Weinstein Calculations for Excited States

Abstract

Weinstein's principle for an energy level, 〈H〉—Q≤E_j≤〈H〉+Q, gives both a lower and an upper bound and is applicable to both ground and excited states. Here, Q²=〈(H—〈H〉)²〉, all expectation values being calculated with respect to the same approximate wavefunction. An iterative method is proposed for solution of the necessary nonlinear equations. In tests on H₂⁺ and H₂, it is shown that the iterative method is workable, and that fair lower bounds are obtained. A formal perturbation theory is used to partly explain our results. Comparisons are made with the conventional variation method.