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.

13 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.

The pair that cancels only at zero overlap

The interference factor the average leaves, one per angular kind. Their first zeros are at 0.66π, π and 1.43π.

Nitrogen’s profile along the bond and averaged over orientations, with two free atoms for comparison. The directional minimum at π/R has no counterpart.

The averaged profile less two free atoms, for the six first-row diatomics. Not a small oscillation: a third of lithium’s profile at zero momentum.

A floor no charge transfer explains

The σ profile for three pairs of atoms, each with equal coefficients on the two atoms — no charge transfer at all. Only the homonuclear pair has a zero.

The depth already present at equal coefficients, with each pair’s difference in effective charge and its bond length. It orders by neither.

Carbon monoxide’s depth against the charge imbalance, with the Cauchy–Schwarz bound underneath. At equal coefficients the bound is 4 per cent of the depth.

The third function fills the floor

The interference term each pair of functions contributes at q = π/R, carbon’s against oxygen’s, per unit coefficients. The two s–p pairs are exactly zero; the s pair cancels and the p pair, in the sense that maximises its overlap, adds.

Carbon monoxide’s σ profile at equal coefficients for three bases — s on each atom, a quarter p mixed into carbon’s function, and a quarter p on both atoms in the sense that maximises their overlap — with the value at π/R marked on each.

The minimum at π/R as a fraction of the peak against the p weight, for p on carbon only, on oxygen only, and on both in each of the two relative senses. Dashed: the smallest value any coefficient ratio allows with p on carbon.

The lone pair sits where the floor was

The momentum profile along the bond of each occupied σ orbital of carbon monoxide, each scaled to its own peak, against momentum in units of π/R.

Mulliken populations of carbon 2s, carbon 2pσ, oxygen 2s and oxygen 2pσ in each occupied σ orbital of carbon monoxide.

Per electron, carbon monoxide’s 5σ profile along the bond against the swept picture’s two bonds: one s function on each atom, and a quarter p on both in the maximum-overlap sense.

The minimum belongs to the heavy atom's node

Lithium hydride’s one occupied valence σ orbital, as computed and with a quarter of it on hydrogen, each scaled to its own peak. Only the second has a minimum, and it is not at π/R.

The populations of the heavy atom’s 2s and 2pσ and hydrogen’s 1s in every occupied σ orbital of the seven first-row hydrides, per electron.

Hydrogen fluoride’s two occupied σ orbitals, each scaled to its own peak. Neither has a minimum.

The node a nodeless orbital has

Three lithium 2s functions, drawn as r·R® so that the outer lobe appears at its true weight. The Hartree–Fock function turns over at 0.83 bohr; the hydrogenic one at 1.54; the Slater function never does.

The same three functions in momentum, weighted by $p^2$, each scaled to its own largest magnitude, against momentum in units of lithium hydride’s π/R. All three have exactly one node — including the function with no node in position.

The two ends of each transform: the sign of the function’s sum, which is its momentum function’s sign at zero; its value and slope at the nucleus; the sign of the far tail; and the count of momentum nodes. Where sum and slope agree, there is a node.