Multibody potentials

Among the systems for which one cannot reasonably employ pairwise additive potentials are metals. It is well known that only a small fraction of the binding energy of a metal can be accounted for by pairwise potentials [25]. Furthermore, the use of pairwise additive potential functions leads to incorrect relationships between the components of the bulk modulus [15]. The simulation of a metal requires a truly multibody potential function. 

Metals can also be simulated using empirical multibody potential functions developed from quantum-mechanical results. One procedure that has been successful is the embedded-atom method [30,31], which focuses on the quantum-mechanically-derived energy required to introduce an atom into the host metal. This energy is considered to be a function only of the electron density at the point of insertion, and is written as a sum of two terms A... 

The proper approach to employ depends upon what properties are to be calculated or what processes are to be investigated. If significant redistribution of electronic density is involved, more purely quantum-mechanical methods will be required. However if mechanical or thermodynamic properties are the primary focus of interest, then empirical multibody potential functions may suffice. 

Our ultimate goal is the simulation of alloys and their behavior under conditions of elevated temperature. Accordingly, empirical multibody potentials present an attractive combination of physical accuracy and computational efficiency. To facilitate simulation under the widest possible variety of conditions of temperature, pressure, and surface tension, we decided to incorporate a multibody potential function for copper into a widely used, commercially available molecular dynamics program. We chose CHARMM [35], because of its widespread use, constant pressure/ temperature/surface tension capabilities, and reliability. 

We are using the multibody potential energy function of Rosato et al [33,34], equation (2) ... 

The notation Z)/ in the first term indicates a summation over all atoms in the system and Vi(r/) represents the potential in an external force field. The second term, usually called the pair potential, is probably the most important energy term in a molecular dynamics simulation. The pair potential sums over all distinct atom pairs i and j without counting any pair twice. The function V2(r/, Vj) depends only on the separation between atoms i and j and hence can also be expressed as V2(r,y). Three-body and other multibody potentials are normally avoided in molecular dynamics simulations since they are not easy to implement and can be extremely time consuming. The multi-body effects are usually taken into account by modifying the pair potential, i.e., using an effective pair potential, which is not the exact interaction potential between the two... 

The most extensive use of multibody potentials for the simulation of materials by means of ionic pair potentials has been the addition of bond angle bending terms to two-body potentials used to describe silicate and framework structured materials. - s jn addition, harmonic planarity restraining terms have been employed in the simulation of polyatomic anions (in, e.g., inorganic carbonates). Here, as in the organic molecular mechanics methodology described latei structural distortions about an expected geometry are realized at an enei etic cost. 

The important feature in the simulations using the multibody potential is that the overcoordinated species are allowed to form, but are at a higher energy state and are therefore less stable than the tetrahedrally coordinated Si. Thus, within the timeframe of a simulation, the pentacoordinated Si relaxes back to the 4-coordinated state. 

We shall see in S.6 a proof of their equivalence for a particular model. The first published attempt at a geneitd proof was that of Jhon, Desai, and Dahler " which led to an inequality tr from (4.104) is equal to or larger than cr from (4.173). Hieir methods were taken further by Schofield who first established the equality of these two results for a system of pair potentials, and whose treatment we follow here. The extension to multibody potentials was made independently by Schofield and by Grant and Desai, and the extension to molecular fluids by the latter. ... 

The two other routes to virial equation of 4.4 can be extended to the multibody potential of this model, and leads to an expression for o- whidi, in the mean-field approximation, is a double integral over p(z) ... 

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