Molecular Geometry & Bonding
what the shape decides
Figure

Every sign, lost

Formaldehyde in its own principal axes. Open circles are the atoms where they are; filled ones are where Kraitchman's equations put them, from the change in the three moments when each atom in turn is made heavier. The two agree to 7.6e-8 ångström — the equations are an identity for a rigid structure — but they return the square of each coordinate, so the two hydrogens at b = ±0.9348 both come back at +0.9348 and land on the same point.
Every sign, lost. Formaldehyde in its own principal axes. Open circles are the atoms where they are; filled ones are where Kraitchman's equations put them, from the change in the three moments when each atom in turn is made heavier. The two agree to 7.6e-8 ångström — the equations are an identity for a rigid structure — but they return the square of each coordinate, so the two hydrogens at b = ±0.9348 both come back at +0.9348 and land on the same point.

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.

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.

In the essays

The coordinate an isotope reports

The factor of thirteen was generous. Two error models for the same inversion — an empirical rule and a computed correction — put the uncertainty in a small coordinate an order of magnitude apart, and the computed one is the larger. Every coordinate here is measured in a principal-axis frame the equations return, even though every moment fed to them was measured in a different one.

The four atoms where they are, and where the moments say they are. The two agree to a hundred-millionth of an ångström. The two hydrogens do not: they sit at b = +0.9348 and −0.9348 and both come back at +0.9348, because what the equations return is a square.

The zero-point correction, computed rather than assumed. Formaldehyde is planar, so its third moment is the sum of the other two exactly — which means one of its three coordinates is not independent, and the correction has to be applied to the two that are before the third is inferred from them.

The correction that was invented

The invented error model against the computed one, on the three quantities the error model reported. None of the three moves in the direction that helps.

The computed corrections, with each molecule’s three moments drawn either side of zero. Water’s first bar goes the other way from the other two.

The original picture of the cancellation, which is not changed here — only the number that goes into it. What correlation buys is a real effect measured against a mismatch, and the mismatch was the invented part.

Every figure · Every orbital, by what it encloses · All essays