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. 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.

Orbital selection and scale: The menu contains all 650 real basis functions with n = 1…12, grouped by principal quantum number. Higher shells use m ≥ 0: cos(mφ) and sin(mφ) distinguish the two normalized real combinations of ±m; these are not individual magnetic-quantum-number eigenstates. For n above 12 or multi-electron atoms this model is not applicable. Fine lobes and radial nodes can be lost with a coarse grid or a high surface level; use a slice or increase quality to inspect them. The 3D scale bar stays at the bottom right and changes with zoom. It measures lengths in the plane perpendicular to the view through the orbital center (perspective makes nearer and farther parts appear at different scales). Optional X/Y/Z axes indicate orientation without ticks. 1 nm = 1000 pm; 1 a₀ ≈ 52.918 pm.

NIST DLMF 14.30

In automatic 3D surface mode, the grid focuses on the region that can reach the selected level, using the radial function and the spherical-harmonic bound with a margin. This preserves small inner lobes in high-n states; it does not include every low-density outer lobe. Clouds and slices retain the full tail region. A manually chosen diameter is used unchanged.