Three bands of one spectrum, and the bond length behind each

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

Two 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 width of a band is a bond length

The two potentials and the two wavefunctions whose overlap decides every intensity, for the strongly bonding case. The ion’s minimum is 77.2 thousandths of an ångström to the right, so the neutral’s ground state sits over the flank of the ion’s well rather than over its bottom, and the states it overlaps best are the ones with amplitude out there.

The three bands as intensity distributions, computed from the bond lengths and vibrational constants above. Each band’s lines add to one. The first keeps 92 per cent of its strength in a single line, the third 88 per cent in one with a shoulder at 12, and the middle one is spread over five, with more intensity in the first excited vibrational state than in the ground one.

The computed intensities of the broad band against the Poisson distribution the harmonic approximation gives. The head of the band agrees to 0.004. The tail does not: at the fourth excited state the anharmonic calculation gives 0.051 against a harmonic 0.034, because a Morse well is wider than a parabola at high energy and the overlaps out there are larger.

A band is a filter on the modes

Which modes a band can show, for five molecules at once, on a logarithmic scale spanning sixteen decades. Removing an electron from a non-degenerate orbital lengthens every bond alike, and only a totally symmetric mode survives that: eight modes across the five molecules carry the whole of the intensity and every other one sits on the floor at arithmetic noise. The other modes are not weak in a band; they are absent, and no improvement in resolution changes it.

Sulfur dioxide’s band as it would appear: a Poisson progression in each of its two totally symmetric modes, with Huang–Rhys factors of 0.145 and 2.089. Its third mode, the antisymmetric stretch at 1382 cm⁻¹, has a factor of 3 × 10⁻³¹ under the symmetric change and 0.470 under an unsymmetric one of the same size.

Water’s band under the same change. Its symmetric stretch has a Huang–Rhys factor of 0.549 and its bend 2.7 × 10⁻⁶ — both totally symmetric, so both permitted, and one of them essentially absent because a change that lengthens both O–H bonds barely changes the angle between them. Permitted and present are different things, and only the first is symmetry’s business.