  We derive continuum limits of atomistic models in the realm of nonlinear
elasticity theory rigorously as the interatomic distances tend to zero. In
particular we obtain an integral functional acting on the deformation gradient
in the continuum theory which depends on the underlying atomistic interaction
potentials and the lattice geometry. The interaction potentials to which our
theory applies are general finite range models on multilattices which in
particular can also account for multi-pole interactions and bond-angle
dependent contributions. Furthermore, we discuss the applicability of the
Cauchy-Born rule. Our class of limiting energy densities consists of general
quasiconvex functions and the class of linearized limiting energies consistent
with the Cauchy-Born rule consists of general quadratic forms not restricted by
the Cauchy relations.
