Helium Atom Variational Method: Effective Nuclear Charge

Trial Wavefunction

Effective nuclear charge Z' 2.00

Energy vs. Z'

Live Energetics (Hartree)

Kinetic T

8.00

Nucleus–e⁻ V_ne

-16.00

e⁻–e⁻ repulsion V_ee

1.25

Total E(Z')

-6.75

Optimal Z'_eff = Z − 5/16

1.6875 → E_min = -2.8477 Ha

How it works

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 point cloud is 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×.

e⁻ density cloud |ψ|² He nucleus (Z=2)

⚠ Couldn't load the 3D engineThree.js failed to load from the CDN. Check your connection and reload.