σ-bond expression for an analytic bond-order potential:: Including π and on-site terms in the fourth moment

An improved expression for sigma bonds in sp-valent systems is derived for the analytical bond-order potential to four levels. The enhancement concerns the evaluation of the fourth moment of the local density of states with on-site and pi terms. The latter introduce a torsional stiffness to the sigma-bond order, previously known only for the pi-bond order. The relative strength of the pi enhancement as compared to the pure sigma terms depends on the ratio of the pi- and sigma-bond integrals of the specific system. In the cubic diamond phase it is large for silicon (48%) and rather small for carbon (5%). The potential parameters are given for silicon and the predicted properties are compared with ab initio calculations and experimental data.