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 ${\mathcal{H}}^2$ magnifies the poor features of trial functions. Moreover, for the two‐electron systems, an evaluation of the three components of $〈{\mathcal{H}}^2〉$ , ${\mathrm{〈T}}^2〉$ , $\mathrm{〈TV\hspace{0.17em}+\hspace{0.17em}VT〉}$ , and ${\mathrm{〈V}}^2〉$ , 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.