Molecular Geometry & Bonding
what the shape decides
Figure

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.
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.

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.

In the essays

The moment that is the sum of the other two

Seven molecules with their three principal moments, the difference Ic − Ia − Ib, the asymmetry parameter and the top classification. The defect column separates the table into two groups with fourteen orders of magnitude between them, and the separation is exactly the separation between planar structures and the rest.

The top that reports all three

Water’s levels up to J = 4, each J diagonalised as a matrix of side 2J+1. The constants are computed from water’s own atomic positions and masses: A = 27.3773, B = 14.5868, C = 9.5164 cm⁻¹. A symmetric top would show these levels in degenerate pairs above K = 0; here every pair is split, and by an amount that is not small.

Every level of water up to J = 4 with how far it moves when A is raised by one per cent, divided by the change made. Twenty-three of the twenty-five move; the two that do not are the ground state and the level at B + C, and no level above them is blind to A at any J.

The count, against Ray’s asymmetry parameter, with two of the constants held and the third swept between them. The number of blind levels is five at the prolate end and two everywhere else — including at asymmetries far too small to see in a spectrum. Symmetry is not a limit that is approached; it is a condition that either holds or does not.

Three numbers is not a structure

Inertial defects for five molecules, computed from their accepted structures. Every planar one comes out at zero to machine precision, because the relation is exact for a rigid planar structure — which is what makes the measured defect a measurement of the vibration rather than of the geometry.

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