Variational Monte Carlo

Jastrow wavefunction Ψ_J(r) = Φ_0(r) exp(-½Σᵢⱼ u(rᵢⱼ)) encodes electron correlations. VMC samples |Ψ_J|² via Metropolis and estimates ⟨H⟩. Optimize variational parameters to minimize energy — watch convergence in real time.

VMC Parameters

Initializing...

Jastrow Factor Physics

Jastrow wavefunction:
Ψ_J = exp(-½Σᵢ u(r) = a·r/(1+b·r) (Padé form)

Metropolis sampling: Propose x→x', accept with probability min(1, |Ψ(x')|²/|Ψ(x)|²). Samples the exact wavefunction density.

Local energy: E_L(R) = Ψ⁻¹HΨ. VMC energy ⟨E⟩ = ∫|Ψ|² E_L dR / ∫|Ψ|²dR. Variance decreases when Ψ → exact eigenstate.

Pair correlation g(r): Measures how electrons avoid each other. Jastrow creates a correlation hole at r→0 (exchange-correlation).