4s and 3d from Sc to Zn

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.

Five 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

The 4s and 3d orbitals from potassium to zinc, each drawn at the nuclear charge that shell actually feels. The 3d is smaller at every element and by a widening margin. It is also, by every one-electron energy estimate available, lower — and it is the 4s that fills first.

The third period, where the two shells being compared are in the same row and the order never changes. Both contract steadily and the 3s stays inside the 3p at every element, so the filling order across this row is fixed and nothing about it is contested — which is the ordinary case the transition series is an exception to.

The fourth row’s own s and p shells, which behave as ordinarily as the third row’s. The crossing this essay is about is not between an s and a p at all: it is between the 4s and the 3d, two shells of different principal quantum number, and it happens because one of them penetrates the core and the other does not.

What an electron actually feels

The mean radius of the 4s and 3d orbitals across the first transition series, each drawn at the nuclear charge that shell actually feels. The 3d is smaller than the 4s at every element and by a widening margin — a factor of two at scandium and nearly five at zinc — and it is the 4s that fills first. Size and filling order are different questions, and this is the picture that separates them.

The second period, with the 2s and 2p orbitals drawn at the charge each feels. Both contract steadily from lithium to neon: eight protons are added and eight electrons with them, but the added electrons go into the same shell and screen each other at 0.35 apiece, so more than half of every added proton survives. The atom gets smaller as it gets heavier, which is the opposite of the naive expectation and the reason a period has a trend at all.

What the screening model cannot see

The mean radius of the 2s and 2p orbitals across boron to neon, at the effective charge Slater’s rules give each. The two traces are parallel because the two charges are identical at every element: the rules put 2s and 2p in one group. The radii differ only because the closed form for the mean radius contains l, and it does so at a fixed ratio of exactly 1.2.

The same calculation one period down. Silicon to argon, 3s against 3p, with the two screened charges again identical at every element and the radii again in a fixed ratio — 1.08 this time, because the ratio is [3n² − l(l+1)] with n = 3 rather than 2.

Gallium to krypton, 4s against 4p. Slater’s rules put both in one group again, so the two traces lie on top of each other for the whole period and the two mean radii are equal at every element — the same coincidence the second period produced, at a shell where the real 4s–4p gap is several electronvolts. A constant fitted differently would move both traces together; nothing available inside the scheme separates them, because the scheme has one number per group and these are one group.

The radius that was tabulated

Screening as computed from Slater’s rules. The rules track the shape of the periodic table well enough to explain it, which is what they were built for; the comparison above is about a different use of the same numbers, one they were not built for and are not adequate to.

The surface a table draws

The screening constant and the effective charge across the ten-electron series, from the rules rather than from a table. The screening does not move because the electron count does not move; the effective charge rises by exactly one at each step because the nucleus does. That is the whole of what distinguishes one member of an isoelectronic series from another in this model, and it is enough to make their sizes differ by a factor of two.