Atomic Orbitals

Choose a named real orbital or custom quantum numbers, select a view, then Render. Changing calculation inputs cancels old work and removes its result. Cancel and navigation terminate the worker; worker failure does not start a hidden calculation in the page. A hidden tab stops the job and releases the 3D view; Render restores a cached result when available. A point probe evaluates the orbital belonging to the current result.

The model is a normalized, nonrelativistic one-electron Coulomb wavefunction ψ = Rₙₗ(r;Z)Yₗᵐ(θ,φ) in Bohr units. It neglects finite nuclear mass, spin, relativity, screening and electron interactions. Z is a hydrogenic scale parameter (0.1–118), not an accurate many-electron atom model. The supported integers obey 1 ≤ n ≤ 12, 0 ≤ l < n and |m| ≤ l. Named px/py/d orbitals are real combinations and need not be eigenstates of Lz. Custom nonzero m uses complex spherical harmonics with the Condon–Shortley convention. Overall sign is conventional.

For Z = 1, ψ₁ₛ = exp(−r)/√π and ψ₂ₚ𝓏 = z exp(−r/2)/(4√(2π)). Thus 1s density at the origin is 1/π a₀⁻³; 2pz has a node in the xy plane and opposite signs above/below it. 2s has a radial node at r = 2 a₀. The number of radial nodes is n − l − 1. Angular nodes depend on the chosen real combination or complex state. The probe reports Re ψ, Im ψ and |ψ|² without rescaling to the view.

Density surfaces use a fraction of the estimated maximum |ψ|². Signed surfaces use a fraction of the estimated maximum |Re ψ|, with separate positive and negative levels. This is a level, not enclosed probability. Both 3D surfaces and point clouds use yellow/blue for the sign of Re ψ, not electric charge or the full complex phase. In slices, density uses a dark-to-light scale relative to that slice; signed brightness follows √(|Re ψ|/peak). The optional white shell is |Re ψ| at 12% of the selected level, not an exact node; it can also enclose low-amplitude tails.

The automatic extent is orientation independent: a radial scan finds the last R² value above 10⁻⁴ of its sampled peak and adds 5%. It does not specify enclosed probability. Manual domain diameter replaces that extent. Meshes/slices use a box of that width; cloud samples lie in the inscribed sphere. Tails are omitted, manual domains can cut through a surface, and high states require suitable grid resolution. Grid quality sets 48, 64, 112 or 144 samples per edge; Custom allows 16–160. The chosen grid is used directly.

Cloud radii use a 4096-bin numerical CDF of r²R², normalized inside the sphere. Directions use rejection sampling with the spherical-harmonic addition-theorem bound. The sample is reproducible for identical model and point-count inputs. It is a finite numerical approximation, not a trajectory or measurement sequence. The nucleus marker is enlarged. 3D axes use meters; model/probe and slice coordinates use a₀ ≈ 5.29177×10⁻¹¹ m. Plain wheel scrolls; Ctrl/Command + wheel zooms.

References: Richard Fitzpatrick, Hydrogen Atom and Spherical Harmonics.