Calculation of Elastic Constants Using Molecular Dynamics

Abstract

Ray and Rahman have developed a useful method to determine elastic constants in molecular dynamics computer simulations. The adiabatic elastic constants are contained in a formula involving fluctuations in the microscopic stress tensor. M_ij, in the microcanonical or EhN ensemble, whereas the isothermal elastic constants are contained in a fluctuation formula of the same form in the canonical or ThN ensemble. Here, E is the system energy, h is a 3x3 matrix constructed from the three vectors spanning the periodically repeating computational cell: h=(a,b,c), N is the particle number, and T is the system temperature. For a potential U which depends only upon the distances between the particles (and is not necessarily pairwise additive) this formula gives the elastic constants as a sum of three terms: a fluctuation term, a kinetic term and the Born contribution which depends upon the potential U. In the static Born method of calculating elastic constants, we have only the Born term evaluated at the static lattice positions of the atoms. The fluctuation equation furnishes a practical method of calculating elastic constants which introduces temperature contributions to the static Born values, producing a significant difference.

We shall give results of our calculations for a nearest neighbor Lennard-Jones system, for which independent Monte Carlo data is available, and for silicon using the 2- and 3-body Stillinger-Weber potential.