  We explore systematically the ground state properties of one dimensional
fermions with long-range interactions decaying in a power law $\sim1/r^\alpha$
through the density matrix renormalization group algorithm. By comparing values
of Luttinger liquid parameters precisely measured in two different ways, we
show convincing evidence that Luttinger liquid theory is valid if $\alpha$ is
larger than some threshold, otherwise the theory breakdown. Combining analysis
on structure factor, charge gap and charge stiffness, we determine how the
metal-insulator transition point develops as the interaction range is
continuously tuned. A region in the range of $0\leq\alpha\leq1$ and small
interactions has finite charge gaps, but, interestingly, it shows metallic
nature at the same time. From these, we obtain approximate phase diagrams for
the entire parameter space and for band fillings equal 1/2 and 1/3. Finally, we
compare certain bosonization and field theory formula with our quasi-exact
numerical results, from which disagreements are found.
