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In recent years, remarkable progress has been achieved in developing novelnon-perturbative techniques for constructing valence space shell modelHamiltonians from realistic internucleon interactions. One of these methods isbased on the Okubo--Lee--Suzuki (OLS) unitary transformation applied to no-coreshell model (NCSM) solutions. In the present work, we implement thecorresponding approach to solve for valence space effective electromagneticoperators. To this end, we use the NCSM results for $A=16-18$, obtained at$N_{\rm max}=4$, to derive a charge-dependent version of the effectiveinteraction for the $sd$ shell, which allows us to exactly reproduce selectedNCSM spectra of $^{18}$O, $^{18}$F and $^{18}$Ne within the two valence nucleonspace. We then deduce effective single-particle matrix elements of electricquadrupole ($E2$) and magnetic dipole ($M1$) operators by matching them to theelectromagnetic transitions and moments for $^{17}$O and $^{17}$F from the NCSMat $N_{\rm max}=4$. Thus, effective $E2$ and $M1$ operators are obtained assets of single-particle matrix elements for the valence space ($sd$ shell)which allow us to reproduce the NCSM results for $A=17$ exactly. Systematiccomparison of a large set of $sd$ shell results on quadrupole and magneticdipole moments and transitions for $A=18$ using effective $E2$ and $M1$operators that we derive from the full NCSM calculations demonstrates aremarkable agreement.

Effective operators for valence space calculations from the {\itshape ab initio} No-Core Shell Mode