Effective Shell-Model Interactions from the Valence Cluster Expansion

The {\em ab initio} No-Core Shell Model (NCSM) has had considerable success in describing the binding energies, excitation spectra and other physical properties of light nuclei, $A\le12$ {\it e.g.} [1]. However, it becomes rather challenging to produce converged results for nuclei with $A>12$. Following the idea of Ref. [2], we develop a valence cluster expansion to construct effective 2- and 3-body Hamiltonians for the $0p$-shell by performing $12\hbar\Omega$ NCSM calculations for $A=6$ and 7 nuclei and explicitly projecting the many-body Hamiltonians onto the $0\hbar\Omega$ space. We separate these effective Hamiltonians into 0-, 1- and 2-body contributions (also 3-body for $A=7$) and analyze the systematic behavior of these different parts as a function of the mass number $A$ and size of the NCSM basis space. The role of effective 3- and higher-body interactions for $A>6$ will be discussed. [1] P. Navratil, J.P.Vary, B.R.Barrett, {\it Phys. Rev. C.} {\bf 62}, 054311 (2000). [2] P. Navratil, M. Thoresen, and B. R. Barrett, Phys. Rev. C. {\bf 55}, R573 (1997).