Error Bounds for the Energy and Overlap of Approximate Wavefunctions

A sequence of variational principles for upper and lower bounds to the energy of the ground state of a quantum-mechanical system is constructed. The first approximations in the sequence are simply the Ritz upper bound and the Temple lower bound. Higher approximations give improved upper and lower bounds, at the cost of computing additional integrals of the form 〈φ | Hm | φ〉 over the approximate wavefunction. As in the Temple formula, the lower bounds to the ground-state energy also depend on a knowledge of the energy of the first excited state. If in addition one knows an accurate energy for the ground state (say from experiment), then it is shown that one can construct a sequence of upper and lower bounds to the overlap between the approximate function and the true (but unknown) ground-state wavefunction. The first approximation in this sequence is the Eckart lower bound to this overlap, and the higher approximations provide improved upper and lower bounds to the overlap.