Truncation of the valence space in the shell model and effective Hamiltonian

An explicit numerical estimate of the many-body contributions (specifically, three-body and four-body contributions) to the effective Hamiltonian, introduced by the truncation of the shell-modell valence space, is presented. On the average, they are found to be rather small. In particular, in the case of four nucleons, it turns out that three-body and four-body contributions have generally opposite signs and therefore cancel each other. Only rarely they add and become significant. This holds for realistic and phenomenological interactions and also for empirical set of interaction matrix elements, and therefore lends support to the conventional practice of using effective two-body interactions in nuclear shell-model calculations. This is confirmed, in the case of six and eight nucleons, by the results obtained in the one-broken-pair approximation.