Variational principles for first-order wave functions

Abstract

Complementary variational principles are developed for approximate solutions of the first-order Rayleigh- Schrodinger perturbation correction to the wave equation, yielding upper and lower bounds for the second-order energy correction. The upper bound is the same as Hylleraas's; the complementary lower bound is related to Temple's result for eigenvalues, and (unlike previous lower bounds) is shown to be unconditional. The analysis extends to cover the first-order Brillouin-Wigner correction. As a by-product of the theory it is shown how the Rayleigh-Ritz upper bound and the Temple lower bound for eigenvalues arise in a complementary manner.