Molecular Geometry & Bonding
what the shape decides
Concept

Symmetric top — where it appears

A molecule with two of its three principal moments of inertia equal, which any axis of order three or higher forces. Its rigid rotational spectrum reports only one of its two constants: the other cancels from every line.

Named by 5 essays across one field — each of them below, with the objects they name alongside it.

carbonyl sulfide: a linear. The principal axes of carbonyl sulfide drawn at its centre of mass, with the moment of inertia about each. Which kind of top this makes it is decided by comparing three numbers, and the point group forces the same answer independently: an axis of order infinity means a linear molecule, with one moment at zero forbids an asymmetric top.

The rotational spectrum is a moment of inertia

Every line in a microwave spectrum sits at a multiple of one number, and that number is a conversion constant divided by a sum of mass times distance squared. No bonding argument appears anywhere in it.

spectra · Rotation ammonia's rotational levels, sorted by K. The rigid rotational levels of ammonia up to J = 4, each drawn at its computed energy and grouped by J. Within a group the levels are pushed apart by the second rotational constant, so it is plainly there in the level pattern.

The constant a spectrum cannot see

A symmetric top has two rotational constants and its microwave spectrum reports one of them. Not badly, not with difficulty: ammonia's A of 6.3406 wavenumbers appears in none of its lines at any J and any K, because the term it belongs to cancels exactly out of every transition. The molecule turns about that axis, the energy is real, and the measurement is blind to it.

spectra · Rotation water: every level up to J = 4, from a matrix. The rotational levels of water at κ = -0.4322, each J diagonalised in the symmetric-top basis. A symmetric top would show one level per K with everything above K = 0 doubly degenerate; here every degeneracy is split, and the size of each splitting is what the third constant is measured from.

The top that reports all three

A symmetric top hides one of its two rotational constants in every line of its spectrum. Break the symmetry and the hiding stops: for water, twenty-three of the twenty-five levels up to J = 4 move when A is changed, and the two that do not are the ground state and the one at B + C. There is no formula for any of them.

spectra · Rotation The coefficient each principal axis needs. The dimensionless coefficient the molecule-averaged expression requires, evaluated on each principal axis separately rather than on the mean of the three. Across the twelve axes the positive ones span a factor of 2.33, against the 1.36 the molecule-averaged version spans — and one of them is negative, which no positive constant can be.

The axis that goes the other way

One dimensionless coefficient turned a zero-point correction into an expression a spectroscopist could evaluate, and the three principal axes were averaged over to get it. Split by axis it gets worse, not better — the coefficients span 2.3 where the molecule-averaged ones span 1.36 — and water's smallest moment does not grow at all. It shrinks.

spectra · Rotation The frame H₂O → HDO turns. H₂O → HDO: the parent's principal axes and the daughter's, drawn on the same nuclei. Two of the three turn by 21.12° and the third does not move at all, because it is the normal to a plane no substitution can tilt. A per-axis comparison between these two species is comparing moments about lines this far apart, which is a rotation of the frame rather than a correction to a number.

Two moments about two different lines

Asked axis by axis, the substitution method's near-cancellation gives numbers as large as 163 per cent. The arithmetic is the smaller half of the answer. A principal axis is an eigenvector of a tensor built from the masses, so one deuterium turns water's frame by 21.12° — and the two moments being compared are not moments about the same line.

spectra · Rotation

Named alongside it

The objects these essays reach for when they reach for this one.

DegeneracyMoment of inertiaRotational constantClosed formConventionModel limitApproximationEigenvalueIsotope substitutionPoint groupRigid-rotorSelection rules

All concepts