Effective interactions in $sd$-shell from \textit{ab-initio} shell model with a core*

We perform \textit{ab-initio} no-core shell model calculations for $A=18$ and $19$ nuclei in a $4\hbar\Omega$ model space using JISP16 and CD-Bonn nucleon-nucleon potentials and project the many-body Hamiltonians onto the $0\hbar\Omega$ model space to construct the effective $A$-body Hamiltonians in the $sd$-shell. We separate the effective $A$-body Hamiltonians with $A=18$ and $A=19$ into inert core, one- and two-body pieces. Then, these core, one- and two-body pieces are used to perform a standard shell model calculations for the $A=18$ and $A=19$ systems. Finally, we compare the standard shell model results with the exact no-core shell model results for the $A=18$ and $A=19$ systems.