  In the present Letter we use the Wannier function basis to construct lattice
approximations of the nonlinear Schr\"{o}dinger equation with a periodic
potential. We show that the nonlinear Schr\"{o}dinger equation with a periodic
potential is equivalent to a vector lattice with long-range interactions. For
the case-example of the cosine potential we study the validity of the so-called
tight-binding approximation i.e., the approximation when nearest neighbor
interactions are dominant. The results are relevant to Bose-Einstein condensate
theory as well as to other physical systems like, for example, electromagnetic
wave propagation in nonlinear photonic crystals.
