Simulations of rigid bodies in an angle-axis framework

Abstract

A general prescription for deriving the first and second derivatives of the energy for a system of rigid bodies interacting via site-site pair potentials with respect to the rigid-body coordinates is suggested for use in geometry optimisation techniques with coarse-grained models. The scheme is based on an angle-axis representation of the rigid-body rotational coordinates and employs a matrix formulation. This framework can be used to treat a variety of systems composed of rigid bodies with arbitrary symmetry, interacting via site-site isotropic, single-site anisotropic, and site-site anisotropic potentials. Illustrative examples are provided. Analytic expressions for the Hessian eigenvectors corresponding to overall rotation, or rotation about the local symmetry axis for an axially symmetric rigid body, are also derived within this scheme for use in geometry optimisation to characterise transition states and pathways. For a 55-molecule TIP4P water cluster, the matrix formulation results in an order of magnitude enhancement in the performance of geometry optimisation over our previous implementation using the angle-axis representation, and is comparable in performance with one that exploits the specific molecular symmetry using an Euler angle representation. The angle-axis scheme in its matrix formulation allows new site-site rigid-body potentials to be coded rapidly and efficiently in a very flexible way.