  How the nuclear force behaves in cluster states, in particular those
consisting of the $\alpha$ clusters, has been investigated so far, but not yet
elucidated. Today the chiral effective field theory is established and it would
shed new light on the microscopic understanding of the cluster states. We aim
to address a possible source of the attraction in the cluster states of
$^8\mathrm{Be}$ in view of the pion exchange. Namely, we investigate whether
the two-pion-exchange interaction acts as a dominant attraction in the
$\alpha+\alpha$ system as predicted by a previous work. We describe
theoretically the cluster structure of $^8\mathrm{Be}$ by the Brink model, for
which the effective interaction is designed from the realistic nuclear force
derived through the chiral effective field theory. The two-body matrix elements
of the chiral interaction with the local-Gaussian bases are formulated within
the approximation of the spin-isospin saturation forming an $\alpha$ particle.
Introducing a global prefactor to the chiral interaction phenomenologically,
the ground and low-lying excited states of $^8\mathrm{Be}$, the scattering
phase shift of the $\alpha$-$\alpha$ system as well, are satisfactorily
depicted. The attraction in the cluster states is found to be stemming from the
two-pion-exchange contributions dominantly, along with nonnegligible
short-range terms. The present work can be the foundation towards constructing
realistic cluster models, by which the cluster states will be revealed
microscopically in the next step.
