  We address the formulation and analysis of energy and momentum conserving
time integration schemes in the context of particle dynamics, and in particular
atomic systems. The article identifies three critical aspects of these models
that demand a careful analysis when discretized: first, the treatment of
periodic boundary conditions; second, the formulation of approximations of
systems with three-body interaction forces; third, their extension to atomic
systems with functional potentials. These issues, and in particular their
interplay with Energy-Momentum integrators, are studied in detail. Novel
expressions for these time integration schemes are
  proposed and numerical examples are given to illustrate their performance.
