CALCULATION OF LOWER BOUNDS TO ENERGIES OF MOLECULAR SYSTEMS .I. MATHEMATICAL METHODS AND ENERGY VARIANCES FOR SIMPLE SYSTEMS

The energy variance, which provides a sensitive test for any proposed wavefunction, has been calculated for a series of Gaussian-type wavefunctions for the systems H2+, H2, HeH+, He2++, and H3+. The integrals that arise over the square of the Hamiltonian can be expressed either in closed form, or involving a single numerical integration. The magnitude of the values obtained for the energy variance, which should be zero for an exact solution, are consistent with other suggestions that the operator ℋ2 magnifies the poor features of trial functions. Moreover, for the two-electron systems, an evaluation of the three components of 〈ℋ2〉 , 〈T2〉, 〈TV+VT〉, and 〈V2〉, shows by their lack of convergence that they all contribute to this result. A further examination of these wavefunctions is made by a calculation of the net forces acting on the nuclei.