OPOpenPithMake curiosity visibleChemistry lab / Atomic orbitals and electron clouds

About this lab

An electron has no path, only a probability of where it will be found.

The first three experiments look at a single electron: the dot cloud is a record of repeated position measurements, the isosurface is a boundary that encloses most of the probability, and the radial distribution tells you at what distance from the nucleus the electron is most likely to be found. The last four move to many-electron atoms: the aufbau principle sets the order in which electrons enter orbitals, the Pauli principle and Hund’s rule decide how a box is filled, and the quantum numbers give every orbital an address.

Dot clouds, isosurfaces, curves and levels all come from the same exact hydrogen wavefunctions, not from pre-drawn pictures. One dot in the cloud is not one electron; the colours show the positive and negative phase of the wavefunction ψ, not electric charge.

What the seven experiments show

What the seven experiments show

  • 01 Electron cloud: the 1s dots are densest near the nucleus, yet only 32.3 % of them lie within 1 a₀ of it. Dense dots mean a high probability per unit volume, not an electron circling there.
  • 02 Orbital shapes: an s orbital is a sphere, a p orbital is a two-lobed dumbbell, and most d orbitals are four-leaved. The isosurface encloses 90 % of the probability; at 50 % the boundary shrinks towards the nucleus and the lobes keep their directions.
  • 03 Radial distribution: the 1s r²R² peaks at r = a₀ = 52.9 pm. 2s has one radial node at r = 2 a₀ and 3s has two (1.90 a₀ and 7.10 a₀); there are n − l − 1 radial nodes and l angular nodes.
  • 04 Aufbau: electrons enter subshells in order of increasing n + l, and for equal n + l the smaller n goes first. So 4s (n + l = 4) fills before 3d (n + l = 5).
  • 05 Electron configuration: an orbital holds at most two electrons with opposite spins (Pauli principle); within a subshell electrons first occupy separate orbitals with parallel spins (Hund’s rule). Nitrogen’s three 2p electrons sit in three boxes with parallel spins. Chromium is 3d⁵4s¹ and copper is 3d¹⁰4s¹: half-filled and filled exceptions.
  • 06 Quantum numbers: l can only be 0 to n − 1 and m only −l to +l. In hydrogen the energy depends on n alone: E₁ = −13.6 eV, E₂ = −3.40 eV, E₃ = −1.51 eV, E₄ = −0.85 eV.
  • 07 Orbital library: shell n has n² orbitals and holds at most 2n² electrons; there are 30 orbitals with n ≤ 4. Shell 4 has 16 orbitals and holds at most 32 electrons.

Model and sources

Model and sources

  • Hydrogen wavefunctions: ψ = R(r)·Y(θ, φ) in atomic units, lengths in Bohr radii a₀ = 52.9 pm. The radial part uses generalised Laguerre polynomials and the angular part real spherical harmonics; numerical integration confirms ∫|ψ|² dV = 1 to better than 10⁻⁶.
  • Dot cloud: |ψ|² splits into a radial part r²R² and an angular part |Y|²; each is rejection-sampled with a fixed random seed and the dots accumulate in order, so the same settings always give the same dots.
  • Isosurfaces: ψ is sampled on a 44 × 44 × 44 grid and marching tetrahedra extract the two surfaces ψ = +c and ψ = −c; c is the value that encloses 50 %, 70 % or 90 % of the probability.
  • Energy levels: hydrogen Eₙ = −13.6 eV / n². The level order of many-electron atoms follows the Madelung (n + l) rule; the height of the level ladder shows order only, not true energy. The ground states of chromium and copper are taken from measured data.

Many-electron orbitals borrow the hydrogen shapes, a teaching approximation; in real atoms the electrons repel each other, so orbitals contract and energies split.

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