water: a asymmetric top

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.

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 rotational spectrum is a moment of inertia

Carbonyl sulfide, with its principal axes drawn at its centre of mass and the moment about each. Being linear, one of its moments is zero — there is no mass off the axis to resist rotation about it — and the other two are equal. One number, 82.85 in units of unified mass times ångström squared, is the whole of what its rotational spectrum reports.

Benzene, an oblate symmetric top. Its two in-plane moments are equal to the last bit, and the moment about the sixfold axis is exactly their sum — which is the perpendicular axis theorem and holds for any planar body. That sum rule is an independent check on the tensor, because it follows from the definition of a moment rather than from the diagonalisation that produced these three.

Methane, a spherical top: all three moments identical to the last bit, because two independent threefold axes leave no freedom at all. Its rotational levels are as evenly spaced as any linear molecule’s, and it has no dipole, so none of them can be reached by absorbing a photon.

The moment that is the sum of the other two

Water’s three principal axes drawn at its centre of mass, with the moment about each. The centre of mass sits 0.07 Å from the centre of the atoms — enough to matter and easy to get wrong — and the smallest moment is about the axis bisecting the H–O–H angle, which is where most of the mass lies.

Formaldehyde: planar, asymmetric, with moments of 1.794, 12.976 and 14.771 u Ų. The defect is 1.8×10⁻¹⁵, which is arithmetic noise rather than a small physical quantity — the same kind of statement as the symmetry-forbidden overlap of exactly zero, which comes out at 10⁻¹⁷.

The planar sum rule at its cleanest, with the axes drawn. Benzene’s two in-plane moments are equal to the last bit a double holds and the perpendicular one is exactly twice either — so the defect is zero for a reason that is arithmetic rather than approximate, and any departure from it in a measured structure is the vibration rather than the shape.

The constant a spectrum cannot see

Ammonia’s three principal moments, computed from its atom positions and the masses of its nuclei. Two are equal to a part in a million — the residual is a rounding in the stored coordinates, not a physical asymmetry — and one is half again as large.

The top that reports all three

Where water’s three constants come from: its principal axes and the moments about them, computed from the coordinates and the isotopic masses. Three different moments is the definition of an asymmetric top, and every result in this essay follows from those three numbers.

The axis that goes the other way

The dimensionless coefficient each principal axis needs, for the four molecules with force fields. One of the twelve is below zero.

The coefficient the molecule-averaged expression needs beside the three the per-axis version needs, for each molecule.

The correction as a percentage of each principal moment. Eleven of the twelve grow. Water’s smallest falls by 0.46 per cent.

Two moments about two different lines

Water’s principal axes and monodeuterated water’s, on the same nuclei. Two of the three turn; the third is the normal to the plane and cannot.

The parent’s moments, the daughter’s own, and the daughter’s about the parent’s axes — the quantity a per-axis comparison assumes it has.

Monodeuterated water’s inertia tensor in ordinary water’s principal frame. A frame that diagonalised it would have zeroes off the diagonal.