Perturbative approximations to the effective hamiltonian in the presence of an intruder state

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Perturbative approximations to the effective Hamiltonian for J π = 0 +, T = 1 states in 18O are studied through 4th order, in the context of an 8 × 8 Hamiltonian matrix. Comparison of the results with exact effective matrix elements indicates that low-order perturbation theory provides a good approximation, even though the perturbation series is divergent. We explain this success by showing that the singularity which causes the divergence has a negligible influence on low-order perturbation theory, because the intruder state and the uppermost model state interact weakly.