What an orbital is

An orbital is a function. It takes a position and returns a number, and that number can be positive or negative. It is not a region, not a shell, not a container, and not a path.

Almost every persistent confusion in this subject is a confusion about that sentence, so it is worth taking apart.

Where it comes from #

Solve the Schrödinger equation for one electron moving in the field of a nucleus of charge $Z$ and the solutions are a discrete family of functions, each labelled by three integers: $n$, $l$ and $m$.

Those functions are the orbitals. They are exact — genuinely, analytically exact — for that problem, which is one electron and one nucleus. Hydrogen, He⁺, Li²⁺.

For anything else they are not solutions of anything. They remain extremely useful as a basis: a set of functions in which an approximate answer can be expressed. That distinction between an exact solution and a useful basis is the whole content of the caution that runs through this site, and it gets its own essay.

What the number means #

The wavefunction’s value at a point has no direct physical interpretation. Its square does: $|\psi|^2$ is a probability density, so $|\psi|^2\,dV$ is the probability of finding the electron in a small volume there.

Two consequences follow that people find uncomfortable and that are simply true.

The electron is not anywhere in particular until it is measured, and the density is the complete description of what can be said about where it might be found.

The density is non-zero almost everywhere. It falls off exponentially, but it never reaches zero at any finite distance, so there is no surface beyond which the electron is not. That is why an orbital picture has to be a contour: the function has no edge to draw.

Why the sign matters #

If only $|\psi|^2$ is observable, why does anybody care about the sign of $\psi$?

Because orbitals are added. When two atoms approach, the combinations that describe the resulting system are sums and differences of atomic orbitals, and a sum and a difference are different functions with different energies. Where two lobes of the same sign meet, the wavefunction reinforces and density builds between the nuclei — a bonding combination. Where opposite signs meet, it cancels, density is expelled from between the nuclei, and the combination is antibonding.

So the sign is not observable in isolation and is entirely observable in its consequences, which is a perfectly ordinary situation for phase to be in. A picture that drew only the magnitude would look almost identical and would have thrown away the thing that makes chemistry happen.

That is why every lobe here is coloured by sign, and why the two colours chosen separate cleanly for every common form of colour vision.

What the picture is #

Given a function with no edge, drawing it means choosing a value and drawing the surface where the function takes it.

That is an isosurface, and the value is a choice made by whoever draws. Almost no source says which choice was made, which is why two textbooks can print the same orbital at visibly different sizes with the same caption. Saying what the contour encloses is the discipline this site is built around, and every orbital figure here states the fraction of the density inside it.

Three things it is not #

Not an orbit. The word is a deliberate softening of “orbit” and the softening is the point: there is no trajectory, no period, and nothing going round. Bohr’s orbits were circles with definite radii, and they were abandoned because they cannot be made consistent with what is measured.

Not a shell. A shell is a set of orbitals sharing a principal quantum number, and it is a bookkeeping category rather than a place. The 2s and 2p orbitals overlap in space substantially.

Not the electron’s home. An orbital is a function describing a possible state; an electron in that state has a density everywhere the function is non-zero. Asking which orbital an electron is “in” is meaningful only in the approximation where the electrons are treated one at a time, which is exactly the approximation that has no exact justification for more than one of them.

The quantum numbers, and what each controls #

Three integers label an orbital and each does one job.

$n$, the principal quantum number, sets the energy — for hydrogen, entirely — and roughly the size. It runs from 1 upward.

$l$, the angular momentum quantum number, sets the shape and runs from 0 to $n-1$. The letters are historical: $l = 0$ is s, from sharp; 1 is p, from principal; 2 is d, from diffuse; 3 is f, from fundamental. Those were descriptions of spectral lines long before anybody knew what they meant.

$m$, the magnetic quantum number, sets the orientation and runs from $-l$ to $+l$, which is why there are three p orbitals and five d orbitals.

Together they fix the number of nodes exactly, which is a countable claim and one of the most useful checks available on whether a drawing is right.

Real orbitals and complex ones #

A detail that causes real confusion when a reader meets both conventions.

The exact solutions with definite $m$ are complex functions. The familiar $p_x$, $p_y$, $p_z$ and the five d shapes are real linear combinations of them — equally valid descriptions of the same subspace, chosen because they are easier to draw and easier to relate to the axes of a molecule.

Nothing physical distinguishes the two sets. Which is used is a basis choice, and a basis is not a thing — the same point that will matter a great deal when hybrid orbitals appear.

The one place it bites is magnetism: the complex set are eigenfunctions of the angular momentum operator and the real ones are not, so in a magnetic field the complex set is the natural one. Everywhere else the real set is more convenient and equally correct.

The density, and where it actually is #

An orbital’s contour says where a chosen proportion of the density lies. It does not say where the electron is most likely to be found, and the two answers differ.

Three numbers for one orbital, all correct and all answering different questions. Where the electron is takes that apart properly, and it is the clearest case on the site of a picture being asked a question it does not answer.

What is computed here, and how #

Every orbital here is built from its quantum numbers rather than traced from a book.

The radial part comes from an associated Laguerre polynomial evaluated by its three-term recurrence, and the angular part from the real spherical harmonics written out explicitly. The density is integrated on a product Gauss–Legendre grid, and three things are then checked before anything is drawn: that the density integrates to one, that any two distinct orbitals on the same centre are orthogonal, and that the node counts are $n - l - 1$ radial and $l$ angular.

Those checks are cheap and they earn their place. A convention mismatch in the Laguerre index — and textbooks use at least two — produces a function with the right shape and the wrong number of nodes, which no amount of looking at the picture would reveal.

What it costs #

The whole apparatus is about six hundred lines and runs in a second or two per build, and the interesting part of the bill is not the arithmetic.

The polynomials cost a recurrence. An associated Laguerre polynomial evaluated by its three-term recurrence costs a handful of multiplications and is numerically stable, where the closed-form sum with alternating factorials is neither. That choice matters by 3d: the alternating sum loses several significant figures to cancellation, and the loss shows up as an orbital whose density integrates to 0.9997 rather than 1.

The integrals cost a grid. Normalisation, orthogonality and enclosed probability are all integrals of the density, evaluated by Gauss–Legendre quadrature. These functions decay exponentially, and a product Gauss rule converges several orders of magnitude faster than an evenly spaced sum with the same number of points — which is what makes an accuracy of a part in $10^6$ affordable in advance rather than a compromise.

The checks cost about as much again, and that is the ratio worth noticing. Roughly half the computation behind the pictures here is verification of the other half. Every orbital’s density is integrated to confirm it comes to one; every pair on a centre is integrated to confirm they are orthogonal; every node is counted by walking the function rather than by reading $n-l-1$ off the quantum numbers.

And the pictures cost far less than expected, which was the surprise of building this site. A three-dimensional isosurface is normally a volume grid and a marching-cubes pass, and the first plan here was exactly that. It was never needed. A hydrogenic wavefunction separates as $\psi = R(r)\,Y(\Omega)$, so the surface where $|\psi| = c$ meets each direction wherever $|R(r)| = c/|Y(\Omega)|$ — a one-dimensional root-find along a ray, repeated over a few hundred directions. No volume grid, no resampling, no bitmap anywhere on the site.

That is not merely cheaper. It produces a picture the grid method gets wrong: because each ray is solved separately, an orbital with a radial node comes out as nested shells rather than as one blob, and the shell count is exactly the radial node count plus one. Almost every published 2s picture is a single ball, which is what the surface looks like only if the contour is high enough to lose the inner region entirely.

The reason for spending half the budget that way is specific rather than general. Each of these functions is plausible over a wide range of wrongness. A radial polynomial with a mistaken coefficient still decays, still has lobes, still looks like an orbital, and still produces a picture nobody would question. The three checks above are the only things capable of noticing, and each of them is worth feeding a deliberately wrong input — a contour claiming the wrong fraction, an orbital declared orthogonal to itself, a surface claimed to have one shell when it has two — because a check that has never rejected anything is not evidence of anything.

The three things this makes possible #

It is worth naming what the effort in the previous section buys, because on its own an accurate hydrogenic function is not obviously worth six hundred lines.

Contours that state what they enclose. The site’s central discipline needs the density integrable to a part in a million, because a contour level is found by bisecting on that integral and a level accurate to a per cent would make the whole exercise pointless.

Overlaps that vanish exactly. An interaction that symmetry forbids returns around $10^{-17}$ here rather than a small number, and that is only meaningful because the same integrator agrees with the closed-form 1s overlap to six parts in a hundred thousand. Without the second fact the first could be dismissed as a defect of the method.

Node counts that can contradict the formula. Walking the computed function and finding $n-l-1$ radial nodes is a check on the radial polynomial; reading $n-l-1$ off the quantum numbers is not a check on anything.

Each of the three is a claim this site makes elsewhere, and each rests entirely on the functions here being right rather than merely being plausible.

Where the model stops #

Two limits, both standing.

One electron. Everything here solves the one-electron problem. A many-electron atom is not exactly soluble, its electrons are not independent, and the orbital description of it is an approximation whose quality is a real question. Where a many-electron system is discussed, that is said plainly.

No relativity, no spin-orbit coupling, no nuclear motion. These are non-relativistic solutions for a point nucleus of infinite mass. For hydrogen the corrections are small; for a heavy element they are not, and the shapes of the orbitals of gold are genuinely different from the hydrogenic ones.

The orbitals most calculations use are defined as fictitious #

The essay’s caution — that an orbital is a one-electron function and for any atom but hydrogen not an exact anything — has a modern form that is sharper than the historical one, because the orbitals in most widespread use today are defined to belong to a system that does not exist.

The construction is this. Take the real molecule, with its interacting electrons, and ask what non-interacting system would have exactly the same electron density. Such a system exists, its electrons move independently in a potential chosen to make the densities match, and because they are non-interacting it has orbitals in the exact sense — genuine one-electron eigenfunctions of a genuine one-electron Hamiltonian.

Those are the orbitals a density-functional calculation produces, and they are the orbitals plotted in most published pictures of molecular bonding.

Their status is unusually clear and unusually awkward. They are exact solutions of a fictitious problem, chosen so that one property — the density — comes out right, and nothing guarantees that any other property of them corresponds to anything. Their energies are not ionisation energies, their shapes are not measured, and the system they solve was constructed rather than found.

So the essay’s warning does not soften with better methods; it changes form. The older difficulty was that a many-electron atom has no exact orbitals. The current one is that it has a set of exact orbitals belonging to an atom that is not the one being studied — which is a more precisely stated fiction and no less a fiction.

Who worked it out #

Schrödinger published the equation and its hydrogen solutions in 1926, in four papers over six months — one of the more remarkable bursts of productivity in the history of physics.

The interpretation came separately and less comfortably. Born proposed later that year that $|\psi|^2$ is a probability density, in a footnote added in proof; Schrödinger disliked the probabilistic reading and spent much of his life arguing against the direction it took. The word “orbital” is later still, coined by Mulliken in 1932 precisely to avoid the word “orbit” and its Bohr-model associations.

That the vocabulary was designed to prevent a misunderstanding, and that the misunderstanding is nevertheless the commonest one in the subject, is worth noticing. Naming a thing carefully does not stop people picturing it the old way.

What each orbital looks like #

A tour, because the shapes are the part everybody remembers and the reasons for them are worth attaching.

The 1s is the only one of the shapes with no directional information in it at all, which is why it is the one that can be drawn as a single number — a radius. Everything below has a direction, and the pictures start carrying signs.

The dz² shape is worth a sentence because it is the one students find suspicious. There are five d orbitals and the five real combinations cannot all be made to look alike — four come out as four-lobed and the fifth as a doughnut with two lobes. Nothing physical distinguishes it; a different choice of real combinations would spread the oddity differently, and the complex set from which they are built is perfectly uniform.

That is a basis choice showing through, and it is the first place most people meet one without being told that is what it is.

What follows from an orbital #

The immediate consequence for every orbital picture is saying what the contour encloses, which is the discipline every honest drawing of one needs.

The immediate structural feature is the node — countable, exact, and the fastest way to catch a wrong drawing.

And the immediate caution, worth reading early, is that these are not where the electron is.

What the pictures here cannot show. An orbital has no edge, and every figure on this page draws a contour that does. The density outside the surface is not zero; the picture shows where a chosen proportion lies and cannot show where the rest goes.