carbon dioxide: every mode, and whether it can be seen

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

Three 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.

A dipole is not what an infrared spectrum sees

Five molecules, with the dipole moment of a point-charge model, the number of modes each has and the number whose dipole derivative does not vanish. Three of the five have no dipole whatever and between them hold most of the active bands in the table.

Every mode of carbon dioxide with its dipole derivative. The two bends at 673 cm⁻¹ and the antisymmetric stretch at 2396 have identical derivatives of 0.0703; the symmetric stretch at 1354 has a derivative of exactly zero. The molecule’s dipole moment is zero throughout.

Methane’s nine modes. The three T2 deformations and the three T2 stretches have finite derivatives; the two E modes and the totally symmetric stretch have derivatives of 10⁻¹⁷ and 10⁻¹⁶. The last column is what the character table says, computed separately, and it agrees on every row.

The mode that moves least radiates most

Boron trifluoride’s five infrared-active modes, with the band strength and the zero-point amplitude of each scaled to its own largest. The strongest band is at 719 cm⁻¹, where the amplitude is the third of five; the mode with the largest amplitude is at 480 cm⁻¹ and is nine and a half times weaker. The rank correlation between the two columns is 0.20.

How well a band’s strength predicts how far anything moves, across the active modes of five molecules. A value of one would mean the strongest band is the largest motion. Two of the five are negative — the two orderings run backwards — and the one at +1.00 is methane, which has only two distinct active frequencies to order and so cannot disagree with anything.

The same comparison for ammonia, where the two columns are nearly the reverse of one another. The umbrella mode moves the atoms furthest and is not the strongest band; the highest-frequency stretch moves them least and is. Nothing about the molecule is unusual — this is what happens when one quantity is a length and the other is a derivative of a charge distribution.

The table that could not have mattered

The intensity–motion correlation computed with charges from four published tables. The four marks in each row coincide exactly, and the coincidence is a theorem rather than a numerical accident.

Four molecules of three elements or more, each with the dipole its charges give on four tables anchored identically. The atoms named beside each row are the ones whose charge changes sign from one table to another.

The underlying quantity: how the dipole changes as each mode is walked through, which is what an infrared intensity is. Its ranking across modes is what the theorem protects and its size is what a change of table moves.