  While the Hubbard model is the standard model to study Mott metal-insulator
transitions, it is still unclear to which extent it can describe
metal-insulator transitions in real solids, where non-local Coulomb
interactions are always present. By using a variational principle, we clarify
this issue for short- and long-ranged non-local Coulomb interactions for
half-filled systems on bipartite lattices. We find that repulsive non-local
interactions generally stabilize the Fermi-liquid regime. The metal-insulator
phase boundary is shifted to larger interaction strengths to leading order
linearly with non-local interactions. Importantly, non-local interactions can
raise the order of the metal-insulator transition. We present a detailed
analysis of how the dimension and geometry of the lattice as well as the
temperature determine the critical non-local interaction leading to a
first-order transition: for systems in more than two dimensions with non-zero
density of states at the Fermi energy the critical non-local interaction is
arbitrarily small; otherwise it is finite.
