On Optimization Problem Arising in Computer Simulation of Crystal Structures

Gradient optimization methods are often used to solve problems of computer simulation of the crystal structures of materials. In this case it becomes necessary to calculate the partial derivatives of the total atoms' system energy according to different parameters. Frequently the calculation of these derivatives is an extremely time-consuming and difficult problem. When describing and modeling the crystal structure of a material characterized by chemical composition, geometry, and type of chemical bond, interatomic interaction potentials are used. In this paper a special multi-step process is constructed to calculate the energy of the atoms' system in the case when the interaction of atoms is described by the Tersoff Potential. On the basis of the constructed multi-step process an algorithm for calculation the second derivatives (Hessian) of the atoms' system energy with respect to the coordinates of the atoms is presented. The above-mentioned second derivatives are provided both for the case when the material under study has a three-dimensional structure, and for the case when a two-dimensional model of a multilayer piecewise-homogeneous material is considered.