H₂O: 3 distinct modes

One of the figures on spectra: How many bands there can be, where they sit, and what an absent one proves.

Ten essays draw this figure, each at the values its own argument needs rather than at the setting shown above. What each one uses it to show is below, in the words of its own caption.

Normal modes are not bond stretches

The three vibrations of water, drawn from the eigenvectors of its mass-weighted Hessian. The arrows are the actual displacements of the atoms, not the mass-weighted coordinates, which is why the oxygen barely moves: it is sixteen times heavier than what is pulling on it. Under each mode is the internal coordinate that holds the largest share of the motion, and for the two stretches no coordinate holds more than half.

A planar molecule with one motion that is genuinely localised. Boron trifluoride’s out-of-plane bend is the only mode in the collection where a single internal coordinate does all the work, and it is localised for a reason that has nothing to do with bond strengths: there is one out-of-plane coordinate and nothing to share it with.

Another bent triatomic, where the same three motions come out with the heavy atom in the middle barely moving. The shapes are set by the masses and the force constants together, and neither of the two stretches belongs to one bond any more than water’s does — a fact about the arrangement rather than about the particular atoms.

Group frequencies, and where they stop

What a localised mode looks like when a molecule has one. HOD’s upper stretch moves the hydrogen and essentially nothing else; its lower stretch moves the deuterium and essentially nothing else. Compare water’s two stretches, in which both hydrogens move equally in both, and the difference is a symmetry rather than a bond.

The six motions that are not modes

The three motions of water that are vibrations, with the displacement of each atom drawn. The other six move the molecule without deforming it, and this figure has nothing to show for them — which is exactly the property that makes them identifiable.

A linear molecule, where the count is 3N−5 rather than 3N−6. One of the three rotations moves no atom at all — a rotation about the molecular axis is the identity on a set of collinear points — so the six motions that are not modes are five, and the missing one is missing for a geometric reason rather than a dynamical one.

Four atoms and six vibrations, with the same six rigid motions projected out. The count is 3N−6 again and the projection is the same projection, so nothing about the argument depends on the molecule having three atoms — which is what makes the six a property of space rather than of the structure.

The atoms are not at the points

The three motions the spreads are built from. Each mode has a frequency and a shape, and the shape says how much of each nucleus’s displacement comes from it. The lowest frequency contributes the most, because a spread goes as the inverse square root of the frequency.

A dipole is not what an infrared spectrum sees

The three kinds of motion, drawn. The symmetric stretch keeps the molecule symmetric at every instant and is the silent one; the antisymmetric stretch and the bend break the symmetry as they go and are the visible ones. Nothing about the equilibrium dipole enters that distinction.

How much of a band is a bond stretch

The motions themselves, which is the thing the percentages are trying to summarise. A picture has the advantage of not needing a convention at all, and the disadvantage that it cannot be tabulated or compared across a series of a hundred compounds — which is why the percentages exist.

The mode that moves least radiates most

Boron trifluoride’s six vibrations. The out-of-plane bend at 719 cm⁻¹ is the one in which the boron moves and the fluorines do not — 89.9 per cent of the motion belongs to the boron, which carries a fifth of the molecule’s mass — and it is the strongest band in the spectrum. The pair at 480 cm⁻¹ are the reverse, at 14.2 per cent boron, and they are the weakest.

More coordinates than motions

Methane’s nine vibrations, which are what the force field above reproduces. Every one of them is unchanged by the flat direction, and none of them can be used to determine where along it the true constants lie — because there is no where: every point on the line reproduces every observable this model has.

A ratio that squares what it measures

One of the modes being watched: a degenerate E mode of ammonia, whose mean polarisability derivative is zero by symmetry and stops being zero when a bond is pulled.

Ten directions no frequency can see

What the whole argument is about the constants of: methane’s nine vibrations, which every one of these force fields reproduces exactly. The disagreement is invisible here, and that is the point.