Quantum Monte Carlo, ingredients and basic steps

• 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 $H$

$$E[H]=\langle H \rangle = \frac{\int d\boldsymbol{R}\Psi^{\ast}_{T}(\boldsymbol{R},\boldsymbol{\alpha})H(\boldsymbol{R})\Psi_{T}(\boldsymbol{R},\boldsymbol{\alpha})} {\int d\boldsymbol{R}\Psi^{\ast}_{T}(\boldsymbol{R},\boldsymbol{\alpha})\Psi_{T}(\boldsymbol{R},\boldsymbol{\alpha})}.$$

• Thereafter we vary $\alpha$ according to some minimization algorithm and return to the first step.

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