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Pictures of atomic orbitals are some of the most recognisable images in chemistry: fuzzy spheres, dumbbells and four-leafed clovers. But what do they actually represent, and why do they have those particular shapes? This article explains each orbital type, what the shapes mean physically, and why they matter for the way atoms bond.
What an orbital picture shows
An electron in an atom doesn’t follow a path. Quantum mechanics describes it with a wavefunction, ψ (psi), whose square, ψ², gives the probability of finding the electron at each point in space.
An orbital “shape” is usually drawn as a boundary surface: a surface enclosing the region where there’s about a 90% chance of finding the electron. The electron could be outside it; it’s just much less likely.
Orbital shapes come from solving the Schrödinger equation for the hydrogen atom. They depend on two quantum numbers:
- l (angular momentum quantum number) determines the shape: l = 0 is s, 1 is p, 2 is d, 3 is f.
- mₗ (magnetic quantum number) determines the orientation in space.
See quantum numbers explained.
s orbitals: spheres
- Shape: spherical, centred on the nucleus.
- Number per shell: one (1s, 2s, 3s…).
- Orientation: none; a sphere looks the same from every direction.
The electron density is highest near (but not at) the nucleus and falls off with distance. Larger shells give larger spheres: 2s is bigger than 1s, and 3s is bigger still.
Radial nodes
The 2s orbital has a spherical node: a shell-shaped surface inside it where the probability of finding the electron is zero. The 3s orbital has two such nodes. In general, an ns orbital has n − 1 radial nodes. These nodes are a sign of the electron’s wave nature, like the still points on a vibrating string.
p orbitals: dumbbells
- Shape: two lobes on opposite sides of the nucleus, like a dumbbell or a figure of eight in 3D.
- Number per shell: three (from n = 2 onward), because mₗ can be −1, 0 or +1.
- Orientation: along the three axes: pₓ, p_y and p_z, at 90° to each other.
Each p orbital has a nodal plane through the nucleus, where the probability is zero. For pₓ, it’s the y–z plane.
The two lobes have opposite phases (often shown as different colours, or + and −). Phase doesn’t mean charge; it’s the sign of the wavefunction, and it matters when orbitals overlap to form bonds.
In an isolated atom, the three p orbitals of the same shell have equal energy (they’re degenerate).
d orbitals: four lobes (mostly)
- Number per shell: five (from n = 3 onward), with mₗ = −2, −1, 0, +1, +2.
- Shapes:
- d_xy, d_xz, d_yz: four lobes lying between the axes in the xy, xz and yz planes.
- d_x²−y²: four lobes lying along the x and y axes.
- d_z²: two lobes along the z axis with a doughnut-shaped ring around the middle.
Each d orbital has two angular nodes (nodal planes or, for d_z², nodal cones).
d orbitals are central to transition metal chemistry. In complexes, ligands approach along the axes and interact more strongly with d_x²−y² and d_z² than with the other three. This splits the d orbitals into two energy groups, and the energy gap corresponds to visible light, which is why so many transition metal compounds are coloured. See transition metals.
f orbitals: complex shapes
- Number per shell: seven (from n = 4 onward).
- Shapes: complex, with up to eight lobes and three angular nodes each. They’re usually drawn from computer calculations.
f orbitals are filled in the lanthanides and actinides. They’re buried deep inside the atom, shielded by outer s and p electrons, which is why the lanthanides are so chemically similar to one another and so hard to separate. See rare earth elements and actinides.
Summary table
| Orbital type | l | Orbitals per subshell | Angular nodes | Typical shape |
|---|---|---|---|---|
| s | 0 | 1 | 0 | sphere |
| p | 1 | 3 | 1 | dumbbell |
| d | 2 | 5 | 2 | cloverleaf (and one dumbbell with a ring) |
| f | 3 | 7 | 3 | multi-lobed |
Total nodes in an orbital = n − 1 (radial nodes + angular nodes). For example, 3p has 2 nodes: 1 angular and 1 radial.
How size changes with n
For each type, orbitals get larger and higher in energy as n increases: 1s < 2s < 3s; 2p < 3p < 4p. The outer electrons of larger atoms are in larger orbitals, further from the nucleus. This underlies the atomic radius trend down a group.
Why shapes matter
Bonding
Covalent bonds form when orbitals on neighbouring atoms overlap:
- Sigma (σ) bonds form by head-on overlap along the line between the nuclei (for example, s–s, s–p or p–p end-to-end).
- Pi (π) bonds form by sideways overlap of parallel p orbitals, above and below the bond axis. Double bonds contain one σ and one π bond.
The directional shapes of p and d orbitals give bonds their directions, which is why molecules have definite shapes.
Hybridisation and molecular shape
In many molecules, s and p orbitals mix to form hybrid orbitals (sp, sp², sp³) pointing towards the corners of a line, triangle or tetrahedron. Methane’s tetrahedral shape (bond angles 109.5°) is explained this way. See VSEPR and molecular geometry.
Colour and magnetism
The energies and shapes of d orbitals explain transition metal colours and magnetism. See magnetic elements and why is gold yellow?
How do we know the shapes are real?
Orbital shapes come from calculations, but they’re supported by experiment. The directions of chemical bonds, measured by X-ray crystallography and spectroscopy, match what overlapping p and hybrid orbitals predict. The splitting of d-orbital energies in complexes, which follows directly from their shapes, explains measured colours and magnetic properties with great accuracy. Researchers have even used advanced microscopy techniques to image the electron density of individual molecules, showing features that resemble calculated orbital shapes.
Common misconceptions
- “The electron travels along the figure-of-eight.” The shape shows where the electron is likely to be found, not a path.
- “The two lobes of a p orbital hold one electron each.” A single electron in a p orbital is described by the whole orbital, both lobes together.
- “Orbitals have hard edges.” They fade out gradually; the drawn surface is just a 90% probability boundary.
- “How can an electron cross a node?” It doesn’t travel from lobe to lobe like a particle; as a wave, it occupies the whole orbital at once.
Key takeaways
- Orbital shapes are 90% probability boundaries derived from quantum mechanics.
- s orbitals are spherical; p orbitals are dumbbells along x, y and z; d orbitals mostly have four lobes; f orbitals are more complex.
- The number of each type per subshell is 1, 3, 5 and 7, and each can hold two electrons.
- Orbitals have nodes where electron probability is zero; an orbital has n − 1 nodes in total.
- Orbital shapes and orientations explain bond directions, molecular shapes and transition metal colours. See shells vs subshells vs orbitals.
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