We start with the variational principle. Given a hamiltonian H and a trial wave function ΨT, the variational principle states that the expectation value of ⟨H⟩, defined through
E[H]=⟨H⟩=∫dRΨ∗T(R)H(R)ΨT(R)∫dRΨ∗T(R)ΨT(R),is an upper bound to the ground state energy E0 of the hamiltonian H, that is
E0≤⟨H⟩.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.