6 molecules, counted

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.

How many frequencies, not how many modes

Six molecules, counted three ways. The first column is 3N−6; the second is how many distinct frequencies those modes have, which is smaller wherever a species is degenerate; and the rest count frequencies by what can see them. Every column after the first counts frequencies rather than modes, because a spectrum shows lines and a degenerate pair is one line.

Two structures, two spectra

Four molecules, two pairs. Carbon dioxide is a linear XY₂ and sulfur dioxide a bent one; boron trifluoride is a planar XY₃ and ammonia a pyramidal one. Within each pair the atom count is the same and every column differs. Nothing in this table uses a force constant or a measured frequency.

Tetrahedral XY₄ against square planar XY₄. Both have nine vibrations. The tetrahedral molecule’s degeneracies collapse them to four frequencies; the square planar molecule has only one degenerate species and gets seven. Two infrared bands against three, four Raman against three, four coincidences against none, and one silent frequency against none.

What an absence proves

Six molecules and the frequencies each has that neither experiment can reach. The last column is not an experimental limitation: those frequencies are forbidden by the same theorem that permits the others, and the number rises with symmetry — water, at the bottom of the table, has no forbidden frequency at all.

The counts those arguments are made on. The “both” column is the coincidence count, and it is zero for exactly the molecules with a centre of inversion — a positive observation in that column is what refutes a centre, and the refutation needs one line rather than a full assignment.

A spectrum that changes when only a mass does

The five methane isotopologues from one force field. The operations column counts how many of methane’s twenty-four survive each mass pattern; the group is named from the surviving operations’ own conjugacy classes; and the last four columns are counts of frequencies rather than of modes.

The four ammonia isotopologues. C3v’s six operations fall to two for the mixed cases, the group becomes Cs, and the four distinct frequencies become six. As with methane, the substituted molecules that keep the full symmetry are the ones with the fewest lines.

When a mode becomes a bond stretch

Every distinct frequency in this collection with how localised it is. The rule the census shows is that localisation follows isolation in frequency, and the isotopic case is the one where the isolation is put there deliberately.

A band is a filter on the modes