Quantum Monte Carlo algorithm

The Algorithm for performing a variational Monte Carlo calculations runs thus as this

• Initialisation: Fix the number of Monte Carlo steps. Choose an initial $\boldsymbol{R}$ and variational parameters $\alpha$ and calculate $\left|\psi_T^{\alpha}(\boldsymbol{R})\right|^2$.
• Initialise the energy and the variance and start the Monte Carlo calculation.
• Calculate a trial position $\boldsymbol{R}_p=\boldsymbol{R}+r*step$ where $r$ is a random variable $r \in [0,1]$.
• Metropolis algorithm to accept or reject this move $w = P(\boldsymbol{R}_p)/P(\boldsymbol{R})$.
• If the step is accepted, then we set $\boldsymbol{R}=\boldsymbol{R}_p$.
• Update averages
• Finish and compute final averages.

Observe that the jumping in space is governed by the variable step. This is Called brute-force sampling. Need importance sampling to get more relevant sampling, see lectures below.

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