Analytic bond-order potentials beyond Tersoff-Brenner. I. Theory

Analytic bond-order potentials (BOP's) are derived for the sigma and pi bon d orders by approximating the many-atom expansion for the bond order within the two-center, orthogonal tight-binding (TB) model. The analytic expressi on, BOP4, is obtained by retaining terms to four levels in the continued fr actions for the appropriate Green's functions and describes the sigma bonds in the dimer C-2, the tetrahedral methane molecule CH4 and the trigonal me thyl radical CH3 exactly. A simplified, but accurate, variant, BOP4S, depen ds only on the two recursion coefficients b(1) and b(2) that characterize t he root-mean-square width and the unimodal versus bimodal shape of the sigm a bond eigenspectrum, respectively. An analytic expression for the pi bond order, BOP2M, is obtained by performing matrix recursion to two levels, the reby ensuring that the expression is independent of the choice of coordinat e axes, depending only on neighboring bond integrals, bund angles and dihed ral angles. A simple analytic expression for the promotion energy is also p resented. Advantages of these BOP's over the empirical Tersoff-Brenner pote ntials are, first, their analytic form is predicted by the theory, second, the sigma bond order expression BOP4S includes the very important shape par ameter (b(2)/b(1))(2), and third, the pi bond order expression BOP2M descri bes the breaking of saturated pi bonds both on radical formation and under torsion. The following paper examines the accuracy of these BOP's for model ing the energetics of diamond, graphite, and hydrocarbon molecules. [S0163- 1829(99)03313-5].