About

This is a growing collection of illustrated essays about molecular shape and bonding. Each takes a single idea and draws it until the argument is visible — and every orbital in every picture was computed and then made to state what it encloses.

Every orbital says what it encloses

An orbital picture is an isosurface: the surface on which the wavefunction takes one chosen value. The value is a choice, and almost no source says which choice was made. "The ninety per cent probability surface" is the stock caption and is very often not what has been plotted, which is why two textbooks can draw the same orbital at visibly different sizes with the same words underneath.

Here the level is found by integrating. The density is integrated over the region inside a candidate contour, and the contour is solved for by bisection until the enclosed fraction is the one claimed. The figure then asserts it: a picture captioned ninety per cent that encloses eighty-five throws, and the build stops.

What else is checked

• Normalisation and orthogonality. The density integrates to one, and two different orbitals on the same centre have zero overlap. A wrong constant or a wrong sign fails here and nowhere else.
• Node counts. An orbital with quantum numbers n and l has exactly n − l − 1 radial nodes and l angular ones, counted from the computed function. A wrong radial polynomial has the right shape and the wrong number of nodes.
• Overlaps that symmetry forbids are exactly zero. Not small — zero. The integrand is odd about a plane and the contributions cancel in pairs, and the number that comes out is arithmetic noise.
• The integrator is checked against a closed form. The overlap of two 1s orbitals has an exact expression, so the numerical integral is validated rather than trusted.
• Point groups round-trip. A molecule's group is detected from its coordinates by searching for the operations that permute its atoms, and must match the caption.

A figure that fails any of these throws and the build stops. That is not a formality: three real errors were caught this way before a single essay existed, and none of them looked wrong on screen.

Vector, not raster — and why that is possible

The usual reason for rendering orbitals as bitmaps is that they are three-dimensional scalar fields. That is true of a field and false of the object being drawn, because an orbital picture is a surface rather than a volume. And for a hydrogenic orbital the surface is easy, because the wavefunction separates: along any direction the contour sits wherever the radial function crosses a threshold, which is a one-dimensional root-find solved exactly.

So there is no volume grid, no marching cubes and no resampling, and the surface is as accurate as the root-find. There is a second benefit: where the radial function has a node the equation has several roots along the same ray, which is the correct answer rather than an inconvenience. A 2s orbital's ninety-per-cent surface is a shell inside a shell, and drawing it as a single ball — which is what almost every published 2s picture does — hides the node that is the entire difference between 1s and 2s.

Where the models stop

Three places, and the first is the one that matters most.

These are one-electron orbitals. Every wavefunction drawn here is a hydrogen-like solution. A many-electron atom has no exact orbitals at all — the orbital picture is a basis for an approximation, and a very good one, but the objects on this site are hydrogenic functions and nothing else. No figure implies otherwise, and where an essay discusses a many-electron system it says which approximation it is talking about.

A basis is not a thing. Hybrid orbitals, localised bonds and canonical molecular orbitals are related by transformations that leave every observable unchanged. Arguing about which is real is arguing about a coordinate system, and several confident textbook claims are exactly that argument made without noticing.

Geometry is not energy. The repulsion minimisation behind the shape figures says where points on a sphere sit to minimise a simple potential. It is not a calculation of a molecule's energy, it takes no account of bonding, and where it agrees with observation it agrees for reasons worth being suspicious about.

The wrong explanations are tested here, not ignored

This subject has an unusual number of explanations that are memorable, widely taught and false. That hybridisation explains a shape rather than describing it. That hypervalent molecules use d orbitals. That a dipole moment is a sum of bond vectors. Each is stated fairly here and then tested — and the rule is that a refutation must be a computation, not a contrary assertion.

On being wrong

Corrections are welcome and will be made. An orbital picture with a wrong caption looks exactly like one with a right caption, which is the whole argument for computing the caption.