Molecular Geometry & Bonding
what the shape decides
Figure

Orbitals at the 90 per cent contour

Several orbitals drawn at the same enclosed fraction and, unless stated otherwise, at the same scale — so the sizes on the page are the sizes. Each contour was solved for separately by integrating that orbital's own density. 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; 3dz2 at 90% of its density, |ψ| = 3.60e-3.
Orbitals at the 90 per cent contour. Several orbitals drawn at the same enclosed fraction and, unless stated otherwise, at the same scale — so the sizes on the page are the sizes. Each contour was solved for separately by integrating that orbital's own density. 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; 3dz2 at 90% of its density, |ψ| = 3.60e-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.

Seven 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

Orbitals are not where the electron is

Three hydrogenic orbitals at one enclosed fraction and one scale. These are exact solutions for one electron and one nucleus. For any atom with more than one electron they are not solutions of anything, and their role changes from description to basis.

Three orbitals whose relative energies and shapes the approximation gets right, and whose radial details it gets approximately. Everything a first course does with these — filling order, bonding, spectra — rests on the robust part.

The two functions the penetration argument compares, at one enclosed fraction and one scale. The 2s is a shell outside a shell, and the inner one is what reaches inside the 1s density — an argument about many electrons made entirely out of one-electron functions, which is the compromise this essay is about.

Where the electron is

The same trend as surfaces: 1s, 2s and 3s at one enclosed fraction and one scale. The shells nest, the outer ones are much larger, and each has one more radial node than the last.

The radial distribution across the periodic table

Three s orbitals at one enclosed fraction and one scale, so the sizes on the page are the sizes. Each has one more inner peak than the last — the node count again — and each reaches substantially further out, which is the shell structure that the periodic table’s rows are.

Complex harmonics against real ones

The three real p orbitals, at one enclosed fraction and one scale. These point along the Cartesian axes and are what everybody draws. Two of the three are combinations of the complex solutions rather than solutions themselves.

A filled shell has no shape

The three functions whose densities add to a constant. Each is drawn at a contour enclosing ninety per cent of its own density and each is unmistakably a dumbbell. Their sum, in every direction, is 0.238732415 — the same number this figure’s three panels could not look less like.

The same three functions at half the density rather than nine tenths. Each is a smaller version of itself and each still has its nodal plane exactly where it was, so the sum of the three is still spherical — the theorem is about the angular part alone, and no choice of contour can make it fail or make it visible.

Four of the five, at the contour enclosing ninety per cent of each. Four is not a shell, and the sum of these four is not a constant — which is what makes the fifth’s absence something a ligand can notice.

The surface a neighbour moves

A ninety per cent contour with and without a neighbour’s field. The dashed circle is the surface with nothing beside it; the closed curve is the same surface in a field of the size a sodium ion produces at a fluoride ion’s distance. It is not a displaced sphere — it is fatter on one side and thinner on the other, which is what a first-order mixing of an s function with a p one looks like.

The displacement as a share of the radius, against the field, for four effective charges. Every curve is straight, which is the statement that the response is linear — and the shaded band is where a linear response is describing a displacement too large to be called small. Two of the ions in a real rock-salt structure sit inside it.

A control that outranked the mechanism

The candidate already ruled out, drawn: an orbital in a neighbour’s field, with the polarisation the correction was built from. It is real, it is computable, and it ranks the wrong way.

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