  Non-perturbative aspects of the quantum many-body problem are revisited,
discussed and advanced in the equation of motion framework. We compare the
approach to the two-fermion response function truncated on the two-body level
by the cluster expansion of the dynamical interaction kernel to the approach
known as time blocking approximation. Such a comparison leads to an extended
many-body theory with non-perturbative treatment of high-order configurations.
The present implementation of the advanced theory introduces a new class of
solutions for the response functions, which include explicitly
beyond-mean-field correlations between up to six fermions. The novel approach,
which includes configurations with two quasiparticles coupled to two phonons
(2q$\otimes$2phonon), is discussed in detail for the particle-hole nuclear
response and applied to medium-mass nuclei. The proposed developments are
implemented numerically on the basis of the relativistic effective
meson-nucleon Lagrangian and compared to the models confined by two-fermion and
four-fermion configurations, which are considered as state-of-the-art for the
response theory in nuclear structure calculations. The results obtained for the
dipole response of $^{42,48}$Ca and $^{68}$Ni nuclei in comparison to available
experimental data show that the higher configurations are necessary for a
successful description of both gross and fine details of the spectra in both
high-energy and low-energy sectors.
