Mathematical Physics - Volume II - Numerical Methods

6.2 Molecular Dynamics

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can be obtained from the interatomic potential Φ which is, in general, a function of the position of all atoms of the system ( r i j = | r i j | = | r i − r j | being the intensity of the distance vector between the atoms i and j , that is, the interatomic distance, Figure (6.4b)). Thus, when the initial conditions and the interaction potential are defined, Equations (6.1) 1 can be solved numerically. Namely, the positions and velocities of all atoms of the system as a function of time are obtained as a result of solving a system of algebraic equations that approximates the system of differential Equations (6.1) 1 . Thus, the motion of each individual atom (and each ensemble of atoms) is completely deterministic. In most cases, analysts are not interested in the trajectories of individual atoms but in the macroscopic properties of materials that result from the motion of a multitude of atoms. The information resulting from computer simulations can be averaged at certain time intervals for all (or selected) atoms of the system to obtain thermodynamic parameters (Chapter 6.2.4). 6.2.2 Empirical Interatomic Potentials Empirical potentials used in materials science and mechanics of materials are called interatomic potentials. The role of interatomic forces (6.2) is crucial since the MD simulation is realistic only insofar as the interatomic forces are similar to those operating between real atoms in the corresponding atomic configuration [16]. As already noted, the classical definition of interatomic interaction, based on empirical potentials, represents the rigorous quantum mechanical nature of materials in a limited way through impromptu approximations. The interatomic potentials depend on the states of the electrons, thus, the electrons are the origin of the interatomic forces. Nonetheless, the electrons are not directly present in the traditional MD model – their influence is introduced indirectly through analytical functions that define potential energy solely on the basis of the atomic (nuclei) positions (6.3). The creation of the analytical function of potential energy and the choice of input parameters is often based on the fitting of the available experimental data that are of greatest interest for the specific problem being studied (e.g., modulus of elasticity, cohesive energy, phase transition temperature, vibration frequencies). When forming an MD model, the interatomic potential is adopted either on the basis of knowledge of the atomic nature of the simulated material or a priori. The construction of interatomic potential is as much an art as it is a science, but from the point of view of users, the choice is, nowadays, quite simplified thanks to the available literature. This choice is essential not only because the adequacy and accuracy of the potential dictate the quality of the simulation results but also due to the fact that its complexity determines the efficiency of the code in terms of simulation duration. Although some compact potentials may seem inadequate, many fundamental, generic aspects of a physical phenomenon can be observed thanks to the advantages provided by their simplicity. The empirical interatomic potential Φ = Φ ( r 1 , r 2 , . . . , r N ) = = ∑ i Φ 1 ( r i )+ ∑ i ∑ j > i Φ 2 ( r i , r j )+ ∑ i ∑ j > i ∑ k > j > i Φ 3 ( r i , r j , r k )+ · · · (6.3)

represents the potential hypersurface of the non-bonded interactions [5]. In expression