The 2pz orbital

One of the figures on orbitals and their contours: Isosurfaces solved for at a stated enclosed fraction, their nodes, and the shells a radial node cuts them into.

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.

What an orbital is

A 2p orbital, drawn as what a picture of one always is — a contour surface, at a level enclosing ninety per cent of the density. The two colours are the two signs of the function, which is the single most important thing the picture carries.

The 2s orbital at the same enclosed fraction and the same scale as the 2p above it. Size grows with $n$ and shape is set by $l$, so this one is larger than a 1s and round where a 2p is not — and the size on the page is the size, which panel-by-panel fitting would conceal.

1s: spherical, no nodes of any kind, and the simplest wavefunction there is. The contour is a sphere because the function depends on radius alone.

Orbitals are not where the electron is

The orbital both helium electrons are said to occupy. It is an exact solution for one electron and a hydrogen nucleus, and helium’s electrons are in neither of those situations — the nuclear charge is different and each electron is screened by the other.

Say what it encloses

A 2p orbital at the ninety-per-cent contour, with the level printed beside it. The number is not decoration: it is what makes the picture a statement rather than an impression.

Complex harmonics against real ones

The one p orbital that is a solution as it stands: $m = 0$, real, and drawable directly. Nothing about this picture distinguishes it from the other two — which is precisely the difficulty, since the other two are not solutions.

And $p_x$, which is a combination. The picture is the same shape rotated, the enclosed fraction is the same, and no feature of the drawing records that this one was assembled rather than solved for.

The orbital everybody draws with a doughnut. Its shape is not a physical peculiarity of the $m=0$ state — it is the consequence of having to build five real functions out of a five-dimensional space that would prefer six symmetric ones.

What an electron actually feels

A 2p orbital at a nuclear charge of one — the hydrogen case, and the only one that is exact. The contour encloses ninety per cent of the density, at a level found by integrating that density rather than chosen to look right.

The same orbital at the charge a carbon 2p electron feels. The shape is identical — the angular part knows nothing about the nucleus — and the size is not: every length has been divided by 3.25, so the ninety-per-cent contour sits at a wavefunction value six times higher. The number under the picture is the honest way to state that, and it is why every orbital figure here carries one.

And at neon’s 5.85. Three pictures of one function at three charges, all drawn at the same enclosed fraction, so the sizes on the page are the sizes. Comparing pictures drawn at different fractions would say nothing at all, which is the argument of say what it encloses.