  We propose an importance truncation scheme for the no-core shell model, which
enables converged calculations for nuclei well beyond the p-shell. It is based
on an a priori measure for the importance of individual basis states
constructed by means of many-body perturbation theory. Only the physically
relevant states of the no-core model space are considered, which leads to a
dramatic reduction of the basis dimension. We analyze the validity and
efficiency of this truncation scheme using different realistic nucleon-nucleon
interactions and compare to conventional no-core shell model calculations for
4He and 16O. Then, we present the first converged calculations for the ground
state of 40Ca within no-core model spaces including up to
16\hbar\Omega-excitations using realistic low-momentum interactions. The scheme
is universal and can be easily applied to other quantum many-body problems.
