A subdivision of space into discrete cells underlies the traditional discrete dipole model. This model presumes that only nonlocal electric interactions between cells govern the electromagnetic response of a condensed matter system. Apart from the case of simple dielectrics, this is not realistic. Cells can also influence each other directly through the wave functions, when those extend across cell boundaries. In general, such nonlocal quantum mechanical interaction requires the use of nonlocal polarizabilities. In this paper it is shown how existing discrete dipole descriptions for clusters, slabs and (semi)-infinite systems have to be altered to incorporate the effects of nonlocal polarizabilities. The modified method is called discrete cellular method.