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

Abstract

Analytic bond-order potentials (BOP's) are derived for the $\ensuremath{\sigma}$ and $\ensuremath{\pi}$ bond orders by approximating the many-atom expansion for the bond order within the two-center, orthogonal tight-binding (TB) model. The analytic expression, BOP4, is obtained by retaining terms to four levels in the continued fractions for the appropriate Green's functions and describes the $\ensuremath{\sigma}$ bonds in the dimer ${\mathrm{C}}_{2},$ the tetrahedral methane molecule ${\mathrm{CH}}_{4}$ and the trigonal methyl radical ${\mathrm{CH}}_{3}$ exactly. A simplified, but accurate, variant, BOP4S, depends only on the two recursion coefficients ${b}_{1}$ and ${b}_{2}$ that characterize the root-mean-square width and the unimodal versus bimodal shape of the $\ensuremath{\sigma}$ bond eigenspectrum, respectively. An analytic expression for the $\ensuremath{\pi}$ bond order, BOP2M, is obtained by performing matrix recursion to two levels, thereby ensuring that the expression is independent of the choice of coordinate axes, depending only on neighboring bond integrals, bond angles and dihedral angles. A simple analytic expression for the promotion energy is also presented. Advantages of these BOP's over the empirical Tersoff-Brenner potentials are, first, their analytic form is predicted by the theory, second, the $\ensuremath{\sigma}$ bond order expression BOP4S includes the very important shape parameter ${(b}_{2}{/b}_{1}{)}^{2},$ and third, the $\ensuremath{\pi}$ bond order expression BOP2M describes the breaking of saturated $\ensuremath{\pi}$ bonds both on radical formation and under torsion. The following paper examines the accuracy of these BOP's for modeling the energetics of diamond, graphite, and hydrocarbon molecules.