Many-body effects in empirical potentials

Three-body effects can significantly affect the dispersion interaction. For example, it is believed that three-body interactions account for approximately 10% of the lattice energy of crystalline argon. For very precise work, interactions involving more than three atoms may have to be taken into account, but they are usually small enough to be ignored. A potential that includes both two- and three-body interactions would be written in the following 

Axilrod and Teller investigated the three-body dispersion contribution and showed that the leading term is  

The effect of the Axibod-Teller term (also known as the triple-dipole correction) is to make the interaction energy more negative when three molecules are linear but to weaken it when the molecules form an equilateral triangle. This is because the linear arrangement enhances the correlations of the motions of the electrons, whereas the equilateral arrangement reduces it. 

The three-body contribution may also be modelled using a term of the form 

The computational effort is significantly increased if three-body terms are included in the model. Even with a simple pairwise model, the non-bonded interactions usually require by far the greatest amount of computational effort. The number of bond, angle and torsional terms increases approximately with the number of atoms (N) in the system, but the number of non-bonded interactions increases with N. There are N(N — 1)/2 distinct pairs of 

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The pair potential functions for the description of the intermolecular interactions used in molecular simulations of aqueous systems can be grouped into two broad classes as far as their origin is concerned empirical and quantum mechanical potentials. In the first case, all parameters of a model are adjusted to fit experimental data for water from different sources, and thus necessarily incorporate effects of many-body interactions in some implicit average way. The second class of potentials, obtained from ab initio quantum mechanical calculations, represent purely the pair energy of the water dimer and they do not take into account any many-body effects. However, such potentials can be regarded as the first term in a systematic many-body expansion of the total quantum mechanical potential (dementi 1985 Famulari et al. 1998 Stem et al. 1999). 

Atomistic MD implements predetermined potential functions, also referred as force fields (FFs), either empirical or derived from independent electronic stmcture calculations. An important aspect in any MD simulation is how to describe or approximate the interatomic interactions. Usually, the potentials that describe these interactions are determined a priori and the full interaction is partitioned into two-, three-, and many-body contributions, long- and short-range terms, and so on, for which suitable analytical functional forms are devised. There are arguments that empirically fitted potentials are more reliable since they include in an implicit way the many-body effects. In principle, by comparing the results of computer and laboratory experiments, we can improve and refine the theoretical model. 

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 ... 

Beside the empirical or semiempirical models described above, the need for inclusion of many-body effects, polarizability at least, in water-water potentials has also been recognized in the development of more recent ab initio potentials [45,75,103-108]. 

Several studies have focused on extensive MD simulations of Pt nanoparticles adsorbed on carbon in the presence or absence of ionomers [109-113]. Lamas and Balbuena performed classical molecular dynamics simulations on a simple model for the interface between graphite-supported Pt nanoparticles and hydrated Nation [113]. In MD studies of CLs, the equilibrium shape and structure of Pt clusters are usually simulated using the embedded atom method (EAM). Semi-empirical potentials such as the many-body Sutton-Chen potential (SC) [114] are popular choices for the close-packed metal clusters. Such potential models include the effect of the local electron density to account for many-body terms. The SC potential for Pt-Pt and Pt-C interactions provides a reasonable description of the properties of small Pt clusters. The potential energy in the SC potential is expressed by... 

The exchange-correlation potential represents all many-body effects due to electron electron interaction that are not included in the Hartree potential. The correlation problem in metals remains largely unsolved, and in heavy-fermion materials the many-body correlation is thought to be the cause of all the observed anomalous properties. On the other hand, it has been found empirically that one can perform extremely good band calculations for simple as well as transition metals by using the local-density-functional approximation (LDA) for (Kohn and Sham 1965) ... 

There are many empirical and semi-empirical pair potentials which describe quite satisfactorily the properties of liquids and solids, see chapter 5 in book The parameters in these potentials are not real parameters of a true two-body interaction, their values depend upon properties of a medium. So these effective two-body potentials include nonadditive interactions through their parameters. The latter can not be directly related to the definite physical... 

Three-Body Interactions. - While the three-body interaction term (see equation (1)) can be neglected in some systems, it can be very important in others. In addition, the importance of the inclusion of three-body terms varies with the property considered.87 In most classical liquid state simulations today, two-body effective empirical potentials are used. Since many of the parameters used in these potentials are determined using liquid state data, it is presumed that some of the effects due to the three-body terms will be accounted for. Because determination of forces between molecules is the most computationally intensive part of a simulation, and inclusion of three-body interactions increases the time by a factor of order N, it is often assumed that effective pair potentials provide adequate representations of the system. 

Although there have been a fairly large number of first-principle simulations of condensed phase published to date, this number is completely dwarfed by simulations based on empirical potentials. The popular empirical pair potentials for water [76-78] have been used in many (thousands) research projects. The empirical potentials are usually fitted in simulations for liquids to reproduce measured properties of this phase. Thus, these potentials mimic the nonadditive effects by distortions of two-body potentials... 

The problem with empirical many-body potentials lies in the choice of the parameters used to define the interatomic force fields. [89] Since empirical potentials are usually derived by fitting bulk properties, their application to finite systems can lead to incorrect answers. However, successful examples of empirical potential functions which incorporate duster size dependent effects, such as the sp hybridization in Be dusters, [90] have been reported. 

KLEIN - We do indeed use a semi-empirical model for the various interaction potentials. First, we model the ammonia inter molecular potential with an effective pair potential which ignores many body polarization. Models of this type are remarkably successful in explaining the physical properties of polar fluids. Of course, we really should include many body forces, but at this stage we ignore them. The ammonia potential is fitted to the heat of evaporation and the zero-pressure density. The electron-alkali metal (Lithium) potential is represented by the Shaw pseudo potential fitted to the ionization energy. This is the simplest and crudest model possible. We have explored the effect of using (a) Heine-Abarenkov, (b) Ashcroft, and (c) Phillips-Kleinman forms. Our results are not very sensitive to the choice of pseudo potential. (In the case of Cs metal, which I did not discuss, the sensitivity to the potential is crucial). 

Consider a diatomic, AB, interacting with a surface, S. The basic idea is to utilize valence bond theory for the atom-surface interactions, AB and BS> along with AB to construct AB,S For each atom of the diatomic, we associate a single electron. Since association of one electron with each body in a three-body system allows only one bond, and since the solid can bind both atoms simultaneously, two valence electrons are associated with the solid. Physically, this reflects the ability of the infinite solid to donate and receive many electrons. The use of two electrons for the solid body and two for the diatomic leads to a four-body LEPS potential (Eyring et al. 1944) that is convenient mathematically, but contains nonphysical bonds between the two electrons in the solid. These are eliminated, based upon the rule that each electron can only interact with an electron on a different body, yielding the modified four-body LEPS form. One may also view this as an empirical parametrized form with a few parameters that have well-controlled effects on the global PES. 

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