Calculating the Mechanical Properties of Materials from Interatomic Forces

Determining the mechanical properties of materials from microscopic models of their electronic structure is difficult, first because of the difference between microscopic and macroscopic length scales, and second because the linear relation between stress and strain fails in the plastic limit. The purpose of this paper is to describe how these difficulties may be avoided by use of Green’s functions and the black body theorem. Although the relation between stress and strain can be non-linear, the density of atomic positions and momenta in phase space evolves linearly, and consequently can be expressed using Green’s functions whose singular frequency dependence describes the behavior of the systems on macroscopic time scales. The black body theorem says that for quantities which obey wave equations, the local density of modes is insensitive to distant parts of system. The Green’s functions describing the evolution of local disturbances of the atomic position and momentum distributions obey just such a wave equation and are hence insensitive to the structure of the system on large length scales.