Molecular Geometry & Bonding
what the shape decides
Figure

1s, 2s, 2pz, 3s, 3dz2, 4s at one level and at one fraction

The orbitals 1s, 2s, 2pz, 3s, 3dz2, 4s, each with the contour level that encloses 90 per cent of its own density, and beside it what each one encloses when they are instead all drawn at 1s's level. The first column is a set of pictures that can be compared; the second is what a plate drawn at a single contour value actually shows. Contours drawn: 1s at 90% of its density, |ψ| = 3.94e-2; 2s at 90% of its density, |ψ| = 7.40e-3; 2pz at 90% of its density, |ψ| = 9.48e-3; 3s at 90% of its density, |ψ| = 2.64e-3; 3dz2 at 90% of its density, |ψ| = 3.60e-3; 4s at 90% of its density, |ψ| = 1.24e-3.
1s, 2s, 2pz, 3s, 3dz2, 4s at one level and at one fraction. The orbitals 1s, 2s, 2pz, 3s, 3dz2, 4s, each with the contour level that encloses 90 per cent of its own density, and beside it what each one encloses when they are instead all drawn at 1s's level. The first column is a set of pictures that can be compared; the second is what a plate drawn at a single contour value actually shows. Contours drawn: 1s at 90% of its density, |ψ| = 3.94e-2; 2s at 90% of its density, |ψ| = 7.40e-3; 2pz at 90% of its density, |ψ| = 9.48e-3; 3s at 90% of its density, |ψ| = 2.64e-3; 3dz2 at 90% of its density, |ψ| = 3.60e-3; 4s at 90% of its density, |ψ| = 1.24e-3.

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.

Two 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

One level is not one comparison

Six orbitals with the contour level each one needs in order to enclose ninety per cent of its own density, and beside it what each encloses when they are all drawn at the 1s orbital’s level instead. The left column is a comparison. The right column is what a single-contour plate actually shows.

One isovalue against many fractions, and one fraction against many isovalues, on the same axes. The two readings are inverses of each other and neither is flat: a level chosen for one orbital encloses a quite different fraction of another, and a fraction fixed across a set requires a different level for every member of it.

The same arithmetic at half the density and with a different reference. At the level that draws half of a 2p, a 2s encloses 4.7 per cent, a 3s 0.99, and a 3d again nothing — its wavefunction peaks at 2.77×10⁻² and the level is 3.16×10⁻².

The isovalue nobody chose

Seven orbitals of hydrogen, four conventional isovalues, and what each surface encloses. Reading down a column shows what one number means for different orbitals; the last column is the level each orbital needs to enclose ninety per cent, which is the same information the other way round.

The four s orbitals of hydrogen, drawn twice: once at a level chosen so that each encloses ninety per cent of its own density, and once at one shared isovalue. At the shared level the 4s is a speck and the 1s fills the frame, and neither picture is wrong — they answer different questions, and only one of them was asked.

A 2s orbital drawn at four fixed isovalues, for nuclear charges one to eight. Every curve rises: screening pulls the orbital in, the density at every radius goes up, and a fixed level encloses more and more. The identical picture is a fifty-two per cent surface for hydrogen and a ninety-nine per cent surface for oxygen.

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