Atomic Orbitals Visualiser

What an atomic orbital is, and what this draws

An atomic orbital is a one-electron wavefunction, ψ = R(r) Y(θ, φ): a radial part that sets how far out the electron is likely to be and an angular part that sets the shape. This visualiser draws the s, p, d and f orbitals of hydrogen and other one-electron ions in three dimensions from that exact solution of the Schrödinger equation, as a cloud of points whose density is |ψ|².

Two numbers decide everything you see. The shell n sets the size and the energy; the subshell l (0 for s, 1 for p, 2 for d, 3 for f) sets the shape. The orbital has n − l − 1 radial nodes, spheres where the wavefunction is zero, and l angular nodes, planes or cones through the nucleus, so n − 1 nodes in all.

How to use it

• Choose the shell and the orbital. Orbitals with l up to n − 1 exist, so p needs n of 2 or more, d 3 or more and f 4 or more. Pick one the shell cannot hold and the other control moves to the nearest pair that exists, with a line saying so.
• Turn the cloud by dragging it or with the arrow keys. The axes are labelled, so you can check that 2px points along x and 3dxy sits between the x and y axes.
• Switch to the cross-section to see the nodes. The cloud of a 2s or 3p orbital hides its inner lobe inside its outer one; a slice through the nucleus shows both, with the radial nodes as dashed circles.
• Read the plot under the scene. It is the radial distribution P(r) = r²R(r)², the chance of finding the electron at distance r from the nucleus whatever the direction, with the radial nodes marked where it touches zero.

Worked example: the 3p orbital

For 3p, n = 3 and l = 1, so there is 3 − 1 − 1 = 1 radial node and 1 angular node. The radial function of hydrogen is R₃₁ ∝ (1 − r/6a₀) r e^(−r/3a₀), which is zero at r = 6 a₀, or 318 pm. The angular node of 3pz is the xy plane, where z is zero.

The radial distribution is then P(r) ∝ r⁴(6 − r)² e^(−2r/3) with r in Bohr radii. Away from the node, setting its derivative to zero gives r² − 15r + 36 = 0, so it peaks at 3 a₀ and at 12 a₀. The outer peak is the taller one, so the most probable radius is 12 a₀ (635 pm), and the mean radius (3n² − l(l + 1))/2 is 12.5 a₀. The energy, −13.6 eV / n², is −1.51 eV, the same as 3s and 3d. Set the shell to 3 and the orbital to pz and the readouts show each of these numbers.

Nodes and radii for every subshell to n = 4

The pattern worth seeing is within a shell. The subshell with the most angular nodes has the fewest radial ones and the smallest mean radius, yet the s orbital, with the largest mean radius, also has the innermost lobe. 3s holds 1.4 percent of its probability inside its first node at 1.9 a₀, where 3d holds only 0.03 percent. That inner lobe is what lets s electrons penetrate the inner shells of a many-electron atom, where the same shapes are filled in a different order of energy.

What the two colours mean

Blue is where ψ is positive and orange where it is negative. The sign is not a charge and it cannot be measured on its own, because only |ψ|² is a probability. It matters when orbitals combine: lobes of the same sign overlap to build up electron density between two nuclei and form a bonding orbital, and lobes of opposite sign cancel. A node is exactly where the colour changes.

The p_x, d_xy and other named orbitals are real combinations of the complex solutions with +m and −m. The complex ones have the same probability in every direction around the z axis, so the complex 2p with m = 1 is a ring, not a dumbbell; the real ones point along axes, which is why chemistry draws those. For f the set shown is the general one built from m = 0, ±1, ±2 and ±3; some inorganic texts use a cubic set, a different mixture of the same seven.

What this does not cover

• Atoms with more than one electron. Their orbitals have the same shapes and node counts, but the radial functions shrink and the energies split by subshell, which no closed form captures. The periodic table shows the configurations that result.
• Spin, relativity and fine structure. The model is the textbook one-electron atom with an infinitely heavy nucleus, the Schrödinger (1926) solution. The reduced-mass correction would stretch hydrogen by about one part in 1836, below anything the scene can show.
• Orbitals beyond f. g orbitals and higher exist for n of 5 and up but are not occupied in the ground state of any known element, so the selector stops at f.
• Molecules. How orbitals on neighbouring atoms combine into bonds and shapes is what the VSEPR molecular geometry explorer takes up.

Common mistakes

• Reading the boundary as a hard edge. An orbital has no surface. The textbook outline encloses a chosen share of the probability, commonly 90 percent, which is what the 90% radius readout reports.
• Confusing the peak of |ψ|² with the most probable radius. For 1s, |ψ|² is largest at the nucleus itself, yet the most probable distance is 1 a₀, because the shell of radius r has an area that grows as r². The plot shows r²R² for exactly that reason.
• Counting nodes from the letter alone. 2p and 3p are both p, but 3p has one radial node that 2p lacks. The total is always n − 1.
• Treating the colours as charge. Orange is not negative charge; the electron density is positive everywhere, and only the sign of the wavefunction changes.
• Thinking an orbital is an orbit. The Bohr orbit of hydrogen has radius n² a₀, which matches the most probable radius of 1s, 2p, 3d and 4f only. The electron is not on a path at all, and the de Broglie picture of a standing wave around an orbit is where the de Broglie wavelength calculator comes in.

The energy differences between these levels are what an atom emits and absorbs as light; the photon energy calculator turns a level gap into a wavelength.