Use of Numerical Integration in the Computation of the Expectation Value of H2 with Applications to H2

The use of numerical integration for evaluating the expectation values of the Hamiltonian and the square of the Hamiltonian is difficult due to the presence of singularities in the integrand. A basis set is discussed which removes one of the principal singularities from Hψ in the case of the hydrogen molecule. This basis set is used and all integrals are evaluated numerically to obtain variational upper bounds and Weinstein and Temple lower bounds for the ground state of H2. First-excited-state upper bounds are obtained by a method involving the expectation value of the square of the Hamiltonian and the possibility of computing excited-state wavefunctions by minimization of the width, W=〈H2〉—〈H〉2, is discussed.