A structure out of a spectrum

One of the figures on rotation and vibration: Moments of inertia, normal modes, isotope shifts, and the frequencies a force field does and does not fix.

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 bond length out of a spectrum

The inversion twice over. Above, on moments computed from a structure this site chose, where the recovered lengths must be the ones supplied and come back to twelve figures. Below, on rotational constants a spectrometer measured, three pairs of isotopologues at a time — and the three answers disagree with each other by seven thousandths of an ångström, which is not arithmetic error.

The two better-conditioned pairs on their own. Dropping the sulfur substitution — the one an instinct for heavy atoms would reach for first — leaves the oxygen and carbon pairs, whose independence is four and one and a half times better, and whose recovered lengths still differ from each other by seven thousandths of an ångström. The disagreement is not conditioning; it is the zero-point average.

The rotor that stretches

The inversion that turns two measured moments into two bond lengths, run over three isotopic pairs. Every number in it is a rigid-rotor number, the three pairs disagree by a fraction of a milliångström, and the corrections above are what stands between any of those answers and an equilibrium geometry.

Three numbers is not a structure

Seven structures of formaldehyde, each of which reproduces the measured A and B to the last digit the solver carries. The C=O length is chosen and the other two parameters solved for. Across the family the C–H length falls from 1.557 to 0.952 ångström while the HCH angle opens from 74.6 degrees to 164.5, and the two fitted constants are identical in every column.

Each member of the family’s prediction for the deuterated molecule. The B constant varies from 0.917 to 1.182 across the family and the measured 1.1215 picks a member out. The A constant is 4.7063 for every one of them.

The method that does work, on a molecule where the counting is favourable: carbonyl sulfide, a linear triatomic with two bond lengths, recovered from three different pairs of isotopologues. Linear molecules have one moment and no planarity relation, so the counting is clean — and the top row is the inversion run on moments computed from a known structure, which returns it to twelve figures. The three measured pairs return three answers spanning 7.4 milliangstrom, which is not the arithmetic and is the difference between a ground-state average and an equilibrium geometry.