The moment that does not depend on where the origin is

One of the figures on point groups from coordinates: A molecule turned in space with its group recovered from the turned coordinates, and what the group settles on its own.

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.

What a dipole cannot tell apart

Six molecules with point charges on their atoms, and their two lowest moments computed about the centre of mass and again about an origin moved 1.6 bohr away. The dipole column never changes. The quadrupole column changes for the two polar molecules and not for the four non-polar ones, which is the theorem in one table.

The bond sum, the measurement, and what is left over. Three pyramidal molecules with the same shape and quite different moments: the sum of bond dipoles accounts for part of each and the residue is not the same fraction in any two of them — which is the arithmetic form of the claim that a dipole is not a decomposition.

Two molecules whose moments survive a shift of origin, with the shift applied. A dipole is origin-independent whenever the total charge is zero, and every higher moment is not — so the quadrupole quoted for a polar molecule is a quantity with a convention attached, and the dipole is the one moment in the table that is not.

The lone pair is not the missing term

The bond sum, the measured total, and the difference between them, for four pyramids. Positive is towards the lone pair. Ammonia’s bonds point that way and the trifluoride’s point the other, which is the reversal. The third bar in each group is what the additive model does not contain.

The moment that does not depend on where the origin is, beside the charges it is computed from. A molecule with a dipole has a dipole at every origin; only the higher moments move when the origin does. That is the property the bond-moment arithmetic below is trying to reproduce, and the one it does not need a lone pair to explain.

Two molecules with the same number of bonds to hydrogen and quite different moments: ammonia has one and methane has none, by symmetry. Whatever the lone pair is worth, it is not the difference between these two rows — the difference is that one group leaves a direction fixed and the other does not.

Zero dipole is not no interaction

The moments of six molecules, with the lowest non-vanishing one marked in each case and the origin-dependence made explicit. A molecule’s lowest surviving moment is the same at every origin and every higher one is not, which is why the quadrupole moment is a well-defined quantity for a molecule with no dipole and an ill-defined one for a molecule with a dipole.