The radial function of 2s

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.

What an orbital is

The radial function of a 1s orbital and its radial distribution. The wavefunction is largest at the nucleus and the probability of finding the electron at a given radius is largest at one bohr — because there is more volume in a shell further out. The ninety-per-cent contour is at 2.661.

Orbitals are not where the electron is

The feature that explains the ordering: the inner peak of the 2s distribution, which penetrates the screening of the 1s shell. That is a many-electron argument made with one-electron functions, and it works.

Nodes

The radial function of a 3s orbital, which crosses zero twice. Those crossings are its two radial nodes, and their positions were found by walking the computed function rather than looked up.

The 2s radial distribution, showing the small inner peak inside the node. That peak is the penetration: density close to the nucleus, where the other electrons screen it least, and it is why 2s lies below 2p in every atom but hydrogen.

Where the electron is

The radial function of a 1s orbital and its radial distribution. The first is largest at the nucleus and the second peaks at one bohr, and the difference between them is entirely a matter of how much volume there is at each radius.

The 2s radial distribution, with two peaks separated by the node. Most of the density is in the outer peak, and the small inner one is close to the nucleus — much closer than any part of a 2p.

The 2p radial distribution for comparison: a single peak, no inner lobe, and nothing close to the nucleus. That absence is the reason 2p lies above 2s once there is more than one electron.

The radial distribution across the periodic table

The radial distribution of a 3s orbital. Most of the density is in the outer peak, and two small inner peaks sit much closer to the nucleus than anything a 3d orbital has. Those inner peaks are what decides the filling order.

A 3p distribution: two peaks, with the inner one much smaller and further out than 3s’s. Less penetration, so less stabilisation, so 3p lies above 3s in every atom with more than one electron.

And 3d: a single peak with nothing inside it at all. Zero radial nodes, no inner structure, and no density close to the nucleus. That absence is the reason 3d lies highest of the three.

How big is an orbital

The 1s radial function and its radial distribution. R® is largest at the nucleus and never changes sign; the distribution vanishes at the nucleus, because there is no volume there, and peaks at exactly one bohr. Two curves from one function, answering two different questions, and the disagreement between them is where three of the four measures above come from.

A filled shell has no shape

The half of the wavefunction the theorem does not touch. The radial function of a 2p orbital, which is the same for all three members of the shell and which carries every statement about size, penetration and where the density is.

The atom does not bring its own orbital

The function whose one parameter is being varied. Changing the exponent from 1 to 1.238 pulls this curve inwards by a quarter and raises its maximum by the same proportion — the entire content of the variational improvement.

A slice is not the surface

The radial distribution of a 2s, which is the density weighted by the volume of the shell it sits in. This is the function the three-dimensional enclosed fraction integrates and the function a plane through the nucleus does not see — the plane samples the density at each radius and gives it the weight of a circle rather than a sphere.

Closer is not more overlap

The radial function the first case turns on. It crosses zero at two bohr and is negative outside — so its product with anything is positive in one region and negative in another, and the overlap integral is a difference rather than a sum.

The orbital in momentum space

The radial distribution of a 2s in position, with its radial node. The node is what makes the 2s and the 2p differ so much in ⟨r²⟩ — 42 against 30 — while leaving ⟨p²⟩ identical at a quarter, because the node costs curvature at short range and buys reach at long range in exactly compensating amounts.

A function that is already there

The function the whole exercise is approximating: a 1s orbital, with a cusp at the nucleus and an exponential tail. Neither end is a Gaussian’s shape, which is why a basis of them needs several functions and therefore runs into the problem this essay is about.

How nearly a broken symmetry survives

Why the two orbitals of one shell feel a radial screening differently: an s function has density at the nucleus and a p function does not, so a term that grows as the inverse square of the distance reaches one of them and barely touches the other.