Atomic Radial Probability Visualizer
Mode 1: Hydrogen-like (exact). Mode 2: Multi-electron (Slater+δ, approx.). Mode 3: He (2e) variational total distribution.
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Mode Selection
Hydrogen-like (exact)
Multi-electron (approx.)
He (2e) – variational
Nuclear charge Z
Principal quantum number n
Angular momentum l (0=s,1=p,2=d,3=f)
Samples (N)
Element
Subshell (valence)
Z
eff
(Slater estimate)
n* (quantum defect: n* = n − δ)
Samples (N)
Estimate Z
eff
(Slater)
Rough estimate for the selected element/subshell; you can tweak it manually.
Quantum defect δ (optional)
n* = n − δ. In alkali atoms δ
s
is typically large (e.g., Na: ~1.37).
For He (two electrons): variational 1s×1s with ζ* total radial distribution.
Samples (N)
Comparison curves
He (ζ*), He (naive Z=2), H (reference)
Only He (ζ*)
He (ζ*) + He (naive)
r unit
Bohr (a₀)
Ångström (Å)
E₀(He, var.) ≈ —
∫P(r)dr ≈ —
Model: uncorrelated spin-singlet 1s×1s; P₂e(r,ζ)=2·P₁e(r,ζ). Variational optimum ζ*=1.6875.
P(r) = 4π r² |R|²
|R(r)|
|R(r)| (scaled)
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n=1, l=0, Z=1
Plot
Explanation