Every orbital, by what it encloses

"The ninety per cent probability surface" is the stock caption under an orbital, and it is very often not what has been plotted, because nobody did the integral. Two textbooks print the same orbital at visibly different sizes with the same words underneath, and neither says which surface it drew.

Here the level is found by integrating the density and bisecting until the enclosed fraction is the one claimed, and the integral is run again on the level that is drawn — so a picture captioned ninety per cent encloses ninety per cent, not eighty-five.

Every number below was read back out of the finished drawing, not from the request that produced it. The fraction is the one each essay's caption states; the level is the value of the wavefunction that encloses that fraction, found by the integral rather than chosen.

45 contours, sorted by shell and then by how much of the density each encloses. Levels are in atomic units, as |ψ| — the value of the wavefunction on the surface, not the density. Enclosing more of the density always needs a lower contour, and every row obeys that. All of it is one-electron and hydrogen-like; a many-electron atom has no exact orbitals at all.

Every contour above is solved the same way — by integrating the density inside a candidate surface and bisecting on the level — whether it appears as a single orbital, a set of orbitals at several levels, an enclosed-fraction curve or an overlap picture. Every figure lists the diagrams by kind.