We start with the variational principle. Given a hamiltonian \( H \) and a trial wave function \( \Psi_T \), the variational principle states that the expectation value of \( \langle H \rangle \), defined through
$$ E[H]= \langle H \rangle = \frac{\int d\boldsymbol{R}\Psi^{\ast}_T(\boldsymbol{R})H(\boldsymbol{R})\Psi_T(\boldsymbol{R})} {\int d\boldsymbol{R}\Psi^{\ast}_T(\boldsymbol{R})\Psi_T(\boldsymbol{R})}, $$is an upper bound to the ground state energy \( E_0 \) of the hamiltonian \( H \), that is
$$ E_0 \le \langle H \rangle . $$In general, the integrals involved in the calculation of various expectation values are multi-dimensional ones. Traditional integration methods such as the Gauss-Legendre will not be adequate for say the computation of the energy of a many-body system.