Variational Helium: Screened Nuclear Charge Z*
He atom (Z = 2) — trial wavefunction ψ ∝ e
−Z*r
— Hartree atomic units
⟨H⟩(Z*) — Expectation Value vs Screened Charge
⟨H⟩ total
⟨T⟩ = (Z*)²
⟨V_ne⟩ = −4Z*
⟨V_ee⟩ = (5/8)Z*
Exact E = −2.9037
Screened nuclear charge: Z* =
1.69
0.5
1.0
1.5
2.0
2.5
3.0
Minimize → Z* = 27/16
State at Current Z*
Z* (screened charge)
1.6875
⟨T⟩ kinetic
—
⟨V_ne⟩ e–nucleus
—
⟨V_ee⟩ e–e repulsion
—
⟨H⟩ total
—
Exact E (He)
−2.9037 Eh
Error (⟨H⟩ − exact)
—
⟨H⟩ ≥ E_exact (variational principle satisfied)
Minimum Point (Z* = 27/16)
Optimal Z*
27/16 = 1.6875
E_var = −(27/16)²
−2.8477 Eh
Exact E_He
−2.9037 Eh
Error
+56.0 mEh (1.93%)
Screen effect Z−Z*
2 − 1.6875 = 0.3125
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