  We present the exact solution, obtained by means of the Transfer Matrix (TM)
method, of the 1D Hubbard model with nearest-neighbor (NN) and
next-nearest-neighbor (NNN) Coulomb interactions in the atomic limit (t=0). The
competition among the interactions ($U$, $V_1$, and $V_2$) generates a plethora
of T=0 phases in the whole range of fillings. $U$, $V_1$, and $V_2$ are the
intensities of the local, NN and NNN interactions, respectively. We report the
T=0 phase diagram, in which the phases are classified according to the behavior
of the principal correlation functions, and reconstruct a representative
electronic configuration for each phase. In order to do that, we make an
analytic limit $T\to 0$ in the transfer matrix, which allows us to obtain
analytic expressions for the ground state energies even for extended transfer
matrices. Such an extension of the standard TM technique can be easily applied
to a wide class of 1D models with the interaction range beyond NN distance,
allowing for a complete determination of the T=0 phase diagrams.
