  We present a new scheme to select the most relevant phonons in the
phonon-coupling model, named here time-blocking approximation (TBA). The new
criterion, based on the phonon-nucleon coupling strengths rather than on
$B(EL)$ values, is more selective and thus produces much smaller phonon spaces
in TBA. This is beneficial in two respects: first, it curbs down the
computational cost, and second, it reduces the danger of double counting in the
expansion basis of TBA. We use here TBA in a form where the coupling strength
is regularized to keep the given Hartree-Fock ground state stable. The scheme
is implemented in an RPA and TBA code based on the Skyrme energy functional. We
first explore carefully the cutoff dependence with the new criterion and can
work out a natural (optimal) cutoff parameter. Then we use the freshly
developed and tested scheme to a survey of giant resonances and low-lying
collective states in six doubly magic nuclei looking also on the dependence of
the results when varying the Skyrme parametrization.
