Atomic-Scale Simulation in Materials Science
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The second principal class of atomistic simulation techniques are the Monte Carlo methods.5 These methods are all based on approximation of timeaveraged thermodynamic properties by averaging over a statistical selection of representative states from the appropriate t h e r m o d y n a m i c e n s e m b l e . Although the exact dynamics of the system is not simulated, the Monte Carlo method is often more efficient at calculating the time-averaged properties of a system than is the molecular dynamics technique. The states from the canonical ensemble are generated, in the most common procedure, by creating new configurations by randomly varying the atomic positions, and accepting or rejecting the new configurations based on a Boltzmann-weighted energy criterion. This procedure defines a Markov chain of states characteristic of the equilibrium ensemble. In the limit of infinite chain length, the average properties of the states making up the chain are identical to the equivalent time-averaged properties. Although the above description is strictly valid only for simulations within the canonical ensemble, the procedure is typical of the general class of Monte Carlo methods.
Realistic Many-Body Interatomic Potentials The potential energy of a structure composed of N atoms is simply a function of the 3N atomic coordinates. It is convenient to expand the potential energy U in terms of a series of n -body interatomic potentials,
+ 2 * 4 («;«)•..
(1)
where *„ is the «-body interatomic potential, which is a function of the positions of the n atoms ijkl..., and the sums are taken over all combinations of n atoms in the structure. The above expansion is formally exact when interactions of all orders in n are considered. In practice, however, some degree of simplification is needed. A common procedure is to approximate the sum of n-body interactions appearing in Eq. 1 by restricting n to very small values or to special subclasses of higher order interactions. Until recently, simple pairwise (n =2) potentials have dominated the field. The Lennard-Jones (L-J) potential was developed for materials (such as rare gases and some organic compounds) which interact primarily through van der Waals attraction and core-core repulsion. This class of potential takes the form *2(r) = Ar° - B /•-",
(2)
where the standard L-J potential is defined by
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