  The short answer to the question in the title is 'no'. We identify classes of
truncated configuration interaction (CI) wave functions for which the
externally corrected coupled-cluster (ec-CC) approach using the three-body
($T_{3}$) and four-body ($T_{4}$) components of the cluster operator extracted
from CI does not improve the results of the underlying CI calculations.
Implications of our analysis, illustrated by numerical examples, for the ec-CC
computations using truncated and selected CI methods are discussed. We also
introduce a novel ec-CC approach using the $T_{3}$ and $T_{4}$ amplitudes
obtained with the selected CI scheme abbreviated as CIPSI, correcting the
resulting energies for the missing $T_{3}$ correlations not captured by CIPSI
with the help of moment expansions similar to those employed in the completely
renormalized CC methods.
