  Interatomic potentials approximate the potential energy of atoms as a
function of their coordinates. Their main application is the effective
simulation of many-atom systems. Here, we review empirical interatomic
potentials designed to reproduce elastic properties, defect energies, bond
breaking, bond formation, and even redox reactions. We discuss popular two-body
potentials, embedded-atom models for metals, bond-order potentials for
covalently bonded systems, polarizable potentials including charge-transfer
approaches for ionic systems and quantum-Drude oscillator models mimicking
higher-order and many-body dispersion. Particular emphasis is laid on the
question what constraints ensue from the functional form of a potential, e.g.,
in what way Cauchy relations for elastic tensor elements can be violated and
what this entails for the ratio of defect and cohesive energies, or why the
ratio of boiling to melting temperature tends to be large for potentials
describing metals but small for short-ranged pair potentials. The review is
meant to be pedagogical rather than encyclopedic. This is why we highlight
potentials with functional forms sufficiently simple to remain amenable to
analytical treatments. Our main objective is to provide a stimulus for how
existing approaches can be advanced or meaningfully combined to extent the
scope of simulations based on empirical potentials.
