molecule has two O–H bonds and two stretching frequencies. The bonds are equivalent, the frequencies are 3,832 and 3,943 wavenumbers, and the arrangement invites an obvious reading: one frequency for each bond, differing for some reason to be explained later.

That reading is wrong in a way worth taking seriously, because it survives well past the point where it should. Every mode of the molecule involves both bonds, in equal measure, and the difference between the two frequencies is not a difference between two bonds. It is the difference between two ways of moving both of them.

What has to be built before a frequency exists #

Symmetry alone says how many bands a molecule may have and cannot say where any of them is. Those are different kinds of statement and the second needs a calculation the first does not.

The counting statement comes from symmetry alone. Selection rules are one theorem sets out the whole of it: the 3N Cartesian displacements span a representation, the translations and rotations span two more, and what is left after subtracting them is the vibrations. For water that is 2a₁ ⊕ b₂ — three modes, two of one species and one of another — and the derivation touches no force constant, no mass and no energy.

A position needs four more things, and it is worth being explicit about which of them are computed and which are assumed.

A potential. The molecule is given a quadratic energy surface in its own internal coordinates: every bond is a stretch, every pair of bonds at a common atom is a bend, and the energy is a quadratic form in those displacements. This is the general valence force field, and it is the only model here.

A reference geometry. The reference values of the coordinates are the structure’s own, so the molecule sits at a stationary point of that potential by construction. Nothing has to be relaxed before anything can be computed, and — more usefully — the six motions that deform nothing must come out at exactly zero curvature.

A Hessian, mass-weighted. The second derivatives of the potential with respect to the 3N Cartesian coordinates, each divided by the square roots of the two masses involved. This is where mass enters, and it is the only place it does.

A diagonalisation. The same cyclic Jacobi solver Hückel theory uses for a matrix of ones and zeroes. Its eigenvalues are the squares of the angular frequencies; its eigenvectors are the motions.

The six that must vanish #

Of the nine eigenvalues water’s Hessian produces, six have to be zero, and requiring it is the cheapest check available on the whole apparatus.

Three of the six are translations. Move every atom the same distance in the same direction and no bond length or angle changes, so the potential — which depends only on internal coordinates — does not change either. Three more are rotations, for the same reason: turning the molecule bodily leaves every internal coordinate exactly where it was.

The largest of water’s six comes out at about one part in ten million of the largest genuine eigenvalue. That is the accuracy of a four-point central difference and not the accuracy of the physics, and the distinction matters: the zeroes are exact in the model and approximate only in the arithmetic that finds them. An error anywhere — a mis-signed derivative, a mass attached to the wrong atom, a coordinate whose value is discontinuous — shows up here first, as a seventh non-zero frequency or a rotation that costs energy.

A linear molecule has five rather than six, and the missing one is worth naming: rotation about the molecular axis moves no atom at all, so it is not a motion of the molecule in the first place. Carbon dioxide accordingly has four vibrations rather than three, which is the count the band that is not there turns on.

The composition, which settles the question #

With the modes in hand, the question “which bond is this frequency?” can be asked precisely rather than rhetorically. Displace the molecule along a mode, measure how much each internal coordinate changed, square them, and normalise. What comes out is the share of the motion each coordinate holds.

Water’s bend is 100 per cent in the angle, and that is a genuine localisation: only one coordinate can change when the two bond lengths stay put. Its two stretches are 50 per cent in each bond, exactly, and the exactness is not a coincidence. The two O–H bonds are related by a symmetry operation of the molecule, so any normal mode has to be either symmetric or antisymmetric under exchanging them, and both possibilities put equal magnitude in each. Symmetry forbids a localised stretch in water before any force constant is chosen.

That is a stronger statement than the arithmetic alone, and it generalises: wherever two bonds are symmetry-equivalent, no mode of the molecule can be a motion of one of them. Methane’s four C–H bonds are all equivalent, and every one of its four stretching modes comes out at a quarter in each bond.

Methane’s nine modes reduce to four distinct frequencies, and each of its four stretching motions is exactly a quarter in each of the four C–H bonds. The degenerate modes’ shares are averaged over their partners, because the individual members of a degenerate set depend on an arbitrary choice of basis within it and their average does not.

The degenerate case needs that caveat and repays it. Methane’s three t₂ stretches all sit at the same frequency, so any three orthogonal combinations of them are equally valid normal modes; one run of the solver reports 61 per cent of a mode in one bond and another reports 45 per cent in a different bond, and both are right. Averaged over the degenerate block the answer is a quarter in each bond and stays a quarter however the solver rotates within the block. A number that changes when nothing changes is not a property of the molecule, and a figure printing one would be printing an artefact.

Two routes to one answer #

The mode shapes are the part of this computation that nothing was fitted to, so they carry the weight. There is a check available on them that costs nothing and is genuinely independent.

Apply a symmetry operation to a mode’s displacement field — turn each atom’s arrow with the operation and move it to wherever the atom went — and project the result back onto the mode. That gives its character. Do it for every operation, sort into classes, and reduce: the mode’s symmetry species falls out.

Nothing in that calculation knows what the group-theoretic count said. The eigenvectors came from a matrix of second derivatives of a force field; the count came from counting atoms that stay put under each operation. They must agree, and for every molecule here they do — water’s modes come out A₁, A₁, B₂ against the group’s 2a₁ ⊕ b₂, methane’s come out T₂, E, A₁, T₂ against a₁ ⊕ e ⊕ 2t₂.

The way this check fails is worth stating because it is not obvious. A force field that gave two symmetry-equivalent bonds slightly different constants would still diagonalise, would still return 3N−6 positive frequencies, and would look entirely reasonable. What it would not do is produce modes that transform as irreducible representations of the group, and the reduction formula would return fractional multiplicities — which have no meaning at all and expose the error at once. Character tables and reduction is where that arithmetic is set out; here it is being used as a tripwire.

Where the force constants come from, and what that costs #

Nothing above derived a force constant. They are fitted, here, by least squares against observed frequencies, and every figure drawn from them prints the residual.

Water’s field has four constants: one for the O–H stretch, one for the bend, one coupling the two stretches to each other, and one coupling a stretch to the bend. Fitted to the six harmonic frequencies of H₂O and D₂O together they come out at 8.456, 0.763, −0.100 and 0.248 mdyn per ångström, with a residual of seven thousandths of a per cent. Six numbers reproduced by four is not nothing, but it is nothing like a prediction.

The honest form of the claim is a division. The positions were fitted. The shapes, the species, the compositions and every other isotopologue were not. The force field is not in the spectrum takes the first half of that apart — the fit is much less determined than it looks — and the isotope shift is arithmetic takes the second half, where a field fitted to one molecule predicts another with no further freedom at all.

Breaking the symmetry on purpose #

If symmetry is what forbids water’s stretches from localising, then removing the symmetry should let them localise. It does, completely, and the case is easy to arrange: replace one hydrogen with deuterium.

The numbers are stark. In H₂O the two stretches sit at 3,833 and 3,943 and are each half in each bond. In HOD they sit at 2,824 and 3,890, and the first is 100 per cent in the O–D bond while the second is 100 per cent in the O–H. Not approximately: to the precision the composition is computed at.

Nothing about the bonds changed. The force constants are identical — they are properties of the electronic energy surface, which does not know which isotope is sitting on it — and the geometry is identical. What changed is that the two hydrogens stopped being interchangeable, and the modes stopped being obliged to treat them alike.

This is the cleanest available demonstration that mode delocalisation is a symmetry effect rather than a bonding one. It also explains a practical fact: HOD is used in preference to H₂O in a great deal of spectroscopy precisely because its O–H stretch is a genuine O–H stretch and can be treated as one.

Carbon dioxide, where two identical bonds give frequencies a thousand apart #

Water’s two stretching frequencies differ by 110 wavenumbers, which is small enough that the wrong reading survives on the grounds that the two bonds are nearly the same. Carbon dioxide removes that defence.

Its two C=O bonds are equivalent by symmetry and carry a single fitted force constant of 16.0 mdyn per ångström. Its symmetric stretch is at 1,354 wavenumbers and its antisymmetric stretch at 2,396. The gap is a thousand wavenumbers, the bonds are the same bonds, and the force constant is the same number in both modes.

What separates them is the interaction constant — 1.25 mdyn per ångström, coupling one stretch to the other — together with the kinematics: in the symmetric stretch the carbon stays put and only the oxygens move, while in the antisymmetric stretch the carbon moves against both. Different effective masses, different restoring forces, same bonds. The frequency is not the bond strength is where that separation is taken apart properly.

What the picture cannot show #

Three limits, stated where they bite rather than at the end as a disclaimer.

Everything here is harmonic. The potential is quadratic by construction, which means every overtone is exactly twice its fundamental and no band ever shifts with temperature. Real molecules are not harmonic: water’s antisymmetric stretch is observed at 3,756 wavenumbers and its harmonic frequency is 3,943, a gap of nearly five per cent that is real physics rather than error. Every observed number quoted here is a harmonic frequency, obtained by spectroscopists who measured the overtones as well, and fitting a harmonic model to anharmonic band centres would push that discrepancy into the force constants where it would look like a bond being weaker than it is.

The compositions depend on the coordinate set. “Half in each bond” is a statement about a chosen set of internal coordinates, and a different set — symmetry coordinates, say — would report the same modes differently. What does not depend on the choice is the fact that no single coordinate of any reasonable set can hold a symmetric molecule’s stretching mode, because the symmetry operations move the coordinates into one another.

No intensity appears anywhere. Whether a mode is active is a symmetry question and is answered; how strong its band is depends on how the dipole moment changes along the mode, which this calculation does not compute and does not pretend to. A stick in these figures marks a position and says nothing about a height.

The same molecule’s stretches become bond stretches higher up #

Each mode being exactly half in one bond and half in the other is a statement about the fundamental — the first excited level of each vibration. Climb to higher levels and the same molecule stops behaving that way, which is the sharpest available demonstration that the delocalisation is a property of a state rather than of the bonding.

Two identical oscillators coupled together give symmetric and antisymmetric combinations, and how strongly the combinations are preferred depends on the coupling against whatever separates the two oscillators. At the fundamental level nothing separates them — the two bonds are equivalent — so the mixing is complete and the split between the two frequencies, 99 wavenumbers for water, is the coupling itself.

Anharmonicity changes that with every quantum added. A real bond’s levels are not evenly spaced, so exciting one bond twice costs less than twice exciting it once, and that shortfall grows with each level climbed. By the third or fourth overtone the anharmonicity of a single bond far exceeds the coupling between the two, and the states that diagonalise the problem are no longer symmetric and antisymmetric combinations — they are states with all the excitation in one bond.

The measurement shows it directly. The symmetric and antisymmetric partners, 99 wavenumbers apart at the fundamental, draw together as the overtone order rises until they are separated by a few wavenumbers — a near-degenerate pair, which is what two states differing only in which bond is excited must look like.

So water’s stretching states are normal modes low down and local modes high up, in one molecule with one force field and one geometry. The essay’s finding is exact where it is made, and the reason it is exact there is that nothing yet distinguishes the two bonds — which is the same reason an isotopic substitution undoes it, arrived at by adding energy instead of by adding a neutron.

Who worked this out #

The apparatus is E. Bright Wilson’s, and the standard reference is the book he wrote with Decius and Cross in 1955, Molecular Vibrations — where the GF matrix method, the internal coordinate formalism and the symmetry classification of modes are all set out in the form still used. The idea that a molecule’s vibrations are the normal modes of a coupled system is older and is mechanics rather than chemistry; what Wilson supplied was the method for doing it with the coordinates a chemist actually thinks in.

The specific point of this essay — that a normal mode is not a bond stretch — was already clear to him and has been getting lost ever since, because the language of spectroscopy is full of phrases like “the carbonyl stretch” that are useful, are approximately true in the cases they are used for, and are false in general. Group frequencies, and where they stop is about exactly when the approximation earns its keep and when it does not.

Where to read on #

This essay establishes the method and one negative result. What follows is about what the method can and cannot be asked.

The next question is what a spectrum determines about a force field, and the answer is much less than the count of numbers suggests: the force field is not in the spectrum computes a whole family of quite different fields that reproduce water’s three frequencies exactly. After that comes what happens when only the masses change, which is the one situation where this calculation makes a genuine prediction — the isotope shift is arithmetic, and it is held to an exact identity while it does. From there the argument moves outward: how many frequencies, not how many modes counts what a spectrum can show at all, and two structures, two spectra turns the whole apparatus round and uses it to decide a geometry.