Bird's eye view on Variational MC
The basic procedure of a Variational Monte Carlo calculations consists thus of
- 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 \( \boldsymbol{\alpha}=(\alpha_1,\dots ,\alpha_M) \).
- Then we evaluate the expectation value of the hamiltonian \( H \)
$$
\overline{E}[\boldsymbol{\alpha}]=\frac{\int d\boldsymbol{R}\Psi^{\ast}_{T}(\boldsymbol{R},\boldsymbol{\alpha})H(\boldsymbol{R})\Psi_{T}(\boldsymbol{R},\boldsymbol{\alpha})}
{\int d\boldsymbol{R}\Psi^{\ast}_{T}(\boldsymbol{R},\boldsymbol{\alpha})\Psi_{T}(\boldsymbol{R},\boldsymbol{\alpha})}.
$$
- 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.