Orbitals and Subshells

An orbital is not a path, not an orbit, and not a container. It is a mathematical function. Its square, at any point in space, gives the probability of finding the electron in a small volume around that point — and that is the entire content of the idea.

Every difficulty with this topic comes from trying to hold on to a picture of the electron going somewhere. It does not go anywhere. Between measurements it has no position at all, only a distribution, and the balloon-shaped drawings in your textbook are contour surfaces of that distribution rather than objects.

Four numbers that address every electron

Electrons in an atom are organised by a strict nesting: shells contain subshells, subshells contain orbitals, orbitals contain at most two electrons. Four quantum numbers pick out exactly one electron by walking down that hierarchy.

• n, the principal quantum number: 1, 2, 3, … It sets the shell — roughly the size and the energy.
• ℓ, the azimuthal or angular momentum quantum number: any whole number from 0 up to n − 1. It sets the subshell and therefore the shape. ℓ = 0 is called s, ℓ = 1 is p, ℓ = 2 is d, ℓ = 3 is f.
• mₗ, the magnetic quantum number: every whole number from −ℓ to +ℓ. It distinguishes the individual orbitals within a subshell, which differ in orientation rather than shape.
• mₛ, the spin quantum number: +½ or −½, and nothing else.

The constraint on ℓ explains a fact that otherwise looks arbitrary. Shell 1 has only ℓ = 0, so only an s subshell — there is no such thing as 1p. Shell 2 permits ℓ = 0 and 1, so 2s and 2p and no 2d. Shell 3 is the first that can hold d orbitals.

Why 2, 6, 10 and 14

Count the allowed values of mₗ for a given ℓ and you get 2ℓ + 1 orbitals in the subshell. Each orbital takes two electrons, one of each spin, so a subshell holds 2(2ℓ + 1):

These four numbers are the widths of the four regions of the periodic table, and the connection is not a coincidence but a consequence — it is worked through on the blocks page.

Adding up across a whole shell gives n² orbitals and 2n² electrons. Shell 3 has one s, three p and five d orbitals: nine altogether, eighteen electrons. Committing 2n² to memory is fine; knowing it comes from counting mₗ values means you can rebuild it if it slips.

The letters are spectroscopists' adjectives

s, p, d and f are not initials of anything structural. They come from nineteenth-century descriptions of the appearance of spectral lines: sharp, principal, diffuse and fundamental. Spectroscopists had classified series of lines by how they looked long before anyone knew what produced them, and when the quantum theory arrived the existing labels were kept and attached to the angular momentum states that turned out to be responsible.

Past f the naming gives up and continues alphabetically — g, h, i — which is why the sequence looks random at the start and orderly afterwards. No known element in its ground state uses a g orbital, though calculations suggest elements beyond about 120 would.

Shapes, nodes, and the places an electron never is

An s orbital is spherical: the probability depends only on distance from the nucleus, not on direction. A p orbital has two lobes on opposite sides of the nucleus, and the three p orbitals in a subshell point along three perpendicular axes.

Four of the five d orbitals are four-lobed cloverleaves. The fifth, d(z²), looks entirely different — two lobes along one axis with a ring around the middle — and students reasonably suspect a mistake. There is no mistake, but there is an accounting subtlety worth knowing: six "natural" cloverleaf shapes can be drawn, only five of which are mathematically independent, and d(z²) is what you get when the redundant sixth is folded into the fifth.

A node is a surface where the wavefunction is zero, so the probability of finding the electron there is exactly zero. Two kinds exist, and their counts follow simple rules:

• angular nodes — flat planes or cones through the nucleus — number ℓ;
• radial nodes — spherical shells at fixed distances — number n − ℓ − 1;
• total nodes therefore come to n − 1.

A 3p orbital has one angular node and 3 − 1 − 1 = 1 radial node. A 2s orbital has no angular node and one radial node, which is why its cross-section shows a small sphere inside a larger shell.

Now the genuinely strange part. An electron in a 2s orbital is found on both sides of that radial node but never on it. If you insist on picturing a particle moving, you have to explain how it crosses a surface it can never occupy — and you cannot, because that picture is the mistake. The electron is a standing wave, and a standing wave on a guitar string has nodes for the same reason without anything having to travel past them.

The plus and minus signs, or the two colours, on textbook orbital pictures indicate the phase of the wavefunction. They are not charges. Phase matters enormously when orbitals combine into bonds, and not at all for a single atom.

Two rules for putting electrons in

The Pauli exclusion principle says no two electrons in an atom may have the same set of all four quantum numbers. Since n, ℓ and mₗ together specify an orbital, and mₛ has only two possible values, this is where "two electrons per orbital, opposite spins" comes from — it is a consequence, not a separate rule.

Hund's rule of maximum multiplicity governs subshells with more than one orbital: electrons occupy them singly, with parallel spins, before any orbital takes a second electron. Nitrogen's 2p³ is three orbitals with one electron each, all spins aligned. Oxygen's 2p⁴ is forced to pair one of them, leaving two singly occupied.

Two reasons lie behind it. Electrons in separate orbitals stay further apart and repel each other less, and parallel spins gain an additional quantum-mechanical stabilisation called exchange energy that has no classical analogue. The consequence is measurable: pairing that fourth electron into an already-occupied 2p orbital costs enough that oxygen's first ionisation energy is lower than nitrogen's, breaking the trend across period 2 — an anomaly explored on the ionisation energy page.

In hydrogen, 2s and 2p have the same energy. Nowhere else.

This is the single most useful thing on this page for understanding the periodic table, and most courses state it backwards.

For a one-electron atom, energy depends only on n. Hydrogen's 2s and 2p orbitals are exactly degenerate — identical in energy, differing only in shape.

Add a second electron and that degeneracy is destroyed. Inner electrons screen the nucleus, and how effectively they screen depends on the shape of the orbital in question. An s orbital has significant probability density very close to the nucleus, inside the inner shells, where it feels nearly the full nuclear charge; a p orbital penetrates less, a d orbital less still. So in every atom from helium onwards the ordering is 2s below 2p, 3s below 3p below 3d, and so on.

That penetration ordering is what makes the filling sequence non-obvious, and eventually makes 4s drop below 3d. The procedure for using it is on the electron configuration guide.

What the pictures leave out

Two caveats keep the model honest.

The surfaces drawn in textbooks usually enclose 90% of the probability. The wavefunction itself never reaches zero at any finite distance, so an atom has no boundary — a fact with direct consequences for how atomic radius has to be defined.

More fundamentally, exact orbitals are solutions for an atom with one electron. For everything else they are an approximation, because each electron's behaviour depends on where all the others are. It is an extraordinarily good approximation and the whole of chemistry is taught in its language, but a real many-electron atom is not literally a set of hydrogen-like orbitals with electrons dropped into them.