Part A · What stuff is

3 · Electrons as clouds

Heat sodium and it glows a pure yellow-orange, the colour of old street lamps. Heat neon and it glows red. Every element gives off its own set of sharp colours, like a barcode. That barcode is the first clue to what electrons are doing inside atoms.

Chapter 1 treated atoms as fuzzy balls that attract and repel. Chapter 5 already used orbitals to explain bonds. This chapter opens the ball up: what the electrons are doing, why they come in shells, and why the outermost ones do all the chemistry.

Sharp colours mean fixed energies

If an electron in an atom could have any energy, a hot gas would glow in a smooth rainbow. Instead it emits only a few exact colours. The explanation: an electron bound to an atom can only have certain energies. When it drops from a higher level to a lower one, it emits a photon carrying exactly the difference, and the photon's colour is set by its energy (λ = hc/ΔE).

For hydrogen, the simplest atom, the allowed energies are remarkably tidy:

E_n = −13.6 eV / n², n = 1, 2, 3, …

Left: hydrogen's energy levels. Click a level to lift the electron there; it then falls back down, level by level, emitting a photon at each step. Right/bottom: the spectrum those photons build up, on a log wavelength scale. Only the drops that end on level 2 land in the visible range.

Which lower level the electron drops to is chosen at random here. Real atoms have definite probabilities for each jump, so real line strengths differ. The wavelengths are exact.

The four visible lines (red 656 nm, blue-green 486, violet 434 and 410) are the Balmer series, all ending on n = 2. Astronomers see them everywhere: the pink-red glow of nebulae such as Orion is mostly hydrogen's 656 nm line. Starlight also shows the same barcodes as dark lines, where atoms in a star's outer layers absorb exactly those colours. That's how we know what stars are made of. Helium was discovered in the Sun's spectrum in 1868, decades before anyone found it on Earth.

Standing waves, not orbits

Why only certain energies? Because an electron bound to a nucleus behaves like a wave trapped in a box, and a trapped wave can only form standing patterns, like a guitar string that can only play its fundamental and its harmonics. Each allowed pattern is an orbital, and its square is the probability cloud of where you'd find the electron.

The 2s orbital has more energy than 1s. What do you think its cloud looks like?

A ball with a shell around it, separated by an empty sphere. Higher-energy standing waves have more nodes, places where the wave passes through zero, just as a guitar string's second harmonic has a still point in the middle. In 2s the node is a sphere; in 2p it's a plane through the nucleus, giving two lobes of opposite sign. Pick orbitals below and count the nodes.

A slice through the nucleus of hydrogen's orbitals. Brightness is the probability density |ψ|² (square-root scaled so faint regions show). Blue and orange are the two signs of the wave, ψ > 0 and ψ < 0. Sign is what decides in-phase versus out-of-phase overlap in chapter 5's bonds. The scale bar is 1 Å.

These are exact hydrogen orbitals, shown as a 2-D slice. The real clouds are 3-D: 2p is a dumbbell with two lobes, and 3d_z² is a dumbbell through a doughnut. Other atoms' orbitals have the same shapes but different sizes and energies, because the other electrons shield part of the nucleus.

Common trap

"Electrons orbit the nucleus like planets." That picture (Bohr's 1913 model) happens to give hydrogen's energies correctly, but it's wrong in every other way. Electrons don't follow paths. An s orbital has no rotation at all and is densest right at the nucleus, and the planetary picture can't explain bonding, shapes or any atom beyond hydrogen. The school-book drawing of rings with dots on them is an energy-level diagram, not a picture.

Filling up: shells and the exclusion principle

An atom with many electrons is a much harder problem, since the electrons repel each other. But a good approximation keeps hydrogen's orbital shapes and fills them from the lowest energy up, following two rules:

The Pauli exclusion principle: each orbital holds at most two electrons, with opposite spins. This is why atoms don't all collapse into 1s, and ultimately why matter takes up space.
Hund's rule: within a set of equal-energy orbitals, electrons spread out one per orbital before pairing up, because electrons in separate orbitals repel each other less.

Atomic number Z

Orbitals as boxes, arranged by filling order (lowest at the bottom). Each arrow is an electron and its spin. The outermost shell (the valence electrons, which do the chemistry) is highlighted.

Notice the 4s boxes fill before 3d. In many-electron atoms the energy ordering isn't simply by n: the rule of thumb (the Madelung rule) orders subshells by n + ℓ, then by n. It mostly works. But try chromium and copper: each moves an electron from 4s into 3d, giving a half-full or full set of d boxes. About twenty elements break the simple filling rule in some way.

For chemistry the punchline is this: the electrons in the outermost shell, the valence electrons, are the ones that get shared, stolen and swapped. The inner ones sit close to the nucleus, held far too tightly to take part. Atoms with the same arrangement of valence electrons behave alike. Chapter 4 turns that into a map.

Physics / ML bridge

Orbitals are eigenfunctions of the hydrogen Hamiltonian, labelled by quantum numbers n, ℓ, m, plus spin. Each shell holds 2n² states (2, 8, 18, 32), and those are exactly the lengths of the periodic table's rows. Because of the filling order, each length after the first appears twice: 2, 8, 8, 18, 18, 32, 32. Filling is a priority queue: pop the lowest-energy free state, insert an electron, repeat.

For many electrons, the standard approach (Hartree–Fock) gives each electron the average field of all the others and iterates to self-consistency. That's a mean-field approximation, the same idea as mean-field variational inference: replace a hard joint distribution with a product of independent factors, each optimised against the others' averages.

The mean-field picture misses electron correlation, which is often what decides chemistry's fine details (most of quantum chemistry is about putting it back). The filling order shifts with charge: transition-metal ions lose their 4s electrons before 3d. For heavy elements, relativity matters, as chapter 1 said about gold.

Tell it to a 6-year-old

Look: A balloon and some tiny bits of torn tissue paper on the table.
Wonder: "Can a balloon pick up paper without touching it?"
Try: Rub the balloon on your hair (or a wool jumper) for a few seconds, then hold it just above the paper bits. Watch them jump up. Look at your hair too!
True sentence: "Every atom has tiny electric bits on its outside, called electrons. Rubbing moves some of them from your hair onto the balloon, and then the balloon can pull on the paper."

This is static electricity: electrons moving from hair to rubber, so the balloon ends up negatively charged and pulls on the paper by inducing charges in it. It's true at her level that electrons are on the outside of atoms and can move. Avoid "electrons go round and round like planets"; "a fuzzy cloud" is fine if she asks what they look like.

Play more

Falstad: hydrogen atom orbitals. Rotate the real 3-D clouds and mix them.
PhET: Models of the Hydrogen Atom. From the plum pudding to Schrödinger, with spectra.
Ptable. Its "Orbitals" view shows every element's electron configuration.
For the 6-year-old: look at a neon sign or an old fluorescent tube through a cheap diffraction grating (or the shiny side of a CD) and spot the separate coloured lines. Compare an ordinary filament bulb, which gives a smooth rainbow.