Interatomic bond-order potentials (BOPS) have recently been derived for covalently bonded systems whose electronic structure is well described by the tight-binding (TB) approximation. This paper introduces the key ideas behind this novel class of interatomic potentials through a case study of the factors controlling the relative structural stability of s-valent four-atom clusters with respect to packing as a linear chain, square, rhombus or tetrahedron. We find that interatomic potentials, which are based on the second-moment approximation to the local density of states (LDOS) or bond order, are unable to predict which structure is most stable. This requires information about the higher moments, which the BOPS include in a systematic way. Analytic expressions are given for the LDOS and bond orders within the so-called four-level approximation, reproducing exactly the results of our case study. Simplified expressions are then obtained for the sigma and pi bond orders of sp-elements with half-full valence shells. We show that these not only reproduce the relative stability of open versus close-packed phases of silicon, but also quantify the ubiquitous concept of single, double, triple and resonant bonds in carbon systems. These analytic BOPS are, therefore, the only `classical' interatomic potentials which include both structural differentiation and radical formation naturally within their remit. (C) 2003 Elsevier Ltd. All rights reserved.