This paper presents a way of improving second-order perturbation theory calculations by summing contributions of uncoupled excitations to infinite order. For problems involving molecular vibrations, the new theory is shown to give similar results to conventional second-order perturbation theory when the system treated has no near resonances but also to give accurate and stable results even very close to resonance. The new theory is tested by comparison to converged variational calculations for vibrational energy levels of formaldehyde, formaldehyde-d(2), and two two-dimensional model subsystems based on formaldehyde.