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.