  The energy gap of correlated Hubbard clusters is well studied for
one-dimensional systems using analytical methods and
density-matrix-renormalization-group (DMRG) simulations. Beyond 1D, however,
exact results are available only for small systems by quantum Monte Carlo. For
this reason and, due to the problems of DMRG in simulating 2D and 3D systems,
alternative methods such as Green functions combined with many-body
approximations (GFMBA), that do not have this restriction, are highly
important. However, it has remained open whether the approximate character of
GFMBA simulations prevents the computation of the Hubbard gap. Here we present
new GFMBA results that demonstrate that GFMBA simulations are capable of
producing reliable data for the gap which agrees well with the DMRG benchmarks
in 1D. An interesting observation is that the accuracy of the gap can be
significantly increased when the simulations give up certain symmetry
restriction of the exact system, such as spin symmetry and spatial homogeneity.
This is seen as manifestation and generalization of the "symmetry dilemma"
introduced by L\"owdin for Hartree--Fock wave function calculations.
