  The reformulated coupled-cluster method (CCM), in which average many-body
potentials are introduced, provides a useful framework to organize numerous
terms appearing in CCM equations, which enables us to clarify the structure of
the CCM theory and physical importance of various terms more easily. We
explicitly apply this framework to $^4$He, retaining one-body and two-body
correlations as the first illustrating attempt. Numerical results with using
two modern nucleon-nucleon interactions (AV18 and CD-Bonn) and their
low-momentum interactions are presented. The characters of short-range and
many-body correlations are discussed. Although not considered explicitly, the
expression of the ground-state energy in the presence of a three-nucleon force
is given.
