Molecular Dynamics as a Mathematical Mapping. I. Differentiable Force Functions

Abstract

The molecular dynamics technique can be viewed as a deterministic mathematical mapping between, on one side, the force field parameters that describe the potential energy interactions and the input macroscopic conditions, and, on the other, the calculated macroscopic properties of the bulk molecular system. The differentiability of such a mapping in the conventional molecular dynamics calculations is affected by the discontinuities in particle positions introduced by the periodic boundary conditions and the discontinuities introduced by the minimum image convention and other methods commonly employed to approximate the calculation of interparticle potential and force. This paper proposes an alternative molecular dynamics framework based on modified force functions which are almost everywhere continuous and differentiable, and exhibit a natural periodicity. These characteristics obviate the need for both the periodic boundary conditions and the minimum image convention, as well as for any corrections for long-range interactions. They also make it possible to apply standard methods of variational calculus for the computation of partial derivatives of the molecular dynamics mapping. The modified framework is first introduced for the case of simple monoatomic fluids where the nature of the forces exerted between any pair of two particles is identical. A more general model describing the interactions of flexible molecules is then developed. We describe the application of this approach to mixtures of alkane molecules interacting via the NERD force field.