Bond-order potentials: bridging the electronic to atomistic modelling hierarchies

Novel analytic bond-order potentials (BOPs) may be derived for atomistic simulations by coarse-graining the electronic structure within the orthogonal two-centre tight-binding (TB) representation. We show that these BOPs allow the concept of single, double, triple and conjugate bonds in carbon systems to be quantified, so that they provide the first ‘classical’ interatomic potentials that handle both structural differentiation and radical formation naturally within their remit. Finally, we indicate that this recently developed BOP formalism allows explicit, analytic expressions for the environmental dependence of the TB bond integrals to be derived, thereby providing a systematic methodology for bridging from the electronic to atomistic modelling hierarchies. 2002 Elsevier Science B.V. All rights reserved. The modelling and simulation of materials is a challenging task as it often involves linking the world of the electron theorist to the world of the continuum engineer, thereby spanning at least twelve orders of magnitude in either length or time. This linking is achieved by successively coarse-graining the problem [1]. Firstly, the electronic degrees of freedom may be removed by imagining the atoms to be held together by some sort of glue or interatomic potential, thereby allowing large scale atomistic simulations to be performed on fracture at crack tips, for example [2]. Secondly, the atoms themselves may be coarsegrained by grouping them together into cells which interact via deterministic or stochastic rules, thereby allowing, for example, the simulation of microstructure evolution during the growth of single crystal turbine blades [3]. Finally, the continuum world may be reached by averaging over the microstructure and describing the material through a set of constitutive relationships [4]. In this paper we outline the derivation of novel analytic bond-order potentials (BOPs) that are obtained directly by bridging the electronic to atomistic modelling hierarchies within a tightbinding (TB) framework. Coulson [5] had shown in 1939 that the global description of energy bands with an associated band energy could be broken down into a local description of covalent bonds with associated individual bond energies. In Computational Materials Science 23 (2002) 33–37 www.elsevier.com/locate/commatsci * Corresponding author. Tel.: +44-1865-283325; fax: +441865-273764. E-mail address: [email protected] (I.I. Oleinik). 0927-0256/02/$ see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S0927-0256 (01 )00204-X particular, for the case of a single orbital per site, the bond energy could be written