  The no-core shell model (NCSM) is an \textit{ab initio} method that solves
the nuclear many-body problem by expanding the many-particle wave function into
a (typically) harmonic oscillator basis and minimizing the energy to obtain the
expansion coefficients. Extensions of the NCSM, such as its coupling with
microscopic-cluster basis states, further allow for an \textit{ab initio}
treatment of light-ion nuclear reactions of interest for both astrophysics and
nuclear technology applications. A downside of the method is the exponential
scaling of the basis size with increasing number of nucleons and excitation
quanta, which limits its applicability to mass $A\lesssim 16$ nuclei, except
for variants where the basis is further down-selected via some truncation
scheme. We consider a basis selection method for the NCSM that captures the
essential degrees of freedom of the nuclear wave function leading to a
favorable complexity scaling for calculations and enabling \textit{ab initio}
reaction calculations in $sd$-shell nuclei. The particle configurations within
the NCSM basis are ordered based on their contribution to the first moment of
the Hamiltonian matrix that results from the projection onto the many-body
basis. The truncation scheme then consists in retaining only the
lowest-first-moment configurations, which typically contain only few many-body
basis states (Slater determinants). We present calculations for $^7$Li and
$n+^{12}$C scattering using nucleon-nucleon interactions derived from chiral
effective field theory and softened using the similarity renormalization group
method. The obtained energy levels invariably demonstrate exponential
convergence with the size of the basis, and we find improved convergence in
scattering calculations. To demonstrate the possibilities enabled by the
approach, we also present a first calculation for the scattering of neutrons
from $^{24}$Mg.
