Hydrogen 4f Electron Orbital Probability Distribution

Here is a 3D view of a hydrogren atom in the 4f state. The “gas-like” cloud is composed of a volume of 2D cross-sectional layers. The animations were made in POV-Ray using DF3 density files. The right animation shows what a hypothetical 12o orbital might look like.

POV-Ray has a built-in internal function for the 3d orbital:
// runtime: 4 seconds
camera{location 16*z look_at 0} #declare P=function{internal(53)}; #declare P0=P(0,3,0,0);
box{-8,8 pigment{rgbt t} hollow interior{media{emission 0.5 density{function{(P(x,y,z,0)-1.2)/(P0-1.2)} color_map{[0 rgb 0][1 rgb 1]}}}}}
Mathematica 4.2, 8/22/04; animated version: C++, POV-Ray 3.6.1, 11/27/05

Links

• HydrogenLab – animations of hydrogen energy transitions

Hydrogen Electron Orbital Probability Distribution Cross-Sections

Here are some different shapes that a hydrogen atom can take. These plots are based on the same function as the vibrating balloon :
P = 4 π r²ψ_r²ψ_θ², ψ_r = e^-ρρ^lL_n-1-l^2l+1(ρ), ψ_θ = Y_l^m(θ,φ), ρ = 2r/n
(* runtime: ggg second *)
P[n_, l_, m_, {x_, y_, z_}] := Module[{r = Sqrt[x^2 + y^2 + z^2]}, 4 Pi r^2(Exp[-r/n]r^l LaguerreL[n - 1 - l, 2l + 1, 2 r/n])^2 Abs[SphericalHarmonicY[l, m, ArcCos[z/r], ArcTan[x, y]]]^2];
DensityPlot[P[3, 1, 0, {x, 0, z}], {x, -20, 20}, {z, -20, 20}, Mesh -> False, PlotPoints -> 275, Frame -> False, Compiled -> False]