Helium's two 1s electrons repel each other, so neither one feels the bare nuclear charge Z=2 — each is partly shielded by the other. The variational method models this with a trial wavefunction built from two hydrogen-like 1s orbitals that share one adjustable effective charge Z' instead of the true Z:
ψ(r₁,r₂) = (Z'³/π) e^(−Z'r₁) e^(−Z'r₂)
Evaluating ⟨ψ|Ĥ|ψ⟩ for this trial function gives a closed-form energy (atomic units, Z=2 for helium):
E(Z') = Z'² − 2·Z·Z' + (5/8)·Z'
= T(Z') + V_ne(Z') + V_ee(Z')
The variational principle guarantees E(Z') ≥ E_true for every Z', so the best trial state is the one that minimizes this parabola. Setting dE/dZ' = 2Z' − 2Z + 5/8 = 0 gives the textbook shielding result:
Z'_eff = Z − 5/16 = 1.6875
E_min = -2.8477 Hartree (experiment: -2.9037 Ha, ~2% off from just one parameter)
- Z' slider — sets the trial charge directly; the disc is a 2D cross-section through the two-electron density |ψ|² (both electrons occupy the same shrinking/swelling 1s cloud, opposite spins) and the energy marker moves along the parabola.
- Auto-minimize — runs a live gradient-descent step, Z' ← Z' − η·(2Z' − 2Z + 5/8), each frame until it settles at 1.6875 — the same idea (without the calculus) that self-consistent-field methods use for real many-electron atoms.
- 5/16 = 0.3125 is the screening constant: each electron shields about 0.3 of a full nuclear charge from the other, which is why He's ionization energy is far higher than hydrogen's but well below Z²=4×.
- Drag to pan the cloud, scroll (or pinch) to zoom — the radial distribution itself only depends on Z', not on the view.