  A systematic diagrammatic expansion for Gutzwiller-wave functions (DE-GWF)
proposed very recently is used for the description of superconducting (SC)
ground state in the two-dimensional square-lattice $t$-$J$ model with the
hopping electron amplitudes $t$ (and $t'$) between nearest (and next-nearest)
neighbors. On the example of the SC state analysis we provide a detailed
comparison of the method results with other approaches. Namely: (i) the
truncated DE-GWF method reproduces the variational Monte Carlo (VMC) results;
(ii) in the lowest (zeroth) order of the expansion the method can reproduce the
analytical results of the standard Gutzwiller approximation (GA), as well as of
the recently proposed "grand-canonical Gutzwiller approximation" (GCGA). We
obtain important features of the SC state. First, the SC gap at the Fermi
surface resembles a $d_{x^2-y^2}$-wave only for optimally- and overdoped
system, being diminished in the antinodal regions for the underdoped case in a
qualitative agreement with experiment. Corrections to the gap structure are
shown to arise from the longer range of the real-space pairing. Second, the
nodal Fermi velocity is almost constant as a function of doping and agrees
semi-quantitatively with experimental results. Third, we compare the doping
dependence of the gap magnitude with experimental data. Fourth, we analyze the
$\mathbf{k}$-space properties of the model: Fermi surface topology and
effective dispersion. The DE-GWF method opens up new perspectives for studying
strongly-correlated systems, as: (i) it works in the thermodynamic limit, (ii)
is comparable in accuracy to VMC, and (iii) has numerical complexity comparable
to GA (i.e., it provides the results much faster than the VMC approach).
