The Monte Carlo algorithm
The Algorithm for performing a variational Monte Carlo calculations runs 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 often referred to as the brute-force sampling and is normally replaced by what is called importance sampling, discussed in more detail next week..