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Bird's eye view on Variational MC

The basic procedure of a Variational Monte Carlo calculations consists thus of

  1. Construct first a trial wave function \psi_T(\boldsymbol{R};\boldsymbol{\alpha}) , for a many-body system consisting of n particles located at positions \boldsymbol{R}=(\boldsymbol{R}_1,\dots ,\boldsymbol{R}_n) . The trial wave function depends on \alpha variational parameters \boldsymbol{\alpha}=(\alpha_1,\dots ,\alpha_M) .
  2. Then we evaluate the expectation value of the hamiltonian H
\overline{E}[\boldsymbol{\alpha}]=\frac{\int d\boldsymbol{R}\Psi^{\ast}_{T}(\boldsymbol{R},\boldsymbol{\alpha})H(\boldsymbol{R})\Psi_{T}(\boldsymbol{R},\boldsymbol{\alpha})} {\int d\boldsymbol{R}\Psi^{\ast}_{T}(\boldsymbol{R},\boldsymbol{\alpha})\Psi_{T}(\boldsymbol{R},\boldsymbol{\alpha})}.
  1. Thereafter we vary \boldsymbol{\alpha} according to some minimization algorithm and return eventually to the first step if we are not satisfied with the results.

Here we have used the notation \overline{E} to label the expectation value of the energy.