  Self-energy corrections due to fermion-magnon interaction are studied in the
antiferromagnetic state of the $t-t'-t''$ Hubbard model within the rainbow
(noncrossing) approximation in the full $U$ range from weak to strong coupling.
The role of classical (mean-field) features of fermion and magnon dispersion,
associated with finite $U,t',t''$, are examined on quantum corrections to
quasiparticle energy, weight, one-particle density of states etc. A finite-$U$
induced classical dispersion term, absent in the $t-J$ model, is found to play
an important role in suppressing the quasiparticle weight for states near ${\bf
k}=(0,0)$, as seen in cuprates. For intermediate $U$, the renormalized AF band
gap is found to be nearly half of the classical value, and the weak coupling
limit is quite non-trivial due to strongly suppressed magnon amplitude. For
finite $t'$, the renormalized AF band gap is shown to vanish at a critical
interaction strength $U_c$, yielding a spin fluctuation driven first-order AF
insulator - PM metal transition. Quasiparticle dispersion evaluated with the
same set of Hubbard model cuprate parameters, as obtained from a recent magnon
spectrum fit, provides excellent agreement with ARPES data for $\rm Sr_2 Cu O_2
Cl_2 $.
