Molecular Geometry & Bonding
what the shape decides
Orbitals

What an orbital is

Not a region the electron occupies, not a path it follows, and for any atom but hydrogen not an exact anything. An orbital is a one-electron wavefunction, and almost every difficulty in this subject comes from forgetting that.
12 min read 7 figures One electron onlySay what it enclosesTaught confidently, and wrong

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.

The 2pz orbitalThe 2pz orbital at the contour enclosing 90 per cent of its density — a level solved for by integration rather than chosen. The two colours are the two signs of the wavefunction, which is what distinguishes a bonding interaction from an antibonding one.2pzencloses 90% of the densitycontour at |ψ| = 9.48e-30 radial nodes1 angular nodecontour solved for by integrating the density1 shell · one electron
Fig. 1 A 2p orbital, drawn as what a picture of one always is — a contour surface, at a level enclosing ninety per cent of the density. The two colours are the two signs of the function, which is the single most important thing the picture carries.

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 ZZ and the solutions are a discrete family of functions, each labelled by three integers: nn, ll and mm.

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: ∣ψ∣2|\psi|^2 is a probability density, so ∣ψ∣2 dV|\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 ∣ψ∣2|\psi|^2 is observable, why does anybody care about the sign of ψ\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.

1s with 1s at 2.8 bohrThe two orbitals in the plane containing both nuclei, with the regions where their product is positive and negative shown faintly. The overlap integral is the signed volume of that product, and where symmetry makes the two regions mirror images it comes out exactly zero.1s · 1sS = 0.38997sigma interactionseparation 2.8 bohrcontours at 50% of each densitythe signed product integrated over all spaceone electron
Fig. 2 Two 1s orbitals close enough to interact, with the regions where their product is positive shown faintly. That product, integrated, is the overlap — and its sign is what decides which of the two combinations is the lower in energy.

So the sign is not observable in isolation and is entirely observable in its consequences, which is a perfectly ordinary situation for a phase. 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 on this site 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.

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

ll, the angular momentum quantum number, sets the shape and runs from 0 to n−1n-1. The letters are historical: l=0l = 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.

mm, the magnetic quantum number, sets the orientation and runs from −l-l to +l+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.

Orbitals at the 90 per cent contourSeveral orbitals drawn at the same enclosed fraction and, unless stated otherwise, at the same scale — so the sizes on the page are the sizes. Each contour was solved for separately by integrating that orbital's own density.one scale across the plate1s0 radial · 0 angular2s1 radial · 0 angular2pz0 radial · 1 angular3dz20 radial · 2 angulareach level solved for separately90% · one electron
Fig. 3 Four orbitals at one enclosed fraction and one scale. Size grows with nn; shape is set by ll; and the sizes on the page are the sizes, which panel-by-panel fitting would conceal.

Real orbitals and complex ones

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

The exact solutions with definite mm are complex functions. The familiar pxp_x, pyp_y, pzp_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.

The radial function of 1sThe radial part of the wavefunction, which changes sign at each node, and the radial distribution, which is the probability of finding the electron in a shell at that radius. The second vanishes at the nucleus and the first does not.most probable radius 1.00 a₀R(r)and 4πr²R²r / bohrmeasured off the computed function0 radial nodes · one electron
Fig. 4 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.

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 on this site 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−1n - l - 1 radial and ll 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.

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 an essay on this site discusses a many-electron system it says so.

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.

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 ∣ψ∣2|\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 orbitalThe 1s orbital at the contour enclosing 90 per cent of its density — a level solved for by integration rather than chosen. The two colours are the two signs of the wavefunction, which is what distinguishes a bonding interaction from an antibonding one.1sencloses 90% of the densitycontour at |ψ| = 3.94e-20 radial nodes0 angular nodescontour solved for by integrating the density1 shell · one electron
Fig. 5 1s: spherical, no nodes of any kind, and the simplest wavefunction there is. The contour is a sphere because the function depends on radius alone.
The 3dz2 orbitalThe 3dz2 orbital at the contour enclosing 90 per cent of its density — a level solved for by integration rather than chosen. The two colours are the two signs of the wavefunction, which is what distinguishes a bonding interaction from an antibonding one.3dz2encloses 90% of the densitycontour at |ψ| = 3.60e-30 radial nodes2 angular nodescontour solved for by integrating the density1 shell · one electron
Fig. 6 3d_z²: two lobes along the axis and a ring around the middle, which looks like an exception among the d orbitals and is not. It is a combination of two of the complex harmonics, and its odd appearance is a consequence of which combination was chosen to make the set real.
The 3dxy orbitalThe 3dxy orbital at the contour enclosing 90 per cent of its density — a level solved for by integration rather than chosen. The two colours are the two signs of the wavefunction, which is what distinguishes a bonding interaction from an antibonding one.3dxyencloses 90% of the densitycontour at |ψ| = 3.78e-30 radial nodes2 angular nodescontour solved for by integrating the density1 shell · one electron
Fig. 7 3d_xy: four lobes between the axes, with two nodal planes. The other three d orbitals with this shape differ only in orientation, and the fifth is the one above.

The d_z² 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.

Where the ladder goes next

The immediate consequence for every picture on this site is saying what the contour encloses, which is the discipline the whole collection is built on.

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.

Where this goes next

50% of the density|ψ| = 1.48e-190% of the density|ψ| = 3.94e-299% of the density|ψ| = 8.44e-31slevels solved for by integration, drawn at one scale Say what it encloses An orbital picture is a contour at a level somebody chose, and almost no source says which. Two textbooks can draw the same orbital at visibly different sizes with the same caption, and both be printed in good faith. node 1.90node 7.10most probable radius 13.10 a₀R(r)and 4πr²R²r / bohrmeasured off the computed function2 radial nodes · one electron Nodes An orbital with quantum numbers n and l has exactly n−l−1 radial nodes and l angular ones. That is a count, it is exact, and it is the fastest way to catch a drawing that is wrong. one scale across the plate1s0 radial · 0 angular2s1 radial · 0 angular2pz0 radial · 1 angulareach level solved for separately90% · one electron Orbitals are not where the electron is A many-electron atom has no exact orbitals at all. The orbital picture is a basis for an approximation — an extremely good one — and treating it as a description of reality is the source of most of the confusion in this subject.