  This work deals with the problem of strongly correlated electrons in
two-dimensions (2D). We give a reduced density matrix (RDM) based tool through
which the ground-state energy is given as a functional of the natural orbitals
and their occupation numbers. Specifically, the Piris Natural Orbital
Functional 7 (PNOF7) is used for studying the 2D Hubbard model and hydrogen
square lattices. The singlet ground-state is studied, as well as the doublet
mixed quantum state obtained by extracting an electron from the system. Our
method satisfies two-index necessary N-representability conditions of the
two-particle RDM (2RDM) and guarantees the conservation of the total spin. We
show the ability of PNOF7 to describe strong correlation effects in these 2D
systems by comparing our results with exact diagonalization, density matrix
renormalization group (DMRG), and auxiliary-field quantum Monte Carlo
calculations. PNOF7 overcomes variational 2RDM methods with two- and
three-index positivity N-representability conditions, reducing computational
cost to mean-field scaling. Consistent results are obtained for small and large
systems up to 144 electrons, weak and strong correlation regimes, and many
filling situations. Unlike other methods, there is no dependence on
dimensionality in the results obtained with PNOF7, and no particular
difficulties have been observed to converge PNOF7 away from half-filling.
Smooth double occupancy of sites is obtained regardless of the filling.
Symmetric dissociation of 2D hydrogen lattices shows that long-range nondynamic
correlation drammatically affects electron detachment energies. PNOF7 compares
well with DMRG along the dissociation curve.
