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 \( \left|\psi_T(\boldsymbol{R},\boldsymbol{\alpha})\right|^2 \).
• Initialise the energy and the variance and start the Monte Carlo calculation by looping over trials.
• 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 \( w = P(\boldsymbol{R}_p,\boldsymbol{\alpha})/P(\boldsymbol{R},\boldsymbol{\alpha}) \).
• 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 often called brute-force sampling. Need importance sampling to get more relevant sampling.