Where the electron is, and how fast it is going

One of the figures on radial functions: The one-dimensional half of a wavefunction: where the density is, what screening does to it, and how far out an orbital reaches.

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.

The orbital in momentum space

The radial distribution in position on the left and in momentum on the right, for the same five orbitals. Both are normalised and both are the same function. The 1s is the most compact in space and the widest in momentum, and every orbital that spreads out in one narrows in the other.

The uncertainty product for five hydrogenic orbitals against its lower bound of 9/4. Nothing hydrogenic reaches it. The line is attained exactly, at every exponent, by a single Gaussian — which is the function every basis set in quantum chemistry is built from.

The three 2p orbitals in momentum rather than in position. They point along three perpendicular axes and are the same function turned, so their momentum distributions are the same function turned as well — and the sum of the three squared angular parts is a constant, in momentum exactly as in position. Nothing about the transform disturbs a closed shell’s sphericity.

The measurement a basis was not fitted to

The exact momentum functions of a 1s and a 2s. Neither has a cusp — the cusp is in position — and both die as a power rather than exponentially, which is the feature no finite sum of Gaussians has, because every Gaussian dies faster than any power.

The property that gets worse

And the exact momentum distributions the profile is built from. A property computed in momentum space weights the cusp — a sharp feature in position is a broad one in momentum — so the errors a Gaussian basis makes are magnified rather than averaged away.

The nodes in the other variable

Every hydrogenic orbital with a radial node, with its position nodes on one axis and its momentum nodes on the other.

The 4s orbital’s two radial functions reduced to the polynomials whose zeros they are, both of degree three.

The products of position and momentum nodes taken one to one, innermost with innermost and innermost with outermost.

Oblate in the picture nobody draws

The two factors the bonding orbital’s momentum density is made of: the atomic density, and the interference between the centres.

The bonding orbital’s directional Compton profile along the bond and across it, with the zeros marked.

The momentum at which the profile along the bond first vanishes, against the separation it was computed at.

The zero belongs to one determinant

The profiles along the bond of the bonding and antibonding combinations at the same separation and exponent.

The ratio of the second moment along the bond to the one across it, for both orbitals, in both pictures.

Each profile divided by the atomic one, so that only the interference is left: the bonding orbital alone, and a filled pair.

The zero is a parity, not a bond

The bonding and antibonding combinations of three atomic functions on nitrogen, each along the bond and each scaled to its own peak.

Which factor each of nitrogen’s six valence combinations carries, set beside its bonding character and its two parities.

The profile along the bond at π/R, per electron, for six first-row diatomics, split by the occupied combination each part comes from.