<i>Ab initio</i>effective interactions for<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>s</mml:mi><mml:mi>d</mml:mi></mml:mrow></mml:math>-shell valence nucleons

We perform ab initio no-core shell-model calculations for $A=18$ and 19 nuclei in a $4\ensuremath{\hbar}\mathrm{\ensuremath{\Omega}}$, or ${N}_{\mathrm{max}}=4$, model space by using the effective JISP16 and chiral N3LO nucleon-nucleon potentials and transform the many-body effective Hamiltonians into the $0\ensuremath{\hbar}\mathrm{\ensuremath{\Omega}}$ model space to construct the $A$-body effective Hamiltonians in the $sd$ shell. We separate the $A$-body effective Hamiltonians with $A=18$ and $A=19$ into inert core, one-, and two-body components. Then we use these core, one-, and two-body components to perform standard shell-model calculations for the $A=18$ and $A=19$ systems with valence nucleons restricted to the $sd$ shell. Finally, we compare the standard shell-model results in the $0\ensuremath{\hbar}\mathrm{\ensuremath{\Omega}}$ model space with the exact no-core shell-model results in the $4\ensuremath{\hbar}\mathrm{\ensuremath{\Omega}}$ model space for the $A=18$ and $A=19$ systems and find good agreement.