A bond's length, and how much of it is uncertain

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

Six 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.

The atoms are not at the points

Every nucleus of four molecules, with its root-mean-square displacement in the vibrational ground state. Every hydrogen is above 0.09 Å; every heavy atom is an order of magnitude closer to where it is drawn. Methane’s hydrogens are the largest at 0.134 Å.

Carbon dioxide drawn with its nuclei the size they actually occupy. The carbon is spread nearly twice as far as either oxygen despite being the heavier of the two, because both of the degenerate bends at 673 wavenumbers swing the central atom and neither swings the ends much. The spread is a property of the modes rather than of the mass.

One bond and one angle from each of five molecules, with the zero-point spread of each beside its value. Every bond is uncertain by about seven per cent of its own length. Every angle is uncertain by more than eight degrees.

The width of a band is a bond length

And the reason the neutral molecule’s lowest state is not a point: a zero-point amplitude, computed for a real molecule. The width of that distribution is what samples the ion’s potential, and a molecule genuinely at rest at its equilibrium geometry would give a single line in every band.

The mode that moves least radiates most

How far ammonia’s nuclei actually move in their zero point, mode by mode. This is a picture of amplitudes and it contains no information about intensities whatever: the same picture would be drawn for a molecule made of neutral atoms, which would have no infrared spectrum at all.

A band is a filter on the modes

The zero-point spread of each internal coordinate for five molecules, which is the denominator every Huang–Rhys factor above is divided by. A band’s progression is long when the ion’s geometry moved by a lot compared with this — so the same change of shape produces a longer progression in a stiff mode than in a soft one.

The correction that was invented

Where the amplitudes come from, applied to the quantities a chemist recognises: one bond and one angle from each of four molecules, with the zero-point spread of each beside its value. Everything above is that spread squared, applied to a moment of inertia instead.

An expression for what was a warning

The computed correction against the expression, on logarithmic axes. The dashed line is the mean coefficient; the corrections span a factor of eleven.

The expression written out and evaluated on each molecule, with the coefficient and the constant it uses.