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VMC Theory

The Metropolis Algorithm

VMC uses the Metropolis-Hastings algorithm to sample the probability distribution \(P(\mathbf{R}) \propto |\Psi_T(\mathbf{R})|^2\).

  1. Initialization: Start with an initial configuration of electrons \(\mathbf{R}\).
  2. Proposal: Propose a move to a new configuration \(\mathbf{R}'\) (e.g., by moving one electron).
  3. Acceptance/Rejection: Accept the move with probability: $$ A(\mathbf{R} \to \mathbf{R}') = \min \left( 1, \frac{|\Psi_T(\mathbf{R}')|^2 T(\mathbf{R}' \to \mathbf{R})}{|\Psi_T(\mathbf{R})|^2 T(\mathbf{R} \to \mathbf{R}')} \right) $$ where \(T(\mathbf{R} \to \mathbf{R}')\) is the transition probability (often symmetric).
  4. Accumulation: Accumulate averages of local observables (e.g., local energy) over the sampled configurations.

Local Energy

The local energy \(E_L(\mathbf{R})\) is a central quantity in VMC.

\[ E_L(\mathbf{R}) = -\frac{1}{2} \sum_i \frac{\nabla_i^2 \Psi_T}{\Psi_T} + V(\mathbf{R}) \]
  • The kinetic energy term involves the Laplacian of the wavefunction.
  • The potential energy term \(V(\mathbf{R})\) includes electron-electron, electron-nucleus, and nucleus-nucleus interactions.

Statistical Error

The error in the VMC energy estimate decreases as \(1/\sqrt{N}\), where \(N\) is the number of independent samples. Because samples in a Markov chain are correlated, the effective number of samples is \(N_{eff} = N / \tau_{corr}\), where \(\tau_{corr}\) is the autocorrelation time.

Blocking analysis is used to estimate the true statistical error by grouping steps into blocks to remove correlation.