This tutorial visualizes atomic orbitals as probability clouds—where the electron is likely to be found—instead of fixed orbits. The 3D shape comes from the wave function ψ of quantum mechanics: the probability density is |ψ|².

Mathematical foundation

1. Wave function and probability

For a hydrogen-like atom, ψ depends on three quantum numbers: n (principal), l (orbital angular momentum), and m (magnetic). The probability of finding the electron in a small volume is |ψ(r, θ, φ)|² dV. In spherical coordinates, dV = r² sin θ dr dθ dφ, so the density we sample is |ψ|² r² sin θ.

2. Radial part R(r)

The radial wave function sets the “shell” structure. With ρ = 2r/n (in units of the Bohr radius): 1s ∝ e^−ρ/2; 2s ∝ (2−ρ)e^−ρ/2; 2p ∝ ρ e^−ρ/2; 3s, 3p, 3d have higher-order polynomials in ρ times e^−ρ/2. Larger n means the electron is on average farther from the nucleus.

3. Angular part (spherical harmonics)

The angular part gives the iconic shapes: s (l=0) is constant (sphere); p (l=1) has lobes along one axis (e.g. p_z ∝ cos θ); d (l=2) has cloverleaf and donut shapes (e.g. d_z² ∝ 3cos²θ−1). Real spherical harmonics are used so the lobes align with x, y, z.

4. Aufbau and Hund’s rule

Electrons fill subshells in order of energy (1s, 2s, 2p, 3s, 3p, 3d…). Hund’s rule: within a subshell, fill each orbital with one electron (same spin) before pairing. The “Aufbau Filling Station” below shows this: ↑ in each box first, then ↓.

Usage

Quantum numbers: Use Principal (n) slider (1–4) and Subshell (l) dropdown (s, p, d, f for l = 0, 1, 2, 3). Orbital (m) selects which orientation (e.g. p_z, p_x, p_y or d_z², d_xz, …). The 3D cloud updates to show the probability density |ψ|² using rejection sampling; points are colored by the phase of ψ (cyan = positive, magenta = negative).

3D view: Drag to rotate, scroll to zoom. Use Iso, Front, Top, Side for preset camera views.

Aufbau Filling Station: Click +1 to add an electron and −1 to remove one. The diagram fills subshells in order (1s, 2s, 2p, 3s, 3p, 3d, 4s, 4p) following Hund’s rule: each orbital gets one spin-up (↑) before any gets a second spin-down (↓). This matches the Pauli exclusion principle (max 2 electrons per orbital).

Guided labs

🔬 Lab 1: The "Spherical Miracle" (Superposition)
Concept: Prove that while individual p-orbitals are directional, a full subshell is perfectly spherical.
Steps: (1) In Aufbau Filling Station, select preset Ne (10e). (2) Set Principal (n) to 2, Subshell (l) to p, Orbital (m) to All. (3) Set Slice to None. (4) Drag to rotate the view. The p_x, p_y, and p_z clouds together form a spherical cloud.

🧪 Lab 2: The 4s vs 3d "Energy Race"
Concept: Visualize why Potassium (Z=19) fills the 4s orbital before the 3d, despite 4 being a higher principal number.
Steps: (1) Select preset Ar (18e), then click +1 to reach 19e (Potassium). (2) Set Slice to Radial. (3) Set n=3, l=d, m=0 to view the 3d orbital (magenta/cyan by phase). (4) Then set n=4, l=s, m=0 to view the 4s orbital. Compare: the 4s has a small "inner lobe" very close to the nucleus (penetration), which lowers its energy below 3d.

☢️ Lab 3: The Node Finder (Cross-Section Lab)
Concept: Discover "dead zones" where electrons can never exist.
Steps: (1) Select preset Na (11e). (2) Set Principal (n) to 3, Subshell (l) to s, Orbital (m) to 0 (3s). (3) Set Slice to Hemisphere (xy) — horizontal. You will see two clear empty concentric rings inside the cloud—the radial nodes.