Hidden configurations and effective interactions: A comparison of three different ways to construct renormalized hamiltonians for truncated shell-model calculations

We discuss in general some criteria and methods for constructing effective Hamiltonians. Then three different methods are compared for constructing an effective Hamiltonian to be used in nuclear shell-model calculations for A = 17-20, allowing (A-16) active nucleons in the d _{5 2}, s _{1 2} vector space. For all three methods, the aim is to obtain a d _{5 2}, s _{1 2} model which will simulate the results of a given full d _{5 2}, s _{1 2}, d _{3 2} model. The three methods for finding the effective Hamiltonian are. 1. (a) conventional low-order perturbation theory; 2. (b) a projection technique, in which we construct a Hamiltonian whose eigenvalues excactly match a selected subset of d _{5 2}, s _{1 2}, d _{3 2} eigenvalues, and whose eigenvectors excatly match the projections of d _{5 2}, s _{1 2}, d _{3 2} eigenvectors on the d _{5 2}, s _{1 2} space; and 3. (c) least-square fit to selected d _{5 2}, s _{1 2}, d _{3 2} energies. For all three methods, we first restrict the effective Hamiltonian to a linear combination of 1-body and 2-body operators. Then for the perturbation and projection techniques, we also calculate the 3-body-operator terms in the effective Hamiltonian. When the effective Hamiltonians are limited to 1-body and 2-body terms, the leastsquare method yields the best overall fit to the low-lying spectrum of d _{5 2}, s _{1 2}, d _{3 2}.