Four measures of size for 6 orbitals

One of the figures on radial functions: The one-dimensional half of a wavefunction: where the density is, what screening does to it, and how far out an orbital reaches.

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 aufbau order is not a property of the atom

Five orbitals by four measures. Reading down the list the sizes rise steadily with n and fall with l, which is the ordering the fixed-order picture assumes. What the picture leaves out is that a 4s and a 3d of the same atom feel different screened charges, so their real sizes are not in the ratio this figure shows at all — hydrogen’s are, and no other atom’s.

How big is an orbital

Four measures of size for six orbitals, each computed by integrating that orbital’s own radial function. The most probable radius, the mean radius, the root-mean-square radius and the radius of the sphere holding ninety per cent of the density. For every orbital drawn here the four disagree by at least a third, and for the 1s by a factor of 2.66.

The same pair of moments one shell out, where the disagreement is wider. ⟨1/r⟩ is identical across the 3s, 3p and 3d and ⟨1/r²⟩ runs five to one across them. The measure that agrees is the one an energy depends on, and the measure that disagrees is the one anything else depends on.

The s orbitals alone, where the trend is clearest: the spread across the four measures falls steadily from 2.66 for 1s to 1.37 for 4s. A single-lobed distribution is badly characterised by any one number; a many-lobed one is dominated by its outer lobe and better behaved.

The degeneracy no group predicts

Two moments of the n = 2 shell. The first, ⟨1/r⟩, is the same for the 2s and the 2p to six figures and matches its closed form Z/n². The second, ⟨1/r²⟩, is 0.250000 and 0.083333 — a ratio of exactly three — and matches its own closed form Z²/n³(l+½).

The two orbitals the accidental degeneracy is about, on four measures of size. They differ on three of the four and agree on the one an energy depends on — which is the whole of what an accidental degeneracy is: an equality in one quantity and not in the others.

The n = 3 shell, where the same computation gives 0.074074, 0.024691 and 0.014815 for the s, p and d — a ratio of five to one across the shell. Every value matches its closed form and the three energies remain identical to 3×10⁻⁸.

The symmetry that is not a rotation

The degeneracy itself, as two numbers. ⟨1/r⟩ is identical for the 2s and the 2p to six figures, which is why they share an energy; ⟨1/r²⟩ differs by a factor of three, which is why anything other than a 1/r potential splits them. Nothing in the rotation group requires the first of those and nothing forbids the second.

The surface a table draws

Four measures of an orbital’s size, for the hydrogenic shells: the most probable radius, the mean radius, the ninety-per-cent contour and the ninety-nine-per-cent contour. They are not proportional to one another, and the ratio between any two of them depends on which orbital is being measured — so the choice of measure is not a choice of units.

How nearly a broken symmetry survives

The radial moments that set the splitting. ⟨1/r²⟩ is the one the first-order form uses, and it differs by a factor of three between the s and p orbitals of one shell — which is the whole of why a radial screening splits a shell at all.