What an orbital is, and is not

Not a track. A standing wave with a shape, and the shapes are where the blocks of the table come from — because solving one equation forces exactly 2, 6, 10 and 14 electrons into the four kinds of subshell.

1913 – 201316 min

By the end you should be able to

Say what an orbital is — a one-electron standing wave — and what it is not
Count the orbitals in a subshell from the allowed values of l and m
Explain why the four blocks of the periodic table are 2, 6, 10 and 14 wide
Read a node off a picture of an orbital, and say why a node rules out a trajectory

Start with the picture you were given first, and take it seriously, because it worked. Niels Bohr, 1913. Put the electron on a circular track around the nucleus, and allow only the tracks whose angular momentum is a whole-number multiple of a fixed unit. Energy is released when the electron drops from one allowed track to another, and the energy of the photon is the difference. Out of that came the wavelengths of the hydrogen spectrum, correct to four figures — and, far more impressively, the Rydberg constant itself, a number Johannes Rydberg had extracted from measured spectral lines in 1888 with no idea what it was made of. Bohr wrote it out of the electron mass, the electron charge and Planck’s constant. That is not a toy. A model that predicts a measured constant from unrelated constants has earned the benefit of the doubt, and it is why the little solar system is still the first thing anyone draws when asked to draw an atom. It is also wrong in a way that cannot be repaired, and the repair is the subject of this lesson.

Four failures, and none of them is the kind you patch. Add one electron and it stops. Nobody has ever got the spectrum of helium out of Bohr orbits. Not approximately — at all. A theory that works for exactly one atom in the universe is describing a coincidence or a special case. It gets hydrogen’s own ground state wrong. Bohr’s lowest orbit has one unit of angular momentum. The ground state of hydrogen has zero, which was measured and which a circular track cannot produce, since a particle going round in a circle necessarily has angular momentum. It says nothing about intensities. Spectral lines differ enormously in brightness, and the model has no opinion on any of them. And the quantisation was assumed. This is the one that mattered to Bohr. He had to postulate that only certain orbits exist. Nothing in the theory says why the others are forbidden, and a postulate that does the entire work of a theory is an admission rather than an explanation.

A black and white portrait photograph of Erwin Schrödinger in 1933, in a suit and round spectacles, facing the camera. — Schrödinger published the wave equation in four papers over six months in 1926, and solved hydrogen in the first of them. He disliked what became of it: he wanted the wave to be the electron, spread out in space, and spent the rest of his life unreconciled to Born’s reading of it as a probability.

An orbital is a standing wave, and standing waves are quantised for free

Louis de Broglie, 1924: the allowed orbits are the ones where a wave of the electron’s own wavelength fits a whole number of times around the loop. That single sentence converts Bohr’s postulate into a consequence. Quantisation is not a rule imposed on a particle. It is what happens to any wave confined to a region. A violin string of a fixed length can vibrate at some frequencies and not others, and nobody finds that mysterious. So stop asking about the orbit. Ask the question a wave allows you to ask: what shapes can a wave take in the field of a nucleus? Schrödinger asked exactly that in 1926, and hydrogen is one of a handful of physical systems where the question has an exact closed-form answer. The solutions are standing waves. Each one is labelled by three whole numbers — n, l and m — and the numbers are not put in by hand. They appear because those are the only shapes that stay finite everywhere and join smoothly onto themselves. Everything else in this lesson, including the shape of the periodic table, is a consequence of that requirement.

Start at n = 1 and look at the sphere. Then step n up to 2 and find the dark ring. Blue is where the wavefunction is positive, amber where it is negative, and dark where it is exactly zero.

Loading the orbitals…

A slice straight through the nucleus, with the wavefunction coloured rather than a surface drawn around it. The usual balloon-shaped picture hides two things: that there is no edge — the density falls away exponentially and never reaches zero — and that there is structure inside. From outside, a 2s is indistinguishable from a 1s. The whole difference is a spherical shell partway in where the electron is never found.

Now the question that the picture is designed to provoke. The electron is found inside the dark ring. It is also found outside it. It is never found on it. So how does it get across? It does not, and the question is malformed. "Getting across" assumes a path, and the node is precisely what rules a path out. A guitar string plucked into its second mode has a point at the centre that never moves, while the two halves move in opposite directions at every instant. Nobody asks how the vibration travels through the still point. There is no travelling vibration. There is one standing wave, with a still point in it. An orbital is that, in three dimensions and with the still points as surfaces. The count is fixed and worth memorising, because it is the easiest check you can run on any picture of an orbital: l angular nodes — flat planes or cones through the nucleus, where the shape changes sign from lobe to lobe. n − l − 1 radial nodes — spherical shells at fixed distances. n − 1 in total, always, however they are divided between the two kinds. A 3s has two radial and no angular; a 3d has none radial and two angular; a 3p has one of each. All three are the third shell, and all three have two nodes.

Change l and the shape changes with it, and this is where chemistry starts caring. l = 0, the s orbitals. Spherical. There is no direction in them at all — an s orbital points nowhere, which is exactly why an atom bonded only through s orbitals would have no shape. l = 1, the p orbitals. Two lobes on opposite sides of the nucleus, of opposite sign, with a flat node through the middle. There are three of them, at right angles: p_x, p_y, p_z. This is where direction enters chemistry, and the reason a water molecule can be bent at all. l = 2, the d orbitals. Four lobes in a plane, alternating in sign, with two angular nodes between them — except for d_z², which is two lobes with a ring around the waist and looks like the odd one out. It is not: the oddity is a consequence of choosing real orbitals with directions in them rather than the complex ones, and the five together are perfectly symmetric. l = 3, the f orbitals. Seven of them, hard to draw and impossible to hold in your head. That does not matter in the slightest, because what shapes the table is not the shapes. It is how many there are. The sign is not decoration, and it is worth flagging now for the act after this one. When two orbitals overlap, whether the lobes that meet carry the same sign decides whether the waves add or cancel — and that is the difference between a bond and no bond.

Drag l across 0, 1, 2, 3 and watch which block lights up. Read the row lengths down the left-hand edge.

Loading the orbitals…

The periodic table in its long form, so the f block is not exiled to a footnote and the widths can be counted. Every width on this diagram is 2(2l+1). The row lengths — 2, 8, 8, 18, 18, 32, 32 — are those widths added up in the order the subshells fill.

The answer to the question this act opened with

Act 5 began by pointing out that periodicity is a fact about measurement. Mendeleev sorted the elements by atomic weight and by how they behaved, found that the behaviour repeated, and built a table with gaps in it. He was completely right and had no idea why. Here is why. The rows are the lengths they are because that is how many ways a standing wave can sit in a Coulomb field, doubled for spin. Two columns of s block. Six of p. Ten of d. Fourteen of f. Those four numbers were not extracted from chemical data and then explained afterwards — they fall out of the constraints on l and m before any chemistry is mentioned at all. It is worth pausing on how strange that is. A table assembled in 1869 out of atomic weights, valencies and the colours of salts turns out to be a picture of the solution space of a differential equation written in 1926. Nobody arranged that. The measurement and the mathematics arrived from opposite directions and met.

The experiment

Hydrogen atoms under magnification

Aneta Stodolna, Arnaud Rouzée, Marc Vrakking and colleagues at AMOLF, with collaborators in Lyon, Ioannina and Auburn · Published in Physical Review Letters, May 2013 · Amsterdam

The question: The nodal structure of an atomic wavefunction is calculated, not seen. Can it be projected onto a detector and photographed directly?
The apparatus: A beam of hydrogen atoms, excited by two lasers into a highly excited state while sitting in a static electric field. The field lets the electron escape; an electrostatic lens then magnifies its outgoing wave by a factor of around twenty thousand onto a two-dimensional position-sensitive detector. The pattern that builds up is an interference pattern between the paths the electron wave can take out of the atom, and it carries the node count of the state it came from.

Theory predicted

The four states selected were calculated to have 0, 1, 2 and 3 nodes in the relevant coordinate. If the nodal structure is real rather than a bookkeeping device, the detector should show that many dark rings, and no others.

They measured

Exactly that. Concentric rings, counted directly off the image: none, one, two, three, in the four successive states. The pattern matched the calculated probability distribution across the whole detector, not merely in the number of rings.

How sure could they be? The magnification is the point: the structure being resolved is a few nanometres across, and the projection expands it to millimetres. Individual electron impacts were accumulated over many shots, so the image builds up one particle at a time — the same way the interference pattern does in the double-slit experiment.

Why it mattered

The nodes are not a calculational artefact. They are a structure in the world that can be magnified and counted. This is as close as anyone has come to a direct image of the interior of an atomic wavefunction, and it agreed with an equation solved eighty-seven years earlier.

Now be careful about what that image is, because it is routinely oversold — including by people who should know better, and occasionally by the press releases of the groups who made it. The states are Stark states. Hydrogen in a strong static electric field, not hydrogen sitting alone. The field is what lets the electron escape to be detected in the first place, and it distorts the states it is imaging. The detector records |ψ|², not ψ. Squaring throws the sign away, and in the next act the sign is the whole story: it is what decides whether two overlapping orbitals add into a bond or cancel into nothing. What was photographed is the probability, and the probability is genuinely less information than the wavefunction. And every picture in this lesson is a surface of constant probability density, or in the simulation above, a slice through one. The convention is usually to draw whatever surface encloses 90% of the probability. That is a choice. Choose 99% and every orbital gets visibly bigger. There is no edge to find, only a threshold to pick.

You might think

An orbital is the path the electron follows — a fuzzy orbit rather than a sharp one.

Actually

There is no path, and the nodes are the cleanest proof of it. A 2s orbital has a spherical surface on which the electron is never found, with regions of substantial probability on both sides of it. Any trajectory from the inner region to the outer one must cross that surface, and it cannot, because crossing means being there. The standing wave has no more of a trajectory than the vibration of a drumhead does. What the orbital gives you is a probability of finding the electron in a given volume, if you look — and looking is a physical process that ends the standing wave. The word "orbital" is itself a fossil: it was coined by Robert Mulliken in 1932 as a deliberate shortening of "one-electron orbital wavefunction", and the resemblance to "orbit" is the whole reason the wrong picture survives.

Has anyone actually seen an orbital?

In 1999 a group at Arizona State — Jian-Min Zuo, John Spence and colleagues, with Michael O’Keeffe — combined convergent-beam electron diffraction with X-ray data to map the electron density in cuprite, Cu₂O, and published it in Nature under a title that said they had directly observed d-orbital holes. The density around the copper atoms was visibly non-spherical, with a deficiency pointing along the copper–copper direction, exactly where a filled d shell should have a hole. The response was a small philosophical war. Eric Scerri argued in the Journal of Chemical Education that orbitals are in principle unobservable — they are one-electron wavefunctions, and a many-electron atom does not have any, so nothing that exists was seen. Others pointed out that the measurement was real, the asymmetry was real, and refusing to call it a d orbital is a semantic position rather than an experimental one. Both sides are right about something, and the disagreement is worth understanding rather than resolving. What is measurable is the total electron density, which is one function of position that includes all the electrons at once. What an orbital is, strictly, is a one-electron wavefunction — an exact object for hydrogen and an approximation everywhere else, and not a uniquely defined one, since the same total density can be built from different sets of orbitals. So the density in cuprite was measured and has the shape a d-orbital picture predicts. Whether that counts as seeing an orbital depends on what you thought an orbital was. This site’s answer: an orbital is the best one-electron description of a many-electron atom, it is extraordinarily useful, it predicts the shape of the periodic table exactly, and it is not a thing sitting inside the atom. All four of those are true at once.

Problem

Where the node is

The 2s orbital of hydrogen has one radial node — a spherical surface on which the electron is never found. Its radial wavefunction is R_2s(r) ∝ (2 − r/a₀) · e^−r/2a₀ How far from the nucleus is that surface, in picometres?

Bohr radius: a₀ = 52.918 pm
Radial nodes: n − l − 1
For 2s: n = 2, l = 0

The motion of particles conforms to the laws of probability, but the probability itself propagates in accordance with the law of causality.
Max BornZur Quantenmechanik der Stoßvorgänge, 1926. Born first wrote that ψ gives the probability, and corrected it to the square of ψ in a footnote added while the paper was in proof. Every picture in this lesson is drawn from that footnote.

1869Mendeleev arranges the elements by weight and behaviour, and leaves gaps for elements nobody has found.
1913Bohr puts the electron on quantised circular orbits and gets the hydrogen spectrum right to four figures.
1924De Broglie: the allowed orbits are the ones a whole number of electron wavelengths fit around. Quantisation becomes a consequence.
1925Pauli’s exclusion principle. Uhlenbeck and Goudsmit propose electron spin, supplying the fourth quantum number it needs.
1926Schrödinger solves hydrogen exactly. Three quantum numbers fall out of the boundary conditions rather than being assumed.
1926Born reads |ψ|² as a probability density, in a footnote added in proof. Schrödinger never accepts it.
1932Mulliken coins the word "orbital", as a shortening of "one-electron orbital wavefunction".
1999Zuo, Spence and O’Keeffe map the charge density in Cu₂O and report seeing d-orbital holes. A ten-year argument follows about whether that is possible even in principle.
2013Stodolna and colleagues magnify the nodal structure of excited hydrogen twenty thousand times onto a detector, and count the rings.

Where this leaves us

The shape of the periodic table is explained, and explained by counting rather than by measuring. The rows are 2, 8, 8, 18, 18, 32 and 32 elements long because that is how many ways a standing wave can sit in a Coulomb well, doubled for the two states of spin. Mendeleev’s empirical arrangement and Schrödinger’s differential equation arrived from opposite directions and met. But notice what has not been explained. The shapes account for the width of a row. They say nothing about what changes as you walk along one. Lithium and fluorine are both in the second shell. They have the same orbitals available, the same shapes, the same nodes. And fluorine holds on to its outermost electron more than three times as hard, is half the size, and will strip an electron off almost anything it meets. Something is changing steadily across a row, and it is not the shapes. There is only one thing left in the atom that it could be.