Empirical Bond-Order Model

Similar to the Finnis-Sinclair potential, the empirical bond-order approach mimics quantum mechanical atomic bonding based on the local chem- 

Rather than present the original derivation, much of which is similar to the discussion above, Eq. [25] can be directly obtained from the Finnis-Sinclair bond energy Eq. [23] by the following  

With this derivation the analytic bond order b.. in Eq. [25] is 

For the case of an ideal lattice with only nearest-neighbor interactions, all bond lengths are identical and the bond-order function reduces to the inverse square root of coordination z. The cohesive energy can then be written as 

Hence for uniform expansion of an ideal lattice, this expression reduces to an 

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The Tersoff potential was designed specifically for the group 14 elements and extends the basic empirical bond-order model by including an angular term. The interaction energy between two atoms i and j using this potential is ... 

The Tersoff potential [Tersoff 1988] is based on a model known as the empirical bond-order potential. This potential can be written in a form very similar to the Finnis-Sinclair potential ... 

The semi-empirical bond polarization model is a powerful tool for the calculation of, 3C chemical shift tensors. For most molecules the errors of this model are in the same order of magnitude as the errors of ab initio methods, under the condition that the surrounding of the carbon is not too much deformed by small bond angles. A great advantage of the model is that bond polarization calculations are very fast. The chemical shift tensors of small molecules can be estimated in fractions of a second. There is also virtually no limit for the size of the molecule. Systems with a few thousand atoms can be calculated with a standard PC within a few minutes. Possible applications are repetitive calculations during molecular dynamics simulations for the interpretation of dynamic effects on 13C chemical shift distribution. 

In the early 1990s, Brenner and coworkers [163] developed interaction potentials for model explosives that include realistic chemical reaction steps (i.e., endothermic bond rupture and exothermic product formation) and many-body effects. This potential, called the Reactive Empirical Bond Order (REBO) potential, has been used in molecular dynamics simulations by numerous groups to explore atomic-level details of self-sustained reaction waves propagating through a crystal [163-171], The potential is based on ideas first proposed by Abell [172] and implemented for covalent solids by Tersoff [173]. It introduces many-body effects through modification of the pair-additive attractive term by an empirical bond-order function whose value is dependent on the local atomic environment. The form that has been used in the detonation simulations assumes that the total energy of a system of N atoms is ... 

In Sec. 2, we review the reactive empirical bond order potentials that we have developed over the last decade which possess these essential characteristics. Our model systems and computational techniques are also described. Simulations using the first of our models to exhibit classic detonation behavior are discussed in Sec. 3, where results involving both detonation and initiation are presented. In Sec. 4, the results of our molecular dynamics simulations are compared in detail with the predictions of continuum theory. Then, in Sec. 5, we review some simulations that raise... 

To overcome this limitation we developed a series of potentials in the late 1980 s and early 1990 s that have become known as reactive empirical bond order (REBO) potentials. These potentials are based on the empirical bond order potential form introduced by Tersoff to describe the static properties of silicon but were tailored by us to incorporate a modicum of chemistry. In Sec. 2.1, after introducing the REBO potential form, we describe our simple models for energetic materials that are based on these potentials. In Sec. 2.2, we provide an overview of the approach taken to implement our simulations of shock-induced chemistry and detonations. 

In Section 3, we showed that large-scale molecular-dynamics (MD) simulations could be used to study the effect of impacts upon perfect crystals of high-explosive diatomic molecules whose interactions are modeled by the reactive empirical bond-order (REBO) potential. We showed that perfect crystal shock simulations lead to detonation above a threshold impact velocity, with characteristics that satisfy the ZND theory of detonations. To see if the threshold for initiation of chemical reaction can be lowered, we also introduced a variety of defects into our samples. 

The first theoretical work providing information on the Debye temperature (Go) of intermetallic clathrates dates back to the year 1999 [33]. Molecular dynamics calculations for the carbon-framework of type-I and type-II clathrates used a Lennard-Jones potential (later on also for Si-based clathrates [34]). 0d for Ci36 [35] and for Siiae [34] were estimated from the calculated elastic constant Cn applying the empirical relation Qd = —11.3964 + 0.3475 x C — 1.6150 x 10 X Cj 1. Moriguchi et al. [36] used an empirical bond-order potential developed by Tersofif for the calculation of several thermodynamic properties, including the heat capacity, for the type-I and type-II Si networks. From the heat capacity data in the temperature range from 0 to 150 K 6d was extracted applying the Debye-model. The heat capacity, Cy, was calculated by the density functional theory (DFT),... 

Recently, semiempirical methods based on DFT calculations have been developed for catalyst screening. These methods include linear scaling relationships [41, 42] to transfer thermochemistry from one metal to another and Brpnsted-Evans-Polanyi (BEP) relationships [43 7]. Here, these methods and also methods for estimation of the surface entropy and heat capacity are briefly discussed. Because of their screening capabilities, semiempirical methods can be used to produce a first-pass microkinetic model. This first-pass model can then be refined using more detailed theory aided by analytical tools that identify key features of the model. The empirical bond-order conservation (BOC) method, which has shown good success in developing microkinetic models of small molecules, has recently been reviewed [11] and will not be covered here. 

The most widely used family of reactive FFs is based on the concept of bond order it is assumed that the strength of a bond between two atoms is not constant, but depends on the local environment. Examples indude the Tersoff potential,the reactive empirical bond order (REBO and REB02) modd, the adaptive intermolecular reactive empirical bond order (AIREBO) model, the second-moment TB potentials, and the bond-based analytic bond order (BOP) potentials. These models allow for bond formation,... 

These results can be interpreted successfully in terms of Pauling s valence bond order concept. In the framework of this model, a chemical bond between X and H in diatomic molecule XH or between H and B in a HB molecule can be characterized by empirical valence bond orders Pxh and Phb decreasing exponentially with bond distance ... 

Metal-phosphine bonds can generally be modeled in much the same way as any other metal-heteroatom bond. The fact that phosphines participate in x-backbonding (filled dn (metal) -> empty d or a (phosphorus) interaction) is only of importance for generic force field parameterization schemes, and half-integer bond orders have been used to describe the effect of x-back-donation[ 153). In the usually adopted empirical force field formalism, x-bonding effects, like most of the other structural/elec-tronic effects, are accommodated by the general parameter-fitting procedure (see Parts I and III). 

In order to obtain more insight into the dynamics, molecular dynamics (MD) simulations of similar slit systems have been performed [62-64], Since ab initio MD methods are not applicable, an effort was made to incorporate the Grotthus mechanism into the MD simulations via a simplified empirical valence bond (EVB) model [65],... 

Recently, Cioslowski and Mixon proposed a method for obtaining bond orders that is more in tune with the spirit of the topological method. Their method eliminates the need for any empirical parameters or arbitrary choice of molecules to serve as a model set and thus can apply to any pair of bonded atoms. 

The classical approach appropriately known as molecular mechanics has been used with conspicuous success to predict molecular geometries, chemical reactivities and even magnetic, electronic and spectral properties of molecular systems. Molecular mechanics functions with no intention or pretence to elucidate the essential nature of molecules it applies concepts that pertain to the nineteenth-century classical model of the molecule, i.e, bond length, bond order, force constant, torsional rigidity and steric congestion. Transferable numerical values are empirically... 

The various methods to calculate the vibrational frequencies and force constants from ab-initio data on diatomic molecules is represented in Sections 5 A to K. It is seen that the various approximations yield results which fluctuate from molecule to molecule, although the order of magnitude is mostly correct. It is clear, however, that it is not at the present moment possible to calculate co and ke of molecules to such a d ee of accuracy that the factors which contribute to the intemudear forces in molecules can be pinpointed and compared. This is perhaps the reason why semi-empirical models continue to be exploited, e.g. the simple bond-charge model (electrostatic) model for P.E.-curves of homonudear diatomic molecules of Parr and Borckmann (114) based upon the Fues potential from which the famous Birge-Mecke relation is derived ... 

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