Molecular dynamics as a mathematical mapping. III. Efficient evaluation of the differentiable force functions and their derivatives

The fully continuous and differentiable framework for performing molecular dynamics calculations introduced in parts I and II of this paper [1,2] requ ires the evaluation of rather complex force functions and their spatial par tial derivatives. This paper presents an efficient interpolation scheme for the evaluation of these quantities over a finite spatial domain. The modified force function is approximated by a linear combination of Herm ite cubic basis functions such that both the interpolant of the force and i ts spatial derivatives are continuous across the grid boundaries. In order to achieve better accuracy for a given grid size, a nonuniform rectilinear grid is constructed via iterative refinement procedure. The latter guarante es the accuracy of the force computed by interpolation within any specified tolerance epsilon > 0. For many potential functions of practical interest, it is possible for poly nomial interpolants to be constructed for parts of the force functions whic h are independent of the potential parameters and system density (the so-ca lled "separable force functions"). In such cases, a single interpolation gr id which is applicable for a wide range of potential parameters and system densities can be constructed a priori.